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Simultaneous heat and mass transfer in wet wood particles Edwards, Wayne Clifford 1977

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SIMULTANEOUS HEAT AND MASS TRANSFER IN WET WOOD PARTICLES by WAYNE CLIFFORD EDWARDS B. Ap. S c . , U n i v e r s i t y o f B r i t i s h Columbia, 1975 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE In the Department o f MECHANICAL ENGINEERING We accept t h i s t he s i s as conforming to the requ i red standard THE UNIVERSITY OF BRITISH COLUMBIA March, 1977 © Wayne C l i f f o r d Edwards, 1977 In p re sen t i ng t h i s t he s i s i n p a r t i a l f u l f i l l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Columbia, I agree tha t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r re ference and study. I f u r t h e r agree tha t permiss ion f o r ex tens i ve copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood tha t p u b l i c a t i o n , i n pa r t o r i n whole, or the copying o f t h i s t he s i s f o r f i n a n c i a l gain s h a l l not be a l lowed w i thout my w r i t t e n permi s s ion . WAYNE CLIFFORD EDWARDS Department o f Mechanical Eng ineer ing The U n i v e r s i t y o f B r i t i s h Columbia, Vancouver, Canada V6T 1W5 i i ABSTRACT A study i s made o f the s imultaneous heat and mass t r a n s f e r processes which occur w i t h i n a f i n i t e wood c y l i n d e r o f c i r c u l a r c r o s s -s e c t i o n when i t i s c o n v e c t i v e l y d r i e d . Governing t r an spo r t equat ions are developed a l l o w i n g f o r the f u n c t i o n a l dependence o f both thermal and moisture d i f f u s i v i t i e s on moisture content and temperature. D e r i v a t i v e type boundary cond i t i on s are i n c l uded i n the a n a l y s i s . The equat ions are formulated us ing a c y l i n d r i c a l coo rd ina te system because i t i s we l l s u i t e d to mode l l i ng wood's an i so t ropy . In the case cons ide red , the a x i a l coord inate d i r e c t i o n i s a l i g ned w i t h the wood-grain. Due to the coup l ing and n o n - l i n e a r i t y present i n the t r an spo r t and boundary equa t i on s , an i m p l i c i t f i n i t e - d i f f e r e n c e s o l u t i o n scheme i s fo rmu la ted. The t h r e e - t i m e - l e v e l scheme uses an equat ion s p l i t t i n g technique to s i m p l i f y i t s s o l u t i o n on the computer. A mathematical wood-model, as a v a i l a b l e i n the l i t e r a t u r e , i s r e f i n e d and used to determine moisture and thermal d i f f u s i v i t i e s , and mass t r a n s f e r boundary c o n d i t i o n s . Resu l t s from t h i s wood-model apply to softwoods below the f i b e r - s a t u r a t i o n moisture content . Com-bined d i f f u s i v i t i e s o f l i q u i d and vapour are c a l c u l a t e d f o r the r a d i a l and a x i a l d i r e c t i o n s and r e s u l t s f o r the r a d i a l d i r e c t i o n are compared to those found from d i f f u s i o n exper iments. Desorpt iona l isotherms are used i n the mass t r a n s f e r boundary c o n d i t i o n equat ions to r e l a t e sur face humidity and moisture content. i i i The wood-model i s used i n the mass t r a n s f e r equat ion to de te r -mine wood d ry ing behaviour under i sothermal c o n d i t i o n s . I n i t i a l moisture contents are uniform and equal to the f i b e r - s a t u r a t i o n va lue . So l u t i on s are presented to g ive l o c a l and average moisture content as a f unc t i on o f t ime as we l l as moisture content p r o f i l e s . The e f f e c t o f d r y i ng temperature was i n v e s t i g a t e d f o r one case o f wood dens i t y and shr inkage. IV ACKNOWLEDGEMENT The author would l i k e to express h i s s i n ce re g r a t i t u d e to Dr. Adams f o r h i s encouragement and adv ice throughout t h i s work and f o r h i s guidance on the approach to s c i e n t i f i c re search. The a s s i s t ance o f Dr. Hauptmann dur ing the f i n a l p repa ra t i on o f the t he s i s i s g r e a t l y app rec i a ted . Thanks a l s o goes to the Nat iona l Research Counci l who s upp l i ed the f i n a n c i a l support which made t h i s t he s i s work p o s s i b l e . Most i m p o r t a n t l y , the author would l i k e to thank h i s f am i l y and f r i e n d s f o r a l l the i n t e r e s t and good times which were an i n t e g r a l pa r t o f t h i s work. V TABLE OF CONTENTS Chapter Page ABSTRACT i i ACKNOWLEDGEMENT i v LIST OF TABLES i x LIST OF FIGURES x I. INTRODUCTION 1 I I . BACKGROUND AND PREVIOUS WORK 5 A. Drying o f Porous S o l i d s 5 1. Per iods o f d ry ing 5 2. Mass t r a n s f e r mechanisms 5 3. Heat t r a n s f e r mechanisms 10 4. Constant ra te pe r i od 10 5. F a l l i n g ra te pe r i od 12 6. Simultaneous mass and heat t r a n s f e r 15 B. Wood S t r u c tu re 16 I I I . DEVELOPMENT OF MASS AND HEAT TRANSFER EQUATIONS 19 A. Problem D e f i n i t i o n 19 B. Mass T ran s fe r Ana l y s i s 19 1. Mass t r a n s f e r equat ion 21 2. Mass t r a n s f e r boundary c ond i t i o n s 23 C. Heat T ran s fe r Ana l y s i s 27 1. Heat t r a n s f e r equat ion 29 2. Heat t r a n s f e r boundary cond i t i on s 31 v i Chapter Page IV. WOOD MODEL 34 A. Sh r i nk i ng and S w e l l i n g o f Wood 34 B. Moisture D i f f u s i v i t i e s 37 1. S i m p l i f i e d wood s t r u c t u r e 3 7 2. D i f f u s i o n network f o r movement o f l i q u i d s and vapours i n the s i m p l i f i e d wood s t r u c t u r e . . 42 3. D i f f u s i v i t y o f l i q u i d i n wood substance . . . . 45 4. D i f f u s i v i t y o f water vapour 49 5. Pathway d i f f u s i v i t i e s f o r d i f f u s i o n network . . 52 6. D i f f u s i v i t y Re la t i on s Used i n the Mass T rans fe r Equation 53 C. Thermal Conduc t i v i t y and S p e c i f i c Heat 60 V. NUMERICAL SOLUTION OF MASS AND HEAT TRANSFER EQUATIONS . . . 62 A. Mass T ran s fe r 63 B. Mass T rans fe r Boundary Cond i t i ons 68 1. I n te rna l boundary 69 2. Externa l boundary 69 C. Heat T ran s fe r 71 D. Heat T ran s fe r Boundary Cond i t ions 74 1. I n te rna l boundary 74 2. Externa l boundary 75 E. So l vab le Form of Mass and Heat T ran s fe r Equations 75 v i i Chapter Page F. C a l c u l a t i o n o f Drying Rate and Average Moisture Content 7 6 G. Computer Program to Solve the Mass T rans fe r Equat ion 7 8 VI. DISCUSSION OF RESULTS 80 A. Mois ture D i f f u s i v i t i e s P r ed i c t ed by the Wood Model 80 B. Numerical S o l u t i o n Method 83 C. Combination o f Wood Model and Mass T rans fe r Equat ion 88 1. E f f e c t o f boundary cond i t i on s • • • ^ 2. Wood d ry ing . 89 D. Heat T rans fe r 97 E. Scope o f Po s s i b l e Future Work 98 VI I . CONCLUSIONS 99 LIST OF REFERENCES "101 APPENDIX A - DEVELOPMENT OF MASS TRANSFER EQUATION 104 APPENDIX B - DEVELOPMENT OF HEAT TRANSFER EQUATION 108 APPENDIX C - WOOD MODEL 110 1. Re l a t i on Between D^  and D g 110 2. Re l a t i on Between D^ and D y I l l 3. Curve F i t f o r Desorpt iona l Isotherms o f S i t k a Spruce 113 v i i i Chapter Page 4. Path D i f f u s i v i t i e s 117 APPENDIX D - NUMERICAL SOLUTION OF MASS TRANSFER EQUATION . . . . 122 APPENDIX E - NUMERICAL SOLUTION OF HEAT TRANSFER EQUATION . . . . 137 APPENDIX F - CONVECTIVE MASS TRANSFER COEFFICIENT 138 APPENDIX G - COMPUTER PROGRAM OPERATION PROCEDURE 141 APPENDIX H - COMPUTER PROGRAM LISTING 144 APPENDIX I - U.B.C. USAGE DOCUMENT FOR TRISLV 162 NOMENCLATURE 164 IX LIST OF TABLES Table Page I. P ropor t i ons o f Wood S t ruc tu re s and Trache id Dimensions . . . 39 I I . S t r u c t u r a l Data f o r P i t s o f American Softwoods 41 I I I . D e f i n i t i o n s and S t r u c t u r a l Dimensions f o r the Wood Model 44 Appendix Table C-I. V a r i a t i o n o f E q u i l i b r i u m Humidity w i th Temperature 115 C- I I . F i be r S a tu r a t i on Po in t as a Funct ion o f Temperature . . . . 116 G-I. Computer Input Va r i ab l e s 143 X LIST OF FIGURES F igure Page 1. Drying Rate as a Funct ion o f Time 2 2. Average Moisture Content as a Funct ion o f Time 6 3. Average Temperature as a Funct ion o f Time 6 4. Magn i f ied Three Dimensional Sketch of a Softwood 17 5. Body Coord inate System and Dimensions 20 6. I n te rna l and Externa l Mass T ran s fe r Processes 20 7. Desorpt ion "Moisture Content - R e l a t i v e Vapour Pressure Curves f o r S i t k a Spruce at D i f f e r e n t Temperatures 25 8. I n te rna l and Externa l Heat T ran s fe r Processes 28 9. Vo lumetr i c Shr inkage of L o b o l l y P ine o f D i f f e r e n t Den s i t i e s 36 10. Magn i f ied Three Dimensional Sketch o f Softwood 38 11. Schematic o f T rache id 40 12. Schematic o f P i t - P a i r 40 13. S i m p l i f i e d Wood S t r uc tu re Used i n Wood Model 43 14. Moisture D i f f u s i o n Network f o r Wood Model 43 15. A x i a l Bound Water D i f f u s i v i t y as a Funct ion o f E f f e c t i v e Moisture Content f o r S i t k a Spruce at 25 ? C . . . . 46 16. Desorptional. Isotherms f o r S i t k a Spruce at D i f f e r e n t Temperatures • 51 17. Radial D i f f u s i v i t y as a Funct ion o f Moisture Content at D i f f e r e n t Temperatures 55 x i F igure Page 18. Radia l D i f f u s i v i t y as a Funct ion o f Moisture Content a t D i f f e r e n t Dry-Wood Dens i t i e s 56 19. A x i a l D i f f u s i v i t y as a Funct ion o f Mois ture Content a t D i f f e r e n t Dry-Wood Dens i t i e s 57 20. F r a c t i on o f D i f f u s i o n Occur r ing Along P r i n c i p a l Pathways 58 21. Radia l D i f f u s i v i t y a t F i b e r - S a t u r a t i o n as a Funct ion o f Temperature 59 22. G r i d System f o r Numerical Method 64 23. Comparison o f Exact and Numerical S o l u t i on s 79 24. Comparison o f P r ed i c t ed and Experimental Radia l D i f f u s i v i t i e s f o r the E f f e c t o f Temperature 82 25. E f f e c t o f S p a t i a l Step S i z e on Numerical S o l u t i o n f o r Drying Curves 84 26. E f f e c t o f S p a t i a l Step S i z e on Numerical S o l u t i o n f o r Mois ture P r o f i l e s 85 27. O s c i l l a t o r y Numerical S o l u t i o n f o r Drying Curves 87 28. Comparison o f Drying Curves from Real and L i n e a r i z e d Boundary Cond i t ions 90 29. Center Mois ture Content As a Funct ion of Time For Softwood a t Temperatures o f 40°C and 60°C 92 30. Average Mois ture Content as a Funct ion o f Time f o r Softwood a t a Temperature o f 60°C 94 x i i Fi gure Page 31. Dimensionless Drying Rate as a Funct ion o f Time f o r Softwood at Temperatures o f 40°C and 60°C 95 32 Moisture P r o f i l e s f o r Softwood at a Temperature of 60°C 96 Appendix F igure C- l Slope o f Desorpt iona l Isotherms as a Funct ion o f Moisture Content f o r Temperatures o f 25°C and 100°C . . . . 118 1 I. INTRODUCTION Simultaneous heat and mass t r a n s f e r i n wet wood occurs when wood i s subjected to a d ry ing process. Ob jec t i ve s of a na l y s i s of such dry ing problems a re : (1) determinat ion o f d ry ing t imes , (2) determin -a t i o n o f d r y i ng r a t e s , and (3) f i n d i n g which v a r i a b l e s con t r o l the d r y i ng process. Drying problems have been i n v e s t i g a t e d f o r severa l decades, but on ly now i s a comprehensive understanding o f the d ry ing process being developed f o r porous ma te r i a l s such as softwoods. Dry ing problems were f i r s t analyzed t h e o r e t i c a l l y dur ing the 1920 ' s . Sherwood (24,25,26)* Newman (20) , T u t t l e (35) and many others were concerned w i th mass t r a n s f e r i n porous m a t e r i a l s . Ma te r i a l s commonly cons idered at t ha t time were wood, soap, and o ther manufactured products . S o l u t i on s to these d r y i ng problems were obta ined by assuming the movement o f moisture through porous ma te r i a l s was a d i f f u s i o n phenomenon and the moisture d i f f u s i v i t y was constant . Experiments were r equ i r ed to determine an e f f e c t i v e d i f f u s i o n co -e f f i c i e n t f o r each d ry ing problem cons ide red . Once i t was known and s u b s t i t u t e d i n t o the mass t r a n s f e r equat i on , d ry ing behaviour could be p r e d i c t e d . I t was dur ing t h i s i n i t i a l work t ha t the now c l a s s i c a l d ry -ing per iods were i d e n t i f i e d and attempts made to e xp l a i n t h e i r cause. The d r y i ng per iods are most r e a d i l y i d e n t i f i e d by the ra te at which Numbers i n parentheses r e f e r to l i t e r a t u r e c i t e d i n the l i s t o f r e fe rence s . 2 the body i s d r y i n g . I f the body i s i n i t i a l l y s a t u r a t e d , the f i r s t pe r i od w i l l be constant rate d r y i ng . This pe r i od occurs when d ry ing i s c o n t r o l l e d by ex te rna l mass t r a n s f e r and a l l evaporat ion takes p lace from a l i q u i d f i l m on the s u r f a ce . Drying cont inues a t a constant r a te u n t i l a c r i t i c a l moisture content i s reached. At t h i s t ime, a l i q u i d f i l m cannot be mainta ined on the sur face and the falling rate pe r i od begins. Drying dur ing t h i s pe r i od i s p r i n c i p a l l y c o n t r o l l e d by i n t e r n a l mass t r a n s f e r . A t y p i c a l d r y i n g - r a t e curve i s shown i n F igure 1. 0 1 0 TIME FIGURE 1 - DRYING RATE AS A FUNCTION OF TIME 3 During the d r y i ng p roces s , heat i s t r a n s f e r r e d from the surroundings to the d ry ing body. There fo re , a complete d ry ing model must con s ide r heat as we l l as mass t r a n s f e r . Equations f o r these s imultaneous t r an spo r t processes are coupled through the d i f f u s i o n co -e f f i c i e n t s and the i n t e r n a l evaporat ion term present i n the heat t r a n s f e r equat ion . Simultaneous t r an spo r t equat ions w i th constant moisture and thermal d i f f u s i v i t i e s have been so lved numer i ca l l y by Berger and Pei (2) and Harmathy ( 9 ) . Although the s o l u t i o n s obta ined are f o r the complete d r y i ng model they s t i l l r e q u i r e , as d i d the prev ious s imp le r models, knowledge o f e f f e c t i v e d i f f u s i v i t i e s . Understanding o f the fundamental mechanisms a f f e c t i n g moisture movement dur ing the constant ra te pe r i od i s good and adequate c a l c u l -a t i on s o f d r y i ng ra te can be made. Because the d ry ing ra te i s e x t e r n a l l y c o n t r o l l e d dur ing t h i s p e r i o d , i t can be c a l c u l a t e d by equat ing the energy requ i red f o r su r face evaporat ion w i th the heat t r a n s f e r r e d from the surroundings to the s u r f ace . Standard methods o f c a l c u l a t i o n are a v a i l a b l e i n Reference (23). The f a l l i n g r a te pe r i od i s the most time consuming o f the two d r y i ng per iods and t he re fo re i s very important i n the p r e d i c t i o n o f o v e r - a l l ' d r y i ng t imes. P r e s e n t l y , d ry ing behaviour must be determined us ing the governing t r an spo r t equat ions w i th e f f e c t i v e moisture and thermal d i f f u s i v i t i e s . The shortage o f exper imental e f f e c t i v e d i f f u s i v i t i e s makes a n a l y s i s o f the f a l l i n g r a te pe r iod u n s a t i s -f a c t o r y . 4 Wood i s a porous, a n i s o t r o p i c and hygroscopic m a t e r i a l . I t s heat and mass t r a n s f e r d i f f u s i v i t i e s are f unc t i on s o f both moisture content and temperature. Use o f e f f e c t i v e d i f f u s i v i t i e s , as i n constant c o e f f i c i e n t a n a l y s i s , leads to s o l u t i o n s o f wood d r y i ng problems which are l i m i t e d by t h e i r dependence on exper imental i npu t s . A l a r ge amount o f research has been performed by Stamm (28) , MacLean (17 ) , and others to p r e d i c t wood p r o p e r t i e s . Stamm's work i n the 1930 ' s , and i t s l a t e r re f i nement s , have produced a mathematical model that can be used to p r e d i c t the d i f f u s i v i t i e s of l i q u i d s and vapours i n softwood. Research by MacLean has produced emp i r i c a l thermal c o n d u c t i v i t y r e l a t i o n s h i p s f o r softwoods. Resu l t s o f t h i s past research can be combined i n t o one complete wood model which then can be app l i ed i n the t r an spo r t equat ions . The o b j e c t i v e s o f the work conta ined i n t h i s t he s i s were to r e f i n e the e x i s t i n g wood model, to develop the heat and mass t r a n s f e r equat ions r e l e v a n t to convect ive d r y i ng o f a f i n i t e c i r c u l a r wood c y l i n d e r , to formulate a numerical method to so lve the s imultaneous t r an spo r t equat ions and f i n a l l y , to s u b s t i t u t e moisture d i f f u s i v i t i e s p r e d i c t e d by the wood model i n t o a computer ized s o l u t i o n o f the mass t r a n s f e r equat ion and ob ta in wood d ry i ng behaviour f o r s e l e c t i sothermal d r y i ng c o n d i t i o n s . 5 I I . BACKGROUND AND PREVIOUS WORK A. Drying o f Porous S o l i d s 1. Per iods o f d ry ing Drying i nvo l ve s the removal of water o r some o the r adsorbed f l u i d from a wet porous s o l i d . In the m a j o r i t y o f d ry ing s i t u a t i o n s the body i s surrounded by a heated f l u i d o f low moisture content. The body i s heated by i t s sur round ings , e i t h e r by convect ion o r r a d i a t i o n , o r both, and moisture i s removed i n the vapour phase by convec t i on . Typ i ca l moisture and temperature curves f o r a s o l i d subjected to convect i ve d r y i ng by a hot gas are shown i n F igures 2 and 3, r e s p e c t i v e l y . 2. Mass t r a n s f e r mechanisms The movement o f moisture through porous s o l i d s i s a complex phenomenon. Past research has r e s u l t e d i n i d e n t i f i c a t i o n o f severa l d r i v i n g p o t e n t i a l s which can cause i n t e r n a l mass t r a n s f e r . For a p a r t i c u l a r d r y i ng problem, moisture con ten t , temperature and cha rac te r o f the s o l i d ' s p o r o s i t y determine which p o t e n t i a l s are most important. In c u r r en t l i t e r a t u r e , porous s o l i d s are i d e a l l y model led e i t h e r as rows o f long cont inuous c a p i l l a r i e s ( c a p i l l a r y porous ) , o r as an assemblage o f non-communicating c a v i t i e s , or a combinat ion o f these. D i f f e r e n t mechanisms o f moisture movement are important f o r each case. ft TIME ' ' FIGURE 3 - AVERAGE TEMPERATURE AS A FUNCTION OF TIME 7 There are f ou r i n t e r n a l mass t r a n s f e r d r i v i n g p o t e n t i a l s which may be important at d i f f e r e n t stages i n the d r y i ng process. These a r e : ( i ) Concent rat ion g rad ien t Mass t r a n s f e r due to a concen t ra t i on g rad ient i s analagous to heat t r a n s f e r due to a temperature g rad ien t . The mass f l u x i s c a l -c u l a t e d us ing F i c k ' s f i r s t law o f d i f f u s i o n , m = mass f l u x (g/cm - s e c ) , 2 D = d i f f u s i o n c o e f f i c i e n t (cm / s e c ) , 4^- = c oncen t ra t i on g rad ien t i n the x d i r e c t i o n (g/cm ). • X F i c k ' s r e l a t i o n was used by Sherwood (25,26) and others (20, 35) i n t h e i r e a r l y a na l y s i s o f d r y i ng problems and has subsequently been used e x t e n s i v e l y . Comparison o f p r e d i c t i o n s w i th experiment has been good and t h i s has l e d to general acceptance of concent ra t i on g rad ient d i f f u s i o n as an important mechanism o f moisture movement. Use o f the equat ion requ i re s knowledge o f both l i q u i d and vapour d i f f u s i v i t i e s . ( i i ) Temperature g rad ien t Mass t r a n s f e r as a r e s u l t o f a temperature g rad ien t was f i r s t d i s covered i n 1934 by Lu ikov (14) and now i s r e f e r r e d to as moisture 8 thermal d i f f u s i o n . The po s tu l a ted r e l a t i o n f o r mass f l u x i s mass f l u x (g/cm -sec) , thermal d i f f u s i o n c o e f f i c i e n t (°C~^) , p d i f f u s i o n c o e f f i c i e n t (cm /sec) , temperature g rad ien t i n the x d i r e c t i o n (°C/cm) . S o v i e t s c i e n t i s t s (14) have i n v e s t i g a t e d moisture thermal d i f f u s i o n as i t a p p l i e s to d r y i ng problems and found tha t i t was not important a t low temperatures. In the case o f wood d r y i n g , i t can be neg lected f o r wood temperatures l e s s than 100°C. ( i i i ) C a p i l l a r y p o t e n t i a l C a p i l l a r y p o t e n t i a l i s important f o r c a p i l l a r y porous s o l i d s near s a t u r a t i o n . For these s o l i d s , l i q u i d e x i s t s i n the l a r g e r c a p i l l a r i e s and can migrate along the c a p i l l a r y i f su r face tens ion fo rces cause a s u f f i c i e n t net l i q u i d pressure g rad ien t . C a p i l l a r y f low i s a p r i n c i p a l mass t r a n s f e r mechanism dur ing the constant ra te pe r i od . Movement o f moisture by t h i s mechanism can cont inue u n t i l the c a p i l l a r -i e s are empty o f l i q u i d . A thorough i n v e s t i g a t i o n o f t h i s type o f mass t r a n s f e r has been made by Lu ikov (15) . Mathematical treatment i s analogous to t h a t f o r concen t ra t i on g rad ien t d i f f u s i o n . m = where m = & = D = J I 3x 9 ( i v ) Lebedev (14) has found by experiment tha t mass t r a n s f e r can occur i n porous s o l i d s because o f a g rad ien t i n t o t a l p res sure. During h i s wood d ry ing t e s t s , t o t a l i n t e r n a l gas pressure was measured at severa l po in t s throughout the sample. S i g n i f i c a n t vapour movement from the wood i n t e r i o r to the per iphery caused by an advantageous pressure g r ad i en t occurred when the wood temperature was g rea te r than 100°C. The mass f l u x r e l a t i o n po s tu l a ted f o r t h i s mechanism i s m = " D p 9x" • where D„ = c o e f f i c i e n t o f molar f low (sec) , p v ^— = pressure g rad ien t i n the x d i r e c t i o n (dyne/cm ) . oX For a l l the above i n t e r n a l mass t r a n s f e r mechanisms, mass t r a n s f e r from the su r face to the surrounding f l u i d i s cons idered to be analogous to convect i ve heat t r a n s f e r . The r e l a t i o n s h i p used i s where m = K G ( H S - H a ) , m = mass f l u x (g/cm -sec) , K n = su r face convect i ve mass t r a n s f e r c o e f f i c i e n t (g/cm -sec) , H„»H, = su r face and ambient r e l a t i v e humid i ty , r e s p e c t i v e l y , s a 3. Heat t r a n s f e r mechanisms The two p r i n c i p a l mechanisms o f heat t r a n s f e r f o r porous ma te r i a l s are conduct ion and convec t i on . Conductive heat t r a n s f e r i s the on ly one commonly assumed f o r the i n t e r i o r o f the body, wh i l e con-v e c t i v e t r a n s f e r i s assumed a t the body su r f ace . Convect ive heat t r a n s f e r w i t h i n a c a p i l l a r y porous s o l i d has been analyzed by Dyer and Sunderland ( 6 ) . They cons idered steady s t a t e s ub l im ina l d r y i ng o f a c a p i l l a r y porous s o l i d c on ta i n i n g f rozen l i q u i d when both i n t e r n a l conduct ion and convect ion occur . 4. Constant ra te pe r i od Constant r a te d r y i ng has been cons idered e x t e n s i v e l y i n the l i t e r a t u r e (7 ,8 ,23) . Mass t r a n s f e r dur ing t h i s f i r s t d ry ing pe r iod i s adequately understood f o r p r a c t i c a l d ry ing problems. The accepted method o f c a l c u l a t i o n w i l l be g iven here, but more complete d i s cu s s i on can be obta ined i n the p r e v i o u s l y c i t e d l i t e r a t u r e . During constant ra te d r y i n g , i n t e r n a l r e s i s t a n c e to mass t r a n s f e r i s small compared to ex te rna l r e s i s t a n c e , consequent ly d r y i ng ra te i s c o n t r o l l e d by ex te rna l v a r i a b l e s : su r face mass t r a n s f e r c o e f f i c i e n t and ambient humid i ty . Mois ture moves from the body i n t e r i o r to the su r face because o f c a p i l l a r y f low and concen t ra t i on g rad ient d i f f u s i o n . As the f l u i d i s removed from the s u r f a ce , the su r face l i q u i d r e t r e a t s s l i g h t l y i n t o the c a p i l l a r i e s and forms concave s u r f ace s . Sur face tens ion and the nea r - su r face concent ra t i on g rad ien t induce s u f f i c i e n t moisture move-11 ment to keep the su r face covered w i th a t h i n l i q u i d f i l m . Other mass t r a n s f e r mechanisms make n e g l i g i b l e c o n t r i b u t i o n s to t o t a l mass f l u x dur ing t h i s p e r i o d . At steady s t a t e , the heat i npu t balances the heat requ i red f o r su r face evaporat ion and from a balance o f these energy terms, the d ry ing ra te i s determined. Al lowance must be made f o r energy requ i red to heat the vapour as i t passes through the thermal boundary l a y e r . The e f f e c t o f t h i s heat - s i nk on ac tua l heat t r a n s f e r to the s o l i d has been i n v e s t i g a t e d by Spa ld ing (p. 72 o f (12)) and severa l o t h e r , researchers (21,22). Spa ld ing recommends a c o r r e c t i o n f a c t o r that would ad ju s t the heat t r a n s f e r c o e f f i c i e n t f o r a dry body to tha t f o r the same body w i th su r face mass f l u x . Assuming t h i s c o r r e c t i o n and equat ing a l l s u r face heat t r a n s f e r terms, the r e l a t i o n s f o r su r face temperature and d ry i ng ra te are 0 K r T s = T a " T T < H s " H a> and where ft = A K G ( H S - H A ) , o = l a t e n t heat o f evaporat ion (ca l/g) , Kg = convect i ve mass t r a n s f e r c o e f f i c i e n t (g/cm -sec) , h = ac tua l convect i ve heat t r a n s f e r c o e f f i c i e n t f o r 2 the s o l i d su r face (cal/cm -sec-°C) , T^.T, = su r face and ambient temperature, r e s p e c t i v e l y ( °C ) , s a H..I-L = su r face and ambient r e l a t i v e humid i ty , r e s p e c t i v e l y , S a 12 A = su r face area (cm11) , M = d r y i ng ra te (g/sec) . The equat ion f o r su r face temperature has to be so lved by i t e r a -t i o n . The su r face temperature w i l l be the wet bulb temperature f o r pure convect i ve d ry ing o r above t h i s i f o ther heat sources, such as r a d i a t i o n , are p resent . 5. F a l l i n g r a te pe r i od I d e a l l y , the moisture content dur ing the constant ra te pe r i od i s uniform throughout the body and moisture content decreases w i th time u n t i l a c r i t i c a l moisture content i s reached. The c r i t i c a l moisture content depends on d ry ing c ond i t i o n s and mate r i a l p r ope r t i e s and i s t he re fo re very d i f f i c u l t to p r e d i c t a n a l y t i c a l l y . Est imates o f c r i t i c a l moisture contents f o r convect i ve d ry ing o f va r ious ma te r i a l s are g iven i n Reference (23). As the m a t e r i a l ' s moisture content approaches the c r i t i c a l v a l ue , r e s i s t a n c e to i n t e r n a l mass t r a n s f e r becomes i n c r e a s i n g l y important . Mass t r a n s f e r r a t e depends on the s e r i e s r e s i s t ance s o f i n t e r n a l and ex te rna l t r a n s f e r . Because each term a f f e c t s the d ry ing r a t e , the t r a n s i t i o n from constant to f a l l i n g r a te d r y i ng i s gradual r a the r than abrupt. As su r face moisture approaches i t s e q u i l i b r i u m va l ue , the d r y i ng r a t e becomes complete ly c o n t r o l l e d by i n t e r n a l t r a n s f e r . 13 I n te rna l mass t r a n s f e r can be caused by any o f the four means p r e v i ou s l y mentioned; concent ra t i on g r ad i en t , c a p i l l a r y p o t e n t i a l , temperature g r ad i en t and pressure g rad ien t d i f f u s i o n . The types r e c e i v i n g most a t t e n t i o n i n the l i t e r a t u r e are concent ra t i on g rad ient d i f f u s i o n and c a p i l l a r y f l ow. Other types o f mass t r a n s f e r are impor-tan t i f temperature above the f l u i d ' s b o i l i n g po i n t are i n v o l v e d . C a p i l l a r y f low requ i re s the ex i s t ence o f l i q u i d i n the c a p i l l a r i e s and i s t he re fo re more a s soc i a ted w i t h the constant ra te than the f a l l i n g r a te pe r i od . was made by Sherwood and Newman. This d i f f u s i o n phenomenon i s analogous to heat conduct ion i n a s o l i d . Using t h i s analogy, a n a l y t i c a l s o l u t i o n s to the governing mass t r a n s f e r equat ion are r e a d i l y a v a i l a b l e i n works on heat t r a n s f e r . The necessary assumptions are constant moisture d i f f u s i v i t y and constant p r o p o r t i o n a l i t y between sur face mois-ture content and humid i ty . As an example, the s o l u t i o n f o r average moisture content o f an i n f i n i t e s l a b , c onvec t i v e l y d r i e d on both su r -faces from a uni form i n i t i a l moisture content , as adapted from r e f e r -ence (15) , i s shown below. Ana l y s i s o f concent ra t i on g rad ient d i f f u s i o n i n porous s o l i d s (1) where average, i n i t i a l and ambient moisture content , D r e s p e c t i v e l y , (mass water/mass dry wood), p moisture d i f f u s i v i t y , (cm /s) , 14 L = s l ab h a l f th i cknes s (cm) . The c o e f f i c i e n t y n i s obta ined from the s o l u t i o n o f the t ranscendenta l equa t i on , y n COt y " " n B i m B n i s determined from the r e l a t i o n s h i p , 2 B i 2 m n m m n Both i n vo l ve the B i o t mass number, B i ' m which i s g iven by the equa t i on , K„L const B i . - G where Dp D c f su r face moisture content su r face r e l a t i v e humidity ' Kg = convect i ve mass t r a n s f e r c o e f f i c i e n t (g/cm - s e c ) , P D = dens i t y o f dry s o l i d (g/cm ) . S o l u t i on s f o r s imple f i n i t e bodies such as b locks and f i n i t e c y l i n d e r s can be obta ined by t a k i n g products o f s o l u t i o n s f o r r e l a t e d i n f i n i t e bod ies . De ta i l ed a n a l y s i s o f these types o f problems are g iven by Luikov (15) . He a l s o t r e a t s cases where i n i t i a l moisture d i s t r i b u t i o n s are p a r a b o l i c . Drying ra te s can be determined by d i f f e r e n t i a t i n g the r e l a t i o n f o r average moisture content w i th re spect to t ime. 15 The d i f f i c u l t y i n app l y i ng constant c o e f f i c i e n t case a n a l y s i s to a c tua l d r y i ng problems i s t ha t actua l d i f f u s i o n c o e f f i c i e n t s may vary w i t h moisture content and temperature. E f f e c t i v e d i f f u s i o n c o e f f i c -i en t s f o r mass t r a n s f e r are determined expe r imen ta l l y then used i n the constant c o e f f i c i e n t case a n a l y s i s . The d i f f i c u l t y w i th t h i s approach i s the dependence on p r e l im i na r y exper imental r e s u l t s . An i n s u f f i c i e n t volume o f exper imenta l r e s u l t s are c u r r e n t l y a v a i l a b l e to dependably determine the e f f e c t i v e c o e f f i c i e n t which app l i e s to a p a r t i c u l a r d ry ing problem. 6. Simultaneous mass and heat t r a n s f e r Determinat ion o f the thermal behaviour o f a d r y i ng body r e -qu i re s s o l u t i o n o f both the heat and mass t r a n s f e r equat ions . The mass t r a n s f e r equat ion i s r equ i red to i n c l ude the hea t - s i nk caused by i n t e r n a l evapora t i on . Because the two equat ions are coup led, s o l u t i o n s are obta ined by numerical procedures. Berger and Pei (2) so lved the s imultaneous t r an spo r t equat ions f o r an i s o t r o p i c a l l y porous i n f i n i t e s l a b . They assumed c a p i l l a r y f low o f l i q u i d and concent ra t i on g rad ien t d i f f u s i o n o f vapour. L i q u i d and vapour concent ra t i ons were coupled us ing a general numerical approx imat ion o f s o r p t i o n a l i sotherms. Moisture and thermal d i f f u s -i v i t i e s were taken to be constant over the f u l l d r y i ng process. S o l u t i on s i n d i c a t e the e f f e c t o f changes i n B i o t heat and mass numbers and Lu ikov and Kossovich numbers on d ry ing ra te f o r both the constant and f a l l i n g r a te pe r i od s . No experimenta' l r e s u l t s were a v a i l a b l e tha t cou ld be compared to t h e i r p r e d i c t i o n s . 16 Harmathy ( 9 ) cons idered d i f f u s i o n a l heat and mass t r a n s f e r i n an i s o t r o p i c a l l y porous i n f i n i t e s l a b . He assumed a l l mass t r a n s f e r occurred i n the vapour phase and tha t moisture and thermal d i f f u s i v i t i e s were cons tant . Input constants were obta ined e i t h e r from handbooks o r exper iments. Theo re t i c a l and exper imental moisture and temperature curves f o r the f a l l i n g ra te d r y i ng o f c l a y b r i c k were compared. Agree-ment between theory and experiment was good f o r mo i s tu re , but poor f o r temperature. B. Wood S t r u c t u r e Wood i s a complex o rgan i c mate r i a l comprised o f many s p e c i a l -i z ed s t r u c t u r e s necessary f o r i t s growth. A sketch o f a softwood ( con i fe rous spec ie s ) i s shown i n F igure 4 a long w i t h the three p r i n -c i p a l s t r u c t u r a l dimensions. The p r i n c i p a l s t r u c t u r e s are t r a c h e i d s , r ay s , parenchyma, r e s i n canals and p i t s . Tracheids extend i n the a x i a l d i r e c t i o n o f a t ree and are long l i n e a r c e l l s w i th a length as much as 75 times i t s d iameter; they taper to a b l un t t i p when viewed r a d i a l l y and to a sharp t i p when viewed t a n g e n t i a l l y . Rays are r i b b o n - l i k e arrangements o f c e l l s p r i n c i p a l l y o r i e n t ed i n the r a d i a l d i r e c t i o n . Resin cana l s are l a r ge t u b u l a r spacings surrounded by n u t r i e n t storage c e l l s and ac t to t r an spo r t n u t r i e n t s i n both the a x i a l and r a d i a l d i r e c t i o n s . P i t s are t h i n areas o f the t r a c h e i d w a l l s which a l l ow f o r i n t e r - t r a c h e i d passage of n u t r i e n t s . Parenchyma are c e l l s used f o r n u t r i e n t storage and they extend i n both the r a d i a l and a x i a l d i r e c t i o n s . 17 FIGURE 4 - MAGNIFIED THREE DIMENSIONAL SKETCH OF A SOFTWOOD: t t , a x i a l s u r f a c e ; t g , t angen t i a l s u r f a c e ; r r , r a d i a l s u r f a c e ; t r , t r a c h e i d s ; wr, wood ray ; c , c a v i t y ; p, p i t ; sp, sprinciwood; sm, summerwood; a r , annual r i n g ; r d , r e s i n duc t . 18 Thorough d i s cu s s i on o f each o f the s t r u c t u r e s mentioned above and other l e s s important ones are a v a i l a b l e i n the l i t e r a t u r e (4,18) . 19 I I I . DEVELOPMENT OF MASS AND HEAT TRANSFER EQUATIONS A. Problem D e f i n i t i o n The problem i s to develop heat and mass t r a n s f e r equat ions f o r d r y i ng o f the f i n i t e c i r c u l a r wood c y l i n d e r shown i n F igure 5. The body i s surrounded by a moving f l u i d o f known temperature, moisture content , v e l o c i t y and p r o p e r t i e s . Wood-grain d i r e c t i o n i s a l i g ned w i th the a x i a l d i r e c t i o n . B. Mass T ran s fe r Ana l y s i s I n te rna l and ex te rna l mass t r a n s f e r processes i nc luded i n the ana l y s i s are shown i n F igure 6. Assumptions used i n the equat ion development a re : ( i ) F i c k ' s f i r s t law o f d i f f u s i v e mass t r a n s f e r a p p l i e s . This equat ion can be app l i ed to a l l the mass t r a n s f e r mechanisms p r e v i o u s l y d i scussed (pp. 7-9 ) and has been used i n prev ious d ry ing a n a l y s i s , ( i i ) Mass t r a n s f e r due to temperature o r pressure g rad ient s i s n e g l i g i b l y s m a l l . These mass t r a n s f e r mechanisms have been i n v e s t i g a t e d by Lebedev (14) f o r wood and were found to make an i n s i g n i f i c a n t c o n t r i b u t i o n to t o t a l mass t r a n s f e r i f the body temperature was below 100°C. ( i i i ) One d i f f u s i v i t y c o e f f i c i e n t f o r combined movement o f both l i q u i d and vapour can be determined us ing the wood model. 21 ( i v ) No coup l i n g e x i s t s between mass t r a n s f e r i n each coo rd ina te d i r e c t i o n , (v) Mass t r a n s f e r from the su r face can be p r o p o r t i o n a l l y r e l a t e d to the d i f f e r e n c e between sur face and ambient vapour den s i t y us ing a su r face convect ion c o e f f i c i e n t . ( v i ) Wood shr inkage and s w e l l i n g can occur . 1. Mass t r a n s f e r equat ion To develop the mass t r a n s f e r equa t i on , the law o f conservat ion o f mass was a p p l i e d to an elemental volume o f the f i n i t e c y l i n d e r . Moisture movement was r e l a t e d to the l i q u i d g rad ien t through a combined d i f f u s i v i t y which i nc ludes the movement o f both l i q u i d and vapour phases. An order o f magnitude es t imate o f l i q u i d and vapour c o n t r i b u t i o n s to moisture capac i tance was made f o r the con t r o l volume. I t was found tha t the vapour c o n t r i b u t i o n was 10"^ times that o f the l i q u i d (pp. 105-106, Appendix A ) . Because o f t h i s , the change i n moisture content o f the elemental volume w i t h time was taken as the change i n l i q u i d content w i t h t ime. The e f f e c t o f wood shr inkage o r s w e l l i n g on the l i q u i d g rad ien t and on the moisture content i n the elemental volume were i n -c luded i n the a n a l y s i s . In dimensional form, the mass t r a n s f e r equat ion (Appendix A) i s l f r ( - * R < U . T ) f ) + f I < A z ( U . T ) f ) - B ( U ) f (2) where U = moisture content (mass water/mass dry wood), 22 T = temperature (°C) , t = time (sec) , r = r a d i a l coord inate (cm) , z = a x i a l coord inate (cm) , Ap(U,T) = d i f f u s i v i t y o f moisture i n the r a d i a l d i r e c t i o n A 7 (U,T) = d i f f u s i v i t y o f moisture i n the a x i a l d i r e c t i o n For the case o f constant d i f f u s i v i t y and no vo lumet r i c changes, t h i s equat ion reduces to the one so lved by Newman (20). Equat ion 2 was made d imens ion less to improve the s i z e o f the terms by us ing i n i t i a l values o f body dimensions and r a d i a l moisture d i f f u s i v i t y . The non-dimensional form o f the equat ion i s as a f u n c t i o n o f moisture content and tempera-ture (cm /sec) , B(U) as a f unc t i on o f moisture content and tempera-2 ture (cm /sec) , Volume o f dry element (cm3) Volume o f wet element (cm3) * ( r *A* (U,T) 3 r * 3 z * 3 (AJ(U.T) = B(U) 3 U (3) where t * m A R ( U , T ) . t r * r/R, 23 z * = z / L . , A * ( U , T ) = A R ( U , T ) / A R ( U , T ) . , A f ( U , T ) = ( ^ ) 2 A z ( U , T ) / A R ( U , T ) i . The i m p o r t a n t p a r a m e t e r a r i s i n g f r o m the two d i m e n s i o n a l i t y o f t he mass t r a n s f e r p r o c e s s i s ( R / L ) 2 ( A Z ( U , T ) / A R ( U , T ) . ) . 2. Mass t r a n s f e r bounda r y c o n d i t i o n s The g o v e r n i n g mass t r a n s f e r e q u a t i o n has i n t e r n a l and e x t e r n a l boundary c o n d i t i o n s i t must s a t i s f y . The e x t e r n a l boundary c o n d i t i o n f o r t he r a d i a l d i r e c t i o n i s o f s t a n d a r d fo rm (13) and i s g i v e n by t he d i m e n s i o n a l e q u a t i o n , where 311 p D VU'T) | ? + K G R ( P V p a ) = ° . r = R ; - L < z < L .(4) 3 P D = d e n s i t y o f d r y wood (g/cm ) , K g R = c o n v e c t i v e mass t r a n s f e r c o e f f i c i e n t f o r the r a d i a l s u r f a c e (cm/sec) , 3 Py = s u r f a c e w a t e r v a p o u r d e n s i t y (g/cm ) , 3 p g = amb ien t w a t e r v apou r d e n s i t y (g/cm ) . 24 Equation 4 i s not c o n s i s t e n t i n moisture content , the de s i r ed dependent v a r i a b l e . I t was necessary to use a r e l a t i o n , s p e c i f i c to the ma te r i a l being con s ide red , tha t would r e l a t e water vapour den s i t y to moisture content . Re l a t i on s which do t h i s are s o r p t i o n a l i sotherms] S ince the f i n a l o b j e c t i v e was to model wood d r y i n g , deso rp t iona l isotherms f o r S i t k a spruce were used. These are shown i n F igure 7. They g ive e q u i l i -brium r e l a t i o n s h i p s between l o c a l moisture content and l o c a l abso lute humidity f o r d i f f e r e n t wood temperatures. To use the de so rp t i ona l isotherms i t i s necessary to assume that the r a t e o f a t t a i n i n g e q u i l i b r i u m between sur face moisture and l o c a l humidity i s much g rea te r than the ra te o f moisture d i f f u s i o n . This assumption i s r equ i red s i nce non - equ i l i b r i um isotherms are not p r e sen t l y a v a i l a b l e . Knowing the r e l a t i o n between moisture content and humidity can be obta ined from the deso rp t i ona l i so therms, Equation 4 was r e -w r i t t e n as 8U x J KGR R p s a t fH,\ „ KGR R p a 8 r * j A R ( U , T ) p D ¥ / U - A R (U,T) P r j r * = 1 ; - 1 < z * < 1 . (5) The equat ion i s now i n d imens ionless form and c o n s i s t e n t i n the moisture content v a r i a b l e , U. L e t t i n g 1 So rp t i ona l isotherms r e f e r to isotherms f o r e i t h e r an adsorpt ion or deso rpt ion p roces s . 25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 J.O REL ATIV E;. VAPOUR PRESSURE, H) FIGURE 7 -C^SS^TlM^MOISTURE'CONTENT^ELATIVE VAPOUR PRESSURE' CURVES FOR SITKA SFRUCE AT DIFFERENT TEMPERATURES §32) 26 K G R R p s a t ,H and KGR R p a  A R " A R ( U , T ) p D ' the boundary c o n d i t i o n equat ion i s | ^ + B i m R ( U , T ) U = XR ; r * = 1 ; -1 <z*<l ; (6) where B i m R ( U , T ) = r a d i a l B i o t mass number as a f unc t i on o f U and T , X R = ambient c ond i t i o n app l y i ng to the r a d i a l su r face . Because o f the symmetry o f both the body and d ry i ng c o n d i t i o n s , i n t e r n a l boundary cond i t i on s have a s imple form. No mass t r a n s f e r w i l l occur across the c y l i n d e r ax i s o r across the r a d i a l plane s i t u a t e d . a t z * = 0. For the r a d i a l d i r e c t i o n , the i n t e r n a l boundary c o n d i t i o n i s g iven by the equa t i on , f p * = 0 ; r * = 0 ; -1 < z * < 1 (7) Boundary cond i t i on s f o r the a x i a l d i r e c t i o n are o f i d e n t i c a l form to those f o r the r a d i a l d i r e c t i o n . The ex te rna l and i n t e r n a l boundary c ond i t i o n s a r e , r e s p e c t i v e l y , 27 | ^ + B 1 m Z ( U , T ) U = X z (8) and Here 9U_ = 0 . (9) Bi m 2 = Biot mass number for the axia l d i rect ion K G Z L p s a t H and A Z(U,T) p D U - ambient condition for the axia l d i rect ion where KGZ L Pa A Z(U,T) p D Kg^ = convective mass transfer coe f f i c i en t for the axial surface (cm/sec) . C. Heat Transfer Analysis The heat transfer equation was developed by performing a heat balance on an elemental volume of the f i n i t e cy l inder. Heat transfer processes included in the analysis are shown in Figure 8. Assumptions used in the equation development are: ( i ) Internal heat transfer occurs by conduction and obeys Four ier ' s law. ( i i ) Thermal conduct iv i t ies are given by the wood model and are e f fec t i ve values which take porosity into account. a 28 HEAT-SINK) FROM INTERNAL SURFACE F I G U R E 8 - . I N T E R N A L AND E X T E R N A L H E A T T R A N S F E R P R O C E S S E S 29 ( i i i ) I n te rna l evaporat ion occurs and has a constant l a t e n t heat. ( i v ) There i s no coup l ing between heat t r a n s f e r i n each coord ina te d i r e c t i o n , (v) Wood shr inkage and s w e l l i n g can occur . ( v i ) Heat t r a n s f e r from the su r face can be p r o p o r t i o n a l l y r e l a t e d to the d i f f e r e n c e between sur face and ambient temperature us ing a su r face convect ion c o e f f i c i e n t . 1. Heat t r a n s f e r equat ion The heat t r a n s f e r equa t i on , i n dimensional form, (Appendix B) ^ ( ^ ( U . T j f l ) + | j ( K Z ( U , T ) | I ) v conduct ive heat t r a n s f e r - w {}IFW-T' f » W - T > f >- B<u> f} v heat - s i nk caused by i n t e r n a l evaporat ion = p D C(U,T)B(U) | I v (10) ra te of change of i n t e r n a l energy where K R (U,T) = thermal c o n d u c t i v i t y i n the r a d i a l d i r e c t i o n (cal/cm-sec-°C) , K Z (U,T) = thermal c o n d u c t i v i t y i n the a x i a l d i r e c t i o n (cal/cm-sec-°C) , 30 fp = f r a c t i o n o f r a d i a l d i f f u s i o n t ha t i s l i q u i d , fj = f r a c t i o n of a x i a l d i f f u s i o n tha t i s l i q u i d , a = l a t e n t heat o f evaporat ion (ca l/g) , C(U,T) = s p e c i f i c heat o f wet wood (ca l/g-°C) . Equat ion 10 was made d imens ion less by us ing i n i t i a l values o f K R (U ,T ) , A R (U,T) and C(U,T), as we l l as the de f ined non-dimensional temperature, V T T* = where and a ^ ambient i ^ i n i t i a l . The non-d imens iona l i zed form of Equat ion 10 i s ^ | ^ ( r * K * R ( U , T ) p + | ^ ( K Z ( U , T ) f £ ) + < K o i L u i » { ^ l ^ ( ' - * f R A R ( U ' T » | ? F ) ' • y ( f / z ( U , T ) | ^ ) - B ( U ) | ^ } m J = B(U)C*(U,T) | £ (11) 31 where K*(U,T) = K R (U ,T )/K R (U ,T ) . , K*(U,T) = ( R ) 2 K z ( U , T ) / K R ( U , T ) i , C*(U,T) = C(U,T)/C(U,T). Ko. = i n i t i a l Kossovich number , ( T a - T . ) C ( U , T ) . , Lu. = i n i t i a l Lu ikov number , = A R ( U , T ) i / a R ( U , T ) i , a D ( U , T ) . = i n i t i a l r a d i a l thermal d i f f u s i v i t y K 1 = K R (U,T) ./C(U,T) . p D , t * = a R ( U , T ) . t/R 2 . 2. Heat t r a n s f e r boundary c ond i t i o n s Equat ion 11, the governing heat t r a n s f e r equa t i on , must s a t i s f y ex te rna l and i n t e r n a l boundary c o n d i t i o n s . The ex te rna l boundary c o n d i t i o n i nc ludes a heat - s i nk term to account f o r l i q u i d to vapour phase-change a t the s u r f ace . For the r a d i a l d i r e c t i o n , the ex te rna l 32 boundary c o n d i t i o n i s where where K R (U ,T ) | I • h R ( T s - T a ) - p D O f R A R ( U , T ) f - 0 v. v v conduct ion convect ion su r face to su r face from evaporat ion su r face r* = R ; -L < z * < IL (12) h D = r a d i a l convect i ve heat t r a n s f e r c o e f f i c i e n t (cal/cm -sec-°C) . In non-dimensional form Equation 12 can be w r i t t e n as |J*r + B i R T * + K o L u L R | ^ = 0 ; r* = 1 ; -1 < z * < 1 ; (13) B i R = r a d i a l B i o t number = h R R/K R (U,T) , Ko = a / C ( U , T ) ( T a - T . ) , L u L R = f R A R ( U , T ) / a R ( U , T ) a R (U ,T ) = K R (U,T) /p D C(U,T ) , The i n t e r n a l boundary c o n d i t i o n f o r heat t r a n s f e r mathematic-a l l y s t a te s tha t temperature p r o f i l e s are symmetric about the c y l i n d e r a x i s and across the r a d i a l plane s i t u a t e d a t z * = 0. The i n t e r n a l boundary c o n d i t i o n f o r the r a d i a l d i r e c t i o n i s 33 |P = 0 ; r* = 0 ; -1 < z * < 1 (14) Ex te rna l and i n t e r n a l boundary equat ions were developed f o r the a x i a l d i r e c t i o n i n the same manner as shown f o r the r a d i a l d i r e c t i o n . These are J £ + B i z T * + K o l _ u L Z | ^ = o ; z * = 1 ; 0 < r * < 1 , . . . . ( 1 5 ) and where 8T* 8z* 0 ; z * = 0 ; 0 < r* < 1 ; (16) B i z = a x i a l B i o t number h z L/K z (U,T) , Ko = a / C ( U , T ) ( T a - T . ) L u L Z = f z A z ( U , T ) / a z ( U , T ) a 7 (U ,T ) = K 7 (U,T)/p n C(U,T) . The mass and heat t r a n s f e r equat ions and boundary cond i t i on s given by Equations 3, 6-9, 11 and 13-16 complete ly desc r ibe the mass and heat t r a n s p o r t w i t h i n the c y l i n d e r . What i s now requ i red are the moisture d i f f u s i v i t i e s and thermal c o n d u c t i v i t i e s f o r wood and a method to so lve these n o n - l i n e a r t r an spo r t equat ions . 34 IV. WOOD MODEL The complete wood model i s developed i n three s e c t i o n s , each concerned w i th a s p e c i f i c wood c h a r a c t e r i s t i c requ i red f o r the prev ious s imultaneous heat and mass t r a n s f e r a n a l y s i s . S h r i n k i ng and s w e l l i n g o f wood i s cons idered f i r s t , f o l l owed s e q u e n t i a l l y by moisture d i f f u s i v i t y , thermal c o n d u c t i v i t y and s p e c i f i c heat. A l l r e s u l t s are r e s t r i c t e d to softwoods w i th a moisture content not g rea te r than the f i b e r - s a t u r a t i o n va lue , U f . A. Sh r i nk i ng and S w e l l i n g o f Wood Wood volume i s dependent on moisture content j" i n t e r n a l s t r u c t u r e and d e n s i t y . There i s comprehensive exper imental research a v a i l a b l e which i n d i c a t e s how these v a r i a b l e s a f f e c t wood volume (4,10, 29). When wet wood i s d r i e d , no shr inkage occurs u n t i l the moisture content i s below the f i b e r s a t u r a t i o n po i n t . I f a l l moisture i s r e -moved the bulk volume change may range from 6% to 20% o f the dry volume. Shr inkage i s g rea te s t i n the t angen t i a l d i r e c t i o n , va ry ing from 5% to 12% (based on dry volume). I t i s u s ua l l y from 2% to 7% i n the r a d i a l d i r e c t i o n and l e s s than 1% i n the a x i a l d i r e c t i o n . The r a d i a l wood r ay s , f o r the r a d i a l d i r e c t i o n , and, the t r a c h e i d s , f o r the a x i a l d i r e c t i o n , decrease shr inkage i n those d i r e c t i o n s by a c t i n g as s t r u c t u r a l re in fo rcement . Local volume change i s i n f l u enced by 35 surrounding as w e l l as l o c a l moisture content . I f a moisture g rad ien t e x i s t s i n the body, d i f f e r e n t i a l shr inkage w i l l occur . Wood substance i s the ma te r i a l forming the t r a c h e i d w a l l s and i s predominately l i g n i f i e d c e l l u l o s e f i b r i l s . Below the f i b e r s a t u r a t i o n p o i n t , water i s adsorbed i n the wood substance producing a s o l i d s o l u t i o n and reduc ing , but not e l i m i n a t i n g , wood-substance cohes ive f o r c e s . As moisture i s adsorbed o r desorbed, the f i b r i l l a r s t r u c t u r e o f the wood substance i n h i b i t e s motion o f the t r a c h e i d i nne r wa l l and subsequently keeps voidage volume r e l a t i v e l y constant . Change i n bulk volume i s caused by the change i n t r a c h e i d wa l l volume. S ince water forms a s o l i d s o l u t i o n when i t i s adsorbed i n the t r a c h e i d w a l l , the change i n bulk volume i s almost d i r e c t l y p r opo r t i ona l to the volume o f l i q u i d adsorbed. A l i n e a r r e l a t i o n between wood volume and moisture content was measured by Stamm (29) from d ry i ng experiments conducted us ing s o f t -woods (F igure 9 ) . The approximat ion used to model t h i s volume change behaviour i s where V s " (I + kU/U f) 3 V Q = volume o f dry wood ( c m ) , 3 V s = volume o f wood a t moisture content U (cm ) , (17) shr inkage f a c t o r 0.20.L 6 0.1 .0.2 0.3 .0.4 0.5 0.6 0.7 0.8 0.9 i).0 u F-1GURE 9 - VOLUMETRIC SHRINKAGE OF LOBLOLLY PINE OF DIFFERENT DENSITIES 37 The shr inkage f a c t o r k i s determined by f i t t i n g Equation 17 to e x p e r i -mental data f o r the p a r t i c u l a r wood spec ie being cons ide red . Because o f the form o f Equation 17, k i s a c t u a l l y the maximum f r a c t i o n a l change i n volume based on dry volume. B. Moisture D i f f u s i v i t i e s 1. S i m p l i f i e d wood s t r u c t u r e A sketch o f a softwood s e c t i o n i s shown i n F igure 10. The m a j o r i t y o f softwoods con ta in a l l the s t r u c t u r e s shown, but quan t i t y w i l l vary acco rd ing to spec i e s . Tracheids are the p r i n c i p a l s t r u c t u r a l component o f softwood occupying g rea te r than 90% o f the volume. Table I g ives the p ropo r t i on o f t o t a l wood volume occupied by the var ious wood components. The c r o s s - s e c t i o n a l area and wa l l th i cknes s o f a t r a c h e i d depends upon the p a r t o f the growing season i n which i t developed. In summer, t r a che i d s have a r ec tangu l a r c r o s s - s e c t i o n and t h i c k w a l l s wh i l e i n s p r i ng they have a square c r o s s - s e c t i o n and t h i n w a l l s . A sketch o f a t r a c h e i d i s shown in F igure 11 to i l l u s t r a t e i t s tapered ends. To s i m p l i f y treatment o f t r a che i d s i n the d i f f u s i o n a n a l y s i s they are modelled as un i fo rmly s i z e d c e l l s o f square c r o s s - s e c t i o n . The i r l ength i s taken as 3/4 o f the mean t i p to t i p l e n g t h , 0.38 cm, measured by Stamm (28) f o r a v a r i e t y o f softwoods. The e f f e c t i v e length i s taken as 3/4 o f the average length to account f o r the tapered ends. T yp i c a l t r a c h e i d dimensions are g iven i n Table I f o r severa l American softwoods. 38 FIGURE 10 - MAGNIFIED THREE DIMENSIONAL SKETCH OF A SOFTWOOD: t t , a x i a l ~ ; " - : s u r f ate:; t g , .tangenti a l s u r f a ce ; r r , r a d i a l . s u r f a c e ; t r , t r a c h e i d s ; wr, wood r ay ; c , c a v i t y ; p, p i t ; sp, springwood; sm, sumrnerwood; a r , annual r i n g ; r d , r e s i n duct . TABLE I P roport ions o f Wood S t ruc tu res and Trache id Dimensions P ropor t ions o f . C e l l s , Percent Volume Long i t ud i na l Trache id Dimensions Length,mm Species Tracheids Rays Long i tud ina l Parenchyma Resin Canals Average Minimum Maximum Tangent ia l Diameter u range Eastern whi te pine 94.0 5.3 0.7 3.0 1.6 5.0 25 - 35 Ponderosa pine 93.0 6.7 0.3 3.6 1.5 5.0 35 - 45 S i t k a spruce 92.5 7.2 0.3 5.6 3.6 7.3 35 - 45 Douglas f i r 92.5 7.3 0.2 3.9 1.7 7.0 25 - 45 Western hemlock 91.2 8.8 4.2 1.8 6.0 30 - 40 Balsam f i r 94.3 5.7 3.5 1.9 5.6 30 - 40 Western red-cedar 93.1 6.9 Trace 3.5 1 .4 5.9 30 - 40 Reproduced from Reference 4. GO 40 41 I n t e r - t r a c h e i d communicating p i t s are t h i n areas i n the t a n -g e n t i a l and r a d i a l w a l l s , u s u a l l y s i t u a t e d near the ends o f the t r a c h e i d . Bordered p i t s , as shown i n F igure 12,,are the type most commonly found i n softwoods. M i c ro scop i c holes c a l l e d p i t pores e x i s t i n the p i t membrane. Relevant p i t membrane and p i t pore dimensions as we l l as es t imates o f the number o f p i t s per t r a c h e i d f o r American softwoods are given i n Table I I . TABLE II S t r u c t u r a l Data f o r P i t s o f American Softwoods Average th i cknes s o f p i t membrane . 5 to 1.2 y Average f r a c t i o n o f c a v i t y wa l l made up o f p i t openings .014 Average number o f p i t s per wood f i b e r 130 Average diameter o f p i t pore 30 my Average diameter o f p i t chamber 2 to 10 y Data obta ined from Reference 28. Stamm (28) has shown tha t the minor c on s t i t uen t s such as r e s i n ducts and wood rays have n e g l i g i b l e e f f e c t on moisture movement. They were t he re fo re not i nc luded i n the s i m p l i f i e d wood-s t ructure model. 42 T rea t i n g the t r ache id s as c e l l s and i n c l u d i n g the p i t s on the t r a c h e i d w a l l s , Stamm (28) proposed the s i m p l i f i e d wood model shown i n F igure 13. This s t r u c t u r e app l i ed to a l l s t r u c t u r a l d i r e c t i o n s , but d i f f e r e n t values f o r i n t e r n a l parameters must be used f o r each. Mean values o f the necessary s t r u c t u r a l dimensions r equ i red to apply the model to the r a d i a l o r a x i a l d i r e c t i o n s are g iven i n Table I I I . 2. D i f f u s i o n network f o r movement o f l i q u i d s and vapours i n the s i m p l i f i e d wood s t r u c t u r e The movement o f l i q u i d and vapour i s modelled as a concent ra -t i o n g rad ien t d i f f u s i o n phenomenon. D i f f u s i o n paths f o r moisture movement through the s i m p l i f i e d wood s t r u c t u r e are shown i n F igure 14. Movement o f l i q u i d through the wood substance, o f ten r e f e r r e d to as bound water d i f f u s i o n , occurs i n the continuous c e l l w a l l s w i th d i f f u s -i v i t y D-|, i n the d i scont inuous c e l l w a l l s w i th d i f f u s i v i t y D^ and i n the p i t membrane w i t h d i f f u s i v i t y Dg. The po s s i b l e paths f o r vapour movement are through the c e l l c a v i t y , w i th d i f f u s i v i t y D£, through the p i t chamber, w i t h d i f f u s i v i t y D^, and through the p i t pore, w i th d i f f u s i v i t y Dg. A general s o l u t i o n to the d i f f u s i o n network can be determined i n terms o f the s i x pathway d i f f u s i v i t i e s . This s o l u t i o n g ives the combined d i f f u s i v i t y o f l i q u i d and vapour through wood. The combined d i f f u s i v i t y , D r , was determined by the author to be TRACHEID CAVITY TRACHEID WALL (WOOD SUBSTANCE) PIT PAIR FIGURE 13 - SIMPLIFIED WOOD SRUCTURE USED IN WOOD MODEL CONTINUOUS WALL (LIQUID) CAVITY (VAPOUR) •rfiTS CONTINUOUS T J A T . T . (T.TQUTT)) PIT CHAMBER (VAPOUR) PTT MEMBRANE-(LIQUID) PIT PORE (LIQUID) FIGURE 14 - MOISTURE DIFFUSION NETWORK FOR WOOD MODEL GO 44 TABLE I I I i n i t i o n s and S t r u c t u r a l Dimensions f o r the Wood Model P roper ty o r Dimension Symbol Value Re l a t i on sh i p Void f r a c t i o n Density o f dry wood e P D .65 to .72 .4 1 - , P D ( 1 + ^ Dens i ty o f wood substance pws 1.46 g/cm3 Dens ity o f adsorbed water p 0 1.11 g/cm3 Average th i cknes s o f double c e l l w a l l s L w 5.2y - 7.2y 1 -n t Average c a v i t y width dc 27 y Average t r a c h e i d length 4 .38 cm Average e f f e c t i v e c a v i t y length .285 cm 3/4 L Average p i t membrane th i cknes s L p 1 y t Number o f f i b e r c a v i t i e s t r a v e r -sed per cm i n the a x i a l d i r e c t i o n " f 3.5 Number o f c e l l w a l l s t r ave r sed per cm in the a x i a l d i r e c t i o n "£ 2.5 n f - 1.0 Number o f c e l l w a l l s t r ave r sed i n the r a d i a l d i r e c t i o n n t 300 E f f e c t i v e f r a c t i o n o f c r o s s -s e c t i o n a l area covered by p i t membrane pores f o r a x i a l d i r e c t i o n q £ .0038 E f f e c t i v e f r a c t i o n o f c r o s s -s e c t i o n a l area covered by p i t membrane pores f o r r a d i a l d i r e c t i o n q t .00052 F r a c t i on o f c r o s s - s e c t i o n a l area covered by p i t s q p .014 Data obta ined from Reference (28). 45 where D A (D, + D f i) The separate pathway d i f f u s i v i t i e s , D-|_g> must be determined w i th respect to the same concen t ra t i on g rad ien t . In t h i s ana l y s i s a g rad ient i n l i q u i d concent ra t i on (mass o f l i q u i d per volume o f wet wood) was used. When D-j i s small and Y i s approx imate ly equal to D^, d i f f u s i o n p r i m a r i l y occurs by an evaporat ion-condensat ion process ; l i q u i d evapor-ates from one c e l l w a l l , d i f f u s e s across the c a v i t y , and condenses on the oppos i te c e l l w a l l . E l i m i n a t i o n o f the p i t system from the wood model removes d i f f u s i v i t i e s to Dg w i t h the r e s u l t t ha t Y = D^. 3. D i f f u s i v i t y o f l i q u i d i n the wood substance D i f f u s i v i t y o f bound water i n wood i s a f unc t i on o f moisture content and temperature. When water i s adsorbed i n the wood substance i t e s t a b l i s h e s weak hydrogen bonds w i th the a v a i l a b l e hydroxyl groups o f the c e l l u l o s e . Bonding fo rces decrease e x p o n e n t i a l l y w i t h d i s t ance from the adso rp t ion s i t e s . Hence, d i f f u s i v i t y increases i n an exponen-t i a l manner w i t h inc reases i n moisture content . The e f f e c t o f moisture content on- bound water d i f f u s i v i t y i n the a x i a l d i r e c t i o n i s shown in F igure 15 (30) . In t h i s f i g u r e d i f f u s i o n c o e f f i c i e n t s , averaged over a complete adso rp t ion exper iment, are p l o t t e d versus an e f f e c t i v e moisture content . The e f f e c t i v e moisture content i s the po i n t a f t e r which the adso rp t i on process i s una f fec ted by i n i t i a l c ond i t i on s and 46 + 't_K05) : <CLjL0> '0^15) <o;r20> ^>25 0.30 <3> AXIAL BOUND WATER DIFFUSIVITY AS A 'FUNCTION1... OF EFFECTIVE MOISTURE CONTENT FOR SITKA SPRUCE AT 25°C 47' was f e l t , by the r e sea r che r s , to be the moisture content to which the measured d i f f u s i v i t y best a p p l i e d . The e f f e c t i v e moisture content was c a l c u l a t e d us ing the r e l a t i o n s h i p , U e = U. + 2/3 ( U u - U . ) , (20) where U.,U ,U = i n i t i a l , u l t i m a t e , and e f f e c t i v e 1 u ' e moisture con tent s , r e s p e c t i v e l y . The r e s u l t s shown i n F igure 15 are the best p r e sen t l y a v a i l a b l e f o r a x i a l d i f f u s i o n and were used i n t h i s work. Because o f the absence o f r e s u l t s f o r r a d i a l bound water d i f f u s i o n , the e f f e c t . o f moisture on a x i a l d i f f u s i o n was assumed to a l s o apply to the r a d i a l d i r e c t i o n . The e f f e c t o f temperature on r a d i a l bound water d i f f u s i o n has been c o r r e l a t e d by Yao (40) us ing the Ar rhen ius r a te equat ion DQ exp (-E/RT) , r a d i a l bound water d i f f u s i v i t y at abso lute temperature T, (cm /sec) , 2 constant (cm /sec) , gas constant (cal/mole°K) , a c t i v a t i o n energy (cal/mole) . where 'BR 'BR R E 48 His r e l a t i o n f o r Dg R , on the bas i s o f a g rad ient i n mass,of water per volume o f wood substance, i s D B R = .353 exp (-8,184/RT)cm 2/sec . (21) A r e l a t i o n was po s tu l a ted by the au tho r . t ha t would account f o r the e f f e c t o f mo i s tu re :content and temperature on bound water d i f f u s i v i t y . I t s general form i s D B = fn (U) fn (T) . The f u n c t i o n o f moisture con tent , f n (U ) , was determined us ing the e x p e r i -mental r e s u l t s f o r a x i a l bound water d i f f u s i v i t y . In F igure 15, the e f f e c t i v e moisture content was taken as a f i r s t approximat ion to ac tua l moisture content and used i n a polynomial f i t to the d i f f u s i v i t y curve. This polynomial was combined w i th Yao ' s r e s u l t g iven i n Equation 21 to ob ta i n the f i n a l r e l a t i o n s h i p , Dg = (46.84 U 2 - 410.93 U 3 + 1106.11 U 4) exp(-8, l84/RT) , (22) where Dg = bound water d i f f u s i v i t y f o r a g rad ient i n mass o f water per volume wood substance (cm /sec) . Equat ion 21 was transformed to apply to a g rad ien t i n mass o f water per volume o f wet wood. The r e l a t i o n s h i p between DX and de s i r ed 49 bound water d i f f u s i v i t y , D R , i s g iven by the f o l l o w i n g equat ion : D, B (23) where P, ws den s i t y o f wood substance (g/cm ) , and e vo id f r a c t i o n . The d e r i v a t i o n o f Equation 23 i s i n Appendix C. 4. D i f f u s i v i t y o f water vapour D i f f u s i o n o f water vapour i n the wood voids was taken to be the same as i n f r ee a i r . This d i f f u s i v i t y was c a l c u l a t e d us ing the emp i r i c a l Equation (11) , = .22(T/273) 1.75 (760/P) (24) where D V = d i f f u s i v i t y o f water vapour i n a i r w i th respect 2 to a dens i t y g rad ien t (cm /sec) , T = abso lute temperature (°K) , P = pressure (mm Hg) . The d i f f u s i v i t y , Dy, app l i e s to a d i f f e r e n t g rad ien t than the one used f o r the bound water d i f f u s i v i t y . To be compatable, Dy was converted to a g rad ien t i n mass o f water per volume o f swo l len wood. The f i n a l 50 d i f f u s i v i t y r e l a t i o n , a s shown i n Appendix C, i s g iven by where Dy = d i f f u s i v i t y o f water vapour w i th respect to a g rad ien t i n mass o f water per volume o f wet 2 wood (cm /sec) , 3 p s a t = s a t u r a t i o n dens i t y o f water vapour (g/cm ) , -|g = l o c a l s lope o f de so rp t i ona l i so therm. The value o f - ^ j was determined us ing the deso rp t i ona l isotherms f o r S i t k a spruce. These were approximated w i t h a r e l a t i o n o f the form, H = f({}-) + g(T) h({{-) (26) u f u f where f(U/U^) and h(U/U^) are f unc t i on s o f the moisture content v a r i a b l e U/U^r and g(T) i s a f unc t i on o f temperature. D e t a i l s o f these three f unc t i on s are given on pages 113 to 116 of Appendix C. Resu l t s of Equation 26 f o r three temperatures, 2 5 ° , 60° and 100°C are compared w i th e x p e r i -mental r e s u l t s f o r S i t k a spruce i n F igure 16. D i f f u s i o n o f water vapour through p i t pores cannot be c a l c u l a t e d by assuming i t i s the same as f o r f ree a i r . Pore diameters and the mean f r ee path o f a water vapour molecule are o f the same order o f magnitude and passage o f gas i s t he re f o re not a pure d i f f u s i o n process. DIFFERENT TEMPERATURES 52 Tarkow and Stamm (33,34) have est imated d i f f u s i v i t i e s f o r water vapour d i f f u s i o n through p i t pores by f i t t i n g the d i f f u s i o n model to e x p e r i -mental r e s u l t s f o r carbon d i o x i de d i f f u s i o n . From t h e i r work they recommended a d i f f u s i v i t y 1/40 o f the f r e e a i r value should be used f o r pore d i f f u s i o n of water vapour. Th is approximat ion was used. 5. Pathway d i f f u s i v i t i e s f o r the d i f f u s i o n network The d i f f u s i v i t i e s f o r each path i n the d i f f u s i o n network (F igure 14) were c a l c u l a t e d us ing the bound water and vapour d i f f u s i v i -t i e s g iven by Equations 23 and 25, r e s p e c t i v e l y . C a l c u l a t i o n s were made by con s i de r i n g the f r a c t i o n a l area and f r a c t i o n a l t o t a l length which app l i ed to the path. The general r e l a t i o n used i s D (Area o f path) (Tota l length) D, path (Tota l area o f c r o s s - s e c t i o n ) (Path length) B,V (27) where D, path = pathway d i f f u s i v i t y (cm /sec) , = d i f f u s i v i t y o f bound water or water vapour, which ever app l i ed (cm /sec) . Path d i f f u s i v i t i e s r e s u l t s f o r the r a d i a l and a x i a l d i r e c t i o n s are g iven i n Appendix C. Stamm1s recommended values o f wood parameters f o r the path d i f f u s i v i t y r e l a t i o n s are g iven i n Table I I I . 53 6. D i f f u s i v i t y r e l a t i o n s used i n the mass t r a n s f e r equat ion Mois ture d i f f u s i v i t i e s as obta ined from the d i f f u s i o n network equat ions , Equations 18 and 19, apply to a g rad ient i n l i q u i d concen-t r a t i o n o f wet wood. The standard mass f l u x r e l a t i o n appropr ia te f o r t h i s d i f f u s i v i t y i s m = -D c (U ,T ) | ^ (p s U) g/cm 2-sec . (30) The den s i t y of wet wood, p , can be r e l a t e d to the den s i t y o f dry wood us ing Equat ion 17. S u b s t i t u t i n g t h i s r e l a t i o n i n t o Equation 30 and us ing the cha in r u l e o f d i f f e r e n t i a t i o n the mass f l u x r e l a t i o n becomes D C(U,T) 2 m = m r TZT g/cm -sec . (31) (1 + k ^ ) 2 8 x L e t t i n g , x D C ( U ' T ) A(U,T) = 2 (1 + kU ) I Equation. 31 has the s imple form U f rii =-A(U,T) | ^ g / c m 2 - s e c . (32) which i s the one used i n the development o f the mass and heat t r a n s f e r equat ions . A R (U,T) and A Z (U,T) apply to Equation 32 when the r a d i a l and a x i a l d i r e c t i o n s , r e s p e c t i v e l y , are being cons idered. The r a d i a l and a x i a l d i f f u s i v i t i e s were c a l c u l a t e d f o r d i f f e r -ent temperatures, dry-wood d e n s i t i e s and shr inkages . The e f f e c t o f temperature on A R (U,T) and A Z (U,T) i s shown i n F igure 1 7 . A Z (U,T) i s a s t rong f u n c t i o n o f temperature as a r e s u l t o f the i n f l u e n c e o f vapour 54 d i f f u s i o n . The maximum i n the curve f o r the a x i a l d i f f u s i v i t y and the lower i n f l e c t i o n po i n t i n the curve f o r the r a d i a l d i f f u s i v i t y both r e s u l t from the combined i n f l u e n c e of vapour d i f f u s i o n and the shape of the de so rb t i ona l i sotherms. These changes i n the curves are s i t u a t e d where there i s a maximum i n 9H/;3U. This i s approx imately a t U/U^ = .25. The e f f e c t o f dry-wood den s i t y i s shown i n F igure 18 f o r the r a d i a l d i r e c t i o n and i n F igure 19 f o r the a x i a l d i r e c t i o n . Each d i f f u s i v i t y behaves d i f f e r e n t l y to changes i n den s i t y . A x i a l d i f f u s i v i t y , because o f i t s dependency on vapour d i f f u s i o n , i s most s e n s i t i v e to changes i n wood d e n s i t y . Several d i f f e r e n t shr inkages ( rang ing from 0% to 12%), were used i n the c a l c u l a t i o n of d i f f u s i v i t i e s f o r a wood temperature o f 25°C. There was a s l i g h t i nc rease i n d i f f u s i v i t y w i th inc reased sh r i nkage , but these r e s u l t s were not p l o t t e d because maximum changes i n d i f f u s i v i t y were l e s s than 4%. R e l a t i v e c o n t r i b u t i o n s o f the d i f f e r e n t paths i n the d i f f u s i o n network are g iven i n F igure 20. From t h i s f i g u r e i t i s ev ident tha t continuous bound water d i f f u s i o n p lays a minor r o l e except near f i b e r s a t u r a t i o n . At h igher temperatures than the one shown, bound water d i f f u s i o n has l e s s e f f e c t . The h igher s e n s i t i v i t y of a x i a l d i f f u s i o n to vapour movement and de so rp t i ona l isotherms i s shown by the curve f o r (1 - D - j /D^ . Near U^, 3H/9U approaches 0 and thereby inc reases the r e l a t i v e c o n t r i b u t i o n made by bound water . The curves f o r D^/Y show the importance o f the p i t system near zero mo i s tu re . A R (U f ,T ) i s shown i n F igure 21. These are the d i f f u s i v i t i e s used to non-d imens iona l i ze the mass t r a n s f e r equat ion . The computer program used to c a l c u l a t e the d i f f u s i v i t i e s i s g iven on pp. 158 and 159 o f Appendix H. The r e l a t i o n s h i p f o r U f i s g iven on p.117 o f Appendix C. u f RADIAL D I F F U S I V I T Y AS A FUNCTION OF MOISTURE CONTENT AT DIFFERENT TEMPERATURES-T- sirs clFXGURE QTU -56 57 FIGURE 19 --AXIAL DIFFUSIVITY AS A FUNCTION OF MOISTURE CONTENT AT DIFFERENT DRY-WOOD DENSITIES . Q5> cn 60 C. Thermal C o n d u c t i v i t y and S p e c i f i c Heat The thermal c o n d u c t i v i t y o f wood depends p r i n c i p a l l y on the three v a r i a b l e s : (1) d i r e c t i o n o f heat t r a n s f e r , (2) moisture content , and, (3) wood den s i t y . Of secondary importance are the r e l a t i v e p ro -p o r t i o n o f s p r i n g and summer wood, de fect s present i n the wood s t r u c t u r e and the k ind and quan t i t y o f chemical substances such as gums, tannins and o i l s . Wangaard (38) , from heat c o n d u c t i v i t y determinat ions on both hardwoods and softwoods, found that f o r hardwoods, thermal c o n d u c t i v i t y was s i g n i f i c a n t l y g rea te r i n the r a d i a l d i r e c t i o n than i n the t angen t i a l d i r e c t i o n , w h i l e f o r softwoods, there was n e g l i g i b l e d i f f e r e n c e between these c o n d u c t i v i t i e s . In o ther work, Wangaard (37) examined the r a t i o between a x i a l and t ransver se ( r a d i a l o r t a n g e n t i a l ) c o n d u c t i v i t i e s . He found the r a t i o f o r Douglas f i r ranged from 2.28 to 3.8 w i th a mean o f approx imately 3.0. MacLean (17), i n t e s t s w i t h Douglas f i r and Red oak, found t h i s r a t i o to be between 2.25 and 2.75. MacLean conducted thermal c o n d u c t i v i t y experiments on 32 spec ies comprised o f both softwoods and hardwoods. Radia l c o n d u c t i v i t i e s were measured then c o r r e l a t e d to wet-wood den s i t y (dry mass/wet volume) and moisture content . The r e l a t i o n s h i p he obta ined app l i e s to hard-woods and softwoods which have a moisture content l e s s than 40%. Equation 17 r e l a t i n g wet and dry wood volumes was used to put h i s r e l a t i o n s h i p i n a form compatable w i th r e l a t i o n s h i p s used i n t h i s work, t ha t i s , i n terms o f dry wood d e n s i t y . The f i n a l equat ion f o r r a d i a l c o n d u c t i v i t y i s 61 K D (U,T) = (.2 + .4 U) + .024 watts m°K (33) In a l a r ge group o f exper imental t e s t s made on samples w i t h moisture contents va r y i ng from 0 to 33%, r a d i a l c o n d u c t i v i t i e s c a l c u l a t e d from MacLean's equat ion were compared to exper imental va lues . The agree-ment was good; 78% o f a l l c a l c u l a t e d c o n d u c t i v i t i e s were w i t h i n 10% o f the exper imental va lue . Equation 31 f o r r a d i a l c o n d u c t i v i t y and Wangaards f i n d i n g tha t the r a t i o o f a x i a l to r a d i a l c o n d u c t i v i t y i s approx imately 3.0. The r e l a t i o n s h i p f o r a x i a l c o n d u c t i v i t y i s t h e r e f o r e , . V Emp i r i c a l equat ions f o r the s p e c i f i c heat o f dry wood are a v a i l a b l e i n re fe rence (27) . These r e l a t i o n s h i p s have been determined expe r imen ta l l y f o r severa l American softwoods and g ive s p e c i f i c heat as a f unc t i on o f temperature. A l so g iven i n t h i s re fe rence i s a method commonly used to account f o r moisture e f f e c t . Moisture i s i n c l uded i n the r e l a t i o n by assuming a s p e c i f i c heat c o n t r i b u t i o n p r opo r t i ona l to water volume. The a x i a l c o n d u c t i v i t y o f wet wood was approximated us ing K Z (U,T) = (.6 + 1.2 U) + .071 watts m°K (34) 62 V. NUMERICAL SOLUTION OF THE MASS AND HEAT TRANSFER EQUATIONS A f i n i t e - d i f f e r e n c e approximation technique was used to formu-l a t e a s o l u t i o n method f o r the two-dimensional t r an spo r t equat ions . A very s t a b l e scheme was de s i r ed because o f the unstable cha rac te r o f coupled equat ions w i t h v a r i a b l e c o e f f i c i e n t s and boundary c o n d i t i o n s . To s a t i s f y the requirement o f good s t a b i l i t y , an i m p l i c i t t h r e e - t i m e - l e v e l scheme was chosen. In t h i s method, moisture content and temperature at advanced time are c a l c u l a t e d us ing t h e i r values a t present and prev ious t ime. A g r i d system was e s t a b l i s h e d w i t h i n the body and the advanced-time values o f moisture content and temperature, f o r each g r i d p o i n t , were obta ined by s o l u t i o n o f a mat r i x equat ion . Boundary c ond i t i o n equat ions are i n c l uded i n the fo rmu la t i on o f the mat r i x equat ion . S o l u t i o n o f the mat r i x equat ion consequently s a t i s f i e s a l l necessary boundary c o n d i t i o n s . The mat r i x equat ion a r i s i n g from an i m p l i c i t numerical s o l u t i o n to a two dimensional problem i s poor ly cond i t i oned and can be d i f f i c u l t to so l ve on a d i g i t a l computer. An a l t e r n a t i n g d i r e c t i o n method as g iven i n M i t c h e l l (19) was used to s p l i t the one mat r i x equat ion i n t o two equat ions . This two-step s o l u t i o n method i nvo l ve s s o l u t i o n o f two w e l l - c o n d i t i o n e d t r i - d i a g o n a l mat r i x equat ions a long l i n e s f i r s t p a r a l l e l to one then p a r a l l e l to the o the r coord inate a x i s . The t r i - d i a g o n a l form makes these equat ions r e a d i l y s o l v a b l e . Symmetry o f the mass and heat t r a n s f e r processes i n the body enables the use o f a r e c t i l i n e a r g r i d system. The g r i d system, 63 as shown i n F igure 22 app l i e s a t any angle o f r e v o l u t i o n about the c y l i n d e r a x i s . G r i d po i n t i n d i c e s , K and KP, are used when s o l v i n g along l i n e s p a r a l l e l to the r a d i a l and a x i a l c oo rd i na te s , r e s p e c t i v e l y . A. Mass T ran s fe r The f i n i t e d i f f e r e n c e form of the non-dimensional mass t r a n s f e r equa t i on , Equation 3 , i s 6 R [ r * A * ( U , T ) 6 R ] U n + 6 Z [ A * ( U , T)S Z] U n = f- B(U) (u n + 1- u"-1) 2 (35) where S R = c en t r a l d i f f e r e n c e operator f o r the r a d i a l d i r e c t i o n , &2 = c e n t r a l d i f f e r e n c e operator f o r the a x i a l d i r e c t i o n , U = U at p r e v i ou s , present and advanced time l e v e l s , r e s p e c t i v e l y , m non-dimensional time step (non-dimensional space step) 2 = At*/S 2 . The term, £ m , i s the mesh r a t i o o f the f i n i t e d i f f e r e n c e approx imat ion. The c e n t r a l d i f f e r e n c e operators determine the d i f f e r e n c e between the z* j A X I A L E X T E R N A L B O U N D A R Y 21 22 23 24. NK (NJ) (10) (15) (20) (NK) 16 . 17 18 19 20 BOUNDAR (4) (9) (14) | -«— % —+• 1 (19) . (24) BOUNDAR1 i — l < 11 12 13. 1 14 15 INTERN (3) (8) (13) (18) (23) EXTERN, <fl l — l 6 7 8 9 10 •J <! i - ( RAD (2) (7) (12) (17) (22) RAD K 2 3 4 (21) NJ (KP)' • (6) (11) ; (16) . 1(^3 , r * A X I A L I N T E R N A L B O U N D A R Y F I G U R E 22 - G R I D S Y S T E M F O R N U M E R I C A L M E T H O D 65 dependent v a r i a b l e one -ha l f space step e i t h e r s i de o f a g r i d po i n t . Mathemat i ca l l y , these are given by the r e l a t i o n s , and 6 Z UKP = U K P + j " U K P - 1 <37> where K and KP are as g iven i n F igure 22 and i s the moisture content a t g r i d po i n t K. For a t h r e e - t i m e - l e v e l scheme, the s u b s t i t u t i o n un = U n + 1 + U n + U n - ] i s used. Use o f t h i s s u b s t i t u t i o n i n the general d i f f e r e n c e fo rmu la , Equation 35, w i th A*(U,T) = A*(U,T) = B(U) = 1 leads to an uncond i t ion-a l l y s t a b l e scheme i n the sense o f von Neumann (19). The t h r e e - t i m e -l e v e l form o f Equat ion 35 i s r 2£ 6R(r*A*(U,T)6R) > , { 1 " WUT C ^ ~ ; — ~ + ¥ ¥ U ' T ) ^ } U K 2?m <5 R ( r *AS (U,T )6 R ) 3BTDT C - ^ — ^ + ¥ ¥ U ' T ) 5 Z > J U K r ZE 6 R ( r * A * ( U , T ) 6 R ) ^ + { 1 + WUJ C " B - ^ " + ¥ A ! ( U , T ) 6 Z ) ] } ug •1 K (38) 66 Equation 38, when expanded to mat r i x form, i s poor l y cond i t i oned and f o r cases where l a r ge numbers o f g r i d po in t s are used, d i g i t a l s o l u t i o n s are e x c e s s i v e l y time consuming. The equat ion s p l i t t i n g technique was used on Equation 38. A f ou r th order operator term was added to both s ides so t ha t the l e f t hand s ide cou ld be f a c t o r e d . Removing the parameter l i s t s from the c o e f f i c i e n t s , the f a c t o r ed form o f the equat ion i s I1 - ^ 6 R ^ A J 5 R ) ] [ 1 -^h^i *Z)K+1 2 V V r * A R < V 3 ^ [ - : R J»R-+-6Z(A|6Z)] U £ f 2 ? m 6 R ( r * A R 6 R ) 1 n 1 f^Mr*W¥AzVK + eUnK_1) (39) The two parameters, a and 3, a r i s e from a d d i t i o n o f the f ou r t h order terms to the r i g h t hand s i de o f Equat ion 38. They are r e l a x a t i o n parameters and must s a t i s f y the c o n d i t i o n , a + 3 = 1 , to have the same order o f accuracy as the o r i g i n a l fo rmula . 67 (n+1)* De f i n i ng an in te rmed ia te value o f U as l r , Equation 39 can be s p l i t i n t o the two de s i r ed equat ions : ^ [ J L J U L + W ] u"K r* 2 ? m "V^RV n 1 + D + _ m [ ; R R ,+ 6 Z ( A * 6 Z ) ] u£ 1 2? + 3 T « Z ( A Z 6 Z ) ( a U K + 3 U K ] • ( 4 0 ) 2? 2. [1 - 3 ^ 6 ^ ) ] = U K p (n+1) * 2? + 3 T 6 Z ( A Z 6 Z ) ( a U K P + 3 U K P 1 } ( 4 1 ) S ince the l e f t s i de o f each equat ion i nvo l ve s on ly one ope ra to r , the matr i ces formed are t r i - d i a g o n a l . The s o l u t i o n f o r moisture content a t the advanced time l e v e l , (n + 1 ) , i s obta ined by s o l v i n g f i r s t f o r l j ( n + 1 ) * then f o r U ^ K Equations 40 and 41 apply on ly to i n t e r n a l g r i d po int s because they do not i n c l ude boundary c o n d i t i o n s . Knowledge of the boundary cond i t i on s a t p rev i ou s , present and advanced time l e v e l s are r equ i red 68 Values f o r the advanced time l e v e l s are not known, but f o r t h i s work, were approximated by those from the present time l e v e l . B. Mass T ran s fe r Boundary Cond i t ions Approximat ion Use o f the cen t r a l d i f f e r e n c e operators a t g r i d po int s a long i n t e r n a l o r e x te rna l boundaries r equ i re s s p e c i a l t reatment. The d i f f i -c u l t y which a r i s e s can be seen by expanding a f i n i t e d i f f e r e n c e equat ion about any g r i d po i n t . For the r a d i a l d i r e c t i o n , us ing the K index, the expansion i s ^ 6R(r*A*6R) U, = ± [r*^ A * ( U K + l J ( U | ( + 1 - U K) - ^4 AR(UK-|)(UK"UK-I)] <42) For the a x i a l d i r e c t i o n , us ing the KP i ndex, the expansion i s 6Z(AZ6Z) U K p = A * ( U K p ^ ) ( U K p + 1 - U K p ) " AZ(UKP-1)(UKP-UKP-1) • <43> When con s i de r i n g a g r i d po in t s i t u a t e d on an i n t e r n a l boundary, terms U^_i and U^_l_, f o r the r a d i a l d i r e c t i o n ( Equa t i on 42) , and terms U^ p -j and U^ p_l_, f o r the a x i a l d i r e c t i o n (Equat ion 43) , are ou t s i de the de f ined g r i d a rea . S i m i l a r l y at the ex te rna l boundar ies , the terms 69 ^K+l ' ^K+^-' ^KP+1 a n c ' \ p 4 J - a r e o u t s i c ' e the def ined g r i d a rea. These terms are necessary f o r the complete mat r i x equat ion and must be est imated us ing approximate i n t e r n a l and ex te rna l boundary c o n d i t i o n equat ions . The equations used f o r each boundary w i l l now be cons idered. 1. I n te rna l boundary Radia l and a x i a l i n t e r n a l boundary cond i t i on s g iven by Equations 7 and 9, r e s p e c t i v e l y , were approximated us ing the f i n i t e d i f f e r e n c e techn ique. The f i n a l r e l a t i o n s are UK_-, = U K ; r * = 0; 0 < z * < 1 , (44) and UKP-1 = UKP ; 0 < r * < 1; z * = 0 (45) Values of the h a l f - s t e p terms, U^JJ.and U K p J _ were determined by averaging adjacent va lue s . For example, 1 = UK-1 + U K = u ( 4 6 ) 2. Ex te rna l boundary The general form f o r the ex te rna l boundary c o n d i t i o n i s g iven by the r e l a t i o n s h i p | ^ + B i m X ( U , T ) U = A x (47) 70 3U where x * i s e i t h e r space coo rd i na te . The d e r i v a t i v e i—^ i s approximated oX as the change i n U across two g r i d spaces. Mathemat i ca l l y , f o r each d i r e c t i o n , these d e r i v a t i v e s are 3U _ U K + 1 - U K - _ 1 ( 4 8 ) 3r 2S and 9U _ U K P + T U I 0 M ^ ( 4 9 ) 9z 2S Equations 48 and 49 were s u b s t i t u t e d i n t o the approp r i a te form o f Equat ion 47, depending on the coord ina te d i r e c t i o n , to ob ta i n the f o l l o w -ing approx imat ions : UK+1 = UK-1 - 2 S B i m R U K + 2 S X R ; r * = 1 ; 0 < z * < 1 , (50) and U KP + 1 = UKP-1 " 2 S B i m Z U K P + 2 S A Z ; 0 < r* < 1 ; z * = 1 . (51) Values o f h a l f - s t e p terms, such as U K +1_ and U K p + l _ , were approximated by averaging values a t adjacent g r i d p o i n t s . Cons ider ing the r a d i a l d i r e c t i o n on l y , U K + 1 U K + U K + 1 71 (52) C. Heat T ran s fe r The governing heat t r a n s f e r equa t i on , i n d imens ion less form, (Equat ion 11) was approximated us ing f i n i t e d i f f e r e n c e ope ra to r s . The r e s u l t , as g iven below, i s very s i m i l a r to the mass t r a n s f e r equa t i on . ^ R ( r * K * ( U , T ) 5 R ) T * n + 6 Z ( K * ( U , T ) 6 Z ) T * n where + (KoLu ) i | i 6 R ( r * f R A * ( U , T ) 6 R ) U n + « z(f zAf(U,T)fi z) U n - M t ^ - O " ' 1 ) } B(U)C*(U,T) ( T * ( n + 1 ) - T * ( n ~ 1 } ) % 2 (53) T * ( n - 1 ) T * n T *(n+1) ^ 1 » i » I = T* a t p rev i ou s , present and advanced time l e v e l s , r e s p e c t i v e l y , and ,2 L e t t i n g I W 6 R ( r * f R A R ( U ' T ) ( S R ) l j n + fiz(fZA|(U,T)6z)Un B(U) ( U n + 1 - l/ 1 " 1 ) and T *( n + D + T * n + T *(n-1) the heat t r a n s f e r equa t i on , neg l e c t i n g c o e f f i c i e n t parameter l i s t s , becomes ] - 3 B ^ [ r* + 6Z(KZ6Z)] } T* 2? h <5 R(r*K *6 R ) 3BC^-V^ +¥ K! 5Z>J T* + 1 + 25 h 6 R ( r * K * 6 ) ^ , n 2 ? h + (KoLu) i I Adding f ou r t h order operator terms Equat ion 55 can be s p l i t i n t o the two equat ions : 1 - U 3BC* ~ J T = 2? 5 R ( r * K*8 R) 7 3 + I 1 + 25* + g ^ ( K o L u ) i I . ( 5 6 ) 2 . [ 1 - 3 B ^ 6 Z ( K | 6 Z ) ] T * " = T 2?. , + 3 3 ^ 6 z ( K * 6 z ) ( a T * n + ( 3T * n _ l ) . ( 5 7 ) As f o r the development o f the mass t r a n s f e r equat i on , a and 3 are r e l a x a t i o n parameters. They must s a t i s f y the c o n d i t i o n a + 6 f o r Equations 5 6 and 5 7 to have the same order of accuracy as Equation 5 5 . The s o l u t i o n f o r T * ^ n + ^ i s found i n two s tep s . F i r s t , ( n + 1 ) * Equat ion 5 6 i s so lved f o r T * v ' then the de s i r ed advanced-time value of temperature, T * ^ n + 1 ^ - , i s found from Equat ion 5 7 . But, before the s o l u t i o n can be c a l c u l a t e d , the two approx imat ing equations must be mod i f ied to i n c l ude boundary c o n d i t i o n s . Approximations f o r i n t e r n a l and ex te rna l boundary cond i t i on s are cons idered i n the next s e c t i o n . 74 The heat - s i nk term, I, accounts f o r i n t e r n a l l i q u i d evapora t i on . I t was i nc luded on the r i g h t hand s i de o f Equation 56 because of the assumption t ha t the mass t r a n s f e r equat ion would have p r e v i ou s l y been s o l ved . D. Heat T ran s f e r Boundary Cond i t i on s Boundary cond i t i on s were i n c l uded i n the heat t r a n s f e r mat r i x equat ion i n the same way they were i nc luded i n the mass t r a n s f e r equat ion . Th is was done to compensate f o r g r i d po in t s used by the c e n t r a l d i f f e r -ence operators a t the boundaries which were out s ide the e s t a b l i s h e d g r i d system. The heat t r a n s f e r i n t e r n a l boundary cond i t i on s are of i d e n t i c a l form to the mass t r a n s f e r i n t e r n a l boundary c o n d i t i o n s . Ex-t e r na l boundary cond i t i on s are of d i f f e r e n t form t o , t h e i r mass t r a n s f e r • coun te r - pa r t s -because o f a term added to account f o r su r face evapora t i on . 1. I n te rna l boundary Approximat ion o f i n t e r n a l boundary c o n d i t i o n Equat ions , 14 and 16, produce the two equat ions , TK-1 = T K ; r * = 0 ; 0 < z * < 1 (58) and T * p ; 0 < r * < 1; ; z * = 0 . (59) T* 1 KP-1 75 2. Ex te rna l boundary Ex te rna l boundary c o n d i t i o n approximations f o r the r a d i a l and a x i a l d i r e c t i o n s , us ing Equations 13 and 15, are TK+1 = TK-1 - 2 SB i R T* - 2 K 0 L u L R ( U K - U ^ ) ; r* = 1 ; 0 < z * < 1 , . . . . . (60 ) and TKP+1 = TKP-1 " 2 S B i Z T K P - 2 K 0 L u L Z ( U K p . U ^ ) ; 0 < r* < 1 ; z * = 1 . (61) E. So l vab le Form o f the Mass and Heat T ran s fe r Mat r i x Equations The mass t r a n s f e r boundary c o n d i t i o n s , Equations 44, 45, 50, and 51, when combined w i th the f i n i t e d i f f e r e n c e approximat ion f o r the mass t r a n s f e r equa t i on , Equations 40 and 41, produce two equat ions f o r each g r i d p o i n t . One equat ion i s f o r the (n+1)* time l e v e l , the other f o r the (n+1) time l e v e l . W r i t i n g these f o r each g r i d po in t produces a se t o f s imultaneous equations which form a mat r i x equat ion . The order o f the matr i ces i s equal to the number o f g r i d p o i n t s . The two matr i x equations f o r mass t r a n s f e r a r e : 1. [F] { U } ( n + 1 ) * = { W } • (62a) 76 [G] { U } ( n + 1 ) = {X} + [Y] { U } ( n + 1 ) * . (62b) Values o f U, f o r each time l e v e l and g r i d p o i n t , are held i n vec t o r s . [F] and [G] are square, t r i - d i a g o n a l a r r a y s . In the above equat ions a l l , but {U }^ n + 1 ^ * and {U }^ n + 1 ^ , are known. D e t a i l s o f the arrays shown are given i n Appendix D. The general mat r i x equat ions f o r heat t r a n s f e r are i d e n t i c a l to those f o r mass t r a n s f e r except f o r the presence o f the heat - s i nk term, {A}. These equations a r e : 1. [FT] { T * } ( n + 1 ) * = {WT} + {A} , (63a) 2. [GT] { T * } ( n + 1 ) = {XT} + [YT] { T * } ( n + 1 ) * . (63b) {T*} terms are vector s w i th the number o f e n t r i e s equal to the number o f g r i d p o i n t s . [FT] and [GT] are square, t r i - d i a g o n a l a r r a y s . A l l terms are known except f o r { T * } ^ n + 1 ^ * and {T * } ^ n + 1 ^ . D e t a i l s o f a l l a r rays i n Equation 63 are l o ca ted i n Appendix E. F. C a l c u l a t i o n o f Drying Rate and Average Moisture In order to ob ta i n complete r e s u l t s f o r wood d r y i n g , d r y i ng r a te and average moisture content were c a l c u l a t e d f o r each success i ve time l e v e l . Drying ra te was c a l c u l a t e d from the ra te o f convect i ve su r face mass t r a n s f e r . Th is c a l c u l a t i o n requ i re s a su r face mass t r a n s f e r c o e f f i c i e n t f o r the r a d i a l and a x i a l s u r f a ce s , and sur face and ambient h u m i d i t i e s . I n t eg r a t i on over the complete su r face was pe r -formed us ing a Simpson 's r u l e approx imat ion (p. 24 o f Reference 5 ) . Drying ra tes f o r the n t n time l e v e l were determined us ing humid i t i e s from the same time l e v e l . Humid i t ie s were c a l c u l a t e d us ing su r face moisture contents i n con junc t i on w i t h deso rp t iona l i sotherms. The computer program w r i t t e n f o r the d ry ing ra te c a l c u l a t i o n i s g iven on pp. 159 to 161 o f Appendix H. Average moisture content was determined, depending on the p a r t i c u l a r d r y i ng problem, by one o f two methods. For the we l l behaved cases , those cases not us ing moisture d i f f u s i v i t i e s obta ined from the wood model, the average moisture content was found by i n t e g r a t i n g the d r y i ng r a t e w i t h re spec t to t ime. Th i s i n t e g r a t i o n was performed f o r each succes s i ve time i n t e r v a l . The amount o f mass removed dur ing the time i n t e r v a l was sub t rac ted from the prev ious moisture content to ob ta in an up-dated va lue . The computer program used f o r these c a l c u -l a t i o n s i s g iven on p. 161 of Appendix H. For the d r y i ng cases where mois ture d i f f u s i v i t i e s from the wood model were used, average moisture content was determined by i n t e g r a t i n g over the i n t e r i o r volume. I n t eg r a t i on was achieved us ing a Trapezo id r u l e approximat ion a t each time l e v e l . D e t a i l s o f the computer program are g iven on p.. 161 o f Appendix H. 78 G. Computer Program to Solve the Mass T ran s fe r Equat ion A computer program to so l ve the mass t r a n s f e r equat ion as g iven by Equation 62 was w r i t t e n i n For t ran IV f o r an IBM 370 d i g i t a l computer. Terms o f the equat ions were generated us ing the method and equat ions g iven i n Appendix D. S o l u t i on s to the mat r i x equat ions were achieved us ing a Computer Sc ience Department standard program c a l l e d TRISLV. This program so lves t r i - d i a g o n a l mat r i x equat ions and a l l necessary c a l l i n g procedures are g iven i n the U n i v e r s i t y o f B r i t i s h Columbia usage document reproduced i n Appendix I. A l i s t i n g o f the complete computer program i s g iven i n Appendix H. De ta i l ed exp lanat ion o f ope ra t i on procedures are g iven i n Appendix G. Exact and numerical s o l u t i o n s to a d ry ing problem were compared. For t h i s comparison, moisture d i f f u s i v i t i e s and boundary cond i t i on s were^assumed constant and moisture content was un i fo rmly d i s t r i b u t e d throughout the body. The exact s o l u t i o n was obta ined by adapt ing r e s u l t s from Luikov (16) f o r t r a n s i e n t heat conduct ion. The comparison, shown i n F igure 23, i n d i c a t e s good agreement between the two s o l u t i o n s . The numerical s o l u t i o n method was a l s o s u c c e s s f u l l y te s ted on a case where B i m D = 20 and B i m 7 = 20. 80 VI. DISCUSSION OF RESULTS A. Moisture D i f f u s i v i t i e s P r e d i c t e d by the Wood Model P r e d i c t e d r a d i a l d i f f u s i v i t i e s were compared to exper imental r e s u l t s o f B i g g e r s t a f f (3) and V e l j k o v i c (36). B i g g e r s t a f f c a l c u l a t e d i n t e g r a l r a d i a l d i f f u s i v i t i e s a t var ious wood temperatures from d ry i ng t e s t s w i t h t h i n Eastern Hemlock s l a b s . I n i t i a l moisture content was uniform and equal to tha t a t f i b e r s a t u r a t i o n . Radia l d i f f u s i v i t i e s were c a l c u l a t e d by comparing the exper imental r e s u l t s to an approximate s o l u t i o n a p p l i c a b l e to constant d i f f u s i v i t y dry ing o f an i n f i n i t e s l a b . This s o l u t i o n app l i e s when d ry ing i s i n t e r n a l l y c o n t r o l l e d and Fou r i e r numbers are l e s s than O i l (19) . This method o f c a l c u l a t i n g d i f f u s i v i t i e s has been used e x t e n s i v e l y (30, 31) because o f i t s s i m p l i c i t y . V e l j k o v i c determined t angen t i a l and r a d i a l d i f f u s i v i t i e s f o r Western Hemlock. In her work, d i f f u s i v i t i e s were c a l c u l a t e d by b e s t - f i t t i n g a constant d i f f u s i v i t y s o l u t i o n to exper imental r e s u l t s . Wood samples, 2 i n x 4 i n x 12 i n c on t a i n i n g 70% to 100% moisture (dry b a s i s ) , were d r i e d i n a f l u i d i z e d bed o f sand. I n te rna l temperature and moisture content were monitored at severa l i n t e r n a l l o c a t i o n s . Wood temperatures, a t the beg inn ing o f the reg ion used to c a l c u l a t e the d i f f u s i v i t i e s , were approx imately 20% below bed temperatures. For purpose o f comparison w i t h wood model r e s u l t s , the bed temperatures were taken as wood temperatures. Radia l d i f f u s i v i t i e s p r e d i c t e d by the wood model are f unc t i on s o f both temperature and moisture content . An e f f e c t i v e moisture content , 81 as obta ined from Equation 20, was used to e l i m i n a t e the i n f l u e n c e o f moisture content . The value used was 1/3 o f f i b e r s a t u r a t i o n moisture content app rop r i a te to d ry ing temperature. work i s shown i n F igure 24. A l s o shown are t h e o r e t i c a l p r e d i c t i o n s made by Wirakusumah (39) who used Stamm1s wood model, but i n a l l c a l c u l a t i o n s cons idered bound water d i f f u s i v i t y to be cons tant . Resu l t s o f our work compare very we l l w i t h the exper imental work shown. This good agreement concern ing the e f f e c t o f temperature i s p r i m a r i l y due to the r e l a t i o n s h i p i nc luded i n the a n a l y s i s which a l lows f o r the dependency o f bound water d i f f u s i o n on temperature. Un fo r tuna te l y , exper imental r e s u l t s were not a v a i l a b l e w i th which the p r ed i c t ed e f f e c t o f moisture content and d i f f u s i o n d i r e c t i o n cou ld be compared. In our r e s u l t s , the r a t i o o f a x i a l to r a d i a l d i f f u s i v i t y eva luated at 1/3U f ranged from 20 a t 25°C to 41 a t 100°C. Bateman e t a l . (40) measured an a x i a l d i f f u s i v i t y o f approximately -5 2 3 8.5 x 10 cm /sec f o r wood w i th a dry wood dens i t y o f 0.4 g/cm d r i e d a t 40°C. This compares reasonably we l l w i t h the p r ed i c t ed value of -5 2 11.5 x 10 cm /sec c a l c u l a t e d at the e f f e c t i v e moisture content . r a d i a l d i f f u s i v i t i e s i n woods o f d i f f e r e n t d e n s i t i e s should be i n v e r s e l y r e l a t e d to the square o f t h e i r d e n s i t i e s . In equat ion form, t h i s r e l a t i o n s h i p i s The comparison o f wood-model p r e d i c t i o n s and p r e v i ou s l y c i t e d Yao (40) , from s t r u c t u r a l c o n s i d e r a t i o n s , determined tha t Dc @ p Dc.@.p (64) 5.0 4 .01 20 30 40 50 60 70 .80 90 T (°C) V.F.I_GURJi "|4 - COMPARISON OF PREDICTED AND EXPERIMENTAL RADIAL DIFFUSIVITIES FOR THE EFFECT OF TEMPERATURE 83 He v a l i d a t e d t h i s r e l a t i o n s h i p by c o m p a r i n g p r e d i c t e d r e s u l t s t o e x p e r i m e n t a l d i f f u s i v i t i e s measured by Stamm ( 3 1 ) . Agreement was w i t h i n a p p r o x i m a t e l y 15%. R e s u l t s f o r r a d i a l d i f f u s i v i t y a t d i f f e r -e n t d e n s i t i e s as g i v e n by t he wood model a l s o have t h e i n t e r r e l a t i o n -s h i p shown i n E q u a t i o n 64. T h i s can be e x p l a i n e d w i t h t he use o f r e l e v a n t te rms o f t h e d i f f u s i v i t y e q u a t i o n s . In t he r a d i a l d i r e c t i o n , bound w a t e r d i f f u s i o n t h r o u g h t h e d i s c o n t i n u o u s c e l l w a l l s c o n t r o l s t h e d i f f u s i o n t h r o u g h t h e c a v i t y - w a l l - c a v i t y wood s t r u c t u r e . W i t h t h i s c o n s i d e r a t i o n and n e g l e c t i n g p i t and c o n t i n u o u s w a l l d i f f u s i o n , E q u a t i o n 18 s i m p l i f i e s t o D c * D 3 . From c a l c u l a t i o n s f o r t he pathway d i f f u s i v i t i e s , D , a ( — ) ^ . T h e r e f o r e , J P D f o r c o n s t a n t p w s , combined d i f f u s i v i t i e s can be r e l a t e d t o wood d e n s i t y as i n E q u a t i o n 64. T h i s r e l a t i o n s h i p does n o t h o l d f o r d i f f u s i o n i n t he a x i a l d i r e c t i o n because o f the i m p o r t a n c e o f v apou r d i f f u s i o n . B. N u m e r i c a l S o l u t i o n Method The mass t r a n s f e r e q u a t i o n w i t h c o n s t a n t d i f f u s i v i t i e s and b o u n -d a r y c o n d i t i o n s was s o l v e d f o r s e v e r a l d i f f e r e n t v a l u e s o f g r i d s p a c i n g . R e s u l t s o f t h e t e s t s a r e shown i n F i g u r e s 25 and 26 . From F i g u r e 25 i t i s e v i d e n t t h a t space i n c r e m e n t e f f e c t s t he a c c u r a c y o f 84 0.3 A., Cf \ A o V 0.2 \ GRID INTERVALS o 3 • 5 • 10 16 . X 13 o \ o . i J . \ \ ( V = 2 5 BimR = 4 - 8 4  B im z =( 0 ; - ? 7 a o \ as S T * = 1 ; (zj* = 1 V O r* = 0 ; z* = 0 -o-0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 t * m FIGURE 25 - EFFECT OF SPATIAL STEP SIZE ON NUMERICAL SOLUTION FOR DRYING CURVES 85 FIGURE 26 - EFFECT OF SPATIAL STEP SIZE ON NUMERICAL SOLUTION FOR MOISTURE PROFILES 86 the s o l u t i o n over the f u l l d r y ing t ime. The s o l u t i o n f o r center mois-ture content i s most a f f e c t e d because o f the approximation used f o r the i n t e r n a l boundary c o n d i t i o n . Although accuracy improves w i th decreased g r i d spac ing , improvements were n e g l i g i b l e f o r cases where the number o f g r i d i n t e r v a l s exceeded twice the l a r g e s t B i o t mass number. Th is e s t imate f o r g r i d spac ing needs more i n v e s t i g a t i o n s i nce i t i s l i k e l y t ha t there i s a more opt imal c r i t e r i o n o f s e l e c t i o n e s p e c i a l l y f o r B i o t numbers g rea te r than 20. Due to space l i m i t a t i o n s the maximum number o f g r i d i n t e r v a l s the computer program was designed to accept i s 44; t h i s s e l e c t i o n produces 1936 g r i d p o i n t s . The e f f e c t o f g r i d spac ing on.moisture p r o f i l e s i s shown i n F igure 26. These r e s u l t s i n d i c a t e the s o l u t i o n f o r the a x i a l d i r e c t i o n i s more s e n s i t i v e to g r i d spac ing than i s the s o l u t i o n f o r the r a d i a l d i r e c t i o n . This d i f f e r e n c e i s caused by the equat ion s p l i t t i n g techn ique. The s p l i t equat ions so l ve f i r s t f o r the r a d i a l d i r e c t i o n then f o r the a x i a l d i r e c t i o n . Th is p r e f e r e n t i a l t r ea tmen t ; i s s u f f i c i e n t to make the s o l u t i o n f o r the r a d i a l d i r e c t i o n more accurate f o r a l l values o f g r i d spac ing . Resu l t s f o r the i n v e s t i g a t i o n o f g r i d spac ing e f f e c t s were s t a b l e and acceptab ly smooth as a r e s u l t o f adequate cho ice o f the time increment. S o l u t i on s become . o s c i l l a t o r y when l a r ge time increments are used. I f very l a r ge ones are used and/or the d i f f e r e n t i a l equat ion i s h i gh l y n o n - l i n e a r , i t i s p o s s i b l e to ob ta in unstab le o s c i l l a t o r y s o l u t i o n s . An example o f an o s c i l l a t o r y d ry ing curve i s shown i n F igure 27. Here i t can be seen t ha t o s c i l l a t i o n s o f the su r face 87 0 0.1 0.2 0.3 0.4 0.5 t * m FIGURE 27 - OSCILLATORY NUMERICAL SOLUTION FOR DRYING CURVE • . 88 s o l u t i o n s may not s i g n i f i c a n t l y a f f e c t the s o l u t i o n f o r the cente r mois-ture content . From s o l u t i o n s generated f o r t h i s work, i t was found tha t smooth s o l u t i o n s were obta ined when the mesh r a t i o , E s was l e s s than u n i t y ; sma l l e r time steps produce smoother s o l u t i o n curves . Numerical s o l u t i o n s o f the mass t r a n s f e r equa t i on , f o r the case o f constant d i f f u s i v i t y and boundary c o n d i t i o n s , were obta ined us ing three pa i r s o f the r e l a x a t i o n parameters, a and g . Keeping w i t h i n the c o n d i t i o n tha t t h e i r sum equal u n i t y , the values chosen were 0.5, 0.5 ; 2, -1 ; 1, 0 . The d i f f e r e n c e between the three s o l u t i o n s was o f the order o f 0.02%. This i s an i n s i g n i f i c a n t d i f f e r e n c e and f o r s i m p l i c i t y , subsequent runs used values o f a = 0.5 and!.g = 0.5. C. Combination o f Wood Model and Mass T rans fe r Equation 1. E f f e c t o f boundary cond i t i on s Real boundary c o n d i t i o n s , as g iven by the S-shaped isotherms o f F igure 16, and l i n e a r i z e d boundary c o n d i t i o n s , as g iven by the s t r a i g h t l i n e s o f F igure 16, were each used to c a l c u l a t e d ry ing curves . Th is was done to determine i f the s o l u t i o n to the mass t r a n s f e r equat ion f o r wood cou ld be s i m p l i f i e d by us ing l i n e a r i z e d boundary cond i t i on s w i thout unduly a f f e c t i n g accuracy. Mois ture d i f f u s i v i t i e s were taken to be cons tant . The d i f f e r e n c e between r e s u l t s f o r the two i sotherm shapes t e s ted was found to be dependent on the magnitude o f the B i o t mass 89 number. F igure 28 shows r e s u l t s f o r r e l a t i v e l y small B i o t mass numbers. S ince f o r a m a j o r i t y o f su r face moisture contents the S-shaped isotherms g ive l a r g e r su r face humid i t i e s than do the l i n e a r i z e d i sotherms, d ry ing ra tes w i l l be cor respond ing ly h igher . I t i s t h i s l a r g e r d ry ing r a t e which causes f a s t e r d ry ing f o r the r e a l boundary c o n d i t i o n s , as shown i n F igure 28. For l a r ge B i o t mass numbers, su r face moisture content q u i c k l y approaches ambient moisture content . When ambient moisture i s low, the d i f f e r e n c e between r e s u l t s us ing the ac tua l and l i n e a r i z e d isotherms should be s m a l l . This behaviour was i n v e s t i g a t e d by con s i de r i n g a case w i th B i f f l R = 21 and B i f f l Z = 210. For t h i s case, there was n e g l i g i b l e d i f f e r e n c e between d ry i ng r e s u l t s us ing the two types o f boundary c o n d i t i o n s . Use o f the l i n e a r i z e d i sotherms has an advantage when con= s i d e r i n g mass t r a n s f e r w i th constant moisture d i f f u s i v i t i e s . The mass t r a n s f e r equat ion can be w r i t t e n i n standard non-dimensional form; moisture content U becomes the new v a r i a b l e U* where U* = ( U - U , ) / ( U . - U ) . a i a Using U*, the mass t r a n s f e r equat ion can be so lved e x a c t l y i n the form of a product s o l u t i o n . Mathematical treatment o f t h i s type o f problem can be adapted from Luikov (16). An example i s g iven i n t h i s work on pages 13 and 14. 2. Wood d ry i ng Wood model r e s u l t s were s u b s t i t u t e d i n t o the numerical s o l u t i o n procedure f o r the mass t r a n s f e r equat ion to s imulate i sothermal d ry ing o f softwood. I n i t i a l moisture contents were set equal to the f i b e r FIGURE 28 - COMPARISON OF DRYING CURVES' FROM REAL AND LINEARIZED BOUNDARY CONDITIONS 91 s a t u r a t i o n value appropr i a te f o r the d ry ing temperature and taken to be un i fo rmly d i s t r i b u t e d . Ambient vapour dens i t y was z e r o ; - Convect ive mass t r a n s f e r c o e f f i c i e n t s were c a l c u l a t e d from an emp i r i c a l equat ion which app l i e s to heat t r a n s f e r from an i n f i n i t e c y l i n d e r p laced pe rpend i cu l a r to a f l ow ing f l u i d (13). Th i s r e l a t i o n assumes heat and mass t r a n s f e r from the su r face are analagous processes. I t was necessary to assume tha t the convect ion c o e f f i c i e n t s c a l c u l a t e d f o r the c y l i n d e r w a l l a p p l i e d a l s o to the c y l i n d e r ends. The f i n a l r e l a t i o n -sh ip (Appendix F) i s a f unc t i on o f Reynolds number and the r a t i o o f the Schmidt and P randt l numbers. In a l l c a l c u l a t i o n s the r a t i o o f the Schmidt and P rand t l numbers was taken as u n i t y and the length to rad ius r a t i o o f the body was 2.0. A p l o t o f center moisture content versus time f o r d r y i ng a t 40°C and 60°C i s shown i n F igure 29. A l s o shown i s a r e s u l t f o r d r y i ng w i th constant moisture d i f f u s i v i t y . The curve f o r constant d i f f u s i v i t y can be cons idered to c o n s i s t o f three reg ions . The f i r s t reg ion ends when the moisture content has decreased to 60% o f i t s o r i g i n a l va lue . This reg ion i s the one where d r y i ng i s s t r ong l y i n f l uenced by i n i t i a l c o n d i t i o n s . The second reg ion i s the l i n e a r p o r t i o n o f the curve. Whi le i n t h i s r e g i o n , d r y i ng i s an exponent ia l f u n c t i o n o f t ime. The s lope o f t h i s l i n e i s commonly used by exper imenters to determine e f f e c t i v e d i f f u s i v i t y because i t i s p ro -p o r t i o n a l to d i f f u s i v i t y . The t h i r d s e c t i o n i n d i c a t e s the body i s coming to e q u i l i b r i u m w i th i t s sur round ings . 93 Con t r a s t i n g the constant and wood model d i f f u s i v i t y cases, i t can be seen tha t the second and t h i r d r eg i on s , f o r the l a t t e r case are not c l e a r l y de f i ned . The f i r s t reg ion s t i l l e x i s t s u n t i l the 60% moisture content p o i n t , but i n s tead o f a l i n e a r second r e g i on , the s lope o f the l i n e cont inuous l y decreases. Th is behaviour i n d i c a t e s the ex i s t ence o f a cont inuous l y decreas ing e f f e c t i v e d i f f u s i v i t y , a r e s u l t which i s i n q u a l i t a t i v e agreement w i t h exper imental f i n d i n g s . Average moisture content versus time f o r 60°C i s shown i n F igure 30. O s c i l l a t i o n s e x i s t e d i n the computer r e s u l t s used to generate t h i s curve because o f l i m i t a t i o n s o f the numerical method. These o s c i l l a t i o n s were e l i m i n a t e d , f o r the purpose o f p l o t t i n g , by averag ing between succes s i ve time i n t e r v a l s . Dimensionless time i n t e r v a l s were l e s s than .002 and t he re f o re j u s t i f y t h i s s imple averaging procedure. Drying r a te curves f o r the two temperatures cons idered are shown i n F igure 31. F a l l i n g ra te d r y i ng occurs over the f u l l d r y ing process. Drying r a te f a l l s o f f r a p i d l y , showing t ha t the d ry ing process i s i n t e r n a l l y c o n t r o l l e d . For t * g rea te r than 0 .1 , su r face moisture content has reached near i t s e q u i l i b r i u m va lue . The e f f e c t o f moisture d i f f u s i v i t i e s on moisture content p r o f i l e s i s shown i n F igure 32. The p r o f i l e i n the r a d i a l d i r e c t i o n has the usual p a r a b o l i c c ha r ac te r . This i s expected s i nce r a d i a l moisture d i f f u s i v i t y , when compared to a x i a l d i f f u s i v i t y , i s r e l a t i v e l y cons tant . The b e l l shaped form o f the a x i a l d i f f u s i v i t y i s the cause o f the reverse i n cu rvature seen i n the a x i a l moisture p r o f i l e . i . o 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 r* and z* FIGURE 32 - MOISTURE PROFILES FOR SOFTWOOD AT A TEMPERATURE OF 60°C 97 Resu l t s o f the wood d r y i ng s imu l a t i on are sub jec t to l i m i -t a t i o n s o f the wood model and the ex te rna l boundary c o n d i t i o n s . The r e l a t i o n s h i p used f o r convect i ve mass t r a n s f e r app l i e s adequately f o r the Reynolds number range 20 to 4000 and requ i re s t ha t convect ive heat and mass t r a n s f e r are complete ly analogous. The r e s t r i c t i o n s on the wood model, o the r than those i nc luded i n the assumptions made f o r i t s development, a re : (1) the wood s t r u c t u r e must s tay s t r u c t u r a l l y i n t a c t ; breakdown o f the wood s t r u c t u r e cou ld occur because o f i n t e r n a l s t r e s s e s ; p y r o l y s i s occurs when wood temperatures are g rea te r than 100°C, (2) the c y l i n d e r dimensions are g rea te r than the minimum values f o r a t r a c h e i d , i . e . , the body i s l a r ge enough to apply the average q u a n t i t i e s used i n the wood model. D. Heat T ran s fe r Numerical s o l u t i o n o f the heat t r a n s f e r equat ion i s very s i m i l a r to the mass t r a n s f e r equat ion s o l u t i o n . The computer program used to so lve the mass t r a n s f e r equat ion cou ld be mod i f ied by adding the necessary steps to c a l c u l a t e the hea t - s i nk term {A} . Appropr i a te s u b s t i t u t i o n s o f c o e f f i c i e n t s between the two cases would a l l ow the r e s t o f the s o l u t i o n method programming to remain the same. The p r i n c i p a l d i f f i -c u l t i e s would be the l ack o f computer storage space and the cost o f computer t ime. 98 E. Scope o f Future Work Experimental r e s u l t s are requ i red to compare w i th the wood d ry ing s i m u l a t i o n . Resu l t s o f such a comparison would enable improve-ment o f the wood model o r , i f s a t i s f a c t o r y agreement was found, the wood model would g r e a t l y a s s i s t i n s p e c i f y i n g softwood d i f f u s i o n co-e f f i c i e n t s and i n understanding wood d ry ing behaviour. A more thorough i n v e s t i g a t i o n o f the behaviour o f the program i s r equ i red so t ha t opt imal s e l e c t i o n o f time and space i n t e g r a t i o n increments can be made. I t would a l s o l ead to a b e t t e r understanding o f the s o l u t i o n s s e n s i t i v i t y to the var ious i nput parameters such as ambient moisture content , Reynolds number and the Schmidt number -P randt l number r a t i o . The computer program should be reorgan ized so t h a t the s e c t i o n which conta ins the numerical s o l u t i o n method i s a separate subrout ine . This would improve the program's g e n e r a l i t y and ease i t s a p p l i c a t i o n to o the r than d r y i ng problems. Cond i t i ona l to v e r i f i c a t i o n o f the mass t r a n s f e r pa r t o f the wood model, the heat t r a n s f e r equat ion cou ld be s o l ved . This would r equ i r e a c e r t a i n amount o f computer programming, but the e x i s t i n g numerical s o l u t i o n scheme cou ld be used i n i t s present form. VI I . CONCLUSIONS The p r i n c i p a l v a r i a b l e s i n the wood model which con t ro l r a d i a l and a x i a l d i f f u s i v i t y . a r e dry-wood d e n s i t y , temperature and moisture content . Wood shr inkage and s w e l l i n g has on l y a s l i g h t e f f e c t on d i f f u s i v i t y . A l l o the r wood parameters which apply to softwoods are i n t e r n a l l y s p e c i f i e d and from the agreement w i th exper imental r e s u l t s (F igure 24) are adequate f o r softwoods i n gene ra l . The wood model d i f f u s i v i t i e s agree we l l w i th exper imental r e s u l t s on the e f f e c t o f temperature, but there are i n s u f f i c i e n t e x p e r i -mental r e s u l t s a v a i l a b l e to s u b s t a n t i a t e the p r e d i c t e d e f f e c t o f moisture content . The dependency o f r a d i a l d i f f u s i v i t y on wood den s i t y agrees w i th exper imental r e s u l t s . The numerical s o l u t i o n method can be used to so lve two-dimensional n o n - l i n e a r p a r a b o l i c p a r t i a l d i f f e r e n t i a l equat ions . For the s o l u t i o n s , the mesh r a t i o should be l e s s than un i t y and the number o f g r i d i n t e r v a l s should , i f p o s s i b l e , be tw ice the l a r g e s t B i o t mass number. Dry ing times f o r the body analyzed i n t h i s work are s e n s i t i v e to d r y i ng temperature. The time to dry the body to the same center moisture content was 40% l e s s f o r the 60°C case than the 40°C case. For bodies w i t h l a r ge length to rad ius r a t i o s , the small magnitude o f A R (U,T) produces a l a r ge B i o t mass number and d ry ing behaviour o f wood w i l l t he re fo re be r e l a t i v e l y i n -s e n s i t i v e to values o f Reynolds number and r a t i o o f Schmidt number to P randt l number. 101 REFERENCES Bateman, E., Hohf, J . P . , and Stamm, A . 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Sc . , Vo l . 23,(1968), pp. 965-970. F u l f o r d , 6.D., "A Survey of Recent Sov i e t Research on the Drying of S o l i d s , " Can. J . Chem. Eng., V o l . 47 (August, 1969), pp. 378-391. Hansen, D., "The Drying P roce s s , " ASTM Spec ia l Techn ica l Pub. No. 385,(1964), pp. 19-27. Harmathy, T .Z . , "S imultaneous Moisture and Heat T rans fe r i n Porous Systems w i t h P a r t i c u l a r Reference to D r y i ng , " Ind. & Eng. Chem. Fund., V o l . 8,(1969), pp. 92-103. H i gg i n s , N.C., "The E q u i l i b r i u m Moisture Content - R e l a t i v e Humidity Re l a t i on sh i p s o f Se l ec ted Nat ive and Foreign Woods," Forest Prod. J . , (October, 1957), pp. 371-377. I n t e r n a t i o n a l C r i t i c a l Tables of Numerical Data, U.S. Nat iona l Research C o u n c i l , Chapter 5, p. 62. Keey, R.B., Dry ing: P r i n c i p l e s and P r a c t i c e , Pergamon Press L t d . , (1972). K r e i t h , F., " P r i n c i p l e s of Heat T ran s fe r 3rd E d . , " Chapter 9, Forced Convect ion over E x t e r i o r Su r f ace s , Chapter 12, Mass  T r an s f e r , I n tex t Educat ional Pub! . , (1973). 102 14. Lebedev, P.D., "Heat and Mass T rans fe r During the Drying o f Moist M a t e r i a l s , " I n t . J . Heat Mass T ran s fe r , V o l . 1, (1961), pp. 294-301. 15. Lu ikov , A.V. , Heat and Mass T ran s fe r i n Cap i l l a r y - Po r ou s Bod ies , Pergamon P re s s , L t d . , (1966). 16. Lu i kov , A.V. , " A n a l y t i c a l Heat D i f f u s i o n Theory, " Chapter 4, pp. 81-166; Nonstat ionary Temperature F i e l d Without Heat Sources: Boundary Cond i t i on of the F i r s t K i nd , Chapter 6, pp. 201-295; Boundary Cond i t i on o f the Th i rd K ind , Academic Pres s , (1967). 17. MacLean, J .D . , "Thermal C o n d u c t i v i t y o f Wood," Heat ing , P i p i n g and  A i r C o n d i t i o n i n g , V o l . 13, (June, 1941), pp. 280-291. 18. Mc intosh, D.C., "Handbook o f Pulp and Paper Technology," Ed. B r i t t , K.W., Chapters 1-5, pp. 37-46, F i be r S t r u c tu re and P r o p e r t i e s , Van Nostrand Reinhold Co., (197171 19. M i t c h e l l , A.R., "Computat ional Methods i n P a r t i a l D i f f e r e n t i a l Equat ions , " Chapter 2, pp. 17-99, P a r a b o l i c Equat ions , Aberdeen U n i v e r s i t y P re s s , (1970). 20. Newman, A .B . , "The Drying of Porous S o l i d s : D i f f u s i o n and Surface Emiss ion Equat ions , " Amer. I n s t . Chem. Eng. Trans . , V o l . 27, (1931), pp. 203-220. 21. Narasimhan, C , and Gauvin, W.H., "Heat and Mass T ran s fe r to Spheres i n High Temperature Surroundings , " Can. J . Chem. Eng., V o l . 45, (August, 1967). 22. P e i , D.C.T., and Gauvin, W.H., "Natu ra l Convection Evaporat ion from Sphe r i ca l P a r t i c l e s i n High-Temperature Sur round inqs , " A . I . Ch. E. J . , V o l . 9, No. 3, (May, 1963), pp. 375-383. 23. Pe r r y , J . , Ed . , "Chemical Engineers Handbook," pp. 15-32 to 15-44. 24. Sherwood, T.K., " A p p l i c a t i o n of Theo re t i c a l D i f f u s i o n Equations to the Drying o f S o l i d s , " A . I .Ch. Eng. Trans . , V o l . 27, (1931), pp. 190-202. 25. Sherwood, T.K., "The Drying o f S o l i d s - I , " Ind. & Eng. Chem., V o l . 21, No. 1, (January, 1929), pp. 12-15. 26. Sherwood, T.K., "The Drying o f S o l i d s - I I , " Ind. & Eng. Chem., V o l . 21, No. 10, (October, 1929), pp. 976-980. 103 27. Skaar, C , "Water i n Wood," Chapter 4, p. 140, Thermodynamics  o f Moisture S o r p t i o n , Syracuse Univ. P re s s , (1972). 28. Stamm, A . J . , "Passage o f L i q u i d s , Vapours, and D i s so lved Ma te r i a l s through Softwoods," U.S. Dept. Agr. B u l l . 929, (October, 1946). 29. Stamm, A . J . , " S h r i n k i n g and S w e l l i n g o f Wood," Ind. & Eng. Chem., V o l . 27, No. 4, (1935), pp. 401-406. 30. Stamm, A . J . , "Bound-Water D i f f u s i o n i n t o Wood i n the F i be r D i r e c t i o n , " For. Prod. J . , V o l . 9, No. 1, (January, 1959), pp. 27-32. 31. Stamm, A . J . , "Comparison Between Measured and Theo re t i c a l Drying D i f f u s i o n C o e f f i c i e n t s f o r Southern P i n e , " For. Prod. J . , V o l . 11, (November, 1961), pp. 536-543. 32. Stamm, A . J . , Loughborough, K.W., "Thermodynamics o f the Swe l l i n g o f Wood," J . Phy s i ca l Chemistry, V o l . 39, (1935), pp. 121-132. 33. Tarkow, H., and Stamm, A . J . , " D i f f u s i o n Through A i r - F i l l e d C a p i l l a r -i e s o f Softwoods - Pa r t I: Carbon D i o x i de , " For. Prod. J . , V o l . 10, (May, 1960), pp. 247-250. 34. Tarkow, H., and Stamm, A . J . , " D i f f u s i o n Through A i r - F i l l e d C a p i l l a r -i e s o f Softwoods - Pa r t I I : Water Vapour, For. Prod. J . , V o l . 10, (June 1960), pp. 323-324. 35. T u t t l e , F., "A Mathematical Theory o f the Drying o f Wood," J . F r a n k l i n I n s t . , (November, 1925), pp. 609-614. 36. V e l j k o v i c , M., "The Drying o f Lumber i n a F l u i d i z e d Bed o f I ne r t S o l i d s , " M.Ap.Sc. Thes i s , Department o f Chemical Eng ineer ing , Univ. o f B r i t i s h Columbia, Vancouver, B r i t i s h Columbia, Canada, (1976). 37. Wangaard, F.F., "Transverse Heat Conduc t i v i t y o f Wood," J . Heating  P i p i n g and A i r Cond i t i o n i n g , V o l . 12, (1940), pp. 459-464. 38. Wangaard, F.F., "The E f f e c t of Wood S t r u c t u r e Upon Heat C o n d u c t i v i t y , " ASME T rans . , V o l . 65, (February, 1943), pp. 127-135. 39. Wirakusumah, S., "Comparison Between the Experimental and Theor-e t i c a l Drying D i f f u s i o n C o e f f i c i e n t s o f a Softwood and a Hardwood,"M.Sc. The s i s , Department o f Wood Product s , North Ca r o l i n a S ta te Co l l e ge , Ra l e i gh , North C a r o l i n a , (1962). 40. Yao, J . , "A New Approach to the Study o f Drying D i f f u s i o n C o e f f i c i e n t s o f Wood," For. Prod. J . , V o l . 16, No. 6, (June, 1966), pp. 61-69. 1 0 4 APPENDIX A DEVELOPMENT OF MASS TRANSFER EQUATION The ana l y s i s uses a c o n t r o l volume o f the f i n i t e c y l i n d e r . No moisture g rad ien t e x i s t s i n the angular d i r e c t i o n so that mass t r a n s f e r occurs i n the r a d i a l and a x i a l d i r e c t i o n s on l y . The unsteady- s tate mass t r a n s f e r equat ion was obta ined by equat ing the net d i f f u s i o n out o f the con t r o l volume to the decrease i n mass o f moisture w i t h i n the con t r o l volume. The r e s u l t i s r " 3 F ( r D R 3r> + 3 ? ( D Z U] = YZ 3 t ( A " ] where V s = volume o f wet con t ro l volume , = 2nrdrdz(cm 3) , c = l i q u i d c oncen t r a t i o n , = mass o f water/volume of wet wood (g/cm ) , M = mass o f moisture w i t h i n the con t r o l volume (g) . L i q u i d c o n c e n t r a t i o n , c , was r e l a t e d to l i q u i d moisture content , U, us ing the wood model r e l a t i o n s h i p which r e l a t e s wood volume and moisture content . The d e f i n i t i o n o f l i q u i d concen t ra t i on i s c = mass o f l i q u i d _ 9 volume of wet wood 105 mass o f l i q u i d mass o f dry wood — x ( A - 2 ) mass o f dry wpod volume of wet wood The wet and dry volumes o f wood were r e l a t e d us ing Equation 17 to produce, from Equation A -2 , the f o l l o w i n g r e l a t i o n s h i p : P D U c = . (A-3) (1 + kU/U ) S u b s t i t u t i n g Equation A-3 i n t o Equation A - l and us ing the cha in r u l e o f d i f f e r e n t i a t i o n , the mass t r a n s f e r equat ion becomes 1 9 ( r °R Ik) + J L ( ° z ^ \ = 1 M , . r 8 r (T + kU/U f ) 2 9 r 9 2 ( l + k U / U f ) 2 9 2 " p D V S 9 t ( } The moisture capac i tance term, ^ , i s de f ined by the r e l a t i o n -sh ip , f = It i P D V D U + £ P V V S > ( A - 5 ) where 3 Pp = den s i t y of dry wood (g/cm ) , 3 V D = volume of dry wood (cm ) , e = voidage f r a c t i o n , o P v = den s i t y o f water vapour (g/cm ) , 3 V<~ = volume of wet c o n t r o l volume (cm ) 1 0 6 The magnitude of the vapour term, epyV s , i n Equation A - 5 , was compared to the magnitude o f the l i q u i d term, P Q V Q U . Using Equat ion 1 7 , t h i s r a t i o , e P V V S P D V D U can be w r i t t e n as e p s a t H n n ( 1 + kU/U f) . ( A . 6 ) P D U T By con s i de r i n g the po s s i b l e magnitudes o f each term i n Equat ion A - 6 , i t was determined tha t £ ^ s a t H ( 1 + k u / U f ) - 0 p s a t / p D . ( A - 7 ) P D Typ i ca l values o f p Q and p g a t are: p s a t @ 1 0 0 ° C = 4 . 5 x 1 0 " 5 g / c m 3 , 3 P D = 0 . 4 g/cm These t y p i c a l values o f p s a t and P r j were s u b s t i t u t e d i n t o Equat ion A-7 and i t was found t ha t the r a t i o o f mass of vapour to mass o f l i q u i d i n the c on t r o l volume i s o f the order of 1 0 ~ 4 . This r e s u l t was used 107 as j u s t i f i c a t i o n f o r neg l ec t i n g the vapour term present i n Equation A-5. Using the s i m p l i f i e d form of Equat ion A-5, the mass t r a n s f e r equat ion (Equat ion A-4) becomes 1 1 _ r r D R 3U\ + 3_ / °Z <HK = 1 9U r 9 r ( l + k U / U f ) 2 9 r 9 z ( l + k U / U f ) 2 9 2 " U + k u / u f ) a t (A-8) o r , i n terms o f a p p r o p r i a t e l y de f ined c o e f f i c i e n t s , t h i s equat ion can be w r i t t e n as ? IF ( R VU-T» IF' + I i < ¥ " - t > i > - B< u) f <fl-9' 108 APPENDIX B DEVELOPMENT OF HEAT TRANSFER EQUATION A n a l y s i s f o r the development o f the heat t r a n s f e r equat ion was analogous to tha t performed f o r the mass t r a n s f e r equat ion . The net heat t r a n s f e r out o f the c on t r o l volume, i n c l u d i n g the heat - s i nk caused by i n t e r n a l e vapo ra t i on , was equated to the decrease i n i n t e r n a l energy o f the con t r o l volume. The r e s u l t was f i r ( r KR ( U' T ) f ' *h (KZ<U>T> I> - % a p " PDB<U>CS<u-T> I (B-n where C<.(U,T = s p e c i f i c heat of wet wood (cal/g°C) , R/..x _ volume of dry c on t r o l volume * ' volume of wet c on t r o l volume = l / ( l + k U / U f ) , m = r a te o f evaporat ion w i t h i n the c on t r o l volume 3 (g/sec-cm ) , a = l a t e n t heat o f evaporat ion f o r water (ca l/g) . The r a te of e vapo ra t i on , iii was c a l c u l a t e d by s u b t r a c t i n g v evap J 3 the net accumulat ion o f l i q u i d w i t h i n the con t ro l volume from the net d i f f u s i o n o f l i q u i d i n t o the c on t r o l volume. In equat ion form, the r e l a t i o n s h i p de s i r ed f o r Equation B-l i s 1 0 9 m evap where B(U) | £ \ ( B-2) f R = f r a c t i o n o f r a d i a l d i f f u s i o n tha t i s l i q u i d and fj = f r a c t i o n o f a x i a l d i f f u s i o n tha t i s l i q u i d The terms f R and f^ are determined us ing r e s u l t s o f the wood-model d i f f u s -i v i t y equat ions . They equal the f r a c t i o n o f moisture d i f f u s i o n which doe sn ' t pass through the p i t system. To ob ta i n the complete heat t r a n s f e r equat i on , Equation B-2 must be s u b s t i t u t e d i n t o Equation B - l . no APPENDIX C WOOD MODEL 1. Re l a t i on between Dg and Dg The bound water d i f f u s i v i t y , Dg, app l i e s when a g rad ient i n mass o f water per volume wood substance i s used. This d i f f u s i v i t y i s the one shown i n F igure 15 and given by Yao ' s emp i r i c a l r e l a t i o n s h i p , Equation 21. To determine mass f l u x us ing Dg the r e l a t i o n s h i p used i s • _ n 1 — ( mass water » ( r , x " • 3x 1 volume wood substance ; ' The concen t r a t i on term of Equation C- l can be w r i t t e n as t /mass water > , mass dry wood w vo l ume wet wood , ^mass dry wood' ^volume wet wood'^volume wood subs tance ' * The re fo re , i n equat ion form, the r e l a t i o n s h i p f o r l i q u i d concent ra t i on i n Equation C - l i s mass water _ .. 1 volume wood substance P S (1 - e) Voidage, e, i s g iven by the r e l a t i o n s h i p i n Table I I I . S u b s t i t u t i n g f o r e g ives mass water _ U , volume wood substance , .. * \^-^) ( — + — ) pws p 0 I l l S u b s t i t u t i n g Equation C-2 i n t o Equat ion C - l and expanding, the f o l l o w i n g r e l a t i o n s h i p was obta ined: m = -D' Q - 5 - . (C-3) B (1 + pws u\2 dX PO The de s i r ed form f o r the mass t r a n s f e r equat ion uses a d i f f u s -i v i t y DR and a g rad ien t i n moisture c oncen t r a t i on . This appropr i a te mass f l u x r e l a t i o n s h i p i s Using Equation A-3 , Equat ion C-4 becomes, • _ D B P D 9U f r , s k77 ^ ' ( c"5) (1 + £V u f Now, equat ing Equations C-3 and C-5 and rear rang ing g ives D1 Pn B D B = -5 . (C-6) pws P p _ 2 ( _ L + J L ) 2 ( 1 + k U / U f ' ^ Pws PQ } Using the equat ion f o r vo idage, Equat ion C-6 becomes, 112 2. Re l a t i on between Dy and D v The d i f f u s i v i t y o f water vapour i n a i r g iven by Equation 24 was mod i f i ed to be a p p l i c a b l e to a g rad ien t i n l i q u i d c oncen t r a t i on . The r e l a t i o n f o r u n i d i r e c t i o n a l vapour phase d i f f u s i o n w i th re spect to a vapour dens i t y g rad ien t i s * - - °i Psa t IS • ( C - 8 ) where 2 m = mass f l u x (g/cm -sec) , 2 Dy = d i f f u s i v i t y o f water vapour i n a i r (cm /sec) , H = abso lute humidity 3 p s a t = s a t u r a t i o n water vapour dens i t y (g/cm ) The de s i r ed r e l a t i o n s h i p has the form, M " " D V ax" ' where C i s as g iven by Equation A -3 . S u b s t i t u t i n g f o r C and expanding g ives • _ D V P D dU l r Q \ m . (C-9) (1 + kU/U fr 113 Rewr i t i ng Equat ion C-8 i n terms o f a g rad ien t i n U , * " " V s a t w f • ( C-1 0> Equate Equations C-9 and C - l 0 to get Dv • ^ " + k u / u f » 2 W D i ' ( c " n ) The term | g was c a l c u l a t e d from deso rp t i ona l i sotherms. In t h i s work, 3H isotherms f o r S i t k a Spruce were used. The r e l a t i o n f o r i s developed i n the next s e c t i o n . 3. Curve f i t f o r de so rp t i ona l isotherms o f S i t k a spruce. The expe r imen ta l l y determined isotherms used are reproduced from data o f Loughborough i n F igure 16. In these curves the humid i ty , H, i s a f u n c t i o n o f both moisture content and temperature. To s i m p l i f y c u r v e - f i t t i n g , the isotherms were r e - p l o t t e d as humidity versus the f r a c t i o n o f f i b e r s a t u r a t i o n moisture content , U/U^ . A f u n c t i o n was po s tu l a ted to f i t these curves. Th is f unc t i on has the form, H = f (U/U f) + g(T)h(U/U f ) (C-12) 114 The f u n c t i o n f (U/U f) was determined by f i t t i n g the i sotherm f o r a temperature o f T = 25°C. The r e s u l t was 5 f (U/U f) = E P.(\]/UfV , (C-13) where i=0 P Q = 7.464428 x 1 0 " 3 , P 1 = 3.32111 x 10 " 1 , P 2 = 9.870049 , P 3 = -2.362482 x 10 1 P 4 = 2.110410 x 10 1 P 5 = -6.68861 The f unc t i on h(U/U^) i s the d i f f e r e n c e i n humid i ty , a t any moisture content , between the 100°C and 25°C i sotherm. A 7th o rder polynomial was f i t t e d to the i sotherm f o r 100°C to o b t a i n : H (T = 100°C) = I ( U U / l U 1 , (C-14) i=0 1 T where QQ = 1.601415 x 1 0 " 4 , Q 1 = 1.376967 x 10 1 , 115 Q 2 = 1.093299 x 101 Q 3 = = -4.669788 x 10 % -= -8.222108-x 10 % -= -7.635399 x 10 % -= 3.665102 x 101 °7 s = -7.130367 . S ub t r a c t i n g Equat ion C-T4 from Equation C-15 g ives the de s i r ed r e l a t i o n , 7 . 5 h (U/u» = E O - d V l U 1 - Z P.-CU/Uf) 1 . (C-15) T i=0 1 T i=0 The temperature f unc t i on g(T) was determined us ing the data g iven i n Table C- I . Th is data was taken from the exper imental isotherms o f F igure 16. A good f i t f o r g(T) was obta ined us ing the po l ynomia l , TABLE C-I V a r i a t i o n o f E q u i l i b r i u m Humidity w i t h Temperature Temperature T(°C) H(T) - H(T = 25°C) H(T = 100°C) - H(T = 25°C) 25 0.0 40 0.09 50 0.18 60 0.29 80 0.60 90 0.80 100 1.00 116 where 3 g(T) = E J . d - 2 5 ) 1 (C-16) i= l J 1 = .3815 x 1 0 " 2 , J 2 = .13636 x I O " 3 , J 3 = T . 1 1 7 3 x I O " 6 , and T was measured i n degrees C e l s i u s . Exper imental va lues f o r the f i b e r s a t u r a t i o n mois ture content a t temperatures o f 2 5 ° , 4 0 ° , 5 0 ° , 6 0 ° , 80° and 100°C were obta ined from the data o f Loughbourough (p. 59 o f Reference 28). Th is data i s reproduced i n Table C-II a long w i th p r e d i c t i o n s o f the l i n e a r app rox i -mation used i n t h i s work. TABLE C-II F i be r S a t u r a t i on Po in t as a Funct ion o f Temperature Temperature T(°C) F i be r S a tu r a t i on Po i n t Experimental (g/g) P r e d i c t e d (g/g) 25 .306 .308 40 .292 .292 50 .283 .281 60 .272 .270 80 .251 .250 100 .229 .228 117 U f = .308 - .00107 ( T - 2 5 ° C ) g/g . (C-17) A s i m i l a r approximat ion was used by Berger and Pei [2] . The s lope o f the isotherms was r equ i red f o r the moisture d i f f u s -i v i t y c a l c u l a t i o n s , - ^ j was determined by d i f f e r e n t i a t i n g Equation C-12 w i th re spect to U. The r e s u l t was fj = u^ j ? 1 P i ^ / U f ) 1 " 1 + 9(T)( £ 1 Qi(u/uf)1_1 5 I i P , ( U / U f ) l r l ) \ , (C-18) i= l where P., Q. and g(T) are as p r e v i o u s l y g i ven . -rrr f o r temperatures o f 1 1 dU 25°C and 100°C are shown i n F igure C-I. 4. Path D i f f u s i v i t i e s Both the r a d i a l and a x i a l d i f f u s i v i t y equat ions were de r i ved us ing Stamm's s i m p l i f i e d wood model. A sketch o f t h i s model i n c l u d i n g important s t r u c t u r a l dimensions i s shown i n F igure 13. A l l softwood s t r u c t u r a l data and d e f i n i t i o n s necessary f o r the de r i ved equat ions are g iven i n Table I I I . 118 FIGURE C - l 119 The wood c e l l s , as viewed a x i a l l y , are cons idered to have a square c r o s s - s e c t i o n . D o u b l e - c e l l - w a l l t h i c k n e s s , L , i s g iven by w the r e l a t i o n s h i p , L w = . (C-19) Path d i f f u s i v i t i e s were determined us ing Equation 27. The m u l t i p l i e r terms o f Equation 27 are i d e n t i c a l l y p o s i t i o n e d i n the equat ions below. Path 1: Continuous c e l l w a l l . Radia l -D1 = ( l - ^ x l x Dg . A x i a l D1 = (1 - e) x 1 x DE Path 2: C e l l c a v i t y . Radial -D 2 = ( 1 - n t L J ( l - n ^ V ' = • D v . A x i a l -D 2 = e D v . Path 3: D iscont inuous c e l l w a l l Radia l -D 3 = (Je- qt - q ) x ^ D g , L W & - q t - q D 1 - JE A x i a l -^ - V % n  UB n £ / n t ( l - i/e) Path 4: P i t chamber. Radia l D4 = q p X T ^ T T - r D V • q p (1 - ^ - n t L ) A x i a l 120 7^  (1 - v ^ ) - n " t Path 5: P i t membrane Radia l -A x i a l -1 Path 6: P i t pore Radia l A x i a l D 6 = q t x n T T r ( V 4 ° ) t p D = n x _L_ (DV/40) 6 ^ V p 122 APPENDIX D NUMERICAL SOLUTION OF MASS TRANSFER EQUATION The mat r i x terms present i n Equation 62 are themselves comprised o f seve ra l ma t r i c e s . Separat ion o f the mat r i x terms i n t h i s way helped to min imize redundant c a l c u l a t i o n s i n the computer program. The f i r s t equat ion o f the two step s o l u t i o n method, [F] { U , } ( n + 1 ) * = {W} (D - l ) w i l l be cons idered f i r s t . Th i s equat ion was developed by app l y i ng the operator form o f the f i n i t e d i f f e r e n c e mass t r a n s f e r equat ion to each g r i d po i n t . Boundary c o n d i t i o n equat ions were i n c l u d e d . (n+1)* In Equation 1, {U K1 i s a moisture content vec to r w i th an ent ry f o r each g r i d p o i n t . The e n t r i e s are numbered us ing the K index o f F igure 22 i n ascending order from 1 to NK. The o the r two terms o f Equat ion D-l are g iven by the r e l a t i o n -sh ips : [F ] = [FIR] - [FF] + [ 6 ] n (D-2) and {W} = ( [FF] - [ 6 ] n ) { U K } n + ( [F IR] + [FF] - [ e ] n _ 1 ) { U K } n _ 1 + ([G] - U ] n ) ( l + a ) { U K } n + ([G] - [ 4 ) ] n " 1 ) ( l + 3 ) {U^} n " 1 + { V X } (D-3) 123 [FIR] i s a square diagonal a r r a y , NK by NK. In the computer program, t h i s a r ray was t r e a t ed as a vec to r o f length NK. The ar ray form of [FIR] i s [FIR] = 2 ^ 0 B(U 2 )S B(U 3 )2S B (U N J ) R B ( U N J + 2 ) S o B ( U 2 N J ) R B (U N K )R (D-4) [FF] i s the square (NK x NK) t r i -d i .agona l a r r a y , [FF] BF(AF + CF) AF BF CF AF BF CF o o AF+ CF BF 0 0 BF AF + CF AF BF CF AF BF CF (AF + CF) BF - r a d i a l i n t e r n a l boundary r a d i a l e x te rna l boundary (D-5) 124 where AF = t l A j t U - . T " ) . (D-6) CF = r* + ^A*<u£. T + K) , (D-7) and BF = - (AF + CF) . (D-8) The necessary d e f i n i t i o n s f o r Equations D-6, D-7 and D-8 are as f o l l o w s : r * 1 = Radius o f g r i d po i n t (K ± S/2) , (D-9) K ± 2 U* = ^ ± , (D-10) 2. T± = Vli±i . (D-n) [G] i s a square t r i - d i a g o n a l a r ray NK by NK. The d iagona l s are not s i de by s i de as i n the a r ray [ FF ] . I f the rows are i n d i c a t e d by the i index and columns by the j i ndex, the d iagonals are l o ca ted at s i t e s where i = j » i = j + NJ and j = i + NJ . [G] i s g iven by the r e l a t i o n s h i p , 125 [G] where: BG AG + CG BG O AG BG ^ CG ' O ' AG BG CG AG + CG BG o •. BG AG + CG BG AG = r * A* (U~, T~J , CG = r * A* (U * . T+) , a x i a l i n t e r n a l boundary a x i a l ex te rna l boundary BG = - (AG + CG) Mo i s ture contents and Uj^  and temperatures and are g iven by Equations D-10 and D - l l . In Equations D-13 and D-14, r * = Radius o f g r i d po i n t K . [9] and [(J)] ho ld i n fo rmat ion on the g r i d po in t B i o t mass numbers f o r the r a d i a l and a x i a l s u r f a c e s , r e s p e c t i v e l y . Present and prev ious values o f [9] and [(J)] are used i n the method. The ar rays are square w i t h a l l e n t r i e s on the p r i n c i p a l d i agona l . 126 In the computer program these ar rays are each held i n NK x 2 vecto r s . In a r ray form they are [ e r = 2S 0 o o CF Bi .A m'R 0 0 CF Bi o I m'R 0 0 CF Bi I m'R r a d i a l ex te rna l boundary and [<j>]A=-= 2S o o CG Bi .1 mZ CG B i ^ z O CG Bi ml a x i a l e x te rna l boundary where I i n d i c a t e s the n o r (n - 1) time l e v e l . 127 {VX} = 2S < 0 CF ( 2 X ^ + X ^ 1 ) 0 CF (2x£ + X ^ ' 1 ) CG((1 + a) X^ + (1 + 3) X5 _ 1 ) r a d i a l e x te rna l boundary CF (2XJJ + X J J ' V CG ( ( l + a ) X z + (1 + 3 ) X Z " 1 ) a x i a l e x te rna l boundary (D-18) r a d i a l and a x i a l e x te rna l boundary The second pa r t o f the two step s o l u t i o n i s [ J ] ( U K p } n + 1 = (X) + [Y] ( U K p } ( n + 1 ) * (D-19) In Equation D-19 , [ J ] = [ F l ] - ( [ J J ] - [ d n ) , (D-20) {X} = ( [ J J ] - [ d n ) a { U K p } n + ( [ J J ] - [ d n " 1 ) 6 { U K p } n " 1 (D-22) 128 and [Y] = [F I ] (D-22) { IL D } i s a vec to r c on ta i n i n g moisture contents f o r g r i d po in t s sequent ia l from 1 to NK i n the KP index. [F I ] = B(U 9 ) o o (D-23) U-j, U 2 > U K p , i n the above and f o l l o w i n g a r r a y s , i s moisture content f o r a g r i d po i n t us ing the KP index. ]29 [ J J ] = BJ (AJ + CJ) AJ BJ CJ • • o • • • A?J BJ CJ (AJ + CJ) BJ 0 0 BJ AJ + CJ. AJ BJ CJ • • • • • • o •• •• •• AJ + CJ BJ a x i a l i n t e r n a l boundary a x i a l ex te rna l boundary a x i a l i n t e r n a l boundary a x i a l ex te rna l boundary where AJ - A z ^ K P ' "^KP^ •••• CJ - (^j^p» "'"KP^ • • • • BJ = - (AJ + CJ) U^p and T^p are determined by r e p l a c i n g K w i th KP. i n Equations D-10 and D-11. [ C J n and [ ? ] n _ 1 are square a r r a y s , NK by NK, w i t h a l l non-zero e n t r i e s on the p r i n c i p a l d i a gona l . They con ta in the a x i a l mass B i o t numbers f o r the n and n-1 time l e v e l s , r e s p e c t i v e l y , and, are held i n one NK by 2 v ec to r i n the computer program. 130 [?]^ = 2S 0 CJ Blj^ 0 0 CJ Bi mZ 0 C J B i h Z a x i a l ex te rna l boundary (D-28) where H = n - l or n The boundary c o n d i t i o n vec to r [n] i s g iven by the equa t i on , {n} = 2S 0 C J ( ( l - o ) X z - $ A Z _ 1 ) 0 0 CJ((l-a)xJ - P A ^ " 1 ) 0 0 C J ( ( l - a ) X z - 3 X Z _ 1 ) a x i a l ex te rna l boundary (D-29) 131 APPENDIX E NUMERICAL SOLUTION OF'HEAT TRANSFER EQUATION The f i n i t e d i f f e r e n c e s o l u t i o n equat ions f o r heat t r a n s f e r , 63a and 63b, are i n a form very s i m i l a r to the equations f o r mass t r a n s f e r . Use was made o f t h i s s i m i l a r i t y to reduce d u p l i c a t i o n o f ar rays as much as p o s s i b l e . The f i r s t equat ion o f the two step s o l u t i o n method f o r heat t r a n s f e r was cons idered f i r s t . This equat ion i s [FT] { T * } ( n + 1 ) * = {WT} +•{A} . ( E - l ) (n+1)* {T£ } v i s a vecto r con ta i n i n g non-d imens iona l i zed temperatures f o r g r i d po in t s from 1 to NK i n terms o f the K index. These values are entered i n ascending order s t a r t i n g a t K = l . The other a r rays present are determined us ing the f o l l o w i n g equat ions : 1. [FT] - [FIRT] - [FFT] + [ 9 T ] n . (E-2) 2. {WT} = [(FFT) - ( © T ) n J {T*} n + [(FIRT) + ^ (FFT) - ( e T ) n _ 1 ] { T * } n _ 1 + [{(GT) - ( f T ) n } ( l + a ) ] {T*} n + [{(GT) - ( f O n - 1 } ( l +3)] { T * } n _ 1 - {VAT} (E-3) 132 { A } = 3(KoLu). ( < D ) n ] N l f R ] [ ( F F ) - ( 6 ) n ] + [ f z ] [ ( G ) {U K } n + {a}- l/3[F IR ] {u£ + 1 - U j^ - 1 }| (E-4) [FIRT] i s o f i d e n t i c a l form as [FIR] and can be c a l c u l a t e d by s u b s t i t u t i n g 5m " ?n and B(U K ) = B(U K ) C * (U K , T K ) (E-5) (E-6) i n t o Equat ion D-4. S i m i l a r i l y f o r [FFT] , the equat ion f o r [FF] can be used. The s u b s t i t u t i o n s to make i n Equat ion D-5 are and AF = r * _ l . K* (U " , T~) , CF - r*^ K* (U + K , T + K) , BF = -(AF + CF) (E-7) (E-8) (E-9) and Tj^ are c a l c u l a t e d us ing Equations D-10 and D - l l . [GT] i s o f the same form as [G] and can be c a l c u l a t e d from Equation D-12 by l e t t i n g AG = r* K* (U~, TJj) , (E-10) CG = r* K* (U+, T+) 133 ( E - l l ) and BG (AG + CG) (E-12) r£ i s the rad ius o f g r i d po i n t K. [ 9 T ] n and [ 6 T ] n - 1 can be c a l c u l a t e d from the r e l a t i o n s f o r [ e ] n and [ 9 ] n _ 1 , r e s p e c t i v e l y , by us ing CF as g iven by Equation E-8 and r e p l a c i n g Bi mR w i th B i R , where l i n d i c a t e s the app rop r i a te time l e v e l . S i m i l a r i l y f o r [<£T]nand [<f>T]n"\ Equation D-l7 f o r [<)>] can be used. In t h i s case, CG i s g iven by Equat ion E - l l and B i ^ i s rep laced by B i ^ • The other terms present i n Equation E-3 and E-4 are not o f the same form as any terms i n the mass t r a n s f e r equat ion and w i l l now be cons idered i n d e t a i 1 . In the r e l a t i o n s h i p f o r {A} three terms, not p r e v i ou s l y con-s i d e r e d , are p resent . These are { f R } , and {f^} and ' {a} . For t h e i r development, the values o f f R and f^ were assumed constant throughout the elemental volume and to have a value equal to tha t a t the g r i d po i n t . Using t h i s assumption, the ar rays are KR]'" VU2> O O (E-13) [ f z ] fz(u2) o o fz(uNK) ..(E-H) 134 and {a} = 2S • < 0 f R ( U N J ) CFXnR 0 0 fR(u2NJ) CFAS R 0 - r a d i a l ex te rna l boundary f R ( U K J . CGX Z f z ( U K + 1 ) CG V U N K - 1 > C G * Z f R ( U N K ) CFAnR + f z ( U N K ) CGA, (E-15) a x i a l e x te rna l boundary r a d i a l and a x i a l ex te rna l boundary In Equat ion E-15, CF and CG f o r each g r i d po i n t are obta ined from Equations D-7 and D-14. X R and A z are determined f o r each g r i d po i n t from Equations 6 and 8 , r e s p e c t i v e l y . 135 The f i na l term, {VAT}, i s calculated using the fol lowing equation: {VAT} = 2S 0 CF(2u R 1 + y ; ; - ' 1 ) 0 0 CF(2y^ + y ^ 1 ) 0 0 CF ( 2 y R + y^ " 1 ) CG((l+a)y 7 + (1+B)y 7 _ 1) CF(2y^ + y j " 1 ) + CG((l+a)y5 + ( l+BjyJ" 1 ) radia l external boundary (E-16) axial external boundary radia l and external boundary where = KoluR(XR - B i ^ ) ] ' (E-17) and y* = [KoLu z(A z - B i m Z U K ) ] £ (E-18) I = n or (n-1) time level In Equation E-16, CF i s calculated from Equation E-8 and CG' i s calculated from Equation E - l l . The second part of the heat transfer equation solut ion i s given by the equation, 136 [JT] { T * p } n + 1 = {XT} + [YT] { T * p } ( n + 1 ) * . (E-19) In Equation E-19, [JT] = [FIT] - [ J JT ] + [ ? T ] n , (E-20) {XT} = ( [ J J T ] - [cT] n)a { T * p } n + ( [ J JT ] - • [ ? T ] n " 1 ) 3 { T j 5 p } n " 1 * {nT} (E-21) and [YT] = [FIT] . (E-22) [F IT] i s determined from Equation D-23 f o r [F I ] by making the s u b s t i t u -t i o n s , £ = £ u (E-23) sm ^h and B(U K ) = B ( U K p ) C * ( U K p , T K p ) . (E-24) [ J JT ] i s determined from Equation D-24 f o r [ J J ] us ing the r e l a t i o n s h i p s , AJ = K* ( U ~ p , T~ p ) , (E-25) CJ = K* ( U + K p , T+ p) , ( E " 2 6 ) and BJ = - (AJ + CJ) . * (E-27) 137 [ ? T ] n and [ c T ] n _ 1 are o f the same form as [ c ] n and [ ? ] n _ 1 g iven by .n-1 'mZ u , , u u lm"Z Equation D-28. They are determined by r e p l a c i n g B in7 and B i " ^ w i th Bn' z and B i z ~ \ r e s p e c t i v e l y . {nT} i s determined by the f o l l o w i n g r e l a t i o n . CJ((l-a)y5 + 3P2 1" 1) 0 {nT} = 2S < CJ ( ( l -a ) yJ + ByJ"1) 0 CJ(( l-a )y z + $y z - 1 ) a x i a l ex te rna l boundary (E-28) CJ i s g iven by Equation E-26 wh i l e y z i s g iven by Equation E-18. 138 APPENDIX F CONVECTIVE MASS TRANSFER COEFFICIENT General convect i ve heat and mass t r a n s f e r r e l a t i o n s f o r a c i r c u l a r c y l i n d e r were obta ined from Reference 13. These r e l a t i o n s use the Col burn j f a c t o r s . For mass t r a n s f e r , .1/3 and f o r heat t r a n s f e r Sh = j m Re S c l / J ( F - l ) Nu = j h Re P r 1 / 3 (F-2) where Sh = Sherwood number, KgL/Dy , Sc = Schmidt number, v/Dy , Nu = -Nu s se l t number, hL/K , Pr = P randt l number, v/a , Re = Reynolds number, 2VR/v , j = Col burn f a c t o r f o r mass t r a n s f e r , m = Col burn f a c t o r f o r heat t r a n s f e r . In many cases j = j h [13] and Equations F - l and F-2 can be used to r e l a t e Sh and Nu. Th is was done to produce the r e l a t i o n s h i p Sh = ( S c / P r ) 1 / 3 Nu • (F-3) A r e l a t i o n f o r average Nusse l t number of a c i r c u l a r c y l i n d e r p laced normal to f l u i d f l ow was obta ined [13 ] ; Nu C Re' 139 •(F-4) where C and n are l i s t e d i n the t a b l e below. Re D C n 4 - 40 .821 .385 40 - 4,000 .615 .466 4,000 - 40,000 .174 .618 The c o e f f i c i e n t s f o r the Reynolds number range 40 - 4,000 was used i n Equation F-4 to g ive the f o l l o w i n g form of Equation F-3: Sh Sc ,1/3 < 6 1 5 R e . 4 6 6 ( — ) (F-5) us ing the d e f i n i t i o n o f the Sherwood number, Sh = and the d i f f u s i v i t y r e l a t i o n f o r d i f f u s i o n o f water vapour i n f r e e a i r , n „ , T ,1.75 , 760 , 2, Dy = .22 ( ) ( — ) c m /s (F-6) where T and P are i n un i t s o f °K and mm Hg, r e s p e c t i v e l y , the convect i ve c o e f f i c i e n t was determined. The f i n a l r e l a t i o n s h i p f o r both the r a d i a l and a x i a l sur faces i s Sh D, 2R 140 (F-7) 141 APPENDIX G COMPUTER PROGRAM OPERATION PROCEDURE The computer program i s s u f f i c i e n t l y documented throughout to f o l l o w a l l the a r i t h m e t i c manipu lat ions and the program l o g i c . Input and o ther important v a r i a b l e s are de f ined i n the program. There are three d i f f e r e n t cases which can be handled by t h i s computer program. The de s i r ed case i s s e l e c t ed through the assignment o f an app rop r i a t e va lue to the v a r i a b l e ICASE. The th ree cases cons idered and the necessary value o f ICASE f o r each are g iven below: 1. Humidity = U/U f Shrinkage > 0 ICASE = 0 D i f f u s i v i t i e s D R and D^ are constant . 2. Humidity = f (U/U f ) + g(T)h(U/U f ) Shr inkage _> 0 D i f f u s i v i t i e s D R and D^ are constant. ICASE = 1 3. F u l l wood model ICASE > 1 There are three a l t e r n a t i v e cho ices f o r program output and thes are s e t by the con t r o l cha rac te r ISPEW. D e t a i l s of the e f f e c t o f ISPEW are g iven i n the comments s i t u a t e d a t the s t a r t o f the program (L ines 138 to 150). 142 For a l l cases , 21 v a r i a b l e s must be read i n t o the computer program. This reading process i s commanded by l i n e s 179 through 185 o f the program. Input v a r i a b l e s , t h e i r For t ran symbol ic name, t h e i r number type and t h e i r format i nput code are given i n Table G-I. For the cases where ICASE _< 1, the program w i l l read the r a d i a l d i f f u s i v i t y , RDIFFN, and the a x i a l d i f f u s i v i t y , ZDIFFN. Un i t s o f both are cm/sec and they are read us ing s c i e n t i f i c n o t a t i o n . The format code i s E l0 .4 and the exponent must be p laced at the f a r r i g h t o f the f i e l d . Output q u a n t i t i e s are l a b e l l e d , i n c l u d i n g u n i t s , when p r i n t e d and are exp l a i ned i n the program comments (L ines 170 to 177). 143 TABLE 6-1 Computer Input Va r i ab l e s V a r i a b l e For t ran Symbolic Name Number Type Format Code I n i t i a l rad ius (cm) R Real F10.4 1 I n i t i a l h a l f l e n g t h (cm) SCRL Real F10.4 I n i t i a l moisture content (g/g) UO Real F10.4 Isothermal temperature (°K) T Real F10.4 Re laxa t ion parameter, a ALPHA Real F10.6 Re laxa t i on parameter, 3 BETA Real F10.6 I n i t i a l d imens ion less time step DT Real F F10.6 i 0 Number o f g r i d i n t e r v a l s NH Integer 15 2 Program con t r o l c ha rac te r I CASE Integer 15 Program con t r o l c ha rac te r ISPEW Integer 15 Reynolds number o f f low RENLD Real F10.4 Schmidt number/Prandtl number SCPR Real F10.4 Ambient water vapour dens i t y (g/cm 3) RHOINF Real F10.4 Dens i ty o f drywood (g/cm 3) RHODW Real F10.4 Shr inkage f a c t o r , k FK Real F10.4 Time end c o n d i t i o n RUUF Real F10.4 Secondary time step STEP1 Real F10.6 T e r t i a r y time step STEP2 Real F10.6 F i na l time step STEP3 Real F10.6 Number o f i t e r a t i o n s to end o f STEP! LOCI Integer 15 Number o f i t e r a t i o n s to end o f STEP2 L0C2 Integer 15 10 i n d i c a t e s a v a i l a b l e f i e l d w i d t h ; 4 i n d i c a t e s the number o f d i g i t s to the r i g h t o f the decimal p o i n t . 5 i n d i c a t e s the f i e l d w i d th . The i n t e ge r value must be p laced at the f a r r i g h t o f the f i e l d . 144 APPENDIX H COMPUTER PROGRAM LISTING 1 C **** S O L U T I O N FOR T H S G E N E R A L P A R T I A L D I F F E R E N T I A L E Q U A T I O N 2 c **** G O V E R N I N G I S O T H E R M A L C O N V E C T I V E D R Y I N G O F A F I N I T E 3 c C I R C U L A R C Y L I N D E R OF S O F T W O O D . T H E E Q U A T I O N IS O F 4 c r T H E F O R M : J 6 7 c r ( 1 / R ) D / 0 R ( R AA1 OU/DR ) < - 0 / D Z ( A A 2 D U / D Z ) = 8 B DU /o t f 8 L. c AA1 = F U N C T I 0 N OF M O I S T U R E , U , AND T E M P E R A T U R E , t . 9 c A f t * * AA2 = F U N C T I G N OF M O I S T U R F f U , AND T E M P E R A T U R E , t . 10 c **** BB= F U N C T I O N OF M O I S T U R E , U , . 1 1 c T H c S O L U T I O N OF T H E G E N E R A L D I F F E R E N T I A L E Q U A T I O N I S 1 2 c NOT D E P E N D E N T ON THS FORM OF T H E S E F U N C T I O N S S O 1 3 c **** WILL G I V E A S O L U T I O N O N C £ T H E R E FORM IS S P E C I F I E O . 1 4 c **** BOUNDARY C O N D I T I O N S FOR E A C H E X T E R N A L S U R F A C E A R E 1 5 c **** O U / D R + Q G A R * U = L A M I B R 16 c **** D U / D Z + Q * U = LAM1 145 1 7 C 1 8 D I M E N S I O N A F ( 2 0 0 0 ) , A C ( 2 0 0 0 ) , A J ( 2 0 0 0 ) , B F ( 2 0 0 0 ) , B G ( 2 0 0 0 ) , 1 9 * B J ( 2 0 0 0 ) , C F ( 2 0 0 0 ) , C G I 2 0 0 0 ) , 2 0 * C J 1 2 0 0 0 ) , V L A M 1 2 0 0 0 ) , V N E T A ( 2 U 0 0 ) , H ( 2 0 0 ) 2 1 D I M E N S I O N F ( 2 0 0 0 , 3 ) , G ( 2 0 0 0 , 3 ) , R A J ( 2 0 0 0 , 3 ) , 2 2 * F K 2 0 0 0 ) , F I R ( 2 0 0 0 ) 2 3 D I M E N S I O N L A M K 2 0 0 , 2 ) , L AM1BR ( 20 0 , 2) , Q ( 2 0 0 , 2 ) » Q B A R { 2 0 0 , 2 ) , 2 4 * U I 2 0 0 0 . 4 ) , T H E T A ( 2 0 0 0 , 2 ) , P H I ( 2 0 0 0 , 2 ) , Z 2 T A ( 2 0 0 0 , 2 ) 2 5 c * * * * * ( A F , B F , C F ) , ( A G , B G , C G I , ( A J , B J , C J ) , A R E T H E T E R M S 2 6 C * * * * * WHICH MAKE UP M A T R I X E S F , G , J R £ S P E C T I V F L Y . 2 7 C * * * * * I, J , I N D I C A T E G R I D P O I N T P O S I T I O N BY L O N G . , R A D I A L 2 8 C * * * * * D I R E C T I O N S R E S P E C T I V E L Y . 2 9 C * * * * * K IS T H E GRID POINT I N D E X , KP IS AN A L T E R N A T I V E I N D E X . 3 0 C * * * * * F I , F I R A R E T H E I D E N T I T Y M A T R I X T I M E S S C A L E R S 3 1 L O G I C A L G C l , B C 2 , BC3» 8 C 4 , F I R S T T 3 2 c * * * * * 3 C 1 = L O G I C A L I N T E R N A L B C U N D & R Y , R A D I A L C I R E C T I O N 33 C * * * * * BC2 = L O G I C A L E X T E R N A L B O U N D A R Y , R A D I A L D I R E C T I O N 3 4 C * * * * * B C 3 = L O G I C A L I N T E R N A L B O U N D A R Y , A X I A L D I R E C T I O N 3 5 C * * * * * B C 4 = L O G I C A L E X T E R N A L B O U N D A R Y , A X I A L D I R E C T I O N 3 6 C * * * * * E X A M P L E GRID N K = 1 6 , N I = N J = 4 3 7 C > <Z> < 3 8 C > B C 4 < 3 9 C > 13 14 15 16 < 4 0 C > BC1 9 10 11 12 BC2 < 4 1 C > 5 6 7 8 < 4 2 C > K = l 2 3 4 - ->> R < 4 3 C > BC 3 < 4 4 C > < 4 5 . R E A L L A M O , L A M O B R , L AM 1 , L AM 1 B R , L M 1 B R 0 , LAM 10 4 6 C * * * * S O L U T I O N SCHEME E Q U A T I O N S : 4 7 C N { N + l ) * N N 4 8 C 1 . ( F I R - ( F - T H E T A ) ) U = ( F - T H E T A ) U + 4 9 C N-1 N-1 N N 5 0 C ( F I R + F - T H E T A ) U + ( G - P H I ) ( i + A L P H A ) U + 5 1 C N - 1 N-1 5 2 C ( G - P H ! ) ( 1 + B E T A ) U +VLAM 5 3 C 5 4 C N N + l ( N + l ) * 5 5 C 2 . ( F I - ( J - Z E T A ) ) U = F I * U 5 6 C N N N-1 N-1 5 7 C ( ( J - Z E T A ) A L P H A * U + ( J - Z 6 T A ) B E T A * U ) +VN5TA 5 8 C * * * * D E F I N I T I O N S : 5 9 . C N-1 I N D I C A T E S P R E V I O U S T I M E 6 0 C N " P R E S E N T " 6 1 C ( N + l ) * " I N T E R M E D I A T E T I M E 6 2 C N + l " ADVA NC EO T I M E 6 3 C 6 4 COMMON / B C / B C 1 , BC2 , B C 3 , B C 4 , I, J , N I , N J , F I R S T T 6 5 COMMON / A C O N / P,D I F FN , Z DI F F N , I C A S E 6 6 COMMON / C H / S T E P 1 , S T E P 2 , S T E P 3 , L 0 C 1 , L 0 C 2 6 7 COMMGN / A / C 1 , C 2 , GT , U F , R H O S A T , A PI 6 8 COMMON / M / F K C R , F K C Z , R H O I N F , R O . 6 9 COMMGN / P A P . / R , S CRL , FK , RHOOW 7 0 COMMON / A V E R G / R M A S S , R L A S T , T I M C O R 7 1 1 0 5 0 F 0 R M A T ( 4 F 1 0 . 4 ) 72 1 0 5 1 F G R M A T ( 2 E 1 0 . 4 ) 7 3 1 0 5 2 F 0 R M A T ( 3 F 1 0 . 6 , 3 I 5 ) 7 4 1 0 6 0 F O R M A T ( « 1 » , 4 X , * K • , T i l , 2 X , « F ( K , 1 ) ' , T 2 2 , 3 X , « F ( K , 2 ) , » 7 5 * T 3 3 , 2 X , ' F ( K , 3 P , T 4 4 , I X , * R A J ( K , 1 ) • » T 5 5 , 2 X , 7 6 * ' R A J ( K , 2) ' , T 6 6 , I X , • R A J { K , 3 ) ' , T 7 7 , 2 X , ' G ( K , 1 ) » , 146 7 7 * T 8 8 , 3 X , ' G I K . 2 ) ' , T 9 9 , I X , ' G ! K , 3 ) ' , T 1 1 0 , 2 X , 'FI(K)', 7 8 * v T 1 2 1 , 2 X , • ' F I R ( K ) • / ) 7 9 1061 FORMAT ( ' • , T i l , 6 1 F 1 0 . 6 , IX)) 8 0 1062 F O R M A T ( • ' , T i l , 2 (11 X , 2 { F 1 0 . 4 , I X ) ) ) a i 1 0 6 3 FORM A T ( • • , T i l , 2 I 2 ( F 1 0 . 4 , 1 X ) , 1 1 X ) ) 8 2 1 C 6 4 F O R M A T ( ' + ' , T 7 7 , 3 J F 1 0 . 4 , I X ) ) 8 3 1 0 6 5 ' F O R M A T ! • + ' , T 8 8 , 2 1 F 1 0 . 4 , I X ) ) 8 4 1 0 6 6 F O R M A T { * + * , T 7 7 , 2 ( F 1 0 . 4 , I X ) ) 8 5 1 0 6 7 FORMAT I • + ' , 3 X , 1 4 , T U O , 2 1 F 1 0 . 4 , I X ) ) 8 6 1 0 6 8 F O R M A T ( * 1 * , 4 X , « K ' , T i l , 3 X , ' V L A M ( K ) ' , T 2 2 , 2X , 8 7 * ' V N E T A ! K) • , 3 8 * T 3 3 , I X , * T H E T A ( K , 1 ) ' , T 4 4 , I X , 1 T H E T A ( K » 2 ) ' , T 5 5 , 2 X , 8 9 * ' P H K K . l l ' , T 6 6 , 2 X , * PHI ( K , 2 ) ' , T 7 7 , I X , ' Z E T A ! K , 1 ) ' , T 8 8 , 9 0 * I X , » Z E T A ! K , 2 ) • / ) 9 1 1 0 6 9 F O R M A T ( ' « , 2 X , 1 5 , T i l , 8 { F 1 0 . 4 , I X ) ) 9 2 1 6 6 0 F O R M A T ( » i , 3X , 1 5 , T i l , 8 { F 1 0 . 4 , I X ) ) 9 3 1661 FORM A T { * ' , 3 X , 1 5 , T 1 1 , 2 ( 1 1 X , 2 1 F 1 0 . 4 , 1X) ) , 2 ( F 1 0 . 4 , IX)) 9 4 1 6 6 2 F O R M A T ! • ' , 3 X , 1 5 , T i l , 2 ! 2 ! F 1 0 . 4 , I X ) , 1 IX ) , 2! F 1 0 . 4 , IX) ) 9 5 1 6 6 3 F O R M A T ( ' 1 * , 4 X , « K ' , T i l , » F ( K , 1 ) ' , T 2 2 , 3 X , ' F ! K , 2 ) « , 9 6 * T 3 3 , 2 X , ' F ( K , 3 ) « , T 4 4 , I X , ' R A J ( K , 1 ) » , T 5 5 , 2 X , 9 7 * ' R A J ( K , 2 ) ' , T 6 6 , I X , ' R A J ( K , 3 ) • , T 7 7 , 1 X , 9 8 * ' B G ( K ) • , T 8 8 , 1 X , ' C G ( K ) ' / ) 9 9 1 6 6 4 F O R M A T ( ' ' , T 2 1 , 1 5 , T 3 0 , 3 1 F 1 0 . 4 , I X ) ) 10 0 1 6 6 5 F O R M A T ( * • , / 5 X , • T I M E ( A R l T / R 2 ) • , T 2 3 , • K ' , T 3 4 , • U ( N + 1 ) * ' , 101 * T 4 5 , ' U ( N + 1 ) • , T 5 6 , ' U / U O ' , 10 2 * T 6 5 , ' M . T . R A T E A C T / I N T ' , T 3 5 , ' A V G U / U O * ) 10 3 1 6 6 6 F Q R M A T ( ' + « , 5 X , F 1 0 . 6 ) 104 1671 F ORMAT ( • NO S O L U T I O N FOR E Q U A T I O N 1 FOUND BY T R I S L V ) 105 1672 FORMAT ( • NO S O L U T I O N FOR E Q U A T I O N 2 FOUND BY T R I S L V ) 10 6 1 6 7 3 F O R M A T ! • • , « H U M ID I T Y = U / U F , F K = 0 OR >0 : • / , ' R D I F F N = • , E l 0.4, 1 0 7 * • Z D I F F N = ' , E 1 0 . 4 ) 108 1 6 7 4 FORM A T ( * ' , ' C Y L I N D E R SHAPE AND I N I T I A L C O N D I T I C N S : • / . ' R = • , 1 0 9 * F 1 0 . 4 , ' S C R L = ' . F 1 0 . 4 , ' U 0 = ' , F 1 0 . 4 , » T = ' , F 1 0 . 4 ) 110 1 675 F O R M A T ! ' • , ' P R O G R A M CONTROL PARA M E T E R S : • / , • A L P H A = • , F 1 0 . 4 , l i i * ' B E T A = ' , F 1 0 . 4 , • D T = ' , F 1 0 . 4 , ' N H = « , I 5 , ' I C A S E=',I5, 112 * , I S P ~ W = , , I 5 ) 113 1676 FORM A T ( * ' , ' B O U N D A R Y C O N D I T I O N S : * / » * R E N L D = ' , F 1 0 . 4 , • S C P R = ' , 1 1 4 * F 1 0 . 4 , ' R H C I N F = * , F 1 0 . 4 ) 115 1 6 7 7 F O R M A T ! • ' , ' W O O D P A R A M E T E P S : ' , • R H O D W = • , F 1 0 . 4 , ' F K = ' , F 1 0 . 4 ) 1 1 6 1 6 7 8 F O R M A T ( ' « , ' S T O P C O N D I T I O N : ' , ' R U U F = • , F 1 0 . 4 ) 11 7 1 6 7 9 F O R M A T ! ' » , » T I M E INCREMENT C O N T R O L : • / , ' S T E P 1 = • , F 1 0 . 4 , 118 * • S T E P 2 = ' , F 1 0 . 4 , ' S T E P 3 = ' , F 1 0 . 4 , ' L 0 C 1 = ' , I 5 , » L0C2=',I5/) 119 1 6 8 0 F O R M A T ( ' • , « H U M I D I T Y = F ( U / U F ) + G ( T ) H ! U / U F ) , F K = 0 0 R > 0 : « , 1 2 0 * » A C T U A L D R = » , E 1 0 . 4 , ' A C T U A L D Z = ' , E 1 0 . 4 ) 121 C * * * * * D E F I N I T I O N OF P R I N C I P A L V A R I A B L E S : 12 2 C * * * * * A L L V A R I A B L E S NOT NOTED AS I N T E G E R S ARE R E A L + 4 . 12 3 c * * * * * R = I N I T I A L R A D I U S , C M 12 4 c * * * * * S C R L = S C R I P T L = 1 / 2 I N I T I A L LENGTH,CM 12 5 c * * * * * U F = M O I S T U R E C O N T E N T AT F I B E R S A T U R A T I O N 12 6 c * * * * * u o = • I N I T I A L M O I S T U R E C O N T E N T , G W A T E R / G DRYWOOO 12 7 c * * * * * I F t U O . G T . U F ) U0=UF IN P R O G R A M . 12 8 c * * * * * A L L WOOD MODEL R E L A T I O N S D E F I N E D ONLY FOR 0 . G E . U / U F . L E * 1 12 9 c * * * * * U = L O C A L M O I S T U R E C O N T E N T , GRAM W A T E R / G R A M DRYWOOD 13 0 c * * * * * T = I S O T H E R M A L T E M P E R A T U R E , OEG K 13 1 c * * * * * C O N T I N U E D B E L O W . 13 2 1 6 8 4 F O R M A T ! • % • I N I T I A L R A D I A L B I O T MASS T R A N S F E R NUMBER= » , 13 3 * F 1 0 . 4 , / ' I N I T I A L L O N G I T U D I N A L B I O T MASS T R A N S F E R N U M B E R * 13 4 * F 1 0 . 4 , / ' I N I T I A L N O N - D I M E N . A M B I E N T C O N D . FOR R A D I A L WALL • t 1 3 5 * . , / ' I I I  - I . I  .  C Y L . E N D S ' , 13 6 * F 1 0 . 4 , / ' I N I T I A L MASS T R A N S F E R R A T E G / ! A R I * T / R * * 2 ) = , , E 1 0.4> 147 13 7 C 13 8 C ***** ALPHA AND BETA ARE RELAXATION PARAMETERS. AL PHA+ BFTA=1. 139 C ***** DT = INITIAL DIMENSIONLE SS TIME INCREMENT, ARI*TIME/R**2 140 C ***** NH = NUMBER OF INTERVALS FOR R* CR Z*, (INTEGER) 141 C ***** PROGRAM CONTROL CHARACTERS, ICASE AMD I SPEW: 142 C ***** ICASE=0 HUMIP!TY=U/UF FK=0,FK>0, (I NTEGER) 143 C ***** ICASE=1 HUMIDITY=F(U/UF)+G(T)*H!U/UF) FK=0,FK>0,(INTEGER) 144 c ***** I C A S E M FULL WOOD MODEL 14 5 c ***** ISPEW<0 DON'T WRITE MATRICES 146 c ***** ISPSW=0 WRITE MATRICES FOR FIRST TIME 147 c ](l * * * * ISPEW>0 WRITE MATRICES EVERY TIME 14 8 c **#** ISPEW=1 STOP PROGRAM AFTER FIRST TIME 149 c *#*rt* II SPEW 1 = 10 WRITE OUT U ( l ) AND U(NK) ONLY FOR EVERY 150 c ***** TIME STEP AND ALL U FOR EVERY 100TH. 15 1 1685 FORMAT(' ' , ' I N I T I A L DIMENSIONLESS MOISTURE CONTENT = 1.0', 15 2 * / '•INITIAL RADIAL DIFFUSIVITY ARI= ',E10.4,'CM**2/SEC•) 153 c 154 c ***** RENLD= REYNOLD'S NUMBER OF FLOW PAST CYLINDER 15 5 c ***** SCPR= RATIO OF SCMIDT NO. TO PRANDTL NO. 15 6 c ***** RHOINF = AMBIENT VAPOUR DENS ITY,G/CM**3 15 7 c 158 c ***** RHODW DENSITY CF DRY WCOD, G/CM**3 159 c ***** FK = SHRINKAGE PARAMETER (WET VOL/DRY V0L=1.+FK*UO/UF 16 0 c ***** RUUF=TIME END CONDITION 16 1 c 16 2 c ***** STEP1=TIME STEP FOR LOCI ITERATIONS 163 c ***** STEP2=TIME STEP FOR (L0C2-L0C1)ITERAT IONS 164 c ***** STEP3=TIM5 STEP FOR ALL FURTHER ITERATIONS 16 5 c 16 6 c ***** R 01FFN= ACTUAL RADIAL DIFFUSIVITY, CM**2/S 16 7 c ***** ZDIFFN^ ACTUAL LONGITUDINAL DIFPUS IVITY,CM**2/S 16 8 c 16 9 1686 FORMAT!' + ' , T72 , 2 ((=10 .4 , I X ) / ) 170 C ***** OUTPUT: 171 C ***** 1. ALL INPUTS WITH LABELS. 17 2 C ***** 2. INITIAL RADIAL DIFFUSIVITY, CM**2/SEC. 173 c ****:!< 3. INITIAL BICT NUMBERS AND NON-DIMENSIONAL AMBIENT 174 c ***** CONDITIONS. 17 5 c ***** 4. INITIAL MASS TRANSFER RATE, G/SEC. 176 c ***** 5. TIME, GRID POINT, U(N+1)*, U(N+1), U/UO,MASS 17 7 c ***** TRANSFER RATE (ACT/INT), AVERAGE U. 17 8 c 179 READ(5,1050) R, SCRL, UO, T 18 0 READ(5,1052) ALPHA, BETA,DT,NH,ICASE»I SPEW 181 REA0(5,1050) RENLD, SCPR, RHOINF 182 READ(5,1050) RHODW, FK 18 3 RE AD(5,1050) RUUF 18 4 READ( 5,1052 ) STEP1 ,STEP2 ,STEP3,LCC1,L0C2 18 5 IF( ICASE .LE. 1) REAO<5,1051)RDIFFN,ZD IFFN 18 6 IF(ICASF.GT.O) GOTO 4 187 WRITE(6,1673)RDIFFN,ZDIFFN 18 8 GOTO 5 18 9 4 CONTINUE 190 I F ( I CASE .GT.l) GOTO 5 191 WR ITE(6,1680 )R0IFFN»ZDIFFN 192 5 CCNTINUE 193 WRITE(6,1674)R,SCRL,U0,T 194 WRITE(6, 1675)ALP HA,BETA,DT,NH,ICASE,I SPEW 195 WRI7E(6,1676)RENLD,SCPR,RHCINF 196 WRITE16,1677)RH0DW,FK 148 197 WRITE(6,1678)RUUF 19 8 WRITE(6,1679)STEP1,STEP2,STEP3,LOC1,LOC2 199 NI=NH+1 20C NJ=NH+l 201 NK=NI*NJ 202 C * * * * * CALCULATE NCN-DIMENSIONAL SPACE STEP. 20 3 HH=1./NH 204 10 CONTINUE 20 5 C * * * * * CALCULATE SURFACE NASS TRANSFER COEFFICIENTS 20 6 c * * * * * FKCR=COFFFICIENT FOR CYLINDER WALLS 20 7 c * * * * * FKCZ=CGEFFICIENT FOR CYLINDER ENDS 20 8 CALL SURFTR(T,RENLD,SCPR,R,FKCR,FKCZ) 209 NT IME=1 21 C IFLAG=0 21 1 c * * * * * START REAL TIME 212 RLTIME=0 . 2 i3 FIR STT = .TRUE.. 2 i 4 c * * * * * POSITIONS IN U ( K , J ) : (N + l)DT>>U(K , 1) , ( N+l ) *DT»U( K, 2) 2i 5 c * * * * * (N)DT>>U(K,3), (N-1)DT>>U(K,4) 216 Y=T-298. 16 217 c * * * * * U F = MOISTURE CONTENT AT FIBER SATURATION 21 8 UF=.308-.00107*Y 219 IF(UO.GT.UF) U0=UF 22 0 c * * * * * CALCULATE MASS CONSTANT FOR SUBROUTINE AVGU 221 FK=FK/UF 22 2 RMASS=12.56637*R**2*SCRL*RHQDW*U0/(1.+FK*U0J 22 3 c * * * * * INITIALIZE BOUNDARY CONDITION AND MOISTURE CONTENT VECTORS 22 4 DO 20 1=1,NK 22 5 VLAM(I)=0. 22 6 VNETA(I)=0. 22 7 DO 18 L=3,4 22 8 U ( I , L)=U0 22 9 THE TA(I,L-2)=0. 2.>0 PHK I,L-2) = 0. 231 ZETA(I,L-2)=0. 23 2 18 CONTINUE 23 3 20 CONTINUE 23 4 DC 22 JJ=1,NJ 23 5 L AM I (JJ,2)=0.0 23 6 22 CONTINUE 23 7 DO 24 11=1,NI 23 8 LAM1BP.( I I,2)=0.0 23 9 24 CONTINUE 24 0 c * * * * RHOSAT= DENSITY OF SATURATED VAPOUR AT TEMPERATURE T, 241 c * * * * (G/CM**3). USE CLAUSIOUS-CLAPEYRON EQUATION. 24 2 RHOSAT=(l.E-3/(461.5 2 0 9 * T ) ) *EXP(61.0 22-68 5 2.49 3/T 243 * -5.262*AL0G(T)) 24 4 c ****•« TEMPERATURE EFFECT FOR DESORBT IONAL ISOTHERM 24 5 GT= . 38 15E-2* Y+ . 13636E-3*Y*-*2-. 1173E-6*Y**3 24 6 c * * * * * RHOWS= DENSITY OF WOOD SUBSTANCE AS DE TE RMINED BY STAMM 24 7 c * * * * * USING HELIUM DISPLACEMENT,G/CC. 248 c * * * * * RHO0= DENSITY OF ADSORBED WATER, G/CC. 24 9 RH0WS=1.46 25 0 RH00=1.11 251 C1=RH0WS/RH00 25 2 C2=R>0DW/RH0WS 253 RADFAC=1.+FK+U0 25 4 TIMC0R=1. 255 c * * * * ARI= NONDIMENSIONALIZING RADIAL DIFFUSIVITY 25 6 ARI=1. 149 25 7 ARI=AA(1»UO »T) 258 RO=R 259 SCRLO=SCRL 260 25 CONTINUE 261 K = 0 262 RR=DT/(HH*HH) 263 RAT1=3./(2*RR) 264 DO 180 1=1,NT 265 DO 180 J=l ,NJ 266 K = K + 1 267 KP=( J-D+NI + I . 268 C * * * * * DETERMINE IF ON BOUNDARY 269 CALL BOUND 270 IF (.NOT. (BC2 .OR. BC4)I GO TO 50 271 IF( ICASE .GT. 0) GOTO 30 27 2 C 273 C * * * * S5T UP BOUNDARY CONDITIONS FOR ICASE = 0 274 C 27 5 IK=I+NJ-J 276 H(IK)= U(K,3)/UF 27 7 GO TO 34 278 30 CONTINUE 27 9 C 280 C * * * * * SET UP BOUNDARY CONDITIONS FOR I CASE > 0 231 C * * * * * CALCULATE LAMl, LAM BR, C, AND QBAR. 28 2 C 283 IK=I+NJ-J 284 IF(FIRSTT) GOTO 31 285 C * * * * * REAPPROXIMATE VALUES OF U ON EXTERNAL BOUDAR IE S. 286 IF(.NOT.8C4)U(K, 3)=(UCK-l,3)+HH*LAM18R(I,1))/ 287 * (l.+HH*Q3AR( I ,1) ) 28 8 IF(BC4)U(K,3)=(UIK-NJ,3)+HH*LAM1(J,1))/(l.+HH*Q(J,l)) 239 31 CONTINUE 290 X=U(K,3)/UF 291 C * * * * * MINIMIZE FIT ERROR FCR HUMIDITY. 292 IFIX.GT. .01) GOTO 32 293 C * * * * * SAVE H FOP. DRYING RATE CALCULATION 294 H(IK) = X 295 GOTO 34 29 6 32 CONTINUE 297 C * * * * * CALCULATE DE SORBTIONAL ISOTHERMS 298 HT25=7.464428E-03+ 3.32111 E-01*X «-9.870049*X**2 299 * -2.362482E + 0l*X**3+ 2.110410c+01*X**4- 6 .688610*X*#5 300 HT100=1.601415E-04+ 1. 376967*X + 1.093299S+01*X**2 301 * -4.669788E+01*X**3+ 8.222108F+01*X**4- 7.6353 99E+0l+X**5 302 * +3.665102E+01*X**6- 7.130367*X**7 303 HRUUF= HT100-HT25 304 IFIHRUUF .LT . 0.0) HRUUF=0.0 305 C * * * * * SAVE H. 306 H( I K) =HT25*GT*HRUUF 307 C * * * * * SET UP APPROPRIATE BOUNDARY CONDITIONS 308 34 CONTINUE 309 IF ( .NOT. BC4) GOTO 35 310 C * * * * * DU/DZ + 0*U = LAMl 311 ZDIF=AA(2,U(K,3),T) 312 RAT2=R**2*FKCZ/(ARI*SCRL*RHODW*ZDI F) 313 LAMK J,1)=RAT2*RHQINF 314 Q(J,1)=RAT2*RH0SAT*H(IK)/U(K,3) 315 35 CONTINUE 316 IF(.NOT. BCZY GOTO 50 150 317 C * * * * * OU/DP + QfiAR*U = LAM18R 318 RDIF-AA(1,U(K,3),T) 319 RAT3=R*FKCR/(ARI*RHODW*ROIF) 320 LAM1BR(I,1)=RAT3*RH0!NF 32 1 QBAR(I,1)^RAT3*RH0SAT*H(IK)/U(K,3) 322 50 CONTINUE 32 3 C 324 C * * * * * GENERATE A, B, C, FOR EACH MATRIX * * * * * 325 C ***** B IS ON OIAGCNAL, OTHERS RELATE AS WRITTEN. 326 C 32 7 IF IBC2) GO TO 60 328 CF(K)=(J-1./2)*HH*AA(1,(U(K,3)+U{K+1,3)>/2,T) 329 AF(K*1)=CF(K) 330 IF (BCD GO TO 55 331 GO TO 65 332 55 CONTINUE 333 AF(K)-AF(K+1) 334 GO TO 65 33 5 60 CONTINUE 33 6 IF(.NOT. FIRSTT ) GOTO 6 2 337 C * * * * * SUDOUF(J) IS U AT GRID POINT OUTSIOE BODY. 338 SUDQUF=U(K,3) 339 GOTO 64 340 62 CONTINUE 341 SUDDUF=U(K-l,3)+U(K,3)*(1.-2.*HH*QBAR{1,1)) 342 SUDOUF=(SUDOUF+2.*HH*LAMIBR(I,1))/2. 343 .64 CONTINUE 344 CF(K)=(J-1./2)*HH*AA(1,SUDCUF,T) 34 5 THE TA(K,1) = 2*CF(K)*HH*QBAR(I,1) 346' 65 CONTINUE 347 IF (3C4) GO TO 75 348 CJ(K)=AA(2,(U(K,3)+U(K+NJ,3))/2,T) 349 AJ{K+NJ)=CJ(K) 350 CG(K)=(J- l ) *HH*CJ(K) 351 AG(K+NJ)=CG(K) 35 2 IF (BC3) GO TO 70 353 GO TO 80 354 70 CONTINUE 355 AJ(K)=AJ(K+NJ) 356 AG(K)=AG(K+NJ) 357 GO TC 80 358 75 CONTINUE 359 IF(.NOT. FIRSTT ) GOTO 77 360 SLD0UJ=U(K,3> 361 GOTO 79 362 77 CONTINUE 36 3 SUD0UJ=U(K-NJ,3)+U(K,3)*(1.-2.*HH*Q(J,I))+2.*HH*LAM1(J,I) 364 SUD0UJ=SUD0UJ/2. 365 79 CONTINUE 366 CJ(K )=AA(2,SU00UJ,T) 367 CG(K) = U-1)*HH*CJ(K1 368 PHI (K,1)=2*CG(K)*HH*Q< J , l ) 36 9 ZFT A(KP,1) = 2*CJ(K)*HH*Q(J ,1) 37 0 8 0 CONTINUE 371 BF(K)=-(AF(K)+CF(K)) 372 BG(K)=-JAG(K)+CG(K)) 373 BJ(K)=-(AJ(K)+CJ(K)) 374 C 375 C ***** GENERATE F, G, J , MATRICES FOR SOLUTION METHOD * * * * * 376 C 151 377 F(K,2)=BF(K) 3/8 IF (BCD GO TO 85 379 IF (BC2) GO TO 90 380 F(K,l>=AF(K) 381 F(K,3)=CF(K) 382 GO TO 95 383 85 CONTINUE 384 F(K,3)=-BF(K) 38 5 GO TO* 95 386 90 CONTINUE 387 F(K,1)=-BF(K) 38 8 9 5 CONTINUE 389 G(K,2)=BG(K) 390 RAJ(KP,2)=BJ(K) 391 IF { BC 3) GO TO 100 392 IF (BC4) GO TO 105 393 G(K,1)=AG(K) 394 G(K,3)=CG(K) 395 RAJ(KP,1)=AJ(K) 396 RAJ(KP,3)=CJtK) 397 GO TO 110 398 100 CONTINUE 399 G(K,3)=-BG(K) 400 RAJ(KP,3)=-BJ(K) 401 GO TO 110 402 105 CONTINUE 40 3 G(K,1)=-8G(K) 404 G(K,2)=BG(K) 405 RAJ (KP, 1 )=-BJ<K) 40 6 110 CONTINUE 40 7 C 408 C * * * * * GENERATE BOUNDARY CONDITION VECTOR VLAM * * * * * 40 9 C 410 IF(FIRSTT) GOTO 118 411 IF(RHOINF „EQ. 0) GOTO 162 412 118 CONTINUE 413 IF (BC4) GO TO 130 414 IF (BC2) GO TO 125 415 GO TO 162 416 125 CONTINUE 417 VLAM(K)=CF(K)*(2*LAM1BR(I,1)+LAM1BRC1,2)) 418 GO TO 160 419 130 CONTINUE 420 IF (BCD GO TO 162 421 IF (BC2) GO TO 140 42 2 VLAM(K)=CG(K)*(<1+ALPHA)*LAM1(J,1) + (1 +BET A)*lAMI<J,2)) 42 3 GO TO 160 424 140 CONTINUE 42 5 VLAM(K)=CF( K) *(2*LAM1BR( I, D+LAM1BP. ( I , 2 ) ) *CG(K) * ( ( 1+ALPHA )* 426 * LAMK J , 1>+<l + BETA)*LAMll J ,2) ) 427 GO TO 160 428 160 CONTINUE 429 VLAM(K)=VLAM(K)*2*HH 430 162 CONTINUE 431 C 432 C * * * * * GENERATE BOUNDARY CONDITION VECTOR VNETA * * * * * 43 3 C 434 IF(FIRSTT) GOTO 164 435 IF(RHOINF .EO. 0.) GOTO 179 436 164 CONTINUE 152 43 7 IF ( BC4) GO TO 170 438 GO To 179 ' 439 170 CONT INUE 44 0 VNETA(KP )=CJ (K ) * ( LAM1(J ,1 ) * (1 -ALPHA ) -BETA*LAM I ( J .2 ) ) 441 175 CONTINUE 442 VNETA(KP)=VNETA(KP)*2*HH 443 179 CONTINUE 444 180 CONTINUE 44 5 C 446 C * * * * * GENERATE FI AND FIR * * * * * * 44 7 C 448 185 CONTINUE 449 K=0 450 DO 190 1=1,NI 451 DO 190 J=1,NJ 452 K=K +1 453 KP=(J-1)*NI+ I 454 F I{KP)=BB(U(K,3) ,FK)*RAT1 455 F IR(K) = F K K P ) * ( J - D * H H 45 6 190 CONTINUE 45 7 C 438 C * * * * * WRITE OUT ALL THE MATRICES THAT EXIST TO NOW ( IF DESIRED) 45 9 C 460 IF (ISPEW) 300,200,205 461 2CC IF (.NOT. FIRSTT) GO TC 300 462 205 CONTINUE 463 WRITE(6,1060) 464 K = 0 465 DO 285 1=1,NI 466 DC 285 J=1,NJ 467 K = K + 1 46 8 CALL BOUND 469 IF ( B C D GO TO 255 470 IF (BC2) GO TO 260 471 WRITE(6,1061) ( F ( K , I Y ) , IY=1,3), ( R A J ( K , I Y ) , IY=1,3) 472 GO TO 265 473 255 CONTINUE 474 WRITE(6,1062) ( F ( K , I Y ) , IY=2,3), ( R A J ( K . I Y ) , IY=2,3) 475 GO TO 265 476 260 CONTINUE 477 WRITEC.6,10631 ( F ( K , I Y ) , IY=1,2), ( R A J ( K , I Y ) , IY= l ,2 ) 478 265 CONTINUE 479 IF (BC3) GO TO 270 430 IF (BC4) GO TO 275 481 WRITE(6,1064 ) ( G ( K , I Y ) , IY = 1,3) 482 GO TG 280 483 270 CONTINUE 484 WRITE16,1065) ( G ( K , I Y ) , IY=2,3) 485 GO TG 28G 436 275 CONTINUE 4o7 WRITE(6,1066) (G(K, IY>, IY=1,2) 48 8 280 CONTINUE 489 WRITE(6,1067) K, F I ( K ) , F IR(K) 490 285 CONTINUE 491 WRITE(6,10681 492 DO 290 K=1,NK 493 WRITE(6,1069) K, VLAM(K), VNETA(K) , (THETA(K,L ), L = l , 2 ) , 494 * ( P H K K . L ) , L = l , 2 ) , ( ZET A ( K, L ) , 1 = 1,21 495 290 CONTINUE 496 ' 300 CONTINUE 153 497 C * * * * * MANIPULATE ALL MATRIX ENTRIES TO PUT MATRIX EQUATION 1 493 C * * * * * j N SOLVABLE FORM ANO EQUATION 2 IN MOST REDUCED FORM. 499 K=0 500 DO 370 1=1,NI 501 DO 370 J=1,NJ 502 K=K+1 503 KP=(J-l)*N!+I 504 CALL BOUND 505 IF (BCD GO TO 310 50 6 IF (BC2) GO TO 3 25 507 IF (BC3) GO TC 340 508 IF (8C4J GO TO 345 50 9 8 G(K)= G(K,1)*(1 + ALPHA)*U(K-NJ,3)+ F (K , 1 ) *U (K - l , 3 ) + 510 * (F(K,2)+G(K,2)*(1 + ALPHA) )*U(K,3) + 511 * F(K,3)*U(K+I,3)+G(K,3)*(1+ALPHA)*U(K+NJ,3)+ 512 * G{K,1)*(1+8FTA)*U(K-NJ,4)+F(K,1)*U(K-1,4)+ 513 * (FIR(K)+F(K,2.)+G(K,2)*(l + BETA))*U(K,4) + 514 * F(K,3)*U(K + 1,4)+G(K,3)M1+BETA)*U(K+NJ,4) 515 305 CONTINUE 516 CG(KP)=(RAJ(KP,1)*U(K-NJ,3)+RAJ(KP,2)* 517 * U(K,3)+RAJ(KP,3)*U(K+NJ,3))*ALPHA+(RAJ(KP,1)*U(K-NJ,4)+ 518 * RAJ(KP,2)*U(K,4)+RAJ(KP,3)*UCK+NJ,4))* 519 * BETA 520 GO TO 350 521 310 CONTINUE 52 2 BG(K) = (F(K,2)-PHI (K, l ) *(1+ALPHA))*U(K,3) + 523 * F(K,3>*U(K+1,3)+ 524 * (FIR(K)+F(K,2)-PHI(K,2)*(1+BETA))*U(K,4)+ 525 * F(K,3)*U(K+I,4) 526 IF (EC3) GO TO 315 52 7 IF (BC4J GO TO 320 528 GO TO 305 529 315 CONTINUE 53 0 CG(KP)=(RAJ(KP»2)*U(K»3)+RAJ(KP,3)*U<K+NJ , 3 ) ) * 531 * ALPHA+ 532 * (RAJ(KP,2)*U(K ,4)+RAJ<KP,3)*U(K+NJ,4))* 533 * BETA 534 GO TO 350 535 320 CONTINUE 53 6 CG(KP)=((RAJ(KP,2)-ZETA(KP,U)*U(K,3)+RAJ(KP,1)*U(K-NJ,3>)* 5J7 * ALPHA+ 53 8 * ( (RAJ(KP,2)-ZETA(KP,2)) *U(K,4)+RAJ(KP,1)*U(K-NJ,4)) *BETA-539 * VNETA(KP) 540 GO TO 350 541 325 CONTINUE 542 IF (BC3) GO TO 330 543 IF (BC4) GO TO 335 5*4 BG(K)=G(K,1)*(1+ALPHA)*U(K-NJ,3)+F(K,1)*U(K-l,3)+ 545 * (F(K,2)-THETA(K,1)+G(K,2)M1+ALPHA))*U(K,3)+ • 546 * G(K,3)*(1+ALPHA )*U(K + NJ,3) + 547 * G ( K , l ) M l + BETA) *U(K-NJ,4 )+F(K, l ) *U(K- l ,4 ) + 548 * (FIR(K)+F(K,2)-TH=TA(K,2)+G(K,2)*(1+BETA))*U(K,4)+ 549 * G(K,3)*(1+8ETA)*U(K+NJ,4)+VLAM(K) 550 GO TO 305 551 330 CONTINUE 552 BG(K)=F(K,1)*U(K-1,3)+ 553 * (F(K,2)-THETA(K,1)+G(K,2)*(1+ALPHA))*U(K,3>* 554 * G(K,3)*(1+ALPHA)*U(K+NJ,3)+ 555 * F(K, U*U(K-1 ,4 ) + 556 * (FIR(K)+F(K,2)-THETA(K,2)+G(K,2)*(1+BETA))*U(K,4)+ 154 557 * G(K,3)*(H-8ETA)*U(K<NJ,4)+VLAMIK) 55 8 GO TO 315 559 335 CONTINUE 56 0 BG(K)=G(K,1)*(1+ALPHA)*U(K-NJ,3)+F(K,1)+U(K-l,3>+ 561 * (F( K,2) -THETMK.l ) -4 - (G(K,2 ) -PH I (K,l))*(l *ALPHA) ) *U(K,3 ) + 562 * G(K,1>*(1+6ETA)*U(K-NJ,4)+F(K,1)*U(K-1,4)+ 563 * (F I R (K )+F (K ,2 ) - T H S T A(K ,2 ) + (G(K,2)-PHI(K,2))*(1+BETA >)* 56 4 * U(K,4)+VLAM(K) 56 5 GO TO* 32 0 566 340 CONTINUE 567 BG(K)= C (K,1)*U(K-1,3)+ 568 * (F(K,2)+G(K,2)*(1+ALPHA))*U(K,3)+ 569 * F(K,3)*U(K + 1,3)+G(K,3)*(1+A.LPHA)*U(K + NJ,3) + 570 * F(K,1)*U<K-1,4)+ 571 * (F IR(K)+F(K,2)+G(K,2)*t l + BF.TA))*U(K,4) + 572 * F(K,3)*U(K+1,4)+G(K,3)*(1+BETA)*U(K+NJ,4) 57 3 GO TO 315 574 345 CONTINUE 57 5 BG(K)=G(K, l ) * (1+ALPH a ) *U(K-NJ,3)+F(K»1) *U(K- l , 3) +• 576 * (F(K,2)+(G(K,2)-PHI (K , 1 ) ) * (1 + ALPHA))*U(K,3) + 577 * F ( K,3)*U(K>1, 3) • 578 * G(K, 1)*( H-BETA)+U(K-NJ,4>+F{K,1)*U(K~l,4) + 579 * (F IR(K)+F( K,2)+(G (K,2)-PHI(K, 2) )*( 1+BETA ) )*U( K,4) «• 580 * F(K,3)*U(K+1,4)+VLAM(K) 581 GO TO 320 582 350 CONTINUE 583 IF (K .EQ. NK) GO TO 355 584 F(K,3)=-F(K,3) 585 RAJ(KP,3)=-RAJ(KP,3) 586 CF(K)=F(K,3) 587 CJ(KP)=RAJ(KP,3) 588 355 CONTINUE 589 IF (K .EQ. 1) GO TO 360 59 0 F ( K , 1) =- F ( K , 1) 591 RAJ(KP,1)=-RAJ(KP,1) 592 AF(K)=F(K,1) 593 AJ(KP)=RAJ(KP,1) 594 360 CONTINUE 59 5 F(K,2)=FIR(K)-F(K,2)*THETA(K, 1) 596 RAJ(KP,2) = FI(KP)-P.AJ(KP,2)+ZETA(KP,1) 597 C 598 C * * * * * EQUATIONS ARE NOV* IN THE FORM: 599 C 1. F * U(N+1)*= BG 600 C 2. J * UCN+1) = FI * U(N + D * - CG 60 1 C * * * * * SET UP FOR TRISLV * * * * * 60 2 C 603 BF(K)=F(K,2) 604 BJ(KP)=RAJlKP,2) 605 370 CONTINUE 606 C 6 0 7 c * * * * * WRITE OUT MATRICES TO BE SOLVED (IF DESIRED) * * * * * 608 C 609 IF (ISPEW) 400,375,380 610 375 IF ( . NOT.FIRSTT) GO TO 400 611 380 CONTINUE 612 WPITE(6,1663) 613 K=0 614 DO 395 1=1,NI 615 DO 395 J=1,NJ 616 K=K+1 155 617 CALL BOUND 618 IF (BCD GO TO 385 619 IF (BC2) GO TO 390 62 0 WRITEI6, 1660) K , ( F (K , I Y ) , IY=1,3), (R AJ{K » IY) , IY = 1,3) » 621 *BG(K),CG(K) 62 2 GO TO 395 62 3 385 CONTINUE 62 4 KRI TE(6, 1661) K, ( F (K , I Y ) , IY = 2 ,3 ) , (RAJIK, !Y) , IY = 2,3) » 62 5 * 8G(K),CG(K) 62 6 GC TO 395 62 7 390 CONTINUE 62 8 WRITE(6,1662) K, ( F (K . I Y ) , IY=1,2), (R A J (K , IY) , IY = 1,2) 62 9 *BG(K) ,CG(K) 63 0 395 CONTINUE 631 C 63 2 c * * * * * SOLVE FIRST EQUATION TO FIND U(N + D* 63 3 c * * * * * AF= LOWER DIAGONAL 63 4 c * * * * * BF= .01 AGONAL 63 5 c ***** cp= UPPER DIAGONAL 63 6 c * * * * * BG= RIGHT HAND SIDE VECTOR ON ENTRANCE 63 7 c * * * * * = U(N+U* ON EXIT 63 8 c 63 9 400 CONTINUE 640 CALL TRISL V(NK» AF, BF, CF, BG, 2, £405) 641 GO TO 410 642 405 CONTINUE 64 3 WRITE(6,1671) 6*4 STOP 64 5 410 CONTINUE 64 6 c 64 7 c * * * * * ADD U(N+D* TO CG TERM TG REDUCED FORM OF EQUATION 2 ***** 64 8 c 64 9 K=0 65 0 DO 415 1=1,NI 651 DO 415 J - D N J 652 K = K+1 65 3 KP=( J-D*NI+I 65 4 U(K,2)=BG(K) 65 5 CG(KP)=-CG(KP)+FI(KP)*BG(K) 65 6 415 CONTINUE 65 7 c 65 8 c * * * * * SOLVE SECOND EQUATION * * * * * 65 9 c 66 0 CALL TRISLV(NK, A J , BJ , C J , CG, 2, 6,420) 661 c * * * * * SOLUTION FOR UIN+1) HELD IN CG 66 2 GO TG 425 66 3 420 CONTINUE 664 WRITE(6,1672) 66 5 STOP 66 6 425 CONTINUE 66 7 c 66 8 c * * * * CALCULATE MASS TRANSFER RATE, RMDOT * * * * 66 9 c 67 0 CALL MDOTT(H,RMDCT1,RMDOT) 671 IF(FIRSTT) GOTO 182 672 c 673 c * * * * CALCULATE AVER AGE MOISTURE CONTENT IN ONE OF TWO WAYS. 674 c USE AVGU IF ICASE .LE . 1 (ITS FASTEST) OTHERWISE USE USUM. 67 5 c -676 . IF(ICASE .LE. 1) GOTO 427 156 67 7 DO 426 1=1,NK 67 8 AG(I)=U(I,3) 679 426 CONTINUE 68 0 CALL USUM(AG,NH,UO,UAVG) 681 GOTO 428 68 2 427 CONTINUE 68 3 CALL AVGUIDTLAST,RMDCT1,RMCOT,UAVG) 634 428 CONTINUE 68 5 WRITEC6,1686)RMD0T,UAVG 68 6 GOTO 183 68 7 182 CONTINUE 68 8 UAVG=1. 68 9 RLAST=1. 69 0 WRITE(6,1685)ARI 691 WRITE(6,1684)QBAR(1,1) ,0(1 ,1),LAM1BR(1,1),LAM1{1 ,1) , RMDOT 692 183 CONTINUE 693 IF(FK .EQ.O) GOTO 429 694 IFtUAVG .LT.O.) GOTO 429 69 5 C 696 c * * * * * ALLOW FOR CHANGE OF BODY DIMENSIONS CAUSED BY SHRINKAGE. 697 c ASSUME ISOTROPIC SHRINKAGE. 69 8 c 69 9 0Z= ( (l.+ FK*UO'!>UAVG)/RACFAC)**( 1./3.) 700 R=R0*OZ 701 SCRL=SCRLO*QZ 702 TIMCOR= 0Z**2 70 3 429 CONTINUE 704 IFLAG=IFLAG+1 70 5 K = 0 706 IF(FIRSTT) WRITE(6,1665) 70 7 DO 44 5 I=1,NI 70 8 DO 445 J=1,NJ 70 9 K = K+1 710 KP=(J-1)*NI+I 711 U(K,l)=CG(KP) 712 UU0=U(K, u/uo 713 c * * * * * WRITE OUT MOISTURE CONTENT 714 IF(IABSlISPEW) ,E0 . 10) GO TO 430 715 WRITF(6,1664) K, U (K ,2 ) , U(K,1),UU0 716 GO TO 445 717 43C CONTINUE 718 IF( I FLAG .NE. 100) GOTO 435 719 IF( I .EQ.l.OR.I.EO.NI ) KRITE(6,1664)K,U(K,2),U(K, 1) , UUO 72 0 I F( J.EQ. l.OR. J.EQ.NJ) WRI TE < 6 , 1664 ) K. ,U { K, 2 ) ,U ( K, 1)  UUO 721 GOTO 445 72 2 435 CONTINUE 72 3 IF (K.FQ. l ) WRITF(6,1664)K,U(K,2),U(K,1),UU0 72 4 IF(K.EQ.NK) WRITE!6,1664)K,U(K,2),U(K,1),UUO 72 5 445 CONTINUE 72 6 c * * * * * COMPENSATE TIME TO ALLOW FOR SHRINKAGE. 72 7 RLTIME=RLTIMF+{CT*TIMCOR) 72 8 WRITE16,1666) RLTIME 72 9 IF ( ISPEW.EG.1) STOP 73 0 c * * * * * SET UP U(K,J) FOR NEXT TIME STEP 731 00 450 11=1,NK 732 U( I I ,4)=U( I I ,3) 73 3 U( II ,3)=U(II,1) 73 4 450 CONTINUE 73 5 c * * * * * SHIFT BOUNDARY CONCITION TERMS 73 6 IZ=NK-(NJ-l) 157 73 7 DO 451 I = I Z , N K 73 8 P H I { I , 2 ) = P H I < 1 , 1 ) 73 9 451 C O N T I N U E 740 DO 452 I = N J , N K , N J 741 T H E T A ( I , 2 > = T H E T A ( I , 1 ) 74 2 452 C O N T I N U E 74 3 DO 453 I = N I , N K , N I 7 4 4 Z E T A ( I , 2 ) = Z E T A ( I , 1 ) 74 5 453 C O N T I N U E 74 6 I F t R H O I N F . E Q . O . A N D . . N O T . F I R S T T ) GOTO 4 5 9 ' 74 7 DO 4 5 6 1 = 1 , N I 74 8 L AMI B R ( I , 2 ) = L A M I BR( I , 1 ) 74 9 4 5 6 C O N T I N U E 75 0 CO 4 5 8 J = 1 , N J 751 L A M 1 ( J , 2 ) = L A N , 1 ( J , 1 ) 75 2 4 5 8 C O N T I N U E 75 3 4 5 9 C O N T I N U E 75 4 I F ( I F L A G . E G . 1 0 0 ) I F L A G = 0 75 5 F I R S T T = . F A L S E . 75 6 N T I M E = N T I M E + 1 75 7 C * * * * * T E S T T I M E 75 8 IF ( R L T I M E . G E . R U U F ) S T O P 75 9 c * * * * * S c E jp S P E C I A L C A S E R E D U C E C A L C U L A T I O N S FOR 76 0 c * * * * * L A T E R R U N S . 7 6 1 I F ( I C A S E . G T . O . O R . FK . G T . 0 . ) GO TO 4 8 0 76 2 K = 0 76 3 DO 4 7 0 1 = 1 , N I 764 DO 4 7 0 J = 1 , N J 7 o 5 K=K + 1 76 6 C A L L BOUND 76 7 ! F ( . N O T . ( B C 2 . 0 R . B C 4 ) ) GOTO 4 6 0 76 8 I K = I + N J - J 76 9 HI I K ) = U ( K , 3 ) /UF 77 0 460 C O N T I N U E 771 I F ( K . E Q . N K > GO TO 4 6 2 772 F ( K , 3 ) = - F ( K , 3 ) 77 3 R A J ( K , 3 ) = - R A J ( K , 3 ) 77 4 462 C O N T I N U E 77 5 I F ( K . E O . l ) GO T O 4 6 5 7 7 6 F ( K , 1 ) = - F ( K , 1 ) 77 7 R f i J I K . l ) = - R A J ( K , l ) 77 8 465 C O N T I N U E 7 7 9 F ( K , 2 > = F I R ( K ) - F ( K , 2 ) + T H E T A { K , 1) 78 0 R A J ( K , - 2 ) = F I ( K ) - R A J ( K , 2 ) + Z E T A ( K , 1 1 73 1 4 7 0 C O N T I N U E 73 2 D T L A S T = D T 78 3 DTP= C H G T I M ( N T I M E ) 73 4 I F ( D T P . E O . C T ) GOTO 4 7 5 7 3 5 DT=DTP 73 6 R R = D T / { H H * H H ) 73 7 RAT1=3./(2*RR> 7 8 8 c 78 9 c * * * * A A 1 , A A 2 , B B , A N D BOUNDARY C O N D I T I O N S ARE C O N S T A N T . 79 0 c **** pK- o. 79 1 c 79 2 GO TO 185 793 4 7 5 C O N T I N U E 79 4 c 79 5 c * * * * DT , A A 1 , A A 2 , B B , A N D , BOUNDARY C O N D I T I O N S A R E C O N S T A N T 79 6 c * * * * F K = 0 . 158 79 7 C 79 8 GO TO 300 • 79 9 C 80 0 C * * * * I C A S G > 0 OR FK >0 80 1 C 802 4 8 0 C O N T I N U E 803 O T L A S T = O T 80 4 C ***** IF D E S I R E D C H A N G E T I M E S T E P . 80 5 D T = C H G T I M ( N T I M E ) 806 GO TO 25 80 7 END 80 8 F U N C T I O N A A ( L , U , T ) 80 9 C 8 1 0 C * * * * C A L C U L A T I O N OF C O M B I N E D D I F F U S I V I T I E S OF WATER IN 8 i 1 c * * * * WOOD. A A l IS T H E R A D I A L D I F F U S I V I T Y , AA2 IS T H E 812 c * * * * L O N G I T U D I N A L D I F F U S I V I T Y . O N E X I T : 813 c AA1 = A A l / A R I 8 1 4 c A A 2 = A A 2 * ( R / S C R L ) * * 2 / A R I 8 1 5 COMMGN / P A R / R , S C R L , F K , R H O D W 816 COMMON / A / C I , C 2 , G T , U F , R H O S A T , ARI 8 i 7 COMMON / A C O N / R D I F F N . Z D I F F N , I C A S E 8 1 8 I F ( I C A S E . E C . 0 . O R . I C A S E . 5 0 . 1) GO TO 3 0 8 1 9 I F ( U . L T . 0 . ) U = 0 . 82 0 V 0 I D = 1 - C 2 M ( l + C l * U ) / ( 1 + F K * U ) ) 821 S V O I D = S Q R T ( V O I D ) 822 D B P R I = ( 4 6 . 8 4 * U * * 2 - 4 1 0 . 9 3 * U * * 3 + 1 1 0 6 . 1 1 * U * * 4 ) * 82 3 * E X P I - 4 1 2 0 . 4 3 / T ) 82 4 D B = C 2 * D B P R I / ( l - V 0 I D ) * * 2 82 5 X = U / U F 82 6 c 82 7 c * * * * * U S E F I T FOR H V E R S U S U / U F TO F I N D 82 8 c S L O P E OF I S C T H E R M , C H O U . 82 9 DHDU2 5 = 3 . 3 2 1 1 1 E - 0 1 + 1 9 . 7 4 0 0 9 8 * X - 7 . 0 8 7 4 4 6 E + 0 1 * X * * 2 8 3 0 * + 8 . 4 4 1 6 4 E + 0 1 * X * * 3 - 3 2 . 4 4 3 0 5 * X * * 4 831 D E L H = 1 . 0 4 485 6 + 2 . 1 2 5 8 8 2 * X - 6 9 . 2 1 9 1 8 * X * * 2 + 2 4 4 . 4 6 7 9 2 * X * * 3 83 2 * - 3 4 8 . 3 2 6 9 * X * * 4 + 2 1 9 . 9 0 6 1 * X * * 5 - 4 9 . 9 1 2 5 7 * X * * 6 833 D H D U = D H D U 2 5 + G T * D E L H 8 3 4 D H O U = D H D U / U F 83 5 D V P R I = . 2 2 * < T / 2 7 3 )**1 . 7 5 83 6 DV= R H O S A T / R H O D W * ( 1 + F K * U ) * * 2 * D V P R I * D H D U 83 7 I F ( L . E Q . 2 ) GO T O 10 83 8 c 83 9 c * * * * C A L C U L A T E D I F F U S I O N C O E F F I C I E N T S FOR R A D I A L D I R E C T I O N * * * 840 c 84 1 D l = ( 1 - S V C I D ) * D B 84 2 C2=0V 8 4 3 D 3 = < ( S V O I D - . 0 1 4 5 2 ) / ( 1 - S V O I D ) ) * D 8 84 4 D 4 = ( 4 . 6 6 E - 5 / 1 ( ( 1 - S V O I D ) / 3 0 0 > - 1 . E - 4 ) ) * D V 84 5 D 5 = . 4 6 6 * D B 84 6 D 6 = 4 . 3 E - 4 * D V 84 7 GO T G 20 84 8 c 849 c * * * * C A L C U L A T E D I F F U S I O N C O E F F I C I E N T S FOR A X I A L D I R E C T I O N * * * 85 0 c 85 1 10 C O N T I N U E 852 D l = ( l - V O I 0 ) * D 8 8 5 3 D2=VCI0*DV 8 5 4 D 3 = ( ( S V 0 I D - . 0 1 7 8 ) / ( ( 1 - S V O I D ) * . 0 0 83) *DB 85 5 4 . 0 1 4 * D V / ( ( - S V O I D ) * . 0 0 8 3 - 2 . 5 E - 4 ) 8 5 6 D5=56*DB 159 85 7 D 6 = . 3 8 * D V 85 8 20 C O N T I N U E 8 5 9 Z 0 T l = D 4 + 0 5 + C 6 86 0 Z 0 T 2 = D 4 * { 0 5 + 0 6 1 86 1 D D = ( D 3 * Z Q T l + Z O T 2 ) * D 2 / ( ( 0 2 + 0 3 ) * Z O T l + Z O T 2 ) 862 DD=DD+01 863 GO TO 50 8 6 4 C * * * * C A L C U L A T E D I F F U S I V I T I E S FOR C O N S T A N T C O E F F E C I E N T C A S E 865 30 C O N T I N U E 866 I F ( L . E Q . 2) GO T O 40 8 6 7 C * * * * C A L C U L A T E R A D I A L D I F F U S I V I T Y 868 D D = R D I F F N 8 6 9 GO TO 50 87 0 4 0 C O N T I N U E 871 C * * * * C A L C U L A T E A X I A L D I F F U S I V I T Y 872 DD = Z O I F F N 873 5C C O N T I N U E 8 7 4 IF ( L . E d . 2 ) D D = D D * ( R / S C R L ) * * 2 875 A A = D C / ( 1 + F K * U ) * * 2 876 I F ( ! F ! X ( A R ! 1 . E O . 1 . A N D . I C A S E . L E . I ) AA=DD 8 7 7 A A = A A / A R I 8 7 8 R E T U R N 8 7 9 END 8 8 0 F U N C T I O N B B ( U , F K ) 8 8 1 B B = 1 . / ( 1 . + F K * U ) 882 R E T U R N 83 3 END 884 S U B R O U T I N E BOUND 8 8 5 C * * * * * D E T E R M I N E I F ON 3 0 U N D A R Y ( S E E GRID A T S T A R T CF P R O G R A M . ) 8 3 6 - COMMON / O C / B C 1 , BC2 , B C 3 , B C 4 , I , J , N I , N J , F I R S T T 837 L O G I C A L B C 1 , B C 2 , BC 3 , B C 4 , F I R S T T 838 B C 1 = . F A L S E . 8 3 9 B C 2 = . F A L S E . 8 9 0 B C 3 = . F A L S E . 891 8 C 4 = . F A L S E . 8 9 2 I F ( J . E G . 11 B C 1 = . T R U 5 . 893 I F ( J . E Q . N J ) B C 2 = . T R U E . 894 I F ( I . E Q . 1) B C 3 = . T R U E . 895 I F ( I . F Q . N I ) B C 4 = . T R U E . 8 9 6 R E T U R N 697 END 8 9 8 F U N C T I O N C H G T I M ( N X ) 8 9 9 C * * * * * C H A N G E T I M E I N C R E M E N T IN S T E P S , ( S E E D E F I N I T I O N S A T S T A R T . 9 0 0 COMMON / C H / S T E P 1 , S T E P 2 , S T E P 3 , L G C I , L 0 C 2 9 0 1 I F ( N X . G T . L O C l ) GO TO 10 9 0 2 C H G T I M = S T E P 1 90 3 GO TO 40 9 0 4 10 C O N T I N U E 905 I F ( N X . G T . L 0 C 2 ) GO TO 2 0 9 0 6 C H G T I M = S T E P 2 9 0 7 GO TO 4 0 908 20 C O N T I N U E 9 0 9 C H G T I M = S T E P 3 9 1 0 4 0 C O N T I N U E 9 1 1 R E T U R N 912 END 9 1 3 S U B R O U T I N E M O O T T ( H , R M D O T 1 , P M 0 0 T 1 9 1 4 C * * * * S I M P S O N ' S R U L E I N T E G R A T I O N OF S U R F A C E M A S S T R A N S F E R 9 1 5 C TO F I N D T O T A L R A T E OF MASS T R A N S F E R FROM BODY (RMDOT 1 9 1 6 COMMON / M / F K C R » F K C Z » R H Q I N F , R O 160 9 1 7 COMMON / A / C 1 , C 2 , G T , U F , R H O S A T , A R T S18 COMMON / P A R / R , S C R L , F K , R H O O W 9 1 9 COMMON / E C / B C 1 , B C 2 , 8 C 3 , 3 C 4 , I , J , N I , N J , F I R S T T . 9 2 0 L O G I C A L B C 1 , B C 2 , B C 3 , B C 4 , F I R S T T 921 D I M E N S I O N H ( 2 0 0 ) 922 C * * * * H C O N T A I N S V A L U E S OF H U M I D I T Y FOR C Y L I N D R I C A L S U R F A C E 9 2 3 C FROM 1 T O MI AND FOR' END S U R F A C E FROM P E R I M E T E R TO 9 2 4 C C E N T E R . C E N T E R H H E L D AS H ( N I K ) 92 5 C MINIMUM NI OR N J I S 3 9 2 6 N I K = N I + N J - 1 9 2 7 N J 1 = N J - 1 9 2 8 N J 2 = N J - 2 9 2 9 DO 1 I J = 1 , N J 2 9 3 0 I K = M + I J 9 3 1 H ( I K ) = ( 1 . - 1 . * ! J / N J 1 ) * H ( ! K ) 93 2 1 C O N T I N U E 93 3 N I 2 = N I - 2 9 3 4 N I 1 = NI + 1 9 3 5 N I K 2 = N I K - 2 9 3 6 N I l l = N I + 2 9 3 7 C * * * * T E S T FOR ODD OR E V E N NUMNBER OF I N T E R V A L S 9 3 8 C * * * * H B A R R = I N T E G R A L FROM 0 T O 1 OF H+DZ 9 3 9 C * * * * HBARZ = INTi;-GRAL FROM 0 TO 1 OF R * H * D R 9 4 0 I F { N I / 2 * 2 . E C . N I ) GO TO 10 94 1 GO TO 20 9 4 2 5 C O N T I N U E 9 4 3 I F ( N J / 2 * 2 . E G . N J ) GC TO 30 94 4 GO TO 40 94 5 10 C O N T I N U E 9*6 S U M = H l 1 ) * 2 + H ( N I ) 9 4 7 DO 15 I K = 2 , N I 2 , 2 948 S U M = S U M + 2 * H ( I K ) + 4 * H ( I K + 1) 9 4 9 15 C O N T I N U E 9 5 0 H B A R R = ( 1 . / ( N I - 1 ) ) / 3 * S U M 951 GO T O 5 9 5 2 20 C O N T I N U E 9 5 3 S U M = H ( 1 ) + 4 * H ( 2 ) + H ( N I ) 95 4 DO 21 ! K = 3 , M 2 » 2 9 5 5 IF ( I K . G T . N I 2 ) GO TO 21 9 5 6 S U M = S U M + 2 * H ( I K ) + 4 * H < I K + 1 ) 9 5 7 21 C O N T I N U E 9 5 8 H B A R R = ( 1 . / ( N I - 1 > ) / 3 * S U M 9 5 9 GO TO 5 9 6 0 30 C O N T I N U E 961 S U M = H ( N I ) * 2 9 6 2 DO 31 IK = N U , N I K 2 , 2 963 S U M = S U M + 2 * H ( I K ) + 4 * H < I K + 1 ) 9 6 4 31 C O N T I N U E 9 6 5 HEAR Z = ( 1 . / N J 1 ) / 3 * S U M 9 6 6 GO TC 50 96 7 40 C O N T I N U E 9 6 8 S U M = H ( N I ) + 4 * H ( N I l ) 9 6 9 00 41 I K = N ! 1 1 , N I K 2 , 2 9 7 0 I F ( I K . G T . N I K 2 ) GO T O 41 971 S U M = S U M + 2 * H ( I K ) + 4 * H ( I K + 1 ) 97 2 41 C O N T I N U E 9 7 3 H E A R Z = ( 1 . / N J 1 ) / 3 * S U M 9 7 4 50 C O N T I N U E g 7 5 C * * * * RMOOT= T O T A L MASS T R A N S F E R FROM F I N I T E C Y L I N D E R S U R F A C E 9 7 6 C * * * * FORMULA A P P L I E S FOR C O N S T A N T T E M P E R A T U R E C A S E ONLY 161 9 7 7 P I = 3 . 1 4 i 5 9 3 9 7 8 R K O N = 2 * P I * R * R H O I N F * ( 2 * S C R L * F K C R + R * F K C Z ) 97 9 R M D 0 T ' = 4 * P I * R * R H 0 S A T * ( S C R L * F K C R * H B A R R + R * F K C Z / ! ' H B A R Z } - R K O N 9 3 0 C * * * * N O N D I M E N S I C N . A L I Z E RMDOT U S I N G AR I AND I N I T I A L R A D I U S 9 8 1 R M D Q T = R M D Q T * R 0 * * 2 / A R I 9 8 2 I F ( F I R S T T ) R M D 0 T 1 = R M D 0 T 983 I F ( F I R S T T ) R F T U R N 9 8 4 R M D 0 T = R M D 0 T / R M D 0 T 1 9 3 5 R E T U R N 9 3 6 END 9 3 7 S U B R O U T I N E S U R F T R ( T , R E N L D , S C P R , R , F K C R , F K C Z ) 98 8 C 939 C * * * * C A L C U L A T E S U R F A C E MASS T R A N S F E R C O E F F I C I E N T S , F K C R » F K C Z . 99 0 C 991 D M O L E = . 2 2 * ( ( T / 2 7 3 . ) * * 1 . 7 5 ) 992 S H B A R = . 6 1 5 * { S C P R * - * ( 1 . / 3 . ) ) * ( R E N L D * * . 4 6 6 ) 9 9 3 SH= SHB AR 9 9 4 F K C R = D M 0 L E * S H B A R / ( 2 . * R ) 9 9 5 F K C Z = D M O L E * S H / ( 2 . * R ) 9 9 6 R E T U R N 9 9 7 END 998 S U B R O U T I N E A V G U ( D T , R M D O T l , R M D O T , U A V G ) 9 9 9 COMMON / B C / B C 1 , BC2 , BC3 , B C 4 , 1 , J . N I . N J , F I R S T T 1 0 0 0 COMMON / A V E R G / R M A S S , R L A S T , T I M C C R 1 0 0 1 C * * * * * I C A L C U L A T E T H E A V E R A G E M O I S T U R E C O N T E N T A T 1 0 0 2 C * * * * * ANY T I M E BY I N T E G R A T I N G D R Y I N G R A T E ONE S T E P AT A T I M E . 1 0 0 3 C * * * * * U A V G I S T H E A V E R A G E U ON E N T R A N C E AND T H E NEW 1 0 0 4 C * * * * * ONE ON FX I T 100 5 U A V G = U A V G - R M O O T l * O T * T I M C O R M RL AS T+RMDOT ) / R M A S S 1006 . R L A S T = R M D O T 1 0 0 7 R E T U R N 100 8 END 1 0 0 9 S U B R O U T I N E U S U M ( P , N H , U 0 , U A V G ) 1 0 1 0 C * * * * I N T E G R A T E U OVER V O L U M E U S I N G T R A P E Z O I D R U L E IN B O T H 1 0 1 1 C * * * * D I R E C T I O N S , F I R S T I N T H E Z D I R E C T I O N T H E N I N 1 0 1 2 C * * * * R D I R E C T I O N . 1 0 1 3 D I M E N S I O N P ( 2 C 0 0 ) 1 0 1 4 NI=NH+1 1 0 1 5 NJ=NI 1 0 1 6 N E N D I = N H 1 0 1 7 HHH = 1 . / N H 1018 NLA ST = N E N D I * N J 1 0 1 9 R I N T = 0 * 1 0 2 0 DO 10 J = 2 , N J 1 0 2 1 S U M = 0 . 0 1 0 2 2 DG 20 I = 2 , N E N D I 1 0 2 3 K = ( T - l ) * N J + J 1 0 2 4 SUM=SUM+P(K) 1 0 2 5 20 C O N T I N U E 1 0 2 6 S U M = S U M + ( P { J ) + P ( N L A S T + J ) ) / 2 . 102 7 SUM=SUM*HHH 1 0 2 8 I F ( J . E O . N J ) S U M = S U M / 2 . 1 0 2 9 R I N T = ( J - 1 ) * H H H * S U M + R I N T 1 0 3 0 10 C O N T I N U E 1 0 3 1 R I N T = R I . N T * H H H * 2 . 1 0 3 2 U A V G = R I N T / U 0 1 0 3 3 R E T U R N 1 0 3 4 E N D APPENDIX I T S I S L V 162 Purpose To f i n d the solution to a system of l i n e a r equations of the form Ax=b where A i s a t r i - d i a g o n a l matrix (a form often found in the numeric solution of p a r t i a l d i f f e r e n t i a l equations). Available i n the public f i l e *NUMLIB. CALL TRISLV (N,DL, DD,DU, RH, MODE, Snn) where: H i s the number of equations. DL i s a one-dimensional array, dimensioned at l e a s t N. Its type depends on the value of MODE. On entry, i t contains the lower diaqonal c o e f f i c i e n t s s t a r t i n g at DL(2). The contents of DL are not changed by TRISLV and DL(1) i s not referenced. DD i s a one-dimensional array, dimensioned at l e a s t N. Its type depends on the value of MODE. On entry, i t contains the diagonal c o e f f i c i e n t s . DO i s a one-dimensional array, dimensioned Its type depends on the value of MODE, contains the upper diagonal set equal to zero by TRISLV. at On c o e f f i c i e n t s , l e a s t N. entry, i t DO (N) i s Ril i s a one-dimensional array, dimensioned at le a s t M. Its type depends on the value of MODE. On entry, i t contains the rig h t hand side, and on e x i t , the solution vector. MODE ind i c a t e s the type of the arrays DL, DD, DU, and RH and the kind of matrix being entered. =0 i f the arrays are REAL*8 and the o r i g i n a l matrix, A, i s entered. =1 i f the arrays are REAL*8 and the decomposion of the matrix, A, i s entered. -2 i f the arrays are REAL*4 and the o r i g i n a l matrix, a, i s entered. 1 6 3 = 3 i f the arrays are REAL*4 and the decoiaposion of the matrix, A, i s entered. nn i s the statement number in the c a l i i n q program to which con t r o l i s returned i f no solution i s found. Comment For the I°th equation (1 < I < N) and MODE even: DL (I) *X (1-1) +DD (I) * X (I) +DD (I) * X (1 + 1) = H H ( I ) Accuracy. The accuracy depends on the condition of the matrix. The routine performs best on a diagonally dominant matrix. The sample output i l l u s t r a t e s t h i s point. Timing An estimate of the CPU time on the 370/168 in microseconds i s : MODE Time 0 1 2 3 16*N 8*N 1 1 *N 6*N NOMENCLATURE 164 A(U,T) d i f f u s i v i t y o f moisture w i th re spec t to a g rad ien t i n mass of water per volume dry wood (cm 2 /sec) Ag(U,T) A R (U ,T ) /A R (U ,T ) . A*(U,T) (R/L ) 2 A z ( U , T ) / A R ( U , T ) i B(U) r a t i o o f drywood volume to wet wood volume as given by Equation 17 Bi K(LUT) ' ^ 1 0 t n u m ' 3 e r KpL Po-,4- u Bi -.„/,, T\ — Ti , B i o t mass number A(U,T) PD U m C(U,T) s p e c i f i c heat o f wet wood (cal/g-°C) C*(U,T) C (U ,T ) /C (U ,T ) . D. i = 1 , 2 , - «» v6 pathway d i f f u s i v i t i e s Dg d i f f u s i v i t y o f l i q u i d i n wood substance w i t h re spect to a g rad ient i n mass o f water per volume o f wet wood (cm 2 / sec) d i f f u s i v i t y o f moisture w i th respect to a g rad ien t i n mass o f water per volume o f wet-,wood (crrr/sec) Dy d i f f u s i v i t y o f water vapour w i th re spect to a g rad ient i n mass o f water per volume wet wood (cm 2 /sec) f f r a c t i o n o f moisture d i f f u s i o n tha t i s l i q u i d h convect i ve mass t r a n s f e r c o e f f i c i e n t (cal/cm sec°C) , r e l a t i v e humidity p s a t I net i n t e r n a l evaporat ion per u n i t volume as de f ined by Equat ion 54 K g r i d po i n t index k shr inkage f a c t o r as de f ined by Equation 17 K(U,T) thermal c o n d u c t i v i t y o f wet. wood (cal/cm-sec-°C) 1 6 5 Kft(=U,T) K R (U,T)/K R (U,T) n . K*(U,T) (R/L ) ?K z (U ,T )/K R (U ,T ) i Kg convect i ve mass t r a n s f e r c o e f f i c i e n t (cm /sec) a Ko 7^ j ) C ( y T ) 5 Kossovich number a i KP g r i d po i n t index L c y l i n d e r h a l f l ength (cm) L c h a r a c t e r i s t i c l e n g t h ; L. f o r the a x i a l d i r e c t i o n and Rj f o r the r a d i a l d i r e c t i o n (cm) Lu A(U,T) / a(U,T), Luikov number m mass f l u x (g/cm sec) M* ac tua l d r y i ng r a t e / i n i t i a l d r y i ng r a te NJ number o f g r i d po in t s i n the r a d i a l o r a x i a l d i r e c t i o n s NK t o t a l number o f g r i d po in t s P pressure r r a d i a l coord inate (cm) r* r/Rj R c y l i n d e r rad ius (cm) S 1/ (NJ - 1 ) , d imens ion less g r i d spacing t time (sec) t * A R (U,T) 1 - t/R.j, d imens ionless time f o r mass t r a n s f e r 2 t * K R (U,T) 1 . t/R.j, d imens ion less time f o r heat t r a n s f e r T temperature (°C) T a " T T* j , d imens ionless temperature a i U mass water/mass dry wood, moisture content z * a a (U ,T ) . K(U,T)/ • P DC(U,T) 6 6 A e A Q P a Subsc r i p t s a B D h i K KP 166 moisture content a t f i b e r s a t u r a t i o n volume (cm ) a x i a l coo rd ina te (cm) z / L . r e l a x a t i o n parameter thermal d i f f u s i v i t y o f wood (cm /sec) r e l a x a t i o n parameter c e n t r a l d i f f e r e n c e operator d i f f e r e n c e increment i n quan t i t y vo id f r a c t i o n (volume void/volume wetwood) dimension!ess ambient c o n d i t i o n as given by Equations 5 and 8 o f o rder o f 2 At*/S , mesh r a t i o o f numerical method o den s i t y (g/cnn) l a t e n t heat o f evaporat ion (ca l/g) ambient c o n d i t i o n o f d r y i ng media bound water dry wood i n d i c a t e s v a r i a b l e app l i e s to heat t r a n s f e r i n i t i a l c o n d i t i o n g r i d po i n t l o c a t i o n g r i d po i n t l o c a t i o n L m R S sa t V ws Z Supe r s c r i p t s n-1 n (n+1)* (n+1) v a r i a b l e app l i e s to l i q u i d v a r i a b l e app l i e s to mass t r a n s f e r r a d i a l d i r e c t i o n wet wood sa tu ra ted water vapour water vapour wood substance a x i a l d i r e c t i o n average quan t i t y prev ious time present time in te rmed ia te time advanced time 

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