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Thermodynamics of sodium aluminosilicate formation in aqueous alkaline solutions relevant to closed-cycle… Park, Hyeon 1999

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Thermodynamics of Sodium Aluminosilicate Formation in Aqueous Alkaline Solutions relevant to Closed-cycle Kraft Pulp Mills by HYEON PARK B . S . , H a n y a n g Unive r s i ty , K o r e a , 1989 M . S . , H a n y a n g Un ive r s i ty , K o r e a , 1993 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U E S T M E N T S F O R T H E D E G R E E O F D O C T O R O F P H T L O S P H Y i n T H E F A C U L T Y O F G R A D U A T E S T U D I E S Department o f C h e m i c a l and B i o - R e s o u r c e Eng inee r ing W e accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A Augus t , 1999 © H y e o n P a r k , 1999 in presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of 0)CVY\ Iced? W fifa-JxW-^u^ Krg<h ^jzr-l^j The University of British Columbia Vancouver, Canada Date / U j.J^, DE-6 (2/88) 11 ABSTRACT Accumulation of Al and Si ions in the recovery cycle of a kraft pulp mill may cause sodium aluminosilicate scale formation. This glossy scale forms on process equipment and is very hard to remove. Thus, formation of the scale can create several operational problems in mills moving towards progressive system closure and should be prevented. The purpose of this study is to supply: (a) data on the precipitation conditions of sodium-aluminosilicates in green and white liquors of the recovery cycle; and (b) a model to predict such conditions. The data can either be used directly or to test process models for the design and optimization of progressive system closure strategies. The precipitation conditions of sodium aluminosilicates in synthetic green and white liquors at 368.15 K (95 °C) were determined. In the experiments, the effects of varying the Al/Si ratio and concentrations of OH", C0 3 2", SO4 2 ' , and H S " were studied. The structure of the precipitates was identified by X-ray diffraction, thermogravimetry and chemical analysis. The precipitates were found to have the structure of hydroxysodalite dihydrate (Na8(AlSi04)6(OH)2-2H20) except in a simulated green liquor system with low OH" and high Cf concentrations where sodalite dihydrate (Nag(AlSi04)6Cl2-2H20) was formed. The precipitation conditions in mill green and white liquors at 368.15 K were also measured. The effects of varying the Al/Si ratio, NaOH, Na2C03, and Na2S concentrations were studied. The precipitates were found to have the structure of hydroxysodalite dihydrate. A thermodynamic model for sodium aluminosilicate formation in aqueous alkaline solutions was developed. Pitzer's method was adopted to calculate the activity of Ill water and the activity coefficients of the other species in solution. The system under consideration contained the ions of Na + , Al(OH)4', Si0 3 2", OH", C0 3 2", S0 4 2', Cl', HS" dissolved in water and in equilibrium with two possible solid phases (sodalite dihydrate : Na8(AlSi04)6Cl2-2H20 and hydroxysodalite dihydrate : Na 8(AlSi0 4) 6(OH)2- 2H 20) at 368.15 K. The equilibrium constants of sodalite dihydrate and hydroxysodalite dihydrate formation reactions were determined using the thermodynamic properties of the species involved. Property values that were not available in the literature were estimated by group contribution methods. The model calculates the molality of all species at equilibrium including the amount of solid precipitates. The calculations were compared with published data and were found to be in good agreement. Meanwhile, since the system contains the Si032" and Al(OH)4" ions, knowledge of the relevant Pitzer's model parameters is required. Osmotic coefficient and water activity data for Na2Si03 and mixed Na2Si03-NaOH aqueous solutions were obtained at 298.15 K by employing an isopiestic method. The binary Pitzer's parameters, P(0), f3(1), and C*, for Na2Si03 and the mixing parameters, Q 0 H - S i 0 2- and ¥ N a + O H - s i o i - , were estimated using the osmotic coefficient data. In addition, osmotic coefficient data were obtained for the aqueous solutions of NaOH-NaCl-NaAl(OH) 4. The solutions were prepared by dissolving AICI36H2O in aqueous NaOH solutions. The osmotic coefficients of the solutions were measured by the isopiestic method at 298.15 K. The osmotic coefficient data were used to evaluate the unknown binary and mixing parameters of Pitzer's model for the aqueous NaOH-NaCl-NaAl(OH) 4 system. The experimental osmotic coefficient data were correlated well with Pitzer's model using the parameters obtained. iv TABLE OF CONTENTS A B S T R A C T i i T A B L E O F C O N T E N T S i v L I S T O F T A B L E S v i i i L I S T O F F I G U R E S x i A C K N O W L E D G E M E N T S x i v C H A P T E R 1. I N T R O D U C T I O N 1 1.1. Env i ronmenta l Impact o f the Kra f t P u l p i n g Industry 3 1.2. C losed -cyc l e M i l l Operat ion 4 1.3. Non-process E lements in the Kra f t P u l p i n g Process 5 1.3.1. Sources and profi le o f non-process elements 5 1.3.2. Adve r se effects o f non-process elements 9 1.4. Recent Deve lopments o f C losed -cyc l e Techno logy 10 1.5. Scale Fo rma t ion i n the R e c o v e r y C y c l e 13 1.5.1. S o d i u m A l u m i n o s i l i c a t e Scale Fo rma t ion 15 1.6. Research Object ives 19 C H A P T E R 2. T H E O R E T I C A L B A C K G R O U N D 21 2.1 . The rmodynamic E q u i l i b r i u m Constant 21 2.2. E q u i l i b r i u m Constant Ca lcu la t ion at Specif ic Temperature 23 2.3. A c t i v i t y o f Water , Osmot ic Coeff ic ient , and A c t i v i t y Coeff ic ient 24 2.4. P i tze r ' s A c t i v i t y Coeff ic ient M o d e l 28 2.5. Isopiestic M e t h o d 32 C H A P T E R 3. M A T E R I A L S A N D M E T H O D S 35 3.1. So lub i l i ty Exper iments us ing Synthetic L i q u o r s o f System A 35 3.1.1. Exper imenta l design 3 5 3.1.2. Exper imenta l procedure 3 7 3.1.3. Identification o f the precipitates 39 V 3.2. Solubility Experiments using Synthetic Liquors of System B 40 3.2.1. Experimental design 40 3.2.2. Experimental procedure and analysis 42 3.3. Solubility Experiments using Mill Liquors 42 3.3.1. Experimental design 43 3.3.2. Experimental procedure and analysis 44 3.4. Osmotic Coefficient Measurement for Na2Si03 and Na2Si03-NaOH 45 Systems 3.4.1. Identification of silicic species by a titration method 45 3.4.2. Apparatus and chemicals 46 3.4.3. Experimental procedure 49 3.5. Osmotic Coefficient Measurement for NaOH-NaCl-NaAl(OH) 4 System 50 3.5.1. Apparatus and chemicals 50 3.5.2. Experimental procedure 51 CHAPTER 4. THERMODYNAMIC MODELING OF SODIUM 53 ALUMINOSILICATE FORMATION 4.1. Model Equations 53 4.2. Structure of the Thermodynamic Model 55 CHAPTER 5. OSMOTIC COEFFICIENT D A T A FOR Na 2Si0 3, Na 2Si0 3-NaOH, 57 AND NaOH-NaCl-NaAl(OH) 4 AQUEOUS SYSTEMS 5.1. Identification of Metasilicic Species by Titration 57 5.2. Mole Fraction of Metasilicic Species 63 5.3. Osmotic Coefficient Data for Na2Si0 3 Aqueous System 63 5.4. Osmotic Coefficient Data for Na 2Si0 3-NaOH Aqueous System 65 5.5. Osmotic Coefficient Data for NaOH-NaCl-NaAl(OH) 4 Aqueous System 70 CHAPTER 6. DETERMINATION OF PITZER' S PARAMETERS FOR Na 2Si0 3, 73 Na 2Si0 3-NaOH, AND NaOH-NaCl-NaAl(OH) 4 AQUEOUS SYSTEMS 6.1. Pitzer's Parameters for Na2Si0 3 and Na2Si0 3-NaOH Aqueous Systems 73 v i 6.2. P i t ze r ' s Parameters for N a O H - N a C l - N a A l ( O H ) 4 Aqueous System 76 6.3. R e l i a b i l i t y o f the P i t ze r ' s Parameter Determinat ion 81 C H A P T E R 7. A P R I O R I D E T E R M I N A T I O N O F M O D E L P A R A M E T E R S 84 7.1. P i t ze r ' s Parameters 84 7.2. E q u i l i b r i u m Constants 86 7.3. Es t ima t ion o f The rmodynamic Properties 88 7.3.1. AHf° and AGf° o f sodalite dihydrate and hydroxysodal i te dihydrate 88 7.3.2. C p ° o f Sodali te dihydrate and hydroxysodal i te dihydrate 91 7.3.3. S° o f S i 0 3 2 ' ( a q ) 93 7.3.4. A H f ° o f Si032"(aq) 95 7.3.5. C p ° o f S i 0 3 2 - ( a q ) 96 7.3.6. Ca lcu la t ion o f K s o d at 368.15 K 97 7.3.7. Ca lcu la t ion o f K h s o d at 368.15 K 98 7.4. Change o f In Khsod w i t h Temperature 99 C H A P T E R 8. S O L U B I L I T Y M A P S O F A l A N D S i I N G R E E N A N D W H I T E 101 L I Q U O R S 8.1. Synthetic Green and W h i t e L i q u o r s o f Sys tem A 101 8.2. Synthetic Green and W h i t e L i q u o r s o f Sys tem B 106 8.3. M i l l G reen and W h i t e L i q u o r s 112 8.4. Structure o f Precipitates f rom synthetic l iquors 118 8.5. Structure o f Precipitates f rom m i l l l iquors 122 8.6. M o r p h o l o g y o f Precipitates 122 C H A P T E R 9. P R E D I C T I O N O F T H E P R E C I P I T A T I O N C O N D I T I O N S O F 125 S O D I U M A L U M T N O S I L I C A T E S 9.1. N a + - A l ( O H ) 4 " - S i 0 3 2 - - OFT - C 0 3 2 " - S 0 4 2 - - C f - H 2 0 (System A ) 126 9.2. N a + - A l ( O H ) 4 " - S i 0 3 2 " - O H " - C 0 3 2 _ - C l ' - H S " - H 2 0 (System B ) 135 C H A P T E R 10. C O N C L U S I O N S A N D R E C O M M E N D A T I O N S 139 vii N O M E N C L A T U R E 142 BIBLOGRAPHY 146 APPENDIX I. Chemical Analysis by Atomic Absorption Spectrophotometer 159 APPENDIX II. Standard Errors and Confidence Intervals for Experimental 161 Solubility Data APPENDIX III. Tables of Solubility Data 162 APPENDIX IV. Calculation of Titration Curve 165 APPENDIX V. Uncertainty of the Measured Osmotic Coefficient 167 APPENDIX VI. Uncertainty of the Estimated Thermodynamic Properties and 170 Equilibrium Constants APPENDIX VII. Comparison of Pitzer's Parameters with Published Values 174 APPENDIX VIII. Sensitivity Analysis of Pitzer's Parameters 176 APPENDIX IX. Computational Source Codes in FORTRAN 77 178 Determination of Pitzer's binary parameters for single electrolyte system using 178 osmotic coefficient data Calculation of osmotic coefficient and activity coefficient using Pitzer's binary 183 parameters for single electrolyte system Determination of Pitzer's mixing parameters for multi-component electrolyte 186 system using osmotic coefficient data Calculation of osmotic coefficient and activity coefficient using Pitzer's binary 193 and mixing parameters for multi-component electrolyte system Computation of equilibrium state for NI^-NI^OH-Ff-HCl-NELiCl-Cr-Na*- 199 NaCl-K +-KCl system at 573.15 K Computation of equilibrium state for sodium aluminosilicate formation 205 v i i i LIST OF TABLES Table 1.1 A m o u n t o f non-process elements from each source. 6 Table 1.2. M e t a l contents i n the b leach effluent. 8 Table 1.3. Concentra t ion range o f process and non-process elements i n green and 8 whi te l iquors . Table 3.1. So lub i l i ty experiment design for synthetic l iquors o f system A . 36 Tab le 3.2. So lub i l i ty experiment design for synthetic l iquors o f system B . 41 Table 3.3. A n a l y s i s results o f m i l l l iquors. 43 Table 3.4. So lub i l i ty experiment design for m i l l l iquors . 44 Table 5.1. Osmot i c coefficients and water activities for the N a 2 S i 0 3 aqueous 66 system at 298.15 K . Table 5.2. Osmot i c coefficients and water activities for the N a 2 S i 0 3 - N a 0 H 68 aqueous system at 298.15 K . Table 5.3. Osmot i c coefficients o f N a C l and K C 1 as reference and standard 71 solutions at 298.15 K . Table 5.4. Osmot i c coefficients o f N a O H - N a C l - N a A l ( O H ) 4 aqueous solutions at 72 298.15 K . Table 6.1. The P i t ze r ' s parameters o f N a 2 S i 0 3 and N a 2 S i 0 3 - N a O H systems at 75 298.15 K . Table 6.2. The P i t ze r ' s parameters at 298.15 K available i n the literature. 79 Table 6.3. The P i t ze r ' s parameters o f N a O H - N a C l - N a A l ( O H ) 4 aqueous system at 80 298.15 K obtained i n this study. Table 6.4. The P i t ze r ' s parameters o f N a T c 0 4 , N a T c 0 4 - N a C l , and N a B r - N a C 1 0 4 82 systems at 298.15 K . Table 6.5. C o m p a r i s o n o f measured osmotic coefficients w i t h calculated those for 82 the N a T c 0 4 system at 298.15 K . ix Table 7.1. Pitzer's binary parameters for the modeling of sodium aluminosilicate 85 formation. Table 7.2. Pitzer's mixing parameters for the modeling of sodium aluminosilicate 85 formation. Table 7.3. Thermodynamic data at 298.15 K and 1 bar published in the literature. 87 Table 7.4. Estimations of AH f° and AG f° of anhydrous sodalite (Na 8(AlSi0 4) 6Cl 2). 89 Table 7.5. Estimated thermodynamic data at 298.15 K and 1 bar. 90 Table 7.6. Estimation of C p° of anhydrous sodalite (Na 8(AlSi0 4 )6Cl 2 ) . 92 Table 7.7. Contribution by 2H 20 for C° p estimation. 92 Table 7.8. Contributions by C l 2 and (OH) 2 for C° p estimation. 93 Table 7.9. Prediction of the entropy of aqueous ions at 298.15 K, cal/mol K. 94 Table 7.10. Thermodynamic data of Na2SiC>3(S), Na + (aq), and S i03 2 " (aq) for the 95 calculation of the AG 0, AH°, and AS 0 of reaction (7.11). Table 7.11. Heat capacities of aqueous oxy-anions at 298.15 K, cal/mol K. 96 Table 8.1. Repeatability of the experiments in three runs. 102 Table 9.1. Comparison of the calculation results for the NH, - N H j O H - H* - H C l - 126 NH4CI - C f - Na + - NaCl - K + - KC1 system at 573.15 K. Table 9.2. Example of the modeling calculation results for the Na + - Al(OH) 4" - 129 Si0 3 2" - O f f - C0 3 2" - S0 4 2 _ - C f - H 2 0 system at 368.15 K. Table A. 1.1. Analytical data for the analysis of liquid phase alkaline samples. 159 Table A. 1.2. Analytical data for the analysis of samples from the solid precipitates. 160 Table A.3.1. Solubility data for synthetic liquors of system A. 162 Table A.3.2. Solubility data for synthetic liquors of system B. 163 Table A.3.3. Solubility data for mill liquors. 164 Table A.7.1. Comparison of Pitzer's parameters. 174 Table A.8.1. Sensitivity analysis of Pitzer's parameters for Na 2Si0 3 aqueous 176 system. X Table A . 8 . 2 . Sensi t ivi ty analysis o f P i t ze r ' s parameters for Na2SiC«3-NaOH aqueous 177 system. Table A . 8 . 3 . Sensi t ivi ty analysis o f P i t ze r ' s parameters for N a O H - N a C l - 177 N a A l ( O H ) 4 aqueous system. x i LIST OF FIGURES Figure 1.1. A diagram o f the kraft pu lp ing process. 2 F igu re 1.2. M e t a l profi le o f pre-oxygen stage kraft pulp. 7 F igu re 3.1. E q u i l i b r i u m vessel for the so lubi l i ty experiments. 38 F igu re 3.2. The isopiestic apparatus. 47 F igu re 4 .1 . A b l o c k diagram o f the thermodynamic mode l ing . 56 F igure 5.1. Ti t ra t ion curve for the Na2SiC»3-NaOH solut ion w i t h H C 1 solut ion. 58 F igure 5.2. Calcu la ted titration curve for the N a 2 S i 0 3 - N a O H and N a 2 C 0 3 - N a O H 62 solutions w i t h H C 1 solution. F igu re 5.3. M o l e fraction o f metas i l ic ic species w i t h p H . 64 F igu re 5.4. Osmot i c coefficients o f N a 2 S i 0 3 and K C I aqueous solut ions at 298.15 67 K . F igu re 5.5. Osmot i c coefficients o f m i x e d N a 2 S i 0 3 - N a O H and K C I aqueous 69 solutions at 298.15 K . F igu re 6.1. M e a n act ivi ty coefficients o f N a 2 S i 0 3 in N a 2 S i 0 3 aqueous solut ion at 77 298.15 K . F i g u r e 6.2. M e a n act ivi ty coefficients o f N a 2 S i 0 3 and N a O H i n m i x e d N a 2 S i 0 3 - 78 N a O H aqueous solut ion at 298.15 K . F igu re 7.1. Change o f In K h s o d wi th temperature. 99 F igu re 8.1. A l and S i ions approaching equ i l ib r ium obtained f rom the base 102 experiment us ing synthetic green l iquor o f system A . F igu re 8.2. So lub i l i ty map o f A l and S i in synthetic green and whi te l iquors o f 104 system A . F igure 8.3. Effect o f h y d r o x y l ions on the so lubi l i ty l imi t i n synthetic green and 105 whi te l iquors o f system A . F igu re 8.4. Effect o f carbonate ions on the solubi l i ty l i m i t i n synthetic green and 107 Xll white liquors of system A. Figure 8.5. Effect of sulfate ions on the solubility limit in synthetic green and white 108 liquors of system A. Figure 8.6. Solubility map of Al and Si in synthetic green and white liquors of 110 system B . Figure 8.7. Effect of Na2S on the solubility limit in synthetic green and white 111 liquors of system B . Figure 8.8. Solubility map of Al and Si in mill green and white liquors. 113 Figure 8.9. Effect of NaOH on the solubility limit of Al and Si in mill liquors. 114 Figure 8.10. Effect of N a 2 C 0 3 on the solubility limit of Al and Si in mill liquors. 115 Figure 8.11. Effect of Na2S on the solubility limit of Al and Si in mill liquors. 116 Figure 8.12. Solubility map comparing solubility limit of Al and Si in varying 117 liquors. Figure 8.13. X-ray diffraction pattern of precipitates in synthetic green liquor of 119 system A. Figure 8.14. Thermogravimetric analysis of hydroxysodalite dihydrate, 121 Na8(AlSi04)6(OH)2-2H20, precipitated in synthetic white liquor of system A. Figure 8.15. Scanning electron micrographs of precipitates. 123 Figure 9.1. Equilibrium concentration of aluminum and silicon species. 127 Figure 9.2. Effect of hydroxyl ion concentration changes on the equilibrium 128 concentration of aluminum and silicon species. Figure 9.3. Effect of carbonate ion concentration changes on the equilibrium 130 concentration of aluminum and silicon species. Figure 9.4. Effect of sulfate ion concentration changes on the equilibrium 131 concentration of aluminum and silicon species. Figure 9.5. Effect of hydroxyl ion concentration changes on the equilibrium 132 concentration of aluminum and silicon species. xiii Figure 9.6. Effect of carbonate ion concentration changes on the equilibrium 133 concentration of aluminum and silicon species. Figure 9.7. Effect of sulfate ion concentration changes on the equilibrium 134 concentration of aluminum and silicon species. Figure 9.8. Effect of hydrosulfide ion concentration changes on the equilibrium 135 concentration of aluminum and silicon species. Figure 9.9. Effect of hydrosulfide ion concentration changes on the equilibrium 136 concentration of aluminum and silicon species. Figure 9.10. Comparison of model predictions with published correlations and 137 industrial data. Figure A.7.1. Osmotic coefficients for NaCl, Na 2S0 4, and Na 2S 20 3 systems. 175 xiv ACKNOWLEDGEMENTS The author wishes to thank those who supplied the ideas, the time, and the friendship for this work. In particular, the author wishes to express his sincere appreciation to his supervisor, Dr. Peter Englezos, who guided him to the sea of electrolyte thermodynamics. His advice, patience, instruction and support are the principal contributions to the successful completion of this study. Sincere gratitude is extended to the thesis committee members, Dr. Charles Haynes for his fruitful discussions about the osmotic coefficient measurement as well as Dr. A. Paul Watkinson, and Dr. Chad P. J. Bennington for their valuable comments. The author is deeply indebted to Dr. Mati Raudsepp in the Department of Earth and Ocean Sciences for his kind help in performing the X-ray diffraction and S E M analysis. The author would like to acknowledge the help and support provided by Mike Towers, Vic Uloth, Brian Richardson, and Jim Wearing from PAPRICAN. Thanks are also expressed to Mr. Horace Lam, Ms. Rita Penco, Mr. Tim Paterson, Mr. Peter Taylor, Ms. Brenda Dutka, Mr. Ken Wong and all the staff in the Department of Chemical Engineering and Pulp and Paper Centre. John Bates Centennial fellowship as well as University Graduate Fellowship are gratefully acknowledged. On a more personal level, I wish to thank my best friends met in Canada: Tazim Rehmat for her kind friendship including a fortune chip from Las Vegas, Khizyr Khoultchaev for his unique instruction of titration methods, Peter Pang for his kind help in the lab, Geoff Bygrave for his useful discussion about thermodynamics, Isabelle XV Pineaul t for sweet cookies and muffins, O k H y u n A h n for his great help w i t h computers, A h H y u n g Pa rk for her kindness to m y fami ly , D a l H o o n L e e for enjoying coffee breaks together, Jong C h o o n L i m and W o o k D o n g K w o n for sharing fresh atmosphere o n the green at M c C l e e r y . I a m also grateful to Prof. K y o n g O k Y o o , Prof . D o o Sub K i m , Prof . H e e T a i k K i m , Prof . Sang June C h o i , Prof. D a e W o n Park, Prof. Tae Joo C h o i , D r . D o n g H y u n L e e , and D r . B y o u n g M o o M i n for their grateful encouragement. I w o u l d pay special tribute to m y mother, brothers, and parents i n l a w for their uncondi t ional l ove and support f rom m y home, K o r e a . T w i n k l i n g eyes o f m y son, S i Y o u n g , have been a catalyst for this work . M o s t o f a l l , I w o u l d l ike to express special thanks to m y wife , Sun M e e L e e for her love, sacrifice and best dinners. She never doubted m y dec i s ion to study abroad for many years. I w o u l d l i ke to dedicate this w o r k to m y father in heaven. H i s confidence and encouragement i n me have never weakened. Sincerely hope he is n o w proud o f his son. 1 CHAPTER 1. INTRODUCTION Kraft pulp is made by reacting wood chips with a strongly alkaline solution. This process is called digestion in the pulp industry. The pulp fibres are then subjected to bleaching for brightness improvement. The kraft pulp process consists of three streams; brown fibre line, bleach plant fibre line and chemical recovery cycle (Smook, 1992). A simplified diagram of the kraft pulping process is shown in Figure 1.1. Wood chips are delignified in the digester by using a solution of NaOH (sodium hydroxide) and Na 2S (sodium sulfide). After digestion, the delignified chips are discharged into a blow tank to be disintegrated into individual fibres. Separated weak black liquor from the brown stock washer is evaporated and burnt in the recovery boiler. Most of organic materials are incinerated. The remaining inorganic chemicals are entered into the smelt dissolving tank. Major portion of the smelt is a solid form of N a 2 C 0 3 (sodium carbonate), Na 2S and NaOH. The smelt is dissolved in water which forms green liquor. The dissolved N a 2 C 0 3 in green liquor is converted to NaOH in white liquor by the reaction (1.1) in the causticizers. N a 2 C 0 3 + Ca(OH) 2 -> 2NaOH + CaC0 3 (1.1) White liquor containing the cooking chemicals, NaOH and Na 2S, is recycled to the digester and used for cooking chips (Smook, 1992; Grace etal, 1989). 2 3 The fibres separated at the brown stock washer do not have enough brightness. The lignin content of the fibres is reduced by 50 % in the oxygen delignification stage. Dissolved solids are recovered by washing the oxygen-delignified stock and they are recycled to the recovery boiler. The pulp is discharged to the bleach plant to increase the brightness (Grace et al, 1989). Several different sequences of bleach stages are used in modern kraft pulp mills. The conventional bleaching method using the chlorine gas is not used any more because of the dioxin formation problem. Most of kraft mills have E C F (Elementary Chlorine Free) or TCF (Totally Chlorine Free) bleach stages (Teder et al, 1990; Chandra, 1997). The kraft pulp is used to make high quality papers such as printing paper because of its better strength and brightness compared to mechanical pulp (Grace et al, 1989). 1.1. Environmental Impact of the Kraft Pulping Industry Dioxin and furan detection in effluents from kraft mills has been reported (Servos et al, 1996; Norstrom et al, 1988; Bohn, 1998). In 1983, U.S. government reported that dioxin was found in the downstream of pulp mills in Maine, Wisconsin and Minnesota. Biologists of Canadian Wildlife Service found some of the highest dioxin levels in the egg shells near the pulp mill at Crofton on Vancouver Island in 1987 (Bohn, 1998). Norstrom et al. (1988) reported that ecologically significant amounts of chlorinated dioxins and furans were present in sediments and fish in the vicinity of B.C. mills. As a result of these findings, stringent pollution standards were set by the government and more efforts to minimize the environmental impact have been requested from the kraft pulp mill industry (Martin, 1998). 4 The large amount of water consumption is another environmental concern of the kraft pulp mills. A few decades ago, it was common for a bleached kraft mill to use up to 400 m3/ton pulp. In spite of the significant improvements in the water consumption, total water use still ranges from 30 to 150 m3/ton pulp (Bihani, 1996). 1.2. Closed-cycle Mill Operation Closed-cycle mill operation has been considered as the ultimate method to achieve further effluent reduction and to minimize the environmental impact. Closed-cycle in the context here means no liquid effluents from the mill. Of course, there might be some emissions as gaseous and solid wastes which can be handled in an environmentally safe manner. This is the concept of reducing pollution at the source by reusing more of the water, fibre, chemicals and energy contained in the waste stream. This may also reduce the load of the effluent treatment (Galloway et al, 1994). The first attempt at closed-cycle at the Great Lakes Forest Products Ltd. bleached kraft pulp mill at Thunder Bay, Ontario, Canada was unsuccessful due to corrosion in the white liquor evaporators, the recovery boiler superheater tubes, and the black liquor evaporator tubes, pitch deposition, scaling in the salt recovery process and other miscellaneous problems (Patrick et al, 1994). Several trials of closed-cycle kraft pulp production have been attempted but it has not been achieved yet (Galloway et al, 1994). The gradual move towards a closed mill is what the industry refers to a progressive system closure. Although closed-cycle operation of any kraft mill has not been achieved yet, a number of mills have stated this as their goal, and are working towards it. It should be noted that closed-cycle CTMP (chemi-thermomechanical pulp) mills are already a 5 commerc ia l reali ty (Barnes et al, 1995). Such technological development o f c losed-cycle C T M P m i l l s opened the poss ibi l i ty o f v i r tua l ly e l imina t ing l i q u i d discharges from bleached kraft pulp mi l l s . 1.3. Non-process Elements in the Kraft Pulping Process One o f the issues that prevent c losed-cycle kraft m i l l operation is related to the problems caused by the bui ld-up o f non-process elements ( N P E s ) . N P E s are ions such as a l u m i n u m ( A l ) , s i l i con (Si ) , ca l c ium (Ca) , ba r ium (Ba) , potassium ( K ) , chlor ide ( C l ) , magnes ium ( M g ) , phosphorus (P), manganese ( M n ) , i ron (Fe), and copper (Cu) . They enter the system as part o f the w o o d chips, water, and chemicals and are not required to manufacture pulp. In an open m i l l N P E s are wi thdrawn but they accumulate i n a c losed one ( G a l l o w a y et al, 1994). The bui ld-up o f N P E s can cause severe process problems such as scal ing, corrosion, and interference o f b leaching ( G a l l o w a y et al, 1994; G l e a d o w et al, 1997). The proper management o f N P E s is prerequisite for the successful implementa t ion o f c losed-cycle technology. 1.3.1. Sources and profile of non-process elements The major sources o f the non-process elements are w o o d , make-up l ime, and water. Table 1.1 shows the amount o f non-process elements in t roduced to the process. M o s t o f the non-process elements tend to concentrate in the bark, roots, and foliage. Therefore, unbarked w o o d o r who le tree chips w i l l contain higher concentrations o f non-process elements. Sand, clay and contaminat ion attached on the logs contain s ignif icant 6 amounts o f s i l i con and a luminum. L o g s transported by sea add extremely large amounts o f chlor ide. Table 1.1. A m o u n t o f non-process elements from each source ( G a l l o w a y et al, 1994). E lemen t W o o d chips M a k e - u p l ime Wate r (mg/kg a.d. pulp) (mg/kg a.d. pulp) (mg/kg a.d. pulp) A l 29 23 <10 S i 724 400 120 C a 2310 13400 86 K 1100 <10 22 C l 629 <10 74 M g 352 139 32 M n 159 <10 <10 F e 76 29 16 There are no major differences in N P E s between hardwoods and softwoods except for the ca l c ium and potassium content. C a l c i u m is m u c h higher i n hardwoods, especial ly i n aspen. Potass ium appears to be a litt le higher i n ha rdwood . C a l c i u m and magnes ium are integral components o f the w o o d (Magnusson et al, 1979). M a k e - u p l ime is one o f the major sources for ca lc ium. S i l i c o n and magnesium contents i n the make-up l i m e are not negl ig ib le . A b o u t 5 ~ 12.2 kg/ ton a.d. pulp o f the make-up l i m e is used for the kraft pulp ing. D i s s o l v e d organic and inorganic materials i nc lud ing non-process elements are washed i n b r o w n stock washer and enter into the recovery line. A por t ion o f non-process elements are attached to the pulp, w h i c h is introduced to the b leaching l ine w i t h pulp. The metal profi le o f the pre-oxygen stage kraft pulp o f 15.3 % consistency and p H 10.3 from a m i l l ( in P r ince George , B . C . ) was determined i n our laboratory. The results are shown i n F igu re 1.2. The pulp sample o f 20 g was dried in an oven at 105 °C and ashed in a furnace at 575 °C . The ash was digested i n 6 N H C 1 solution. The digested solut ion was di luted w i t h dis t i l led and de ionized water and filtered through 0.45 p m membrane filter. The filtrate was analyzed by the I C P (Inductively C o u p l e d P lasma , A C M E A n a l y t i c a l L a b . Vancouver , B . C . ) to determine the metal contents i n the pulp. M e a n w h i l e , a por t ion o f the pulp sample was squeezed to extract the l iquor . The l iquor was filtered through 0.45 p m membrane filter. The filtrate sample was analyzed by I C P to determine the metal content i n the l iquor (shaded bars i n F igu re 1.2). M e t a l content i n the fibres (white bars in F igure 1.2) was calculated by subtracting the metal content i n the l iquor from that in the pulp. F igu re 1.2. M e t a l profile o f pre-oxygen stage kraft pulp. 25000 3 20000 Q. -ri o I? 15000 E, § 10000 c o o •jjj 5000 Na 2000 1500 1000 500 600 500 400 300 A 200 100 A T Pulp 1 Liquor Fibre K Ca Mg Si Al Fe Mn P Cu Ba In the pre-oxygen stage kraft pulp, large amounts o f N a , K , C a , and M g elements were present. A m o u n t s o f S i , A l , Fe , M n , P , C u , and B a were not negl ig ib le . M o s t o f 8 elements were distributed in fibers and their content in liquor was very low except for sodium and potassium. Table 1.2 shows metal content in acid and alkaline effluents o f an E C F bleaching plant. Metal ion level in the acid bleach effluent is higher than that in alkaline effluent. Table 1.2. Metal contents in the bleach effluent, mg/kg oven dried pulp (Towers, 1995). N a A l Si C a K M g M n Fe A c i d 1060 40 400 1650 250 300 50 125 Alkaline N D 35 290 350 60 75 7 40 N D : not detected Table 1.3. Concentration range o f process and non-process elements in green and white liquors, mol/L. mol/L element Green liquor White liquor N a 3 . 2 - 4 . 5 1 5 1 2.53 - 4 . 2 6 1 1 1 A l - 0.0029 [ 3 ] 0 . 0 0 0 1 6 - 0 . 0 0 0 5 6 m Si 0.002 - 0.009 1 3 1 0.003 - 0.006 [ 1 ] C a 0.0001 - 0.0006 [ U 1 - 0.0004 1 2 1 M g - 0.00003 [ 1 ] - 0 .000012 [ 1 ] K 0.11 - 0 . 3 2 [ 1 ] 0 . 1 2 - 0 . 2 2 m c o 3 2 - 0 . 7 - 1 . 5 1 6 1 0 . 1 8 - 0 . 4 2 l 4 ' 6 ] S 0 4 2 " 0.05 - 0.2 [ 4 5 ] 0 . 0 5 - 0 . 1 5 [ 4 > 6 ] HS" 0.3 - 0.8 [ 5 ] 0.7 t5> OH" 0.25 - 1.0 [ 4 ' 5 ] 2.3 - 2.75 1 3 , 4 1 c r 0.0513 - 0 . 5 9 1 m 0 . 0 5 3 8 - 0 . 6 2 3 [ 1 ] [1] Towers 1995, [2] Ulmgren 1995, [3] Ulmgren 1987, [4] Magnusson etal. 1979, [5] Lindberg and Ulmgren 1982, [6] Grace etal. 1989. Table 1.3 shows the concentration range o f non-process elements as well as process elements in molarity. The A l , Si, and K are soluble elements in alkaline solutions and their concentration levels are higher than those o f C a and Mg. 9 1.3.2. Adverse effects of non-process elements A l u m i n u m and s i l i con are found part icular ly i n the bark o f the p u l p w o o d . These elements are a concern i n the recovery cyc le since no large purge points are available. A c c u m u l a t i o n o f a l u m i n u m and s i l i con i n the sodium cyc le can result in the format ion o f sod ium a luminos i l ica te scales. Format ion o f these scales lowers heat transfer eff ic iency and reduces evaporat ion capacity o f the b lack l iquor evaporators (U lmgren , 1982). A n increase i n the s i l i con content o f l ime m u d lowers the react ivi ty o f the burnt l ime because a lka l i - s i l i con compounds cou ld melt on the surface o f the l ime pellets thus reducing the porosi ty o f l ime (Magnusson et al, 1979). C a l c i u m is considered a non-process element except i n the recaust ic iz ing and l i m e k i l n area. C a l c i u m can lead to scal ing problems i n a number o f m i l l areas ( G l e a d o w et al, 1996). R e c y c l i n g o f ac id ic effluents to the alkal ine browns tock increases c a l c i u m concentrations, w h i c h can cause equipment scales such as c a l c i u m carbonate, c a l c i u m oxalate and c a l c i u m sil icate. W h e n the ca l c ium concentration in ac id b leaching stages is h igh, c a l c ium sulfate scales are formed because c a l c i u m sulfate is re lat ively insoluble at l o w p H (Bryant and Edwards , 1996). A c c u m u l a t i o n o f ba r ium results i n ba r ium sulfate deposit problems i n a E C F or T C F bleach plant (U lmgren , 1996). Po tass ium enters the m i l l w i t h the w o o d chips. It combines w i t h chlor ide and leads to p lugging related problems caused by st icky recovery boi le r dust. Po tass ium has the greatest tendency to accumulate i n the sodium cyc le under c losed-cycle operation. There is no effective purge for potassium since it is soluble in a lka l ine solutions. The major undesirable impact o f potassium is i n the recovery furnace where, i n combina t ion w i t h chlor ide, it reduces the mel t ing point o f sodium salts that are entrained i n the flue 10 gas and accumulate on the tubes o f the boiler . S o d i u m and potassium produce s labbing and r ing format ion i n the l ime k i l n at h igh concentrations because o f the l o w mel t ing points o f sod ium and potassium carbonate i n the ca l c ium cyc le (Gi lbe r t and Rapson , 1980). H i g h chlor ide levels i n the recovery cyc le cause excessive cor ros ion o f the superheater tubes in the recovery boi ler , decreasing recovery bo i le r capacity, and a l l o w i n g accumula t ion o f dust on the boi ler tubes ( B l a c k w e l l and H i t z r o t h , 1992) M a g n e s i u m also enters as MgSCu used in the fiber l ine and w i t h the make-up l ime . M a g n e s i u m and phosphorus levels are important in the c a l c i u m cyc le . A c c u m u l a t i o n o f these elements causes increased deadload, reduced settling rates, and poor filterability in the c a l c i u m cyc le ( G l e a d o w et al, 1996). Fo rma t ion o f Mg(OH )2 , a gelatinuous material , causes p lugg ing problems in whi te l iquor and l ime m u d filters ( G a l l o w a y et al, 1994). Trans i t ion metals such as M n , Fe , and C u interfere w i t h b leaching by catalysing decomposi t ion o f hydrogen peroxide under a lka l ine condi t ions (U lmgren , 1996). 1.4. Recent Developments of Closed-cycle Technology A key element i n progressing towards either a c losed-cycle or effluent-free m i l l is the r ecyc l ing o f b leach plant effluents. G l e a d o w and his coworkers (1993) suggested a method to achieve complete recyc l ing o f the effluents. In this method, the a lkal ine b leach effluent is used i n browns tock wash ing and the ac id ic effluent is concentrated and added to the b lack l iquor . Evapora tor condensates are treated for reuse as process water supply. I f the ac id ic b leach effluent is recycled to the b r o w n stock washers as shower water, some o f the metals can be redeposited onto the pulp. T h i s recycle o f metals results 11 i n an increase in transit ion metal content for the pulp that is sent to the b leach plant. In addi t ion, the concentration o f scale- inducing metal ions can be increased throughout the fiberline and i n the weak b lack l iquor (Bryant and Edwards , 1996). R icha rdson et al. (1995) suggested the r ecyc l ing o f bleach-plant effluents to the green l iquor l ine. A l t h o u g h fresh water o f 50 % was replaced w i t h ac id ic b leach ing effluents(Dioo stage), caust ic iz ing eff ic iency was on ly decreased from 81.7 % to 77.4 % . The replacement o f 50 % fresh water w i t h a lkal ine b leaching effluents ( E 0 stage) just decreased the caus t ic iz ing eff iciency from 81.7 % to 76.0 % . H o w e v e r , the r ecyc l ing o f b leaching effluents resulted i n the b u i l d up o f non-process elements such as s i l i con , a luminum, ca l c ium, magnesium, and potassium. M a p l e et al. (1994) reported the development o f the b leach filtrate recycle ( B F R ™ ) process for bleach plant closure. The important aspects o f the process are chlor ide r emova l and metal removal . A l k a l i n e bleach effluents are recyc led for b r o w n stock washing . D i s s o l v e d sol ids i n the effluents are transported into the b lack l iquor evaporators and fired i n the recovery boiler . The chlor ide component in the electrostatic precipitator dust can be removed. D u r i n g the chloride removal process, the dust is m i x e d w i t h hot water. Because o f greater solubi l i ty o f sod ium chlor ide than that o f sod ium sulfate, the sodium sulfate i n the dust is precipitated out and recycled to the b lack l iquor . S o d i u m chlor ide i n aqueous form is disposed. M e t a l ions i n ac id b leach effluent are removed by chemica l precipi tat ion w i t h sod ium hydroxide and sod ium carbonate o r i o n exchange. Treated effluent is recycled to the bleach plant. U l m g r e n (1996) suggested a recyc l ing strategy for a closed bleach plant. The a lkal ine b leach effluents are taken to the b r o w n stock washer. The organic material i n the 12 alkal ine effluents can be removed by burning i n the recovery boi le r v i a the b r o w n stock washer. The ac id ic effluents can be taken into the chemica l recovery cyc le for whi te l iquor preparation. In the case o f the acid effluent f rom an E C F bleach plant, the effluent contains a h igh level o f chlor ide . These ac id ic effluents, r i ch i n chlor ide , should be treated separately to prevent the intake o f chlor ide ion . G l e a d o w et al. (1998) presented three c losed-cycle case studies us ing process s imulat ions fo l lowed by the analysis o f elemental component behaviour i n the kraft pu lp ing process. Par t ic ipat ing mi l l s were a 1965 vintage B . C . coastal E C F kraft m i l l , a modern (1990s vintage) B . C . coastal E C F m i l l , and a 1965 vintage B . C . interior m i l l . The amount o f b leach effluent, 24-54 m 3 / a .d . metric ton, is a b i g por t ion o f the total m i l l effluent, 64-99 m 3 / a .d . metric ton. They suggested c losed-cycle designs i n w h i c h a part or al l o f the b leach plant effluent was concentrated i n a bleach plant evaporator and the solids incinerated i n the exis t ing b lack l iquor recovery boi ler . R e c y c l e o f a lka l ine filtrate to b r o w n stock was also considered. A c c u m u l a t i o n o f K and C l can be contro l led by purg ing precipitator catch in the recovery boiler . S imula t ion w i t h these design options showed no increase i n the corrosiveness o f the recovery boi ler deposits and the potential for p lugging . Potass ium and chlor ide levels were constant or decreased as a result o f process modif icat ions. C a l c i u m and magnesium levels, however , in b lack l iquor raised a concern. The concentration o f these elements were increased up to 300 % . A new bleaching plant evaporator and a recovery boi ler precipitator catch leaching system are required. A d d i t i o n a l b leaching stages are be required to compensate for the decrease o f b leach ing eff ic iency in some cases. Capi ta l cost requirements are s imi lar to those o f effluent treatment facili t ies. 13 1.5. Scale Formation in the Recovery Cycle Several types o f sca l ing occur throughout the recovery cyc le o f kraft pulp mi l l s . Mate r ia l s insoluble in b lack l iquor are fibres, sand, o i l soap (at 2 5 - 3 0 % solids) , l i g n i n (at p H b e l o w 11), ca l c ium carbonate scale (at h igh temperature), sodium carbonate-sodium sulfate double salt (usually about 5 0 % solids), and sod ium a luminos i l i ca te scale. Sand and fibres are usual ly found to be combined w i t h deposits w h i c h p l u g tubes (Grace et al, 1989). C a l c i u m scal ing is ve ry sensitive to temperature i n b lack l i quo r evaporators. A complex is formed between ca l c ium ion and l i gn in sub-groups i n the l iquor . A t h igh temperature, the complex breaks down. The released c a l c i u m ions combine w i t h carbonate ions on the hot surface, where they form ca l c ium carbonate scale (Westervel t et al, 1982). Increased amount o f c a l c ium oxalate in b lack l iquor s ignif icant ly increases the tendency o f the ca l c ium carbonate scale formation (Grace et al, 1989). S o d i u m carbonate-sodium sulfate scales are also c o m m o n in b lack l iquor evaporators. The scales are soluble i n b lack l iquor . I f their concentration exceeds the so lub i l i ty l i m i t , they form by c rys ta l l i z ing from a supersaturated solution. The scales are usual ly found to have a structure o f burkeite, 2Na2S04-Na2CC>3. The so lubi l i ty at h igh temperature is s l ight ly lower than that at l o w temperature i n the range 100 ~ 140 °C Organ ic compounds i n l i quo r do not greatly inf luence the so lub i l i ty (Grace et al, 1989). The sod ium carbonate-sulfate scal ing can be aggravated by soap. Soap has a strong tendency to precipitate on fibre to fo rm a s t icky mass. Thus, soap and fibre should be kept out o f the b lack l iquor as m u c h as possible ( U l o t h and W o n g , 1985; Grace et al, 1989). 14 S o d i u m oxalate (Na2C204) scal ing occurs sometimes in the kraft pulp m i l l s us ing ha rdwood species, especial ly if weak b lack l iquor oxida t ion is used to prevent odorous emissions o f H 2 S gas . The sodium oxalate is moderately soluble in water (6.0 g/100 g at 100 °C) and shows a normal so lub i l i ty temperature relationship. Thus , so lub i l i ty increases w i t h increasing temperature. So lub i l i ty o f the sodium oxalate varies w i t h amounts o f other sod ium salts. It forms scales i n b lack l iquor evaporators by precipi ta t ion as the so l id content i n the l iquor increases (Grace et al, 1989). S o d i u m aluminosi l ica te scales are hard and glossy. Since they have a very l o w thermal conduct iv i ty , a very th in layer is enough to cause a serious reduct ion o f the evaporation capacity. A l t h o u g h these scales g r o w s lowly , they are very tenacious and diff icul t to remove (Grace et al, 1989). So lub i l i ty increases s l ight ly w i t h increasing temperature i n the range 95 ~ 150 °C (Streisel, 1987). A higher solids content (decomposi t ion products o f l ign in) i n the b lack l iquor results i n a higher so lubi l i ty . Thus , the l iquor can tolerate more soluble A l and S i before sod ium a luminos i l ica te precipitates. T h i s is probably due to the increased concentration o f decomposi t ion products o f l ign in . These organic compounds form strong chelate complexes w i t h A l . Such structures occur i n b l ack l iquor as decomposi t ion products o f l i gn in (U lmgren , 1985). Green l iquor systems also have sca l ing problems in transfer l ines, the clarifiers, and the d i s so lv ing tanks. These scales are usual ly found to have structures o f pirsonnite ( N a 2 C 0 3 N a 2 C 0 3 - 2 H 2 0 ) and gaylussite ( N a 2 C 0 3 N a 2 C 0 3 - 5 H 2 0 ) . Scales seem to be favored by decreased temperatures (Frederick and Kr i s h n an , 1990). Increased intake o f a l u m i n u m and s i l i c o n can cause sodium aluminosi l ica te formation in green and whi te l iquor lines (Wannenmacher et al, 1996). 15 1.5.1. Sodium aluminosilicate scale formation The sodium aluminosilicate scales may form in the recovery cycle of a kraft pulp mill and during the Bayer process for aluminum production (Gasteiger et al, 1992; Swaddle et al, 1994; Zheng et al, 1997; Gerson and Zheng, 1997). In the case of the kraft pulping process, aluminum and silicon ions enter the process with wood chips, water, and make-up lime as seen in Table 1.1. In particular, significant amounts of aluminum and silicon enter from unwashed chips since bark and soil on the chips are a major source of those ions (Galloway et al, 1994). If aluminum and silicon ions are introduced to the recovery cycle, those ions can not be purged out easily from the cycle since they are soluble in alkaline aqueous solutions. The build up of aluminum and silicon results in sodium aluminosilicate scale formation in the recovery cycle. The scale is glossy and hard to remove. It may lower the efficiency of heat exchangers and evaporation capacity of a kraft pulp mill (Ulmgren, 1982). The scale formation becomes more severe in closed-cycle plant operation because the effluent is recycled and the aluminum and silicon ion concentrations increase rapidly. Recycling of bleach plant effluents carries a lot of non-process elements to the recovery cycle as seen in Table 1.2. The Al and Si are not removed easily by precipitator dust removal or green and white liquor clarifications (Gleadow et al, 1998). Ulmgren (1982) studied the Na-Al-Silicate scale formation in black liquors of the recovery cycle. He measured the solubilities of Al and Si using the synthetic black liquors made of NaOH, Na 2 C0 3 , Na 2 S0 4 , and CH 3 COONa (sodium acetate) at 120 - 150 °C. The scale samples were found to have structures of NaAlSi04-l /3Na 2 C03 and/or 16 NaAlSi04-l/3Na2S04 which look like cancrinite. According to his results, one of the main factors governing formation of aluminosilicates was the Off concentration. More soluble species formed in solution as the Off concentration increased in the range of the Off concentration from 0.03 to 1.6 mol/L. In addition, high concentrations of Off caused scale formation on the surface of vessels and low concentrations resulted in precipitation as particles. Addition of HS" of 0.4 mol/L did not make any significant change in the solubility of Al and Si. The effect of temperature on the formation of sodium aluminosilicate was not significant at 120 °C, 135 °C, and 150 °C. Solubility increased slightly with increasing temperature. Ulmgren also presented a model for the formation of Na-Al-Silicate scale in evaporators for black liquor in kraft pulp mills. For the chemical modeling, he assumed the Na-Al-Silicate in the evaporators were formed by precipitation from a solution saturated with ions involved such as Na + , Al(OH)4\ Si0 2 (0H) 2 2 \ SiO(OH) 3\ Off, C 0 3 2 \ S 0 4 2 \ Cf and HS". Finally, he determined the equilibrium constant of the cancrinite formation using the experimental solubility data to correlate the precipitation conditions. Ulmgren (1987) suggested a method to reduce the concentration of aluminum in the recovery cycle by adding a magnesium salt such as magnesium sulfate and magnesium hydroxide carbonate to the smelt dissolving tank. The insoluble salt, hydrotalcite (Mgi. xAl x(0H) 2(C0 3)x/ 2nH 20, 0.10 <x< 0.34), was formed which could be removed from the process together with the dregs by the green liquor clarification. In this way, sodium aluminosilicate scale formation by the build up of Al and Si could be avoided. He observed a decrease in the aluminum concentration from 1.5 to 0.5 mmol/L by the addition of magnesium salts with molar ratio of Mg/Al=6 into the mill green 17 liquor. He also observed that removal of aluminum by the addition of the magnesium salts could be expedited by reducing the OH" concentration and/or temperature of the liquor. Streisel (1987) studied the Na-Al-Silicate formation using synthetic liquors. He prepared the liquors using NaOH and NaCl. He measured the precipitation conditions of the sodium aluminosilicates by varying the following parameters: the Al/Si molar ratio from 0.08 to 275, the hydroxyl ion concentration from 0.1 to 4 N, and the ionic strength from 1 to 4 N. The temperature was set at 95 and 150 °C. According to the results, as the hydroxide ion concentration increased, more soluble aluminum and silicon species formed in solution. The apparent solubility product of Al and Si decreased with increasing ionic strength. The apparent solubility product, [Al][Si] at 150 °C was higher than the solubility product at 95 °C. He found that the precipitates had a structure of sodalite (Na8(AlSi04)6Cl2) by the X-ray diffraction analysis. Streisel (1987) also studied the effects of the cationic ions (K, V, Fe, Mn, Ca, Mg) addition on the solubility of Al and Si. The potassium and vanadium are very soluble in alkaline solution and neither potassium or vanadium had any significant influence on the apparent solubility product of Al and Si. Iron and manganese are insoluble in alkaline solution. Iron and manganese precipitated as iron oxide (Fe203) and manganese hydroxide (Mn(OH)2). Neither iron or manganese had any significant influence on the apparent solubility product of Al and Si. Calcium is nearly insoluble in alkaline solution. The calcium has an effect on the rate of precipitation of Al and Si but not on the apparent solubility product. When the molar ratio of Al/Si is less than one, aluminum tobermorite (Ca5Si5Al(OH)n5H20) was mainly formed. When the molar ratio was equal to one, 18 sodalite and calcium hydroxide were formed. By the addition of MgS0 4 , hydrotalcite was formed. The higher Mg/Al ratios resulted in significantly more precipitation of aluminum. Streisel (1987) developed a model to predict the precipitation conditions in the concentration range of Off from 0.1 to 1.0 N for Na + - Cl" - Al(OH)4" - HSi0 4 3" - OH* -H2O system. He adopted Meissner's method (Zemaitis et al, 1986) to calculate the activity coefficients of species and extracted Meissner's model parameters from his experimental solubility data. His model was unable to calculate the concentration of aluminum at higher hydroxide concentrations because of a convergence problem. Gasteiger et al. (1992) upgraded Streisel's model. Pitzer's method (Pitzer, 1991) was adopted to calculate activity coefficients of the species and activity of water. The model contained five adjustable parameters, an equilibrium constant of sodalite/hydroxysodalite formation and four Pitzer's parameters of J3(0) and for NaHSi0 4 and p ( 0 ) and p ( 1 ) for NaAl(OH) 4. They obtained the five adjustable parameters by correlating Streisel's experimental solubility data. They found that the precipitates had the structure of sodalite (Na8(AlSi04)6Cl2) and/or hydroxysodalite (Nas(AlSi04)6(OH)2) by X-ray diffraction analysis. Wannenmacher et al. (1996) reported the effect of CaO and M g S 0 4 addition on the precipitation conditions of sodium aluminosilicate scale in an unclarified kraft mill green liquor at 95 °C. For the 1:1 addition of CaO ([CaO]:[Si]), no change in the aluminum and silicon solubility was observed. The addition of M g S 0 4 was found to lower the solubility of Al. They presented graphs showing the precipitation conditions of the Na-Al-Silicates for black, green and white liquors based on their experimental and 19 literature data. They also reported several equations to calculate apparent so lubi l i ty product, [Al][Si] at equ i l ib r ium w i t h sodalite (Na8(AlSiC»4)6Cl2) and/or hydroxysodal i te (Na8(AlSi04 )6 (OH)2) , cancrinite, natrolite, and hydrotalcite. The correlations were based o n experimental so lub i l i ty data. A c t i v i t y coefficients were not considered for the calculat ions. 1.6. R e s e a r c h Ob jec t i ve s K n o w l e d g e o f the sodium aluminosi l ica te scale precipi tat ion condi t ions is required to design c losed-cycle pulp mi l l s . A l t h o u g h the format ion o f sod ium aluminosi l ica te scales i n a lkal ine solutions has been studied i n the past (U lmgren , 1982; U l m g r e n , 1987; Streisel , 1987; Wannenmacher et al, 1996), the effects o f N a 2 C 0 3 , Na2S04 and ISfoS concentration on the precipitat ion condi t ions i n green and whi te l iquors have not yet been investigated. Thus, one objective o f this w o r k was to measure the so lubi l i ty o f A l and Si in h igh ly a lka l ine solutions at 368.15 K (95 °C) and determine the effects o f N a 2 C 0 3 , Na2S04 and Na2S addi t ion on the precipi tat ion condit ions. W e used synthetic l iquors o f two different systems, A and B , that s imulated m i l l green and whi te l iquors . The synthetic green and whi te l iquors o f system A were prepared us ing N a O H , N a 2 C 0 3 , Na2S04, and N a C l . Those o f system B were prepared us ing N a O H , N a 2 C 0 3 , and Na2S. The precipi tat ion condi t ions o f sodium aluminosi l ica te scale in m i l l green and whi te l iquors were also measured. The effects o f N a O H , N a 2 C 0 3 and Na2S at different concentrations o n the precipi tat ion condit ions i n m i l l l iquors were observed. M i l l green and whi te l iquors received f rom a m i l l i n P r ince George , B . C . were used for the study. In 20 addition, the structure of the precipitates were identified by X-ray, chemical and thermogravimetric analysis. The other objective of this work was to present a thermodynamics-based model for sodium aluminosilicate formation in aqueous alkaline solutions. Pitzer's activity coefficient method was used to calculate the activity of water and the activity coefficients of ions in the solution because it performs well even at high molalities (Pitzer, 1991; Zemaitis et al, 1986). All parameters needed by the model were obtained from independent experimental data or available property estimation methods. The effects of the anions of OH - , C O 3 2 , SO42", and HS - were taken into account. The model predicts the precipitation conditions of sodalite dihydrate and/or hydroxysodalite dihydrate. Since the system contains the Si032' and Al(OHV ions, knowledge of the relevant Pitzer's model parameters is required. Osmotic coefficient data for the Na2Si03 single electrolyte system and the Na2Si03-Na0H multi-component electrolyte system were obtained at 298.15 K by an isopiestic method. The binary parameters, P ( 0 ) , and C*, for Na 2 Si0 3 and the mixing parameters, 9 O H - s i o l - and ^ Na+0H-Si02-, for Na2Si0 3-Na0H system were determined using the osmotic coefficient data. In addition, the binary parameters, f3(0), P ( 1 ) , and C*, for NaAl(OH) 4 and the mixing parameters, © O H - ^ ^ - , e c r A i ( o H ) ; . * N . - O H - A I ( O H ) ; » A N D ^ N . - C T A K O H H a r e n e e d e d - 0 s m o t i c coefficient data for NaOH-NaCl-NaAl(OH)4 aqueous system were also obtained at 298.15 K. The unknown Pitzer's parameters relevant for the description of the system were obtained using these osmotic coefficient data. 21 CHAPTER 2. THEORETICAL BACKGROUND T h i s chapter presents the fundamental thermodynamic relations that govern the behaviour o f electrolyte solutions as w e l l as P i t ze r ' s ac t iv i ty coefficient model . In addit ion, the isopiest ic method for the measurement o f osmot ic coefficients is also presented. 2.1. T h e r m o d y n a m i c E q u i l i b r i u m C o n s t a n t L e t us consider the f o l l o w i n g reaction in an aqueous electrolyte solut ion. a A + b B = c C + d D (2.1) The condi t ion o f chemica l equ i l ib r ium can be denoted by au, A + bu, B = cue + duo (2.2) where p. is a chemica l potential. It is convenient i n aqueous solut ion thermodynamics to describe the chemica l potential o f a species i i n terms o f its act ivi ty ai. u i = H ? + R T t a ( a i ) (2.3) where u,° is a chemica l potential at an arbitrari ly chosen standard state. W h e n the compos i t ion o f the solut ion is described in terms o f the mola l i ty scale, the standard state o f i o n species is that o f a hypothetical one molar solut ion at the same temperature and pressure as the solut ion (Pitzer, 1991). F o r the solvent (water), the standard state refers to pure solvent (water) at the same temperature and pressure as the solution. The act iv i ty is 22 a measure of the difference between the component's chemical potential at the state of interest and its standard state. The deviation from ideality is defined by the activity coefficient, y = a/m. Substituting the activity of species i, ai to yjmj in equation (2.3) gives ^ - u r + R T l n C y . m J (2.4) Thus, the general expression of equation (2.2) for the equilibrium can be expanded as follows a(u° +RTln( Y A m A ) ) + b ( K +RTln(y Bm B)) = c(p.° + RTln(y c m c ) ) + d(u D +RTln(y D m D )) (2.5) By combining terms a u ° A + b u ° B - c n ° - d u ° D = R T l n ( Y cm c ) C ( V D m D ) D ( Y A m A ) A ( Y B m B ) B (2.6) Since the partial molar Gibbs free energy is also defined as the chemical potential, the u,°, can be substituted by AG° ; (standard Gibbs free energy of formation). aAGf A + bAG° B - c A G ° c - d A G ° D = RTln ( Y c m c ) C ( Y D m D ) D ( Y A m A ) A ( y B m B ) B (2.7) The solubility of A, B, C, and D species at equilibrium are expressed in molality, ITIA, me, mc, and mo respectively. Since the thermodynamic equilibrium constant for this reaction is defined as K = exp aAG° A +bAG f ° B - c A G ? c - d A G f ° t RT (2.8) the equation (2.8) can be written as follows 23 K = ( Y cm c ) C ( Y D m D ) D - A G ° - exp R T . ( Y A m A ) A ( Y B m B ) B . where A G 0 is the standard G i b b s free energy change o f reaction (2.1). (2.9) 2 . 2 . Equilibrium Constant Calculation at Specific Temperature The thermodynamic equ i l ib r ium constant K i n equation (2.9) can also be wri t ten as fo l lows . A G 0 I n K R T Different ia t ing the equation (2.10) w i t h T at constant P and compos i t ion gives D d l n K d ( A G ° / T ) d T The G i b b s - H e l m h o l t z equation is g iven by d T d_ 5T f A G 0 > A H 0 (2.10) (2.11) (2.12) where A H 0 is the standard enthalpy change o f reaction. Thus, the equation (2.11) can be wri t ten as fo l lows . R d l n K = A H : d T T 2 The A H ° can be expressed as a function o f temperature i n terms o f heat capacity, ACp. A H ° = A H ° + fT A C P d T (2.14) «r0 A s s u m i n g constant ACp i n the range o f the To and T gives A H 0 = A H ° + A C ° ( T - T 0 ) (2.15) 24 C o m b i n i n g (2.13) and (2.15) gives R d t a K = A H I + ^ n i ] d T T 2 \T I2) (2.16) Integrating the equation (2.16) between the l imi t s o f the reference temperature, To, and T gives A H 0 l n K - l n K n = °-i n T X AC; f 1 1 N l n T - l n T 0 + T 0 ( - - — ) V 1 Ao j (2.17) It can be wri t ten as fo l l ows since In K 0 = - A G Q / R T 0 . l n K = A G J V R T o J A H o c R I__L A C o f R l n ^ - 1 V T 0 T (2.18) Equa t ion (2.18) is used to calculate the equ i l ib r ium constant at any temperature, T (Ander son and Crerar , 1993). 2.3. Activity of Water, Osmotic Coefficient, and Activity coefficient A c t i v i t y and act ivi ty coefficients were introduced to describe the non-ideal behaviour o f a component i n a solution. Based o n equation 2.3, the act ivi ty o f solvent (water), a w , can be expressed by: exp R T (2.19) where p w is the chemica l potential o f the solvent, p^, is the standard state chemica l potential o f the solvent, R is the gas constant, and T is the absolute temperature. T h e standard state refers to pure solvent (water) at the same temperature and pressure as the solut ion (Pi tzer 1991). 25 A c t i v i t y , however, is not sensitive at l o w molal i t ies and requires several s ignificant digi ts to express the behaviour accurately. The practical osmot ic coefficient was int roduced to avoid this p roblem by exaggerating the devia t ion between real and ideal behaviour (Pitzer 1991, Zemai t i s et al. 1986). The practical osmot ic coefficient , <j>, is defined as fo l lows -10001naw M w 2 > i m i (2.20a) where, M w is the molecular weight o f the solvent (water), Vi is number o f ions produced by 1 m o l o f solute i , and m; is the mola l i ty o f solute i . F o r a single aqueous electrolyte solut ion w i t h a solute mola l i ty equal to m s and v ions produced when the electrolyte is d isso lved the above equation is wri t ten as - l O O O l n a . , , M „ , v m . (2.20b) L e t us n o w consider single electrolyte aqueous solution. The total G i b b s energy n G is a function o f T , P , and the numbers o f moles o f the chemica l species present (solute and water). Thus, the n G is g iven by n G = f ( T , P , n s , n w ) (2.21) where n is number o f moles, subscript w stands for solvent (water), and subscript s denotes the solute (electrolyte). The total differential o f n G is d ( n G ) = d ( n G ) ap dP + T,n d ( n G ) err d T + JP,n d(nG) an. d n . + P.T,n„ a(nG) an... d n w (2.22) P.T.n, Since the partial molar G i b b s energy o f species i is defined as fo l lows , G : = a(nG) an; (2.23) JT.P.tij 26 the equation (2.22) can be rewritten as fo l lows. d ( n G ) = ~5(nG)~ dP + "d(nG)~ T,n 5T d T + G „ d n + G . „ d n S 3 W W (2.24) P,n M e a n w h i l e , a general differential equation for (nG) at constant T and P is g i v e n by d ( n G ) = 2 n i d G , + 2 _ G , d n l and d ( n G ) = n s d G s + n w d G w + G s d n s + G w d n w (2.25) C o m p a r i s o n o f this equation (2.25) w i t h (2.24) y ie lds the G i b b s - D u h e m equation. g ( n G ) cP d P + - lT ,n d ( n G ) 5 T d T - n s d G s - n w d G w = 0 (2.26) P.n F o r the case o f 1 k g o f solvent (water) at constant T and P , the equation (2.26) becomes - m . d G - m , d G , „ = 0 W W and m s d G s = - m w d G w (2.27) where m s and m w are the molal i t ies o f the electrolyte (solute) and water (solvent) respectively. S ince the chemica l potential is defined as the partial molar G i b b s energy, the equation (2.27) becomes m s d p s = - m w d p w (2.28) T a k i n g the general formula o f solute (electrolyte) to be C v A V j , the chemica l potential o f the solute is u . = v c u . e + v . u . _ = v c p : + v . u . + v . R T l n ( a c ) + v a R T ln(a.) = v c p : + v 8 p : + v c R T l n ( m c y c ) + v a R T l n ( m a Y a ) (2.29) where subscript c stands for cat ion and subscript a denotes the anion. 27 Different ia t ing the equation (2.29) gives dn s = R T d l n ( m ^ m : - y : ' y : - ) (2.30) The mean mola l i t y and mean act ivi ty coefficient are defined as m ± = ( m > i - r = ( ™ > : c ™ > : - Y v = ^ > : - Y v (2.31) Y ± - ( Y .V - Y . V , ) , , V (2-32) where v = v c + v a . Thus , equation (2.30) becomes d n s = v R T d l n ( m ± Y ± ) (2.33) The chemica l potential o f solvent (water) is g iven b y U w - U ° + R T l n a w (2.34) Different ia t ing the equation (2.34) gives dp . w = R T d l n a w (2.35) Substi tut ing equations (2.33) and (2.35) to equation (2.28) gives v m s R T d l n ( m ± Y ± ) = - m w R T d In a w and v m s d l n ( m ± Y ± ) = - m w d l n a w ( 2 3 6 ) Ano the r expression for equation (2.20b) is g iven as fo l lows for single electrolyte system - M w v m s l n a w = - -<b 1000 and l n a w = ^ m ^ ( j ) (2.37) m w where m w = 1000 /M W =55 .509 moles /kg H 2 0 Thus, substituting the equation (2.37) to the equation (2.36) gives v m s d l n ( m ± Y ± ) = - m w d 28 m s d l n ( m ± Y ± ) = d(m3<t>) and m s d In (m±y±) = (|)dms + msd<|) (2.38) E q u a t i o n (2.38) becomes d l n ( m ± y ± ) = — d m s +d<t> m , d l n m ± + d l n y ± = - ^ - d m s +d<J) d m Since d l n ( m ± ) = d l n ( m s ) = L f rom equation (2.31), the above equation gives d l n y ± = — d m s +d<|> (2.39) m s The above relat ion between mean act ivi ty coefficient o f solute, y±, and osmotic coefficient, <|>, can be obtained by integration o f equation (2.39) from m s = 0 (where y± = 1 and <|) = 1) to m (Pi tzer 1991). l n Y ± = r ^ d m s + ( d ) - l ) (2.40) J o m 2.4. Pitzer's Activity Coefficient Model Pi tze r developed a mode l to provide improved estimation o f electrolyte so lu t ion properties by t ak ing into account the effect o f short-range forces. The P i t ze r ' s mode l is based o n the v i r i a l expansion o f the excess G i b b s free energy. T h e first term in P i t ze r ' s equation is a modi f ica t ion o f the D e b y e - H u c k e l mode l for the electrostatic effect. The second term represents short range interaction i n the presence o f solvent between t w o solute species ( ion-dipole interaction). The th i rd term represents tr iple interaction 29 between solute species i n solvent. P i t ze r ' s mode l can predict the act ivi ty coefficients o f ions accurately even at h igh mola l i ty o f ions (Pitzer, 1991; Zemai t i s et al, 1986). P i tzer expressed the excess G i b b s energy as a series o f terms in increasing powers o f mola l i ty to derive his equation (Pitzer, 1991). — — - f ( I ) + XSm1mJX i j(I) + X E Z m . m J m ^ . j k + -W,„K1 j j i j k (2.41) where, W w : number o f K g o f water mi mj nik : mola l i ty o f solute i j k I : i on ic strength ( I = ~^ m;zf ) f(I) : includes the D e b y e - H i i c k e l l i m i t i n g l aw Xij(I) = X,ji(I) : short-range interaction between solute species i and j in solvent Uijic : t r iple interaction between solute species i , j and k i n solvent The act ivi ty coefficient, Yi o f solute species i and osmot ic coefficient, <|) are obtained by differentiating equation (2.41) as fo l lows . l n y i = [ 3 ( G V W w R T ) / a n i L 4_l = _ ( a G e * / a w j n i / ( R T £ m i ) (2.42) (2.43) R e w r i t i n g equation (2.41) i n terms o f experimental ly determinable quantities B and C instead o f the ind iv idua l i o n quantities X and p. gives the f o l l o w i n g equation (2.44). G e W R T f( I ) + 2 £ £ m c m ( m r z r r C C • Z Z m m . + Z Z m m , a a aa / J c caa (2.44) +2Y Y m n ml„ c + 2V Y mnra,L + 2V V m„mB,Ln, + y m X m +• / J / J n c nc / J / J n a na f J / tl f n n nn / * n nm 30 Pitzer derived semi-empirical equations for y and by taking the derivative of equation (2.44). For single electrolyte system, y± and are given by l n Y ± = |z M z x | f T +m(2v M v x /v )B^ + m 2 [ 2 ( v M v x ) 3 / 2 / v ] C M X (2.45) 4 . - l H z M z x | f * + m ( 2 v M v x / v ) B t K + m 2 [ 2 ( v M v x ) 3 / 2 / v ] C * l x (2.46) where, y± : mean activity coefficient z : charge M : cation X : anion v : number of ions produced by 1 mole of the solute f = -A4[I1/2/(l + bl 1 / 2) + (2/b)ln(l + bl1 / 2)] (2.47) f* = -A4 1 / 2 / ( l + bl 1 / 2) (2.48) A4 : Debye-Huckel osmotic coefficient parameter b : universal parameter with the value of 1.2 (kg.mol)1/2 BY = B M X + B*MX (2.49) BMX =PSc+P (^g(a,I , / 2)+PScg(a 2I , / 2) (2.50) B L =P^x+3l!_ < exp(-a i r / 2 ) + PS c exp(-a 2 I 1 / 2 ) (2.51) =3C*x/2 (2.52) The P(0)MX, 3(1)MX, C'MX are tabulated binary parameters specific to the salt M X . The P(2)MX is a parameter to account for the ion pairing effect of 2-2 salts. When either 31 cation M or anion X is univalent, cti = 2.0. For 2-2, or higher valence pairs, cti = 1.4. The constant a 2 is equal to 12. For multi-component electrolyte solutions, y and § are given by 1 ° Y M _ Z M a c 20Mc+2>/F1 Mca a a' In y x = zxF + Z mc [2BcX + ZC c X ] + Z m c a +ZSmcmJcc,x + | Z x | _ T _ > > c m a C 2 0 X a + £ m c ¥ ( cXa lnYMx = | z M z x | F + — Z m < 2B M a +ZC M a +^0 X a V , 2B c X +ZC c X +^0 M c vx + J J m e m , v '[2vMZ c a (2.54) (2.55) (2.56) d)-l = (2/Zmi)[f*I + Z Z m o m a ( B l +ZCJ i c a + ZZmcmc,(Ot + X m X a ) + Z Z m a ^ a « . + Z ^ c a a ) ] (2.57) c c' The quantity F in the first term of equations (2.54), (2.55), and (2.56) includes the Debye-Huckel term given by F = f Y + Z Z m c m a B ; a + Z Z m c m c , 0 : c , + Z Z n i a m a . O ; a . (2.58) c a c < c' a < a' Also, z = Z m i l z i (2.59) 32 Bc,=fc)g'(a1I1/2)+3L2)g'(a2I1/2)]/I (2.60) * i j = e i j + Be i j(o (2.61) (2.62) <D+=0> + K> IJ IJ IJ (2.63) Gij are tabulated mixing parameters specific to the cation-cation or anion-anion pairs. The is also tabulated mixing parameter specific to the cation-anion-anion or anion-cation-cation pairs. The parameters E6ij(I) and E0'ij(I) represent the effects of unsymmetrical mixing. These values are significant only for 3-1 or higher electrolytes (Pitzer, 1975). The g(x) and g'(x) are functions accounting for the ionic strength dependence of B M x and B'MX given by The binary parameters P ( 0 ) , and C* have larger effects than the mixing parameters, *F and 9 (Pitzer, 1991). 2.5. I sopies t ic M e t h o d There are several methods to determine the electrolyte thermodynamic properties such as activity coefficient, activity of solvent, and osmotic coefficient. The activity coefficient can be determined by the electromotive force (e.m.f.) method. The activity or osmotic coefficient of the solvent can be determined from the measurement of the vapor pressure of the solvent and from the isopiestic method. The e.m.f. method and vapor g(x) = (2.64) g'(x) = (2.65) 33 pressure measurement are not suitable for this study. T h e e.m.f. method is one o f the most precise measurements, but, proper reversible electrodes are not a lways avai lable for many ions i nc lud ing s i l i c i c ions. In the case o f vapor pressure measurements by the static method, ve ry precise control o f temperature is required because the vapor pressure o f the solvent varies rapid ly w i t h even very smal l temperature change. The isopiest ic method is s impler to perform and more accurate than direct vapor measurements. There are, however , some l imita t ions for the isopiestic method. The solvent should be the on ly vola t i le component i n the system. Furthermore, measurements b e l o w a mola l i ty o f 0.1 do not show good re l iabi l i ty (Pitzer, 1991; Thiessen and W i l s o n , 1987). T h e osmot ic coefficient is part icularly useful for treating isopiestic data (Pitzer, 1991). Sample solutions and one or more reference solutions are prepared i n separate open containers. F o r this study, t w o sample solutions, one reference solut ion ( N a C l ) , and one standard solut ion (KC1) were prepared. The in i t ia l concentrat ion and mass o f each solut ion should be k n o w n . The reference and standard solutions are chosen as the electrolytes whose osmot ic coefficient data are already k n o w n at that temperature. T h e data for the reference solut ion are used to determine the osmot ic coefficients o f the sample solutions. The standard solut ion is used to check the re l iab i l i ty o f the experiments and calculate experimental errors. The containers are placed i n a c losed chamber where a l l solutions share the same vapor phase. The vapor space o f the chamber is evacuated to contain on ly water vapor. Solvent is transferred through the vapor phase. The chamber conta in ing the solutions is kept at isothermal condi t ions at specific temperature unti l no more change in the concentrat ion o f the solut ion is observed, thus, thermodynamic equ i l i b r ium is reached. 34 When the solutions are in thermodynamic equilibrium, then p*w M" w M*w M"w (2.66) where p.w is the chemical potential of water and superscripts, 1, 2, R, and S, refer to sample solution 1, sample solution 2, reference solution, and standard solution, respectively. Thus, the activity of the solvent is the same. In a'w = In a* = In a* = In a* (2.67) Substituting equation (2.67) to equation (2.20) gives the following relationship for the osmotic coefficients: (fr'^v.m, =f £ V j m . = ^ 2 v k m k = 4>85>1m1 (2.68) i j k 1 where, <|> is the osmotic coefficient, v is the number of ions produced by 1 mole of the solute, and m is the molality of the solute. The subscripts, i, j , k, and 1, refer to each component in sample 1, sample 2, reference, and standard solutions. Following isopiestic equilibration, the solutions are weighed. The molality of each solution is calculated from the measured weight. Subsequently, the activity of solvent or osmotic coefficient of the standard solution at that molality is calculated by an activity coefficient model using known parameters from the literature. The activity of solvent and the osmotic coefficient of the other solutions at that molality can be determined using equations (2.67) and (2.68). 35 CHAPTER 3 . MATERIALS AND METHODS Solubility experiments were performed using two types of synthetic liquors (system A and B) and mill liquors to measure the precipitation conditions of sodium aluminosilicates at 368.15 K (95 °C). The concentration levels of NaOH, Na2C03, Na2S04, NaCl, and Na 2S in synthetic green and white liquors were designed based on those in various mill liquors in Table 1.3. The structure of the precipitates was identified by X-ray diffraction, chemical and thermogravimetry analysis. An isopiestic method was used to measure the osmotic coefficients for the Na 2 Si0 3 , Na 2Si0 3-NaOH, and NaOH-NaCl-NaAl(OH) 4 aqueous systems at 298.15 K . 3.1 Solubility Experiments Using Synthetic Liquors of System A The major difference in the chemical composition of green and white liquors is the concentration level of NaOH and N a 2 C 0 3 (Smook, 1992). White liquor contains a higher concentration of O H - and lower of C 0 3 2 ' than those in green liquor as seen in Table 1.3. Two synthetic green and white liquor systems were prepared to study the precipitation conditions of sodium aluminosilicate complex in the systems. 3.1.1 Experimental design System A was prepared by dissolving the following six salts (reagent grade, Fisher Scientific, Vancouver, B.C.) in water: A1C13-6H20, Na 2Si0 3-9H 20, NaOH, NaCl, Na 2 C0 3 , and Na 2 S0 4 . Distilled water was used in all experiments after it was deionized 36 and filtered through a 0.05 um filter in an E L G A S T A T U H P water purification apparatus (Fisher Scientific, Vancouver, B.C.). Based on typical mill concentration levels, a base concentration level was formulated and then perturbations around it gave the other compositions. The experimental design is shown in Table 3.1. The effects of the input molar ratio, Al/Si, and concentrations of N a O H , Na2C03, and N a 2 S 0 4 on the solubility of aluminum and silicon were determined by using system A. Table 3.1. So lub i l i ty experiment design for synthetic l iquors o f system A . Synthetic green l i quo r Exper iment Concentrat ions i n in i t ia l solutions ( m o l / k g H2O) A I C I 3 6 H 2 C » N a 2 S i O r 9 H 2 0 N a O H N a 2 C 0 3 N a 2 S 0 4 N a C l A l * 0.025 0.025 1.0 1.0 0.1 0.25 A 2 * 0.01 0.04 1.0 1.0 0.1 0.25 A 3 * 0.04 0.01 1.0 1.0 0.1 0.25 A 4 0.025 0.025 0.25 1.0 0.1 0.25 A 5 0.025 0.025 2.0 1.0 0.1 0.25 A 6 0.025 0.025 1.0 0.5 0.1 0.25 A 7 0.025 0.025 1.0 1.5 0.1 0.25 A 8 0.025 0.025 1.0 1.0 0.05 0.25 A 9 0.025 0.025 1.0 1.0 0.2 0.25 Synthetic whi te l iquor A 1 0 * 0.05 0.05 2.5 0.3 0.1 0.1 A l l * 0.02 0.08 2.5 0.3 0.1 0.1 A 1 2 * 0.08 0.02 2.5 0.3 0.1 0.1 A 1 3 0.05 0.05 2.0 0.3 0.1 0.1 A 1 4 0.05 0.05 3.0 0.3 0.1 0.1 A 1 5 0.05 0.05 2.5 0.1 0.05 0.1 A 1 6 0.05 0.05 2.5 0.5 0.15 0.1 A 1 7 0.05 0.05 2.5 0.3 0.05 0.1 A 1 8 0.05 0.05 2.5 0.3 0.15 0.1 * base concentrat ion leve l experiments 37 The amounts o f AICI36H2O and N a 2 S i 0 3 - 9 H 2 0 that were added into whi te l iquor were double those added in green l iquor i n order to obtain enough precipitate samples for X - r a y diffraction, thermogravimetr ic , and chemica l analysis. 3.1.2 Experimental procedure T w o solutions were prepared by d i s so lv ing measured amounts o f A l C l 3 - 6 H 2 0 and Na2Si03-9H_C) i n 1 k g o f deionized water respectively. P roper amounts o f N a O H , Na2C03, Na2S04, and N a C l salts were added to the solutions to make the desired in i t ia l condit ions for the experiments. The solutions were stirred unt i l a l l salts were d issolved complete ly and kept at 368.15 K (95 °C) . The solutions were then poured into a vessel made o f stainless steel to create a supersaturated solut ion at the beg inn ing o f the experiments. E q u i l i b r i u m vessels o f capacity 4 L and 2.5 L were buil t for the experiments. A schematic o f the vessel is shown in F igu re 3.1. The inside o f each vessel was coated w i t h Te f lon to prevent any chemica l attack from the alkal ine solutions. A var iable speed stirrer was attached to the vessel . The stirrer shaft and blade were made o f Tef lon-coated steel. A thermocouple probe and a sampl ing port were buil t into the l i d o f the vessel . A T e f l o n -coated thermocouple probe was used. A sampl ing tube, w h i c h was also made o f Tef lon , was passed through the sampl ing port dur ing sampl ing. A n o-r ing was placed between the l i d and the vessel to prevent evaporation o f the experimental solut ion. A Te f lon pack ing g land was used around the stirrer shaft to prevent evaporation. The vessel conta in ing the solut ion was covered w i t h a l i d . The l i d o f one vessel was made o f stainless steel and coated w i t h Te f lon and that o f the other vessel was made o f plexiglass . The vessel was 38 then placed in a water bath at 368.15 K. The air inside the vessel was replaced with nitrogen gas to prevent undesirable reactions such as those with the carbon dioxide in the air. The solution was stirred during the experiments. Figure 3.1. Equilibrium vessel for the solubility experiments. Thermometer Stirrer Packing Gland Sampling Tube O-Ring Shaft and Blade 39 Liquid phase samples were obtained from the vessel using a syringe. Each sample went through a Teflon pipe and a 0.5 um polyvinylidene fluoride membrane filter (Millipore Canada, Nepean, Ontario). It was found by dynamic light scattering (DLS) that all of the precipitate particles were larger than 3 um, meaning all portions of the precipitate were trapped by the filter. The DLS experiments were conducted by using a Malvern Zetasizer III apparatus (Malvern Instruments Inc., Malvern, UK). The filtrates were then analyzed by a GBC 904 Atomic Absorption Spectrophotometer (GBC Scientific Equipment Pty Ltd., Victoria, Australia) using the nitrous oxide-acetylene flame to measure the total soluble Al and Si. The procedure of Atomic Absorption analysis is explained in detail in APPENDIX I. The samples were taken over a period of time until no change in Al and Si concentration was observed. This was taken to be the equilibrium state of the system. It takes about seven days to reach equilibrium. The other approach of solubility experiment by adding sodium aluminosilicate precipitates into an unsaturated solution of Al and Si was not tried because dissolution of the solid precipitate was too slow as it was found by preliminary experiments. In particular, a small amount, 0.005 g, of precipitates could not be dissolved significantly in 100 mL of alkaline solution containing NaOH of 1 mol/kg H 2 0 after four weeks. 3.1.3. Identification of the precipitates After equilibrium was reached, precipitates, which deposited on the wall, stirrer blade, and bottom of the vessel were collected, washed with deionized water several times and dried in an oven at 105 °C. The structure of the dried precipitates was identified by X-ray diffraction (D5000 Diffractometer, Siemens Aktiengesellschaft, Germany) and 40 thermal analysis using a TGS-2 thermogravimetric analyzer (Perkin-Elmer, Norwalk, Connecticut, USA). The software, FIFFRAC / AT v.3.1 (Siemens Industrial Automation, Inc., Madison, Wisconsin, USA), was used to compare the X-ray patterns of samples with those of reference minerals. A portion of the solid materials was dissolved in nitric acid solution for analysis to obtain the molar ratios of Na, Al, and Si using the Atomic Absorption Spectrophotometer (see APPENDIX I). The presence of chlorine and sulfate in the solid were tested by mixing samples of dissolved solid with a AgNC>3 and BaCl 2 solution respectively (Greenberg et al, 1992; Masterton and Slowinski, 1972). The presence of carbonate in the structure of the solids was tested by dissolving the solid in an acidic solution to observe formation of bubbles of CO2 gas (Masterton and Slowinski, 1972). Finally, images of the precipitates were obtained on carbon-coated samples using a Philips XL30 scanning electron microscope (Philips Electronics, Eindhoven, Netherlands). 3.2. Solubility Experiments using Synthetic Liquors of System B 3.2.1. Experimental design The synthetic liquors of system B were prepared to study the effect of Na 2S. The liquor concentrations were chosen to resemble closely mill liquors from a Kraft pulp mill in B.C., Canada. Two salts, N a 2 S 0 4 and NaCl, were not added into the liquors of system B in order to make the system simpler with only major anions. It is also noted that the concentrations of SO42- and Cl" were very small in the mill liquors compared with those of other anions such as OH", HS", and CO3 2 ' (Magnusson et al, 1979). 41 System B was prepared similarly with the following salts: AICI36H2O, Na2Si03-9H20, NaOH, Na2C03, and Na2S. The perturbations around the base concentration level gave the other compositions at which experiments were conducted as seen in Table 3.2. The liquors were prepared by controlling the added amount of Na2S for both green and white liquors. The amount of input Na2S was varied from 0 to 1.0 mol/kg H2O to observe the effects of H S - and OH - concentrations in synthetic green liquor. It is known that Na2S dissociates to Na+, H S - and OH" in H 2 0 according to the following reaction (Smook, 1992): Na2S + H 2 0 ->• 2 N a + + H S " + O H " . Table 3.2. So lub i l i ty experiment design for synthetic l iquors o f system B . Synthetic green liquor Exper iment Concentrat ions i n in i t ia l solutions ( m o l / k g H2O) AICI36H2O N a 2 S i 0 3 - 9 H 2 0 N a O H N a 2 C 0 3 N a 2 S B l * 0.05 0.05 0.25 1.0 0.5 B 2 * 0.1 0.05 0.25 1.0 0.5 B 3 * 0.05 0.1 0.25 1.0 0.5 B4 0.05 0.05 0.25 1.0 0.0 B5 0.1 0.05 0.25 1.0 0.0 B6 0.05 0.1 0.25 1.0 0.0 B7 0.05 0.05 0.25 1.0 1.0 B8 0.1 0.05 0.25 1.0 1.0 B9 0.05 0.1 0.25 1.0 1.0 Synthetic whi te l i quor B I O * 0.05 0.05 2.0 0.25 0.5 B l l * 0.1 0.05 2.0 0.25 0.5 B 1 2 * 0.05 0.1 2.0 0.25 0.5 B 1 3 0.05 0.05 2.0 0.25 0.0 B 1 4 0.1 0.05 2.0 0.25 0.0 B 1 5 0.05 0.1 2.0 0.25 0.0 B 1 6 0.05 0.05 2.0 0.25 1.0 B 1 7 0.1 0.05 2.0 0.25 1.0 B 1 8 0.05 0.1 2.0 0.25 1.0 base concentration level experiments 42 3.2.2. Experimental procedure and analysis Two solutions were prepared by dissolving measured amounts of A I C I 3 6 H 2 O and Na2Si03-9H20 in 1 kg of deionized water respectively. Proper amounts of NaOH, Na2C03, and Na 2S salts were added to the solutions to make the desired initial conditions for the experiments. The subsequent experiments and analysis procedures are similar to those in sections 3.1.2 and 3.1.3. 3.3. Solubility Experiments using Mill Liquors Prior to the solubility experiments, basic information on the mill liquors was obtained and is presented in Table 3.3. The liquor samples were passed through a 0.5 urn polyvinylidene fluoride membrane filter (Millipore Canada, Nepean, Ontario) to remove particles. The concentrations of Al, Si, and K were determined using a GBC 904 Atomic Absorption Spectrophotometer (GBC Scientific Equipment Pty Ltd., Victoria, Australia). The chlorine content was measured by titration with 0.1 M AgN0 3 solution using a Mettler Toledo DL25 titrator (Fisher Scientific, Vancouver, B.C.). A Mettler DM141 silver electrode (Fisher Scientific, Vancouver, B.C.) was used for the titration. The concentrations of NaOH, Na2C03, and Na2S were obtained from the A B C test (Grace et al, 1989) done by the Prince George Lab of the Pulp and Paper Research Institute of Canada. The A B C test is a titration method (TAPPI Standard Method T624 os-68) using HC1 solution to determine the amount of titratable alkali such as NaOH, Na2S, and Na2C03 in green and white liquors (Grace et al, 1989). 43 3.3.1. E x p e r i m e n t a l des ign Exper iments were performed w i t h samples prepared us ing the m i l l green and whi te l iquors and adding appropriate chemicals to simulate: (a) progressive system closure: and (b) observed variat ions i n the concentrations o f carbonate and other ions i n m i l l l iquors . Table 3.3. A n a l y s i s results o f m i l l l iquors . C aemical concentrations (mo l /kg H 2 0 ) A l S i K N a O H N a 2 C 0 3 N a 2 S C l Green liquor N . D . 0.0008 0.245 0.29 (NaOH* 0.85) 1.05 0.56 (NaHS* 0.56) 0.084 White liquor N . D . 0.0008 0.244 2.06 (NaOH* 2.63) 0.23 0.57 (NaHS* 0.57) 0.076 N . D . : not detected * : The concentrations o f N a O H and N a 2 S can also be expressed as N a O H * and N a H S * since 1 m o l o f N a 2 S dissociates to 1 m o l o f N a H S and 1 m o l o f N a O H in water by the f o l l o w i n g reaction : N a 2 S + H 2 0 -> N a H S + N a O H (Smook , 1992). The detailed experimental design is shown i n Table 3.4. Firs t , experiments w i t h base concentration levels were performed us ing the green l iquors (experiments M 1 , M 2 , and M 3 ) i n Table 3.4. Perturbations around the base-case concentration level gave the compos i t ion for experiments M 4 to M 1 2 in order to observe the effects o f N a O H , N a 2 C 0 3 , and N a 2 S on the precipi tat ion condit ions o f sodium aluminosi l ica te . In the case o f experiments w i t h the m i l l whi te l iquor , the base concentration level is the compos i t ion o f l iquors in experiments M 1 3 , M 1 4 , and M 1 5 . A g a i n , perturbations around this base concentration level were done and gave the composi t ions for experiments M l 6 to M 2 4 . 3.3.2. Experimental procedure and analysis The subsequent experiments and analysis procedures are s imi lar to those i n sections 3.1.2 and 3.1.3. The sodium, a luminum, s i l i con , and potassium contents in the so l id precipitates were analyzed us ing the A t o m i c Abso rp t i on Spectrophotometer after d i s so lv ing the so l id samples i n ni tr ic ac id solution. Table 3.4. So lub i l i ty experiment design for m i l l l iquors . v l i l l green l iquor E x p . Concentrat ions i n in i t ia l solutions (mol /kg H 2 0 ) A 1 C 1 3 - 6 H 2 0 N a 2 S i 0 3 - 9 H 2 0 N a O H N a 2 C 0 3 N a 2 S M l * 0.05 0.05 0.29 1.05 0.56 M 2 * 0.10 0.05 0.29 1.05 0.56 M 3 * 0.05 0.10 0.29 1.05 0.56 M 4 0.05 0.05 1.29 1.05 0.56 M 5 0.10 0.05 1.29 1.05 0.56 M 6 0.05 0.10 1.29 1.05 0.56 M 7 0.05 0.05 0.29 1.55 0.56 M 8 0.10 0.05 0.29 1.55 0.56 M 9 0.05 0.10 0.29 1.55 0.56 M 1 0 0.05 0.05 0.29 1.05 1.06 M i l 0.10 0.05 0.29 1.05 1.06 M 1 2 0.05 0.10 0.29 1.05 1.06 M i l l w h i l te l iquor M 1 3 * 0.05 0.05 2.06 0.23 0.57 M 1 4 * 0.10 0.05 2.06 0.23 0.57 M 1 5 * 0.05 0.10 2.06 0.23 0.57 M 1 6 0.05 0.05 2.56 0.23 0.57 M 1 7 0.10 0.05 2.56 0.23 0.57 M 1 8 0.05 0.10 2.56 0.23 0.57 M 1 9 0.05 0.05 2.06 0.48 0.57 M 2 0 0.10 0.05 2.06 0.48 0.57 M 2 1 0.05 0.10 2.06 048 0.57 M 2 2 0.05 0.05 2.06 0.23 1.07 M 2 3 0.10 0.05 2.06 0.23 1.07 M 2 4 0.05 0.10 2.06 0.23 1.07 * base concentrat ion level experiments 45 3.4. Osmotic Coefficient Measurement for Na2Si03 and Na2Si03-Na0H Systems A q u e o u s alkal ine si l icate solutions contain a variety o f s i l i c i c ions, or thos i l ic ic ( S i 0 4 4 \ H S i 0 4 3 _ , H 2 S i 0 4 2 \ H 3 S i 0 4 \ and F L j S i O ^ or metas i l ic ic species ( S i 0 3 2 \ H S i 0 3 ' , and H 2 S i 0 3 ) . Several reactions related to the s i l i c i c species take place in aqueous m e d i u m (Jendoubi et al, 1997). A l t h o u g h thermodynamic calculat ions have been done to check the stabili ty between or thos i l ic ic and metas i l ic ic species (Babushk in , 1985), the nature o f stable s i l i c i c species i n a lkal ine solutions, when metasil icate salt is d issolved in water, has not yet been elucidated experimental ly. Unders tanding the nature o f s i l i c i c ions i n alkal ine solutions is a prerequisite to the study o f electrolyte thermodynamic properties o f those ions. A ti tration method was used to identify the species present when the sodium metasil icate is d issolved i n aqueous alkal ine solut ion. 3.4.1. Identification of silicic species by a titration method A n aqueous solut ion o f N a 2 S i 0 3 - N a O H was prepared and titrated us ing H C 1 solut ion. The concentration o f N a 2 S i 0 3 was 0.25 m o l / L and that o f N a O H was 1.00 m o l / L . The solut ion was prepared by d i s so lv ing proper amounts o f N a 2 S i 0 3 - 9 H 2 0 (sodium metasil icate, Reagent grade, F i sher Scient if ic , Vancouve r , B . C . ) and N a O H (Reagent grade, F isher Scient i f ic , Vancouver , B . C . ) in d is t i l led and de ionized water. The solut ion was then stirred us ing a magnetic bar coated w i t h Te f lon for a week at room temperature to a l l ow for any possible reaction such as convers ion o f metas i l ic ic species to or thos i l ic ic species to occur (Gasteiger, 1988). The H C 1 solut ion o f 1 ± 0.005 m o l / L (Fisher Scient i f ic , Vancouver , B . C . ) was used as a titrant. The titration was conducted by 46 using a Mettler Toledo DL25 titrator (Fisher Scientific, Vancouver, B.C.) equipped with a Mettler DG111-SC pH electrode (Fisher Scientific, Vancouver, B.C.). The pH electrode was calibrated using two buffer solutions of pH 4 and pH 7 purchased from Fisher Scientific. The measured pH values were 12.85 for NaOH solution of 0.1 mol/L and 11.92 for that of 0.01 mol/L respectively. Calculated pH values using Pitzer's equation (2.46) were 12.89 and 11.96 respectively. This result shows that a reliable measurement was available in high pH range using the calibrated electrode. The titration proceeded until the pH reached a value of 2.5 with an increment of the titrant of 0.05 mL. 3.4.2. A p p a r a t u s a n d chemica l s The isopiestic apparatus of four-neck type for osmotic coefficient measurement was designed by modifying a known three-neck type apparatus (Thiessen and Wilson, 1987). One set of the isopiestic apparatus is shown in Figure 3.2. The neck-type apparatus has been used by other researchers also (Lin et al, 1996; Ochs et al, 1990). This neck-type apparatus has several advantages compared to the conventional one. It has smaller capacity (140 mL). The solutions are in good thermal contact since the heat can be transferred through the glass from the water of the water-bath to the sample solutions. In the case of the conventional isopiestic apparatus, the heat is transferred through the vacuum, and it takes long time to reach the equilibrium. The time to reach the equilibrium can be substantially reduced by using a neck-type apparatus. 47 F igure 3.2. The isopiest ic apparatus. sample bottle c a P 48 Four sets of the isopiestic apparatus were built by the Canadian Blowing Company (Richmond, B.C.) in order to speed up the work. One set of the isopiestic apparatus consists of a four-neck flask equipped with a vacuum stopcock, four sample bottles, and their caps. The capacity of the four-neck flask is 100 mL and each neck has a 14/20 joint of female connector. The four-necks are attached symmetrically to the flask. The high vacuum stopcock is attached to the top center of the flask. The capacity of the flat-bottom sample bottle is 10 mL and the neck has a 14/20 compatible joint of male connector. The water-bath (Model 1187, VWR Scientific, Richmond, B.C.) equipped with a digital temperature controller was used to keep the temperature constant. A circulation unit of the water-bath can minimize the temperature gradient inside the water-bath. The temperature stability of the water-bath was + 0.01 °C. The readability of the temperature controller was 0.001 °C. Reagent grade salts (Fisher Scientific, Vancouver, B.C.) were used without further purification. The NaCl, KCI, and NaOH salts were dried in a oven at 105 °C for 24 hours before use. Sodium metasilicate, N a 2 S i 0 3 - 9 H 2 0 , was used without further drying. Distilled water was used in all experiments after it was deionized and filtered through a 0.05 um filter in an E L G A S T A T UHP water purification apparatus (Fisher Scientific, Vancouver, B.C.). The NaCl solution was used as a reference whereas the KCI solution was used as the standard to compare our results with data from the literature. 49 3.4.3. E x p e r i m e n t a l p r o c e d u r e The salt solutions were prepared to the desired concentrations by weighing and dissolving. The first bottle of each apparatus contained the reference (NaCl) solution. The second bottle contained the standard (KG) solution. The remaining two bottles contained the sample solutions of same initial concentration. The sample solution of Na2Si03 was prepared by dissolving the sodium metasilicate, Na.Si03-9H20, salt in the water. The sample solution of Na2SiC>3-NaOH was prepared by dissolving proper amounts of Na2Si03-9H20 and NaOH salts. The molalities of Na2Si03 and NaOH were the same in the sample solution. The presence of Si032', metasilicic ion in alkaline aqueous solution, when the sodium metasilicate is dissolved, will be explained in section 5.1. The ground-glass surfaces of each sample bottle were slightly coated by a silicon vacuum grease. The sample bottles were put into the four-neck flask. The apparatus was evacuated slowly to remove the air and the dissolved gas in the sample solutions using a vacuum pump. Four sets of the apparatus were available. In order to determine the equilibration time, the following procedure was used. The four sets were placed in the water-bath at 298.15 K. Aiter two days, one of them was taken out from the water-bath. The solutions of each sample bottle were weighed. The remaining sets were taken out after two or three more days until no change in the solute concentration was observed. It was then assumed that the total elapsed time is the required equilibrium time. It was found that the equilibrium was reached in five days. It was then decided to set the equilibrium time to seven days. Subsequently, all four sets were used to obtain data. Following equilibration, the solutions were weighed. The molality of each solution was calculated from the measured weight. The osmotic coefficient of the sample 50 solut ion was calculated from the molal i t ies o f the sample solut ion and o f the reference ( N a C l ) solut ion and from the osmotic coefficient o f the reference ( N a C l ) solut ion. The osmotic coefficient o f the reference N a C l solut ion was calculated by P i t ze r ' s mode l us ing equation (2.54) i n section 2.5 w i t h P i t ze r ' s parameters, p ( 0 ) Naci = 0.0765 , P ( 1 ) Naci = 0.2664 , and c W i = 0.00127 at 298.15 K (Pitzer, 1991). The measured osmot ic coefficient o f the K C 1 solut ion was compared to calculated data us ing publ ished P i t ze r ' s parameters to ensure the accuracy o f the experiments (Pitzer, 1991). 3.5. Osmotic Coefficient Measurement for NaOH-NaCl-NaAl(OH)4 System 3.5.1. Apparatus and chemicals The apparatus was the same one used i n the previous osmot ic coefficient study for the Na2Si03 and Na2Si03-NaOH systems. Reagent grade salts (Fisher Scient i f ic , Vancouve r , B . C . ) were used without further purif icat ion. The N a C l , K C 1 , and N a O H salts were dried i n an oven at 105 °C for 24 hours before using. A l u m i n u m chlor ide hydrate, A I C I 3 6 H 2 O , was used wi thout further drying. D i s t i l l e d water was used i n a l l experiments after it was de ionized and filtered through a 0.05 p m filter i n an E L G A S T A T H H P water pur i f icat ion apparatus (Fisher Scientif ic , Vancouve r , B . C . ) . The N a C l solut ion was used as reference whereas the K C 1 solut ion was used as standard to compare our results w i t h data from the literature. A l u m i n u m ion speciation i n aqueous solutions changes w i t h both p H and concentration. In di lute solutions at p H above 9, it is c o m m o n l y assumed that aluminate ion , A l ( O H V , is the predominant one. The formation o f other a l u m i n u m ions such as 51 A l 2 0 ( O H ) 6 2 ' and A l ( O H ) 6 3 " was not considered for this study since these ions can be present on ly at concentrations o f a l u m i n u m above 1.5 m o l / L and at extremely h igh p H (Swaddle etal, 1994; P o k r o v s k i i a n d H e l g e s o n , 1995). The sample N a O H - N a C l - N a A l ( O H ) 4 aqueous solut ion was prepared by d i s so lv ing a proper amount o f A1C13-6H20 in a N a O H aqueous solut ion. T h e concentrations o f a l u m i n u m and h y d r o x y l ions were such that format ion o f other a l u m i n u m ions except for A l ( O H ) 4 ' was prevented. In al l solutions, the input mola l i ty ratio o f AICI36H2O to N a O H was less than 1 : 5 so that a l l a l u m i n u m salts were complete ly d isso lved and were present as A l ( O H ) 4 " ion . Thus, the species present i n the sample solutions were assumed to be N a + , O H " , C l " , and A l ( O H ) 4 " on ly . 3.5.2. E x p e r i m e n t a l p r o c e d u r e The experimental procedure for N a O H - N a C l - N a A l ( O H ) 4 system was s imi la r to that o f the previous osmot ic coefficient study for the Na2Si03 and Na2Si03-NaOH systems. B r i e f l y , an experiment proceeded as fo l lows. The solutions were prepared to the desired concentrations by w e i g h i n g and d isso lv ing . The first bottle o f each apparatus contained the reference ( N a C l ) solution. The second bottle contained the standard ( K C I ) solut ion. The remain ing two bottles contained the sample solutions o f the same in i t ia l concentration. The ground-glass surfaces o f each sample bottle were s l ight ly coated by a s i l i con v a c u u m grease. The sample bottles were put into the four-neck flask. T h e apparatus was evacuated in order to remove the air and the d issolved gas in the sample solutions us ing a v a c u u m pump. The solutions in the apparatus were equil ibrated i n a w e l l thermostated condi t ion at 298.15 K . 52 A t equ i l ib r ium, the chemica l potential o f water is the same in the t w o sample solutions and i n the reference and standard solut ion. T h i s condi t ion may be expressed as fo l l ows (Pitzer , 1991) • T v i m i = * 2 I v i m i = * R I v A = 4 S _ _ > . m . (2-54) i j k I where, <J> is the osmot ic coefficient, v is the number o f ions produced by 1 mole o f the solute, and m is the mola l i ty o f the solute. The superscripts, 1, 2, R , and S, refer to sample solut ion 1, sample solut ion 2, reference solut ion, and standard solut ion. T h e subscripts, i , j , k, and 1, refer to each component i n sample 1, sample 2, reference, and standard solutions. E q u i l i b r i u m in each experiment was reached in seven days. The measured osmot ic coefficient o f the K C 1 solut ion was compared to calculated data us ing publ ished P i t ze r ' s parameters to ensure the accuracy o f the experiments (Pitzer , 1991). F o l l o w i n g equi l ibrat ion, the solutions were weighed . The mola l i ty o f the solutes i n each solut ion was calculated from the measured weight . The osmot ic coefficients o f the sample solutions were then calculated from the measured mola l i t ies o f the sample solutions and o f the reference ( N a C l ) solut ion, and from the osmot ic coefficient o f the reference ( N a C l ) solut ion. The osmot ic coefficient o f the reference N a C l solut ion was calculated by P i t ze r ' s mode l us ing equation (2.54) i n section 2.5 w i t h P i t ze r ' s parameters, P ( 0 ) Nad= 0.0765 , P ( 1 ) N aci = 0.2664 , and C * N a c i = 0.00127 at 298.15 K (Pitzer, 1991). The measured osmot ic coefficient o f the K C 1 solut ion was compared to calculated data us ing publ ished P i t ze r ' s parameters (Pitzer, 1991) to ensure the accuracy o f the experiments. 53 CHAPTER 4. THERMODYNAMIC MODELING OF SODIUM ALUMINOSILICATE FORMATION A thermodynamic equ i l ib r ium model has been developed to predict sod ium aluminosi l ica te precipi tat ion condit ions i n a lkal ine solutions at 368.15 K . A method based on the equ i l ib r ium constants o f the so l id formation reactions was used for this mode l ing (Ander son and Crerar , 1993). P i tze r ' s method was adopted to calculate the act ivi ty o f water and ion act ivi ty coefficients. 4 .1 . M o d e l E q u a t i o n s The predominant a luminum and s i l i con species in solut ion were assumed to be A l ( O H ) 4 ' and S i 0 3 2 ' respectively because the system is very a lka l ine (pH>13) (Babushk in et al, 1985; P o k r o v s k i i and Helgeson , 1995; Swaddle et al, 1994). Thus, the mode l considers that eleven species are present at equ i l ib r ium at 368.15 K and 1 atm: two so l id species ( N a 8 ( A l S i 0 4 ) 6 C l 2 - 2 H 2 0 and N a 8 ( A l S i 0 4 ) 6 ( O H ) 2 - 2 H 2 0 ) , one l i qu id ( H 2 0 ) , and eight ions ( N a + , A l ( O H ) 4 \ S i 0 3 2 ' , O H ' , C 0 3 2 ' , S 0 4 2 " , HS" , and C l ) . The mode l equations consist o f the reaction equ i l ib r ium equations for sodalite dihydrate and hydroxysodal i te dihydrate, one charge balance equation, and the mass balance equations for the elements o f N a , A l , S i , C l , O , H , S, and C . Formulas for sodalite dihydrate and hydroxysoda l i t e dihydrate formation i n a lkal ine aqueous solutions can be wri t ten as f o l l o w i n g (Zheng et al, 1997). 54 8Na + ( a q )+ 6Al(OH) 4 ( a q )+ 6Si0 3 2- ( a q )+ 2Cr ( a q ) <-> Na_(AlSi04)6Cl2-2H20(8)+ 4H 2 0 ( 1 ) + 120H- ( a q ) (4.1) 8Na + ( a q )+ 6Al(OH)4_(aq)+ 6Si0 3 2- ( a q )+ 20H"(aq) Na.(AlSi0 4) 6(OH) r2H 2Ow+ 4H20 (,)+ 120H-(aq) (4.2) Assuming the solids to be pure then their activity is unity and the thermodynamic equilibrium constant of sodalite dihydrate, K s o _ , can be written as follows K a H p ( m Q H - ' V ' o H - ) ^ 3) « a - ) « ( O H ) , YL ( 0 H ) , )«O 3 >- ylo? X m c r ylr ) where, a is the activity, m is the molality, and y is the activity coefficient of the species. Similarly, the equilibrium constant of hydroxysodalite dihydrate, Khsod, can be written as follows K A H 2 Q ( " W V o H - ) (A 4 N (mNa^N a + ) ( M A 1 ( O H ) , Y A K O H K XmSiO>-YSiO?- ^ OW ) The mass balance equations for the Na, Al, Si, Cl, O, H, S, and C elements are given by ntoui = ( " ^ o ) ( ^ ^ 1000/18.015 N a s o d h s o d = ^ O O O n S . O l S ^ ^ ^ + 6 m - + 6 m « } ( 4 6 ) n ^ = ( i d 7 ^ ) ( m c . - + 2 m - ) ^ " r = + ^OOo7£oi5 ) ( 4 m^°"><- + 3 " W + m o » - + 3 m c o i - + 4 m s o J - (4.9) + 26m s o d+28m h s o d) n r = 2 n H 2 0 + ( 1 0 Q Q n / H 1 2 ° 0 1 5 ) ( 4 m A 1 ( O H ) , + m o H _ + m H s . +4m s o d + 6 m h s o d ) (4.10) 55 .OUl = , V w v ( 4 J j x s 1000/18.015 soj HS v ; tow = , " H ^ O ) ( ^ ( 4 1 2 ) c V 1 0 0 0 / 1 8 . 0 1 5 A c o ' ^ V ' The charge balance equation can be wri t ten as fo l lows . m + = m , ^ t i N . +2 rn + n i _ + 2 m ,_ + 2 m + m. _ + n i (4.13) N a + A l ( O H ) 4 SiOJ OH C O | SOJ HS C l v ' where, n is the number o f moles, superscript total stands for total number o f moles o f element, subscript sod indicates sodalite dihydrate, subscript hsod indicates hydroxysodal i te dihydrate, and m is the mola l i ty o f the species. 4.2. S t r u c t u r e o f the T h e r m o d y n a m i c M o d e l A structure o f the mode l ing is shown in F igu re 4 .1 . The total amount o f chemicals , P i t ze r ' s parameters, and equ i l ib r ium constants o f sodalite dihydrate and hydroxysodal i te dihydrate formations were g iven for the mode l ing ca lcula t ion . M o d e l parameters such as P i t ze r ' s parameters and equ i l ib r ium constants are discussed in more detail i n Chapter 7. A n ini t ia l guess for the mola l i ty o f each species was also g iven as input. Wate r act ivi ty and act ivi ty coefficients o f species were calculated us ing the P i t ze r ' s equation i n the subroutine A C T C O F . P i tze r ' s parameters used for this mode l ing w i l l be discussed i n section 7.1. The molal i t ies , act ivi ty coefficients, and water act ivi ty were substituted into the model equations i n the subroutine F U N C V . The mode l equations were a set o f non-linear equations. A N e w t o n - R a p h s o n method (Numer i ca l R e c i p i e s ™ , 1995) was used to solve the non-linear equations. A constraint that the mola l i ty o f any species can not be negative was inc luded in the model . The mola l i t ies o f 56 species at equ i l ib r ium were calculated by so lv ing the mode l equations. The mola l i t ies o f A J ( O H V and S i 0 3 2 ' at equ i l ib r ium define the precipi ta t ion condi t ions. A computat ional source code (kshsod.for) o f this mode l ing is g iven i n A P P E N D I X I X together w i t h examples o f input and output. F igure 4 .1 . A b l o c k diagram o f the thermodynamic mode l ing . INPUT • Total amount of chemicals • Pitzer's parameters • Equilibrium constants ( K ^ and K ^ ) • Initial guess for molalities of species FUNCV • Model equations, (4.3)~{4.13) OUTPUT • Molalities of species (Na \ AI(OH)4", Si032-, OH", C 0 3 2 - , S0 4 2 ", HS", CI-, H 2 0, hsod, sod) ACTCOF • Calculation of activity of water and activity coefficients using Pitzer's equations (2.40)~(2.43) 57 CHAPTER 5. OSMOTIC COEFFICIENT DATA FOR Na2Si03, Na2Si03-NaOH, AND NaOH-NaCl-NaAI(OH)4 AQUEOUS SYSTEMS Prior to the determination of Pitzer's parameters relevant to SiC»32" and Al(OH)4', the osmotic coefficient data for the Na2SiC»3, Na 2Si0 3-NaOH, and NaOH-NaCl-NaAl(OH)4 aqueous systems were measured at 298.15 K and the identification of silicic species in solution was done. 5.1. Identification of Metasilicic Species by Titration A titration curve of Na2Si03-NaOH solution is shown in Figure 5.1. In the figure, 20 mL of the solution containing Na2SiC«3 of 0.25 mol/L and NaOH of 1.00 mol/L was titrated with a HC1 solution of 1+0.005 mol/L. If we assume that metasilicic ion, Si032", and hydroxyl ion, OH', are the predominant ions in the solution and orthosilicic ion, HSi043', is not present, then the strong base, OH', should be titrated first. The Si032" ion, then, should be converted to HSi0 3" ion which in turn should be converted to H2Si03 by the reactions (5.1) and (5.2) with the addition of the titrant (Babushkin et al, 1985; Harris, 1991). Thus, three equivalence points should be displayed on the titration curve. Si0 3 2 ' + It -> HSi0 3" (5.1) HSi0 3 ' + It -> H 2 Si0 3 (5.2) The first equivalence point, VeNaOH of 20 mL should correspond to the OFT since the OH" of 1.00 mol/L in the sample solution of 20 mL is titrated with 20 mL of HC1 58 solution of 1.00 mol/L. The second equivalence point, V e i should be observed at 25 mL since 5 mL of HC1 solution of 1.00 mol/L is needed to convert 20 mL of SiC^2' of 0.25 mol/L to the same amount of HSi03'. The third equivalence point, Ve2 should be observed at 30 mL since 5 mL of HC1 solution of 1.00 mol/L is needed to convert 20 mL of HSiCV of 0.25 mol/L to the same amount of F^SiC^. Figure 5.1. Titration curve for the Na2Si03-NaOH solution with HC1 solution. 14 5 10 15 20 25 30 HCI solution (mL) 59 The pH values corresponding to the volumes of HC1 at which [Si032"] equals [HSi03'] and [HSi03'] equals [H 2Si0 3] are the pK a values of the reactions (5.1) and (5.2) according to the Henderson-Hasselbalch equation (Harris, 1991). The pK a values for the reactions (5.1) and (5.2) were calculated using the data of standard Gibbs free energy of formation, AGf° (Babushkin, 1985). The standard Gibbs free energy change, A G 0 of the reaction (5.1) is equal to AG f °(HSi0 3 ' ) - AG f ° (Si0 3 2 ' ) = -240.7 kcal/mol - (-224.6 kcal/mol) = -16.1 kcal/mol. The pK a 2 of the reaction (5.1) is equal to log K(5.i) = -AG7(2.3RT) = -(-16.1 kcal/mol)/(2.3 x 0.001987 kcal/mol K x 298.15 K) = 11.82. Similarly, the standard Gibbs free energy change, A G 0 of reaction (5.2) is equal to AGf°(H 2 Si0 3 ) - AG f °(HSi0 3 ' ) = - 253.9 kcal/mol - (-240.7 kcal/mol) = -13.2 kcal/mol. The pK a i of the reaction (5.2) is equal to log K<5.2) = -AG7(2.3RT) = -(-13.2 kcal/mol)/(2.3 x 0.001987 kcal/mol K x 298.15 K) = 9.69. Thus, the titration curve should pass from the following two points : (22.5 mL, pH 11.82), (27.5 mL, pH 9.69). If we assume that the metasilicic ion, Si0 3 2 ' is converted to orthosilicic ion, HSi0 4 3 ' by the reaction, Si03 2" + OH' -> HSi0 4 3" (Gasteiger, 1988), then the HSi0 4 3 ' of 0.25 mol/L and the OH* of 0.75 mol/L are present. In this case, the strong base, OH" should be titrated first and the HSi0 4 3 ' ion should be converted to H 2 Si0 4 2 " ion by the reaction (5.3) with the addition of the titrant. Subsequently, the H 2 Si0 4 2 " ion is converted to H 3 Si0 4 ' ion which in turn is converted to H 4 S i 0 4 according to the reactions, (5.4) and (5.5) with the addition of the titrant (Babushkin, 1985; Harris, 1991). Thus, four equivalence points should be present on the titration curve. 60 H S i 0 4 3 " + r f -> H 2 S i 0 4 ; 2- (5.3) H 2 S i 0 4 2 " + H+ -> H 3 S i 0 4 ' (5.4) H 3 S i 0 4 " + I T H 4 S i 0 4 (5.5) The first equivalence point o f 15 m L corresponds to the OFT since the O H " o f 0.75 m o l / L i n the sample solut ion o f 20 m L is titrated w i t h 15 m L o f H C l so lu t ion o f 1.00 m o l / L . The second equivalence point should correspond to 20 m L since 5 m L o f H C l solut ion o f 1.00 m o l / L is needed to convert 20 m L o f H S i 0 4 3 " o f 0.25 m o l / L to the same amount o f H 2 S i 0 4 2 " . The th i rd equivalence point should be observed at 25 m L accord ing to the convers ion o f H 2 S i 0 4 2 " to H 3 S i 0 4 " and the fourth equivalence point should be shown at 30 m L accord ing to the convers ion o f H 3 S i 0 4 " to H 4 S i 0 4 . The p K a values o f the reactions (5.3), (5.4), and (5.5) are 12.00, 11.70, and 9.77 respectively (Babushk in , 1985). Thus, the t i tration curve should pass from the f o l l o w i n g three points: (17.5 m L , p H 12.00), (22.5 m L , p H 11.70), and (27.5 m L , p H 9.77). A s seen from F igu re 5.1, the titration curve d id not pass f rom the points: (17.5 m L , p H 12.00), (22.5 m L , p H 11.70) w h i c h correspond to the p K a values o f or thos i l ic ic species ( H S i 0 4 3 " , H 2 S i 0 4 2 " , H 3 S i 0 4 " , and H » S i 0 4 ) . The curve, however , passed f rom (22.5 m L , p H 11.82) and (27.5 m L , p H 9.69) w h i c h correspond to the p K a values o f metas i l ic ic species ( S i 0 3 2 " , H S i 0 3 " , and H 2 S i 0 3 ) . O n l y t w o equivalence points are shown that correspond to the vo lumes o f 25 m L and 30 m L . The first equivalence point at 20 m L is not significant. It can be expla ined by the reaction: S i 0 3 2 ' + H2O <-» H S i 0 3 " + O H " . There are t w o predominant anions, O H " dissociated from the N a O H , and S i 0 3 2 ' from the N a 2 S i 0 3 , i n the in i t ia l solut ion. The concentration o f the strong base, O H " decreases w i t h 61 the addition of the titrant, hydrochloric acid. As the concentration of OH" decreases, more Si032_ ions react with H_0 and produce HSiCV and OK to establish the equilibrium. The standard Gibbs free energy change, AG°, of the reaction: SiC>32' + H_0 <-> HSiCV + OH" is equal to AG f°(HSi0 3") + AGf°(OH") - AG f °(Si0 3 2 ") - AG f ° (H 2 0) = -240.7 kcal/mol + (-37.59 kcal/mol) - (-224.6 kcal/mol) - (-56.69 kcal/mol) = +3.0 kcal/mol. The log K of the reaction is equal to - A G 0 /(2.3RT0) = -3.0/(2.3 x 0.001987 kcal/mol K x 298.15K) = -2.2017. Thus the equilibrium constant, K is io' 2 2 0 1 7. Since the equilibrium constant, 10" 2.2on Qf t n e r e a c t j o n i s high, the amount of OH" ions produced by the above reaction is enough to hide an inflection of the titration curve at the equivalence point. A calculated titration curve by a method available in the literature (Harris, 1991) is shown in Figure 5.2 with explanations in APPENDIX IV. As seen in the figure, the inflection at the equivalence point is not significant. Another calculated titration curve, when 20 mL of a solution containing Na 2C03 of 0.25 mol/L and NaOH of 1.00 mol/L was titrated with a HCl solution of 1 mol/L, is shown in the same figure. This curve shows the effect of the equilibrium constant on the inflection point. The equilibrium constant, 10"3 6 7 6 9, of the reaction, C0 3 2 " + H 2 0 <-> HC0 3 " + OH" gave a significant inflection at the equivalence point. Based on the above analysis, we conclude that when sodium metasilicate (Na2Si03-9H20) is dissolved in water containing OH", the predominant species is metasilicic ion, Si032". Furthermore the metasilicic ion is not converted to orthosilicic ion (HSi043") under these conditions. 62 F i g u r e 5.2. Calcula ted titration curve for the N a 2 S i 0 3 - N a O H and N a 2 C 0 3 - N a O H solutions w i t h H C 1 solution. 63 5.2. Mole Fraction of Metasilicic Species In addition to SiC>32', other anions such as OFT and HSiGV are also present, when the metasilicate salt (Na2Si03-9H20) is dissolved in water. The following equilibrium between each metasilicic species in the solution is established. Si032" + H 2 0 <-> HSiCV + OH* (5.6) Mole fractions of metasilicic species at various pH levels for the two solutions used in the osmotic coefficient experiments, Na2Si03 and mixed Na2Si03-NaOH, are shown in Figure 5.3. The pH values of the solutions were measured. The mole fractions of metasilicic species with varying pH were calculated by the method described in the literature (Lindsay, 1979) using the calculated equilibrium constants for the reactions (5.1) and (5.2). The mole fractions of H2Si03 and HSi03* are very low compared to that of Si032' in the concentration range of the osmotic coefficient experiments. Due to this reason, the existence of H2Si03 and H S K V can be ignored. Thus, Na2Si03 solution can be assumed to behave as the single electrolyte system of Na+-Si032" and the mixed Na2Si03-NaOH solution can be assumed to behave as a ternary electrolyte system of Na+-Si03-*-OH\ 5.3. Osmotic Coefficient Data for N a 2 S i 0 3 Aqueous System Tables 5.1 contains the osmotic coefficient and water activity data and their uncertainties (standard deviations, a$) from eight experiments respectively for KCI (standard solution), Na2Si03 (sample solution 1), and Na2Si03 (sample solution 2). Figu re 5.3. M o l e fraction o f metas i l ic ic species w i t h p H . pH pH range of osmotic coefficient experiments l l l l Na 2 Si0 3 system Na 2Si0 3-NaOH system 65 The uncertainty of the measured osmotic coefficient was calculated in APPENDIX V from the uncertainty of the weight measurement, ±0.00005 g, and uncertainty in osmotic coefficient of the reference NaCl solution, 0.01 (Pitzer, 1991), by a method available in the literature (Baird, 1995; Holman, 1994). The uncertainty was found to increase with decreasing molality of the standard KC1 and the sample solutions. The osmotic coefficient data and predictions by the Pitzer's model are shown in Figure 5.4. Comparison of the osmotic coefficient data of KC1 standard solutions with prediction by Pitzer's model using the published parameters shows the accuracy of the isopiestic method and our four-neck apparatus. The values of the published Pitzer's parameters are: P(0)KCI= 0.04835, P(1)KCI = 0.2122, and C*Kci = -0.00084 at 298.15 K (Pitzer, 1991). The relative percent error, Rel Err %, was calculated by comparing the measured osmotic coefficients of the KC1 solutions with the calculated ones. For the entire set of experiments, the average relative percent error in the osmotic coefficients of KC1 (standard solution) was 0.26 %. 5.4. Osmotic Coefficient Data for Na2Si03-NaOH Aqueous System Table 5.2 contains the osmotic coefficient and water activity data and their uncertainties from thirteen experiments respectively for KC1 (standard solution), Na2Si03 - NaOH (sample solution 1), and Na2Si03 - NaOH (sample solution 2). The uncertainties were also calculated and found to be increasing with decreasing molality. The osmotic coefficients are shown in Figure 5.5 together with values computed by using Pitzer's model. For the entire set of experiments, the average relative percent error in the osmotic coefficients of KC1 (standard solution) was 0.17 %. 66 Table 5.1. Osmotic coefficients and water activities for the Na2Si03 aqueous system at 298.15 K. Salt Molality °* * Rel Err, % a w cole a w * * Rel Err, % 1 NaCl 0.0864 0.9343 0.9971 KCI 0.0863 0.9352 0.0067 0.9293 0.6309 0.9971 0.9971 0.0018 Na 2 Si0 3 0.0603 0.8923 0.0096 0.8837 0.9638 0.9971 0.9971 0.0028 Na 2 Si0 3 0.0603 0.8926 0.0096 0.8837 0.9971 0.9971 0.9971 0.0029 2 NaCl 0.4990 0.9212 0.9836 KCI 0.5125 0.8969 0.0065 0.8998 0.3233 0.9836 0.9835 0.0054 Na 2 Si0 3 0.3674 0.8339 0.0091 0.8401 0.7435 0.9836 0.9835 0.0123 Na 2 Si0 3 0.3690 0.8304 0.0090 0.8398 1.1320 0.9836 0.9834 0.0187 3 NaCl 0.7048 0.9258 0.9768 KCI 0.7287 0.8955 0.0064 0.8977 0.2457 0.9768 0.9767 0.0058 Na 2 Si0 3 0.5313 0.8188 0.0088 0.8158 0.3664 0.9768 0.9768 0.0086 Na 2 Si0 3 0.5313 0.8188 0.0088 0.8158 0.3664 0.9768 0.9768 0.0086 4 NaCl 1.0750 0.9388 0.9643 KCI 1.1226 0.8990 0.0064 0.8992 0.0222 0.9643 0.9643 0.0008 Na 2 Si0 3 0.8637 0.7790 0.0083 0.7796 0.0770 0.9643 0.9643 0.0028 Na 2Si03 0.8629 0.7797 0.0083 0.7797 0.0000 0.9643 0.9643 0.0000 5 NaCl 1.4424 0.9552 0.9516 KCI 1.5278 0.9018 0.0063 0.9044 0.2883 0.9516 0.9514 0.0143 Na 2 Si0 3 1.2063 0.7614 0.0080 0.7618 0.0525 0.9516 0.9515 0.0026 Na 2 Si0 3 1.2059 0.7617 0.0080 0.7618 0.0131 0.9516 0.9516 0.0007 6 NaCl 1.7545 0.9709 0.9405 KCI 1.8747 0.9086 0.0062 0.9106 0.2201 0.9405 0.9403 0.0135 Na 2 Si0 3 1.4928 0.7607 0.0078 0.7594 0.1709 0.9405 0.9406 0.0105 Na 2 Si0 3 1.4927 0.7608 0.0078 0.7594 0.1840 0.9405 0.9406 0.0113 7 NaCl 2.1187 0.9910 0.9271 KCI 2.2808 0.9206 0.0062 0.9193 0.1412 0.9271 0.9272 0.0107 Na 2 Si0 3 1.8213 0.7685 0.0078 0.7679 0.0781 0.9271 0.9272 0.0059 Na 2 Si0 3 1.8241 0.7674 0.0077 0.7680 0.0782 0.9271 0.9271 0.0059 8 NaCl 2.7716 1.0308 0.9022 KCI 3.0501 0.9367 0.0061 0.9385 0.1922 0.9022 0.9020 0.0198 Na 2 Si0 3 2.3745 0.8021 0.0078 0.8029 0.0997 0.9022 0.9021 0.0103 Na 2 Si0 3 2.3725 0.8028 0.0078 0.8028 0.0000 0.9022 0.9022 0.0000 * Relative Percent Error (Rel Err, %) = I f " " - f H x 100 / ** Relative Percent Error (Rel Err, %) = | aw"* - a w I x 100 / a w ' 67 Figure 5.4. Osmotic coefficients ofNa2Si03 and KC1 aqueous solutions at 298.15 K. Molality (mol/kg H20) 68 Table 5.2. Osmotic coefficients and water activities for the Na2Si03-NaOH aqueous system at 298.15 K. Salt Molality o6 ^calc * Rel Err, % aw calc aw *• Rel Err,% 1 NaCl 0.1206 0.9293 0.9960 KC1 0.1216 0.9215 0.0066 0.9225 0.1085 0.9960 0.9960 0.0004 Na 2SiO rNaOH 0.0466 0.9625 0.0104 0.8993 6.5662 0.9960 0.9962 0.0265 Na 2SiO rNaOH 0.0468 0.9572 0.0103 0.8993 6.0489 0.9960 0.9962 0.0244 2 NaCl 0.2439 0.9215 0.9919 KC1 0.2475 0.9084 0.0066 0.9094 0.1101 0.9919 0.9919 0.0009 Na_SiO.-NaOH 0.0970 0.9267 0.0101 0.8891 4.0574 0.9919 0.9923 0.0329 Na 2SiO rNaOH 0.0973 0.9239 0.0100 0.8891 3.7698 0.9919 0.9922 0.0305 3 NaCl 0.3547 0.9200 0.9883 KC1 0.3636 0.8976 0.0065 0.9034 0.6462 0.9883 0.9882 0.0076 Na 2Si0 3-NaOH 0.1481 0.8816 0.0096 0.8805 0.1248 0.9883 0.9883 0.0015 Na 2SiO rNaOH 0.1465 0.8910 0.0097 0.8808 1.1448 0.9883 0.9884 0.0135 4 NaCl 0.5460 0.9220 0.9820 KC1 0.5604 0.8983 0.0065 0.8991 0.0891 0.9820 0.9820 0.0016 Na2SiOj-NaOH 0.2335 0.8624 0.0094 0.8663 0.4522 0.9820 0.9819 0.0082 NajSiOj-NaOH 0.2338 0.8611 0.0093 0.8663 0.6039 0.9820 0.9819 0.0110 5 NaCl 0.7996 0.9287 0.9736 KC1 0.8282 0.8967 0.0064 0.8976 0.1004 0.9736 0.9736 0.0027 Na 2Si0 3-NaOH 0.3557 0.8352 0.0090 0.8481 1.5445 0.9736 0.9732 0.0413 Na 2Si0 3-NaOH 0.3519 0.8440 0.0091 0.8487 0.5569 0.9736 0.9735 0.0149 6 NaCl 1.0411 0.9375 0.9654 KC1 1.0831 0.9012 0.0064 0.8989 0.2552 0.9654 0.9655 0.0090 Na 2SiO rNaOH 0.4731 0.8253 0.0088 0.8361 1.3086 0.9654 0.9650 0.0460 Na 2SiO rNaOH 0.4740 0.8236 0.0088 0.8350 1.3842 0.9654 0.9650 0.0487 7 NaCl 1.1478 0.9418 0.9618 KC1 1.2008 0.9003 0.0064 0.9000 0.0333 0.9618 0.9618 0.0013 Na 2SiO rNaOH 0.5263 0.8216 0.0087 0.8308 1.1198 0.9618 0.9614 0.0436 Na 2Si0 3-NaOH 0.5289 0.8175 0.0087 0.8306 1.6024 0.9618 0.9612 0.0624 8 NaCl 1.6712 0.9666 0.9435 KC1 1.7735 0.9108 0.0063 0.9087 0.2306 0.9435 0.9436 0.0134 Na 2Si0 3-NaOH 0.7859 0.8222 0.0085 0.8239 0.2068 0.9435 0.9433 0.0120 Na 2SiO rNaOH 0.7846 0.8236 0.0085 0.8239 0.0364 0.9435 0.9434 0.0021 9 NaCl 2.0263 0.9858 0.9306 KC1 2.1824 0.9152 0.0062 0.9171 0.2076 0.9306 0.9304 0.0149 Na 2SiO rNaOH 0.9599 0.8323 0.0084 0.8313 0.1201 0.9306 0.9306 0.0086 Na2Si03-NaOH 0.9582 0.8338 0.0085 0.8312 0.3118 0.9306 0.9308 0.0224 10 NaCl 2.1558 0.9932 0.9258 KC1 2.3274 0.9199 0.0062 0.9203 0.0435 0.9258 0.9257 0.0034 Na2SiOj-NaOH 1.0186 0.8408 0.0085 0.8357 0.6066 0.9258 0.9262 0.0468 Na 2SiO rNaOH 1.0181 0.8412 0.0085 0.8357 0.6538 0.9258 0.9262 0.0505 11 NaCl 2.8345 1.0349 0.8997 KC1 3.1171 0.9410 0.0061 0.9403 0.0744 0.8997 0.8998 0.0079 Na 2Si0 3-NaOH 1.3269 0.8843 0.0085 0.8736 1.2100 0.8997 0.9009 0.1280 Na 2Si0 3-NaOH 1.3246 0.8858 0.0086 0.8732 1.4224 0.8997 0.9011 0.1504 12 NaCl 3.1888 1.0583 0.8855 KC1 3.5465 0.9516 0.0060 0.9522 0.0631 0.8855 0.8854 0.0077 Na 2Si0 3-NaOH 1.4954 0.9027 0.0085 0.9036 0.0997 0.8855 0.8854 0.0121 Na 2SiO rNaOH 1.4967 0.9019 0.0085 0.9038 0.2107 0.8855 0.8853 0.0256 13 NaCl 3.5748 1.0850 0.8696 KC1 4.0204 0.9647 0.0059 0.9658 0.1140 0.8696 0.8694 0.0159 Na 2SiO rNaOH 1.6635 0.9326 0.0086 0.9394 0.7291 0.8696 0.8687 0.1018 Na 2Si0 3-NaOH 1.6629 0.9329 0.0086 0.9393 0.6860 0.8696 0.8688 0.0958 * Relative Percent Error, (Rel Err, %) = | f ** - f H x 100 / f ** ••Relative Percent Error, (Rel Err, %) = | a we x p - a w I x 100 / a w ' 69 Figure 5.5 Osmotic coefficients of mixed Na2Si03-NaOH and K C I aqueous solutions at 298.15 K. / Na2Si03-NaOH • o measured calculated by Pitzer's model with mixing parameters calculated by Pitzer's model (mixing parameters = 0) — | — i — i i i — | — i — i i i | i i i I | I I I I | I I I—I—| i i i i | i — i — i i — | — i i— i— r - p 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Molality (mol/kg H20) 70 5.5. O s m o t i c Coe f f i c i en t D a t a fo r N a O H - N a C l - N a A I ( O H ) 4 A q u e o u s S y s t e m Nine teen experiments were performed. The osmotic coefficient data for the K C I standard solut ion and the t w o sample solutions were obtained f rom each set o f experiments. Table 5.3 shows the osmot ic coefficients for the reference ( N a C l ) and standard ( K C I ) solutions. Table 5.4 shows the results for t w o sample solutions o f interest. The uncertainty (standard deviat ion, o f the measured osmotic coefficient in Tables 5.3 and 5.4 was calculated f rom the uncertainty o f the weight measurement, ± 0 . 0 0 0 0 5 g, and uncertainty i n osmot ic coefficient o f the reference N a C l solut ion, 0.01 (Pitzer, 1991), by a method avai lable i n the literature (Ba i rd , 1995). In both tables, the calculated osmot ic coefficients are also shown. In Table 5.3, compar ison o f the osmot ic coefficient data o f K C I standard solutions w i t h calculated values obtained by P i t ze r ' s mode l shows the accuracy o f the isopiest ic method o f this study. F o r the entire set o f experiments, the overa l l average relative percent error i n the osmot ic coefficients o f K C I was about 0.20 % . The f o l l o w i n g values o f publ ished P i t ze r ' s parameters, P(0)KCI = 0.04835, p ( 1 ) K c i = 0.2122, and C * K c i = -0.00084 at 298.15 K were used (Pitzer, 1991) to calculate the osmot ic coefficient for the K C I solution. Table 5.3. Osmot i c coefficients o f N a C l and K C I as reference and standard solutions at 298.15 K . Exp. No. Salt Molality °* c^alc *Rel Err, % 1 NaCl 0.9909 0.9355 KCI 1.0288 0.8985 0.0064 0.9010 0.2782 2 NaCl 1.0014 0.9359 KCI 1.0417 0.8986 0.0064 0.8997 0.1224 3 NaCl 0.9766 0.9350 KCI 1.0151 0.8984 0.0064 0.8995 0.1224 4 NaCl 1.9501 0.9815 KCI 2.0850 0.9149 0.0062 0.9180 0.3388 5 NaCl 1.9344 0.9806 KCI 2.0789 0.9148 0.0062 0.9125 0.2514 6 NaCl 1.9341 0.9806 KCI 2.0722 0.9147 0.0062 0.9153 0.0656 7 NaCl 11.9259 0.9802 KCI 2.0627 0.9145 0.0062 0.9152 0.0765 8 NaCl 1.9235 0.9800 KCI 2.0654 0.9145 0.0062 0.9127 0.1968 9 NaCl 2.9268 1.0409 KCI 3.2185 0.9430 0.0061 0.9465 0.3712 10 NaCl 2.8991 1.0391 KCI 3.1965 0.9424 0.0060 0.9424 0.0000 11 NaCl 2.8846 1.0381 KCI 3.1622 0.9415 0.0061 0.9470 0.5842 12 NaCl 2.8915 1.0386 KCI 3.1796 0.9420 0.0061 0.9445 0.2654 13 NaCl 2.8605 1.0365 KCI 3.1472 0.9411 0.0061 0.9421 0.1063 14 NaCl 3.9089 1.1089 KCI 4.4280 0.9779 0.0059 0.9789 0.1023 15 NaCl 3.9353 1.1108 KCI 4.4575 0.9788 0.0059 0.9807 0.1941 16 NaCl 3.8825 1.1070 KCI 4.3859 0.9766 0.0059 0.9799 0.3379 17 NaCl 3.8681 1.1059 KCI 4.3722 0.9762 0.0059 0.9784 0.2254 18 NaCl 3.7891 1.1002 KCI 4.2877 0.9737 0.0059 0.9723 0.1438 19 NaCl 3.8034 1.1013 KCI 4.2962 0.9740 0.0059 0.9749 0.0924 * Relative Percent Error (Rel Err, %) = I f " " - f H x 100 / tf* 72 Tab le 5.4. Osmot ic coefficients o f N a O H - N a C l - N a A l ( O H ) 4 aqueous solutions at 298.15 K . Exp. No. **input ratio Molality *Rel Err, % NaOH NaCl NaAl(OH)4 1 0.05 1 0.8080 0.1513 0.0504 0.9180 0.0098 0.9223 0.4684 0.05 1 0.8051 0.1508 0.0503 0.9213 0.0098 0.9222 0.0977 2 0.10 1 0.6204 0.3110 0.1037 0.9054 0.0097 0.9060 0.0663 0.10 1 0.6171 0.3093 0.1031 0.9013 0.0097 0.9059 0.5104 3 0.15 1 0.4041 0.4571 0.1524 0.9008 0.0096 0.8977 0.3441 0.15 1 0.4042 0.4572 0.1524 0.9006 0.0096 0.8977 0.3220 4 0.10 2 1.5964 0.2997 0.0999 0.9590 0.0098 0.9635 0.4692 0.10 2 1.5943 0.2993 0.0998 0.9602 0.0098 0.9633 0.3228 5 0.15 2 1.3996 0.4518 0.1506 0.9476 0.0097 0.9480 0.0422 0.15 2 1.3970 0.4509 0.1503 0.9494 0.0097 0.9478 0.1685 6 0.20 2 1.2019 0.6019 0.2006 0.9462 0.0096 0.9391 0.7504 0.20 2 1.2030 0.6024 0.2008 0.9454 0.0096 0.9392 0.6558 7 0.25 2 1.0081 0.7556 0.2519 0.9366 0.0096 0.9373 0.0747 0.25 2 1.0076 0.7553 0.2518 0.9370 0.0096 0.9373 0.0320 8 0.30 2 0.8014 0.9056 0.3019 0.9384 0.0096 0.9414 0.3197 0.30 2 0.8036 0.9080 0.3027 0.9359 0.0095 0.9417 0.6197 9 0.10 3 2.5581 0.2948 0.0983 1.0323 0.0099 1.0398 0.7265 0.10 3 2.5569 0.2947 0.0982 1.0327 0.0099 1.0397 0.6778 10 0.15 3 2.3657 0.4435 0.1478 1.0187 0.0098 1.0199 0.1178 0.15 3 2.3678 0.4439 0.1480 1.0178 0.0098 1.0200 0.2162 11 0.20 3 2.1751 0.5940 0.1980 1.0092 0.0097 1.0060 0.3171 0.20 3 2.1722 0.5932 0.1977 1.0106 0.0097 1.0058 0.4750 12 0.25 3 1.9960 0.7529 0.2510 1.0011 0.0096 0.9994 0.1698 0.25 3 1.9918 0.7513 0.2504 1.0032 0.0097 0.9990 0.4187 13 0.30 3 1.7811 0.8985 0.2995 0.9953 0.0096 0.9955 0.0201 0.30 3 1.7802 0.8980 0.2993 0.9958 0.0096 0.9954 0.0402 14 0.10 4 3.4814 0.2901 0.0967 1.1205 0.0101 1.1264 0.5266 0.10 4 3.4813 0.2901 0.0967 1.1206 0.0101 1.1264 0.5176 15 0.15 4 3.3572 0.4446 0.1482 1.1067 0.0100 1.1091 0.2169 0.15 4 3.3585 0.4448 0.1483 1.1062 0.0100 1.1092 0.2712 16 0.20 4 3.1613 0.5927 0.1976 1.0876 0.0098 1.0904 0.2574 0.20 4 3.1609 0.5927 0.1976 1.0878 0.0098 1.0904 0.2390 17 0.25 4 2.9570 0.7420 0.2473 1.0840 0.0098 1.0766 0.6827 0.25 4 2.9560 0.7418 0.2473 1.0844 0.0098 1.0765 0.7285 18 0.30 4 2.7236 0.8778 0.2926 1.0706 0.0097 1.0649 0.5324 0.30 4 2.7237 0.8779 0.2926 1.0705 0.0097 1.0650 0.5138 19 0.35 4 2.5645 1.0397 0.3466 1.0602 0.0096 1.0671 0.6508 0.35 4 2.5636 1.0393 0.3464 1.0606 0.0096 1.0670 0.6034 * Relative Percent Error (Rel Err, % ) = | f " - f H x 100 / f " ** Input ratio : Input molality ratio of A1C13-6H20 to NaOH 73 CHAPTER 6. DETERMINATION OF PITZER'S PARAMETERS FOR Na2Si03, Na2Si03-NaOH, AND NaOH-NaCl-NaAl(OH)4 AQUEOUS SYSTEMS Since the sodium aluminosilicate formation system contains the Si032" and Al(OH)4* ions, knowledge of the relevant Pitzer's model parameters is required for the thermodynamic modeling. Unknown binary and mixing parameters of Pitzer's model were obtained using the osmotic coefficient data described in the previous chapter 5. Pitzer derived the semi-empirical equations of the osmotic coefficients and activity coefficients for the single and multi-component electrolyte systems (Pitzer, 1991). Pitzer's method is one of the most popular ones that can be used to calculate electrolyte thermodynamic properties for single and multi-component systems. This method works well even at high molalities (Pitzer, 1991; Zemaitis et al., 1986). For this reason, the Pitzer's method was adopted for the modeling. 6.1. Pitzer's Parameters for Na2SiOs and NaiSiOs-NaOH Aqueous Systems In Pitzer's model, there are four binary parameters, P(0)ca, P(1) ca, P(2) c a , and C* c a and eight mixing parameters, (W, B^-, E6cc'(I), E9aa'(I), E6'cc'(I), ^'^'(I), ^cc*a, and ^ caa-. The parameter, P(2) is important only for 2-2 or higher valence electrolyte (Pitzer, 1991). The effects of the parameters, E9CC<I), ^-(I), E9'cc<I), and ^ - ( l ) , are significant only for 3-1 or more unsymmetrical types of mixing (Pitzer, 1975). These five 74 parameters, B ^ , E0CC<I), EQ^(l), E9'cc(I), and E0'aa-(I), were ignored for the modeling of the systems Na 2 Si0 3 and Na 2Si0 3-NaOH. The three Pitzer's parameters, p(0), P(I), and C*, were determined by least squares optimization using the experimental osmotic coefficient data of Table 5.1 for the Na 2 Si0 3 system. The Gauss-Newton method (Bard, 1974; Englezos, 1996) was used to minimize the following least squares objective function S(P(0),p(1),C+) = £ ( ( | ) d 2 ( 6 1 > d=l where <|)exp is the measured and <J>calc is the calculated osmotic coefficient using equation (2.46). is the uncertainty. Sensitivity of Pitzer's parameters for Na 2 Si0 3 aqueous system is described in APPENDIX VIII. The computational code (siwpam.for) for the optimization is given in APPENDIX IV together with input and output files. Two sets of the binary parameters of NaOH and Na 2 Si0 3 systems and two mixing parameters, 6„„-ov,i- and *F T +-„.c..j. are required for the modeling of Na 2 Si0 3 -NaOH OH S1O3 Na OH S1O3 system. The binary parameters of Na2Si03 were already obtained as explained above and those of NaOH, pl NaOH = 0.0864, P ( 1 )NaOH - 0.253, and C* N ,OH = 0.0044 at 298.15 K are available in the literature (Pitzer, 1991). The remaining two Pitzer's mixing parameters, 0 O H_ s j oi_ and ^ N l l + O H - s i o i - were determined by the least squares optimization method using the osmotic coefficient data of Table 5.2 for the Na2Si03-NaOH system. The computational code (nasiwpam.for) for the optimization is given in APPENDIX IX together with input and output files. The binary and mixing parameters that were obtained are given in Table 6.1 with their standard deviations. The standard deviations were calculated according to methods described in the literature (Box et al, 1978). The 75 calculated osmotic coefficients using Pitzer's parameters have an average relative percent error of 0.33 % for Na2SiC<3 and 1.74 % for the Na2Si03-NaOH system respectively. The computational source codes (siwcal.for and nasiwcal.for) used for the calculations are available in APPENDIX IX. In Figures 5.4 and 5.5 found in the previous chapter, the solid lines are the calculated osmotic coefficients using the Pitzer's parameters obtained in this work. Minor inflections on the curves of calculated osmotic coefficients are observed at low molalities, 0 to 0.1 mol/kg H2O in the figures. The osmotic coefficient curve of K_Pt(CN)4 by Pitzer's model also shows a similar inflection at low molalities (Pitzer, 1991). Furthermore, it is known that the isopiestic method does not give reliable results below a concentration of 0.1 mol/kg FfeO (Pitzer, 1991). Table 6.1. The Pitzer's parameters of Na 2 Si0 3 and Na 2 Si0 3 -NaOH systems at 298.15 K Binary parameters for the Na^SiOs system Parameter Standard Deviation p ( 0 ) = 0.0577 0.0039 p ( 1 ) = 2.8965 0.0559 C* = 0.00977 0.00176 Mixing parameters for the Na2SiOs-NaOH system Parameter Standard Deviation t_i n . - = -0.2703 OH SiOJ 0.0384 ¥ + . = 0.0233 Na OH Siof 0.0095 76 Pitzer introduced the mixing parameters, 0 and *F to account for the differences of ion interactions between multi-component and single salt electrolyte solutions. Thus, the mixing parameters have smaller effects than the binary parameters, p ( 0 ), P(1), and C * have (Pitzer and Kim, 1974). The contribution of the mixing parameters, however, was not negligible for the NaOH-Na2Si03 system. The dashed line in Figure 5.5 represents the calculated osmotic coefficients when the values of the mixing parameters, 0__.-_.__- and ^ N a + o H - s i o f - a r e s e t e c l u a l t 0 z e r o - As shown in this Figure, the mixing parameters should be included for an improved calculation of the osmotic coefficient. Mean activity coefficients of Na2SiC»3 in water at 298.15 K were calculated and are shown in Figure 6.1. Mean activity coefficients of Na2SiC>3 and NaOH in the Na2Si03-NaOH binary system were also calculated and are shown in Figure 6.2. Sensitivity of Pitzer's parameters for Na2Si03-Na0H aqueous system is described in Appendix VIII. 6.2. Pitzer's Parameters for NaOH-NaCI-NaAl(OH)4 Aqueous System Among the Pitzer's parameters, the parameter, P ( 2 ) c a, is important only for 2-2 or higher valence electrolytes. The parameters, E0ee(I), E0aa*(I), E0'cC(I), and E0'__'(I), are significant only for 3-1 or for cases with more unsymmetrical mixing (Pitzer, 1975). Thus, these five parameters, P(2)c_, E0ec'(I), E0__'(I), E0'cc'(I), and E0'__'(I), were ignored for the modeling of the system N a O H - N a C l - N a A l ( O H ) 4 in water. The Gauss-Newton method was used to minimize the following least squares objective function N / i e x p i c a k \ 2 S ( p W > p ( ' ) . C * ) = _ r ^ d 2 ) (6-1) d=i cr. 77 Figu re 6.1. M e a n act ivi ty coefficients o f Na2SiC«3 in Na2SiC«3 aqueous solut ion at 298.15 K . 1.2 _ 1 0 C Q) O § 0.8 O O 0.6 c CU 5 0.4 H Na 2Si0 3 0.2 " i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — | — i — i — i — i — | — i — i — f -0.0 0.5 1.0 1.5 2.0 2.5 3.0 Molality (mol/kg H20) Molality of Na2Si03-NaOH (mol/kg H20) 79 where <J>exp is the measured and (j)03'0 is the calculated osmot ic coefficient us ing equation (2.46). c$ is the uncertainty. Three sets o f the binary parameters, B ( 0 ) , p ( 1 ) , and C * o f N a O H , N a C l , and N a A l ( O H ) 4 systems and s ix m i x i n g parameters, Q , 0 Q OH"Al(OH); ' Cl-Al(OH)7 ' ^ o H - c r > VQH-AKOHK' A N D "WAKOH); are required for the mode l ing o f the N a O H -N a C l - N a A l ( O H ) 4 aqueous system. The binary parameters o f the N a O H and N a C l systems and t w o m i x i n g parameters and are avai lable i n the literature (Pitzer, 1991) as shown i n Table 6.2. Table 6.2. The P i t ze r ' s parameters at 298.15 K available i n the literature (Pitzer, 1991). Binary parameters S y s t e m P a r a m e t e r N a O H 3 ( 0 ) = 0.0864 p (1) = 0.253 C * = 0.0044 N a C l p(0) = 0.0765 p(1) = 0.2664 C * - 0.00127 Mixing parameters 0 = - 0 . 0 5 o i r c r Y + = - 0 . 0 0 6 Firs t , the three P i tze r ' s parameters, P(0), p(1), and C* for the N a A l ( O H ) 4 system were determined by least squares opt imiza t ion us ing the experimental data o f Table 5.4 80 and taking into account the published Pitzer's parameters shown in Table 6.2. The estimated parameters are shown in Table 6.3 with their standard deviations. The mixing parameters, QOH-M(OH)-, QCi-M(0H)-, ^ o i i - M m ) V and V C R A 1 ( 0 H ) - were set equal to zero during this optimization. Table 6.3. The Pitzer's parameters of NaOH-NaCl-NaAl(OH)4 aqueous system at 298.15 K obtained in this study. Binary parameters of NaAl(OH)4 Parameter Standard deviation P ( 0 ) = - 0.0083 0.0417 p ( 1 ) = 0.0710 0.3362 C* = 0.00184 0.00977 Mixing parameters Parameter Standard deviation 0 =-0.2255 OH-Al(OH); 0.0469 0 = - 0.2430 0.0816 ¥ + =-0.0388 Na+OH~AI(OH)i 0.0147 ¥ + = 0.2377 Na+CPAl(OH)7 0.0332 After the determination of the binary parameters, the four Pitzer's mixing parameters, Q0H-M(0H)-, 6 c r A 1 ( O H ) - , V^-^-, and ^ N a + c r A 1 ( O H ) - were determined by 81 least squares optimization method using the osmotic coefficient data of Table 5.4 for the aqueous NaOH-NaCl-NaAl(OH)4 system. It was necessary to follow this approach because the simultaneous minimization search for the binary and ternary parameters did not converge. The mixing parameters that were obtained are also given in Table 6.3 with their standard deviations. The standard deviations were calculated according to methods described in the literature (Box et al, 1978). The calculated osmotic coefficients using Pitzer's parameters have an average relative percent deviation of 0.37 % from the experimental values. Sensitivity of Pitzer's parameters for NaOH-NaCl-NaAl(OFf)4 aqueous system is described in APPENDIX VIII. 6 . 3 . Reliability of the Pitzer's Parameter Determination In order to check the reliability of the optimization procedure, Pitzer's parameters for the NaTcC«4, NaTcCVNaCl, and NaBr-NaC104 systems were obtained using the osmotic coefficient data from the literature (Hernandez-Luis, 1996; Konnecke, 1997). The parameters that we obtained were compared with the published ones. The published parameters for the systems are tabulated in Table 6.4 together with those obtained in this work. Except for the binary parameters for the NaTcC«4 system, all parameters obtained agree well with the published ones. Table 6.5 displays the published osmotic coefficient data and the calculated values for NaTcC>4 system. As seen from the table, the osmotic coefficients calculated by using the parameters obtained in this work match the experimental data slightly better than those obtained by using the published parameter values. 82 Table 6.4. The Pitzer's parameters of NaTcCu, NaTc0 4-NaCl, and NaBr-NaClC>4 systems at 298.15 K. Binary parameters for the NaTc04 system Parameter Konnecke, 1997 This work p ( 0 ) 0.01111 0.01510 3d) 0.1595 0.1145 c* 0.00236 0.00187 Mixing parameters for the NaTcO^NaCl system Parameter Konnecke, 1997 This work 6 TcO<Cl 0.067 0.067* »P N a + T c O i C r -0.0085 -0.0085* Mixing parameters for the NaBr-NaCl04 system Parameter Hernandez-Luis, 1996 This work e Br'ClOi 0.0350 0.0350 Na +Br"C10; -0.0058 -0.0058 * The parameters were obtained using the binary parameters, P(' = 0.01 111, p ( 1 ) = 0.1595, and C* = 0.00236 determined by Konnecke et a/.(1997). Table 6.5. Comparison of measured osmotic coefficients with calculated those for the NaTc0 4 system at 298.15 K. Konnecke, 1997 This work ( P(0)= 0.01111, P(0)= 0.1595, C*= 0.00236) (P(0)= 0.01510, P(0)= 0.1145, C*= 0.00187) Molality tf*f Rel Err, % c^alc Rel Err, % 7.3278 0.9598 0.9639 0.4272 0.9653 0.5730 6.5487 0.9395 0.9341 0.5748 0.9375 0.2129 6.4555 0.9376 0.9308 0.7253 0.9343 0.3520 5.2349 0.9021 0.8923 1.0864 0.8973 0.5321 3.9632 0.8680 0.8629 0.5876 0.8677 0.0346 2.6048 0.8407 0.8463 0.6661 0.8487 0.9516 1.1159 0.8519 0.8545 0.3052 0.8523 0.0470 0.5353 0.8749 0.8739 0.1143 0.8703 0.5258 Ave Rel Err,% 0.5608 Ave Rel Err,% 0.4036 Relative Percent Error (Rel Err, %) = | f * - f H x 100 / 83 Calcu la t ions were also performed for the N a C l , Na2SC>4, Na2S2C>3 and N a C l -N a 2 S C » 4 (see A P P E N D I X V I I ) . T h e calculated parameters were found to g ive exact ly the same values for the osmot ic coefficient o f N a C l and s l ight ly better values for the other systems. 84 CHAPTER 7. A PRIORI DETERMINATION OF MODEL PARAMETERS M o d e l parameters such as P i t ze r ' s parameters and equ i l i b r ium constants o f sodalite dihydrate and hydroxysodal i te dihydrate formation reactions are required for the thermodynamic mode l ing o f the sod ium aluminosi l ica te formation i n a lka l ine solutions. 7.1. Pitzer's parameters Pitzer 's b inary and m i x i n g relevant parameters for our system are shown i n the Tables 7.1 and 7.2. A value o f 0.455 at 368.15 K was used for the D e b y e - H u c k e l parameter (Pitzer, 1991). P i tzer introduced the m i x i n g parameters, 9 and *F to account for the differences o f ion interactions between in mul t i -component and single salt electrolyte solutions. Thus, the m i x i n g parameters have smaller effects than the binary parameters, p ( 0 ) , p ( 1 ) , and C * have (Pi tzer and K i m , 1974). The binary parameters for N a O H , N a C l , and N a 2 C 0 3 for 368.15 K were determined us ing the numerica l expressions avai lable i n the literature (Pitzer, 1991; Si lvester and Pi tzer , 1977; Peiper and Pi tzer , 1982). A l t h o u g h numerica l expressions for the binary parameters for Na2S04 were available i n the literature, they were not used because they were based on the value o f 1.4 for the " a . parameter" instead o f the usual value o f 2.0 for a 1-2 electrolyte (Pitzer , 1991; Pabalan and Pi tzer , 1988). Osmot i c coefficient data for Na2S04 are avai lable at 298.15, 323.15, 348.15, 373.15, and 398.15 K (Pabalan and Pi tzer , 1988). 85 Table 7.1. Pitzer's binary parameters for the modeling of sodium aluminosilicate formation. ion pair 3(0) 3d) c* Na + OH" 0.0847 1 1 1 0.3882 m 0.00021 1 1 1 Na + Cl" 0.1008 P 1 0.3207 [ 2 ] - 0 . 0 0 3 3 7 1 2 1 Na + C 0 3 2 ' 0.05811 [ 3 ] 1.2419 [ 3 ] 0.0052 1 3 1 Na + S 0 4 2 ' 0.0988 [ 4 ] 1.4325 1 4 1 - 0 .01462 [ 4 1 Na + HS" 0.1396 [ 1 ] 0.0 w - 0 . 0 1 2 7 1 1 1 Na + S i 0 3 2 " 0.0577 [ 4 ] 2.8965 [ 4 ] 0.00977 [ 4 ] Na + A 1 ( 0 H ) 4 " - 0.0083 1 4 1 0.0710 [ 4 ] 0.00184 [ 4 ] [1] Pitzer (1991), [2] Silvester and Pitzer (1977), [3] Peiper and Pitzer (1982), [4] this study Table 7.2. Pitzer's mixing parameters for the modeling of sodium aluminosilicate formation. ion pair aa'Na* OH"Cl" -0.05 PI -0.006 [ 1 ] OH" C 0 3 2 " 0.1 m -0.017 1 1 1 OH" S 0 4 2 " -0.013 [ 1 ] -0.009 1 1 1 OH" S i 0 3 2 " -0.2703 1 2 1 0.0233 [ 2 ] OH"Al(OH)4" -0.2255 [ 2 ] -0.0388 1 2 1 cr co32" -0.02 [ 1 ] 0.0085 m Cl" S 0 4 2 " 0.03 [ 1 ] 0.000 1 1 1 C1" Al(OH)4" -0.2430 [ 2 ] 0.2377 [ 2 ] C 0 3 2 " S 0 4 2 ' 0.02 m -0.005 [ 1 ] [1] Pitzer (1991), [2] this study These data were interpolated at 368.15 K and gave a set of values. The binary parameters of N a 2 S 0 4 at 368.15 K were then determined by using these interpolated osmotic coefficient data and minimizing the following least squares objective function 86 s(p(o),p(i),c*)=|;(ct-(|)r1)2 (7.i) d=l where the (J)"* is the interpolated and the (J)0*1 is the calculated osmotic coefficients using equation (2.46). Based on a study with NaCl, Pitzer noted that change in the parameter values from 298.15 to 573.15 K was very small. This justified the use of the values at 298.15 K whenever values at 368.15 K were not available. However, the Debye-Huckel parameter, A$, changes significantly and the value at 368.15 K was used (Zemaitis et al, 1986). In the case that no Pitzer's parameters at 368.15 K were available, those at 298.15 K were used. Binary parameters for NaHS and mixing parameters for OH"-Cl', O H -CO3 2 ' , OFT-S0 4 2' , C1'-C032", C1'-S042", and C0 3 2"-S0 4 2" are available at 298.15 K in the reference of Pitzer 1991. Finally, it is noted that binary parameters of Na2SiC>3 and NaAl(OH) 4 and mixing parameters of Na 2 Si0 3 -NaOH and NaAl(OH) 4-NaOH-NaCl systems at 298.15 K were determined from the osmotic coefficient measurements using isopiestic method as explained in chapters 5 and 6. 7.2. E q u i l i b r i u m C o n s t a n t s Values of the equilibrium constants for sodalite dihydrate and hydroxysodalite dihydrate formation at 368.15 K were not available in the literature. Assuming that heat capacity change of reaction, ACP°, is not a function of temperature, the equilibrium constants at specific temperature can be calculated using by equation (2.18) (Anderson and Crerar, 1993) lnK = AG° RT, AH" 0 j R 1 1 T T, AC" R T T In ___ + _____! V T 0 T (2.18) 87 where K is equ i l ib r ium constant at temperature T , AGJJ is the G i b b s free energy change o f react ion at reference state, AH° is the enthalpy change o f reaction at reference state, AC° is the heat capacity change o f reaction at reference state, R is the gas constant, and To is the reference temperature. Va lues for A G £ , A H ^ , and A C " can be obtained by the f o l l o w i n g equations A G ° = E v A G f products " E v A G f reactants (7.2) AH° = EvAHf°products - EvAHfVeactants (7.3) A C " = I V C P c products - I v C P c reactants (7.4) where the v is the number o f moles o f reactants or products. Table 7.3 shows property values for the species required to calculate the above equ i l ib r ium constants. Proper ty values that were not available were estimated in this w o r k as fo l lows . Table 7.3. The rmodynamic data at 298.15 K and 1 bar publ ished in the literature. AG f° ( k J / m o l ) AH f° (kJ /mol ) CP° (J/mol-K ) Na + (aq) -262.0 ± 0 . 1[ 1 ] -240.34 ± 0 .06 [ 1 ] +46.43 [ 2 ] Cl'(aq) -131.2 ± 0 . 1 [ 1 ] -167.1 ± 0 . 1 t l ] -136.4 1 2 1 O H (aq) -157.2 ± 0 . 1 ( 1 ] -230.01 ± 0 . 0 4 m -148.5 1 1 1 H2O (1) -237.14 ± 0 . 0 4 [ 1 ] -285.83 ± 0 . 0 4 m +75.351 ± 0 . 0 8 m A l ( O H ) 4 " ( a q ) -1311.684 ± 1.255 [ 3 ] -1488.92 [ 4 ] +241.44 [ 4 ] Si03 2 (8q) -939.73 [ 2 ] N / A N / A N a 8 ( A l S i 0 4 ) 6 C l 2 - 2 H 2 0 w N / A [ 5 ] N / A 1 5 1 N / A [ 5 ) N a g t M S i O A C O r D ^ r t O f l , N / A [ 5 ] N / A t 5 ] N / A 1 5 1 N / A : not available in the literature [1] Nordstrom and Munoz (1994), [2] Babushkin et al. (1985), [3] M a y et al. (1979), [4] Raizman (1985), [5] Hemingway (1997). 88 7.3. Estimation of Thermodynamic Properties 7.3.1. A H f ° and A G r ° of sodalite dihydrate and hydroxysodalite dihydrate The AHf° and AGf° o f sodalite dihydrate and hydroxysodal i te dihydrate were estimated by the method o f Mos t a f a et al. (1995). A mean error o f their method was reported as 2.57 % for the est imation o f standard enthalpy o f format ion and 2.60 % for that o f standard G i b b s free energy o f formation. T h e y used the f o l l o w i n g functional forms for the standard enthalpy o f formation and standard G i b b s free energy o f formation o f inorganic salts: A H ° = £ n j A H j (7.5) j A G ^ J X A o . (7.6) j where, nj is the number o f groups o f the j th type and AHJ and AQJ are contributions for the j th a tomic or molecular group in the enthalpy and G i b b s free energy o f format ion respectively. T h e est imation procedure is quite s imple. The molecular structural fo rmula is b roken into appropriate cationic, anionic, or l igand molecular structural groups. Then the numer ica l contr ibut ion o f each group is obtained and mul t ip l i ed by the number o f occurences o f the same group in the molecular structural formula. S u m o f the numer ica l values o f the var ious groups y ie lds estimation for AHf° and AGf°. Table 7.4 shows the sequence o f calculat ions for anhydrous sodalite. The publ ished AHf° o f anhydrous sodalite ( N a g l ^ A l S i O ^ C b ) is -13457.04 k J / m o l ( K o m a d a et al, 1995) and predicted one is -13110.72 k J / m o l . Thus, an error o f the est imation is 2.57 % . The publ ished AGf° o f 89 anhydrous sodalite (Na 8(AlSi0 4) 6Cl 2) is -12702.96 kJ/mol (Komada et al, 1995) and predicted one is -12369.05 kJ/mol. An error of the estimation of the Gibbs free energy is 2.63%. Table 7.4. Estimations of AHf° and AGf° of anhydrous sodalite (Na8(AlSi04)6Cl2). N a 8 ( A l S i 0 4 ) 6 C l 2 group ni A H j ns Aci N a + 8 X -241.688 8 X -199.801 A l 3 + 6 X -553.115 6 X -420.023 su + 6 X -575.556 6 X -411.036 o 2 24 X -173.650 24 X -229.836 cr 2 X -118.794 2 X -134.110 AHf° = -13110.72 kJ/mol AG f° = -12369.05 kJ/mol Since the AHf° of anhydrous sodalite (Na8(AlSi04)6Cl2) is available and the only difference in the chemical formula of anhydrous sodalite (Na8(AlSi04)6Cl2) and that of sodalite dihydrate (Na8(AlSi04)6Cl2-2H20) is two water molecules, the AHf° of Na 8(AlSi0 4)6Cl 2- 2H 2 0 can be calculated by adding the contribution term of 2H 2 0 to the published AH f° of Na 8 (AlSi0 4 ) 6 Cl 2 . The contribution by 2H 2 0 is equal to 2 x -298.933 = -597.866 kJ/mol. The AH f° of Na 8(AlSi0 4) 6Cl 2- 2H 2 0 is equal to AH f° (Na 8(AlSi0 4) 6Cl 2) + Contribution by 2H 2 0 = -13457.04 + (-597.866) = -14054.91 kJ/mol. The AG f° of Na 8 (AlSi0 4 ) 6 Cl 2 -2H 2 0 can be calculated by the same logic. The contribution by 2H 2 0 for the AG f° is equal to 2 x -244.317 = -488.634 kJ/mol. The AG f° of Na 8(AJSi0 4) 6Cl 2-2H 20 is equal to AG f° (Na 8(AlSi0 4) 6Cl 2) + Contribution by 2H 2 0 = -12702.96 + (-488.634) = -13191.59 kJ/mol. The uncertainties of the AH f° and AG f° of the Na 8(AJSi0 4) 6Cl 2-2H 20 were calculated from the uncertainties of the AHf° and AGf° of the Na 8(AlSi0 4)6Cl 2, ± 15.80 kJ/mol and + 16.63 kJ/mol (Komada et al, 1995) and the uncertainties of the 90 contributions by 2H 2 0 by a method available in the literature (Baird, 1995; Holman, 1994). The uncertainties are shown in Table 7.5 together with the estimated values of the thermodynamic properties. Detailed calculations of the uncertainties are given in APPENDIX VI. Table 7.5. Estimated thermodynamic data at 298.15 K and 1 bar. AGf° ( kJ/mol) AHf° ( kJ/mol) C P ° (J/molK ) Si032"(aq) N a 8 ( A l S i 0 4 ) 6 C l 2 - 2 H 2 0 ( s ) Na8(AlSi04)6(OH)2-2H20 (. ) -1075.38 ± 1 . 1 4 -326.27 ± 8 2 . 7 1 -13191.59 ± 2 0 . 9 2 -14054.91 ± 2 2 . 0 4 8 8 6 . 2 8 ± 4 . 6 9 -13384.23 ± 2 5 . 1 0 -14283.02 ± 25.82 895.01 ± 5 . 3 7 The AHf° and AG f° of hydrated hydroxysodalite dihydrate (Na 8(AlSi0 4)6(OH) 2-2H 20) were calculated by the same method. The AHf° and AGf° of Na 8(AlSi0 4)6(OH) 2-2H 20 can be calculated by subtracting the contribution term of C l 2 and adding the contribution term of (OH) 2 and 2H 2 0 to the published AHf° and AGf° of Na 8 (AlSi0 4 ) 6 Cl 2 . The contribution of C l 2 for the AH f° is equal to 2 x -118.794 = -237.588 kJ/mol and that of (OH) 2 is equal to 2 x -232.849 = -465.698 kJ/mol. Thus, the A H / of Na 8(AlSi0 4)6(OH) 2-2H 20 is equal to AH f° (Na 8(AlSi0 4) 6Cl 2) - Contribution by C l 2 + Contribution by (OH) 2 + Contribution by 2H 2 0 = -13457.04 - (-237.588) + (-465.698) + (-597.866) = -14283.02 kJ/mol. The contribution of C l 2 for the AG f° is equal to 2 x -134.110 = -268.220 kJ/mol and that by (OH) 2 is equal to 2 x -230.428 = -460.856 kJ/mol. Thus, the AG f° of Na 8(AlSi0 4) 6(OH) 2-2H 20 is equal to AG f° (Na 8(AlSi0 4) 6Cl 2) -Contribution by C l 2 + Contribution by (OH) 2 + Contribution by 2H 2 0 = -12702.96 - (-268.220) + (-460.856) + (-488.634) = -13384.23 kJ/mol. The uncertainties of the AH f° 91 and AGf° of the Na8(AlSi04)6(OH)2-2H20 were also calculated and are shown in Table 7.5 from the uncertainties of the AHf° and AGf° of the NagfAlSiO^eCb and the uncertainties of the contributions by 2H2O, CI2, and (OFfy groups. 7.3.2. C p° of sodalite dihydrate and hydroxysodalite dihydrate Another group contribution technique was proposed by Mostafa et al. (1996) to predict the C p for solid inorganic salts. A mean error of their method was reported as 3.18 % when predicted values were compared with literature values for heat capacity at 298.15 K. They used the following functional form for the heat capacities of solids: r \ f \ C P =2>Ai + l Z n i A b J x i 0 ' 3 M2>A* x l ° 6 IT^+I S"J A D J X L ° 6 V 1 J T (7.7) V J J where j is atomic or molecular group, nj is the number of groups of the jth type, A is the contributions for the a, b, c, or d coefficient and T is in kelvin. A stepwise procedure for the estimation of C p is similar that of AGf° and AHf° estimations. The molecular structural formula for the solid inorganic salt is broken into appropriate cationic, anionic, or ligand molecular structural groups. Then the numerical contributions of each group are calculated and multiplied by the number of occurrences of the same group in the molecular structural formula. The sum of the numerical values of the various groups yields estimation for EpjAaj, SjnjAbj, EjnjAcj, and SjnjAdj. The C p can then be calculated by the equation (7.7) at the temperature of interest. Table 7.6 shows the sequence of calculations for the C p ° estimation of anhydrous sodalite (Na8(AlSi04)6Cl2). The published C p° of anhydrous sodalite is 812.28 J/mol K at 298.15 92 K (Komada et al, 1995) and predicted one is 808.39 J/mol K. Thus, the estimation error is 0.48 %. Table 7.6. Estimation of C p ° of anhydrous sodalite (Na8(AJSi04)6Ci2). step 1 group A * Acj Adj step 2 Na+ 8 x 14.186 8 x 9.665 8 x 0.529 8 x 4.851 A l 3 + 6 x 10.306 6 x 4.518 6 x -0.623 6 x -3.701 Si 4 + 6 x -2.308 6 x 4.382 6 x -0.041 6 x -3.301 O 2 24x28.152 24 x 12.043 24 x -0.747 24 x -4.023 Cl 2 x 26.609 2 x 10.376 2 x-0.251 2 x 0.657 step 3 ZjnjAij 890.342 440.504 -18.182 -98.442 step 4 C 0 = 890.342 + 440.504xl0-3x298.15 + (-18.182)xl0<7298.152 + (-98.442)xl0'6x298.152 = 808.39 J/mol K Since the chemical formula of Na8(AlSi04)6C-2 differs from that of Na8(AlSi04)6Cl2-2H_0 by two water molecules, the heat capacity of Nag(AlSi04)6Cl2-2H20 can be calculated by adding the contribution term of 2H 20 to the published heat capacity value of Na8(AlSi04)6C.2. The contribution by 2H2O is equal to 73.996 as shown in Table 7.7. The C p° of Na 8(AlSi0 4)6Cl 2 is 812.28 J/mol K. Thus, the C p° of Na8(AlSi04)6Cl2-2H20 is equal to C p° (Na8(AlSi04)6Cl2) + Contribution by 2H 2 0 = 886.28 J/mol K. Table 7.7. Contribution by 2H 2 0 for C p° estimation Group Aaj Abj Acj A_j H 2 0 2 x 15.458 2 x66.593 2 x0.47 2 x-40.518 ZjnjAij 30.916 133.186 0.94 -81.036 Contribution of 2H 2 0 = 30.916 + 133.186xl03x298.15 + 0.94xl06/298.152 +(-81.036)xl0^x298.152 = 73.996 J/mol K 93 Similarly, the heat capacity of Na 8(AlSi04)6(OH) 2-2H 20 can be calculated by subtracting the contribution term of C l 2 and adding the contribution term of (OH) 2 and 2 H 2 0 to the published heat capacity of Na8(AlSi04)6Cl2. The contributions by C l 2 and (OH) 2 are described in Table 7.8. The C p° of Na 8(AlSi0 4)6(OH) 2-2H 20 is equal to C p° (Na8(AlSi04)6Cl2) - Contribution by C l 2 + Contribution by (OH) 2 + Contribution by 2 H 2 0 = 812.28 - 53.875 + 62.608 + 73.996 = 895.01 J/mol K. The estimated C p° values of Na 8(AlSi0 4)6Cl 2-2H 20 and Na 8(AlSi0 4)6(OH) 2-2H 20 are shown in Table 7.5 together with their uncertainties. The uncertainties were calculated from the uncertainty of C p° (± 4.06 J/mol K) for Na8(AlSi04)eCl2 and the uncertainty of the contributions by the 2H 2 0, C l 2 , and (OH) 2 groups following the method used before (Baird, 1995; Holman, 1994). Table 7.8. Contributions by C l 2 and (OH) 2 for C p° estimation. group Ajj Ay Acj Adj Cl 2 2 x 26.609 2 x 10.376 2 x-0.251 2 x 0.657 EjnjAij 53.218 20.752 -0.502 1.314 Contribution of Cl 2 = 53.218 + 20.752xl03x298.15 + (-0.502)xl06/298.152 + 1.314xl0^x298.152 = 53.875 J/mol K (OH)2 2 x28.917 2 x30.73 2 x-0.628 2 x 3.257 ZjnjAij 57.834 61.46 -1.256 6.514 Contribution of (OH)2 = 57.834 + 61.46xl0"3x298.15 + (-1.256)xl06/298.152 +6.514xlQ-6x298.152 = 62.608 J/mol K 7.3.3. S° o f S i Q 3 % * ) Prior to determining the A H / and Cp° of Si032*, the S° of SiC>32" was estimated by the method of Couture and Laidler (1957). They found that the entropy of oxy-anions (XOn"m) in aqueous solution at standard state is given by the following empirical relationship 94 S° = 5.5z + 40.2 + - R l n M w — ? 1 ^ L _ (7 8 ) 2 0.25n r • oxy where, z is the charge o f ion , R is the gas constant (1 .987cal /mol K ) , M w is the molecular weight o f ion , noxy is the number o f oxygen and r is the radius o f a sphere that complete ly c i rcumscr ibes the anion (r = rn + 1.40). The 1.40 A is the v a n der W a a l s radius o f oxygen. T h e rn is the distance between the center o f the central a tom and the center o f the surrounding oxygen atoms. The units o f the calculated entropy by equation (7.8) are ca l /mol K . The predicted entropies o f several aqueous ions are compared w i t h publ ished data i n the literature i n Table 7.9. The publ ished entropy data o f aqueous ions were obtained f rom B a b u s h k i n et al. (1985). The average error o f the estimation was 18 .41%. The rn data except for that o f SiC«3 2 ' are avai lable i n the literature (Couture and La id l e r , 1957). The rn o f S i 0 3 2 ' is also avai lable i n the literature (Ba i le r et al, 1973). The entropy o f the S i C b 2 ' aqueous ion was calculated as -4.99 ca l /mol K ( = -20.87 J /mol K ) . The uncertainty o f the entropy o f the S i 0 3 2 ' was assumed to be, 18 .41% ( ± 3.84 J /mol K ) , w h i c h is the average error o f this est imation method f rom Tab le 7.9. Tab le 7.9. Pred ic t ion o f the entropy o f aqueous ions at 298.15 K , ca l /mol K . r n r M w fro xy z Calculated s° Published S° error CIO 1.70 3.10 51.452 1 -1 11.35 10 13.48% N 0 3 1.24 2.64 62.004 3 -1 33.26 35.1 5.23% C 0 3 2 1.26 2.66 60.008 3 -2 -13.13 -13.6 3.44% so32 1.39 2.79 80.057 3 -2 -9.73 -7 39.04% S e 0 4 2 1.65 3.05 142.956 4 -2 8.32 12.9 35.51% P 0 4 3 1.55 2.95 94.97 4 -3 -45.71 -53 13.75% S i 0 3 2 1.68 3.08 76.082 3 -2 -4.99 95 7.3.4. A H f ° Of Si03 2"(aq) The enthalpy of formation of SiC>32' aqueous ion was calculated by considering the following equation AH° = A G 0 + TAS° (7.9) where, AG° is the standard Gibbs free energy change of reaction, A H 0 is the standard enthalpy change of formation, and AS 0 is the change of standard entropy. Let us consider the dissociation reaction of sodium metasilicate, Na2Si03(s) at 298.15 K (Babushkin etal, 1985). Na2Si03 (s) -> 2Na+(aq) + Si0 3 2'( a q ) (7.10) Table 7.10 shows the thermodynamic data for Na2Si03(S), Na+(aq), and Si032"(aq) needed in the calculation of the A G 0 , AH°, and AS° of the reaction (7.10). Substituting the calculated AH°, AG°, and AS° into equation (7.9) gives a value of -1075.38 kJ/mol for AHf° of S.O3 2". The uncertainty in the calculation of AHf° for Si(_>32" was found to be ± 1.14 kJ/mol following a method available in the literature (Baird, 1995; Holman, 1994). Table 7.10. Thermodynamic data of Na2Si03(S), Na+(aq), and Si032"(aq) for the calculation of the A G 0 , AH°, and AS 0 of reaction (7.10). AGf° (kJ/mol) AHf° (kJ/mol) S° (J/molK) Na2Si03(s) -1469.67 m -1556.7 1 , 1 113.8 1 1 1 Na+(aq) - 2 6 2 . 0 ± 0 . 1 [ 2 ] -240.34 ± 0 . 0 6 1 2 1 58.45 [ 2 ] Si032"(aq) -939.73 [ 1 ] -20.87131 [1] Babushkin etal. (1985), [2] Nordstrom and Munoz (1994), [3] calculated in section 7.3.3. 96 7.3.5. C p° Of Si032"(aq) T h e est imation equation (7.11) for the heat capacities in c a l / m o l K for ion ic solutes was suggested by C r i s s and C o b b l e (1964) C p ° = C p ° a b s - 28 .0z = a + b ( S ° - 5.0z) - 28 .0z (7.11) where, a is equal to -145 and b is equal to 2.20 for oxy-anions ( X O n " m ) at 298.15 K , and z is the charge o f the ion . The Cr i s s and C o b b l e ' s heat capacity est imation method was tested for several oxy-anions. The calculated values are compared w i t h the publ ished data in Tab le 7.11. The publ ished entropy and heat capacity data o f aqueous ions were obtained f rom B a b u s h k i n et a/ .(1985). The estimation error was not negl ig ib le for some ions such as N b 0 3 " and C O 3 2 " . A n average error o f the estimations was 25.35 % . The C p ° o f SiC>32" i on is equal to -77.98 ca l /mol K (= -326.27 J /mol K ) . The uncertainty o f the C p ° o f the S i 0 3 2 " was assumed as the same, 25.35 % ( ± 82.71 J /mol K ) , as the average error o f this est imation method from Table 7.11. Tab le 7.11. Heat capacities o f aqueous oxy-anions at 298.15 K , ca l /mol K . Published Calculated Published z S° r 0 abs c 0 C 0 error N 0 3 -1 35.10 -56.78 -28.78 -20.7 39.03% C K V -1 39.10 -47.98 -19.98 -19.7 1.42% N b 0 3 -1 36.00 -54.80 -26.80 -18.4 45.65% C 0 3 2 -2 -13.60 -152.92 -96.92 -59.7 62.35% S 0 3 2 -2 -7.00 -138.40 -82.40 -64.1 28.55% S e 0 3 2 -2 -1.70 -126.74 -70.74 -67.6 4.64% S 0 4 2 -2 4.20 -113.76 -57.76 -71.6 19.33% A s 0 4 3 -3 -39.40 -198.68 -114.68 -116.9 1.90% S i 0 3 2 -2 -4.99 -133.98 -77.98 97 7.3.6. C a l c u l a t i o n o f at 368.15 K Prior to calculating the equilibrium constant, K s o d , of sodalite dihydrate formation, the A G Q , A H Q , and AC° of the reaction (4.1) were calculated with the thermodynamic data in Tables 7.3 and 7.5. The AG° of reaction (4.1) is equal to AGf0(Na8(AlSi04)6Cl2-2H 20) + 4AG f ° (H 2 0) + 12AG f °(OH) - SAG/fNa*) - 6AG f°(Al(OH) 4") - 6AG f °(Si0 3 2 ") -2AG f°(Cr) = -13191.59 + 4x(-237.14) + 12x(-157.2) - 8x(-262.0) - 6x(-1311.684) - 6x(-939.73) - 2x(-131.2) = -159.666 kJ/mol from equation (7.2). The AH° is equal to AHf°(Na 8 (AlSi04)6Cl 2 -2H 2 0) + 4AH f ° (H 2 0) +. 12AH f°(OH-) - 8AH f °(Na + ) -6AHf0(Al(OH)4") - 6AHf°(Si0 3 2") - 2AH f°(Cl') = -14054.91 + 4x(-285.83) + 12x(-230.01) - 8x(-240.34) - 6x(-1488.92) - 6x(-1075.38) - 2x(-167.1) = -315.63 kJ/mol from equation (7.3). The AC; is equal to C p°(Na 8(AlSi0 4)6Cl 2-2H 20) + 4 C P ° ( H 2 0 ) + 12Cp°(OrT) -8C p °(Na + ) - 6C P °(A1(0H) 4 -) - 6C p °(Si0 3 2 ") - 2C p °(Cf) = 886.28 + 4x(75.351) + 12x(-148.5) - 8x(46.43) - 6x(241.44) - 6x(-326.27) - 2x(-136.4) = -183.976 J/mol K = -0.183976 kJ/mol K from equation (7.4). Substituting AG°, AH°, and AC° into equation (2.18) at T = 368.15 K and T 0 = 298.15 K gives In K s o c i = 39.74. Thus, the equilibrium constant of sodalite dihydrate formation, K s o d , is equal to 1.82E+17. The uncertainty in the calculated values for In K s o _ was found to be ± 8.71 from the uncertainties of the estimated thermodynamic properties in Table 7.5 by the method of Baird (1995). The major contribution in the uncertainty of K S o d was due to the uncertainties in the estimated thermodynamic data, especially that of the AGf°(Na8(AlSi0 4 )6Cl 2 -2H 2 0). 98 7.3.7. C a l c u l a t i o n o f K h s o d at 368.15 K The AG° of reaction (4.2) is equal to AG f 0(Na 8(AlSi04)6(OH)2-2H 20) + 4AG f ° (H 2 0) + 12AGf0(OFT) - 8AG f °(Na + ) - 6AG f °(AJ(OH) 4 ) - 6AG f °(Si0 3 2 ") - 2AGf°(OH") = = -13384.23 + 4x(-237.14) + 12x(-157.2) - 8x(-262.0) - 6x(-1311.684) - 6x(-939.73) -2x(-157.2) = -300.306 kJ/mol. The AH° of reaction (4.2) is equal to AH f 0(Na 8(AlSi0 4)6(OH) 2-2H 20) + 4AH f ° (H 2 0) + 12AHf°(OFT) - 8AH f °(Na + ) -6AH f °(Al(OH) 4 ) - 6AH f °(Si0 3 2 ' ) - 2AH f°(OH-) = -14283.02 + 4x(-285.83) + 12x(-230.01) - 8x(-240.34) - 6x(-1488.92) - 6x(-1075.38) - 2x(-230.01) = -417.92 kJ/mol. The AC° of reaction (4.2) is equal to C p 0(Na 8(AlSi0 4) 6(OH) 2-2H 20) + 4 C P ° ( H 2 0 ) + 12Cp0(OFT) -8C p °(Na + ) - 6C P°(A1(0H) 4-) - 6 C p ° ( S i 0 3 2 ) - 2Cp°(OH") = 895.01 + 4x(75.351) + 12x(-148.5) - 8x(46.43) - 6x(241.44) - 6x(-326.27) - 2x(-148.5) = -151.046 J/mol K = -0.151046 kJ/mol K . Substituting AG°0, AH°, and AC° into equation (2.18) at T = 368.15 K and T 0 = 298.15 K gives In K h s o d = 88.71. Thus, the equilibrium constant of hydroxysodalite dihydrate formation, K h s o d , is equal to 3.38E+38. The uncertainty in the calculated values for In K h s o d was found to be ± 10.41. Detailed calculation of the uncertainty is described in APPENDIX VI. Again major contribution in the uncertainty of K n s o d was due to the uncertainties of the estimated thermodynamic data, especially that of the AGfO(Na 8(AlSi0 4) 6(0H) 2-2H 20). 99 7.4. C h a n g e o f In Khsod w i t h T e m p e r a t u r e 1 Values of In K h s o d were calculated with varying temperature from 298.15 K to 368.15 K using equation (2.18). The results are shown in Figure 7.1. A solid line in the figure represents In K h s o d at the temperature of interest. As seen in the figure, the In K h s o d decreases with increasing temperature. Figure 7.1. Change of In K h s o d with temperature. 130 g Q I i i i i I i i i i I i i i i I i i I 298.15 318.15 338.15 358.15 368.15 Temperature (K) The AC° of the hydroxysodalite dihydrate formation reaction (4.2) is known at 298.15 K as described in section 7.3.7 but not at higher temperature. In this case, it is 1 0 0 generally better to assume that AC° is constant as temperature increases from 2 9 8 . 1 5 K to 3 6 8 . 1 5 K (Anderson and Crerar, 1 9 9 3 ) . A dotted line in the Figure 7 . 1 represents In K h s o d at the reference temperature, 2 9 8 . 1 5 K . The area between the dotted line and the dashed line represents the contribution by the AC° term in equation ( 2 . 1 8 ) on In K h S O d . The area between the dashed line and the solid line represents the contribution by the A H Q term in the equation. As seen in the figure, the AH° term has the largest contribution on the value of In K h s o d - The contribution by the AC° term is very small compared to that by AH°. Thus, the assumption of constant A C ° in the range of temperature from To to T in the equation (2.18) is acceptable. 101 CHAPTER 8. SOLUBILITY MAPS OF Al AND Si IN GREEN AND WHITE LIQUORS Data of the solubility experiments using synthetic liquors (systems A and B) and mill liquors are shown in Tables A.3.1, A.3.2, and A.3.3. in APPENDIX Ul together with their standard error. 8.1. Synthetic Green and White Liquors of System A A solid phase started forming in a supersaturated solution of Al and Si within several hours from the beginning of the experiment. The solid precipitates were formed on the wall, stirrer blade, and bottom of the vessel. During the experiment, the concentrations of the soluble Al and Si decreased with time but after four of five days, they reached a constant value. After seven days, the samples were taken and considered to be in their equilibrium state (Figure 8.1). The base case experiment (Experiment A l ) of synthetic green liquor of system A was carried out three times to check the repeatability of the experiments. The results of the experiment are tabulated on the Table 8.1. The standard deviations for Al and Si concentrations were found to be 6.37xl0'5 and 6.52xl0"5, respectively. 102 Figure 8.1. Al and Si ions approaching equilibrium obtained from the base experiment using synthetic green liquor of system A. (Experimental data were obtained from experiment A l using synthetic green liquors of system A at 368.15 K and latm.) 13 8 c o o 800 CL 700# Q_ U tn 600 \-c o •(5 500 h 400 o V ) c o 300 •t. 200 100 O • Total soluble Si O Total soluble Al o 3 4 5 Time, (days) Table 8.1. Repeatability of the experiments in three runs run Al Si [Al][Si] moles/kg H2O moles/kg H2O (moles/kg H 2 0 ) 2 1 4.49E-3 5.01E-3 2.25E-5 2 4.51E-3 5.06E-3 2.28E-5 3 4.61E-3 4.93E-3 2.28E-5 103 It is customary to present precipitation results in the form of a graph which is known as the solubility map (Wannenmacher et al, 1996). Figure 8.2 shows the solubility map for the synthetic green and white liquors of system A. In the solubility maps, the 95 % confidence interval for each data point has been plotted together with the experimental data points. The confidence interval calculations are explained in detail in APPENDIX II. This was calculated using the standard error of the five Atomic Absorption Spectrophotometer measurements for each sample. If the concentrations of Al and Si in the liquors correspond to a point on the right side of each line, precipitation will occur for this solution. The precipitation conditions in synthetic white liquor are higher than those in the synthetic green liquor. This result is attributed to the effect of [OFT]. White liquor contains more OFT. Higher [OFT] content makes Al and Si more soluble as we see in Figure 8.3 which shows the effect of changing the OFT concentration on the precipitation conditions. The change in the OH" concentration is accomplished by varying the input amount of NaOH during the preparation of the solution. The base concentration, given in Table 3.1, is 1.0 mol/kg H 2 0 in synthetic green liquor. The perturbations around the base case are also given in the figure. As seen from the plot, as the hydroxyl ion concentration increases, precipitation occurs at higher Al and Si concentrations. Zheng and coworkers also studied sodium aluminosilicate crystallisation in relation to the Bayer process at 100 °C (Zheng et al, 1997). Their result is also plotted in Figure 8.3. The chemical compositions of the Bayer liquors are not exactly the same as in this work. The synthetic Bayer liquor contains 4.53 mol/L of NaOH. 104 Figure 8.2. Solubility map of Al and Si in synthetic green and white liquors of system A. (Experimental data were obtained from experiments A l , A2, and A3 using green liquors and from experiments A10, A l 1, A12 using white liquors at 368.15 K and latm.) 105 Figure 8.3. Effect of hydroxyl ions on the solubility limit in synthetic green and white liquors of system A. (Experimental data were obtained at 368.15 K and latm. Initial concentrations of NaOH in green liquors were 0.25 mol/kg H 2 0 for experiment A4, 1.0 mol/kg H 2 0 for experiments A l , A2, and A3, and 2.0 mol/kg H 2 0 for experiment A5. Those in white liquors were 2.0 mol/kg H 2 0 for experiment A13, 2.5 mol/kg H 2 0 for experiments A10, A l 1, and A12, and 3.0 mol/kg H 2 0 for experiment A14.) Si concentration (mol/kg H 20) 106 The effect o f changing the concentration o f the carbonate anion is shown i n F igu re 8.4. Increasing the in i t ia l carbonate ion concentration i n the synthetic green l iquor g ives a l ower solubi l i ty o f A l and S i . In the case o f synthetic whi te l iquor , no significant change i n the so lubi l i ty was observed for the concentrations studied (0.1 to 0.5 m o l / k g H _ 0 ) . It is noteworthy that even though the synthetic green l iquor contains more carbonate, it is sensitive to a perturbation i n the carbonate concentrations whereas the whi te l iquor is not, at least for a ± 0.2 m o l / k g H2O change. T h e concentrations o f the sulfate ions i n real green and whi te l iquors f rom kraft m i l l s are i n the range o f 0.05 to 0.2 m o l / L and 0.05 to 0.15 m o l / L (Magnusson et al, 1979). The effect o f the sulfate ions on the so lubi l i ty o f A l and S i was also measured at input sod ium sulfate concentrations o f 0.05, 0.1, and 0.2 m o l / k g H2O for the synthetic green l iquor and 0.05, 0.1, and 0.15 m o l / k g H2O for the synthetic whi te l iquor . A s seen from F igu re 8.5, no significant change i n the so lubi l i ty was observed. 8.2. S y n t h e t i c G r e e n a n d W h i t e L i q u o r s o f S y s t e m B The precipi tat ion condi t ions i n synthetic green and whi te l iquors o f system B are shown i n F igu re 8.6. The synthetic whi te l iquor o f higher [OFT] compared w i t h the green l iquor shows that precipi tat ion occurs at higher A l and S i concentrations. The effect o f input amount o f Na2S on the precipi tat ion condi t ion is shown i n F igu re 8.7. A s more Na2S is added into the system, precipitat ion occurs at higher A l and S i concentrations. 107 Figure 8.4. Effect of carbonate ions on the solubility limit in synthetic green and white liquors of system A. (Experimental data were obtained at 368.15 K and latm. Initial concentrations of Na2CC>3 in green liquors were 0.5 mol/kg H2O for experiment A6, 1.0 mol/kg H2O for experiments A l , A2, and A3, and 1.5 mol/kg H 2 0 for experiment A7. Those in white liquors were 0.1 mol/kg H2O for experiment A15, 0.3 mol/kg H2O for experiments A10, A l 1, and A12, and 0.5 mol/kg H2O for experiment A16.) 0.1 0.001 0.01 0.1 Si concentration (mol/kg H20) 108 Figure 8.5. Effect of sulfate ions on the solubility limit in synthetic green and white liquors of system A. (Experimental data were obtained at 368.15 K and latm. Initial concentrations of Na2S04 in green liquors were 0.05 mol/kg H 2 O for experiment A8, 0.1 mol/kg H 2 O for experiments A l , A2, and A3, and 0.2 mol/kg H 2 O for experiment A9. Those in white liquors were 0.05 mol/kg H 2 O for experiment A17, 0.1 mol/kg H 2 O for experiments A10, A l l , and A12, and 0.15 mol/kg H 2 0 for experiment A18.) 0.1 Si concentration (mol/kg H20) 109 Hence, it becomes more difficult to precipitate solids. In other words, the window for normal operation expands. This may be attributed to the fact that Na 2S dissociates when dissolved in water and releases OH" by the reaction: Na 2S + H 2 0 -> 2Na + + HS' + OH" (Smook, 1992), that affects the precipitation conditions in a manner seen in Figure 8.3. The effect of [OFT] might be explained by considering the sodium aluminosilicate formation reaction as seen next. The sodium aluminosilicate precipitates were found to be hydroxysodalite dihydrate (Na 8(AlSi0 4) 6(OH) 2-2H 20) and/or sodalite dihydrate (Na 8(AlSi0 4) 6Cl 2-2H 20). The identification of the precipitates is explained in section 8.4. The chemical reaction for sodium aluminosilicate formation of this study can be written as 8Na+ + 6A1(0H)4" + 6Si032" + 2X" <- Na8(AlSi04)6X2-2H20(s) + 4H 20 + 120H" (8.1) where X" can be OH' or Cl" (Zheng et al, 1997). The solubility product of Al and Si can be written as (a (8.2) AI(OH); Ke<,(*W)8(ax-)2 J where a is the activity and K e q is the equilibrium constant". The solubility product increases with increasing [OH"] at constant concentrations of the other ions. 110 Figure 8.6. Solubility map of Al and Si in synthetic green and white liquors of system B. (Experimental data were obtained from experiments B l , B2, and B3 using green liquors and from experiments BIO, B l 1, B12 using white liquors at 368.15 K and latm.) 0.1 o CM D ) O 0.01 E c o > -•—< <D O C O 0.001 \ \ \ \ \ Precipitation \ \ \ < \ \ Q \ \ \ \ \ X \ \ Input ratio Al/Si green Iq. white Iq. 1/2 • A 1/1 • O 2/1 • • \ \ \ N \ 0.001 0.01 0.1 Si concentration (mol/kg H 20) I l l Figure 8.7. Effect of NajS on the solubility limit in synthetic green and white liquors of system B. (Experimental data were obtained at 368.15 K and latm. Initial concentrations of NazS in green liquors were 0.0 mol/kg H2O for experiments B4, B5, and B6, 0.5 mol/kg H2O for experiments B l , B2, and B3, and 1.0 mol/kg H2O for experiments B7, B8, and B9. Those in white liquors were 0.0 mol/kg H2O for experiments B13, B14, and B15, 0.5 mol/kg H 2 0 for experiments B10, B l l , and B12, and 1.0 mol/kg H 2 0 for experiments B16, B17, and B18.) CM 0.1 \ \ \ \ • - * \ green Iq. white Iq. Input Na2S • 0.0 O 0.0 (moles/kg H-O) 0.5 (base} 0.5 (base) • 1.0 • 1.0 O 0.01 q I c U c g 0.001 \ \ o \ \ m \ \ \ \ \ \ \ \ * \ _ l I ' I I I I I ' ' I I I I I I I I I I 'Ns. I I I I 0.001 0.01 0.1 Si concentration (mol/kg H20) 112 8.3. M i l l G r e e n a n d W h i t e L i q u o r s Figure 8.8 shows the solubility map for the green and white liquors. In the solubility maps, the 95 % confidence interval for each data point has been plotted together with the experimental point. As seen from the graph, the data at three different A/Si ratios fall on the same line. If the concentrations of Al and Si in the liquors correspond to a point located on the right side of each line, precipitation will occur in this liquor. As seen, the precipitation conditions in white liquor are higher than that in the green liquor. This result is attributed to the effect of OFT. Higher OFT concentration makes Al and Si more soluble as seen in Figure 8.9. White liquor contains more OH" (2.63 mol/kg H 2 0) than the green liquor (0.85 mol/kg H 2 0). Figure 8.9 shows the effect of changing the OH" concentration on the precipitation conditions. As seen from the plot, higher concentrations of Al and Si are required to induce precipitation as the hydroxyl ion concentration increases. Thus, the liquor can tolerate increased levels of dissolved Al and Si. Increasing initial carbonate ion concentration in the mill liquors gave a lower solubility of Al and Si, which can be seen in Figure 8.10. In other words, as the carbonate ion concentration increases, precipitation occurs at lower Al and Si concentrations. The effect of Na 2S on the precipitation conditions is shown in Figure 8.11. As more Na 2S is added into the system^ precipitation occurs at higher Al and Si concentrations. This effect may be attributed to the fact that Na 2S dissociates when dissolved in water and releases OH" (Ulmgren, 1982; Smook, 1992). 113 Figure 8.8. Solubility map of Al and Si in mill green and white liquors. (Experimental data were obtained from experiments M l , M2, and M3 using green liquors and from experiments M13, M14, M15 using white liquors at 368.15 K and latm.) 0.1 CM o E c o TO Precipitation 0.01 O c O 0.001 o Green liquor (base) White liquor (base) 0.001 0.01 0.1 Si concentration (mol/kg H 20) 114 Figure 8.9. Effect of NaOH on the solubility limit of Al and Si in mill liquors. (Experimental data were obtained at 368.15 K and latm. Initial concentrations of NaOH in green liquors were 0.29 mol/kg H2O for experiments M l , M2, and M3, and 1.29 mol/kg H2O for experiments M4, M5, and M6. Those in white liquors were 2.06 mol/kg H2O for experiments M13, M14, and M15 and 2.56 mol/kg H2O for experiments M16, M17, andM18.) 0.001 0.01 0.1 Si concentration (mol/kg H20) 115 Figure 8.10. Effect of Na2C03 on the solubility limit of Al and Si in mill liquors. (Experimental data were obtained at 368.15 K and latm. Initial concentrations of Na2CC«3 in green liquors were 1.05 mol/kg H 2 0 for experiments M l , M2, and M3, and 1.55 mol/kg H_0 for experiments M7, M8, and M9. Those in white liquors were 0.23 mol/kg H 2 0 for experiments M13, M14, and M15 and 0.48 mol/kg H2O for experiments M19, M20, andM21.) 116 Figure 8.11. Effect of Na 2S on the solubility limit of Al and Si in mill liquors.(Experimental data were obtained at 368.15 K and latm. Initial concentrations of Na2S in green liquors were 0.56 mol/kg FfeO for experiments M l , M2, and M3, and 1.06 mol/kg H2O for experiments M10, M i l , and M12. Those in white liquors were 0.57 mol/kg H 2 0 for experiments M13, M14, and M l 5 and 1.07 mol/kg H 2 0 for experiments M22, M23, and M24.) Green liquor 0.1 0.01 CM C O o E, c o '~t—> CO 8 c o 0.001 Na 2S [OH-] (mol/kg H20) • 1.06 1.35 0.56 0.85, base White liauor Na 2S [OH"] (mol/kg H20) O 1.07 3.13 0.57 2.63, base _1 1 1 I I I I _1 I I I I I I I J I I I ' N l I I 0.001 0.01 0.1 Si concentration (mol/kg H20) 117 The results for the mill liquors with base concentrations were compared with data available in the literature and found in good agreement as seen in Figure 8.12. The straight line on the graph represents a correlation of the precipitation conditions obtained from experiments with mill green liquors (Wannenmacher et al, 1996). The grey square in Figure 8.12 represents the concentrations of AJ and Si in unsaturated mill white liquors. These liquors contain Al and Si at a concentration level that is near the precipitation conditions. Figure 8.12. Solubility map comparing solubility limit of Al and Si in varying liquors. (The data were obtained at 368.15 K and 1 atm) 0.1 Green liquor D) __. O E c o I 8 c 8 0.01 0.001 : • This work (base concentration) - Wannenmacher et. al.1996 -• White liquor • This work (base concentration) • H Wannenmacher etal. 1996 -\ D - • \ -I ' • ' • 0.001 0.01 0.1 S i concentration (mol/kg H 2 0 ) 118 8.4. Structure of the Precipitates from Synthetic Liquors X-ray diffraction analysis of the precipitates from the solutions of system A and B showed that only sodalite (Na8(AlSi04)6Cl2nH20) and/or hydroxysodalite (Nag(AlSi04)6(OH)2nH20) were present (Figure 8.13). There is no significant difference in the X-ray diffraction patterns for the sodalite and hydroxysodalite (Gasteiger et al, 1992). Ulmgren (1982) reported that samples of scales from Swedish pulp mills consist of NaAlSi04l/3Na 2 C0 3 and/or NaAlSi0 4 l /3Na 2 S0 4 which look like cancrinite. Although the system of this study contained carbonate and sulfate ions, the X-ray diffraction patterns of the precipitates did not match to those of any minerals containing the carbonate and sulfate ions. A portion of the precipitates in the synthetic liquors of system A and B was dissolved in IN nitric acid and analyzed to determine the molar ratios of Na, Al, Si, Cl, and O H in the solid. If the precipitates contain carbonate in the structure, dissolving the solid in acid solution can cause release of carbonate ions to combine with hydrogen ions and be converted to C 0 2 gas. However, no bubbles of CO2 gas were detected during the dissolution. Sulfate was not present in the precipitates obtained in the synthetic green and white liquors because the precipitation of B a S 0 4 was not observed by the addition of BaCb solution to the dissolved precipitate solutions. Chloride was not present in the white liquor precipitates of system A and green and white liquor precipitates of system B because the precipitation of AgCl was not observed by the addition of AgN0 3 solution to the dissolved precipitate solutions. The green liquor precipitates of system A, however, had chloride since AgCl precipitated when AgN0 3 solution was added. The amount of 119 chloride ions in the dissolved precipitate solution was measured quantitatively using a titration method with a 0.1 N AgNC«3 solution. Figure 8.13. X-ray diffraction pattern of precipitates in synthetic green liquor of system A. 100 60 00 c 20 Hydroxysodalite (reference) I • i i Sample r p T r r p ' | ' ' ' r ' j ' r r r ' l T * r r T ! i ' V T ! " i | i i 1 r p ' n ' i p T ' n p i r r ^ ^ T r r ^ r r r T ^ ' r r r p - T T T T T T T i n r T r r r p 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 2 0 120 Thus, X-ray diffraction analysis and chemical analysis revealed that the precipitates of the synthetic green liquor of system A contain the structure of sodalite and hydroxysodalite whereas those of the synthetic white liquor of system A and green and white liquors of system B contain the structure of hydroxysodalite only. In the case of the synthetic green liquor of system A, approximately 25 % of the precipitates on a molar basis are sodalite which has the Cl group and 75 % of the precipitates are hydroxysodalite which has the O H group. The molar ratio of Na : Al : Si is 8 : 6.06 ± 0.05 : 6.05 ± 0.06. In the case of the synthetic white liquor of system A, the precipitate is hydroxysodalite. The molar ratio of Na : Al : Si is 8 : 5.96 ± 0.09 : 6.07 ± 0.07. The precipitates of synthetic green and white liquors of system B are hydroxysodalite. The molar ratios of Na : Al : Si are 8 : 6.23 ± 0.20 : 6.15 ± 0.31 and 8 : 6.18 ± 0.13 : 5.89 ± 0.09, respectively. Thermogravimetry analyses were performed for four precipitates, each from the synthetic green and white liquor, of systems A and B. The analyses proceeded from 105 to 900 °C at a heating rate of 10 °C/min under a N 2 atmosphere. The data for synthetic white liquor precipitates from system A are shown in Figure 8.14. Total weight loss was 5.73 % by heating up to 900 °C and the loss is equivalent to 3.1 H 2 0 (« 3 H 2 0) molecules per unit cell of the precipitate. Engelhardt et al. (1992) studied the hydroxysodalite system with thermal analysis. According to their result, a total of n+1 water molecules per unit cell leave the structure of Na8 (AlS i04 )6 (OH) 2 nH 2 0 by heating up to about 700 °C. First of all, n water molecules leave to form Nag(AlSi04)6(OH)2 and another water molecule from the remaining O H groups is released to change to a carnegieite-type phase 121 Na20-6NaAlSi04. Therefore, the total water loss, three water molecules per unit cell in Figure 8.14, indicates that the structure of precipitate in synthetic white liquor is Na8(AlSi04)6(OF£)2-2H 20, hydroxysodalite dihydrate. Other precipitates showed similar weight loss, 5.46 % of synthetic green liquor precipitate of system A, 5.22 % and 5.55 % of synthetic green and white liquor precipitates of system B. This fact means that all precipitates have the same number of hydrated water molecules, 2H2O, in their structure. Figure 8.14. Thermogravimetr ic analysis o f hydroxysodal i te dihydrate, Na8(AlSi04)6(OH)2-2FJ.20, precipitated i n synthetic whi te l iquor o f system A . 100 99 -\ 98 H ^ 9 7 H -•—> C O 'CD 9 6 9 5 H 94 93 H X weight loss : 5.73 % • loss of H20/unit cell: 3.1 — > T T T 100 200 300 400 500 600 700 800 900 temperature (°C) 122 8.5. Structure of the Precipitates from Mill Liquors X-ray diffraction analysis revealed that the precipitates could be sodalite (Na 8(AlSi0 4)6Cl 2nH 20) and/or hydroxysodalite (Na 8(AlSiO 4)6(OF0 2nH 2O). This analysis cannot distinguish between these two solids (Gasteiger et al, 1992). Therefore, it was complemented with chemical analysis and thermogravimetry. The chemical tests described earlier showed that there was not any chloride, sulfate or carbonate present in the precipitated solids. In addition, the atomic absorption spectroscopy tests showed that there was not any potassium present in the solid. The molar ratios of Na : Al : Si in the precipitates from the green and white liquors were found to be 8 : 6.17 ± 0.16 : 5.86 ± 0.08 and 8 : 5.97 ± 0 . 1 1 : 5.99 ± 0.10 respectively. Thus, the chemical formula of the precipitate is that of Na8(AJSi04)6(OIT)2nH20. Finally, the number of the hydrated water molecules was found to be equal to two by thermogravimetry. Hence the precipitate is hydroxysodalite dihydrate, Na8(AlSi04)6(OH) 2-2H 20. Structure of precipitates in mill liquors was found to be similar to that in synthetic liquors. 8.6. Morphology of precipitates The precipitates in green and white liquors consist of particles 5 to 30 pm in diameter. Aggregated particles, as shown in Figure 8.15, were found to be suspended in the solution or present as scale on the wall of the vessel and the stir blade. Most of green liquor precipitates were suspended in solution. Homogeneous nucleation might be the dominant nucleation mechanism in the green liquors (Mullin, 1972). The green liquor precipitates formed on the wall of vessel could be easily removed. The image of the green 123 l iquor precipitates in F igu re 8.15 (a), (b), and (c) shows a less packed structure o f the particles. F igu re 8.15. Scanning electron micrographs o f precipitates. (a) m Precipitates in (a) synthetic green liquor of system A (experiment Al ) , (b) synthetic green liquor of system B (experiment Bl), (c) mill green liquor (experiment Ml) , (d) synthetic white liquor of system A (experiment A10), (e) synthetic white liquor of system B (experiment BIO), and (f) mill white liquor (experiment M13). All scale bars in the images have the same length, 10 um. 124 White liquor precipitates were formed mainly on the wall and bottom of the vessel and on the stir blade. Heterogeneous nucleation might be the dominant nucleation mechanism in the white liquors (Zettlemoyer, 1969). The scale on the wall and bottom of the vessel was very glassy. Thickness of the scale was less than 1 mm. Surface of scale consists of spherical cap type deposits. These precipitates could not be easily detached by scraping. The image in Figure 8.15 (d), (e), (f) shows a well packed structure of the particles in the white liquor. More precipitates were formed in green liquors than in white liquors when the same amount of Al and Si salts were added into the liquors. Amount of precipitates was approximately 7 to 8 g/kg H2O in the synthetic green liquors of system B when Na 2 Si0 3 -9H 2 0 of 0.05 mol/kg H 2 0 and A1C13-6H20 of 0.05 mol/kg H 2 0 were used. The amount of precipitates in synthetic white liquors of system B ranges from 5 to 6 g/kg H 2 0 when the same amounts of Na 2Si0 3-9H 20 and A1C13-6H20 were used in the experiments. 125 CHAPTER 9. PREDICTION OF THE PRECIPITATION CONDITIONS OF SODIUM ALUMINOSILICATES The eleven model equations were solved by the Newton-Raphson method (Numerical Recipies™, 1995) to give the molalities of the species. The activity coefficients of species as well as the activity of water were also calculated. The product of the molalities of Al(OH)4" and SiC<32' at equilibrium was taken as the solubility product. In order to check the reliability of the modeling procedure, calculations for the NH4 - N H 4 O H - If - HCl - NH4CI - Cf - Na + - NaCl - K + - KC1 system were performed. The calculation results were compared with the published ones (Anderson and Crerar, 1993). For the exact comparison, the Davies revision of the Debye-Huckel equation (Anderson and Crerar, 1993) was used for the activity coefficient calculation since the same equation was used in the literature. The computational source code (heqbrm.for) is available in APPENDIX IX together with examples of input and output. As seen from the Table 9.1, the values obtained agree well with published ones. Calculations were performed at 368.15 K for the Na + - Al(OH) 4' - Si0 3 2 ' - OH" -C0 3 2 " - S0 4 2 ' - Cl" - H 2 0 system (system A) and the Na + - Al(OH)4* - Si0 3 2" - OH" - CO3 2 ' - C f - HS" - H 2 0 system (system B). The results were compared with the experimental data described in chapter 8. The calculations showed that only hydroxysodalite dihydrate forms in both systems. During the experiments, most of the precipitates were found to have the structure of hydroxy sodalte dihydrate. Only in the solutions of low OH" and high Cl* concentrations of system A, a small amount of the sodalite dihydrate precipitates was 126 found together with hydroxysodalte dihydrate. A comparison of the calculated results with experimental data is presented next. Table 9.1. Comparison of the calculation results for the NFL, - NH4OH - FT - HC1 -NFL,C1 - Cl" - Na + - NaCl - K + - KCI system at 573.15 K Species Calculated molalities (mol/kg H20) Anderson and Crerar, 1993 This work NH4 1.630E-1 1.630E-1 N H 4 O H 2.844E-3 2.876E-3 FT 1.206E-3 1.220E-3 HC1 1.638E-3 1.657E-3 NH4CI 8.416E-2 8.415E-2 cr 4.782E-1 4.782E-1 Na + 1.385E-1 1.385E-1 NaCl 1.115E-1 1.115E-1 K + 1.755E-1 1.755E-1 KCI 7.450E-2 7.450E-2 The molalities of the species at equilibrium were calculated using the following input amounts of chemicals: NH4CI of 0.25 moles, NaCl of 0.25 mol, KCI of 0.25 mol, and H 2 O o f l . 0 k g . 9.1. N a + - A l ( O H ) 4 - S i 0 3 2 - O H - C 0 3 2 - SO4 2 - C I - H 2 0 (Sys t em A ) Figure 9.1 shows data and calculated values (solid line) for the concentration of the aluminum species versus that of the silicon at equilibrium. When the concentration of the Al and Si species in the solution correspond to a point on the right of the line then precipitation occurs. Calculations were also performed to test the sensitivity of the model to the value of K h S O d - These calculations using values K h S O d ± uncertainty are shown as dotted lines in Figure 9.1. As seen the experimental data are located within the boundaries of the precipitation conditions computed by taking into account the uncertainty in Khsod - The value of K n s o d was subsequently adjusted to 4.39E+36 (In K h s o d 127 = 84.37) instead of the calculated 3.38E+38 (In Khsod = 88.71) and was used to give a solubility product calculation in perfect agreement with the data (dotted-dashed line). Figure 9.1. Equilibrium concentration of aluminum and silicon species. (Experimental data were obtained from experiments A l , A2, and A3 using synthetic green liquors of system A at 368.15 K and latm. For the model calculation, input amount of NaOH was 1.0 mol/kg H 2 0 , that of N a 2 C 0 3 was 1.0 mol/kg H 2 0 , that of Na 2 S0 4 was 0.1 mol/kg H 2 0 , and that of NaCl was 0.25 mol/kg H 2 0 . Input amounts of A1C136H20 and Na 2 Si0 3 9H 2 0 were varied from 0.04 to 0.01 mol/kg H 2 0 and from 0.01 to 0.04 mol/kg H 2 0 respectively.) 0.1 • Experimental data Prediction using In K,,^ = 88.71 Prediction using In K h s o d = 88.71±10.41 Prediction using In K,^ = 84.37 J i J i i i 11 i i i i i r-i 11 i i i i i i i 11 0.001 0.01 0.1 Si concentration (mol/kg H20) 128 Figure 9.2 shows the effect of varying the concentration of OFT on the precipitation conditions. As seen from the plot the solubilities of Al and Si increase with increasing hydroxyl ion concentration and the model is able to capture this behaviour. Figure 9.2. Effect of hydroxyl ion concentration changes on the equilibrium concentration of aluminum and silicon species. (Experimental data were obtained from experiments A l , A2, A3, A4, and A5 using synthetic green liquors of system A at 368.15 K and latm. For model calculations, only input amount of NaOH was varied from 0.25 to 2.0 mol/kg H2O. Input amounts of other chemicals were same as those of Figure 9.1. A value of In K h S O d = 84.37 was used.) NaOH (mol/kg H20) 0.1 • 0.25 • 1.0 A 2.0 P r e d i c t i o n j 1 1 L 0.001 0.01 0.1 Si concentration (mol/kg H 2 0) The amount of hydroxysodalite scale was also calculated to be smaller. This calculation result shows that the solutions can tolerate more dissolved Al and Si species 129 by increasing the amount of OH". Varying the total amount of Na + and OH" by changing the input amount of NaOH affects ion-ion interactions reflected as activity of water and activity coefficients of ions in the system. Table 9.2 shows one example of the calculation results with varying amounts of NaOH. The activity of water and activity coefficients showed significant change with varying input amounts of NaOH. Table 9.2. Example of the modeling calculation results for the Na + - Al(OH )4" - Si03 2" -OH" - C0 3 2" - S0 4 2' - GI" - H 2 0 system at 368.15 K . In the table, m is the molality, y is the activity coefficient, a is the activity, and hsod denotes the hydroxysodalite dihydrate. The total amounts of the chemicals except for the NaOH were AICI36H2O of 0.025 moi, N a 2 S i 0 3 9 H 2 0 of 0.025 moi, N a 2 C 0 3 of 1.0 moi, N a 2S0 4 of 0.1 moi, NaCl of 0.25 moi, and H 2 0 of 1 kg in initial solution. Input amount of NaOH 0.25 mol/kg H 2 0 1.0 mol/kg H 2 0 2.0 mol/kg H 2 0 m Y m Y m Y Na + 2.6990 0.5057 3.4489 0.5325 4.4520 0.5782 Al(OH) 4" 0.0010 0.2654 0.0047 0.1694 0.0120 0.0889 Si0 3 2" 0.0010 0.0519 0.0047 0.0389 0.0120 0.0281 OH" 0.1886 0.6262 0.9272 0.6701 1.9083 0.7391 C0 3 2" 0.9930 0.0192 0.9930 0.0180 0.9931 0.0173 S0 4 2" 0.0993 0.0250 0.0993 0.0235 0.0993 0.0222 Cl" 0.3227 0.5706 0.3227 0.6218 0.3228 0.6996 m a m a m a H 2 0 55.9003 0.9452 55.8978 0.9170 55.8929 0.8775 hsod 0.003969 1 0.003347 1 0.002137 1 Figure 9.3 shows the effect of carbonate ion on the precipitation conditions. The experimental data show that the solubility of Al and Si decreases when the amount of carbonate ion increases. The calculations show a very small sensitivity to the concentration changes of the carbonate ion. 130 Figure 9.3. Effect of carbonate ion concentration changes on the equilibrium concentration of aluminum and silicon species.(Experimental data were obtained from experiments A l , A2, A3, A6, and A7 using synthetic green liquors of system A at 368.15 K and latm. For model calculations, only input amount of Na2C03 was varied from 0.5 to 1.5 mol/kg H2O. Input amounts of other chemicals were same as those of Figure 9.1. A value of In K h s o d = 84.37 was used.) 0.1 0.01 o CM C O o S c o '•*-> CO c CD O c g 0.001 N a 2 C 0 3 (mol/kg H 2 0 ) • 0.5 • 1.0 A 1.5 Prediction _i 1 1 1 1 1 1 0.001 0.01 Si concentration (mol/kg H20) 0.1 Figure 9.4 shows that the model agrees well with the data that show that the sulfate ion did not change the precipitation conditions. Calculations at different concentrations of Al, Si and other species have shown similar agreement between the experimental data and the calculated results. For example, Figure 9.5 shows the effect of hydroxyl ion, Figure 9.6 that of the carbonate ion and Figure 9.7 that of the sulfate ion. 131 Figure 9.4. Effect of sulfate ion concentration changes on the equilibrium concentration of aluminum and silicon species.(Experimental data were obtained from experiments A l , A2, A3, A8, and A9 using synthetic green liquors of system A at 368.15 K and latm. For model calculations, only input amount of Na2SC<4 was varied from 0.05 to 0.2 mol/kg H_0. Input amounts of other chemicals were same as those of Figure 9.1. A value of In K h s o d = 84.37 was used.) 0.001 0.01 0.1 Si concentration (mol/kg H20) 132 Figure 9.5. Effect of hydroxyl ion concentration changes on the equilibrium concentration of aluminum and silicon species. (Experimental data were obtained from experiments A10, A l l , A12, A13, and A14 using synthetic white liquors of system A at 368.15 K and latm. For model calculations, Input amount of Na2C03 was 0.3 mol/kg H 20, that of Na2S04 was 0.1 mol/kg H 20, and that of NaCl was 0.1 mol/kg H 20. Input amount of NaOH was varied from 2.0 to 3.0 mol/kg H 2 0 Input amounts of A l C l 3 6 H 2 0 and Na2Si039H20 were varied from 0.08 to 0.02 mol/kg H 2 0 and from 0.02 to 0.08 mol/kg H 2 0 respectively. A value of In K h s o d = 84.37 was used.) 0.1 CM CD O E c o 2 -»—« c 8 C o o 0.01 0.001 h • \ v 3.0 --\ 2.5 - NaOH (mol/kg H20) 2.0 • 2.0 _ • 2.5 - A 3.0 - Prediction i i i i i i i 1 • i I I i 1 1 1 1 i i i i i 1 1 1 0.001 0.01 0.1 Si concentration (mol/kg H20) 133 Figure 9.6. Effect of carbonate ion concentration changes on the equilibrium concentration of aluminum and silicon species. (Experimental data were obtained from experiments A10, A l l , A12, A15, A16 using synthetic white liquors of system A at 368.15 K and latm. For model calculations, only input amount of Na2C03 was varied from 0.1 to 0.5 mol/kg H 2 0 . Input amounts of other chemicals were same as those of Figure 9.5. A value of In K h s o d = 84.37 was used.) 0.1 CM o £ c o CO l _ -«—' c 8 c o o 0.01 0.001 Na 2C0 3 (mol/kg H20) • 0.1 • 0.3 A 0.5 Prediction _i i i i i i i _i i i i i i i i _i i i i i i i i 0.001 0.01 0.1 Si concentration (mol/kg H20) 134 Figure 9.7. Effect of sulfate ion concentration changes on the equilibrium concentration of aluminum and silicon species.(Experimental data were obtained from experiments A10, A l l , A12, A17, and A18 using synthetic white liquors of system A at 368.15 K and latm. For model calculations, only input amount of Na2SC«4 was varied from 0.05 to 0.15 mol/kg H2O. Input amounts of other chemicals were same as those of Figure 9.5. A value of In K h s o d = 84.37 was used.) 0.1 CM X co o E c o 2 8 C o o 0.15 0.01 0.001 Na 2 S0 4 (mol/kg H 2 0) • 0.05 • 0.1 A 0.15 Prediction 0.001 0.01 0.1 Si concentration (mol/kg H 20) 135 9.2. N a + - A l ( O H ) 4 - S i 0 3 2 - O H - C 0 3 2 - C l - H S - H 2 0 (Sys t em B ) In experiments with system B, N a 2 S was used. Sodium sulfide dissociates in water to Na +, OH", and HS" according to the reaction: N a 2 S + H 2 0 -> 2 N a + + HS" + OH" (Smook 1992). The effect o f different HS" concentrations is shown in Figure 9.8. Figure 9.8. Effect o f hydrosulfide ion concentration changes on the equilibrium concentration o f aluminum and silicon species. (Experimental data were obtained from experiments B 1 - B 9 using synthetic green liquors o f system B at 368.15 K and latm. F o r model calculations, input amount o f N a O H was 0.25 mol/kg H 2 0 , and that o f N a 2 C 0 3 was 1.0 mol/kg H 2 0 . That o f N a 2 S was varied from 0.0 to 1.0 mol/kg H 2 0 . Input amounts o f A1C1 36H 20 and N a 2 S i 0 3 9 H 2 0 were varied from 0.1 to 0.05 mol/kg H 2 0 and from 0.05 to 0.1 mol/kg H 2 0 respectively. A value o f In K n S o d = 84.37 was used.) Na2S (mol/kg H 2 0) 0.001 0.01 0.1 Si concentration (mol/kg H20) 136 As seen from the plot, model and data agree that increasing HS" concentration increases the solubility of Al and Si. The same agreement is also seen in Figure 9.9 that shows results at different total concentration of chemicals. Figure 9.9. Effect of hydrosulfide ion concentration changes on the equilibrium concentration of aluminum and silicon species. (Experimental data were obtained from experiments B10-B18 using synthetic white liquors of system B at 368.15 K and latm. For model calculations, input amount of NaOH was 2.0 mol/kg H 2 O , and that of Na2C03 was 0.25 mol/kg H 2 0 . That of Na2S was varied from 0.0 to 1.0 mol/kg H 2 0 . Input amounts of AJC1 3 6H 2 0 and N a 2 S i 0 3 9 H 2 0 were varied from 0.1 to 0.05 mol/kg H 2 0 and from 0.05 to 0.1 mol/kg H 2 O respectively. A value of In Kh S 0d = 84.37 was used.) 0.1 Na2S (mol/kg H 2 0) 0.5 0.0 1.0 8 c • 0.0 • 0.5 A 1.0 0.001 Prediction j 1 1 1 1 1 1 1 1 1 1 1 1 1 1 j 1 1 1 1 0.001 0.01 0.1 Si concentration (mol/kg H20) 137 The thick line in Figure 9.10 represents experimental data at 368.15 K using kraft pulp mill green liquors containing 4.37 mol/L Na + , 0.85 mol/L OFT, 1.49 mol/L CO3 2 ' , 0.44 mol/L HS' and 0.02 mol/L S0 4 2 ' (Wannenmacher et al, 1996). The apparent solubility product, [Al][Si], was calculated by using the correlation for sodalite or hydroxysodalite formation of Wannenmacher et al (1996) and is plotted in Figure 9.10 as dashed line. The [Al][Si] was also determined using a graph from Gasteiger et al (1992) and is shown in the Figure as dotted-dashed line. As seen from the figure, the calculations based on the model presented in this work are closer to the data than the calculations based on published correlations. Figure 9.10. Comparison of model predictions with published correlations and industrial data. 0.1 Experimental data — Wannenmacher et al. (1996) o C N Predictions This work Gasteiger et al. (1992) Wannenmacher et al. (1996) j 1 1 • • I 0.001 0.01 0.1 Si concentration (mol/kg H20) 138 According to the overall results of the modeling calculations, the calculated precipitation conditions agree with the experimental data well. The model takes into account the effects of anions on the precipitation conditions. Thus, it is able to calculate solubilities of Al and Si in the presence of sodium aluminosilicates in green and white liquors of kraft pulp mills. This work can be used to develop an upgraded model to predict the equilibrium state of liquors involving more complicated reactions. For example, magnesium salts are added to the green liquor containing Al species to form hydrotalcite (Mgi. xAl x(OH)2 (C03)x/2-nH20, 0.10 < x < 0.34) precipitates which can be removed by the green liquor clarification (Ulmgren, 1987). Thus, the aluminum content in the recovery cycle can be reduced and sodium aluminosilicate scale formation can be prevented. If the hydrotalcite formation reaction is incorporated into the model, amount of hydrotalcite formed, remaining aluminum concentration, and required input amount of magnesium salts to prevent the sodium aluminosilicate scale formation can be calculated. In order to do these calculations, knowledge of the thermodynamic properties such as equilibrium constant of hydrotalcite formation and parameters for ion activity coefficient calculation (Pitzer's parameters) are required. Calculations of the precipitation conditions with the proposed model can be used as a tool to design progressive system closure kraft mill configurations that prevent sodium aluminosilicate formation in the recovery cycle. 139 CHAPTER 10. CONCLUSIONS AND RECOMMENDATIONS The precipitation conditions of sodium aluminosilicate scale were determined in two types of synthetic solutions which were prepared to simulate kraft pulp mill green and white liquors. The effects of the anions of OH", CO32", and SO42" and the effect of Na2S on the precipitation conditions were investigated. The predominant aluminum and silicon species in solutions were Al(OH)4" and Si032" because the solutions were very alkaline (pH>13). The solutions were found to tolerate more dissolved Al and Si species by increasing the amount of OH" and Na2S. Increasing the concentration of Na2C03 was found to lower the solubility of Al and Si species in the synthetic green liquor thus making the scale precipitation conditions easier to achieve. No significant effect of the carbonate and sulfate ions on the solubility limit of Al and Si in the synthetic white liquor was observed. Most of the precipitates were found to have the structure of hydroxysodalte dihydrate. Only in the solutions of low OH" and high Cf concentrations of system A, a small amount (about 25 % on a molar basis) of the sodalite dihydrate precipitates was found together with hydroxysodalte dihydrate (about 75 % on a molar basis). The above results were based on X-ray diffraction, chemical, and thermogravimetric analysis. The precipitation conditions of sodium aluminosilicate scale were also determined in mill green and white liquors. The effects of varying the Al/Si ratio and the concentration of NaOH, Na2C03, and Na2S on the precipitation conditions were investigated. The solubility limit of Al and Si was found to increase by increasing the 140 amount of NaOH and Na 2S and thus the liquor can tolerate more dissolved Al and Si. In both cases the amount of hydroxyl ion in the system increases. On the other hand, increasing the concentration of N a 2 C 0 3 lowers the solubility limit and the liquor tolerates less Al and Si dissolved. The precipitates were found to have the structure of hydroxysodalite dihydrate, Na 8(AlSi0 4)6(OH) 2-2H 20. A thermodynamics-based model to predict sodium aluminosilicate (hydroxysodalite dihydrate and/or sodalite dihydrate) precipitation conditions in aqueous alkaline solutions was presented. Activity coefficients were calculated by Pitzer's method. Comparison of the model-based calculated results with precipitation data showed good agreement. The effects of the OH", CO3 2 ' , SO42", HS" ions were predicted satisfactorily. The calculations were found to be sensitive to the value for the equilibrium constant for hydroxysodalite dihydrate. A simple isopiestic apparatus using a four-neck flask was built and employed to measure the osmotic coefficients of Na2Si03 and mixed Na2Si03-NaOH solutions at 298.15 K. The average relative percentage error of the osmotic coefficient measurements for the Na 2 Si0 3 and Na 2 Si0 3 -NaOH systems was 0.215 ± 0.045 %. Pitzer's binary parameters of Na+-Si032' and mixing parameters for OH"-Si032" and Na+-0H"-Si032' were estimated using the above osmotic coefficient data. Finally, it was determined that when metasilicate is dissolved in an alkaline aqueous solution, the predominant silicic species is the metasilicic ion of Si032". Osmotic coefficient data of the NaOH-NaCl-NaAl(OH)4 aqueous solutions were also obtained at 298.15 K. The results were successfully correlated with Pitzer's activity coefficient model. Pitzer's binary parameters for Na+-Al(OH)4" and mixing parameters 141 for OFT-Al(OH)4", Cr-Al(OH) 4', Na+-OFr-Al(OH) 4', and Na +-Cl'-Al(OH) 4' were determined using the experimental osmotic coefficient data. The Pitzer's parameters obtained can be used to calculate activity coefficients in the system involving the Al(OH)4" ion. This work provides a tool based on thermodynamics that can be used to predict the equilibrium state of the green and white liquors involving undesirable sodium aluminosilicate formation caused by the accumulation of the non-process elements of Al and Si. Thus, this work can aid the design of progressive system closure mill configurations. Future studies may include the effect of Ca(OH)2 addition on the sodium aluminosilicate formation. This makes possible to study the interactions of Al and Si with Ca species during the causticizing reaction. In the case of the thermodynamic modeling of the sodium aluminosilicate formation, the temperature dependence on the precipitation conditions might be readily included by adding a subroutine to calculate the equilibrium constants of sodalite dihydrate and/or hydroxysodalite dihydrate formation reaction at specific temperature. In addition, more accurate prediction of the precipitation conditions might be possible if accurate thermodynamic properties such as AG£, AH£, and Cp of sodalite dihydrate and hydroxysodalite dihydrate are available. Finally, the model can be incorporated in process simulation package like a CADSIM™ to examine closed mill processing configuration. 142 NOMENCLATURE a activity A4, Debye-Huckel parameter a.d. air dried 1/2 b universal parameter of Pitzer's equation, 1.2 (kg mol) Cp° standard heat capacity, J/mol K Cp°abs absolute heat capacity ACP° heat capacity change of reaction, J/mol K fn) Debye-Hiickel limiting law term of Pitzer's equation G Gibbs free energy G e x excess Gibbs free energy AG0 standard Gibbs free energy change of reaction, kJ/mol AGf° standard Gibbs free energy of formation, kJ/mol AH° standard enthalpy change of reaction, kJ/mol AHf° standard enthalpy of formation, kJ/mol I ionic strength, mol/kg H2O K thermodynamic equilibrium constant Khsod thermodynamic equilibrium constant of hydroxysodalite dihydrate formation reaction Ksod thermodynamic equilibrium constant of sodalite dihydrate formation reaction m molality, mol/kg H2O 143 M w molecular weight n number of moles n o x y number of oxygen nj number of groups of the jth type R gas constant, 8.314 J/mol K r radius of a sphere that completely circumscribes the anion ru distance between the center of the central atom and the center of the surrounding oxygen atoms S objective function S° standard entropy, J/mol K AS 0 standard entropy change of reaction, J/mol K T temperature To reference temperature V e volume at equivalence point W w number of kg of water stderr standard error x average value of x z charge of ion Aaj group contribution for a of jth ionic, atomic, or ligand molecule group, J/mol K Abj group contribution for b of jth ionic, atomic, or ligand molecule group, J/mol K 2 Aq group contribution for c of jth ionic, atomic, or ligand molecule group, 144 JK/mol A<j group contribution for d of jth ionic, atomic, or ligand molecule group, J/mol K3 A H J contribution for group j in the prediction of AHf°, kJ/mol Aqj contribution for group j in the prediction of AG/, kJ/mol Greek letters p(0),P(1),p(2),C* Pitzer's binary parameters 0, E0, E0', \|/ Pitzer's mixing parameters Xij(I) the short-range interaction between solute species i and j in solvent of Pitzer's equation u. chemical potential, J mol"1 Uijic triple interaction between solute species i, j and k in solvent of Pitzer's equation v number of ions produced by 1 mol of solute <j> osmotic coefficient o uncertainty a$ uncertainty of measured osmotic coefficient cti, cti numerical constant of Pitzer's equation y activity coefficient Y± mean activity coefficient Subscripts a, X anion (aq) aqueous c, M cation contrib contribution d index f formation hsod hydroxysodalite dihydrate, Na 8(AJSi04)6(OH) 2-2H 20 i indicates solute i i, j , k, 1 indicate each component in solutions (1) liquid s solute (s) solid sod sodalite dihydrate, Na 8(AlSi0 4)6Cl 2-2H 20 w solvent or water Superscripts calc calculated exp experimental int interpolated o standard state total total number of moles of element in the system 1, 2, R, S indicate solutions 146 BIBLOGRAPHY Anderson, G. 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Smart, "The influence of sodium carbonate on sodium aluminosilicate crystallisation and solubility in sodium aluminate solutions", J. Crystal Growth, 171, 197-208(1997). 159 APPENDIX I. Chemical Analysis by Atomic Absorption Spectrophotometer Two types of samples containing metals were analyzed using a GBC 904 Atomic Absorption Spectrophotometer (GBC Scientific Equipment Pty Ltd., Victoria, Australia). The first type consisted of samples which were obtained from the vessel of solubility experiments. The samples were treated using a 0.5 pm pore sized polyvinylidene fluoride membrane filter (Millipore Canada, Nepean, Ontario) before analysis. In the case that metal content in sample was much higher than optimum concentration range of analysis, the sample solution was diluted with deionized. Standard solutions of Al, Si, and K analysis were prepared by dissolving AICI36H2O, Na2Si03-9H20, and KC1 in deionized water respectively. Since a lot of sodium salts were dissolved in the samples, proper amount of NaOH and Na2C03 were dissolved into the standard solutions to minimize the interference of the back matrix solution. Table A. 1.1 shows analytical data used for each element. Table A. 1.1. Analytical data for the analysis of liquid phase alkaline samples. Element Wavelength (nm) Flame Optimum cone, range (pg/ml) Sensitivity (pg/ml) Al 396.2 Nitrous oxide-acetylene 5 - 100 m 1 Si 251.6 Nitrous oxide-acetylene 5-150 [ I ] 2 K 404.4 Air-acetylene 145 - 580 [ 2 ] 3.2 , J Greenberg etal. (1992),121 Athanasopoulos (1993). 160 The other type consisted of samples prepared from the solid precipitates. Prior to the analysis, a portion (about 0.1 g) of the precipitate was added into 1 g of deionized water. Nitric acid of 1 mol/L was added to the solution. A small amount (about 1 g) of the nitric acid was enough to dissolve the solid precipitate completely. The solution was diluted with deionized water. Table A. 1.2 shows analytical data used for the analysis of precipitate dissolved samples. Commercial standard solutions from Fisher Scientific, Vancouver, B.C. were used for calibrations. Table A. 1.2. Analytical data for the analysis of samples from the solid precipitates. Element Wavelength (nm) Flame Optimum cone, range (pg/ml) Sensitivity (pg/ml) Na 330.2 Air-acetylene 100-380 1 2 1 2.1 Al 396.2 Nitrous oxide-acetylene 5 - 100 m 1 Si 251.6 Nitrous oxide-acetylene 5-150 ^ 2 K 404.4 766.5 Air-acetylene 145 - 580 1 2 1 0.4-1.5 [ 2 ] 3.2 0.008 Greenberg et al (1992), w Athanasopoulos (1993). 161 APPENDIX II. Standard Errors and Confidence Intervals for Experimental Solubility Data Each sample solution of solubility experiments was analyzed five times using an atomic absorption spectrophotometer. The average value of the analysis results was taken and its standard error and 95% confidence interval were calculated For example, Al content in a sample solution of experiment A l in Table 3.1 was analyzed using three standard solutions of 49.96, 131.65, and 201.27 mg Al/kg H2O prepared in our lab. The atomic absorption analysis gave five values, 121.22, 120.61, 120.61, 119.40, and 124.86 mg/kg H 2 0 . The values were converted to molality scale and are 4.49E-3, 4.47E-3, 4.47E-3, 4.42E-3, and 4.62E-3 mol/kg H 2 0 . Average value was given by _ 0.00449 + 0.00447 + 0.00447 + 0.00442 + 0.00462 n n n A A n . . . T T ^ x= = 0.00449 mol/kg H 2 0 (A.2.1) Standard error of the value was given by stderr = 5 > , - x ) 2 1 "(n-1) =K 2 (Xj -0.00449)2 i=l = 3.36E-5 (A.2.2) 5(5-1) A value of t for 4 degree of freedom (n-1) and 95% confidence interval was 2.776 (Box et al, 1978). Thus, 95% confidence interval of the value 0.00449 mol/kg H 2 0 is given by 95% confidence interval = ± txstderr = 9.33E-05 (A.2.3) The calculated 95% confidence interval was plotted together with solubility data in Figure 8.2. 162 A P P E N D I X I H . T a b l e s o f S o l u b i l i t y D a t a Exper imen ta l so lubi l i ty data are shown i n the f o l l o w i n g Tables A . 3 . 1 - A . 3 . 3 together w i t h standard errors calculated by a method in A P P E N D I X II. Tab le A . 3 . 1 . So lub i l i ty data for synthetic l iquors o f system A . Synthetic green l iquor Exper imen t A l stderr S i stderr (mol/kg H 2 0 ) (mol/kg H 2 0 ) A l 4 .49E-03 3 .36E-05 5.01E-03 8 .55E-05 A l 4 .51E-03 1.22E-04 5.06E-03 1.92E-04 A l 4 .61E-03 5 .89E-05 4 .93E-03 9.11 E - 0 5 A 2 1.01E-03 9 .01E-06 2 .37E-02 1.00E-03 A 3 2 .58E-02 9 .17E-04 7 .36E-04 2 .58E-05 A 4 1.76E-03 1.09E-05 1.41E-03 5 .81E-05 A 5 6 .45E-03 2 .68E-05 6 .62E-03 2 .86E-05 A 6 5.91E-03 2 .49E-04 6 .19E-03 2 .17E-04 A 7 3 .79E-03 6 .97E-05 2 .66E-03 1.08E-04 A 8 6 .09E-03 2 .42E-05 4 .79E-03 5 .72E-05 A 9 5.38E-03 2 .88E-05 4 .70E-03 2 .12E-05 Synthetic whi te l iquor Exper imen t A l stderr S i stderr (mol/kg H 2 0 ) (mol/kg H 2 0 ) A 1 0 1.40E-02 1.10E-04 1.47E-02 3 .25E-04 A l l 5 .05E-03 7 .54E-05 3 .72E-02 2 .07E-04 A 1 2 4 .88E-02 3 .45E-04 4 .60E-03 1.30E-04 A 1 3 7 .97E-03 7 .35E-05 8.31E-03 1.89E-04 A 1 4 1.54E-02 8 .93E-05 2 .02E-02 3 .09E-04 A 1 5 1.32E-02 3 .57E-05 1.55E-02 1.20E-04 A 1 6 1.32E-02 1.53E-04 1.61E-02 2 .02E-04 A 1 7 1.37E-02 1.70E-04 1.37E-02 3 .30E-04 A 1 8 1.43E-02 5.61E-05 1.48E-02 1.07E-04 stderr : standard error o f measured mola l i ty 163 Tab le A . 3 . 2 . So lub i l i ty data for synthetic l iquors o f system B . Synthetic green l iquor Expe r imen t A l (mol/kg H 2 0 ) stderr S i (mol/kg H 2 0 ) stderr B l 3 .41E-03 8.99E-05 3 .98E-03 1.79E-04 B 2 3 .20E-02 2 .94E-04 2 .70E-04 2 .63E-05 B 3 4 .85E-04 1.55E-05 2 .54E-02 7 .12E-04 B 4 3 .03E-03 9 .04E-05 2 .56E-03 1.28E-04 B 5 1.93E-02 2 .04E-04 3 .78E-04 1.70E-05 B 6 3 .92E-04 1.07E-05 1.99E-02 6 .80E-04 B 7 4 .02E-03 1.33E-04 4 .52E-03 1.78E-04 B 8 2 .98E-02 2 .17E-04 6 .51E-04 2 .16E-05 B 9 8 .94E-04 1.81E-05 2 .29E-02 4 .91E-04 Synthetic whi te l i quor Exper imen t A l (mol/kg H 2 0 ) stderr S i (mol/kg H 2 0 ) stderr B I O 8.09E-03 2 .77E-04 9 .40E-03 3 .06E-04 B l l 3 .01E-02 4 .21E-04 2 .55E-03 2 .59E-05 B 1 2 2 .78E-03 3 .15E-05 2 .69E-02 4 .36E-04 B 1 3 9 .02E-03 1.44E-04 7 .22E-03 2 .94E-04 B 1 4 3 .52E-02 2 .64E-04 1.95E-03 5 .12E-05 B 1 5 2 .13E-03 2 .04E-05 2 .88E-02 8 .87E-04 B 1 6 1.14E-02 9 .85E-05 8.90E-03 2 .72E-04 B 1 7 3 .57E-02 3 .93E-04 2 .76E-03 8 .06E-05 B 1 8 3 .74E-03 2 .62E-05 2 .78E-02 7 .36E-04 stderr : standard error o f measured mola l i ty 164 Table A . 3 . 3 . So lub i l i ty data for m i l l l iquors . M i l l green l iquor Exper imen t A l (mol/kg H 2 0 ) stderr S i (mol/kg H 2 0 ) stderr M l 2 .65E-03 1.41E-04 5.63E-03 2 .24E-04 M 2 1.87E-02 2 .47E-04 7 .80E-04 2 .24E-05 M 3 6 .44E-04 2 .55E-05 2 .66E-02 5 .37E-04 M 4 5.54E-03 2 .02E-04 5.42E-03 1.89E-04 M 5 2 .14E-02 3 .22E-04 1.48E-03 3 .57E-05 M 6 1.09E-03 3 .00E-05 2 .65E-02 4 .67E-04 M 7 2 .16E-03 1.00E-04 3 .69E-03 1.93E-04 M 8 1.92E-02 3 .94E-04 3 .87E-04 1.82E-05 M 9 3 .88E-04 1.74E-05 1.98E-02 6 .36E-04 M 1 0 5 .54E-03 9 .23E-05 5.33E-03 2 .86E-04 M i l 3 .02E-02 7 .27E-04 7 .78E-04 7 .76E-05 M 1 2 9 .27E-04 3 .09E-05 2 .39E-02 3 .80E-04 M i l l whi te l i quor Exper imen t A l (mol/kg H 2 0 ) stderr S i (mol/kg H 2 0 ) stderr M 1 4 2 .93E-02 4 .42E-04 1.37E-03 5 .37E-05 M 1 5 1.38E-03 3 .61E-05 2 .94E-02 5 .45E-04 M 1 6 8.05E-03 2 .19E-04 7 .30E-03 2 .85E-04 M 1 7 3 .19E-02 4 .80E-04 1.83E-03 7 .01E-05 M 1 8 2 .02E-03 4 .91E-05 2 .99E-02 4 .04E-04 M 1 9 5 .22E-03 1.77E-04 4 .54E-03 1.59E-04 M 2 0 2 .80E-02 5 .38E-04 8 .88E-04 3 .03E-05 M 2 1 8 .35E-04 1.74E-05 2 .52E-02 4 .42E-04 M 2 2 8.65E-03 8.45E-05 6 .77E-03 2 .40E-04 M 2 3 3 .08E-02 3 .99E-04 1.89E-03 1.47E-04 M 2 4 2 .02E-03 8 .58E-05 2 .66E-02 3 .45E-04 stderr : standard error o f measured mola l i ty 165 A P P E N D I X I V . C a l c u l a t i o n o f T i t r a t i o n C u r v e The titration curve shown in Figure 5.2 was constructed by a method available in the literature that shows how the pH changes as titrant is added (Harris, 1991). One example of the calculation for Na2Si03-NaOH solution titration with H C 1 solution is given below. Before any acid is added, the solution of 20 mL (0.02 L) contains 0.25 mol/L Na2Si03 and 1.0 mol/L N a O H . The amount of O H " in the initial solution is 0.02 mol (1.0 mol/L x 0.02 L) from the dissociation of N a O H . The amount of S i 0 3 2 " is 0.005 mol (0.25 mol/L x 0.02 L) from the dissociation of Na2Si03. A portion of S i 0 3 2 " is converted to H S i 0 3 " and O H " by the following reaction. S i 0 3 2 " + H 2 0 <-> H S i 0 3 " + O H " (A . 4 . 1 ) (0.005 mol/0.02 L - x) x mol/L (x + 0.02 mol/0.02 L) A value of O H " concentration at equilibrium can be calculated. Let x be the concentration of H S i 0 3 " . Since equilibrium constant, K(A.4.I) of the reaction (A . 4 . 1 ) is equal to i 0 ' 2 2 0 1 7 . x(x + 0 . 0 2 / 0.02) = 1 0 . M 0 1 7 ( 0 . 0 0 5 / 0 . 0 2 - x ) The value of x is equal to 0 .001559 mol/L by solving equation (A . 4 .2 ) . Thus, pH of the initial solution is equal to pH = 14 + log[OH"] = 14 + log (0.001559 + 0.02/0.02) = 14.001 166 If the 1.0 mol/L HC1 of 10 mL (0.01 L) is added into the solution, the volume of the solution is equal to 0.02 L + 0.01 L = 0.03 L and the strong base OH" of 0.01 mol is titrated by the acid addition. Equation (A.4.1) can be rewritten as follows. Si0 3 2" + H 2 0 <-» HSi0 3" + OH" (A.4.3) (0.005 mol/0.03 L - x) x mol/L (x + (0.02-0.01) mol/0.03 L) The value of x is equal to 0.003058 mol/L as calculated by equation (A.4.4). x(x + (0.02-0.01)/0.03)_ 1 0_ 2 2 0 1 7 (0.005/0.03-x) Thus, pH of the initial solution is equal to pH = 14 + log[OH"] = 14 + log (0.003058 + (0.02-0.01V0.03) = 13.527 Figure 5.2 was plotted with calculated pH at varying amounts of added HC1 varying from 0 to 24 mL. A similar titration curve for the Na2C0 3-NaOH solution is shown in Figure 5.2 and was generated by the same method using an equilibrium constant of 10"3 6 7 6 9 for the reaction: C0 3 2 " + H 2 0 +» HC0 3 " + OH". 167 APPENDIX V. Uncertainty of the Measured Osmotic Coefficient Uncertainty of the measured osmotic coefficient in Tables 5.1-5.4 was calculated by the following equation available in the literature (Holman, 1994; Baird, 1995). If f is a function of the variables xi, \2,x„, f=f(xi, x 2 , x „ ) (A.5.1) the uncertainty in the calculation of f is given by - | l/2 af dx, X , J + df °"x K6\2 j • + df V ^ n J (A.5.2) where, a is uncertainty. An example of uncertainty calculation for the osmotic coefficient for Na2Si03 aqueous solution is given as follows. The Osmotic coefficient of Na2Si03 aqueous solution in Table 5.1 was calculated from the molality of solutes (Na2Si03) in sample solution, that (NaCl) in reference solution, and osmotic coefficient of reference solution by equation (A.5.3) s^ample V N a 2 S i 0 3 m N a 2 S i 0 3 — ^NaCI V N a C i m N a C l (A.5.3) where, <|> is the osmotic coefficient, v is the number of ions produced by 1 mole of the solute, and m is the molality of the solute. Rearranging the equation (A.5.3) gives _ 2 <l>NaCimNaCI 4> sample 3m (A.5.4) Na2Si03 Uncertainty of the osmotic coefficient of the sample solution, c r ^ , is given by sample <t>N«ci NaCl sample a _ m N « c i d m N a C i j f ^frsample 5m m N . l S i o j L " , , N a 2 S i O j y 1/2 168 (A.5.5) where, is the uncertainty of osmotic coefficient of NaCl solution and o m is the uncertainty of molality of the solute. A value of 0.01 was used for (Pitzer, 1991). Differentiating the equation (A.5.4) with <J>Naci, mwaci, ° m N l j M o , g i v e s (A.5.6) (A.5.7) (A.5.8) The molality, mNaci, is calculated from the measured mass of NaCl and H 2 O as follows 8 NaCl sample 2 m N a C l ^ N a C l 3 m N a j S i 0 3 ^'sample 2<l>NaCl ^ N a C l ^ m N a j S i O } sample _ 2 ( l > N a C i m N a C l ^ N a j S i O , ^ m N a 2 S i 0 3 m NaCl 58-44 x g H j Q where, gNaci is mass of NaCl, 58.44 is molecular weight, and g H 2 0 is mass of H 2 0 Differentiating the equation (A.5.9) with g N a C i and g H j 0 gives (A.5.9) 5m NaCl 1 3g Naci 58.44 x g H 2 0 dm NaCl _ g NaCl d g H , 58.44 xg* (A.5.10) (A.5.11) Since readability of the mass balance is 0.0001 g, the uncertainty of gNaci and g H j 0 is equal to 0.00005 and hence the uncertainty of mNaci is given by 169 mN«CI dm NaCl V ( 5e 8"2° V -11/2 1^ 58.44 x g H j •x 0.00005 ( g N a c i x 0.00005 il/2 58.44 xg (A.5.12) H 2 0 By the same logic, the uncertainty of m N a S i 0 j is given by m N l # i 0 3 ^122.06 x g H j 0 x 0.00005 + J g N a , S i O , 122.06 xg x 0.00005 J 1/2 (A.5.13) where, a valuel22.06 is molecular weight of Na2SiC«3. The values of <j>Naci, mN aci, m N a 2 S i 0 j , gN aci, g N a j S i o 3 > a n d 8H 2 O a r e determined from experiment. Substituting the equations (A.5.6), (A.5.7), (A.5.8), (A.5.12), and (A.5.13) to the equation (A.5.5) gives the uncertainty of the osmotic coefficient, , of the sample solution. 170 APPENDIX VL Uncertainty of the Estimated Thermodynamic Properties and Equilibrium Constants Uncertainty of AGf° of sodalite dihydrate (NarfAlSiO^Ch-l^O) The AGf° of sodalite dihydrate was estimated by the group contribution method in section 7.3.1. A simplified equation of the estimation was given by AG f°(sod) = AG f°(usod) + contribution by 2H 2 0 (A.6.1) where, the sod stands for sodalite dihydrate and the usod stands for unhydrous sodalite. Uncertainty of AGf°(sod) can be calculated using the equation (A.6.2). A G f ( s o d ) A 2 f dAG°(sod) j dAG°(sod) dAG f ° (usod) a A G ? ( u s o d > J + [acontrib(2H 2 0 ) a c o n , r i b ( 2 H j 0 ) J 12 (A.6.2) where the contrib standards for contribution. The uncertainty for AGf°(usod) is equal to 16.63 kJ/mol (Komada et al, 1995). That of contrib(2H20) is equal to contribution by 2H 2 0 x uncertainty of the estimation method (%) = -488.634 kJ/molx2.60% = -12.70 kJ/mol. Thus, uncertainty of AGf°(sod) is equal to 20.92 kJ/mol. Uncertainty of AG f ° of hydroxysodalite dihydrate (Nas(AlSi04)6(OU)2 2H20) The AGf° of hydroxysodalite dihydrate was estimated by the group contribution method in section 7.3.1. A simplified equation of the estimation was given by AGf°(hsod) = AGf°(usod) - contribution by C l 2 + contribution by (OH)2 + contribution by 2H20 (A.6.3) where, the hsod stands for hydroxysodalite dihydrate and the usod stands for unhydrous sodalite. Uncertainty of AGf°(hsod) can be calculated using the equation (A.6.4). 171 AG? (hsod) dAG f°(hsod) ^ A G ° ( u s o d ) ° A G ' ° ( u s o d ) aAGf°(hsod) 5contrib(Cl2) 'contrib(Clj) dAG°(hsod) acontrib((OH)2)( V 'contrib((OH)2) J aAG°(hsod) deontrib(2H20) 'contrib(2H20) J 1/2 (A.6.4) where the contrib standards for contribution. The uncertainty for AGf°(usod) is equal to 16.63 kJ/mol (Komada etal., 1995). That of contrib(Cl2) is equal to contribution by Cb x uncertainty of the estimation method (%) = -268.220 kJ/molx2.60% = -6.97 kJ/mol. That of contrib((OH)2) is equal to contribution by (OH) 2 x uncertainty of the estimation method (%) = -460.856 kJ/molx2.60% = -11.98 kJ/mol. That of contrib(2H20) is equal to contribution by 2H2O x uncertainty of the estimation method (%) = -488.634 kJ/molx2.60% = -12.70 kJ/mol. Thus, uncertainty of AG f°(hsod) is equal to (16.632 + (-6.97)2 + (-11.98)2 + (-12.70)2 ) 1 / 2 = 25.10 kJ/mol. The same method was used to calculate the uncertainties of A H / , and C p° of sodalite dihydrate (Na8(AlSi04)6Cl2-2H20) and hydroxysodalite dihydrate (Na8(AlSi04)6(OH)2 -2H20) in Table 7.5. Uncertainty of S° of Si0 3 2 Average error of the S° estimation in Table 7.9 is equal to (13.48% + 5.23% + 3.44% + 39.04% + 35.51% + 13.75%) / 6 = 18.41%. Thus, uncertainty of S° for Si03 2" is equal to estimated S°(Si0 3 2 ") x 18.41% = 3.84 J/mol K =0.00384 kJ/mol K. Uncertainty of C P ° of Si0 3 2 Average error of the C P ° estimation in Table 7.11 is equal to (39.03% + 1.42% + 45.65% + 62.35% + 28.55% + 4.64% + 19.33% + 1.90%) / 8 = 25.35%. 172 Thus, uncertainty of C P 0(Si0 3 2') is equal to estimated CP 0(Si0 3 2") x 25.35% = 82.71 J/mol K. Uncertainty of AHr° of Si032" The AHf° of Si0 3 2 ' was calculated by the following equation in section 7.3.4. AH f ° (S i0 3 2 - ) = 2AGf°(Na +) + AG f ° (S i0 3 2 - ) - AG f ° (Na 2 Si0 3 ) - 2AH f °(Na + ) + AH f ° (Na 2 Si0 3 ) + T (2S°(Na + ) + S0(Si032") - S° (Na 2 Si0 3 ) ) (A.6.5) Uncertainty of AH f °(Si0 3 2 ") from estimated S°(S i0 3 2 ' ) is given by °AHf (S iOf - ) aAH°(SiQ32-) ^ as°(Sio 3 2-) ' -|l/2 S"(SiOj") J = 298.15K x 0.003 84kJ/moi K = 1.14kJ/mol (A.6.6) Uncertainty of In Khsod The equilibrium constant of hydroxysodalite dihydrate formaiton at 368.15 K was calculated using equation (2.18). lnK = f~ \Q° ^ A X J O f 1 1 > \ A r<° f T T "N RT, AH° o J 1 1 T T, AC" R T T In — + -^ -1 Uncertainty of In K is given by a i n K = ainK vaAG° G a g : J + glnK a A H ° a ^ V 'ainK ^ v a A c ; ° A C v 1/2 (2.18) (A.6.7) Differentiating the equation (2.18) with AG°, AH°,and AC° gives 173 r ainK^ RT n = -0.4034mol/kJ f dlnK d A C ° , R 1 1 = 0.0767mol/kJ 2.4963molK/kJ (A.6.8) (A.6.9) (A.6.10) when R is the gas constant (8.314 kJ/kmol K), To is the reference temperature (298.15 K), and T is the temperature of interest (368.15 K). The A G Q of hydroxy sodalite formation reaction (4.2) is given by AG° = AG f°(hsod) + 4AG f ° (H 2 0) + 12AGf°(OFr) - 8AG f °(Na + ) -6AGf°(Al(OH)4') -6AGf°(Si0 3 2") - 2AGf0(OH') (A.6.11) Uncertainty of AG£ from estimated AGf° (hsod) is given as follows with ignoring other uncertainties of published thermodynamic properties. a A G S = dAG° - .1/2 SAG f°(hsod) A G ? ( h s o d ) = 25.10kJ/mol (A.6.12) By the same method, uncertainty of AH° and AC° are calculated as 26.71 kJ/mol and 0.4963 kJ/mol K respectively. Substituting the values of equations (A.6.8), (A.6.9), (A.6.10), (A.6.12) and uncertainty values of AH^ and AC° to equation (A.6.7) gives uncertainty of In K is equal to ((-0.4034 mol/kJx25.10 kJ/mol)2 + (0.0767 mol/kJx26.71 kJ/mol)2 + (2.4963 mol K/kJx0.4963 kJ/mol K ) 2 ) 1 / 2 = 10.41, for hydroxysodalite dihydrate formation reaction. Uncertainty of In K S O d , 8.71, for sodalite dihydrate formation reaction (4.1) was calculated by the same method. 174 APPENDIX VII. Comparison of Pitzer's Parameters with Published Values The reliability of the optimization method for the determination of Pitzer's parameters was checked for four systems, NaCl, N a 2 S 0 4 , N a 2 S 2 0 3 and NaCl-Na2SC>4. The systems were chosen since both experimental data and Pitzer's parameters were available in the literature. The same osmotic coefficient data were used to determine the parameters. The estimated parameters by Pitzer and in this work are given in Table A.7.1 together with standard deviations and sum (SUM) o f O T P - f 3 ' 6 ) 2 . Table A.7.1. Comparison of Pitzer's parameters. system parameter Pitzer (1991) This work Parameter values Standard deviations p(o) 0.0765 0.0767 0.0003 NaCl m 0.2664 0.2626 0.0027 c* 0.00127 0.00121 0.00005 SUM 0.00003 0.00002 0(0) 0.01958 0.01739 0.00092 Na2S04 m 1.113 1.127 0.018 c* 0.00497 0.00773 0.00028 SUM 0.00456 0.00007 P ( 0 ) 0.06615 0.06403 0.00108 Na 2 S 2 0 3 [ 1 ] pd) 1.27575 1.28655 0.01875 c* 0.00497 0.00602 0.00037 SUM 0.00015 0.00006 crsoj- -0.035 -0.029 0.004 NaCl-Na 2S0 4 1 2 1 *P Na+Cl"SoJ" 0.007 0.007 0.001 SUM 0.00007 6.66064 1 References of osmotic coefficient data [1] Robinson and Stokes (1965), [2] Wu etal. (1968). 175 Calculated curves of osmotic coefficient data using both parameters are shown in Figure A.7.1. As seen from the figure and the table, the parameters obtained in this work gave exactly same prediction for the NaCl system and slightly better for the NazSCu and Na2S2C«3 systems. Figure A.7.1. Osmotic coefficients for NaCl, Na2S04, and Na2S203 systems. 1-3 11 i ~ i £ co o o c CO D 0.6 H 0.5 0 1 2 3 4 5 6 molality 176 APPENDIX V I E . Sensitivity Analysis of Pitzer's Parameters Sensitivities of Pitzer's parameters which were obtained in Chapter 6 were checked for Na 2 Si0 3 , Na 2Si0 3-NaOH, and NaOH-NaCl-NaAl(OH) 4 aqueous systems. Values of the following least squares objective function, S, were calculated using parameter, parameter!-10%, and parameter-10%. s=±(*r-*rr (A.8.0 d=i a + d where (|>exp is the measured and (j)03'6 is the calculated osmotic coefficient using equation (2.46). c$ is the uncertainty. The results of the calculations are given in Tables A.8.1, A.8.2, and A.8.3. In the case of Na 2 Si0 3 aqueous system, the objective function value, S was more sensitive on PSjSio, a n d P(Na2sio3than on C * a 2 S i 0 j . The value, S, was very sensitive on 6OH_sio2_ for Na 2 Si0 3 -NaOH aqueous system. For NaOH-NaCl-NaAl(OH)4 aqueous system, the value, S, was more sensitive on < P N a + c l - A I ( O H ) - and ©OH -^^- than by other parameters. Table A.8.1. Sensitivity analysis of Pitzer's parameters for Na 2 Si0 3 aqueous system. Parameter S value ParameteH-10% S value Parameter-10% S value o (0 ) HNa2SiOj 0.0577 0.06347 30.015 0.05193 29.9941 Od) MNa2Si03 2.8965 3.556 3.18615 32.960 2.60685 32.939 ,"Na 2Si0 3 0.00977 0.010747 6.374 0.008793 6.368 177 Table A . 8 . 2 . Sensitivity analysis of Pitzer's parameters for Na2Si03-NaOH aqueous system. Parameter S value Parameter+10% S value Parameter-10% S value O(0) HNajSiOj 0.0577 0.06347 123.896 0.05193 123.840 pO) HNajSiOj 2.8965 3.18615 116.719 2.60685 149.726 r"t> ^NajSiOj 0.00977 110.71 0.010747 113.799 0.008793 113.772 e OH SiOj -0.2703 -0.29733 142.755 -0.24327 142.843 Na+OH"SiOj" 0.0233 0.02563 114.613 0.02097 114.583 Table A . 8 . 3 . Sensi t ivi ty analysis o f P i t ze r ' s parameters for N a O H - N a C l - N a A l ( O H ) 4 aqueous system. Parameter S value Parameter+10% S value Parameter-10% S value O(0) HNaAl(OH)4 -0.0083 -0.00913 7.943 -0.00747 7.939 o(D PNaAl(OH)4 0.0710 0.0781 7.930 0.0639 7.931 P<t> ^NaAl(OH) 4 0.00184 0.002024 7.933 0.001656 7.936 e OH"Al(OH)i -0.2255 7.929 -0.24805 11.743 -0.20295 11.651 6 CPAl(OH); -0.2430 -0.2673 8.822 -0.2187 8.787 Na+OH"Al(OH)i -0.0388 -0.04268 9.190 -0.03492 9.137 Na+CrAl(OH)7 0.2377 0.26147 14.111 0.21393 14.218 178 APPENDIX IX. Computational Source Codes in FORTRAN 7 7 Deternination of Pitzer's binary parameters for single electrolyte system using osmotic coefficient data Example of inputfile (shvpanuin) 'siwpam.in' •Na2Si03 s i n g l e e l e c t r o l y t e system' 'Charges of cat i o n and anion' 1 -2 'stochiometric numbers of cat i o n & anion' 2 1 'Temperature(oC)' 25.00 'Debye-Huckel parameter at T (kg/mol)l/2' 0.3915 ' i n i t i a l guess of P i t z e r parameters' •BethaO Bethal Cphi• 0.0000 0.0000 0.0000 'number of data' 16 'Mol a l i t y , osm., Wr' 0 0603 0 8923 0 0096 0 0603 0 8926 0 0096 0 3674 0 8339 0 0091 0 3690 0 8304 0 0090 0 5313 0 8188 0 0088 0 5313 0 8188 0 0088 0 8637 0 7790 0 0083 0 8629 0 7797 0 0083 1 2063 0 7614 0 0080 1 2059 0 7617 0 0080 1 4928 0 7607 0 0078 1 4 927 0 7608 0 0078 1 8213 0 7685 0 0078 1 8241 0 7674 0 0077 2 3745 0 8021 0 0078 2 3725 0 8028 0 0078 Example of output file (siwpam. out) D a t a f i l e : siwpam.in Na2Si03 s i n g l e e l e c t r o l y t e system Temperature : 298.15 K i n i t i a l gue^s of P i t z e r parameter : bethaO bethal cphi .00000 .00000 .00000 i t e bethaO bethal cphi P(l) P(2) P(3) S 1 .05770 2.89650 .00977 .0577 2.8965 .009811833.6200 2 .05770 2.89649 .00977 .0000 .0000 .0000 3.5560 I t e r a t i o n s : 2 P i t z e r parameters obtained BethaO Bethal Cphi .05770 2.89649 .00977 S value at s o l u t i o n : 3.555965 Err o r variance, sigma~2 = S/(N-P): .273536 Variance-covariance matrix, COV=(inv A)*(sigma A2) COV(l,l)= .000015 COV(l,2)= -.000182 COV(l,3)= -.000007 COV(2,l)= -.000182 COV(2,2)= .003128 COV(2,3)= .000076 COV(3,l)= -.000007 COV(3,2)= .000076 COV(3,3)= .000003 Standard d e v i a t i o n of parameters, STDEV=(diag COVP0.5 stdev(betha0)= .003911 stdev(bethal)= .055932 stdev(cphi) = .001764 m o l a l i t y osmexp osmcal aH20cal 0603 .8923 .8837 .99712 0603 .8926 . 8837 .99712 3674 .8339 . 8401 .98346 3690 . 8304 .8398 .98339 5313 .8188 .8158 .97685 5313 .8188 .8158 .97685 8637 .7790 .7796 .96427 8629 .7797 .7797 .96430 1 2063 .7614 .7618 .95156 1 2059 .7617 .7618 .95157 1 4928 .7607 .7594 .94058 1 4 927 .7608 .7594 .94058 1 8213 .7685 .7679 .92721 1 8241 .7674 .7680 .92709 2 3745 .8021 .8029 .90210 2 3725 .8028 .8028 .90220 Source code (siwpam.for) c siwpam.for Q ********************************************************** c C a l c u l a t i n g P i t z e r ' s parameters(BethaO,Bethal,Cphi) c at given set of m o l a l i t y and osmotic c o e f f i c i e n t data c f o r Single e l e c t r o l y t e s o l u t i o n using a Gauss-Newton method c with Weighting Factor c Hyeon Park c U n i v e r s i t y of B r i t i s h Columbia c March 05, 1998 Q ********************************************************** i m p l i c i t r e a l (a-h, o-z) external g character*50 d a t a f i l e , h e a d l i n e parameter(npara=3,NP=100,MP=100,GT0L=1.OE-5,ITMAX=100) parameter(nunit=l1 r e a l f(NP),mol<NP),x(MP),osmexp(NP),osmcal(NP) r e a l JM (NP, MP) , J J (MP, MP), J f (NP), P (NP) r e a l i s (NP) , f p h i (NP) ,bphi (NP) ,cphi r e a l AI (MP,MP), xg (MP,MP) , ah2o(NP) r e a l errvar,COV(MP,MP),stdev _p(MP) r e a l wr(NP) ccccc c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c Open I/O f i l e s open (5,file='siwpam.in',form='formatted') open (6,file='siwpam.out',form='formatted') ccccc c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c Read input f i l e read(5,*) d a t a f i l e write(6,10) d a t a f i l e 10 f o r m a t ( / l x , ' D a t a f i l e : ',al2) read(5,*) headline write(6,15) headline 15 formatdx, a40) read(5,*) headline. read(5,*) zc, za read(5,*) headline read(5,*) vc, va v = vc + va read(5,*) headline read(5,*) t c tk = t c + 273.15 write(6,20) tk 20 format(lx,'Temperature : ',f6.2,' K') c Debye-Huckel parameter at T oC read(5,*) headline read(5,*) aphi read(5,*) headline read(5,*) headline c I n i t i a l guess of bethaO,bethal,cphi read(5,*) x(l),x(2),x(3) write(6,25) 25 f o r m a t ( / l x , 1 i n i t i a l guess of P i t z e r parameter : ') write(6,30) 30 format(6x,'bethaO',3x,'bethal•,3x,'cphi') write(6,35) x(1 ) , x(2) , x (3) 35 format(5x,f7.5,2x,f7.5,2x,f7.5) c Number of data read(5,*) headline read(5,*) ndata c Ready to read the m o l a l i t y data ! read(5,*) headline do 47 i=l,ndata read(5,*) moi(i),osmexp(i),wr(i) 47 continue cccccccccccccccccccccccccccccccc U n i v e r s i a l constants of P i t z e r ' s eqs b = 1.2 i f ((zc.eq.2).and.(za.eq.2)) then alphal =1.4 alpha2 = 12.0 el s e alphal =2.0 alpha2 = 0.0 end i f w rite(6,49)'iter','bethaO','bethal','cphi', & 'P(l)','P(2)','P(3)','S' 4 9 format(/Ix,a3,3(a9),2x,3(a9),lx,a6) ccccc c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c S t a r t i t e r a t i o n do 500 iter=l,ITMAX c I n i t i a l i z a t i o n of S (value of ob j e c t i v e function) S = O.OeO cccccccccccccccccccccccccccccccccccc c osmotic c o e f f i c i e n t c a l c u l a t i o n do 51 i=l,ndata i s ( i ) = O.OeO i s ( i ) = 0.5*( moi(i)*vc*(zc**2) + moi(i)*va*(za**2) ) f p h i ( i ) = O.OeO f p h i ( i ) = - a p h i * ( i s ( i ) * * 0 . 5 ) / ( l + b * i s ( i ) * * ( 0 . 5 ) ) c I f i t i s not 2-2 s a l t , betha2 i s 0. bethaO = x ( l ) bethal = x(2) betha2 = O.OeO cphi = x(3) bp h i ( i ) = O.OeO bp h i ( i ) = bethaO b p h i ( i ) = b p h i ( i ) + b e t h a l * e x p ( - a l p h a l * ( i s ( i ) * * 0 . 5 ) ) b p h i ( i ) = b p h i ( i ) + betha2*exp(-alpha2*(is(i)**0.5)) osmcal(i)=0.OeO osmcal(i)=abs(zc*za)*fphi(i)+mol(i)*(2*vc*va/v)*bphi(i) & +(mol(i)**2)*((2*(vc*va)**(3/2))/v)*cphi+l f ( i ) = O.OeO f ( i ) = (osmcal(i)-osmexp(i))/wr(i) S = S + f ( i ) * * 2 . c c a l c u l a t e the a c t i v i t y of water ah2o(i) = O.OeO ah2o(i) = 10**(-0.007823*(vc*mol(i)+va*mol(1))*osmcal(i)) 51 continue c Jacobian matrix c a l c u l a t i o n s do 54 i=l,NP do 53 j=l,MP JM(i,j)=0.0e0 53 continue 54 continue do 55 1=1,ndata JM(i,l) = mol(i)*(2*vc*va/v)/wr(i) JM(i,2) = m o l ( i ) * ( 2 * v c * v a / v ) * e x p ( - a l p h a l * ( i s ( i ) * * 0 . 5 ) ) / w r ( i ) JM(i,3) = (moi(i)**2)*(2*((vc*va)**(3/2))/v)/wr(i)• 55 continue c J J i n i t i a l i z a t i o n do 65 i=l,MP do 60 j=l,MP JJ(i,j 1=0. 60 continue 65 continue c J J c a l c u l a t i o n do 130 i=l,npara do 120 j=l,npara do 110 k=l,ndata JJ(i,j)=JJ(i,j)+JM<k,i)*JM(k,j) 110 continue 120 continue 130 continue c J f i n i t i a l i z a t i o n do 150 i=l,MP J f ( i ) = 0 . 150 continue c J f c a l c u l a t i o n do 180 i=l,npara do 170 j=l,ndata J f (i)=Jf ( i ) + J M ( j , i ) * f (j ) 170 continue J f ( i ) = - J f ( i ) 180 continue cccccccccccccccccccccccc P c a l c u l a t i o n (to obtain updated parameters) do 185 i=l,MP do 184 j=l,MP AI(1,j)=0.OeO xg(i,j)=0.0e0 184 continue 185 continue do 200 i=l,npara do 190 j=l,npara A I ( i , j ) = J J ( i , j ) 190 continue 200 continue do 210 i=l,npara x g ( i , l ) = J f ( i ) 210 continue c By passing the subroutine "gaussj", c Matrix AI becomes Inverse matrix of J J ( AI = Inv J J ) c and Matrix xg becomes P matrix. ( JJ*P = J f ) c a l l gaussj(AI,npara,NP,xg,nunit,MP) c New x ( i ) step_length = 1.0 do 240 1=1,npara p ( i ) = x g ( i , l ) x ( i ) = x ( i ) + p ( i ) * s t e p _ l e n g t h 240 continue w r i t e ( 6 , 2 5 0 ) i t e r , x ( l ) , x ( 2 ) , x ( 3 ) , P ( l ) , P(2) , P(3) ,S 250 format(lx,i3,3(f9.5),2x,3(f9.4),fl0.4) ccccc c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c S t a t i s t i c s of parameters c E r r o r variance (errvar,sigma**2) er r v a r = O.OeO errva r = S/(ndata-npara) c Variance-covarlance matrix (V=(Inv A)*(sigma**2)) do 271 1=1,MP do 270 j=l,MP COV(i,j)=0.0e0 270 continue 271 continue do 281 i=l,npara do 280 j=l,npara C O V ( i , j ) = A I ( i , j ) * e r r v a r 280 continue 281 continue do 290 i=l,npara stdev _ p ( i ) = s q r t ( C O V ( i , i ) ) 290 continue if((abs(P(l)).LT.GTOL).AND.(abs(P(2)).LT.GTOL)) go to 600 500 continue p r i n t *, 'Exceeding maximum i t e r a t i o n s . . . ' stop c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c 600 p r i n t *, ' ' w r i t e ( 6 , ' ( / / l x , a , i 5 ) ' ) ' I t e r a t i o n s : ' , i t e r w r i t e ! * , ' ( / / l x , a , i 5 ) ' ) ' I t e r a t i o n s : ' , i t e r w r i t e ( 6 , ' ( / l x , a ) ' ) ' P i t z e r parameters obtained' w r i t e ( * , ' ( / l x , a ) ' ) ' P i t z e r parameters obtained' w r i t e ( 6 , ' ( l x , a 9 , a l O , a l l ) • ) 'BethaO','Bethal','Cphi' w r i t e ! * , ' ( l x , a 9 , a l O , a l l ) • ) 'BethaO','Bethal','Cphi' write(6,610) x(1 ) , x(2) , x(3) 610 format(3x,f9.5,lx,f9.5,lx,f9.5) write!*,620) x(1),x(2),x(3) 620 format(lx,f9.5,2x,f9.5,2x,f9.5) write(6,'(/lx,a,f12.6)') 'S value at soluti o n : ' , S write!*,'(/lx,a,f12.6)') 'S value at solution:',S write(6,•(/lx,a,f12.6)') 'Error variance, sigma~2 = S/(N-P):', & err v a r write!*,'(/lx,a,f12.6)') 'Error variance, sigma"2 = S/(N-P):', & err v a r write(6,'(/lx,2a)')'Variance-covariance matrix, ', S'COV=(inv A) * (sigma~2') ' write(*,'(/lx,2a)')'Variance-covariance matrix, ', &'COV=(inv A)*(sigma~2)' do 631 i=l,npara do 630 j=l,npara w r i t e ( 6 , ' ( l x , a , i l , a , i l , a, f12 . 6) ' ) 'COV( ' , i , ', ' , j , ' ) = ',COV(i,j) w r i t e ( * , ' ( l x , a , i l , a , i l , a , f 1 2 . 6 ) ' ) 'COV(',i,',',j,')=',COV(i,j) 630 continue 631 continue write(6,'(/lx,2a)')'Standard d e v i a t i o n of parameters, ', 5 'stdev=(diag COV)~0.5* write(*,'(/lx,2a)')'Standard d e v i a t i o n of parameters, ', 6 'stdev=(diag COV)~0.5' 'stdev(betha0)=',stdev _ p ( l ) 'stdev(bethaO)=',stdev _ p ( l ) 'stdev(bethal)=',stdev _p(2) 'stdev(bethal)=',stdev _p(2) 'stdev(cphi) =',stdev _p(3) 'stdev(cphi) =',stdev _p(3) 'molality','osmexp','osmcal', w r i t e ( 6 , ' ( l x , a , f l 2 . 6 ) w r i t e ( * , ' ( l x , a , f l 2 . 6 ) w r i t e ( 6 , ' ( l x , a , f l 2 . 6 ) w r i t e ! * , ' ( l x , a , f l 2 . 6 ) w r i t e ( 6 , • ( l x , a , f l 2 . 6 ) ' ] w r i t e ) * , ' ( l x , a , f l 2 . 6 ) 1 w r i t e ( 6 , ' ! / l x , 4 ( a l 2 ) ) • ] & 'aH20cal' do 640 i=l,ndata w r i t e ( 6 , ' ( I x , 3 ( f l 2 . 4 ) , f l 2 . 5 ) ' ) m o l ( i ) , o s m e x p ( i ) , o s m c a l ( i ) , S ah2o(i) 640 continue p r i n t *, ' ' ccccccc c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c Close I/O F i l e s close(5) close(6,status='keep') end c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c Function g(x) r e a l function g(x) r e a l x i f (x.eq.0) then g = 0 els e g = 2*(l-(l+x)*exp(-x))/(x**2) endif end c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c SUBROUTINE gaussj(a,n,np,b,m,mp) c. gaussj.for c :subroutine imported from "Numerical Recipes Fortran 77", c ve r s i o n 2.07 to c a l c u l a t e an inverse matrix of A and solve c a matrix equation, AX=B. return END c Hyeon Park, UBC, 1998 183 Calculation of osmotic coefficient and activity coefficient using Pitzer's binary parameters for single electrolyte system Example of inputfile (sbvcaLin) • s i w c a l . i n ' ' N a 2 S i 0 3 s i n g l e e l e c t r o l y t e s y s t e m ' ' C h a r g e s o f c a t i o n a n d a n i o n ' 1 - 2 ' s t o c h i o m e t r i c n u m b e r s o f c a t i o n 4 a n i o n ' 2 1 ' T e m p e r a t u r e ( o C ) ' 2 5 . 0 0 ' D e b y e - H u c k e l p a r a m e t e r a t T ( k g / m o l ) 1 / 2 ' r 0 . 3 9 1 5 ' P i t z e r p a r a m e t e r s ' ' B e t h a O B e t h a l B e t h a 2 C p h i ' 0 . 0 5 7 7 2 . 8 9 6 5 0 . 0 0 0 0 0 . 0 0 9 7 7 ' n u m b e r o f m o l a l i t y d a t a ' 129 • M o l a l i t y ' 0 . 0 1 0 0 0 . 0 2 0 0 0 . 0 3 0 0 0 . 0 4 0 0 0 . 0 5 0 0 0 . 0 6 0 0 0 . 0 7 0 0 1 . 9 0 0 0 2 . 0 0 0 0 2 . 2 0 0 0 2 . 4 0 0 0 2 . 6 0 0 0 Example of input file (siwcal out) D a t a f i l e : s i w c a l . i n N a 2 S i 0 3 s i n g l e e l e c t r o l y t e s y s t e m T e m p e r a t u r e : 2 9 8 . 1 5 K P i t z e r p a r a m e t e r : b e t h a O b e t h a l b e t h a 2 c p h i . 0 5 7 7 0 2 . 8 9 6 5 0 . 0 0 0 0 0 .00977 m o l a l i t y o s m c a l aH20 a c t . c o e f 0100 . 9158 .99951 . 7 4 1 5 0200 . 9006 . 9 9 9 0 3 . 6852 0300 . 8932 . 9 9 8 5 5 . 6 5 2 3 0400 . 8888 . 9 9 8 0 8 . 6 2 9 3 0500 . 8858 . 99761 . 6 1 1 9 0600 . 8837 .99714 . 5 9 7 9 0700 . 8821 . 99667 . 5862 1 9000 .7714 . 9 2 3 8 5 . 2 9 7 9 2 0000 .7767 . 9 1 9 4 8 .2961 2 2000 . . 7 8 9 5 .91041 .2937 2 4000 .8051 . 9 0 0 8 5 .2931 2 6000 . 8232 . 8 9 0 7 8 . 2941 Source code (siwcalfor) c s i w c a l . f o r c ********************************************************** c C a l c u l a t i n g o s m o t i c c o e f f i c i e n t and a c t i v i t y c o e f f i c i e n t c a t g i v e n m o l a l i t y , t e m p e r a t u r e ( D e b y e - H u c k e l p a r a m e t e r ) , c and P i t z e r ' s p a r a m e t e r s ( B e t h a O , B e t h a l , B e t h a 2 , C p h i ) c f o r S i n g l e e l e c t r o l y t e s o l u t i o n c H y e o n P a r k c U n i v e r s i t y o f B r i t i s h C o l u m b i a c M a r c h 1 0 , 1998 j - * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * i m p l i c i t r e a l ( a - h , o - z ) e x t e r n a l g c h a r a c t e r * 5 0 d a t a f i l e , h e a d l i n e r e a l m o i , i s , l n g a m c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c Open I / O F i l e s o p e n ( 5 , f i l e = ' s i w c a l . i n ' , f o r m = ' f o r m a t t e d ' ) o p e n ( 6 , f i l e = ' s i w c a l . o u t ' , f o r m = ' f o r m a t t e d ' ) c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c Read I n p u t F i l e r e a d ( 5 , * ) d a t a f i l e w r i t e ( 6 , 1 0 ) d a t a f i l e 10 f o r m a t ( / l x , ' D a t a f i l e : ' , a l 2 ) r e a d ( 5 , * ) h e a d l i n e w r i t e ( 6 , 1 5 ) h e a d l i n e 15 f o r m a t d x , a40) r e a d ( 5 , * ) h e a d l i n e r e a d ( 5 , * ) z c , z a r e a d ( 5 , * ) h e a d l i n e r e a d ( 5 , * ) v c , v a v = v c + v a r e a d ( 5 , * ) h e a d l i n e r e a d ( 5 , * ) t c t k = t c + 2 7 3 . 1 5 w r i t e ( 6 , 2 0 ) t k 20 f o r m a t ( l x , ' T e m p e r a t u r e : ' , f 6 . 2 , ' K ' ) c D e b y e - H u c k e l p a r a m e t e r a t T oC r e a d ( 5 , * ) h e a d l i n e r e a d ( 5 , * ) a p h i c R e a d P i t z e r ' s p a r a m e t e r s r e a d ( 5 , * ) h e a d l i n e r e a d ( 5 , * ) h e a d l i n e r e a d ( 5 , * ) b e t h a O , b e t h a l , b e t h a 2 , c p h i w r i t e ( 6 , 2 5 ) 25 f o r m a t ( / l x , ' P i t z e r p a r a m e t e r : ' ) w r i t e ( 6 , 3 0 ) 30 f o r m a t ( 6 x , ' b e t h a O ' , 3 x , ' b e t h a l ' , 3 x , ' b e t h a 2 ' , 4 x , ' c p h i ' ) w r i t e ( 6 , 3 5 ) b e t h a O , b e t h a l , b e t h a 2 , c p h i 35 f o r m a t ( 5 x , f 7 . 5 , 2 x , f 7 . 5 , 2 x , f 7 . 5 , 2 x , f 7 . 5 ) c Number o f d a t a r e a d ( 5 , * ) h e a d l i n e r e a d ( 5 , * ) n d a t a c R e a d y t o r e a d t h e m o l a l i t y d a t a ! r e a d ( 5 , * ) h e a d l i n e c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c U n i v e r s i a l c o n s t a n t s o f P i t z e r ' s e q s b = 1 .2 i f ( ( z c . e q . 2 ) . a n d . ( z a . e q . 2 ) ) t h e n a l p h a l = 1 . 4 a l p h a 2 = 1 2 . 0 e l s e a l p h a l = 2 . 0 a l p h a 2 = 0 . 0 e n d i f w r i t e ( 6 , 5 0 ) ' m o l a l i t y ' , ' o s m c a l ' , ' a H 2 0 ' , ' a c t . c o e f ' 50 f o r m a t ( / l x , 4 ( a l 2 ) ) w r i t e ! * , 5 1 ) ' m o l a l i t y ' , ' o s m c a l ' , ' a H 2 0 ' , ' a c t . c o e f 51 f o r m a t ( / l x , 4 ( a l 2 ) ) c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c S t a r t i t e r a t i o n d o 1000 i t e r = 1 , n d a t a r e a d ( 5 , * ) moi c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c o s m o t i c c o e f f i c i e n t c a l c u l a t i o n i s = 0 . 0 i s = 0 . 5 * ( m o l * v c * ( z c * * 2 ) + m o l * v a * ( z a * * 2 ) ) f p h i = 0 . 0 f p h i = - - a p h i * ( i s * * 0 . 5 ) / ( l + b * i s * * ( 0 . 5 ) ) c I f i t i s n o t 2 - 2 s a l t , b e t h a 2 i s 0 . b p h i = 0 . 0 b p h i = b e t h a O b p h i = b p h i + b e t h a l * e x p ( - a l p h a l * ( i s * * 0 . 5 ) ) b p h i = b p h i + b e t h a 2 * e x p ( - a l p h a 2 * ( i s * * 0 . 5 ) ) o s m c a l = O.OeO o s m c a l = o s m c a l + a b s ( z c * z a ) * f p h i o s m c a l = o s m c a l + m o i * ( 2 * v c * v a / v ) * b p h i o s m c a l = o s m c a l + m o l * * 2 * ( ( 2 * ( v c * v a ) * * { 3 / 2 ) ) / v ) * c p h i o s m c a l = o s m c a l + 1 . 0 c C a l c u l a t e t h e a c t i v i t y o f w a t e r ah2o = O.OeO ah2o = 1 0 * * ( - 0 . 0 0 7 8 2 3 * ( v c * m o l + v a * m o l ) * o s m c a l ) c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c a c t i v i t y c o e f f i c i e n t f g a m = 0 . 0 f g a m = ( i s * * 0 . 5 ) / ( 1 + b * ( i s * * 0 . 5 ) ) fgam = fgam + ( 2 / b ) * l o g ( 1 + b * ( i s * * 0 . 5 ) ) f g a m = f g a m * ( - a p h i ) bmx = O.OeO bmx = b e t h a O bmx = bmx + b e t h a l * g ( a l p h a l * ( i s * * 0 . 5 ) ) bmx = bmx + b e t h a 2 * g ( a l p h a 2 * ( i s * * 0 . 5 ) ) bgam = bmx + b p h i cgam = 3 . 0 * c p h i / 2 . 0 l n g a m = O.OeO l n g a m = a b s ( z c * z a ) * f g a m l n g a m = l n g a m + m o l * ( 2 * v c * v a / v ) * b g a m l n g a m = l n g a m + ( m o l * * 2 ) * ( 2 * ( ( v c * v a ) * * ( 3 / 2 ) ) / v ) * c g a m gam = e x p ( l n g a m ) w r i t e ( 6 , 5 5 ) m o l , o s m c a l , a h 2 o , g a m 55 f o r m a t ( l x , 2 ( f l 2 . 4 ) , f l 2 . 5 , f l 2 . 4 ) w r i t e ! * , 5 6 ) m o l , o s m c a l , a h 2 o , g a m 56 f o r m a t ( l x , 2 ( f 1 2 . 4 ) , f 1 2 . 5 , f 1 2 . 4 ) 1000 c o n t i n u e c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c C l o s e I / O F i l e s c l o s e ( 5 ) c l o s e ( 6 , s t a t u s = ' k e e p ' ) e n d c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c F u n c t i o n g ( x ) r e a l f u n c t i o n g ( x ) r e a l x i f ( x . e q . 0 ) t h e n g = 0 e l s e g = 2 * ( l - ( l + x ) * e x p ( - x ) ) / ( x * * 2 ) e n d i f e n d c H y e o n P a r k , U B C , 1998 186 Deternination of Pitzer's mixing parameters for multi-component electrolyte system using osmotic coefficient data Example of inputfile (nasbvpam.in) ' n a s i w p a m . i n ' ' N a O H - N a 2 S i 0 3 m u l t i - c o m p o n e n t e l e c t r o l y t e s y s t e m ' ' C h a r g e s o f c a t i o n and a n i o n ( N a + , O H - , S i 0 3 2 - ) ' +1 -1 - 2 ' T e m p e r a t u r e ( o C ) ' 2 5 . 0 0 ' D e b y e - H u c k e l p a r a m e t e r a t T ( k g / m o l ) l / 2 ' 0 . 3 9 1 5 ' P i t z e r ' s p a r a m e t e r s f o r s i n g l e e l e c t r o l y t e s y s t e m ' ' N a O H ' • B e t h a O B e t h a l C p h i ' 0 . 0 8 6 4 0 . 2 5 3 0 . 0 0 4 4 • N a 2 S i 0 3 ' ' B e t h a O B e t h a l C p h i ' 0 . 0 5 7 7 2 . 8 9 6 5 0 . 0 0 9 7 7 ' I n i t i a l g u e s s o f P i t z e r ' s p a r a m e t e r s f o r m u l t i c o m p o n e n t s y s t e m ' ' T h e t a ( O H , S i 0 3 ) P s i ( O H , S i 0 3 , N a ) ' 0 . 0 0 0 0 0 . 0 0 0 0 'Number o f d a t a ' 26 • M o l a l i t y ( N a + , O H - , S i 0 3 2 - ) , o s m , wr (No o f d a t a : 2 6 ) ' 0 1397 0 0466 0 0466 0 9625 0 0104 0 1405 0 0468 0 0468 0 9572 0 0103 0 2911 0 0970 0 0970 0 . 9267 0 0101 0 2920 0 0973 0 0973 0 9239 0 0100 0 4 4 42 0 1481 0 1481 0 8816 0 0096 0 4395 0 1465 0 1465 0 8910 0 0097 0 7005 0 2335 0 2335 0 8624 0 0094 0 7015 0 2338 0 2338 0 8611 0 0093 1 0670 0 3557 0 3557 0 8352 0 0090 1 0558 0 3519 0 3519 0 8440 0 0091 1 4192 0 4731 0 4731 0 8253 0 0088 1 4221 0 4740 0 4740 0 8236 0 0088 1 5789 0 5263 0 5263 0 8216 0 0087 1 5868 0 5289 0 5289 0 8175" 0 0087 2 3577 0 7859 0 7859 0 8222 0 0085 2 3537 0 7846 0 7846 0 8236 0 0085 2 8798 0 9599 0 9599 0 8323 0 0084 2 8747 0 9582 0 9582 0 8338 0 0085 3 0557 1 0186 1 0186 0 8408 0 0085 3 0542 1 0181 1 0181 0 8412 0 0085 3 9807 1 3269 1 3269 0 8843 0 0085 3 9739 1 3246 1 3246 0 8858 0 0086 4 4861 1 4954 1 4954 0 9027 0 0085 4 4 901 1 4967 1 4 967 0 9019 0 0085 4 9904 1 6635 1 6635 0 9326 0 0086 4 9888 1 6629 1 6629 0 9329 0 0086 Example of output file (nasiwpam. out) D a t a f i l e : n a s i w p a m . i n N a O H - N a 2 S i 0 3 m u l t i - c o m p o n e n t e l e c t r o l y t e s y s t e m T e m p e r a t u r e : 2 9 8 . 1 5 K i n i t i a l g u e s s o f P i t z e r p a r a m e t e r : t h e t a a p s i a . 0 0 0 0 0 . 0 0 0 0 0 i t e r t h e t a a p s i a P ( l ) 1 - . 2 7 0 3 1 . 0 2 3 3 0 - . 2 7 0 3 1 2 - . 2 7 0 3 1 . 0 2 3 3 0 . 0 0 0 0 0 I t e r a t i o n s : 2 P i t z e r p a r a m e t e r s o b t a i n e d T h e t a ( O H , S i 0 3 ) P s i ( O H , S i 0 3 , N a ) - . 2 7 0 3 1 . 0 2 3 3 0 S v a l u e a t s o l u t i o n : 1 1 0 . 7 0 5 5 0 0 E r r o r v a r i a n c e , s i g m a ~ 2 = S/(N-P): P (2 ) S 0 2 3 3 0 * * * * * * * * * * 0 0 0 0 0 1 1 0 . 7 0 5 5 0 0 4 . 6 1 2 7 2 9 V a r i a n c e - c o v a r i a n c e m a t r i x , C O V = ( i n v A ) * ( s i g m a ~ 2 ) C 0 V ( 1 , 1 ) = . 0 0 1 4 7 2 C O V ( l , 2 ) = - . 0 0 0 3 5 1 C O V ( 2 , l ) = - . 0 0 0 3 5 1 C O V ( 2 , 2 ) = . 0 0 0 0 9 0 S t a n d a r d d e v i a t i o n o f p a r a m e t e r s , s t d e v = ( d i a g COV)"0.5 s t d e v t h e t a a ( O H - , S i 0 3 2 - ) = . 038362 s t d e v p s i a ( N a + , O H - , S i 0 3 2 - ) = . 0 0 9 4 9 5 mNa+ mOH- m S i 0 3 2 - osmexp o s m c a l a H 2 0 c a l 1397 . 0 4 6 6 . 0 4 6 6 . 9 6 2 5 . 8 9 9 3 . 9 9 6 2 3 1405 . 0468 . 0468 . 9572 . 8 9 9 3 . 99622 2911 . 0 9 7 0 . 0 9 7 0 . 9267 . 8891 . 9 9 2 2 6 2920 . 0 9 7 3 . 0 9 7 3 . 9239 . 8891 .99224 4442 .1481 . 1481 . 8 8 1 6 . 8 8 0 5 . 9 8 8 3 3 4395 . 1 4 6 5 . 1 4 6 5 - . 8 9 1 0 .8808 . 9 8 8 4 5 7005 . 2 3 3 5 . 2 3 3 5 .8624 . 8 6 6 3 . 9 8 1 9 5 7015 . 2338 . 2338 .8611 . 8663 . 98192 1 0670 . 3557 .3557 . 8352 . 8481 . 9 7 3 2 0 1 0558 . 3 5 1 9 . 3 5 1 9 . 8 4 4 0 . 8487 . 9 7 3 4 6 1 4192 . 4731 . 4731 . 8 2 5 3 . 8351 .96504 1 4221 . 4740 . 4 7 4 0 . 8 2 3 6 . 8 3 5 0 . 9 6 4 9 8 1 5789 . 5 2 6 3 . 5 2 6 3 . 8 2 1 6 . 8308 . 9 6 1 3 8 1 5868 . 5 2 8 9 . 5289 . 8 1 7 5 . 8 3 0 6 . 9 6 1 2 0 2 3577 . 7 8 5 9 . 7 8 5 9 . 8222 . 8 2 3 9 . 9 4 3 3 5 2 3537 . 7 8 4 6 . 7 8 4 6 . 8 2 3 6 . 8239 .94344 2 8798 . 9599 . 9599 . 8323 . 8313 . 9 3 0 6 5 2 8747 . 9582 . 9582 . 8338 . 8312 . 9 3 0 7 8 3 0557 1 . 0 1 8 6 1 . 0 1 8 6 . 8408 . 8357 . 9 2 6 2 0 3 0542 1 .0181 1 .0181 . 8412 . 8357 .92624 3 9807 1 . 3 2 6 9 1 . 3 2 6 9 . 8 8 4 3 . 8 7 3 6 . 90087 3 9739 1 . 3 2 4 6 1 . 3 2 4 6 . 8858 . 8732 . 90107 4 4861 1 .4954 1 .4954 .9027 . 9036 .88542 4 4901 1 .4967 1.4 967 . 9019 . 9038 . 8 8 5 2 9 4 9904 1 . 6 6 3 5 1 . 6 6 3 5 . 9326 . 9394 .86871 4 9888 1 . 6 6 2 9 1 . 6 6 2 9 . 932 9 . 9393 . 86877 Source code (nashvpanufor) c n a s i w p a m . f o r c *************************************************************** c C a l c u l a t i n g P i t z e r ' s p a r a m e t e r s ( t h e t a and p s i ) a t g i v e n s e t o f c m o l a l i t i e s and o s m o t i c c o e f f i c i e n t d a t a f o r m u l t i - c o m p o n e n t c s y s t e m c Hyeon P a r k c U n i v e r s i t y o f B r i t i s h C o l u m b i a c M a r c h 0 9 , 1998 c i m p l i c i t r e a l ( a - h , o - z ) c h a r a c t e r * 5 0 d a t a f i l e , h e a d l i n e p a r a m e t e r ( n p a r a = 2 , N P = 1 0 0 , M P = 1 0 0 , G T O L = l . O E - 6 , I T M A X = 5 0 0 0 ) p a r a m e t e r ( n u n i t = l ) p a r a m e t e r ( n c a t = l , n a n i = 2 ) i n t e g e r z c ( n e a t ) , z a ( n a n i ) c r e a l a p h i , b , a l p h a l , a l p h a 2 , s r e a l b e t h a O ( n e a t , n a n i ) , b e t h a l ( n e a t , n a n i ) , b e t h a 2 ( n e a t , n a n i ) r e a l c p h i ( n e a t , n a n i ) r e a l t h e t a c ( n e a t , n e a t ) , e t h e t a c ( n e a t , n e a t ) , e t h e t a p e ( n e a t , n e a t ) r e a l t h e t a a ( n a n i , n a n i ) , e t h e t a a ( n a n i , n a n i ) , e t h e t a p a ( n a n i , n a n i ) r e a l p s i c ( n e a t , n e a t , n a n i ) , p s i a ( n a n i , n a n i , n e a t ) r e a l x ( M P ) , m c ( n c a t , N P ) , m a ( n a n i , N P ) , o s m e x p ( N P ) r e a l i s ( N P ) , m t ( N P ) , z ( N P ) , b p h i ( n e a t , n a n i ) r e a l c ( n e a t , n a n i ) , p h i c ( n e a t , n e a t ) , p h i p c ( n e a t , n e a t ) r e a l p h i a ( n a n i , n a n i ) , p h i p a ( n a n i , n a n i ) , p h i p h i c ( n e a t , n e a t ) r e a l p h i p h i a ( n a n i , n a n i ) , o s m l ( N P ) , o s m 2 ( N P ) , o s m 3 ( N P ) r e a l o s m 3 a ( N P ) , o s m 4 ( N P ) , o s m 4 a ( N P ) , o s m c a l ( N P ) r e a l f ( N P ) , ah2o ( N P ) , JM (NP,MP) , J J ( M P , M P ) r e a l J f (MP) , p ( M P ) , A I (MP,MP) , x g ( M P , M P ) r e a l e r r v a r , C O V ( M P , M P ) , s t d e v _p(MP) r e a l wr (NP) c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c Open I / O F i l e s o p e n ( 5 , f i l e = ' n a s i w p a m . i n ' , f o r m = ' f o r m a t t e d ' ) o p e n ( 6 , f i l e = ' n a s i w p a m . o u t ' , f o r m = ' f o r m a t t e d ' ) c C h a r g e o f s p e c i e s d o 10 i = 1 , n e a t z c ( i ) = 0 10 c o n t i n u e d o 20 k = 1 , n a n i z a ( k ) = 0 20 c o n t i n u e c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c Read i n p u t f i l e c R e a d ' ~ . i n ' r e a d ( 5 , * ) d a t a f i l e w r i t e ( 6 , 3 0 ) d a t a f i l e 30 f o r m a t ( / l x , " D a t a f i l e : ' , a l 2 ) c R e a d • - m u l t i - c o m p o n e n t e l e c t r o l y t e s y s t e m ' r e a d ( 5 , * ) h e a d l i n e w r i t e ( 6 , 4 0 ) h e a d l i n e 4 0 f o r m a t ( l x , a ) c R e a d c h a r g e o f c a t i o n and a n i o n r e a d ( 5 , * ) h e a d l i n e r e a d ( 5 , * ) z c ( 1 ) , z a ( 1 ) , z a ( 2 ) c R e a d t e m p e r a t u r e (oC) r e a d ( 5 , * ) h e a d l i n e r e a d ( 5 , * ) t c t k = t c + 2 7 3 . 1 5 w r i t e ( 6 , 4 5 ) t k 45 f o r m a t ( l x , ' T e m p e r a t u r e : ' , f 6 . 2 , ' K ' ) c R e a d D e b y e - H u c k e l p a r a m e t e r a t T oC r e a d ( 5 , * ) h e a d l i n e r e a d ( 5 , * ) a p h i c c c c c c c c c c c c c c c c c c c c c c I n i t i a l i z a t i o n o f P i t z e r ' s p a r a m e t e r s as O.OeO d o 51 i = 1 , n e a t do 50 k = 1 , n a n i b e t h a 0 ( i , k ) = O.OeO b e t h a l ( i , k ) = O.OeO b e t h a 2 ( i , k ) = O.OeO c p h i ( i , k ) = O.OeO 50 c o n t i n u e 51 c o n t i n u e d o 53 i = 1, n e a t d o 52 j = 1 , n e a t t h e t a c f i , j ) = O.OeO e t h e t a c l i , j ) = O.OeO e t h e t a p e f i , j ) = O.OeO 52 c o n t i n u e 53 c o n t i n u e d o 55 k = 1 , n a n i d o 54 1 = 1 , n a n i t h e t a a ( k , 1 ) = 0 .OeO e t h e t a a ( k , l ) = : O . O e O e t h e t a p a ( k , l ) = O.OeO 54 c o n t i n u e 55 c o n t i n u e d o 58 i = 1 , n e a t d o 57 j = 1 , n e a t d o 56 k = 1, n a n i p s i c ( i , j , k ) = O.OeO 56 c o n t i n u e 57 c o n t i n u e 58 c o n t i n u e d o 61 k = 1 , n a n i d o 60 1 = 1 , n a n i d o 59 i = 1 , n e a t p s i a ( k , l , i ) = O.OeO 59 c o n t i n u e 60 c o n t i n u e 61 c o n t i n u e c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c R e a d i n g P i t z e r ' s p a r a m e t e r s c Read ' P i t z e r ' s p a r a m e t e r s f o r s i n g l e e l e c t r o l y t e s y s t e m ' r e a d ( 5 , * ) h e a d l i n e c Read b e t h a O , b e t h a l , c p h i o f Na+ O H -r e a d ( 5 , * ) h e a d l i n e r e a d ( 5 , * ) h e a d l i n e r e a d ( 5 , * ) b e t h a O ( 1 , 1 ) , b e t h a l ( 1 , 1 ) , c p h i ( 1 , 1 ) c R e a d b e t h a O , b e t h a l , c p h i o f Na+ S i 0 3 2 -r e a d ( 5 , * ) h e a d l i n e r e a d ( 5 , * ) h e a d l i n e r e a d ( 5 , * ) b e t h a O ( 1 , 2 ) , b e t h a l ( 1 , 2 ) , c p h i ( 1 , 2 ) c I n i t i a l g u e s s o f t h e t a a ( O H - , S i 0 3 2 - ) & p s i a ( O H - , S 1 0 3 2 - , N a + ) r e a d ( 5 , * ) h e a d l i n e r e a d ( 5 , * ) h e a d l i n e r e a d ( 5 , * ) x ( l ) , x ( 2 ) w r i t e ( 6 , 7 0 ) 70 f o r m a t ( / l x , ' i n i t i a l g u e s s o f P i t z e r p a r a m e t e r : ' ) w r i t e ( 6 , 7 1 ) 71 f o r m a t ( 6 x , ' t h e t a a ' , 6 x , ' p s i a ' ) w r i t e ( 6 , 7 2 ) x ( l ) , x ( 2 ) 72 f o r m a t ( 5 x , f 7 . 5 , 2 x , f 7 . 5 ) c Number o f d a t a r e a d ( 5 , * ) h e a d l i n e r e a d ( 5 , * ) n d a t a c R e a d y t o r e a d t h e m o l a l i t y d a t a ! r e a d ( 5 , * ) h e a d l i n e c M o l a l i t y o f s p e c i e s c m c ( l , n d a t a ) : Na+ c m a ( l , n d a t a ) : O H -c m a ( 2 , n d a t a ) : S 1 0 3 2 -c o s m e x p ( n d a t a ) : o s m o t i c c o e f f i c i e n t d a t a do 80 n = l , n d a t a r e a d ( 5 , * ) m c ( 1 , n ) , m a ( 1 , n ) , m a ( 2 , n ) , o s m e x p ( n ) , w r ( n ) 80 c o n t i n u e c U n i v e r s a l c o n s t a n t s o f P i t z e r ' s e q . ( n o t 2 - 2 e l e c t r o l y t e ) b = 1 .2 a l p h a l = 2 . 0 a l p h a 2 = O.OeO w r i t e ( 6 , 9 0 ) ' i t e r ' , ' t h e t a a ' , ' p s i a ' , • P ( 1 ) ' , ' P ( 2 ) ' , ' S ' 90 f o r m a t ( / l x , a 5 , 5 ( a 9 ) ) c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c S t a r t i t e r a t i o n d o 500 i t e r = 1 , i t m a x c I n i t i a l i z a t i o n o f S ( v a l u e o f o b j e c t i v e f u n c t i o n ) S = O.OeO c U p d a t i n g p a r a m e t e r s t o be o b t a i n e d t h e t a a ( l , 2 ) = x ( l ) p s i a ( l , 2 , l ) = x ( 2 ) do 100 n = l , n d a t a c i o n i c s t r e n g t h and t o t a l m o l a l i t y 110 111 c C a l c u l a t i o n o f z z ( n ) = O.OeO d o 120 i = 1, n e a t z ( n ) = z ( n ) + m c ( i , n ) * a b s ( z c ( i ) ) 120 c o n t i n u e d o 121 k = 1 , n a n i z ( n ) = z ( n ) + m a ( k , n ) * a b s ( z a ( k ) ) 121 c o n t i n u e c c c c c c c c c c c c C a l c u l a t i o n ' o f b p h i ( c a t , a n i ) and c ( c a t , a n i ) c o e f f i c i e n t s d o 131 i = 1 , n e a t do 130 k = 1, n a n i b p h i ( i , k ) = O.OeO b p h i ( i , k ) = b e t h a O ( i , k ) + b e t h a l ( i , k ) * e x p ( - a l p h a l * s q r t ( i s ( n ) ) ) b p h i ( i , k ) = b p h i ( i , k ) + b e t h a 2 ( i , k ) * e x p ( - a l p h a 2 * s q r t ( i s ( n ) ) ) c ( i , k ) = O.OeO c ( i , k ) = c p h i ( i , k ) / ( 2 * ( ( a b s ( z c ( i ) * z a ( k ) ) ) * * 0 . 5 ) ) 130 c o n t i n u e i s ( n ) = O.OeO mt (n ) = O.OeO d o 110 i = 1 , n e a t i s ( n ) = i s ( n ) + m c ( i , n ) * ( z c ( i ) * * 2 ) m t ( n ) = mt (n) + mc ( i , n) c o n t i n u e do 111 k = 1 , n a n i i s ( n ) = i s ( n ) + m a ( k , n ) * ( z a ( k ) * * 2 ) m t ( n ) = m t ( n ) + m a ( k , n ) c o n t i n u e i s ( n ) = 0 5 * i s ( n ) 131 c o n t i n u e c c c c c c c C a l c u l a t i i o n o f p h i c ( c a t , c a t ) , p h i p c ( c a t , c a t ) , p h i a ( a n i , a n i ) , c a n d p h i p a ( a n i , a n i ) d o 141 i = 1 , n e a t d o 14 0 j = 1 , n e a t p h i c ( i , j ) = O.OeO p h i p c ( i , j ) = O.OeO p h i c ( i , j ) = t h e t a c ( i , j ) + e t h e t a c ( i , j ) p h i p c ( i , j ) = e t h e t a p c ( i , j ) 140 c o n t i n u e 141 c o n t i n u e d o 151 k = 1, n a n i d o 150 1 = 1 , n a n i p h i a ( k , l ) - O .OeO p h i p a ( k , l ) = O.OeO p h i a ( k , l ) = t h e t a a ( k , l ) + e t h e t a a ( k , l ) p h i p a ( k , l ) = e t h e t a p a ( k , 1 ) 150 c o n t i n u e 151 c o n t i n u e c c c c c c c c c c c c c c c c C a l c u l a t i o n o f p h i p h i c ( c a t , c a t ) and p h i p h i a ( a n i , a n i ) d o 161 1 = 1 , ( n c a t - 1 ) d o 160 j = ( i + 1 ) , n e a t p h i p h i c d , j ) = O.OeO p h i p h i c ( i , j ) = p h i c ( i , j ) + i s ( n ) * ( p h i p c ( i , j ) ) 160 c o n t i n u e 161 c o n t i n u e d o 171 k = 1 , ( n a n i - 1 ) d o 170 1 = (k+1 ) , n a n i p h i p h i a ( k , l ) = O.OeO p h i p h i a ( k , l ) = p h i a ( k , l ) + i s ( n ) * ( p h i p a ( k , 1 ) ) 170 c o n t i n u e 171 c o n t i n u e c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c C a l c u l a t i n g o s m o t i c c o e f f i c i e n t o s m l ( n ) = O.OeO o s m l ( n ) a p h i * ( i s ( n ) * * ( 3 . / 2 . ) ) / ( 1 + b * { i s ( n ) * * 0 . 5 ) ) osm2(n ) = O.OeO d o 181 i = l , n c a t d o 180 k = 1 , n a n i o s m 2 ( n ) = o s m 2 ( n ) + ( m c ( i , n ) * m a ( k , n ) ) * ( b p h i ( i , k ) + z ( n ) * c ( i , k ) ) 180 c o n t i n u e 181 c o n t i n u e osm3(n ) = O.OeO d o 192 i = 1 , ( n c a t - 1 ) d o 191 j = ( i + 1 ) , n e a t o s m 3 a ( n ) = O.OeO do 190 k = 1 , n a n i o s m 3 a ( n ) = o s m 3 a ( n ) + m a ( k , n ) * p s i c ( i , j , k ) 190 c o n t i n u e osm3(n ) = o s m 3 ( n ) + ( ( m c ( i , n ) * m c ( j , n ) ) * ( p h i p h i c ( i , j ) + o s m 3 a ( n ) ) ) 191 c o n t i n u e 192 c o n t i n u e osm4(n ) = 0 . O e O d o 202 k = 1 , ( n a n i - 1 ) d o 201 1 = (k+1 ) , n a n i o s m 4 a ( n ) = O.OeO d o 200 i = 1 , n e a t o s m 4 a ( n ) = o s m 4 a ( n ) + m c ( i , n ) * p s i a ( k , l , i ) 200 c o n t i n u e osm4(n ) = o s m 4 ( n ) + ( ( m a ( k , n ) * m a ( 1 , n ) ) * ( p h i p h i a ( k , 1 ) + o s m 4 a ( n ) ) ) 201 c o n t i n u e 202 c o n t i n u e o s m c a l ( n ) = O.OeO o s m c a l ( n ) = o s m l ( n ) + osm2(n) + osm3(n ) + osm4(n ) o s m c a l ( n ) = ( 2 . 0 / m t ( n ) ) * o s m c a l ( n ) o s m c a l ( n ) = o s m c a l ( n ) + 1 . f ( n ) = ( o s m c a l ( n ) - o s m e x p ( n ) ) / w r ( n ) S = S + f ( n ) * * 2 . c C a l c u l a t e t h e a c t i v i t y o f w a t e r a h 2 o ( h ) = O.OeO a h 2 o ( n ) = 1 0 * * ( - 0 . 0 0 7 8 2 3 * m t ( n ) * o s m c a l ( n ) ) 100 c o n t i n u e c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c G a u s s - N e w t o n r o u t i n e c J a c o b i a n m a t r i x c a l c u l a t i o n s do 290 n = l , n d a t a J M ( n , l ) = O.OeO J M ( n , 2 ) = O.OeO 290 c o n t i n u e d o 300 n = l , n d a t a J M ( n , l ) = ( 2 . 0 / m t ( n ) ) * ( m a ( 1 , n ) * m a ( 2 , n ) ) / w r ( n ) J M ( n , 2 ) = ( 2 . 0 / m t ( n ) ) * ( m c ( 1 , n ) * m a ( 1 , n ) * m a ( 2 , n ) ) / w r ( n ) 300 c o n t i n u e c J J i n i t i a l i z a t i o n do 311 m = l , n p a r a d o 310 m m = l , n p a r a JJ(m,mm) = O.OeO 310 c o n t i n u e 311 c o n t i n u e c J J c a l c u l a t i o n d o 322 m = l , n p a r a d o 321 m m = l , n p a r a d o 320 n = l , n d a t a JJ(m,mm) = JJ(m,mm) + J M ( n , m ) * J M { n , m m ) 320 c o n t i n u e 321 c o n t i n u e 322 c o n t i n u e c J f i n i t i a l i z a t i o n d o 330 m = l , n p a r a J f ( m ) = O.OeO 330 c o n t i n u e c J f c a l c u l a t i o n d o 341 m = l , n p a r a d o 34 0 n = l , n d a t a J f ( m ) = J f ( m ) + J M ( n , m ) * f ( n ) 340 c o n t i n u e J f ( m ) = - J f ( m ) 341 c o n t i n u e c c c c c c c c c c c c c c c c c c c c c c c c P c a l c u l a t i o n ( t o o b t a i n u p d a t e d p a r a m e t e r s ) d o 351 i = 1 ,MP d o 350 j = 1,MP A I ( i , j ) = O.OeO x g ( i , j ) = O.OeO 350 c o n t i n u e 351 c o n t i n u e d o 361 i = l , n p a r a d o 360 j = l , n p a r a A I ( i , j ) = J J ( i , j ) 360 c o n t i n u e 361 c o n t i n u e d o 365 i = l , n p a r a x g ( i , l ) = J f ( i ) 365 c o n t i n u e c B y p a s s i n g t h e s u b r o u t i n e " g a u s s j " , c M a t r i x A I b e c o m e s I n v e r s e m a t r i x o f J J (AI = I n v J J ) c and M a t r i x x g b e c o m e s P m a t r i x . ( J J * P = J f ) c a l l g a u s s j ( A I , n p a r a , N P , x g , n u n i t , M P ) c New x ( i ) s t e p _ l e n g t h = 1 .0 d o 370 i = l , n p a r a p ( i ) = x g ( i , l ) x ( i ) = x ( i ) + p ( i ) * s t e p _ l e n g t h 37 0 c o n t i n u e w r i t e ( 6 , 3 8 0 ) i t e r , x ( 1 ) , x ( 2 ) , p ( 1 ) , p ( 2 ) , s 380 f o r m a t ( l x , 1 5 , 2 ( f 9 . 5 ) , 2 x , 2 ( f 9 . 5 ) , f 1 0 . 6) c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c S t a t i s t i c s o f p a r a m e t e r s c E r r o r v a r i a n c e ( e r r v a r , s i g m a * * 2 ) e r r v a r = O.OeO e r r v a r = S / ( n d a t a - n p a r a ) c V a r i a n c e - c o v a r i a n c e m a t r i x (V= ( Inv A ) * ( s i g m a * * 2 ) ) d o 391 i = l , M P d o 390 j = l , M P C O V ( i , j ) = 0 . 0 e 0 390 c o n t i n u e 1 9 2 391 c o n t i n u e d o 401 1 = 1 , n p a r a d o 400 j = l , n p a r a C O V ( i , j ) = A I ( i , j ) * e r r v a r 400 c o n t i n u e 401 c o n t i n u e d o 410 1 = 1 , n p a r a s t d e v _ p ( i ) = s q r t ( C O V ( i , i ) ) 410 c o n t i n u e i f ( ( a b s ( p ( l ) ) . L T . G T O L ) . A N D . ( a b s ( p ( 2 ) ) . L T . G T O L ) ) go t o 600 500 c o n t i n u e c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c E n d o f l o o p 500 p r i n t * , ' i t e r = ' , i t e r p r i n t * , ' E x c e e d i n g maximum i t e r a t i o n s . . . ' s t o p 600 p r i n t * , ' • w r i t e ( 6 , ' ( / / l x , a , i 5 ) ' ) ' I t e r a t i o n s : ' , i t e r w r i t e ! * , ' ( / / l x , a , 1 5 ) ') ' I t e r a t i o n s : ' , i t e r w r i t e ( 6 , ' ( / l x , a ) ' ) ' P i t z e r p a r a m e t e r s o b t a i n e d ' w r i t e ( * , ' ( / l x , a ) ' ) ' P i t z e r p a r a m e t e r s o b t a i n e d ' w r i t e ( 6 , • ( l x , a l 5 , 2 x , a l 6 ) • ) ' T h e t a ( O H , S i 0 3 ) ' , ' P s i ( O H , S 1 0 3 , N a ) ' w r i t e ! * , ' ( l x , a l 5 , 2 x , a l 6 ) • ) ' T h e t a ( O H , S i 0 3 ) ' , ' P s i ( O H , S i 0 3 , N a ) ' w r i t e ( 6 , 6 1 0 ) x ( l ) , x ( 2 ) 610 f o r m a t ( l x , f 9 . 5 , 8 x , f 9 . 5 ) w r i t e ( * , 6 2 0 ) x ( l ) , x ( 2 ) 620 f o r m a t ( l x , f 9 . 5 , 8 x , f 9 . 5 ) w r i t e ( 6 , ' ( / l x , a , f 1 2 . 6 ) ' ) ' S v a l u e a t s o l u t i o n : ' , S w r i t e ! * , ' ( / l x , a , f 1 2 . 6 ) ' ) ' S v a l u e a t s o l u t i o n : ' , S w r i t e ( 6 , ' ( / l x , a , f l 2 . 6) ' ) ' E r r o r v a r i a n c e , s i g m a ' , 2 = S / ( N - P ) : ' , & e r r v a r w r i t e ( * , ' ( / l x , a , f 1 2 . 6 ) ' ) ' E r r o r v a r i a n c e , s i g m a ~ 2 = S / ( N - P ) : ' , & e r r v a r w r i t e ( 6 , ' ( / l x , 2 a ) ' ) ' V a r i a n c e - c o v a r i a n c e m a t r i x , ' , S ' C O V = ( i n v A ) * ( s i g m a ~ 2 ) ' w r i t e ( * , ' ( / l x , 2 a ) ' ) ' V a r i a n c e - c o v a r i a n c e m a t r i x , ' , S ' C 0 V = ( i n v A) * ( s i g m a / s 2 ) ' d o 631 i = l , n p a r a d o 630 j = l , n p a r a w r i t e ( 6 , • ( l x , a , i l , a , i l , a , f 1 2 . 6 ) • ) ' C O V ( ' , i , ' , ' , j , ' ) = ' , C O V ( i , j ) w r i t e ( * , ' ( l x , a , i l , a , i l , a , f l 2 . 6 ) • ) • C O V ( ' , i , ' , ' , j , ' ) = ' , C O V ( i , j ) 630 c o n t i n u e 631 c o n t i n u e w r i t e ( 6 , ' ( / l x , 2 a ) ' ) ' S t a n d a r d d e v i a t i o n o f p a r a m e t e r s , ' , & ' s t d e v = ( d i a g C O V ) A 0 . 5 ' w r i t e ( * , ' ( / l x , 2 a ) ' ) ' S t a n d a r d d e v i a t i o n o f p a r a m e t e r s , ' , & ' s t d e v = ( d i a g C O V l ' - O . S ' w r i t e ( 6 , ' ( l x , a , f l 2 . 6 ) ' ) ' s t d e v t h e t a a ( 0 H - , S i 0 3 2 - ) = * , s t d e v _ p ( 1 ) w r i t e ! * , * ( l x , a , f l 2 . 6 ) ' ) ' s t d e v t h e t a a ( O H - , S i 0 3 2 - ) = ' , s t d e v _ p ( 1 ) w r i t e ( 6 , ' ( l x , a , f l 2 . 6 ) ' ) ' s t d e v p s i a ( N a + , 0 H - , S 1 0 3 2 - ) = ' , s t d e v _ p ( 2 ) w r i t e ( * , ' ( l x , a , f l 2 . 6 ) ' ) ' s t d e v p s i a ( N a + , 0 H - , S i 0 3 2 - ) = ' , s t d e v _ p ( 2 ) w r i t e ( 6 , ' ( / l x , 6 ( a 9 ) ) ' ) ' m N a + ' , ' m O H - ' , ' m S i 0 3 2 - ' , ' o s m e x p ' , & ' o s m c a l ' , ' a H 2 0 c a l ' d o 640 n = l , n d a t a w r i t e ( 6 , 6 4 1 ) m c ( l , n ) , m a ( l , n ) , m a ( 2 , n ) , o s m e x p ( n ) , o s m c a l ( n ) , & a h 2 o ( n ) 641 f o r m a t ( l x , 5 ( f 9 . 4 ) , f 9 . 5 ) 640 c o n t i n u e p r i n t * , ' ' c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c C l o s e I / O F i l e s c l o s e ( 5 ) c l o s e ( 6 , s t a t u s = ' k e e p ' ) e n d c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c SUBROUTINE g a u s s j ( a , n , n p , b , m , m p ) c g a u s s j . f o r s u b r o u t i n e i m p o r t e d f r o m " N u m e r i c a l R e c i p e s F o r t r a n 7 7 " , c v e r s i o n 2 . 0 7 t o c a l c u l a t e an i n v e r s e m a t r i x o f A and s o l v e a m a t r i x e q u a t i o n , AX=B. r e t u r n END c H y e o n P a r k , 1998 193 Calculation of osmotic coefficient and activity coefficient using Pitzer's binary and mixing parameters for multi-component electrolyte system Example of inputfile (nashvcaLin) ' n a s i w c a l . i n ' ' N a O H - N a 2 S i 0 3 m u l t i - c o m p o n e n t e l e c t r o l y t e s y s t e m ' ' C h a r g e s o f c a t i o n and a n i o n ( N a + , O H - , S i 0 3 2 - ) ' +1 -1 - 2 ' T e m p e r a t u r e ( o C ) ' 2 5 . 0 0 ' D e b y e - H u c k e l p a r a m e t e r a t T ( k g / m o l ) l / 2 ' 0 . 3 9 1 5 ' P i t z e r ' s p a r a m e t e r s f o r s i n g l e e l e c t r o l y t e s y s t e m ' • N a O H ' ' B e t h a O B e t h a l C p h i * 0 .0864 0 . 2 5 3 0 . 0 0 4 4 • N a 2 S i 0 3 * ' B e t h a O B e t h a l C p h i ' 0 . 0 5 7 7 2 . 8 9 6 5 0 . 0 0 9 7 7 • T h e t a ( 0 H , S i 0 3 ) P s i ( O H , S 1 0 3 , N a ) ' - 0 . 2 7 0 3 0 . 0 2 3 3 'Number o f d a t a ' 80 ' M o l a l i t y ( N a + , O H - , S i 0 3 2 - ) ' 0 . 0 3 0 0 0 . 0 1 0 0 0 . 0 1 0 0 0 . 0 6 0 0 0 . 0 2 0 0 0 . 0 2 0 0 0 . 0 9 0 0 0 . 0 3 0 0 0 . 0 3 0 0 0 . 1 2 0 0 0 . 0 4 0 0 0 . 0 4 0 0 0 . 1 5 0 0 0 . 0 5 0 0 0 . 0 5 0 0 7 . 8 0 0 0 2 . 6 0 0 0 2 . 6 0 0 0 8 . 1 0 0 0 2 . 7 0 0 0 2 . 7 0 0 0 8 . 4 0 0 0 2 . 8 0 0 0 2 . 8 0 0 0 8 . 7 0 0 0 2 . 9 0 0 0 2 . 9 0 0 0 9 . 0 0 0 0 3 . 0 0 0 0 3 . 0 0 0 0 Example of output file (nasiwcal.out) D a t a f i l e : n a s i w c a l . i n N a 0 H - N a 2 S i 0 3 m u l t i - c o m p o n e n t e l e c t r o l y t e s y s t e m T e m p e r a t u r e : 2 9 8 . 1 5 K mNaOH gamNaOHmNa2Si03gamNa2Si03 o s m c a l aH20 gamNa gamOH g a m S i 0 3 . 0 1 0 0 8399 . 0 1 0 0 . 7177 9250 . 99917 . 8 5 6 9 . 8 2 3 3 . 5 0 3 5 . 0 2 0 0 8006 . 0 2 0 0 . 6610 9120 . 9 9 8 3 6 . 8301 . 7721 . 4 1 9 2 . 0 3 0 0 7761 . 0 3 0 0 . 6287 9054 . 9 9 7 5 6 . 8 1 6 3 . 7 3 7 9 . 3 7 3 0 . 0400 7583 . 0 4 0 0 . 6 0 6 5 9013 . 9 9 6 7 6 . 8078 . 7118 . 3 4 2 0 . 0 5 0 0 7442 . 0 5 0 0 . 5 8 9 9 8984 . 9 9 5 9 6 . 8022 . 6 9 0 3 . 3 1 8 9 2 . 6 0 0 0 9347 2 . 6 0 0 0 .5862 1 2334 .74914 1 . 5 7 5 6 . 5 5 4 5 . 0812 2 . 7 0 0 0 9877 2 . 7 0 0 0 .6187 1 2734 . 7 3 3 6 9 1 .6682 . 5848 . 0851 2 . 8 0 0 0 1 0462 2 . 8 0 0 0 . 6 5 4 6 1 3150 . 7 1 7 7 6 1 . 7 6 9 6 . 6 1 8 5 . 0 8 9 6 2 . 9 0 0 0 1 1108 2 . 9 0 0 0 . 6942 1 3581 . 70137 1 . 8 8 0 6 .6561 . 0 9 4 6 3 . 0 0 0 0 1 1822 3 . 0 0 0 0 .7381 1 4028 . 6 8 4 5 3 2 . 0021 . 6 9 8 0 . 1 0 0 3 Source code (nasiwcalfor) c n a s i w c a l . f o r c c C a l c u l a t i n g o s m o t i c c o e f f i c i e n t , w a t e r a c t i v i t y , and a c t i v i t y c c o e f f i c i e n t a t g i v e n s e t o f m o l a l i t i e s and P i t z e r ' s p a r a m e t e r s c f o r m u l t i - c o m p o n e n t s y s t e m c H y e o n P a r k c U n i v e r s i t y o f B r i t i s h C o l u m b i a c M a r c h 1 0 , 1998 i m p l i c i t r e a l ( a - h , o - z ) e x t e r n a l g , g p c h a r a c t e r * 5 0 d a t a f i l e , h e a d l i n e p a r a m e t e r ( N P = 2 0 0 ) p a r a m e t e r ( n c a t = l , n a n i = 2 ) i n t e g e r z c ( n e a t ) , z a ( n a n i ) r e a l b e t h a O ( n e a t , n a n i ) , b e t h a l ( n e a t , n a n i ) , b e t h a 2 ( n e a t , n a n i ) r e a l c p h i ( n e a t , n a n i ) r e a l t h e t a c ( n e a t , n e a t ) , e t h e t a c ( n e a t , n e a t ) , e t h e t a p e ( n e a t , n e a t ) r e a l t h e t a a ( n a n i , n a n i ) , e t h e t a a ( n a n i , n a n i ) , e t h e t a p a ( n a n i , n a n i ) r e a l p s i c ( n e a t , n e a t , n a n i ) , p s i a ( n a n i , n a n i , n e a t ) r e a l m c ( n e a t , N P ) , m a ( n a n i , N P ) r e a l i s ( N P ) , m t ( N P ) , z ( N P ) , b p h i ( n e a t , n a n i ) r e a l c ( n e a t , n a n i ) , p h i c ( n e a t , n e a t ) , p h i p c ( n e a t , n e a t ) r e a l p h i a ( n a n i , n a n i ) , p h i p a ( n a n i , n a n i ) , p h i p h i c ( n e a t , n e a t ) r e a l p h i p h i a ( n a n i , n a n i ) , o s m l ( N P ) , o s m 2 ( N P ) , o s m 3 ( N P ) r e a l o s m 3 a ( N P ) , o s m 4 ( N P ) , o s m 4 a(NP) , o s m c a l ( N P ) r e a l F l , F 2 , F 3 , F 4 , a h 2 o ( N P ) r e a l l n g a m c ( n e a t ) , g a m e ( n e a t ) , l n g a m a ( n a n i ) , g a m a ( n a n i ) r e a l b m x ( n e a t , n a n i ) , b p a c t ( n e a t , n a n i ) r e a l mNaOH(NP) ,gamNaOH(NP) r e a l m N a 2 S 1 0 3 ( N P ) , g a m N a 2 S i 0 3 ( N P ) c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c Open I / O F i l e s o p e n ( 5 , f i l e = ' n a s i w c a O . i n ' , f o r m = ' f o r m a t t e d ' ) o p e n ( 6 , f i l e = ' n a s i w c a O . o u t ' , f o r m = ' f o r m a t t e d ' ) c C h a r g e o f s p e c i e s d o 10 i = 1 , n e a t z c ( i ) = 0 10 c o n t i n u e d o 20 k = 1, n a n i z a ( k ) = 0 20 c o n t i n u e c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c Read i n p u t f i l e c R e a d ' - . i n ' r e a d ( 5 , * ) d a t a f i l e w r i t e ( 6 , 2 5 ) d a t a f i l e 25 f o r m a t ( / l x , ' D a t a f i l e : ' , a l 2 ) c R e a d ' - m u l t i - c o m p o n e n t s y s t e m ' r e a d ( 5 , * ) h e a d l i n e w r i t e ( 6 , 2 6 ) h e a d l i n e 26 f o r m a t ( / l x , a ) c R e a d c h a r g e o f c a t i o n and a n i o n r e a d ( 5 , * ) h e a d l i n e r e a d ( 5 , * ) z c ( 1 ) , z a ( 1 ) , z a ( 2 ) c Read t e m p e r a t u r e (oC) r e a d ( 5 , * ) h e a d l i n e r e a d ( 5 , * ) t c t k = t c + 2 7 3 . 1 5 w r i t e ( 6 , 2 7 ) t k 27 f o r m a t ( l x , ' T e m p e r a t u r e : ' , f 6 . 2 , ' K ' ) c R e a d D e b y e - H u c k e l p a r a m e t e r a t T oC r e a d ( 5 , * ) h e a d l i n e r e a d ( 5 , * ) a p h i c c c c c c c c c c c c c c c c c c c c c c I n i t i a l i z a t i o n o f P i t z e r ' s p a r a m e t e r s as O.OeO d o 31 i = 1 , n e a t d o 30 k = 1 , n a n i b e t h a 0 ( i , k ) = O.OeO b e t h a l ( i , k ) = O.OeO b e t h a 2 ( i , k ) = O.OeO c p h i ( i , k ) = O.OeO 30 c o n t i n u e 31 c o n t i n u e d o 41 i = 1, n e a t d o 40 j = 1 , n e a t t h e t a c d , j ) = O.OeO e t h e t a c d , j ) = O.OeO e t h e t a p c ( i , j ) = O.OeO 40 c o n t i n u e 41 c o n t i n u e d o 51 k = 1 , n a n i d o 50 1 = 1, n a n i t h e t a a ( k , l ) = O.OeO e t h e t a a ( k , l ) = O.OeO e t h e t a p a ( k , l ) = O.OeO 50 c o n t i n u e 51 c o n t i n u e d o 62 i = 1 , n e a t d o 61 j = 1 , n e a t d o 60 k = 1 , n a n i p s i c ( i , j , k ) = O.OeO 60 c o n t i n u e 61 c o n t i n u e 62 c o n t i n u e d o 72 k = 1 , n a n i d o 71 1 = 1 , n a n i d o 70 i = 1, n e a t p s i a ( k , l , i ) = O.OeO 70 c o n t i n u e 71 c o n t i n u e 72 c o n t i n u e c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c e c c c c c c c c c c R e a d i n g P i t z e r ' s p a r a m e t e r s c c a t i o n - a n i o n : b e t h a O ( c a t i o n , a n i o n ) , b e t h a l ( c a t i o n , a n i o n ) c b e t h a 2 ( c a t i o n , a n i o n ) , c p h i ( c a t i o n , a n i o n ) c R e a d ' P i t z e r ' s p a r a m e t e r s f o r s i n g l e e l e c t r o l y t e s y s t e m ' r e a d ( 5 , * ) h e a d l i n e c R e a d b e t h a O , b e t h a l , c p h i o f NaOH r e a d ( 5 , * ) h e a d l i n e r e a d ( 5 , * ) h e a d l i n e r e a d ( 5 , * ) b e t h a O ( 1 , 1 ) , b e t h a l ( 1 , 1 ) , c p h i ( 1 , 1 ) c R e a d b e t h a O , b e t h a l , c p h i o f N a 2 S i 0 3 r e a d ( 5 , * ) h e a d l i n e r e a d ( 5 , * ) h e a d l i n e : r e a d ( 5 , * ) b e t h a O ( 1 , 2 ) , b e t h a l ( 1 , 2 ) , c p h i { 1 , 2 ) c R e a d t h e t a a ( O H - , S i 0 3 2 - ) , p s i a ( O H - , S i 0 3 2 - , N a + ) r e a d ( 5 , * ) h e a d l i n e r e a d ( 5 , * ) t h e t a a ( 1 , 2 ) , p s i a ( 1 , 2 , 1 ) c Number o f d a t a r e a d ( 5 , * ) h e a d l i n e r e a d ( 5 , * ) n d a t a c R e a d y t o r e a d t h e m o l a l i t y d a t a ! r e a d ( 5 , * ) h e a d l i n e c m o l a l i t y o f s p e c i e s c m c ( l , n d a t a ) : Na+ c m a ( l , n d a t a ) : O H -c m a ( 2 , n d a t a ) : S i 0 3 2 -d o 80 n = l , n d a t a r e a d ( 5 , * ) m c ( l , n ) , m a ( l , n ) , m a ( 2 , n ) 80 c o n t i n u e c U n i v e r s a l c o n s t a n t s o f P i t z e r ' s e q . ( n o t 2 - 2 e l e c t r o l y t e ) b = 1 . 2 e 0 a l p h a l = 2 . 0 e 0 a l p h a 2 = O.OeO w r i t e ( 6 , 9 0 ) ' m N a O H ' , ' g a m N a O H ' , ' m N a 2 S i 0 3 ' , • g a m N a 2 S i 0 3 ' , & ' o s m c a l ' , ' a H 2 0 ' , ' g a m N a ' , ' g a m O H ' , ' g a m S i 0 3 ' 90 f o r m a t ( / l x , 3 ( a 8 ) , a l 0 , 5 ( a 8 ) ) c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c S t a r t i t e r a t i o n d o 100 n = l , n d a t a c i o n i c s t r e n g t h and t o t a l m o l a l i t y i s ( n ) = O.OeO mt (n ) = O.OeO d o 110 i = 1 , n e a t i s ( n ) = i s ( n ) + m c ( i , n ) * ( z c ( i ) * * 2 ) mt (n) = mt (n) + mc ( i , n) 110 c o n t i n u e d o 111 k = 1 , n a n i l s ( n ) = i s ( n ) + m a ( k , n ) * ( z a ( k ) * * 2 ) mt (n) = mt (n) + ma (k , n) 111 c o n t i n u e i s (n) = 0 . 5 * i s (n) c C a l c u l a t i n g z z ( n ) = O.OeO do 120 i = 1, n e a t z ( n ) = z ( n ) + m c ( i , n ) * a b s ( z c ( i ) ) 120 c o n t i n u e d o 121 k = 1, n a n i z ( n ) = z ( n ) . + m a ( k , n ) * a b s ( z a ( k ) ) 121 c o n t i n u e c c c c c c c c c c c c c c c C a l c u l a t i n g b p h i ( c a t , a n i ) and c ( c a t , a n i ) c o e f f i c i e n t s do 131 1 = 1 , neat do 130 k = 1, nani bphi(i,k) = O.OeO bphi(i,k) = be t h a O ( i , k ) + b e t h a l ( i , k ) * e x p ( - a l p h a l * s q r t ( i s ( n ) ) ) bphi(i,k) = bphi(i,k)+betha2(l,k)*exp(-alpha2*sqrt(is(n))) c ( i , k ) = O.OeO c ( i , k ) = c p h i ( i , k ) / ( 2*( (a b s ( z c ( i ) * z a ( k ) ) ) * * 0 . 5 ) ) bmx(i,k) = O.OeO bmx(i,k) = bethaO(i,k)+bethal(i,k)*g(alphal*sqrt('is(n) ) ) bmx(i,k) = bmx(i,k)+betha2(i,k)*g(alpha2*sqrt(is(n))) bpact(i,k) = O.OeO bpact(i,k) = b e t h a l ( i , k ) * g p ( a l p h a l * s q r t ( i s ( n ) ) ) bpact(i,k) = bpact(i,k)+betha2(i,k)*gp(alpha2*sqrt(is(n))) bpact(i,k) = b p a c t ( i , k ) / i s ( n ) 130 continue 131 continue ccccccccccc C a l c u l a t i n g p h i c ( c a t , c a t ) , p h i p c ( c a t , c a t ) , p h i a ( a n i , a n i ) , c and phipa(ani,ani) do 141 1 = 1 , neat do 140 j = 1, neat p h i c ( i , j ) = O.OeO p h i p c ( i , j ) = O.OeO p h i c ( i , j ) = t h e t a c ( i , j ) + e t h e t a c ( i , j ) p h i p c ( i , j ) = e t h e t a p c ( i , j ) 140 continue 141 continue do 151 k = 1, nani do 150 1 = 1 , nani p h i a ( k , l ) = O.OeO phipa(k,l) = O.OeO phi a ( k , l ) = thetaa(k,l) + ethetaa(k,l) phipa(k,l) = ethetapa(k,1) 150 continue 151 continue cccccc C a l c u l a t i n g p h i p h i c ( c a t , c a t ) and phiphia(ani,ani) c o e f f i c i e n t s do 161 i = 1, (ncat-1) do 160 j = (i+1), neat p h i p h i c ( i , j ) = O.OeO p h i p h i c ( i , j ) = p h i c ( i , j ) + i s ( n ) * ( p h i p c ( i , j ) ) 160 continue 161 continue do 171 k = 1, (nani-1) do 170 1 = (k+1), nani p h i p h i a ( k , l ) = O.OeO ph i p h i a ( k , l ) = phi a ( k , l ) + is(n)*(phipa(k,1)) 170 continue 171 continue ccccccccccccccccccccccccccccccccccccc C a l c u l a t i n g osmotic c o e f f i c i e n t osml(n) = O.OeO osml(n) = -aphi*(is(n)**(3./2.))/(1+b*(is(n)**0.5)) osm2(n) = O.OeO do 181 i = l,ncat do 180 k = 1, nani osm2(n)=osm2(n) + (mc(i,n)*ma(k,n))*(bphi(i,k)+z(n)*c(i,k) ) 180 continue 181 continue osm3(n) = O.OeO do 192 1 = 1 , (ncat-1) do 191 j = (i+1), neat osm3a(n) = O.OeO do 190 k = 1, nani osm3a(n) = osm3a(n) + ma(k,n) * p s i c ( i , j , k ) 190 continue osm3(n) = osm3(n)+((mc(i,n)*mc(j,n))*(phiphic(i,j)+osm3a(n)) 191 continue 192 continue osm4(n) = O.OeO do 202 k = 1, (nani-1) do 201 1 = (k+1), nani osm4a(n) = O.OeO do 200 1 = 1 , neat osm4a(n) = osm4a(n) + rac(i,n) * p s i a ( k , l , i ) 200 continue osm4(n) = osm4(n)+((ma(k,n)*ma(1,n))*(phiphia(k,1)+osm4a(n))) 201 continue 202 continue osmcal(n) = O.OeO osmcal(n) = osml(n) + osm2(n) + osm3(n) + osm4(n) osmcal(n) = (2.0/mt(n))*osmcal(n) osmcal(n) = osmcal(n) + 1. c C a l c u l a t e the a c t i v i t y of water ah2o(n) = O.OeO ah2o(n) = 10**(-0.007823*mt(n)*osmcal(n)) c C a l c u l a t i n g F term F l = O.OeO F l = (is(n)**0.5)/(1+b*(is(n)**0.5)) F l = F l + (2/b)*log(l+b*(is(n)**0.5)) F l = Fl*(-aphi) F2 = O.OeO do 211 i = l,n c a t do 210 k = l,n a n i F2 = F2 + mc(i,n)*ma(k,n)*bpact(i,k) 210 continue 211 continue F3 = O.OeO do 221 i = 1, (ncat-1) do 220 j = (i+1),ncat F3 = F3 + mc(i,n)*mc(i,n)*phipc(i,j) 220 continue 221 continue F4 = O.OeO do 231 k = 1, (nani-1) do 230 1 = (k+1),nani F4 = F4 + ma(k,n)*ma(k,n)*phipa(k,1) 230 continue 231 continue F = O.OeO F = F l + F2 + F3 + F4 ccccccccccccccccccccccccccccc C a l c u l a t i n g c a t i o n a c t i v i t y c o e f f i c i e n t do 790 i i = 1, neat csuml = O.OeO csuml = ( z c ( i i ) * * 2 ) * F csum2 = O.OeO do 700 k = 1, nani csum2 = csum2 + ma(k,n)*(2*bmx(ii,k)+z(n)*c(ii,k)) 700 continue csum3 = O.OeO do 720 j = 1, neat i f (j .eq. i i ) go to 720 csum3a = O.OeO do 710 k = 1, nani csum3a = csum3a + ma(k,n) * p s i c ( i i , j , k ) 710 continue csum3 = csum3 + mc(j,n)*(2*phic(ii,j)+csum3a) 720 continue csum4 = O.OeO do 740 K = 1, (nani-1) do 730 L = (K+1), nani csum4 = csum4 + ma(k,n) * ma(l,n) * p s i a ( K , L , i i ) 730 continue 740 continue csum5 = O.OeO do 760 i = 1, neat do 750 K = 1, nani csum5 = csum5 + mc(i,n) * ma(K,n) * C(i,K) csum5 = a b s ( Z C ( i i ) ) * csum5 750 continue 760 continue c game() : a c t i v i t y c o e f f i c i e n t of ca t i o n by P i t z e r eq. lngamc(ii) = csuml + csum2 + csum3 + csum4 + csum5 gamc(ii) = exp(lngamc(ii)) 790 c o n t i n u e c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c C a l c u l a t i n g . a n i o n a c t i v i t y c o e f i c i e n t d o 890 kk = 1, n a n i a s u m l = O.OeO a s u m l = ( z a ( k k ) * * 2 ) * F asum2 = O.OeO d o 800 i = 1, n e a t asum2 = asum2 + m c ( i , n ) * ( 2 * b m x ( i , k k ) + z ( n ) * C ( i , k k ) ) 800 c o n t i n u e asum3 = O.OeO d o 820 1 = 1 , n a n i i f ( l . e q . k k ) go t o 820 asum3a = O.OeO d o 810 1 = 1 , n e a t asum3a = asum3a + m c ( i , n ) * p s i a ( k k , l , i ) 810 c o n t i n u e asum3 = asum3 + m a ( 1 , n ) * ( 2 * p h i a ( k k , 1 ) + asum3a) 820 c o n t i n u e asum4 = O.OeO do 840 1 = 1 , ( n c a t - 1 ) d o 830 J = ( i + 1 ) , n e a t asum4 = asum4 + m c ( i , n ) * m c ( j , n ) * p s i c ( i , j , k k ) 830 c o n t i n u e 840 c o n t i n u e asum5 = O.OeO d o 860 1 = 1 , n e a t d o 850 K = 1 , n a n i asum5 = asum5 + m c ( i , n ) * ma{k ,n ) * C ( i , K ) asum5 = a b s ( z a ( k k ) ) * a s u m 5 850 c o n t i n u e 860 c o n t i n u e c gama() : a c t i v i t y c o e f f i c i e n t o f a n i o n b y P i t z e r e q . l n g a m a ( k k ) = a s u m l + asum2 + asum3 + asum4 + asum5 gama(kk ) = e x p ( l n g a m a ( k k ) ) 890 c o n t i n u e c mean a c t i v i t y c o e f f i c i e n t mNaOH(n) =( ( 1 . / 3 . ) * m c ( 1 , n ) * m a ( l , n ) ) * * ( l . / 2 . ) gamNaOH(n) = (game {1) *gama (1) ) * * ( 1 . / 2 . ) m N a 2 S i 0 3 ( n ) = ( ( ( ( 1 . / 3 . ) * m c ( 1 , n ) ) * * 2 . ) * m a ( 2 , n ) ) * * ( l . / 3 . ) g a m N a 2 S i 0 3 ( n ) = ( ( g a m e ( 1 ) * * 2 . ) * g a m a ( 2 ) ) * * ( l . / 3 . ) w r i t e ( 6 , 3 8 0 ) m N a O H ( n ) , g a m N a O H ( n ) , m N a 2 S i 0 3 ( n ) , g a m N a 2 S i 0 3 ( n ) , & o s m c a l ( n ) , a h 2 o ( n ) , g a m e ( 1 ) , g a m a ( 1 ) , g a m a ( 2 ) 380 f o r m a t ( l x , 3 ( f 8 . 4 ) , f 1 0 . 4 , f 8 . 4 , f 8 . 5 , 3 ( f 8 . 4 ) ) w r i t e ! * , 3 9 0 ) m N a O H ( n ) , g a m N a O H ( n ) , m N a 2 S i 0 3 ( n ) , g a m N a 2 S i 0 3 ( n ) , & o s m c a l ( n ) , a h 2 o ( n ) , g a m e ( 1 ) , g a m a ( 1 ) , g a m a ( 2 ) 390 f o r m a t ( l x , 3 ( f 8 . 4 ) , f 1 0 . 4 , f 8 . 4 , f 8 . 5 , 3 ( f 8 . 4 ) ) 100 c o n t i n u e c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c C l o s e I / O F i l e s c l o s e ( 5 ) c l o s e ( 6 , s t a t u s = • k e e p ' ) e n d c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c F u n c t i o n g ( x ) r e a l f u n c t i o n g ( x ) r e a l x i f ( x . e q . 0 ) t h e n g = 0 e l s e g = 2 * ( l - ( l + x ) * e x p ( - x ) ) / ( x * * 2 ) e n d i f e n d c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c F u n c t i o n g p ( x ) r e a l f u n c t i o n g p ( x ) r e a l x i f ( x . e q . 0 ) t h e n gp = 0 e l s e gp 2 * ( 1 - ( l + x + ( x * * 2 ) / 2 . ) * e x p ( - x ) ) / ( x * * 2 ) e n d i f e n d c H y e o n P a r k , 1998 199 Computation of equilibrium state for NH 4 -NH 4 OH-H + -HCl-NH 4 Cl-Cr-Na + -NaCl-K+-KC1 system at 573.15 K This code was written in order to check the reliability of the modeling procedure by comparing the calculation results with the published ones. Example of input and output ************************************************* N H 4 - N H 4 0 H - H + - H C 1 - N H 4 C 1 - C 1 — N a + - N a C l - K + - K C l s y s t e m OCT , 1998 Hyeon P a r k ************************************************* S y s t e m : NH4+ H+ Na+ K+ C l - NH40H H C l NH4C1 N a C l K C l H20 T e m p e r a t u r e : 300 C > I n p u t amount o f NH4C1 ? 0 . 2 5 > I n p u t amount o f N a C l ? 0 . 2 5 > I n p u t amount o f K C l ? 0 . 2 5 ( P r e s s e n t e r k e y t o s e e r e s u l t . . . ) *+****+****+*+++* R e s u l t * * * * * * * * * * * * * * * * * I n p u t c h e m i c a l s ( m o l e s ) NH4C1 . 2 5 0 0 0 0 N a C l . 2 5 0 0 0 0 K C l . 2 5 0 0 0 0 I n i t i a l g u e s s f o r m o l a l i t i e s mNH4 . 1 0 0 0 0 0 mH . 0 0 2 0 0 0 mNa . 1 0 0 0 0 0 mK . 1 0 0 0 0 0 mCl . 2 0 0 0 0 0 mNH40H . 0 0 2 0 0 0 mHCl . 0 0 5 0 0 0 mNH4Cl . 1 5 0 0 0 0 mNaCl . 1 5 0 0 0 0 mKCl . 1 5 0 0 0 0 F v a l u e F l l ) . 0 0 0 0 8 0 F ( 2 ) . 0 0 0 0 0 0 F ( 3 ) . 0 0 0 0 0 0 F ( 4 ) . 0 0 0 0 0 0 F ( 5 ) . 0 0 0 0 0 0 F ( 6 ) . 0 0 0 0 0 0 F ( 7 ) . 0 0 0 0 0 0 F ( 8 ) . 0 0 0 0 0 0 F ( 9 ) . 0 0 0 0 0 0 F ( 1 0 ) . 0 0 0 0 0 0 I o n i c S t r e n g t h : . 4 7 8 1 8 5 M o l a l i t y [ m o l e s / K g H20] mNH4 .162978 gamNH4 . 4 0 4 2 5 2 mH . 0 0 1 2 2 0 gamH .404252 mNa . 1 3 8 4 8 9 gamNa .404252 mK . 175497 gamK . 4 0 4 2 5 2 mCl . 4 7 8 1 8 5 g a m C l . 4 0 4 2 5 2 mNH40H . 0 0 2 8 7 6 mHCl . 001657 mNH4Cl . 0 8 4 1 4 5 mNaCl . 111511 mKCl . 0 7 4 5 0 3 Source code (heqbrntfor) c h e q b r m . f o r c ****************************************************** c N H 4 - N H 4 0 H - H + - H C 1 - N H 4 C 1 - C 1 — N a + - N a C l - K + - K C l s y s t e m c U n i v e r s i t y o f B r i t i s h C o l u m b i a O c t 0 8 , 1998 H y e o n P a r k I n p u t c h e m i c a l s NH4C1 N H 4 C l i n N a C l N a C l i n KCI K C l i n S p e c i e s a t e q u i l i b r i u m C a t i o n (4) A n i o n (1) No c h a r g e s p e c i e s (5) Unknowns NH4+ H+ Na+ K+ C l -NH40H HC1 NH4C1 N a C l KCI m o l a l i t y m o l a l i t y m o l a l i t y m o l a l i t y m o l a l i t y m o l a l i t y m o l a l i t y m o l a l i t y m o l a l i t y m o l a l i t y o f NH4+ o f H+ o f Na+ o f K+ o f C l -o f NH40H o f HC1 o f NH4C1 o f N a C l o f KCI mNH4 mH mNa mK mCl mNH40H mHCl mNH4Cl mNaCl mKCl x l l ) x ( 2 ) x ( 3 ) x ( 4 ) x ( 5 ) x ( 6 ) x ( 7 ) x ( 8 ) x ( 9 ) x ( 1 0 ) f ( N ) , XG(N) M a i n r o u t i n e INTEGER N PARAMETER(N=10) DOUBLE PRECIS ION x ( N ) , DOUBLE PRECIS ION N H 4 C l i n , N a C l i n , K C l i n DOUBLE PRECIS ION g a m N H 4 , g a m H , g a m N a , g a m K , g a m C l DOUBLE PRECIS ION a H 2 0 , i s COMMON / C I N P U T / N H 4 C l i n , N a C l i n , K C l i n COMMON / R E S U L T / a H 2 0 , i s COMMON / G A M / gamNH4,gamH,gamNa,gamK, g a m C l L O G I C A L c h e c k N H 4 C l i n = 0 . 0 D 0 N a C l i n = 0 . 0 D 0 = 0 . 0 D 0 OCT , 1998 Hyeon P a r k K C l i n PRINT * t PRINT •it t PRINT * r PRINT * PRINT + PRINT * PRINT + t PRINT * PRINT * r PRINT * r WRITE{ •it 9 HC1 NH4C1 N a C l KCI H20 • R E A D ( * , * ) N H 4 C l i n N H 4 C l i n = 0 . 2 5 W R I T E ! * , * ) ' > I n p u t amount o f N a C l ? ' R E A D ( * , * ) N a C l i n N a C l i n = 0 . 2 5 W R I T E ! * , * ) ' > I n p u t amount o f KCI ? ' R E A D ! * , * ) K C l i n K C l i n = 0 . 2 5 PRINT * , ' ( P r e s s e n t e r k e y t o s e e r e s u l t . . . ) 1 READ (* , * ) M o l a l i t y o f s p e c i e s a t i n i t i a l s t a t e ( G u e s s v a l u e s ) c • mNH4 x ( l ) c mH x ( 2 ) c mNa x ( 3 ) c mK x ( 4 ) c mCl x ( 5 ) c mNH40H x ( 6 ) c mHCl x ( 7 ) c mNH4Cl x ( 8 ) 2 0 1 10 14 15 mNaCl x ( 9 ) mKCl x ( 1 0 ) 1 0 0 0 g / 1 8 . 0 1 5 M w x ( l ) = 0 .1 -> 5 5 . 5 0 9 x ( 2 ) = x ( 3 ) = x ( 4 ) = x ( 5 ) = x ( 6 ) = x ( 7 ) = x ( 8 ) = x ( 9 ) = x ( 1 0 ) = 0 . 0 0 2 0 .1 0 . 1 0 . 2 0 . 0 0 2 0 . 0 0 5 0 . 1 5 0 . 1 5 0 . 1 5 TO SHOW I N I T I A L GUESS V A L U E S AT RESULT X G ( 1 ) - X G ( 6 ) XG(1 ) = X ( l ) XG(2 ) = X{2) XG(3 ) = X ( 3 ) XG(4 ) = X ( 4 ) XG(5 ) = X ( 5 ) XG(6 ) = X ( 6 ) XG(7 ) = x ( 7 ) XG(8 ) = x ( 8 ) XG(9 ) = x ( 9 ) X G ( 1 0 ) = x ( 1 0 ) c a l l n e w t ( x , N , c h e c k ) c a l l f u n c v ( N , x , f ) i f ( c h e c k ) t h e n w r i t e ! * , * ) ' C o n v e r g e n c e p r o b l e m s . ' e n d i f a t f i n a l s t a t e PRINT PRINT PRINT WRITE WRITE WRITE WRITE ***************** R e s u l t ( * , * ) ( * , 1 0 ) ( * , 1 0 ) ( * , 1 0 ) I n p u t c h e m i c a l s ( m o l e s ) ' F O R M A T ( 3 x , a , p r i n t * , ' ' WRITE ( * , * ) WRITE ( * , 1 4 ) WRITE ( * , 1 4 ) WRITE ( * , 1 4 ) WRITE WRITE WRITE ( * , 1 4 ) ( * , 1 4 ) ( * , 1 4 ) WRITE ( * , 1 4 ) WRITE ( * , 1 4 ) WRITE WRITE ( * , 1 4 ) ( * , 1 4 F O R M A T ( 3 x , a , PRINT * , ' ' PRINT * , ' ( READ (* , * ) WRITE ( * , * ) WRITE ( * , 1 5 ) WRITE ( * , 1 5 ) ' NH4C1 • N a C l • K C l t 2 5 , f l 2 . 6 ) • I n i t i a l ' mNH4 ' mH • mNa ' mK ' mCl ' mNH40H ' mHCl ' mNH4Cl ' mNaCl • mKCl t 2 5 , f l 2 . 6 ) ' , N H 4 C l i n ' , N a C l i n ' , K C l i n g u e s s f o r m o l a l i t i e s ' , X G ( 1 ) • , X G ( 2 ) * , X G ( 3 ) ' , XG (4 ) ' , X G ( 5 ) ' , X G ( 6 ) ' , XG ( 7 ) ' , XG(8 ) * , X G ( 9 ) ' , X G ( 1 0 ) P r e s s e n t e r k e y t o s e e n e x t p a g e . . . ) ' WRITE WRITE WRITE ( * , 1 5 ) ( * , 1 5 ) ( * , 1 5 ) WRITE ( * , 1 5 ) WRITE ( * , 1 5 ) WRITE ( * , 1 5 ) WRITE ( * , 1 5 ) WRITE ( * , 1 5 ) FORMAT ( 3 x , a , PRINT * , ' ' PRINT * , ' ( READ (* , * ) ' F v a l u e ' F ( l ) ' F ( 2 ) ' F ( 3 ) ' F ( 4 ) ' F ( 5 ) ' F ( 6 ) ' F ( 7 ) ' F ( 8 ) ' F ( 9 ) ' F ( 1 0 ) t 2 5 , f l 2 . 6 ) , F ( 1 ) , F ( 2 ) , F ( 3 ) , F ( 4 ) , F ( 5 ) , F ( 6 ) , F ( 7 ) , F ( 8 ) , F ( 9 ) , F ( 1 0 ) P r e s s e n t e r k e y t o s e e n e x t p a g e . 2 0 2 PRINT * , ' ' WRITE ( * , 1 9 ) ' I o n i c S t r e n g t h : ' , i s 19 F 0 R M A T ( l x , a , f l 2 . 6 / / ) WRITE ( * , 2 0 ) ' M o l a l i t y [ m o l e s / K g H 2 0 ] ' 20 F O R M A T ( l x , a ) WRITE ( * , 2 1 ) ' mNH4 : • , X ( 1 ) , ' g a m N H 4 : ' ,gamNH4 WRITE ( * , 2 1 ) ' mH : ' , X ( 2 ) , • g a m H : ' , g a m H WRITE ( * , 2 1 ) • mNa : • , X ( 3 ) , • g a m N a : ' , g a m N a WRITE ( * , 2 1 ) • mK : ' , X ( 4 ) , ' g a m K : ' , g a m K WRITE ( * , 2 1 ) 1 mCl : ' , X ( 5 ) , ' g a m C l : ' , g a m C l 21 F O R M A T ( 3 x , a , t 2 0 , f l 2 . 6 , 5 x , a , f l 2 . 6 ) WRITE ( * , 2 2 ) ' mNH40H : ' , X ( 6 ) WRITE ( * , 2 2 ) • mHCl : ' , X ( 7 ) WRITE ( * , 2 2 ) • mNH4Cl : ' , X { 8 ) WRITE ( * , 2 2 ) • mNaCl : ' , X ( 9 ) WRITE ( * , 2 2 ) * mKCl : ' , X ( 1 0 ) 22 F O R M A T ( 3 x , a , t 2 0 , f l 2 . 6 ) p r i n t * , ' ' END C f u n c v S u b r o u t i n e C =================== SUBROUTINE f u n c v ( n , x , f ) INTEGER n ,m p a r a m e t e r (m=5) DOUBLE PRECIS ION x ( n ) , f ( n ) , g a m ( m ) DOUBLE PRECIS ION N H 4 C l i n , N a C l i n , K C l i n DOUBLE PRECIS ION mNH4, mH, mNa, mK, mCl DOUBLE PRECIS ION mNH40H, m H C l , m N H 4 C l , m N a C l , mKCl DOUBLE PRECIS ION g a m N H 4 , g a m H , g a m N a , g a m K , g a m C l DOUBLE PRECIS ION a H 2 0 , i s DOUBLE PRECIS ION K I , K 2 , K 3 , K 4 , K5 COMMON / C I N P U T / N H 4 C l i n , N a C l i n , K C l i n COMMON / R E S U L T / a H 2 0 , i s COMMON / G A M / g a m N H 4 , g a m H , g a m N a , g a m K , g a m C l mNH4 = x ( l ) mH = x ( 2 ) mNa = x ( 3 ) mK = x ( 4 ) mCl = x ( 5 ) mNH40H = x ( 6 ) mHCl = x ( 7 ) mNH4Cl = x ( 8 ) mNaCl = x ( 9 ) mKCl = x ( 1 0 ) c a l l d a v i e s ( n , m , x , g a m ) gamNH4= gam(1) gamH = gam(2) gamNa = gam(3) gamK = gam(4) g a m C l = gam(5) c R e a c t i o n 1 KI = 4 . 6 6 7 KI = 1 0 * * K 1 f ( l ) = 0 . 0 D 0 f ( 1 ) = mNH4*gamNH4*aH20/(mNH40H*mH*gamH) f ( l ) = f ( l ) - K l c R e a c t i o n 2 K2 = - 0 . 8 2 K2 = 1 0 * * K 2 f ( 2 ) = 0 . 0 D 0 f ( 2 ) = m N H 4 * g a m N H 4 * m C l * g a m C l / ( m N H 4 C l ) f ( 2 ) = f ( 2 ) - K 2 c R e a c t i o n 3 K3 1 . 0 1 3 K3 = 1 0 * * ( K 3 ) f ( 3 ) = 0 . 0 D 0 f ( 3 ) = m N a * g a m N a * m C l * g a m C l / ( m N a C l ) f ( 3 ) = f ( 3 ) - K 3 c R e a c t i o n 4 K4 = 1 .24 K4 = 1 0 * * ( K 4 ) f ( 4 ) = 0 . 0 D 0 f ( 4 ) = m H C l / ( m H * g a m H * m C l * g a m C l ) f ( 4 ) = f ( 4 ) - K 4 R e a c t i o n 5 K5 = - 0 . 7 3 5 K5 = 1 0 * * ( K 5 ) f ( 5 ) = 0 . 0 D 0 f ( 5 ) = m K * g a m K * m C l * g a m C l / ( m K C l ) f ( 5 ) = f ( 5 ) - K 5 M a s s b a l a n c e o f N f ( 6 ) = 0 . 0 D 0 f ( 6 ) = mNH4+mNH40H+mNH4Cl f ( 6 ) = 0 . 2 5 - f ( 6 ) M a s s b a l a n c e o f C l f ( 7 ) = 0 . 0 D 0 f ( 7 ) = mHCl+mNH4Cl+mCl+mNaCl+mKCl f ( 7 ) = 0 . 7 5 - f ( 7 ) M a s s b a l a n c e o f Na f ( 8 ) = 0 . 0 D 0 f ( 8 ) = mNa+mNaCl f ( 8 ) = 0 . 2 5 - f ( 8 ) M a s s b a l a n c e o f K f ( 9 ) = 0 . 0 D 0 f ( 9 ) = mK+mKCl f ( 9 ) = 0 . 2 5 - f ( 9 ) E l e c t r o n e u t r a l i t y f ( 1 0 ) = O.OdO f ( 1 0 ) = mNH4 + mH + mNa + mK - mCl r e t u r n END S u b r o u t i n e : D a v i e s r e v i s i o n o f t h e D e b y e - H u c k e l e q . C a l c u l a t e a c t i v i t y c o e f f i c i e n t s S o s m o t i c c o e f f i c i e n t s s u b r o u t i n e d a v i e s ( n , m , x , g a m ) i m p l i c i t d o u b l e p r e c i s i o n ( a - h , o - z ) i n t e g e r n , m , n e a t , n a n i p a r a m e t e r ( n c a t = 4 , n a n i = l ) d o u b l e p r e c i s i o n x ( n ) , g a m ( m ) i n t e g e r z c ( n e a t ) , z a ( n a n i ) d o u b l e p r e c i s i o n m c ( n e a t ) , m a ( n a n i ) d o u b l e p r e c i s i o n i s , m t , z , AP d o u b l e p r e c i s i o n l o g g a m c ( n e a t ) , g a m e ( n e a t ) , l o g g a m a ( n a n i ) , g a m a ( n a n i ) common / r e s u l t / a H 2 0 , i s Number 1 2 3 4 Number 1 c a t i o n NH4 + H+ Na+ K+ a n i o n C l -and m o l e s m c ( 1 ) = x ( 1 ) m c ( 2 ) = x ( 2 ) me (3) =x (3) mc(4)=x ( 4 ) m o l e s ma (1 )=x (5 ) a n i o n s c h a r g e z c ( l ) = + l z c ( 2 ) = + l z c ( 3 ) = + l z c ( 4 ) = + l c h a r g e z a ( 1 ) = - 1 M o l e s o f c a t i o n mc (1 )=x (1 ) m c ( 2 ) = x ( 2 ) mc (3 )=x (3 ) m c ( 4 ) = x ( 4 ) ma (1 )=x (5 ) C h a r g e o f c a t i o n and a n i o n s z c ( l ) = l z c ( 2 ) = l z c ( 3 ) = l z c ( 4 ) = l z a ( 1 ) = - 1 i o n i c s t r e n g t h and t o t a l m o l a l i t y i s = O.OdO mt = O.OdO d o 110 i = 1, n e a t i s = i s + m c ( i ) * ( z c ( i ) * * 2 ) mt = mt + mc ( i ) 2 0 4 110 c o n t i n u e d o 111 k = 1, n a n i i s = i s + m a ( k ) * ( z a ( k ) * * 2 ) mt = mt + ma (k) 111 c o n t i n u e i s = 0 . 5 * i s c C a l c u l a t i n g z z = O.OdO d o 120 1 = 1 , n e a t z = z + m c ( i ) * a b s ( z c ( i ) ) 120 c o n t i n u e d o 121 k = 1 , n a n i z = z + m a ( k ) * a b s ( z a ( k ) ) 121 c o n t i n u e c A s s u m e t h e a c t i v i t y o f w a t e r a s u n i t ah2o = l . O d O c D e b y e - H u c k e l p a r a m e t e r c Temp. 25 50 100 150 200 250 300 oC c p a r a . 0 . 5 1 0 . 5 3 0 . 6 0 0 . 6 9 0 . 8 1 0 . 9 8 1 . 2 5 6 AP = 1 . 2 5 6 c C a l c u l a t e c a t i o n and a n i o n a c t c o f d o 790 i i = l , n c a t l o g g a m c ( i i ) = O.OdO l o g g a m c ( i i ) = - ( z c ( i i ) * * 2 ) * A P * D S Q R T ( i s } / ( 1 + D S Q R T ( i s ) ) l o g g a m c ( i i ) = l o g g a m c ( i i ) + 0 . 2 * A P * ( z c ( i i ) * * 2 ) * i s g a m c ( i i ) = 1 0 * * ( l o g g a m c ( i i ) ) 7 90 c o n t i n u e d o 890 kk = 1 , n a n i l o g g a m a ( k k ) = O.OdO l o g g a m a ( k k ) = - ( z a ( k k ) * * 2 ) * A P * D S Q R T ( i s ) / ( 1 + D S Q R T ( i s ) ) l o g g a m a ( k k ) = l o g g a m a ( k k ) + 0 . 2 * A P * ( z a ( k k ) * * 2 ) * i s gama(kk ) = 1 0 * * ( l o g g a m a ( k k ) ) 890 c o n t i n u e gam(1 )=gamc(1 ) gam(2 )=gamc(2 ) gam(3 )=gamc(3 ) gam(4 )=gamc(4 ) gam(5)=gama(1) c R e t u r n aH20 and i s v a l u e s b y COMMON / R E S U L T / c R e t u r n gam() t o f u n c s u b r o u t i n e r e t u r n e n d c n e w t , l u d e m p , f d j a c , l u b k s b , l n s r c h c s u b r o u t i n e s i m p o r t e d f r o m " N u m e r i c a l R e c i p e s F o r t r a n 7 7 " , c v e r s i o n 2 . 0 7 t o c a l c u l a t e n o n - l i n e a r e q u a t i o n s SUBROUTINE n e w t ( x , n , c h e c k ) PARAMETER ( N P = 4 0 , M A X I T S = 2 0 0 , T O L F = l . e - 4 , T O L M I N = l . e - 6 , T O L X = l . e - 7 , S T P M X = 1 0 0 . ) c PARAMETER ( N P = 4 0 , M A X I T S = 2 0 0 , T O L F = l . e - 1 0 , T O L M I N = l . e - 6 , T O L X = l . e - 7 , S T P M X = 1 0 0 . ) END SUBROUTINE l u d e m p ( a , n , n p , i n d x , d ) r e t u r n END SUBROUTINE f d j a c ( n , x , f v e c , n p , d f ) r e t u r n END SUBROUTINE l u b k s b ( a , n , n p , i n d x , b ) r e t u r n END SUBROUTINE l n s r c h ( n , x o l d , f o l d , g , p , x , f , s t p m a x , c h e c k , f u n c ) END c f m i n ( x ) c : f u n c t i o n i m p o r t e d f r o m " N u m e r i c a l R e c i p e s F o r t r a n 7 7 " , c v e r s i o n 2 . 0 7 FUNCTION f m i n ( x ) r e t u r n END c H y e o n P a r k , U B C , 1998 205 Computation of equilibrium state for sodium aluminosilicate formation This code was written in order to compute the equilibrium state for the Na+ - Al(OH)4* -Si032* - OH" - C0 3 2* - S0 4 2' - Cl" - H 2 0 system (system A) and the Na+ - Al(OH)4" -Si032" - OH" - C O 3 2 " - Cl" - HS" - H 2 0 system (system B) at 368.15 K. Example of input and output N a - A l - S i l i c a t e f o r m a t i o n s y s t e m N o v . 1998 H y e o n P a r k * * * * + + * * * * * * * * * + * + * * * * * * * * * * * * * * * * * * * * * * S y s t e m : Na+ O H - C l - C 0 3 2 - S 0 4 2 - H S -A l ( O H ) 4 - S 1 0 3 2 - H20 T e m p e r a t u r e : 95 C P r e s s u r e : 1 a tm > I n p u t amount o f A1C136H20 ? ( 0 .01 - 0 . 1 ) 0 . 0 2 5 > I n p u t amount o f N a 2 S i 0 3 9 H 2 0 ? ( 0 . 0 1 ~ 0 . 1 ) 0 . 0 2 5 > I n p u t amount o f NaOH ? ( 0 . 2 5 ~ 3 . 0 m o l e s ) 1 . 0 > I n p u t amount o f N a C l ? ( 0 . 0 - 3 . 0 m o l e s ) 0 . 2 5 > I n p u t amount o f Na2C03 ? ( 0 . 0 - 2 . 0 m o l e s ) 1 . 0 > I n p u t amount o f Na2S04 ? ( 0 . 0 ~ 2 . 0 m o l e s ) 0 .1 > I n p u t amount o f Na2S ? ( 0 . 0 ~ 1 . 5 m o l e s ) 0 . 0 > I n p u t amount o f H20 ? ( 1 . 0 Kg) 1 .0 E n t e r i n i t i a l g u e s s o f msod ( 0 . 0 0 0 1 - 0 . 0 0 2 ) ? 0 . 0 0 1 E n t e r i n i t i a l g u e s s o f mhsod ( 0 . 0 0 1 - 0 . 0 0 9 d 0 ) ? 0 . 0 0 5 a d d _ p 1 a d d _ p 2 a d d _ p 6 m h s o d i n i 4 . 7 0 0 0 0 0 0 0 0 0 0 0 0 0 1 E - 0 0 3 By c o n s i d e r i n g s o d a l i t e S h y d r o x y s o d a l i t e . . . f ( 1 )= . 0 0 0 0 0 0 f ( 2) = . 0 0 0 0 0 0 f ( 3) = . 0 0 0 0 0 0 f ( 4) = . 0 0 0 0 0 0 f ( 5) = . 0 0 0 0 0 0 f ( 6) = . 0 0 0 0 0 0 f ( 7) = . 0 0 0 0 0 0 f ( 8) = . 0 0 0 0 0 0 f ( 9) = . 0 0 0 0 0 0 f ( 1 0 ) = . 460197 f ( 1 1 ) = . 607041 mSOD : - . 4 6 7 5 3 2 mHSOD : . 4 7 1 6 6 9 T h e mSOD h a s n e g a t i v e v a l u e ! R e c a l c u l a t i n g w i t h a s s u m i n g mSOD = 0 . E n t e r i n i t i a l g u e s s o f mhsod (0.001~O.009d0)? 0 . 0 0 5 a d d _ p 1 a d d _ p 2 a d d _ p 17 a d d _ p 18 m h s o d i n i 4 . 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 E - 0 0 3 C a l c u l a t e d s u c c e s s f u l l y ! f ( 1 )= . 0 0 0 0 0 0 0 0 f ( 2 )= . 0 0 0 0 0 0 0 0 f ( 3 )= . 0 0 0 0 0 0 0 0 ft 4) = f( 5) = ft 6) = f t 7) = f t 8) = ft 9) = f (10) = .00000000 .00000000 . 00000000 .00000000 .00000000 .00000000 .00061927 Press ENTER key to see the r e s u l t . Result Input chemicals(moles) A1C13 Na2Si039H20 NaOH Na2C03 Na2S04 Na2S NaCl A c t i v i t y c o e f f i c i e n t s and gamNa+ gamAl(OH)4-gamSi032-gamOH-gamC032-gamS042-gamHS-gamCl-aH20 Ionic strength Press ENTER key to see the M o l a l i t y [moles/Kg H20] mNa+ mAl(OH)4-mSi032-mOH-mC032-mS042-mHS-mCl-msodalite mhydroxysodalite mH20 mAl*mSi .025000 .025000 1.000000 1.000000 .100000 .000000 .250000 Water A c t i v i t y .532522 .169388 .038886 .670091 .017960 .023491 .580598 .621849 .917029 4.545978 next page. . . 3.448883 .004746 .004746 .927208 .993045 .099304 .000000 .322740 .000000 .003347 55.897780 .0000225 Source code (kshsodfor) c kshsod.for Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * c N a - A l - S i l i c a t e formation i n a l k a l i n e s o l u t i o n s c U n i v e r s i t y of B r i t i s h Columbia c Nov.02, 1998 Hyeon Park Q * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * c Input chemicals c A1C136H20 A l C l i n c Na2Si039H20 NaSiin c NaOH NaOHin c Na2C03 NaCOin c Na2S04 NaSOin c Na2S Na2Sin c NaCl NaClin c H20 H20in c Species at e q u i l i b r i u m c c Cation (1) : Na+ c Anion (7) : Al(OH)4- S1032- OH- C032- S042- HS- C l -c S o l i d (2) : Na8(AlSi04)6C122H20 (sodalite) c Na8(AlSi04)6(OH)22H20 (hydroxysodalite) c L i q u i d (1) : . H20 c Unknowns c c m o l a l i t y of Na+ : mNa xb(l) x s ( l ) xh 2 0 7 c m o l a l i t y o f A l ( O H ) 4 - mAl x b ( 2 ) x s ( 2 ) x h ( 2 ) c m o l a l i t y o f S i 0 3 2 - mSi x b ( 3 ) x s (3) x h ( 3 ) c m o l a l i t y o f O H - mOH x b ( 4 ) x s ( 4 ) x h ( 4 ) c m o l a l i t y o f C 0 3 2 - mCO x b ( 5 ) x s (5) x h ( 5 ) c m o l a l i t y o f S 0 4 2 - mSO x b ( 6 ) x s (6) x h ( 6 ) c m o l a l i t y o f H S - mHS x b ( 7 ) x s ( 7 ) x h ( 7 ) c m o l a l i t y o f C l - mCl x b ( 8 ) x s ( 8 ) x h ( 8 ) c m o l a l i t y o f s o d a l i t e msod x b ( 9 ) x s ( 9) c m o l a l i t y o f h y d r o x y s o d a l i t e mhsod x b ( 1 0 ) xh ( 9) c m o l e s o f H20 mH20 x b ( l l ) x s ( 1 0 ) x h ( 1 0 M a i n r o u t i n e INTEGER n b , n s , n h , i P A R A M E T E R ( n b = l l , n s = 1 0 , n h = 1 0 ) INTEGER p s o d , p h s o d DOUBLE PRECIS ION x b ( n b ) , f b ( n b ) , x s ( n s ) , f s ( n s ) , x h ( n h ) , f h ( n h ) DOUBLE PRECIS ION x i ( n b ) DOUBLE PRECIS ION A l C l i n , N a S i i n , N a O H i n , N a C l i n , N a C O i n DOUBLE PRECIS ION N a S O i n , N a 2 S i n , H 2 0 i n DOUBLE PRECIS ION m N a i n , m A l i n , m S i i n , m O H i n , m C l i n DOUBLE PRECIS ION m C O i n , m S O i n , m H S i n , mH20in DOUBLE PRECIS ION mNa, m A l , m S i , mOH, m C l , mCO, mSO DOUBLE PRECIS ION mHS, mSOD, mHSOD, mH20 DOUBLE PRECIS ION g a m N a , g a m A l , g a m S i , g a m O H , g a m C O , g a m S O DOUBLE PRECIS ION g a m H S , g a m C l DOUBLE PRECIS ION a H 2 0 , i s DOUBLE PRECIS ION m h s o d i n i , m s o d i n i , a d d _ v i n t e g e r a d d _ p , a d d _ i n t COMMON / C I N P U T / A l C l i n , N a S i i n , N a O H i n , N a C l i n , N a C O i n , N a S O i n , N a 2 S i n , H 2 0 i n COMMON / P _ V A L U E / p s o d , p h s o d COMMON / r e s u l t / a H 2 0 , i s COMMON / a c t c o e f / g a m N a , g a m A l , g a m S i , g a m O H , g a m C O , g a m S O , g a m H S , g a m C l L O G I C A L c h e c k p r i n t * , ' * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ' p r i n t * , ' N a - A l - S i l i c a t e f o r m a t i o n s y s t e m 1 p r i n t * , • N o v . 1998 H y e o n P a r k ' p^.j_nt * , ' * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * i p r i n t * , • S y s t e m : Na+ O H - C l - C 0 3 2 - S 0 4 2 - H S - ' p r i n t * , • A l ( O H ) 4 - S i 0 3 2 - H20 • p r i n t * , ' T e m p e r a t u r e : 95 C ' p r i n t * , ' P r e s s u r e : 1 atm ' w r i t e ! * , * ) ' > I n p u t amount o f A1C136H20 ? ( 0 .01 ~ 0 . 1 ) ' r e a d ( * , * ) A l C l i n w r i t e ( * , * ) ' > I n p u t amount o f N a 2 S i 0 3 9 H 2 0 ? ( 0 .01 ~ 0 . 1 ) ' r e a d ( * , * ) N a S i i n w r i t e ! * , * ) ' > I n p u t amount o f NaOH ? ( 0 . 2 5 ~ 3 . 0 m o l e s ) ' r e a d ( * , * ) N a O H i n w r i t e ( * , * ) ' > I n p u t amount o f N a C l ? ( 0 . 0 - 3 . 0 m o l e s ) ' r e a d ( * , * ) N a C l i n w r i t e ( * , * ) ' > I n p u t amount o f Na2C03 ? ( 0 . 0 - 2 . 0 m o l e s ) ' r e a d ( * , * ) N a C O i n w r i t e ! * , * ) ' > I n p u t amount o f Na2S04 ? ( 0 . 0 ~ 2 . 0 m o l e s ) ' r e a d ( * , * ) N a S O i n w r i t e ! * , * ) ' > I n p u t amount o f Na2S ? ( 0 . 0 - 1 . 5 m o l e s ) ' r e a d ! * , * ) N a 2 S i n w r i t e ! * , * ) ' > I n p u t amount o f H20 ? ( 1 . 0 Kg) ' r e a d ! * , * ) H 2 0 i n H 2 0 i n = H 2 O i n * 1 0 0 0 / 1 8 . 0 1 5 M o l e s o f s p e c i e s a t i n i t i a l s t a t e m N a i n : N a + m O H i n : O H - m C l i n : C l - m C O i n : C 0 3 2 - m S O i n : S 0 4 2 -m H S i n : H S - m A l i n : A l ( O H ) 4 - m S i i n : S i 0 3 2 - m H 2 0 i n : H 2 0 mNain = 2 * N a S i i n + N a O H i n + 2 * N a C O i n + 2 * N a S O i n + 2 * N a 2 S i n + N a C l i n m A l i n = A l C l i n m S i i n = N a S i i n mOHin = N a O H i n + N a 2 S i n - 4 * A l C l i n mCOin = N a C O i n mSOin = N a S O i n 2 0 8 mHSin = N a 2 S i n m C l i n = 3 * A l C l i n + N a C l i n mH20in= H 2 0 i n + 6 * A l C l i n + 9 * N a S i i n - N a 2 S i n C M o l a l i t y o f s p e c i e s a t i n i t i a l s t a t e ( G u e s s v a l u e s ) c a d d _ p = 1 w r i t e ( * , * ) ' E n t e r i n i t i a l g u e s s o f msod ( 0 . 0 0 0 1 - 0 . 0 0 2 ) ? ' r e a d ( * , * ) mhsod w r i t e ) * , * ) ' E n t e r i n i t i a l g u e s s o f mhsod ( 0 . 0 0 1 - 0 . 0 0 9 d 0 ) ? ' r e a d ( * , * ) mhsod 320 c o n t i n u e m h s o d i n i = mhsod mNa = m N a i n / ( m H 2 0 i n / 5 5 509) - 8 * m s o d - 8 *mhsod mAl = m A l i n / ( m H 2 0 i n / 5 5 509) - 6 * m s o d - 6 *mhsod mSi = m S i i n / ( m H 2 0 i n / 5 5 509) - 6 * m s o d - 6 *mhsod mOH = m O H i n / ( m H 2 0 i n / 5 5 509 )+12*msod+10*mhsod mCO = m C O i n / ( m H 2 0 i n / 5 5 509) mSO = m S O i n / ( m H 2 0 i n / 5 5 509) mHS = m H S i n / ( m H 2 0 i n / 5 5 509) mCl = m C l i n / ( m H 2 0 i n / 5 5 509) - 2 * m s o d mH20 = mH20in+4*msod+4*mhsod x i (1) = mNa x i (2) = mAl x i (3) mSi x i ( 4 ) = mOH x i (5) = mCO x i (6) = mSO x i (7) = mHS x i (8) = mCl x i (9) msod x i ( 1 0 ) = mhsod x i (11) = mH20 x b ( l ) = mNa x b ( 2 ) mAl x b ( 3 ) = mSi x b ( 4 ) = mOH x b ( 5 ) = mCO x b ( 6 ) = mSO x b ( 7 ) mHS x b ( 8 ) = mCl xb ( 9) = msod x b ( 1 0 ) = mhsod x b ( l l ) = mH20 p s o d = 1 p h s o d = 1 c a l l n e w t ( x b , n b , c h e c k ) c a l l f u n c v ( n b , x b , f b ) i f ( c h e c k ) t h e n w r i t e ( * , * ) ' C o n v e r g e n c e p r o b l e m s . ' e n d i f i f ( ( x b ( 2 ) . L T . O . O d O ) . O R . ( x b ( 3 ) . L T . O . O d O ) ) t h e n a d d _ i n t = ( a d d _ p / 2 ) * 2 a d d _ v = a d d _ p - a d d _ i n t i f ( a d d _ v . e q . 1 . O d O ) t h e n p r i n t * , ' a d d _ p ' , a d d _ p mhsod = m h s o d i n i + a d d _ p * ( 1 . 0 d - 4 ) a d d _ p = a d d _ p + 1 g o t o 320 e l s e i f ( a d d _ v . e q . 0 . O d O ) t h e n p r i n t * , ' a d d _ p ' , a d d _ p mhsod = m h s o d i n i - a d d _ p * ( 1 . 0 d - 4 ) a d d _ p = a d d _ p + 1 g o t o 320 e n d i f e n d i f p r i n t * , ' m h s o d i n i ' , m h s o d i n i mNa = x b ( l ) mAl = x b ( 2 ) m S i = x b ( 3 ) mOH = x b ( 4 ) 2 0 9 mCO = x b ( 5 ) mSO = x b ( 6 ) mHS = x b ( 7 ) mCl = x b ( 8 ) msod = x b ( 9 ) mhsod= x b ( 1 0 ) mH20 = x b ( l l ) p r i n t * , ' ' w r i t e ) * , * ) ' B y c o n s i d e r i n g s o d a l i t e S h y d r o x y s o d a l i t e . . . ' d o 601 1 = 1 , n b w r i t e ) * , ' ( 6 x , a , i 2 , a , f l 2 . 6 ) ' ) ' f ( ' , i , ' ) = ' , f b ( i ) 601 c o n t i n u e p r i n t * , ' ' w r i t e ) * , ' ( 6 x , a , f l 2 . 6 ) • ) ' m S O D : ' , m s o d w r i t e ) * , ' ( 6 x , a , f l 2 . 6 ) ' ) ' m H S O D : ' , m h s o d p r i n t * , ' ' c T e s t msod<0 o r mhsod<0 i f ( m s o d . L T . O . O d O ) t h e n p r i n t * , ' T h e mSOD h a s n e g a t i v e v a l u e ! ' p r i n t * , ' R e c a l c u l a t i n g w i t h a s s u m i n g mSOD = 0 . ' c r e a d ) * , * ) p s o d = 0 msod = O.OdO a d d _ p = 1 p r i n t * , ' ' w r i t e ! * , * ) ' E n t e r i n i t i a l g u e s s o f mhsod ( 0 . 0 0 1 - 0 . 0 0 9 d 0 ) ? ' r e a d ( * , * ) mhsod • 321 c o n t i n u e m h s o d i n i = m h s o d x h ( l ) = m N a i n / ( m H 2 0 i n / 5 5 509) - 8 * m s o d - 8 *mhsod x h ( 2 ) = m A l i n / ( m H 2 0 i n / 5 5 509) - 6 * m s o d - 6 *mhsod x h ( 3 ) = m S i i n / ( m H 2 0 i n / 5 5 509) - 6 * m s o d - 6 *mhsod x h ( 4 ) = m O H i n / ( m H 2 0 i n / 5 5 509 )+12*msod+10*mhsod x h ( 5 ) = m C O i n / ( m H 2 0 i n / 5 5 509) x h ( 6 ) = m S O i n / ( m H 2 0 i n / 5 5 509) x h ( 7 ) m H S i n / ( m H 2 0 i n / 5 5 .509) x h ( 8 ) = m C l i n / ( m H 2 0 i n / 5 5 .509) - 2 * m s o d x h ( 9 ) = mhsod x h ( 1 0 ) = mH20in+4*msod+4*mhsod c a l l n e w t ( x h , n h , c h e c k ) c a l l f u n c v ( n h , x h , f h ) i f ( c h e c k ) t h e n w r i t e ) * , * ) ' C o n v e r g e n c e p r o b l e m s . ' e n d i f i f ( ( x h ( 2 ) . L T . O . O d O ) . O R . ( x h ( 3 ) . L T . O . O d O ) ) t h e n a d d _ i n t = ( a d d _ p / 2 ) * 2 a d d _ v = a d d _ p - a d d _ i n t i f ( a d d _ v . e q . l . 0 d 0 ) t h e n p r i n t * , ' a d d _ p ' , a d d _ p mhsod = m h s o d i n i + a d d _ p * ( 1 . 0 d - 4 ) a d d _ p = a d d _ p + 1 g o t o 321 e l s e i f ( a d d _ v . e q . 0 . O d O ) t h e n p r i n t * , ' a d d _ p ' , a d d _ p mhsod = m h s o d i n i - a d d _ p * ( 1 . 0 d - 4 ) a d d _ p = a d d _ p + 1 g o t o 321 e n d i f e n d i f p r i n t * , ' m h s o d i n i ' , m h s o d i n i p r i n t * , ' ' p r i n t * , ' C a l c u l a t e d s u c c e s s f u l l y ! ' d o 603 1 = 1 , n h w r i t e ) * , ' ( 6 x , a , i 2 , a , f 3 0 . 8 ) ' ) ' f ( ' , i , ' ) = ' , f h ( i ) 603 c o n t i n u e mNa = x h ( l ) mAl = x h ( 2 ) mSi = x h ( 3 ) mOH = xh ( 4) mCO = x h ( 5 ) mSO = x h ( 6 ) 2 1 0 mHS = x h ( 7 ) mCl = x h ( 8 ) mhsod= x h ( 9 ) mH20 = x h ( 1 0 ) e l s e i f ( m h s o d . L T . 0 . O d O ) t h e n p r i n t * , ' T h e mHSOD h a s n e g a t i v e v a l u e ! ' p r i n t * , ' P r e s s ENTER t o r e c a l c u l a t e w i t h a s s u m i n g mHSOD = 0 ' r e a d ( * , * ) p h s o d = 0 mhsod = O.OdO a d d _ p = 1 w r i t e ! * , * ) ' E n t e r i n i t i a l g u e s s o f msod ( O . O O O l d O ) ? ' r e a d ( * , * ) msod 322 c o n t i n u e m s o d i n i = msod x s ( l ) = m N a i n / ( m H 2 0 i n / 5 5 5 0 9 ) - 8 * m s o d - 8 *mhsod x s ( 2 ) = m A l i n / ( m H 2 0 i n / 5 5 5 0 9 ) - 6 * m s o d - 6 *mhsod x s ( 3 ) = m S i i n / ( m H 2 0 i n / 5 5 5 0 9 ) - 6 * m s o d - 6 *mhsod x s ( 4 ) = m O H i n / ( m H 2 0 i n / 5 5 509 )+12*msod+10*mhsod x s ( 5 ) = m C O i n / ( m H 2 0 i n / 5 5 509) x s ( 6 ) = m S O i n / ( m H 2 0 i n / 5 5 509) x s ( 7 ) = m H S i n / ( m H 2 0 i n / 5 5 509) x s ( 8 ) = m C l i n / ( m H 2 0 i n / 5 5 5 0 9 ) - 2 * m s o d x s ( 9 ) = msod x s ( 1 0 ) = mH20 in+4*msod+4*mhsod c a l l n e w t ( x s , n s , c h e c k ) c a l l f u n c v ( n s , x s , f s ) i f ( ( x s ( 2 ) . L T . O . O d O ) . O R . ( x s ( 3 ) . L T . O . O d O ) ) t h e n a d d _ i n t = ( a d d _ p / 2 ) * 2 a d d _ v = a d d _ p - a d d _ i n t i f ( a d d _ v . e q . l . 0 d 0 ) t h e n p r i n t * , ' a d d _ p ' , a d d _ p msod = m s o d i n i + a d d _ p * ( 1 . 0 d - 4 ) a d d _ p = a d d _ p + 1 g o t o 322 e l s e i f ( a d d _ v . e q . O . O d O ) t h e n p r i n t * , ' a d d _ p ' , a d d _ p msod = m s o d i n i - a d d _ p * ( 1 . 0 d - 4 ) a d d _ p = a d d _ p + 1 g o t o 322 e n d i f e n d i f p r i n t * , ' m s o d i n i ' , m s o d i n i p r i n t * , * ' p r i n t * , ' C a l c u l a t e d s u c c e s s f u l l y ! ' d o 605 1 = 1 , n s w r i t e ( * , • ( 6 x , a , i 2 , a , f 1 2 . 6 ) • ) • f ( • , i , • ) = • , f s ( i ) 605 c o n t i n u e mNa = x s (1) mAl = x s ( 2 ) mSi = x s ( 3 ) mOH = x s ( 4 ) mCO = x s ( 5 ) mSO = x s ( 6 ) mHS = x s (7 ) mCl = x s ( 8 ) msod= x s ( 9 ) mH20= x s ( 1 0 ) e n d i f C a t f i n a l s t a t e c p r i n t * , ' ' p r i n t * , ' P r e s s ENTER k e y t o s e e t h e r e s u l t . . . ' r e a d ( * , * ) p r i n t * , ' ' p r i n t * , ' R e s u l t p r i n t * , ' ' w r i t e (* , * ) ' I n p u t c h e m i c a l s ( m o l e s ) ' w r i t e ( * , 1 0 ) ' A 1 C 1 3 : ' , A l C l i n w r i t e ( * , 1 0 ) ' N a 2 S i 0 3 9 H 2 0 : ' , N a S i i n 2 1 1 10 16 20 21 22 w r i t e w r i t e w r i t e w r i t e w r i t e f o r m a t p r i n t WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE WRITE FORMAT p r i n t * p r i n t * r e a d (* , w r i t e ( f o r m a t ( * , 1 0 ) 'NaOH * , 1 0 ) ' N a 2 C 0 3 * , 1 0 ) 'Na2S04 * , 1 0 ) ' N a 2 S * , 1 0 ) ' N a C l 6 x , a , t 2 5 , f l 2 . 6 ) ' , N a O H i n ' , N a C O i n ' , N a S O i n ' , N a 2 S i n ' , N a C l i n *,*> * , 1 6 ) * , 1 6 ) * , 1 6 ) * , 1 6 ) * , 1 6 ) * , 1 6 ) * , 1 6 ) * , 1 6 ) * , 1 6 ) * , 1 6 ) A c t i v i t y c o e f f i c i e n t s and W a t e r A c t i v i t y ' gamNa+ ' , g a m N a g a m A l ( O H ) 4 - ' , g a m A l g a m S i 0 3 2 - ' , g a m S i gamOH- ' , g a m O H g a m C 0 3 2 - ' , g a m C O g a m S 0 4 2 - ' , g a m S O gamHS- ' , g a m H S g a m C l - ' , g a m C l aH20 ' , a H 2 0 I o n i c s t r e n g t h ' , i s f l 2 . 6 ) ' P r e s s ENTER k e y t o s e e t h e n e x t p a g e . ) * , 2 0 ) l x , a ) w r i t e ( * , 2 1 ) w r i t e ( * , 2 1 ) w r i t e ( * , 2 1 ) w r i t e ( * , 2 1 ) w r i t e ( * , 2 1 ) w r i t e ( * , 2 1 ) w r i t e ( * , 2 1 ) w r i t e ( * , 2 1 ) w r i t e ( * , 2 1 ) w r i t e ( * , 2 1 ) w r i t e ( * , 2 1 ) f o r m a t ( 6 x , a , t 2 5 , f 1 2 . 6 ) WRITE ( * , 2 2 ) ' m A l * m S i F O R M A T ( 3 x , a , t 2 6 , f l 2 . 7 ) END M o l a l i t y [ m o l e s / K g H 2 0 ] ' •mNa+ • ,mNa ' m A l ( O H ) 4 - ' , m A l • m S i 0 3 2 - ' , mSi •mOH- ' ,mOH ' m C 0 3 2 - • ,mCO ' m S 0 4 2 - • ,mSO •mHS- ' ,mHS • m C l - ' , m C l ' m s o d a l i t e • ,mSOD ' m h y d r o x y s o d a l i t e ' ,mHSOD •mH20 ' , m H 2 0 : ' , m A l * m S i C f u n c v S u b r o u t i n e C =================== C SUBROUTINE f u n c v ( n , x , f ) i n t e g e r n , m , p s o d , p h s o d p a r a m e t e r (m=8) DOUBLE PRECIS ION f s o d , f h s o d DOUBLE PRECIS ION x ( n ) , f ( n ) , g a m ( m ) DOUBLE PRECIS ION A l C l i n , N a S i i n , N a O H i n , N a C l i n , N a C O i n DOUBLE PRECIS ION N a S O i n , N a 2 S i n , H 2 0 i n DOUBLE PRECIS ION mNa, mOH, m C l , mCO, mSO DOUBLE PRECIS ION mHS, m A l , m S i , mSOD, mHSOD, mH20 DOUBLE PRECIS ION g a m N a , g a m A l , g a m S i , g a m O H , g a m C O , g a m S O DOUBLE PRECIS ION g a m H S , g a m C l DOUBLE PRECIS ION a H 2 0 , i s , K s o d , K h s o d COMMON / C I N P U T / A l C l i n , N a S i i n , N a O H i n , N a C l i n , N a C O i n , & N a S O i n , N a 2 S i n , H 2 0 i n COMMON / P _ V A L U E / p s o d , p h s o d COMMON / R E S U L T / a H 2 0 , i s COMMON / a c t c o e f / g a m N a , g a m A l , g a m S i , g a m O H , g a m C O , g a m S O , & g a m H S , g a m C l mNa = x ( l ) mAl = x ( 2 ) mSi = x ( 3 ) mOH = x ( 4 ) mCO = x ( 5 ) mSO = x ( 6 ) mHS = x ( 7 ) mCl = x ( 8 ) i f ( ( p s o d . e q . 0 ) . A N D . ( p h s o d . e q . 1 ) ) t h e n 2 1 2 mSOD = O.OdO mHSOD= x ( 9 ) mH20 = x ( 1 0 ) e l s e i f ( ( p s o d . e q . 1 ) . A N D . ( p h s o d . e q . 0 ) ) t h e n mSOD = x ( 9 ) mHSOD= O.OdO mH20 = x ( 1 0 ) e l s e i f ( ( p s o d . e q . 1 ) . A N D . ( p h s o d . e q . 1 ) ) t h e n mSOD = x ( 9 ) mHSOD= x ( 1 0 ) mH20 = x ( l l ) e l s e w r i t e ! * , * ) " r u n e r r o r i n f u n c v ! " s t o p e n d i f c a l l a c t c o f ( n , m , x , g a m ) gamNa = g a m ( l ) g a m A l g a m S i gamOH gamCO gamSO gamHS g a m C l gam(2) gam(3) gam(4) gam(5) gam(6) gam(7) gam(8) C M a s s b a l a n c e o f Na f ( l ) = O.ODO f ( l ) = 2 * N a S i i n + N a O H i n + 2 * N a C O i n + 2 * N a S O i n + 2 * N a 2 S i n + N a C l i n f ( l ) = f ( l ) - ( m H 2 O / 5 5 . 5 0 9 ) * ( m N a + 8 * m S O D + 8 * m H S O D ) C M a s s b a l a n c e o f A l f ( 2 ) = O.ODO f ( 2 ) = A l C l i n f ( 2 ) = f ( 2 ) - ( m H 2 O / 5 5 . 5 0 9 ) * ( m A l + 6 * m S O D + 6 * m H S O D ) C M a s s b a l a n c e o f S i f ( 3 ) = O.ODO f ( 3 ) = N a S i i n f ( 3 ) = f ( 3 ) - ( m H 2 O / 5 5 . 5 0 9 ) * ( m S i + 6 * m S O D + 6 * m H S O D ) C M a s s b a l a n c e o f C l f ( 4 ) = O.ODO f ( 4 ) = 3 * A l C l i n + N a C l i n f ( 4 ) = f ( 4 ) - ( m H 2 O / 5 5 . 5 0 9 ) * ( m C l + 2 * m S O D ) C M a s s b a l a n c e o f S f ( 5 ) = O.OdO f ( 5 ) = N a S O i n + N a 2 S i n f ( 5 ) = f ( 5 ) - ( m H 2 O / 5 5 . 5 0 9 ) * ( m S O + m H S ) C M a s s b a l a n c e o f O f ( 6 ) = O.ODO f ( 6 ) = 1 2 * N a S i i n + 6 * A l C l i n + N a O H i n + 3 * N a C O i n + 4 * N a S O i n + H 2 0 i n f ( 6 ) = f ( 6 ) - m H 2 O - ( m H 2 O / 5 5 . 5 0 9 ) * ( 4 * m A l + 3 * m S i + m O H + 3 * m C O + & 4*mSO+26*mSOD+28*mHSOD) C M a s s b a l a n c e o f H f ( 7 ) = O.ODO f ( 7 ) = 1 8 * N a S i i n + 1 2 * A l C l i n + N a O H i n + 2 * H 2 0 i n f ( 7 ) = f ( 7 ) - 2 * m H 2 0 -& (mH2O/55.509)*(4*mAl+mOH+mHS+4*mSOD+6*mHSOD) C M a s s b a l a n c e o f C f ( 8 ) = O.OdO f ( 8 ) = N a C O i n - ( m H 2 0 / 5 5 . 5 0 9 ) * ( m C O ) C E l e c t r o n e u t r a l i t y f ( 9 ) = O.OdO f ( 9 ) = mNa - (mOH+mCl+2*mCO+2*mSO+mHS+mAl+2*mSi) c S o d a l i t e & H y d r o x y s o d a l i t e f o r m a t i o n c A c t i v i t y o f p u r e s o l i d c a n be a s s u m e d as 1 . c K s o d = 1 .82d+17 (95 oC) K s o d = 1 .82d+17 f s o d = O.ODO f s o d = d l o g ( a H 2 0 * * 4 ) + d l o g ( m O H * * 1 2 ) + d l o g ( g a m O H * * 1 2 ) f s o d = f s o d - d l o g ( m N a * * 8 ) - d l o g ( g a m N a * * 8 ) f s o d = f s o d - d l o g ( m A l * * 6 ) - d l o g ( g a m A l * * 6 ) f s o d = f s o d - d l o g ( m S i * * 6 ) - d l o g ( g a m S i * * 6 ) f s o d = f s o d - d l o g ( m C l * * 2 ) - d l o g ( g a m C l * * 2 ) f s o d = f s o d - d l o g ( K s o d ) 2 1 3 c K h s o d = 3 .38d+38 (95oC) c 4 .39d+36 ( f r o m f i t t i n g t o e x p . d a t a ) K h s o d = 4 .39d+36 f h s o d = O.ODO f h s o d = d l o g ( a H 2 0 * * 4 ) + d l o g ( m O H * * 1 2 ) + d l o g ( g a m O H * * 1 2 ) f h s o d = f h s o d - d l o g ( m N a * * 8 ) - d l o g ( g a m N a * * 8 ) f h s o d = f h s o d - d l o g ( m A l * * 6 ) - d l o g ( g a m A l * * 6) f h s o d = f h s o d - d l o g ( m S i * * 6 ) - d l o g ( g a m S i * * 6 ) f h s o d = f h s o d - d l o g ( m O H * * 2 ) - d l o g ( g a m O H * * 2) f h s o d = f h s o d - d l o g ( K h s o d ) i f ( ( p s o d . e q . 0 ) . A N D . ( p h s o d . e q . 1 ) ) t h e n f ( 1 0 ) = f h s o d e l s e i f ( ( p s o d . e q . 1 ) . A N D . ( p h s o d . e q . 0 ) ) t h e n f ( 1 0 ) = f s o d e l s e i f ( ( p s o d . e q . 1 ) . A N D . ( p h s o d . e q . 1 ) ) t h e n f ( 1 0 ) = f s o d f ( l l ) = f h s o d e l s e w r i t e ! * , * ) " r u n e r r o r i n f u n c v ! " s t o p e n d i f r e t u r n e n d c S u b r o u t i n e o f P i t z e r ' s M o d e l c C a l c u l a t e a c t i v i t y c o e f f i c i e n t s S o s m o t i c c o e f f i c i e n t s s u b r o u t i n e a c t c o f ( n , m , x , g a m ) i m p l i c i t d o u b l e p r e c i s i o n ( a - h , o - z ) e x t e r n a l g , g p i n t e g e r n , m , n e a t , n a n i p a r a m e t e r ( n c a t = l , n a n i = 7 ) d o u b l e p r e c i s i o n x ( n ) , g a m ( m ) i n t e g e r z c ( n e a t ) , z a ( n a n i ) d o u b l e p r e c i s i o n b e t h a O ( n e a t , n a n i ) , b e t h a l ( n e a t , n a n i ) d o u b l e p r e c i s i o n b e t h a 2 ( n e a t , n a n i ) , c p h i ( n e a t , n a n i ) d o u b l e p r e c i s i o n t h e t a c ( n e a t , n e a t ) , t h e t a a ( n a n i , n a n i ) d o u b l e p r e c i s i o n e t h e t a c ( n e a t , n e a t ) , e t h e t a p c ( n e a t , n e a t ) d o u b l e p r e c i s i o n e t h e t a a ( n a n i , n a n i ) , e t h e t a p a ( n a n i , n a n i ) d o u b l e p r e c i s i o n p s i c ( n e a t , n e a t , n a n i ) , p s i a ( n a n i , n a n i , n e a t ) d o u b l e p r e c i s i o n m c ( n e a t ) , m a ( n a n i ) d o u b l e p r e c i s i o n i s , m t , z , b p h i ( n e a t , n a n i ) d o u b l e p r e c i s i o n c ( n e a t , n a n i ) , p h i c ( n e a t , n e a t ) , p h i p c ( n e a t , n e a t ) d o u b l e p r e c i s i o n p h i a ( n a n i , n a n i ) , p h i p a ( n a n i , n a n i ) d o u b l e p r e c i s i o n p h i p h i c ( n e a t , n e a t ) , p h i p h i a ( n a n i , n a n i ) d o u b l e p r e c i s i o n o s m l , o s m 2 , o s m 3 , o s m 3 a , o s m 4 , o s m 4 a , o s m c a l d o u b l e p r e c i s i o n F l , F 2 , F 3 , F 4 , a H 2 0 , a p h i d o u b l e p r e c i s i o n l n g a m c ( n e a t ) , g a m e ( n e a t ) , l n g a m a ( n a n i ) , g a m a ( n a n i ) d o u b l e p r e c i s i o n b m x ( n e a t , n a n i ) , b p a c t ( n e a t , n a n i ) common / r e s u l t / a H 2 0 , i s c Number c a t i o n m o l e s c h a r g e c 1 Na+ mc (1 )=x (1 ) z c ( l ) = + l c Number a n i o n m o l e s c h a r g e c 1 A l ( O H ) 4 - ma(1 )=x (2 ) z a ( 1 ) = - 1 c 2 S i 0 3 2 - ma(2 )=x (3 ) z a ( 2 ) = - 2 c 3 0 H - ma(3 )=x (4 ) z a ( 3 ) = - l c 4 C 0 3 2 - ma{4)=x(5) z a ( 4 ) = - 2 c 5 S 0 4 2 - ma(5 )=x (6 ) z a ( 5 ) = - 2 c 6 H S - ma(6 )=x (7 ) z a ( 6 ) = - l c 7 C l - ma (7 )=x (8 ) z a ( 7 ) = - l c M o l e s o f c a t i o n and a n i o n s m c ( 1 ) = x ( 1 ) ma (1 )=x (2 ) ma (2 )=x (3 ) ma (3 )=x (4 ) ma (4 )=x (5 ) ma (5 )=x (6 ) ma (6 )=x (7 ) ma (7 )=x (8 ) c C h a r g e o f c a t i o n and a n i o n s z c ( l ) = l z a ( l ) — 1 z a ( 2 ) — 2 z a ( 3 ) — 1 z a ( 4 ) = - 2 z a ( 5 ) = - 2 z a ( 6 ) = - l z a ( 7 ) = - l c I n i t i a l i z a t i o n o f P i t z e r ' s p a r a m e t e r s a s O.OdO d o 31 1 = 1 , n e a t d o 30 k = 1 , n a n i b e t h a 0 ( i , k ) = O.OdO b e t h a l ( i , k ) = O.OdO b e t h a 2 ( i , k ) = O.OdO c p h i ( i , k ) = O.OdO 30 c o n t i n u e 31 c o n t i n u e d o 41 i = 1 , n e a t d o 4 0 j = 1 , n e a t t h e t a c l i , j ) = O.OdO e t h e t a c ( i , j ) = O.OdO e t h e t a p c ( i , j ) = O.OdO 4 0 c o n t i n u e 41 c o n t i n u e d o 51 k = 1 , n a n i d o 50 1 = 1, n a n i t h e t a a ( k , l ) = O.OdO e t h e t a a ( k , l ) = O.OdO e t h e t a p a ( k , l ) = O.OdO 50 c o n t i n u e 51 c o n t i n u e d o 62 i = 1 , n e a t d o 61 j = 1 , n e a t do 60 k = 1 , n a n i p s i c ( i , j , k ) = O.OdO 60 c o n t i n u e 61 c o n t i n u e 62 c o n t i n u e d o 72 k = 1 , n a n i d o 71 1 = 1 , n a n i d o 70 i = 1 , n e a t p s i a ( k , l , i ) = O.OdO 70 c o n t i n u e 71 c o n t i n u e 72 c o n t i n u e c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c R e a d i n g P i t z e r ' s p a r a m e t e r s c c a t i o n - a n i o n : b e t h a O ( c a t i o n , a n i o n ) , b e t h a l ( c a t i o n , a n i o n ) c b e t h a 2 ( c a t i o n , a n i o n ) , c p h i ( c a t i o n , a n i o n ) c R e a d b e t h a O , b e t h a l , c p h i o f N a A l ( O H ) 4 b e t h a 0 ( l , l ) = - 0 . 0 0 8 3 b e t h a l ( l , l ) = 0 . 0 7 1 0 c p h i ( l , l ) = 0 . 0 0 1 8 4 c R e a d b e t h a O , b e t h a l , c p h i o f N a 2 S i 0 3 b e t h a 0 ( l , 2 ) = 0 . 0 5 7 7 b e t h a l ( l , 2 ) = 2 . 8 9 6 5 c p h i ( l , 2 ) = 0 . 0 0 9 7 7 c R e a d b e t h a O , b e t h a l , c p h i o f NaOH a t 95oC b e t h a 0 ( l , 3 ) = 0 . 0 8 4 7 b e t h a l ( l , 3 ) = 0 . 3 8 8 2 c p h i ( l , 3 ) = 0 . 0 0 0 2 1 c R e a d b e t h a O , b e t h a l , c p h i o f Na2C03 a t 95oC b e t h a 0 ( l , 4 ) = 0 . 0 5 8 1 b e t h a l ( l , 4 ) = 1 . 2 4 1 9 c p h i ( 1 , 4 ) = 0 . 0 0 5 2 c R e a d b e t h a O , b e t h a l , c p h i o f Na2S04 a t 95oC b e t h a 0 ( l , 5 ) = 0 . 0 9 8 8 b e t h a l ( l , 5 ) = 1 . 4 3 2 5 c p h i ( l , 5 ) = - 0 . 0 1 4 6 2 c R e a d b e t h a O , b e t h a l , c p h i o f NaHS b e t h a 0 ( l , 6 ) = 0 . 1 3 9 6 b e t h a l ( l , 6 ) = 0 . 0 c p h i ( l , 6 ) = - 0 . 0 1 2 7 c R e a d b e t h a O , b e t h a l , c p h i o f N a C l a t 95oC b e t h a 0 ( l , 7 ) = 0 . 1 0 0 8 b e t h a l ( l , 7 ) = 0 . 3 2 0 7 c p h i ( l , 7 ) = - 0 . 0 0 3 3 7 c R e a d t h e t a a and p s i a c t h e t a a ( A l ( O H ) 4 - , O H - ) p s i a ( A l ( O H ) 4 - , O H - , N a + ) t h e t a a ( l , 3 ) = - 0 . 2 2 5 5 p s i a ( l , 3 , l ) = - 0 . 0 3 8 8 c t h e t a a ( A l ( O H ) 4 - , C l - ) p s i a ( A l ( O H ) 4 - , C 1 - , N a + ) t h e t a a ( l , 7 ) = - 0 . 2 4 3 0 p s i a ( l , 7 , l ) = 0 . 2 3 7 7 c t h e t a a ( S i 0 3 2 - , O H - ) p s i a ( S i 0 3 2 - , O H - , N a + ) t h e t a a ( 2 , 3 ) = - 0 . 2 7 0 3 p s i a ( 2 , 3 , l ) = 0 . 0 2 3 3 c t h e t a a ( O H - , C 0 3 2 - ) p s i a ( O H - , C 0 3 2 - , N a + ) t h e t a a ( 3 , 4 ) = 0 .1 p s i a ( 3 , 4 , l ) = - 0 . 0 1 7 c t h e t a a ( O H - , S 0 4 2 - ) p s i a ( O H - , S 0 4 2 - , N a + ) t h e t a a ( 3 , 5 ) = - 0 . 0 1 3 p s i a ( 3 , 5 , l ) = - 0 . 0 0 9 c t h e t a a ( O H - , C l - ) p s i a ( O H - , C l - , N a + ) t h e t a a ( 3 , 7 ) = - 0 . 0 5 p s i a ( 3 , 7 , l ) = - 0 . 0 0 6 c t h e t a a ( C 0 3 2 - , S 0 4 2 - ) p s i a ( C 0 3 2 - , S 0 4 2 - , N a + ) t h e t a a ( 4 , 5 ) = 0 . 0 2 p s i a ( 4 , 5 , l ) = - 0 . 0 0 5 c t h e t a a ( C 0 3 2 - , C 1 - ) p s i a ( C 0 3 2 - , C l - , N a + ) t h e t a a ( 4 , 7 ) = - 0 . 0 2 p s i a ( 4 , 7 , l ) = 0 . 0 0 8 5 c t h e t a a ( S 0 4 2 - , C 1 - ) p s i a ( S 0 4 2 - , C 1 - , N a + ) t h e t a a ( 5 , 7 ) = 0 . 0 3 p s i a ( 5 , 7 , l ) = 0 . 0 0 0 c U n i v e r s a l c o n s t a n t s o f P i t z e r ' s e q . ( n o t 2 -2 e l e c t r o l y t e ) b = 1 . 2 d 0 a l p h a l = 2 . 0 d 0 a l p h a 2 = O.OdO c i o n i c s t r e n g t h and t o t a l m o l a l i t y i s = O.OdO mt = O.OdO d o 110 i = 1, n e a t i s = i s + m c ( i ) * ( z c ( i ) * * 2 ) mt = mt + mc ( i ) 110 c o n t i n u e d o 111 k = 1 , n a n i i s = i s + m a ( k ) * ( z a ( k ) * * 2 ) mt = mt + ma (k) 111 c o n t i n u e i s = 0 . 5 * i s c C a l c u l a t i n g z z = O.OdO d o 120 i = 1 , n e a t z = z + m c ( i ) * a b s ( z c ( i ) ) 120 c o n t i n u e d o 121 k = 1 , n a n i z = z + m a ( k ) * a b s ( z a ( k ) ) 121 c o n t i n u e c c c c c c c c c c c c c c c C a l c u l a t i n g b p h i ( c a t , a n i ) and c ( c a t , a n i ) c o e f f i c i e n t s d o 131 i = 1 , n e a t d o 130 k = 1 , n a n i b p h i ( i , k ) = O.OdO b p h i ( i , k ) = b e t h a O ( i , k ) + b e t h a l ( i , k ) * d e x p ( - a l p h a l * d s q r t ( i s ) ) b p h i ( i , k ) = b p h i ( i , k ) + b e t h a 2 ( i , k ) * d e x p ( - a l p h a 2 * d s q r t ( i s ) ) c ( i , k ) = O.OdO c ( i , k ) = c p h i ( i , k ) / ( 2 * ( ( a b s ( z c ( i ) * z a ( k ) ) ) * * 0 . 5 ) ) b m x ( i , k ) = O.OdO b m x ( i , k ) = b e t h a O ( i , k ) + b e t h a l ( i , k ) * g ( a l p h a l * d s q r t ( i s ) ) b m x ( i , l c ) = b m x ( i , k ) + b e t h a 2 ( i , k ) * g ( a l p h a 2 * d s q r t ( i s ) ) b p a c t ( i , k ) = O.OdO b p a c t ( i , k ) = b e t h a l ( i , k ) * g p ( a l p h a l * d s q r t ( i s ) ) b p a c t ( i , k ) = b p a c t ( i , k ) + b e t h a 2 ( i , k ) * g p ( a l p h a 2 * d s q r t ( i s ) ) b p a c t ( i , k ) = b p a c t ( i , k ) / i s 130 c o n t i n u e 131 c o n t i n u e c c c c c c c c c c c C a l c u l a t i n g p h i c ( c a t , c a t ) , p h i p c ( c a t , c a t ) , p h i a ( a n i , a n i ) , c a n d p h i p a ( a n i , a n i ) d o 141 1 = 1 , n e a t d o 140 j = 1 , n e a t p h i c f i , j ) = O . O d O ' p h i p c ( i , j ) = O.OdO p h i c f i , j ) = t h e t a c ( i , j ) + e t h e t a c ( i , j ) p h i p c ( i , j ) = e t h e t a p c ( i , j ) 140 c o n t i n u e 141 c o n t i n u e d o 151 k = 1, n a n i d o 150 1 = 1 , n a n i p h i a ( k , l ) = O.OdO p h i p a ( k , l ) = O.OdO p h i a ( k , l ) = t h e t a a ( k , l ) + e t h e t a a ( k , l ) p h i p a ( k , l ) = e t h e t a p a ( k , 1 ) 150 c o n t i n u e 151 c o n t i n u e c c c c c c C a l c u l a t i n g p h i p h i c ( c a t , c a t ) and p h i p h i a ( a n i , a n i ) c o e f f i c i e n t s do 161 i = 1 , ( n c a t - 1 ) d o 160 j = ( i + 1 ) , n e a t p h i p h i c ( i , j ) = O.OdO p h i p h i c ( i , j ) = p h i c ( i , j ) + i s * ( p h i p c ( i , j ) ) 160 c o n t i n u e 161 c o n t i n u e d o 171 k = 1 , ( n a n i - 1 ) d o 170 1 = ( k + 1 ) , n a n i p h i p h i a ( k , l ) = O.OdO p h i p h i a ( k , l ) = p h i a ( k , l ) + i s * ( p h i p a ( k , 1 ) ) 170 c o n t i n u e 171 c o n t i n u e c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c C a l c u l a t i n g o s m o t i c c o e f f i c i e n t c a p h i : 0 . 3 9 1 5 a t 25oC 0 . 4 5 5 a t 95 oC a p h i = 0 . 4 5 5 o s m l = O.OdO o s m l = - a p h i * ( i s * * ( 3 . / 2 . ) ) / ( 1 + b * ( i s * * 0 . 5 ) ) osm2 = O.OdO d o 181 i = l , n c a t d o 180 k = 1 , n a n i o s m 2 = o s m 2 + ( m c ( i ) * m a ( k ) ) * ( b p h i ( i , k ) + z * c ( 1 , k ) ) 180 c o n t i n u e 181 c o n t i n u e osm3 = O.OdO d o 192 i = 1 , ( n c a t - 1 ) d o 191 j = ( i + i ) , n e a t osm3a = O.OdO d o 190 k = 1 , n a n i osm3a = osm3a + ma(k) * p s i c ( i , j , k ) 190 c o n t i n u e osm3 = o s m 3 + ( ( m c ( i ) * m c ( j ) ) * ( p h i p h i c ( i , j ) + o s m 3 a ) ) 191 c o n t i n u e 192 c o n t i n u e osm4 = O.OdO d o 202 k = 1 , ( n a n i - 1 ) d o 201 1 = (k+1 ) , n a n i osm4a = O.OdO d o 200 i = 1, n e a t osm4a = osm4a + m c ( i ) * p s i a ( k , l , i ) 200 c o n t i n u e osm4 = o s m 4 + ( ( m a ( k ) * m a ( 1 ) ) * ( p h i p h i a ( k , 1 ) + o s m 4 a ) ) 201 c o n t i n u e 202 c o n t i n u e 2 1 7 c o s m c a l = O.OdO o s m c a l = o s m l + osm2 + osm3 + osm4 o s m c a l = ( 2 . 0 / m t ) * o s m c a l o s m c a l = o s m c a l + 1 . c C a l c u l a t e t h e a c t i v i t y o f w a t e r a h 2 o = O.OdO ah2o = 1 0 * * ( - 0 . 0 0 7 8 2 3 * m t * o s m c a l ) c C a l c u l a t i n g F t e r m F l = O.OdO '(' F l = ( i s * * 0 . 5 ) / ( 1 + b * ( i s * * 0 . 5 ) ) F l = F l + ( 2 / b ) * l o g ( l + b * ( i s * * 0 . 5 ) ) F l = F l * ( - a p h i ) F2 = O.OdO d o 211 1 = l , n c a t d o 210 k = l , n a n i F2 = F2 + m c ( i ) * m a ( k ) * b p a c t ( i , k ) 210 c o n t i n u e 211 c o n t i n u e F3 = O.OdO d o 221 1 = 1 , ( n c a t - 1 ) d o 220 j = ( i + 1 ) , n e a t F3 = F3 + m c ( i ) * m c ( i ) * p h i p c ( i , j ) 220 c o n t i n u e 221 c o n t i n u e F4 = O.OdO d o 231 k = 1 , ( n a n i - 1 ) do 230 1 = ( k + 1 ) , n a n i F4 = F4 + m a ( k ) * m a ( k ) * p h i p a ( k , l ) 230 c o n t i n u e 231 c o n t i n u e F = O.OdO F = F l + F2 + F3 + F4 c c c c c c c c c c c c c c c c c c c c c c c c c c c c c C a l c u l a t i n g c a t i o n a c t i v i t y c o e f f i c i e n t d o 790 i i = 1 , n e a t c s u m l = O.OdO c s u m l = ( z c ( i i ) * * 2 ) * F csum2 = O.OdO d o 700 k = 1 , n a n i csum2 = csum2 + m a ( k ) * ( 2 * b m x ( i i , k ) + z * c ( i i , k ) ) 700 c o n t i n u e csum3 = O.OdO d o 720 j = 1 , n e a t i f (j . e q . i i ) go t o 720 csum3a = O.OdO d o 710 k = 1 , n a n i csum3a = csum3a + ma(k) * p s i c ( i i , j , k ) 710 c o n t i n u e csum3 = csum3 + m c ( j ) * ( 2 * p h i c ( i i , j ) + c s u m 3 a ) 720 c o n t i n u e csum4 = O.OdO d o 740 K = 1 , ( n a n i - 1 ) d o 730 L = (K+1) , n a n i csum4 = csum4 + ma(k) * m a ( l ) * p s i a ( K , L , i i ) 730 c o n t i n u e 740 c o n t i n u e csum5 = O.OdO d o 760 1 = 1 , n e a t d o 750 K = 1, n a n i csum5 = csum5 + m c ( i ) * ma(K) * C ( i , K ) 750 c o n t i n u e 760 c o n t i n u e csum5 = a b s ( Z C ( i i ) ) * csum5 c l n g a m c t ) : I n a c t i v i t y c o e f f i c i e n t o f c a t i o n b y P i t z e r e q . c gamc( ) : a c t i v i t y c o e f f i c i e n t o f c a t i o n b y P i t z e r e q . l n g a m c ( i i ) = c s u m l + csum2 + csum3 + csum4 + csum5 g a m c ( i i ) = d e x p ( l n g a m c ( i i ) ) 7 90 c o n t i n u e 2 1 8 c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c C a l c u l a t i n g a n i o n a c t i v i t y c o e f i c i e n t d o 890 kk = 1 , n a n i a s u m l = O.OdO a s u m l = ( z a ( k k ) * * 2 ) * F asum2 = O.OdO d o 800 i = 1, n e a t asum2 = asum2 + m c ( i ) * ( 2 * b m x ( i , k k ) + z * C ( i , k k ) ) 800 c o n t i n u e asum3 = O.OdO d o 820 1 = 1 , n a n i i f ( l . e q . k k ) go t o 820 asum3a = O.OdO d o 810 1 = 1 , n e a t < asum3a = asum3a + m c ( i ) * p s i a ( k k , l , i ) 810 c o n t i n u e asum3 = asum3 + m a ( 1 ) * ( 2 * p h i a { k k , 1 ) + asum3a) 820 c o n t i n u e asum4 = O.OdO d o 840 i = 1, ( n c a t - 1 ) d o 830 J = ( i + 1 ) , n e a t asum4 = asum4 + m c ( i ) * m c ( j ) * p s i c ( i , j , k k ) 830 c o n t i n u e 840 c o n t i n u e asum5 = O.OdO d o 860 1 = 1 , n e a t d o 850 K = 1 , n a n i asum5 = asum5 + m c ( i ) * ma(k) * C ( i , K ) 850 c o n t i n u e 860 c o n t i n u e asum5 = a b s ( z a ( k k ) ) * a s u m 5 c l n g a m a ( ) : I n a c t i v i t y c o e f f i c i e n t o f a n i o n b y P i t z e r e q . c gama() : a c t i v i t y c o e f f i c i e n t o f a n i o n b y P i t z e r e q . l n g a m a ( k k ) = a s u m l + asum2 + asum3 + asum4 + asum5 gama(kk ) = d e x p ( l n g a m a ( k k ) ) c 890 c o n t i n u e gam(1 )=gamc(1 ) gam(2)=gama(1) gam(3)=gama(2 ) gam(4)=gama(3 ) gam(5)=gama(4 ) gam(6)=gama(5 ) gam(7)=gama(6) gam(8)=gama(7) c R e t u r n aH20 and i s v a l u e s b y COMMON / R E S U L T / c R e t u r n gam() t o f u n c s u b r o u t i n e r e t u r n e n d c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c F u n c t i o n g ( x ) d o u b l e p r e c i s i o n f u n c t i o n g ( x ) d o u b l e p r e c i s i o n x i f ( x . e q . 0 ) t h e n g = 0 e l s e g = 2 * ( l - ( l + x ) * d e x p ( - x ) ) / ( x * * 2 ) e n d i f e n d c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c F u n c t i o n g p ( x ) d o u b l e p r e c i s i o n f u n c t i o n g p ( x ) d o u b l e p r e c i s i o n x i f ( x . e q . 0 ) t h e n gp = 0 e l s e gp = - 2 * ( 1 - ( l + x + ( x * * 2 ) / 2 . ) * d e x p ( - x ) ) / ( x * * 2 ) e n d i f e n d Q ****************************************************** c n e w t , l u d e m p , f d j a c , l u b k s b , l n s r c h c : s u b r o u t i n e s i m p o r t e d f r o m " N u m e r i c a l R e c i p e s F o r t r a n 7 7 " , c v e r s i o n 2 . 0 7 t o c a l c u l a t e n o n - l i n e a r e q u a t i o n s 2 1 9 c SUBROUTINE n e w t ( x , n , c h e c k ) PARAMETER ( N P = 4 0 , M A X I T S = 2 0 0 , T O L F = l . e - 2 , T O L M I N = l . e - 6 , * T O L X = l . e - 7 , S T P M X = 1 0 0 . ) c PARAMETER ( N P = 4 0 , M A X I T S = 2 0 0 , T O L F = l . e - 1 0 , T O L M I N = l . e - 6 , c * T O L X = l . e - 7 , S T P M X = 1 0 0 . ) END SUBROUTINE l u d e m p ( a , n , n p , i n d x , d ) c PARAMETER (NMAX=500 ,T INY=1 .Od-20 ) PARAMETER (NMAX=500, T INY=1 . 0d-10 : ) r e t u r n END SUBROUTINE f d j a c ( n , x , f v e c , n p , d f ) r e t u r n END SUBROUTINE l u b k s b ( a , n , n p , i n d x , b ) r e t u r n END SUBROUTINE l n s r c h ( n , x o l d , f o l d , g , p , x , f , s t p m a x , c h e c k , f u n c ) END c f m i n ( x ) c : f u n c t i o n i m p o r t e d f r o m " N u m e r i c a l R e c i p e s F o r t r a n 7 7 " , c v e r s i o n 2 . 0 7 FUNCTION f m i n ( x ) r e t u r n END c H y e o n P a r k , 1998 

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