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Heat transfer and pressure drop in fixed beds of wood chips Chow, Bosco 1985

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HEAT TRANSFER AND PRESSURE DROP IN FIXED BEDS OF WOOD CHIPS by BOSCO CHOW B.Sc. The University of Ottawa, 1982 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Chemical Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1985 ® Bosco Chow, 1985 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of CHW(C0-L EA#7 //A/>g:E/f/Al. Qj The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date "7^7/ ABSTRACT Heat t r a n s f e r from a f l o w i n g gas to a f i x e d bed of d r i e d D o u g a s - f i r wood c h i p s has been s t u d i e d by a t r a n s i e n t method. Hot a i r a t about 130°C flowed upward through 0.2 m d i a x 1 m deep beds of c o m m e r c i a l l y prepared wood c h i p s which had been screened f o r t h i c k n e s s . Four d i f f e r e n t wood c h i p s i z e s were used, which v a r i e d i n mean t h i c k n e s s from 2.44 to 7.26 mm. The t h i c k e s t c h i p s were 18.4 mm wide x 36.3 mm l o n g . Gas temperatures were measured at a number of a x i a l p o s i t i o n s as the bed temperature rose from i t s i n i t i a l temperature of about 20°C. Heat 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 by f i t t i n g the a i r temperature p r o f i l e s to a t r a n s i e n t mathemical model f o r p l u g f l o w o f gas through a bed of s l a b - s h a p e d p a r t i c l e s w i t h f i n i t e i n t e r n a l t h e r m a l r e s i s t a n c e . The heat t r a n s f e r model was s o l v e d a n a l y t i c a l l y u s i n g an approach p i o n e e r e d by Amundson (10) f o r f i x e d beds of s p h e r i c a l p a r t i c l e s and based on Rosen's (6,7) f u n c t i o n . T h i s s o l u t i o n has not appeared elsewhere i n the l i t e r a t u r e , and i s shown to converge to t h a t of A n z e l i u s (1) i f the B i o t number f o r the p a r t i c l e approaches z e r o . E x p e r i m e n t s were done at a s e r i e s of a i r v e l o c i t i e s w i t h f o u r wood-chip t h i c k n e s s e s and w i t h s p h e r i c a l c a t a l y s t p a r t i c l e s to p r o v i d e a check on the t e c h n i q u e . The e f f e c t on h e a t i n g r a t e of 30% by volume steam i n the incoming a i r was i n v e s t i g a t e d . For s e l e c t e d e x p e r i m e n t s , s o l i d t e m peratures w i t h i n the wood c h i p s were measured. A c o r r e l a t i o n of the heat t r a n s f e r c o e f f i c i e n t s i s p r e s e n t e d . P r e s s u r e drop was measured as a f u n c t i o n of a i r v e l o c i t y f o r d i f f e r e n t s i z e s of wood c h i p s a t room temperature and the r e s u l t s are compared w i t h p r e d i c t i o n s of the Ergun e q u a t i o n . - i i i -TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF FIGURES vi LIST OF TABLES v i i i ACKNOWLEDGEMENT . . ix 1. INTRODUCTION 1 1.1 Major Objectives of the Research Program... . 1 1.1.1 Background 1 1.2 Experimental Plan • • • • 8 2. MATHEMATICAL MODELLING AND SIMULATION FOR FIXED BED HEATERS 11 2.1 Pressure Drop for Incompressible Flow through Packed Beds 11 2.2 Gas/Solid Heat Transfer to Particles with Negligible Internal Thermal Resistance 13 2.3 Gas/Solid Heat Transfer to Beds of Large and Low Conductivity Spheres 16 2.4 Gas/Solid Heat Transfer to Beds of Low Conductivity and Rectangular Slab-Shaped Part icle 21 3. EXPERIMENT 26 3.1 Apparatus 26 3.2 Part icle Preparation 27 3.3 Pressure Drop Measurements 37 3.4 Heat Transfer Measurements 37 - i v -Page 4. RESULTS AND DISCUSSION . 4 3 4.1 Pressure Drop 43 4.2 Heat Transfer to S p h e r i c a l P a r t i c l e s 67 4.3 Heat Transfer to Wood Chips...... 75 4.3.1 E f f e c t of A i r V e l o c i t y 112 4.3.2 E f f e c t of Wood Chip Thickness 114 4.3.3 E f f e c t of Steam-Air Mixture 116 4.3.4 I n t e r n a l S o l i d Temperature 120 4.4 C o r r e l a t i o n of Results 123 4.5 Biot Number f o r Wood Chips 130 4.6 Comparison with Design Values used by B l a c k w e l l (19,21) 130 5. CONCLUSIONS 134 6. RECOMMENDATIONS 136 REFERENCE. 137 NOMENCLATURE 139 APPENDICES 142 APPENDIX A RAW DATA . 142 APPENDIX B CALCULATION OF WOOD CHIP AND SPHERE CATALYST THERMAL PARAMETER 162 B-l Wood Chip Thermal Parameters C a l c u l a t i o n 162 B-2 Sphere C a t a l y s t Thermal Parameters C a l c u l a t i o n 163 Page APPENDIX C MATHEMATICAL MODEL ANALYTICAL SOLUTION.. 164 C-l Solution for Gas/Solid Heat Transfer to Beds of Large and Low Conductivity Spheres 164 C-2 Solution for Gas/Solid Heat Transfer to Beds of Low Conductivity and Rectangular Slab-Shaped Part icle 170 APPENDIX D COMPUTER PROGRAM 180 D-l Computer Program for Schumann's Model . . . . 180 D-2 Computer Program for Gas/Solid Heat Transfer to Beds of Large and Low Conductivity Spheres 187 D-3 Computer Program for Gas/Solid Heat Transfer to Beds of Low Conductivity and Rectangular Slab-Shaped P a r t i c l e . . . . . 193 D-4 Computer Program for Heat Transfer Coefficient Calculation for Slab Part icle 200 - v i -LIST OF FIGURES Figure No. Page 1- 1 Schematic of apparatus 9 2- 1 T h e o r e t i c a l gas temperature p r o f i l e s of An z e l i u s ' model 15 2-2 T h e o r e t i c a l gas temperature p r o f i l e s f o r a bed of spheres 20 2- 3 T h e o r e t i c a l gas temperature p r o f i l e s f o r s l a b s . . . 23 3- 1 D e t a i l of column 28 3-2 D e t a i l of heat exchanger 29 3-3 C a l i b r a t i o n of steam rotameter 30 3- 4 C a l i b r a t i o n of a i r rotameter 31 3-5 - 3-9 Photographs of p a r t i c l e s 34-36 3- 10 - 3-13 Photographs of thermocouples 39-42 4- 1 - 4-10 Experimental pressure drop versus Ergun equation p r e d i c t i o n f o r spheres and wood chips 45,46 with d i f f e r e n t thickness 51-58 - 4-11 - 26 T h e o r e t i c a l and experimental temperature p r o f i l e s comparisons f o r spheres and wood chips 69-71 with d i f f e r e n t thickness 51-42 4-27 - 4-29 Experimental gas temperature p r o f i l e s f o r wood chips 95-97 4- 30 Temperature p r o f i l e s f or 4.03 mm t h i c k wood chip with d i f f e r e n t heat t r a n s f e r c o e f f i c i e n t . . . . 98 4-31 - 4-32 I n t e r n a l wood chip and gas temperature for 2.44 mm and 4.03 mm t h i c k c h i p s . . 118,119 4-33 - 4-3^ Comparison of temperture p r o f i l e s f or dry and moist a i r for 4.03 mm and 7.26 mm t h i c k chips....121,122 4-35 - 4-36 P l o t s of t h e o r e t i c a l and moist a i r experimental p r o f i l e s f or 4.03 mm t h i c k chips and 7.26 mm t h i c k chips 1 24 ,125 - v i i -Figure No. Page 4-37 Comparison of wood chip and sphere daCa with l i t e r a t u r e values 126 4-38 P l o t of hj: versus Reynolds Number 129 4-39 P l o t of h T versus Reynolds Number 13L - v i i i -LIST OF TABLES Table Mo. Page 3-1 Dimension of experimental apparatus 27 3- 2 P r o p e r t i e s of p a r t i c l e s 33 4- 1 Summary of pressure drop c o n d i t i o n s and data 44 4-2 - 4-3 T h e o r e t i c a l and experimental pressure drop values for sphere 47,48 4-4 Summary of pressure drop parameters f o r Ergun equation 49 4-5 - 4-12 T h e o r e t i c a l and experimental pressure drop values f o r wood chip p a r t i c l e s : 5,6) 2.44 mm t h i c k chips 7,8) 4.03 on t h i c k chips 9,10) 7.26 mm t h i c k chips 11,12) 4.81 mm (mixed) t h i c k chips 59-66 4-13 Heat t r a n s f e r t e s t c o n d i t i o n s and r e s u l t s f o r s p h e r i c a l p a r t i c l e s 68 4-14 P h y s i c a l p r o p e r t i e s of sphere c a t a l y s t p a r t i c l e . . 73 4-15 - 4-32 T h e o r e t i c a l and experimental gas temperature values f o r sphere and wood chip p a r t i c l e s 15,16,17,18,,19) sphere Re = 87 to 228 20,21,22,23) 2.44 ram t h i c k chips at Re = 255, 329, 403, 476 24,25,26,27) 4.0L mm t h i c k chips at Re = 374, 482, 590, 698 28,29,30) 726 nm t h i c k chips at Re = 490, 632, 774 76-80 31,32) 4.81 mm t h i c k chips at Re = 399 , 629 99-111 4-33 Heat t r a n s f e r t e s t c o n d i t i o n s and r e s u l t s f or wood chips 113 4-34 C o r r e l a t i o n of heat t r a n s f e r c o e f f i c i e n t ( h ^ ) . . . . 128 4-35 Summary of r a t i o f o r a-p/an 1-30 4-36 C o r r e l a t i o n of heat t r a n s f e r c o e f f i c i e n t ( h T ) . . . . 132 4-37 Experimental raw data 142-161 - i x -ACKNOWLEDGEMENT The author would l i k e to thank a d v i s o r , Dr. A.P. Watkinson, for h i s guidance and encouragement. Thanks are a l s o due to the u s e f u l advice and suggestions of Dr. B. Bl a c k w e l l during the formation and experimental phases of t h i s work. Furthe r , the author i s g r a t e f u l to Dr. R.M. Miura, Professor of Mathematics Department at U.B.C. who has checked the general mathematical s o l u t i o n . I am g r a t e f u l to Energy, Mines and Resources Canada to provide the fund for the work on t h i s p r o j e c t . - 1 -1. INTRODUCTION 1.1 Major O b j e c t i v e o f the R e s e a r c h Program 1.1.1 Background The p r e h e a t i n g of wood c h i p s p r i o r to p u l p i n g as proposed by B l a c k w e l l et a l . (19,21) has the p o t e n t i a l to save s u b s t a n t i a l q u a n t i t i e s of steam i n pulp and paper m i l l s . The f l u e gas from the r e c o v e r y b o i l e r of K r a f t m i l l s has been suggested as the most a p p r o p r i a t e source of waste heat f o r t h i s use. T h i s f l u e gas t y p i c a l l y c o n t a i n s about 30 mole % H2O and i s at a temperature of about 150°C. A moving packed bed or s i l o type of c o n t a c t o r i n which the gases f l o w upwards through a downward f l o w of c h i p s was recommended f o r the p r e h e a t i n g v e s s e l . As the c o l d or perhaps f r o z e n c h i p s f i r s t c o n t a c t the hot moist gas, water w i l l condense onto the c h i p s u r f a c e . The c h i p s w i l l be heated and a t some p o i n t the m o i s t u r e w i l l tend to e v a p o r a t e as the c h i p s pass down i n t o h o t t e r zones. S i n c e they l e a v e the u n i t at about 70°C i n c o n t a c t w i t h e n t e r i n g wet gas the c h i p s would not be d r i e d . As the moist gas r i s e s up the s i l o , i t w i l l f i r s t be e n r i c h e d i n m o i s t u r e and r e a c h the a d i a b a t i c s a t u r a t i o n c o n d i t i o n and then d e c r e a s e i n e n t h a l p y as water condenses onto the c h i p s . Thus the u n i t would have two zones, one i n which c o n d e n s a t i o n would occur on the c h i p s , and the o t h e r where e v a p o r a t i o n would o c c u r . The h e i g h t of these two zones i s unknown and would depend on many f a c t o r s . In the p r o p o s a l of B l a c k w e l l et a l . (19) the c a l c u l a t e d c h i p r e s i d e n c e time, s i z e of the s i l o and c a p i t a l as w e l l as o p e r a t i n g c o s t s were based on g e n e r a l i z e d p r e s s u r e - 2 -drop and heat t r a n s f e r e q u a t i o n s f o r packed beds of s o l i d s . The r e s i d e n c e time which i n c l u d e d time f o r m e l t i n g of f r o z e n c h i p s and h e a t i n g them to about 70°C was c a l c u l a t e d to be 40 minutes. These c a l c u l a t i o n s were based on a number of assumptions s i n c e no s p e c i f i c d a t a f o r heat t r a n s f e r and p r e s s u r e drop through beds of wood c h i p s were a v a i l a b l e . The major o b j e c t i v e of t h i s p r o j e c t was to g e n e r a t e s p e c i f i c d a t a on the heat t r a n s f e r c o e f f i c i e n t and p r e s s u r e drop f o r gases i n f i x e d beds of wood c h i p s . These d a t a c o u l d p r o v i d e i m p o r t a n t i n f o r m a t i o n f o r the d e s i g n and c o s t of the wood c h i p p r e h e a t i n g v e s s e l . The common f i x e d bed p r e s s u r e drop e q u a t i o n s such as the Ergun e q u a t i o n (14) and the Leva e q u a t i o n (16) are s u i t a b l e f o r f i x e d beds o f i r r e g u l a r shaped s o l i d s i f an a p p r o p r i a t e shape f a c t o r i s used. The Ergun e q u a t i o n i s e x p r e s s e d a s : 2 s 2 uV v AL 1 5 0 £ 3 ( t D } 2 + l ' 7 5 e p f *D p ( 1 0 ) As s t a t e d i n r e f e r e n c e ( 1 5 ) , by i n c l u d i n g the r a t i o of s u r f a c e a r e a of an e q u i v a l e n t volume sphere to the a c t u a l s u r f a c e a r e a of the g r a i n as a m u l t i p l i e r of the Sauter mean p a r t i c l e d i a m e t e r , the Ergun e q u a t i o n i s s u i t a b l e to p r e d i c t the p r e s s u r e drop v a l u e s of f i x e d beds w i t h d i f f e r e n t shapes of p a r t i c l e s . R e f e r e n c e (15) shows the a p p l i c a b i l i t y o f the Ergun e q u a t i o n f o r p a r t i c l e s w i t h 0.319 < i|/ < 0.965. The s p h e r i c i t y was c a l c u l a t e d m a t h e m a t i c a l l y by o b s e r v i n g the f i x e d bed - 3 -h y d r a u l i c g r a d i e n t f o r a bed of u n i f o r m g r a i n s i z e under known flow r a t e and temperature. A l t h o u g h the s p h e r i c i t y of wood c h i p s used i n t h i s p r o j e c t i s a l s o w i t h i n t h i s range, the shape of wood c h i p s i s more i r r e g u l a r than most s o l i d s f o r which these e q u a t i o n s have been t e s t e d and the s p h e r i c i t y i s low. Thus the common s p h e r i c i t y c o r r e c t i o n f o r n o n - s p h e r i c a l shapes may not h o l d . An i n s p e c t i o n of the g a s - t o - p a r t i c l e f i x e d bed heat t r a n s f e r l i t e r a t u r e shows t h a t most work has been done on p a r t i c l e s w i t h h i g h c o n d u c t i v i t y and s m a l l s i z e where i t can s a f e l y be assumed t h a t t h e r e i s no I n t e r n a l thermal r e s i s t a n c e . For l a r g e r , or low c o n d u c t i v i t y p a r t i c l e s such as wood c h i p s the i n t e r n a l r e s i s t a n c e can not be n e g l e c t e d . In the l i t e r a t u r e (3) coke as l a r g e as 6.3 to 7.6 cm i n d i a m e t e r , l i m e s t o n e of 3 cm i n d i a m e t e r and g r a v e l up to 4 cm i n d i a m e t e r have been c o n s i d e r e d to have n e g l i g i b l e i n t e r n a l r e s i s t a n c e i n f i x e d bed heat t r a n s f e r e x p e r i m e n t s , a l t h o u g h c l e a r l y the i n t e r n a l r e s i s t a n c e of such l a r g e p a r t i c l e s s h o u l d be c o n s i d e r e d . The B i o t number = h L c / k s , which i s the r a t i o of the I n t e r n a l r e s i s t a n c e ( c o n d u c t i o n ) to the e x t e r n a l f i l m r e s i s t a n c e ( c o n v e c t i o n ) i s the parameter which d i c t a t e s whether one or o t h e r of these r e s i s t a n c e s can be i g n o r e d . Thus f o r simple g e o m e t r i e s i f B i < 0.1 the e r r o r i n t r o d u c e d by assuming the temperature to be u n i f o r m w i t h i n the p a r t i c l e i s l e s s than 5%. From the e x t e n s i v e d a t a on heat t r a n s f e r to packed beds, f o u r e m p i r i c a l e q u a t i o n s f o r gas to s o l i d s heat t r a n s f e r are summarized below: 1. The heat t r a n s f e r u n i t was developed by C o l b u r n (4) and - 4 -v e r i f i e d by Bernard ( 1 1 ) . h r C p l J 1 2 / 3 M P C o r r e l a t i o n s are a v a i l a b l e f o r d i f f e r e n t p a r t i c l e Reynolds number ranges: D G j . = 1.064 for Re > 350 (1-2) V - 1 0 j h = 18.1 1 , u f o r R e p < 40 (1-3) 2. Furnas (3) e v a l u a t e d the heat t r a n s f e r c o e f f i c i e n t s i n g r a n u l a r beds o f v a r i o u s ores from the e x p e r i m e n t a l d a t a taken a t the U.S. Bureau of Mines to d e v e l o p the f o l l o w i n g e m p i r i c a l e q u a t i o n : A y 0 . 7 T0.3 1 Q ( 1 . 6 8 e - 3.56e 2) h = _ J ( 1 _ 4 a ) T h i s e q u a t i o n was developed f o r the f o l l o w i n g range of c o n d i t i o n s : 0.3 cm < Dp < 7.6 cm 0.2 m/s < Vf < 3 m/s 300°C < T < 1100°C .485 < e < .585 A. d i f f e r e n t e q u a t i o n was s a i d to a p p l y f o r c o o l i n g of coke p a r t i c l e s i n a i r . For t h i s case A v0.7 T0.3 1 Q ( 1 . 6 8 e - 3.56e 2) h = - — — ( l - 4 b ) - 5 -3. K u n i i and L e v e a s p i e l (18) approximated the heat t r a n s f e r e q u a t i o n f o r f i x e d beds of c o a r s e s o l i d s > I mm d i ameter by: Nup = 2 + 1.8 Pr 1/3 Rep 1/2 Rep > 100 (1-5) 4. G.O.G. Lof and R.W. Hawley (12) i n v e s t i g a t e d the u n s t e a d y - s t a t e heat t r a n s f e r c o e f f i c i e n t s from a i r to a bed of g r a n i t i c g r a v e l . I t was found t h a t the r e s u l t s c o u l d be c o r r e l a t e d by the e q u a t i o n : The s i z e of g r a v e l employed ranged from 0.48 cm to 3.81 cm. A i r r a t e s from 12.05 to 66.3 s t a n d a r d c u b i c f e e t per minute per square f o o t of c r o s s - s e c t i o n a l a r e a were u s e d . The e n t r a n c e a i r temperatures were m a i n t a i n e d over the range 100° to 250°F. The g r a v e l was packed i n t o the bed i n such a manner th a t normal v o i d s were o b t a i n e d . They found t h a t changes i n the temperature of the e n t e r i n g a i r had no a p p r e c i a b l e e f f e c t on the c o e f f i c i e n t . From the above s e l e c t i o n of the l i t e r a t u r e , f o u r d i f f e r e n t e m p i r i c a l e q u a t i o n s have been p r e s e n t e d . The f i r s t e q u a t i o n was d e v e l o p e d by assuming the t r a n s f e r p r o p e r t i e s ( i n c l u d i n g heat t r a n s f e r c o e f f i c i e n t ) of the f l o w i n g gas phase were independent of the p r o p e r t i e s of s o l i d s i n the same range of Reynolds number. G e n e r a l l y , t h i s a ssumption i s not q u i t e t r u e s i n c e d i f f e r e n t k i n d s and s i z e s of s o l i d s have d i f f e r e n t c h a r a c t e r i s t i c heat t r a n s f e r p r o p e r t i e s . For i n s t a n c e , e q u a t i o n s ( 1 - 4 ) , (1-5) and (1-6) show t h a t the heat t r a n s f e r c o e f f i c i e n t = 0.79 (G'/Dp 0.7 (1-6) - 6 -d e c r e a s e s w i t h i n c r e a s i n g p a r t i c l e s i z e i n same range of Reynolds number. Moreover, the parameter A of e q u a t i o n (1-4) i s s t a t e d to be a f u n c t i o n of type of o r e . E q u a t i o n (1-4) of Furnas was g e n e r a t e d by u s i n g p a r t i c l e s of s m a l l s i z e or l a r g e p a r t i c l e s w i t h h i g h thermal c o n d u c t i v i t y . I f l a r g e r , or low c o n d u c t i v i t y p a r t i c l e s are used, the c o e f f i c i e n t i s based o n l y on the s u r f a c e temperature, and the i n t e r n a l r e s i s t a n c e must be i n c l u d e d i f the temperature w i t h i n the s o l i d s i s to be c o n s i d e r e d . E q u a t i o n (1-5) was r e s t r i c t e d to the f l o w r e g i o n w i t h Reynolds number > 100 and some k i n d s of c o a r s e s o l i d s > 1 mm d i a m e t e r . However, d i f f e r e n t s i z e s of c o a r s e s o l i d s have d i f f e r e n t heat t r a n s f e r v a l u e s a c c o r d i n g to r e f e r e n c e ( 1 8 ) . A l s o , e q u a t i o n (1-5) g i v e s a poor f i t f o r the c o a r s e p a r t i c l e d a t a from the l i t e r a t u r e . Meanwhile, e q u a t i o n (1-6) summarizes heat t r a n s f e r v a l u e s f o r g r a v e l p a r t i c l e s o n l y . I t i s not an e m p i r i c a l e q u a t i o n f o r g e n e r a l usage. To sum up, none of these e q u a t i o n s i s p a r t i c u l a r l y s u i t a b l e f o r c a l c u l a t i n g the heat t r a n s f e r r a t e s to wood c h i p s . In the p r e s e n t work, t h e r e f o r e , the heat t r a n s f e r r a t e s were to be determined by measuring temperature p r o f i l e s i n f i x e d beds of wood c h i p s . With the use of an a p p r o p r i a t e m a t h e m a t i c a l model, the heat t r a n s f e r c o e f f i c i e n t s were then to be e x t r a c t e d from the e x p e r i m e n t a l r e s u l t s . A s u r v e y of the l i t e r a t u r e on the m a t h e m a t i c a l s i m u l a t i o n of heat t r a n s f e r i n f i x e d beds, showed t h a t a l t h o u g h t h e r e are some models (9) t h a t have been a p p l i e d to l a r g e r p a r t i c l e s ( u s u a l l y s p h e r i c a l ) no model f o r f i x e d beds of s l a b - s h a p e d p a r t i c l e s w i t h s i g n i f i c a n t i n t e r n a l - 7 -thermal r e s i s t a n c e Ls a v a i l a b l e . For example, the f i x e d bed problem w i t h s m a l l diameter and h i g h thermal c o n d u c t i v i t y p a r t i c l e s ( B i -*• 0) has been s o l v e d by A n z e l i u s ( 1 ) , and a n a l y t i c a l s o l u t i o n s p r e s e n t e d g r a p h i c a l l y by Schumann (2) and Furnas ( 3 ) . In these g r a p h s , the f r a c t i o n a l changes i n temperature of gas and s o l i d w i t h d i s t a n c e and time d u r i n g the h e a t i n g or c o o l i n g of the bed are e x p r e s s e d i n terms o f d i m e n s i o n l e s s v a r i a b l e s . Amundson (9) has s i m u l a t e d two-dimensional heat t r a n s f e r i n f i x e d beds w i t h s m a l l and l a r g e p a r t i c l e s . The d i f f e r e n c e between the s m a l l and l a r g e p a r t i c l e model i s t h a t the temperature i s assumed to be u n i f o r m i n s i d e the s m a l l p a r t i c l e . These two models a r e r a t h e r c o m p l i c a t e d i n t h a t they assume heat i s b e i n g g e n e r a t e d w i t h i n the s o l i d s as a f u n c t i o n of time and as w e l l c o n s i d e r heat t r a n s f e r to be o c c u r r i n g at the r e a c t o r w a l l . Both models were s o l v e d a n a l y t i c a l l y by Amundson ( 9 ) . The s o l u t i o n s of gas and s o l i d temperature are f u n c t i o n of c o m p l i c a t e d i n f i n i t e s e r i e s , B e s s e l f u n c t i o n s , e x p o n e n t i a l f u n c t i o n s as w e l l as i n f i n i t e i n t e g r a l s and the temperature f u n c t i o n s are i n terms of thermal p r o p e r t i e s as w e l l as independent v a r i a b l e s . In t h e i r p r e s e n t form these two models are more c o m p l i c a t e d than n e c e s s a r y f o r t h i s r e s e a r c h where ho heat i s g e n e r a t e d w i t h i n the wood c h i p p a r t i c l e s and where the e x p e r i m e n t a l f i x e d bed can be i n s u l a t e d to approach the a d i a b a t i c s i t u a t i o n . In a d d i t i o n , Amundson's model and t h a t of A n z e l i u s d e a l w i t h s p h e r i c a l s o l i d s and w i t h s m a l l p a r t i c l e s of h i g h thermal c o n d u c t i v i t y r e s p e c t i v e l y . N e i t h e r s i t u a t i o n a p p l i e s i n the p r e s e n t c a s e . T h e r e f o r e , i t was d e s i r e a b l e to s i m u l a t e heat t r a n s f e r i n f i x e d - 8 -beds of s l a b - s h a p e d p a r t i c l e s . However, i t was c o n s i d e r e d n e c e s s a r y to v e r i f y the e x p e r i m e n t a l t e c h n i q u e s by i n c l u d i n g measurements w i t h s p h e r i c a l s o l i d s . 1.2 Experimental Plan There are s e v e r a l p h y s i c a l p r o c e s s e s which o c c u r i n a moving bed wood c h i p p r e h e a t e r : m o i s t u r e c o n d e n s a t i o n on to and e v a p o r a t i o n from wood c h i p s , wood c h i p s e n s i b l e h e a t i n g and gas c o o l i n g , wood c h i p f l o w c o u n t e r c u r r e n t to gas flows e t c . In o r d e r to s i m p l i f y t h i s s i t u a t i o n , b a t c h f i x e d bed experiments were c a r r i e d out to a v o i d problems of s o l i d s f e e d i n g and t h i s procedure makes n e c e s s a r y the measurement of t r a n s i e n t temperature v a l u e s . Once heat t r a n s f e r c o e f f i c i e n t s were e x t r a c t e d from the f i x e d bed model, they c o u l d be used to p r e d i c t the performance of c o n t i n u o u s u n i t s by u s i n g a p p r o p r i a t e m a t h e m a t i c a l t e c h n i q u e s . The e x p e r i m e n t a l apparatus s e t - u p i s shown s c h e m a t i c a l l y i n F i g u r e 1-1. A i r or an a i r - s t e a m m i x t u r e was to be heated to about 150°C i n the steam heat exchanger and was to e n t e r the f i x e d bed of wood c h i p s . The t r a n s i e n t temperature response was to be measured by temperature probes throughout the bed. The p r e s s u r e drop was to be measured by manometers i n a s e p a r a t e s e r i e s of t e s t s . A f t e r the m a t h e m a t i c a l m o d e l l i n g was done f o r the heat t r a n s f e r and p r e s s u r e drop c a l c u l a t i o n , as w e l l as n e c e s s a r y c a l i b r a t i o n and t e s t i n g , a s e r i e s of p r e s s u r e drop and heat t r a n s f e r t e s t s were c a r r i e d out u s i n g d r y a i r , and s p h e r i c a l p a r t i c l e s of known p r o p e r t i e s . These t e s t s were used to v e r i f y the measurement t e c h n i q u e s , by comparing r e s u l t s to known ® P M R: M B A i r L H E ST Steam a C o l u m n • T - thermocoup le P - p r e s s u r e tap or gauge P M - p a n e l meter S - t he rmocoup le switch H E - s h e l l & tube heat exchange r R - r o t a meter S T - s t e a m trap D L - da ta logger for T M B - m a n o m e t e r bank •4-i I © Figure 1-1 Schematic of Apparatus - 10 -d a t a from the l i t e r a t u r e . For wood c h i p s , the heat t r a n s f e r d a t a measurements were to be c a r r i e d out by h e a t i n g up dry c h i p s w i t h hot d r y a i r . P r e s s u r e drop experiments were to be done under room temperature c o n d i t i o n s . The prime v a r i a b l e s such as a i r f l o w r a t e and c h i p s i z e were s t u d i e d . In a d d i t i o n a comparison o f h e a t i n g r a t e s between the hot d r y a i r and hot moist a i r was to be made f o r d i f f e r e n t s i z e s of wood c h i p s . - 11 -2. MATHEMATICAL MODELLING AND SIMULATION FOR FIXED BED HEATERS 2.1 P r e s s u r e Drop f o r I n c o m p r e s s i b l e Flow Through Packed Beds There are many kinds of packing materials such as spheres, cyl inders, and various kinds of commercial packings used in f lu id contacting equipment. Generally speaking, there have been two main theoretical approaches for studying pressure drop through packed beds. In one method the packed column is regarded as a bundle of tangled tubes of variable cross section; the theory is then developed by applying previous results for single straight tubes to the col lect ion of crooked tubes. In the second method the packed column is visualized as a col lect ion of submerged objects, and the pressure drop is calculated by summing up the resistance of the submerged part ic les . The tube-bundle theories have been somewhat more successful, and we shall discuss them here. It is assumed that there is no channeling or other wall effect throughout the following discussion. If the diameter of the packed column is greater than 8 times the diameter of par t ic les , wall effects can generally be neglected. The packed bed f r ic t ion factor is defined as P, 0 - P (2-1) 1/2 PV - 12 -Experimental measurements indicate that a theoretical formula for the laminar region can be written as: V = — ^ ^ 2 E (2-2) 0 L 1 5 0 u ( 1 _ E ) 2 This is the Blake-Kozeny equation, which is generally good for void D G N fraction e < 0.5 and (~~—) (l - e) < 1 0 ' W H E R E GQ = pVfj * T h e corresponding f r ic t ion factor is ^ D G J u ( 2 _ 3 ) e p 0| The empirical model for turbulent flow is given by the Burke-Plummer equation, p - P 0 " L ^ 1 1 .,2 I - £ .„ ., L = 3 ' 3 ° D~ " "2 P 0 T ( 2 _ 4 ) P e D GQ X This equation is good for ( P ^ ) Q _ £^ > 1000, and the corresponding f r ic t ion factor is : f = 0.875 ^ (2-5) £ When the Blake-Kozeny equation for laminar flow and the Burke-Plummer equation for turbulent flow are simply added together, the result is - 13 -150 ( DP Goh> 1 - e + 1.75 (2-6) This is the Ergun equation. As stated in Chapter 1, the shape of particles other than the sphere can be accommodated by incorporating the sphericity. It should be emphasized that the Ergun equation is but one of many that have been proposed for describing pressure drop across packed columns. 2.2 Gas/Solid Heat Transfer to Particles with Negligible Internal  Thermal Resistance If particles are small enough and have a high conductivity, their internal temperature can be considered uniform even during the transient state. If a uniform fluid stream is allowed to flow through a bed of solids which are i n i t i a l l y at some uniform temperature lower than the temperature of the stream, and i f the walls of the fixed bed are well insulated, the temperature of the bed wil l eventually arrive at the inlet temperature of the f l u i d . For the adiabatic case, i f the fluid is in plug flow, the velocity is constant and the fluid and solid properties are independent of O S - G - p ) 1 - e + 1.75 (2-7) - 14 -temperature and i f a x i a l c o n d u c t i o n i s n e g l e c t e d , the e n t h a l p y b a l a n c e s f o r f l u i d and s o l i d s a c r o s s an i n c r e m e n t a l depth of bed can be w r i t t e n a s : Heat i n w i t h gas - Heat out w i t h gas - Heat t r a n s f e r r e d to s o l i d s at z a t z + dz = A c c u m u l a t i o n of heat i n the gas " v f c P f p f " h f a ( T - fc> - e C P f p f l ? <2"8> Heat t r a n s f e r r e d from = A c c u m u l a t i o n of heat gas t o s o l i d s . i n s o l i d s h f a ( T - t) = (1 - e) C P g | | (2-9) s For the i n i t i a l c o n d i t i o n t = t± f o r a l l z a t 9 = 0 (2-10a) and the boundary c o n d i t i o n T = T Q a t z = 0 f o r a l l 9 (2-10b) The s o l u t i o n g i v e n by A n z e l i u s (1) and Schumann (2) i s t = ( T Q - t±) /" I Q (2/YZ) e Y e Z dZ + t± (2-11) T = t + ( T Q - t.) I 0 ( 2 / Y Z ) e Y e Z (2-12) h a z h a where Y = — £ — and Z = — - — l r ^ 9 (2-13) P fCp V P f C p (1 - e) f f Figure 2-1 Theoretical temperature profiles of Anzelius' model - 16 -T h i s model has been programmed as shown i n Appendix D-l and the n u m e r i c a l s o l u t i o n has been compared and matched w i t h Schumann's (2) s o l u t i o n . A p l o t of the s o l u t i o n i s shown as F i g u r e 2-1. 2.3 Gas/Solids Heat Transfer to Beds of Large and Low Conductivity  Spheres B a s i c a l l y , the assumptions are the same as d i s c u s s e d above i n S e c t i o n 2.2 except t h a t the s p h e r i c a l p a r t i c l e s now have a l a r g e s i z e and a low c o n d u c t i v i t y so t h a t the temperature w i t h i n the p a r t i c l e can be no l o n g e r assumed to be u n i f o r m . Thus e q u a t i o n s used to s i m u l a t e the system, s h o u l d account f o r i n t e r n a l temperature g r a d i e n t s . The e n t h a l p y b a l a n c e f o r the gas remains the same except the s o l i d s temperature a t i t s s u r f a c e must be used: "Vf C P f Pf £ " V ( T - W = £ C P f Pf To ( 2 ' 1 4 ) The heat t r a n s f e r r e d from the gas to the s o l i d s s u r f a c e i s removed by c o n d u c t i o n h f ( t r = R " T> " ~ k s lr->r=R ( 2 - 1 5 ) and the heat c o n d u c t i o n e q u a t i o n d e s c r i b e s the temperature f i e l d w i t h i n the p a r t i c l e k ( — o + — - 5 - ) = c P TTQ (2-15) s „ 2 r 3 r ps s 39 3 r - 17 -The i n i t i a l and i n l e t c o n d i t i o n s are g i v e n by e q u a t i o n s (2-10a) and (2-10b) and the boundary c o n d i t i o n at the c e n t r e of the sphere i s At r = 0 3 t / 3 r = 0 a l l z, 8 (2-16) These e q u a t i o n s were s o l v e d by Rosen (6,7) i n a m o d i f i e d form f o r a c o r r e s p o n d i n g mass t r a n s f e r problem i n v o l v i n g a f i x e d bed i o n exchange column. He used Duhamel's theorem f o l l o w e d by L a p l a c e t r a n s f o r m a t i o n o f the r e s u l t i n g s i n g l e l i n e a r 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 to o b t a i n a s o l u t i o n . The s o l u t i o n of these 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 s i s important f o r t h i s r e s e a r c h because the model i s used to g e n e r a t e heat t r a n s f e r c o e f f i c i e n t s from experiments which were done w i t h a s p h e r i c a l c a t a l y s t o f low c o n d u c t i v i t y . The experiments were done to compare w i t h l i t e r a t u r e v a l u e s so t h a t the e x p e r i m e n t a l equipment and t e c h n i q u e s c o u l d be v e r i f i e d . These e q u a t i o n s were r e s o l v e d f o r t h i s t h e s i s u s i n g the approach of Amundson (9) which w i l l be used s u b s e q u e n t l y i n the same problem w i t h s l a b - s h a p e d p a r t i c l e s . To s o l v e the above e q u a t i o n s u s i n g t h i s method, the f o l l o w i n g s t e p s are r e q u i r e d : 1. D i m e n s i o n l e s s v a r i a b l e s u b s t i t u t i o n 2. Double L a p l a c e T r a n s f o r m a t i o n 3. S u b s t i t u t i o n and S i m p l i f i c a t i o n 4. I n v e r s i o n of the L a p l a c e T r a n s f o r m F o r t h i s l a s t s t e p , Rosen's (6,7) f u n c t i o n was used. The s o l u t i o n has been programmed (Appendix D-2) and compared n u m e r i c a l l y w i t h Rosen's (6,7) s o l u t i o n . - 18 -The complete solution is given in Appendix C and the resulting expressions are given below: T(z, 9) = (T Q - t ± ) * + t± (2-17) where SI A 1 t . - T + (T. - t.) — — (2-18) surface 0 l B. 3x l $ = f(B i, x, n/a) I + 1 r0 sin[i2 s . a 2 j « ( 2 . 1 9 ) _ ' k x = 3 ( 1 _ e ) |_ 5 (2-20) V f RZ p P f r R HD1 + i - (HD2 + HD2) H, - 1 1 HD HD„ (1 + ) + (-=V B i B i HD2 H2 = HD— H D ~ ( 2 _ 2 2 ) B i B i HD = X ( - s ± n h 2 X + sin2A) _ 1 cosh2X - cos2X - 19 -u n X(sinh2X - sm2X) , 0 o n N 2 coshZA - cos2A X = H (2-24) To evaluate the i n f i n i t e i n t e g r a l , the appropriate l i m i t s at 3 0, 3 < 0.1 and 3 > 64 are r e q u i r e d . These are worked out i n Appendix C-l and are tabulated below: Function l i m 3 -»- 0 3<0.1 3 > 64 Hj • 0 H2 0 s i n f - 2 ^ . - XH ] L a 2j_ 2n 2 — 3 a 3 X HDj (^j) 3 2 /3 - 1 HD2 | 3 /3 Figure 2-2 i s a pl o t of t h i s s o l u t i o n f o r large spheres at .132. o Figure 2-2 Theoretical temperature profiles for a bed of spheres - 21 -2.4 Gas/Solid Heat Transfer to Beds of Low Conductivity and Rectangular  Slab-Shaped Particle Since there was no simulation and solution of the fixed bed heat transfer problem with slab shapes available in the literature, an attempt was made to obtain the analytical solution. Using the same basic assumptions as in the previous section, i.e., plug flow of gas at constant velocities, constant properties, no axial conduction, no heat loss through reactor wall, the equations for the slab-shaped geometry of particle shown below are: Enthalpy balance on gas " V f C P f P f S " h f a ( T " 'r -L> " £ C P f P f 1? ( 2 " 2 5 ) f x f Conduction within the particle (assumed zero through edges and one-dimensional) k s 2 t 3t , n „,N S - 2 = C p s P s 9 6 ( 2 " 2 6 ) 9r Particle surface/gas interface h f ( t r = L " T ) = - k s # - > r - L ( 2 ~ 2 7 ) X X X The i n i t i a l and inlet conditions are those of equations 2-10a and 2-10b and the symmetry condition is assumed at the particle centre-line — =0 at r = 0 (2-28) 3r x - 22 -The s o l u t i o n of t h i s model i s used to g e n e r a t e the heat t r a n s f e r c o e f f i c i e n t f o r the slab-shaped wood c h i p p a r t i c l e s . A gain the approach o f Amundson (9) was used. The s o l u t i o n s t e p s a r e : 1. D i m e n s i o n l e s s v a r i a b l e s s u b s t i t u t i o n 2. Double L a p l a c e T r a n s f o r m a t i o n 3. S u b s t i t u t i o n and S i m p l i f i c a t i o n 4. The i n v e r s i o n of L a p l a c e T r a n s f o r m . ( U s i n g the complex plane The i n v e r s i o n of the L a p l a c e t r a n s f o r m i s e x t r e m e l y d i f f i c u l t and c o m p l i c a t e d . The author i s g r a t e f u l to Dr. R.M. M i u r a , P r o f e s s o r of Mathematics at U.B.C., who has checked the g e n e r a l s o l u t i o n . As an a d d i t i o n a l check, the p r e d i c t i o n s of the s o l u t i o n have been compared n u m e r i c a l l y w i t h those of Schumann (2) by s u b s t i t u t i n g a h i g h c o n d u c t i v i t y v a l u e and a s m a l l s i z e f o r the s l a b p a r t i c l e s . The a g r e e -ment i s v i r t u a l l y p e r f e c t . F i g u r e 2-3 shows r e s u l t s at = .375. The s o l u t i o n has been programmed and i s shown i n Appendix D-3. The complete s t e p s i n the s o l u t i o n are shown i n Appendix C-2. The f i n a l e q u a t i o n s a r e p r e s e n t e d below: method.) T ( z , 6) = ( T Q - t ± ) $ + t . (2-29) t s u r f a c e (2-30) where o o o 1 1 1 I I 1 0.0 10.0 20.0 30.0 40.0 50.0 60.0 DIMENSIONLESS TIME Figure 2-3 Theoretical temperature profi les for a bed of slabs - 24 -* = f ( B i , x, n/a) —YH l x 1 f° 1 , ,2n ., - „ , dg' ... 2 + i J 0 e S i n [ ~ S " x H 2 ] ( 2 _ 3 1 ) x = ( 1 - £ ) f T T T ( 2 _ 3 2 )  f L P f f S = ( F o - v 7 ^ £ > <2~3> (2X) (|^ + HDj) 1 (-^ + HD^ 2 + HD2 i (2X) (HD 2) H = ? x ? (2-34) (•jp + HDj)^ + HD^ i s i n h 2X + s i n 2 X 1 cosh2X - cos2 X un _ s i n h 2 X - s l n 2 X , . H D 2 " cosh2X - cos2X ( 2 _ 3 5 ) X = /S» (2-36) To e v a l u a t e the i n f i n i t e i n t e g r a l , the a p p r o p r i a t e l i m i t s a t 8' •*• 0 are r e q u i r e d . These are worked out i n Appendix C-2 and are - 25 -tabulated below: Function lim S' > 0 HD2 H! 0 H 2 0 s i n [ l g g' - xH 2] 2 n 2x 3 a In order to solve the i n f i n i t e i n t e g r a l numerically, the i n t e g r a l i s computed as the sum of a series of d e f i n i t e i n t e g r a l s . Thus, c i C o Co. c If the value of the l a s t i n t e g r a l i s i n s i g n i f i c a n t compaed to the sum of a l l preceding terms the sum can represent the value of the i n f i n i t e i n t e g r a l . For the evaluation of parameters such as hf,and k s, the Newton Raphson method with i t e r a t i o n technique was applied to the constraint: 9 ( Y — Y ) 2 experimental t h e o r e t i c a l _ 3X 3. EXPERIMENT 3.1 Apparatus The design of the column and arrangement of the apparatus was done by M. Meenakshi. The goal of the experimental work is to carry out unsteady state experiments in which a fixed bed of particles (spherical catalyst or wood chips) is heated by gas entering at about 150°C and the heat transfer rates determined. Pressure drops were also to be determined under isothermal conditions. Thus, the equipment includes the main column for the bed of particles, a steam/air heat exchanger, copper-constantain thermocouples, a data logger, steam and air rotameters, and water-manometers. The heat transfer and pressure drop experiments were conducted in the small pilot plant schematically shown in Figure 1-1. The main items of equipment are listed in Table 3-1. The column and steam-air heat exchanger dimensions are shown in more detail in Figure 3-1 and 3-2. In operation, the particles (spherical nickeloxide on alumina catalyst or screened wood chips) were loaded in the packed column and the pressure along the column was measured at room temperature using the water-manometers. The desired air flowrate was regulated by using the calibrated air rotameter. In the heat transfer tests, the particles were introduced into the preheated fixed bed which was insulated. The heating air at a steady inlet temperature passed through the heat exchanger and flowed into the bottom section which provided a uniform air flow across the column. Thus the unsteady state gas and solid temperature were measured and recorded by the calibrated copper-- 27 -constantan thermocouples and the temperature recorder, as will be described in detail in Section 4. Referring to Figure 1-1, valve 1 controls gas going into the column. Valve 2 can be used at start-up of the heat exchanger to vent the gas while valve 1 is shut until a steady condition is reached. The moisture content of the air was controlled by regulating the calibrated steam rotameter. The steam and air calibration curves are shown in Figure 3-3 and 3-4 respectively. Table 3-1 Particulars of major equipment Fixed Bed: Column Characteristics: Inside diameter: Outside diameter: Height: Material: Wall Thickness: Heat Exchanger: Type: Shell inside diameter: Numbers of tubes: Tube inside diameter: Outside diameter: 3.2 Particle Preparation The particles used in the experiment were spherical catalyst pellets (15% NiO by weight on AI2O3) and Douglas f i r wood chips. The spherical particle density and diameter were found to be 801 kg/m and 4.77 mm respectively. Twenty individual spheres were measured to obtain 8 in., 0.2032 m 8.25 i n . , 0.20955 m 78 in., 1.98 m Stainless steel 1/8 i n . , 0.64 mm Shell and tube 5| in. , 0.146 m 9 1/4 i n . , 6.35 mm 3/8 i n . , 9.5 mm Figure 3-1 Detail of column Shel l 0.203 m 0.94 m Tube bundle T 0.203m 0.825 m Figure 3-2 Detail of heat exchanger o IT) (M CM O GO . in ~ on. X o . i in . O o " T T 0.0 30.0 Figure 3-4 60.0 90.0 120.0 150.0 ROTAMETER SETTING Calibration of air rotameter (Correction factor 1 7+273 ) 25+273 180.0 i - 32 -these average v a l u e s . The e s t i m a t i o n of thermal p r o p e r t i e s such as heat c a p a c i t y and thermal c o n d u c t i v i t y i s d i s c u s s e d i n Appendix B. The packed bed v o i d a g e was c a l c u l a t e d from measurements of bed h e i g h t and the volume of a one l i t e r c y l i n d e r o c c u p i e d , u s i n g the above v a l u e s of the sphere d e n s i t y as summarized i n T a b l e 3-2. F i g u r e 3-5 i s a photograph of the s p h e r i c a l p a r t i c l e s . Douglas f i r wood c h i p s were o b t a i n e d from a l o c a l s u p p l i e r , and s c r e e n e d f o r t h i c k n e s s as recommended by H atton ( 2 0 ) . C l a s s i f i c a t i o n was done w i t h the f o l l o w i n g s c r e e n s : Top: 45 mm round h o l e s Second: 6 mm s l o t t e d p a r a l l e l s t e e l rods T h i r d : 4 mm s l o t t e d p a r a l l e l s t e e l rods F o u r t h : 2 mm s l o t t e d p a r a l l e l s t e e l rods F r a c t i o n s were d e s i g n a t e d as f o l l o w s -45 mm round +6 mm s l o t t e d were d e s i g n a t e d 6 mm c h i p s ; -6 mm +4 mm s l o t t e d were d e s i g n a t e d as 4 mm, -4 +2 mm s l o t t e d were d e s i g n a t e d 2 mm c h i p s . A mixed sample was a l s o r e t a i n e d . Photographs of the c h i p s a r e shown i n F i g u r e 3-6 to 3-9. The volume of the wood c h i p s of a g i v e n t h i c k n e s s was c a l c u l a t e d from t h i c k n e s s , w i d t h and l e n g t h measurements of a l a r g e number of c h i p s from each of the f o u r s i z e s d e s c r i b e d above. The s p h e r i c i t y was c a l c u a t e d as f o l l o w s : . _ Suface A r e a o f a Sphere of Volume E q u a l to the C h i p S u r f a c e Area of the Slab-Shaped Chip The p a r t i c l e d e n s i t y was c a l c u l a t e d from measured dimensions and weight of f i v e c a r e f u l l y s e l e c t e d u n i f o r m c h i p s which had been d r i e d a t 110°C. - 33 -Tab l e 3-2 P r o p e r t i e s of wood c h i p s and s p h e r i c a l p a r t i c l e s Nominal S i z e 2 mm 4 mm 6 mm Mixed Sphere S p h e r i c i t y Voidage D e n s i ty (kg/m 3) 0.468 0.563 352 0.571 0.535 352 0.643 0.482 352 0.552 0.536 352 1.000 0.445 801 Avg. T h i c k n e s s (mm) D e v i a t i o n (%) V a r i a n c e 2.44 32.8 0.0533 4.03 12.39 0.00403 7.26 27.22 0.004 4.81 38.6 0.0072 Avg. Length 3.39 2.54 3.63 3.93 (cm) D e v i a t i o n (%) 37.8 18.4 37.5 40.1 V a r i a n c e 2.55 0.094 3.58 4.73 Avg. Width 0.964 1.881 1.842 1.564 (cm) D e v i a t i o n (%) 46.4 12.5 26.7 35.9 V a r i a n c e 0.272 0.0478 0.377 0.437 Avg. Diameter 1.045 1.53 2.007 1.632 0.476 (cm) - 34 -Figure 3-6 Photograph of 2 mm t h i c k wood chips (Average thickness 2.44 mm) - 35 -i n v WOOO CHIT I I Figure 3-7 Photograph of 4 mm t h i c k wood chips (average thickness 4.03 mm) Figure 3-8 Photograph of 6 mm t h i c k wood chi p s (Average thickness 7.26 mm) Figure 3-9 Mixed wood chips (average t h i c k n e s s 4.81 mm) - 37 -The dense bed voidage was c a l c u l a t e d from the bed h e i g h t , volume of a c y l i n d e r o c c u p i e d and d r i e d c h i p d e n s i t y . The d e t e r m i n a t i o n of the heat t r a n s f e r parameters of the wood c h i p s such as heat c a p a c i t y and thermal c o n d u c t i v i t y i s d e s c r i b e d i n Appendix B - l . The o v e r a l l summary o f wood c h i p p h y s i c a l p r o p e r t i e s i s g i v e n i n T a b l e 3-2. 3.3 Pressure Drop Measurements C h i p s were dumped i n t o the column, and the w a l l was rapped w i t h a hammer to get a dense-packed bed of u n i f o r m v o i d a g e . B e f o r e p r e s s u r e drop v a l u e s were measured, the c h i p s were d r i e d i n hot a i r a t 100°C f o r 3-hrs to c o n s t a n t weight, and then a l l o w e d to c o o l . P r e s s u r e drop measurements were made i n a i r at a t m o s p h e r i c p r e s s u r e and room temperature on the 2 mm, 4 mm, 6 mm and mixed samples, as w e l l as on the s p h e r i c a l p e l l e t s . The p r e s s u r e v a l u e s were read from the manometers connected to taps l o c a t e d at bed depths o f 0.102 m to 1.626 m. A i r f l o w r a t e s were v a r i e d from 1.13 x 1 0 - 3 m 3/s to 2.63 x IO" 3 m 3/s. 3.4 Heat Transfer Measurements B e f o r e the heat t r a n s f e r experiment was s t a r t e d the empty column was pr e h e a t e d w i t h hot a i r u n t i l the gas temperature i n s i d e the column reached a ste a d y s t a t e at the d e s i r e d i n l e t a i r tem p e r a t u r e . T h i s heat-up p e r i o d took a p p r o x i m a t e l y 3 h o u r s . The p a r t i c l e s were then i n t r o d u c e d i n t o the column about two minutes a f t e r the hot a i r flo w was - 38 -d i v e r t e d to the vent v i a v a l v e 2 ( F i g u r e 1-1). The purpose of the p r e h e a t i n g was to minimize the heat t r a n s f e r from the gas to the column w a l l i t s e l f . A l l the temperature r e a d i n g s were measured with c a l i b r a t e d c o p p e r - c o n s t a n t a n thermocouples connected to the D i g i t r e n d 235 d a t a l o g g e r as w e l l as to a m u l t i p l e channel s w i t c h and a d i g i t a l d i s p l a y . In o r d e r to c a l c u l a t e the heat t r a n s f e r c o e f f i c i e n t to the p a r t i c l e s u s i n g the models, the t r a n s i e n t gas temperature must be measured a t d i f f e r e n t bed d e p t h s . Two type of t h e r m o c o u p l e s — b a r e and s h i e l d e d — w e r e compared i n three p r e l i m i n a r y heat t r a n s f e r t e s t s u s i n g the s p h e r i c a l p e l l e t s . The temperature v a l u e s measured by these two types of thermocouples were found to be i d e n t i c a l at the same time and l o c a t i o n . T h e r e f o r e , the bare thermocouples were used throughout a l l the experiments to r e g i s t e r gas temperature. A photo of the type of thermocouples used as w e l l as the dimensions are shown i n F i g u r e 3-10 to 3-12. A l s o , the temperature v a l u e s measured by the D i g i t r e n d 235 which has an a u t o m a t i c i n t e r n a l c a l i b r a t i o n were found to agree w i t h v a l u e s r e g i s t e r e d by the p a n e l meter w i t h an e x t e r n a l c a l i b r a t i o n done by the a u t h o r . In most e x p e r i m e n t s , the temperature v a l u e s were measured at bed depth of 0.406 m, 0.610 m and 0.813 m over the time of the r u n . The hot gas f l o w was d i v e r t e d back to the column a f t e r the s o l i d s were i n t r o d u c e d and the gas t r a n s i e n t temperature v a l u e s were r e c o r d e d e v e r y 20 sec f o r f i r s t t h r e e minutes, e v e r y minute f o r next f i v e m i n utes, e v e r y 2 minutes f o r f o l l o w i n g ten minutes, and f i n a l l y e v e r y 4 minutes u n t i l the s t e a d y s t a t e was r e a c h e d . - V) Figure 3-10 Photograph of bare thermocouple Figure 3-11 Photograph of shielded thermocouple - 41 -In o r d e r to determine the r e l a t i o n between Che heat t r a n s f e r c o e f f i c i e n t and the f l o w r a t e , the a i r mass flow r a t e was v a r i e d from 0.0181 kg/s to 0.0338 kg/s. The i n l e t gas temperature a l s o v a r i e d somewhat w i t h the flow r a t e due to the absence of a temperature c o n t r o l l e r . The i n l e t temperature range was from 115°C to 139°C. T e s t s were done on the f o u r wood c h i p s i z e s and on the s p h e r i c a l p e l l e t s . In s e l e c t e d e x p e r i m e n t s , an attempt was made to measure the temperature at the c e n t r e of a wood c h i p , and i n the gas at i t s s u r f a c e . A f i n e c o p p e r - c o n s t a n t a n thermocouple (30 gauge) I n s e r t e d i n t o a h o l e d r i l l e d i n t o the s o l i d r e g i s t e r e d the s o l i d temperature at i t s c e n t e r - l i n e and a s i m i l a r thermocouple w i r e was a l s o a t t a c h e d to the s o l i d thermocouple w i r e i n o r d e r to measure the c o r r e s p o n d i n g gas temperature a t the same p o s i t i o n . A photo of t h i s s e t - u p i s shown i n F i g u r e 3-13. The c h i p w i t h i t s a t t a c h e d thermocouples was c a r e f u l l y p l a c e d i n the bed a t a h e i g h t of about 0.5 m. A l t h o u g h most runs were done w i t h d ry wood c h i p s and d r y a i r , the e f f e c t o f m o i s t u r e i n the i n l e t a i r was i n v e s t i g a t e d i n a few e x p e r i m e n t s . Hot a i r c o n t a i n i n g 30 mole % m o i s t u r e w i t h a combined f l o w r a t e e q u a l to 0.027742 kg/s was used to heat up the 4 mm and 6 mm wood c h i p p a r t i c l e s . These s t e a m - a i r runs were done to s i m u l a t e the s i t u a t i o n where K r a f t Recovery B o i l e r f l u e gases of h i g h m o i s t u r e c o n t e n t would be used to pre-heat the c h i p s . - a -F i g u r e 3-12 Photograph of wood c h i p temperature measurement - 43 -4. RESULTS AND DISCUSSION 4.1 Pressure Drop The e x p e r i m e n t a l p r e s s u r e drop d a t a f o r the s p h e r i c a l p a r t i c l e s and wood c h i p s are g i v e n i n T a b l e 4-1. In F i g . 4-1 and 4-2 and T a b l e 4-2 and 4-3, the d a t a f o r the s p h e r i c a l c a t a l y s t p a r t i c l e s a r e compared to v a l u e s c a l c u l a t e d from the Ergun e q u a t i o n u s i n g a s p h e r i c i t y of u n i t y and the measured and f i t t e d v o i d a g e . The e x c e l l e n t agreement between e x p e r i m e n t a l and c a l c u l a t e d p r e s s u r e - d r o p (4% d e v i a t i o n ) i s c o n s i d e r e d to c o n f i r m t h a t the equipment and t e c h n i q u e s worked s u c c e s s f u l l y f o r the p r e s s u r e drop t e s t s . For the wood c h i p s , two procedures were used. F i r s t , the p a r t i c l e s i z e , v o i d a g e and s p h e r i c i t y were determined by d i r e c t measurement, and a comparison made between the e x p e r i m e n t a l p r e s s u r e drops and the p r e d i c t e d v a l u e s u s i n g the Ergun e q u a t i o n s f o r the d i f f e r e n t wood c h i p s i z e s . S e c o n d l y , the v o i d a g e and s p h e r i c i t y were assumed to be unknown, and the p r e s s u r e drop d a t a were f i t t e d to the Ergun e q u a t i o n and the b e s t f i t v a l u e s of v o i d a g e and s p h e r i c i t y e x t r a c t e d . The v o i d a g e and s p h e r i c i t y were found by m i n i m i z i n g the sum over a l l e x p e r i m e n t s of 2 ( E x p e r i m e n t a l Ap - C a l c u l a t e d Ap from the Ergun e q u a t i o n ) . T a b l e 4-4 shows t h a t the f i t t e d s p h e r i c i t y i n c r e a s e s as the t h i c k n e s s of the wood c h i p s i n c r e a s e . T h i s s u g g e s t s t h a t the shape of wood c h i p i s r e l a t i v e l y c l o s e r to the shape of a sphere w i t h l a r g e r t h i c k n e s s o f wood c h i p s . The v o i d a g e i n c r e a s e s as the s p h e r i c i t y d e c r e a s e s . The wood c h i p s w i t h lower s p h e r i c i t y have l a r g e r and rougher - 44 -Table 4-1 Summary of pressure drop c o n d i t i o n s and data Pressure Drop Sphere Diameter Re A i r V e l o c i t y Data (mm) m/s (water cm, at height = 0.2m) 4.73 142.22 0.452 0.85 191.59 0.609 I .4 213.05 0.677 1.6 223.78 0.712 1.75 234.5 0.746 1.95 247.4 0.787 2.20 Wood Chip Thickness (mm) 2.44 373.2 0.541 0.905 419.2 0.609 1.20 466.2 0.678 1 .55 513.1 0.746 1.90 560.1 0.814 2.35 607.1 0.882 2.95 4.03 582.1 0.609 1.00 647 .4 0.678 1.25 712.6 0.746 1.50 777 .9 0.814 1.90 843.1 0.882 2.50 7.28 804.9 0.609 0.55 895.2 0.6775 0.75 985.4 0.746 0.85 1075.6 0.814 1.10 1165.7 0.882 1.35 4.81 654.6 0.609 0.85 727 .9 0.6775 1 .05 801.3 0.746 1.30 874 .6 0.814 1 .55 947.9 0.882 1.85 o 0.0 5.0 10.0 15.0 20.0 25.0 30.0 THEORETICAL P. F i g u r e 4-1 E x p e r i m e n t a l p r e s s u r e drop (cm water) v e r s u s Ergun e q u a t i o n p r e d i c t i o n with measured vo i d a g e F i g u r e 4-2 E x p e r i m e n t a l p r e s s u r e drop versus Ergun e q u a t i o n p r e d i c t i o n w i t h f i t t e d v oidage Table 4-2 A C O M P A R I S O N O F T H E O R E T I C A L A N O E X P E R I M E N T A L P R E S S U R E D R O P V A L U E S O F S P H E R I C A L P A R T I C L E S V A R • O.CK'.'i'" i r.:r ATui.'rn > conns i 1 1 - o i iv>o O l . - . . I M I " O . O O I 7 R HE :c.nr = o >cc M I H E O R F I I C A I . ( C M I .1 5 7 2 6 i ? s 7-111 S I M II . HUH 9 . B 1 5 r <PI'B i MI. ri r A L I L N i 7 . r.ro n I U O ' J 1 " 3 0 R E II 1 1 2 . 2 2 I'J I. fill 2 1.1 . 0 5 2 3 3 . 7 8 2 3 1 . 5 1 2 1 7 . 3 9 11 HE n -EM1 i / r . p • 0 0 1 7 - O 0 0 1 - 0 . 0 2 1 - O 0 3 0 - 0 . 0 3 ! ) - 0 . 0 5 6 T H C O R E T I C A I . ( C M 1 O . 8 9 3 1 . 5 3 1 1 . 8 6 1 2 . 0 3 7 2 . 2 2 2 2 . 4 5 4 l ' > P f B I M C H I A l . 1 ''.M 1 H . H Tin l -I'-'O l . 6 0 0 1 . 7 0 0 1 . 9 5 0 2 . 2 0 0 B E II 1 . 1 2 . 2 2 1 9 1 . 5 8 2 1 3 0 5 2 2 3 . 7 8 2 3 1 . 5 1 2 4 7 . 3 9 1 T H E O - E > " I . ' F A P . 0 . 0 5 I O 0 9 4 O . I 6 3 O . 1 6 1 O . 1 3 ' J 0 1 1 5 HE I G U I = O O I J M T H E O R E T I C A L ( C M ) 7 . 1 J J 1 2 . 2 S O 1 4 . 8 8 G I S . 2 9 9 1 7 . 7 7 5 1 9 . 6 3 0 E J i P E R I M E N T A L I C M ) 7 . 5 0 0 1 2 . 5 0 0 1 5 . J O p 1 7 . i 5 0 1 9 . 0 5 0 2 1 . 1 5 0 RE II 1 4 2 . 2 2 1 9 1 . 5 0 2 1 3 0 5 2 2 3 . 7 8 2 3 4 . 5 1 2 4 7 . 3 9 IIHEO. -E>P l / £ » P . - 0 . 0 4 7 • 0 . 0 2 0 - 0 . 0 3 6 - 0 . 0 5 0 - 0 0 6 7 - 0 0 7 2 4> T H E O P E T I C A L I C M I 1 . 7 8 6 3 0 6 2 3 . 7 2 2 4 . 0 7 5 4.444 4 . 9 0 7 E - P E R I M E N T A L 1 ("1.1 I I . P O O 3 . 0 5 0 3 » ~ 3 I ' C O I 5 5 0 5 . ;-.o RE II 1 . 1 2 . 2 2 1 9 1 . 5 8 2 1 3 0 5 2 2 3 7 8 2 3 1 . 5 l 2 1 7 . 3 9 ( I H E O . - E > P . I / £ v P - 0 . 0 0 8 0 . 0 0 I - 0 . 0 2 1 - 0 0 0 6 - 0 . 0 2 3 - O 0 3 0 T H E O R E T I C A L ( C M ) 1 0 . 7 1 7 1 0 . 3 7 5 2 2 . 3 2 9 2 4 4 4 9 3 6 6 6 3 E X P E R I M E N T A L I '.: r.l ) 1 1 . 0 0 0 1 0 1 5 0 . 2 2 . 3 5 ' 1 2 4 . 8 5 0 2 7 . nco R E II 1 4 2 . 2 2 1 3 I . 5 8 2 1 3 . 0 5 2 2 3 . 7 8 2 3 1 5 1 ( T H E O . - E vP . I . 'E >.P . - 0 . 0 2 6 0 . 0 1 2 - 0 . 0 0 1 - 0 . 0 1 6 - 0 . 0 1 1 Table 4-3 HEIGHT = O.-IOfi M A COMPARISON OF THEORETICAL AND EXPERIMENTAL PRESSURE DROP VALUES OF SPHERICAL PARTICLES VAR • 0.013HI I FITTED ! POROSITY • O. .14200 DIA. ( M ) • 0.00478 THEORETICAL { CM I 3.67 I fi . 21)2 7 . 645 8 . 370 9 . 128 10.080 EXPERIMENTAL I CM ) 3 . 750 G . 150 7 . 600 8 . 400 9 250 10.400 RE II 142 22 19 I .58 213.05 223.78 234.51 247.39 (THEO.-ExP.I /EXP. -0.02 I 0.023 0.006 -0.004 -0.013 -0.031 THEORETICAL I CM ) 0.018 1 . 573 1.911 2 .093 2 . 292 2 . 520 EXPERIMENTAL t CM ) 0-850 I . IO0 I .600 I . 750 1 .950 2 . 200 RE II 142.22 19 I.58 213.05 223.78 234.51 247.39 (THEO.-EXP.)/EXP 0.080 O. 12 1 O. 195 O. 196 O. 170 O. 145 HEIGHT • O B 13 M HEIGHT = 0.203 M THEORETICAL ( CM ) 7 . 342 12.584 15.290 16.740 18.256 20.159 EXPERIMENTAL I CM ) 7 . 500 12.600 1 5 . 450 17 150 19.050 2 1. 150 RE II 142.22 1 9 1.58 213.05 223 78 2 34.51 247 39 (THEO.-EXP.I /EXP. -0.021 0.007 -0.010 -0.021 -0.042 -0.047 00 THEORETICAL ( CM ) t . 836 3 146 3.823 4 . 185 4 . 564 5.040 EXPERIMENTAL ( CM I 1 .800 3.050 3.800 4 . 100 4 .550 5.060 RE II 142.22 191.58 213.05 223.78 234.51 247.39 ( THEO . -EXP . I/EXP . O.O30 0.03 1 0.006 0.02 I 0.003 -0.004 HEIGHT = 1.213 H THE ORE TICAL I CM ) 11.013 18.876 22.935 25.111 27.383 EXPERIMENTAL I CM ) It 000 18. 150 22.350 3 1.850 2 7.800 RE II 142.22 19 1 .58 213 05 223.78 234.5 I (THEO.-EXP.I /EXP. 0.00 1 0.040 0.026 0.010 -0.015 - 49 -Table 4-4 Pressure drop data f i t to Ergun Equation Nominal Chip Size 2 mm 4 mm 6 ram Mixed Sphere F i t t e d S p h e r i c i t y 0.488 0.593 0.677 0.593 F i t t e d Voidage . • 0.535 0.500 0.479 0.488 0.442 Variance Ap - Ap T E 0.0369 0.031 0.015 0.0202 0.0138 Std. D e v i a t i o n of Ap T - Ap E 0.0913 0.0898 0.0832 0.062 0.040 - 50 -s u r f a c e s which p r o v i d e more roughness between the wood c h i p s so th a t the vo i d a g e between the wood c h i p s i s h i g h e r . Agreement between e x p e r i m e n t a l and f i t t e d s p h e r i c i t y and v o i d a g e i s e x c e l l e n t ( T a b l e 3-2 and T a b l e 4-4). As can be seen, the f i t t e d v a l u e s o f s p h e r i c i t y a l s o a r e l a r g e r f o r the t h i c k e r wood c h i p s . The 6 mm wood c h i p s have the l a r g e s t v a l u e and the 2 mm wood c h i p s have the s m a l l e s t . In l i n e w i t h the measured v a l u e s , the f i t t e d v o i d a g e v a l u e s a l s o were observed to be l a r g e r f o r the s m a l l e r s i z e s of p a r t i c l e s . The comparisons between measured p r e s s u r e drop and v a l u e s c a l c u l a t e d from the Ergun e q u a t i o n u s i n g measured and f i t t e d s p h e r i c i t y and v o i d a g e a r e shown i n T a b l e s 4-5 to 4-12 and F i g u r e s 4-3 to 4-10. The d e v i a t i o n s between measured and c a l c u l a t e d p r e s s u r e drop v a l u e s were g r e a t e s t a t h i g h a i r flows and s m a l l e s t f o r the l a r g e s t (6 mm) wood c h i p s . Thus the more " s p h e r i c a l " the p a r t i c l e the b e t t e r i s the agreement w i t h the Ergun E q u a t i o n . As can be seen, the c a l c u l a t e d p r e s s u r e drop f o r the 2 mm, 4 mm as w e l l as the mixed wood c h i p s a r e lower than the e x p e r i m e n t a l v a l u e s ; thus the Ergun e q u a t i o n does not appear to g i v e c o n s e r v a t i v e r e s u l t s . The v a r i a n c e between e x p e r i m e n t a l and c a l c u l a t e d p r e s s u r e drop v a l u e s w i t h f i t t e d s p h e r i c i t y and v o i d a g e v a l u e s are a l s o h i g h e r w i t h c h i p s o f lower s p h e r i c i t y . T h i s can a l s o be observed i n T a b l e 4-5 to 4-12. As e x p e c t e d , the Ergun e q u a t i o n i s b e t t e r f o r packed beds w i t h more s p h e r i c a l - s h a p e d p a r t i c l e s . The v a r i a n c e s w i t h f i t t e d s p h e r i c i t y and v o i d a g e are s m a l l e r than c o r r e s p o n d i n g v a r i a n c e s w i t h measured s p h e r i c i t y and v o i d a g e . For example, the v a r i a n c e f o r the 4 mm wood in 15.0 THEORETICAL P. F i g u r e 4-3 E x p e r i m e n t a l p r e s s u r e drop (cm water) versu s Ergun p r e d i c t i o n f o r wood c h i p of t h i c k n e s s 2.44 mm with f i t t e d v o i d a g e and s p h e r i c i t y in a. < W S t—c OS W CU X W o o . BED DEPTH (M) O .203 • .406 • .610 + .813 WOOD CHIP TfflCKNESS=2.44mm MEASURED P0R0STTY=.563 MEASURED SPHERICTTY=.468 REYNOLDS NO. 372. TO 607. 12.5 15.0 THEORETICAL P. F i g u r e 4-4 E x p e r i m e n t a l p r e s s u r e drop (cm water) v e r s u s Ergun e q u a t i o n f o r wood c h i p of t h i c k n e s s 2.44 mm with measured voidage and s p h e r i c i t y cu w i — i H OH w o o . o C D * CD O CM " Q O ' BED DEPTH (M) O .203 • .406 • .610 + .813 WOOD CHIP TmCKNESS=4.03mm FITTED POROSITY=.500 FITTED SPHERICTTY=.593 REYNOLDS NO. 582. TO 843. 0.0 2.0 4.0 6.0 8.0 10.0 12.0 THEORETICAL P. Figure 4-5 Experimental pressure drop (cm water) versus Ergun p r e d i c t i o n f or wood chip of thickness 4.03 mm with f i t t e d voidage and s p h e r i c i t y F i g u r e 4-6 E x p e r i m e n t a l p r e s s u r e drop (cm water) v e r s u s Ergun p r e d i c t i o n f o r wood c h i p of t h i c k n e s s 4.03 mm with measured vo i d a g e and s p h e r i c i t y o 0.0 2.0 4.0 6.0 • 8.0 10.0 12.0 THEORETICAL P. F i g u r e 4-7 E x p e r i m e n t a l p r e s s u r e drop (cm water) v e r s u s Ergun p r e d i c t i o n f o r wood c h i p of t h i c k n e s s 7.26 mm w i t h f i t t e d v oidage and s p h e r i c i t y F i g u r e 4-8 E x p e r i m e n t a l p r e s s u r e drop (cm water) versu s Ergun p r e d i c t i o n f o r wood c h i p of t h i c k n e s s 7.26 mm w i t h measured vo i d a g e and s p h e r i c i t y o 12.0 THEORETICAL P. F i g u r e 4-9 E x p e r i m e n t a l p r e s s u r e drop (cm water) v e r s u s Ergun p r e d i c t i o n f o r mixed wood c h i p of t h i c k n e s s 4.81 mm w i t h f i t t e d v oidage and s p h e r i c i t y o 0.0 2.0 4.0 6.0 8.0 10.0 12.0 THEORETICAL P. F i g u r e 4-10 E x p e r i m e n t a l p r e s s u r e drop (cm water) v e r s u s Ergun p r e d i c t i o n f o r mixed wood c h i p of t h i c k n e s s 4.81 mm w i t h measured voidage and s p h e r i c i t y Table 4-5 A COMPARISON or THEORETICAL AND EXPERlMENIAL PRESSURE DROP VALUES OF ( 2 MM ) WOOL1 CHIPS VAR O.O3G07 1 f I T l C O ( F 1 IT E D ) POROSITY I SPHERICITY = O. 53501 O . 4 8 7 5 0 HEIGHT = 0 . 6 I O H THEORE T ICAL ( CM I 3 . 157 3.911; 4 . 8 2 1 5 . 7 8 ^ 6 . 828 7 . 9G0 E <»f.li IMEN1 Al I CM I :> . H50 3 . G'-iO 4 . S50 5 . 900 7 . :'00 0 . ooo RE II 372 . 23 4 1 9 . 2 0 4 6G . 16 5 13 .13 5 6 0 . 0 9 6 0 7 . 0 6 I THf.O . - E »l> | / f ' O. 108 0 . OB 1 0 . 0 3 7 - 0 . 0 2 0 - 0 . 0 5 2 - O . I 1 6 THEORETICAL ( CM I 1 .05 2 1 .315 1 . G07 1 . 927 2 . 276 2 .653 EXPERIMENTAL ( CM I 0 950 1 2C0 I . 550 1 . SCO 2 350 2 . 950 RE II 3 7 2 . 2 3 4 1 9 . 2 0 4GG . 16 5 1 3 . 13 5 6 0 . 0 9 G07 .OG ( THEO. - E X P . 1/E.vP. O. 108 0 . 0 9 6 0 . 0 3 7 O . O I -i - 0 . 0 3 1 - O . 1 0 1 THEORETICAL ( CM ) •1 . 209 5 . 2G2 6 . 429 7 . 709 9 . 104 10 .6 13 EXPERIMENTAL I f.M I 3 . GOO J . GOO 5 . 800 7 . 400 9 . 0 0 0 1 1 250 RE II 3 7 2 . 2 3 4 19 .20 4 6 6 . 1 6 5 1 3 . 1 3 5 6 0 . 0 9 6 0 7 . 0 6 ( IHEO - E X P l , ' £ < P O. 169 0 . 144 O. 108 0 . 0 4 2 0 . 0 1 2 - 0 . 0 5 7 1_n THEORETICAL ( CM I 2 . 105 2 .631 3 . 2 1 4 3 .055 J . 552 5 . 307 EXPERIMENTAL 1 CM I ? . 050 2 . COO 3 300 ' . 300 5. 150 6 . 400 RE II 3 7 2 . 2 3 1 1 9 . 2 0 4 C 6 . 16 5 13. 1 3 5 6 0 . 0 9 607 OG ( T H E O . - E X P . I / E X P . 0 . 0 2 7 0 . 0 1 2 - 0 . 0 2 G - O . 1 0 4 - 0 . 1 1 6 - 0 . 1 7 1 Table 4-6 A COMPARISON OF ItlF.ORETICAL AND EXPERIMENTAL PRESSURE DROP VALUES OF ( 2 MM I WOOD CHIPS VAR • O. 16781 ( MEASURED I POROSITY > 0.56341 ( MEASURED I SPHERICITY = 0.468t4 HEIGHT • 0.6 10 M THEORETICAL ( CM ) 2 . G2 1 3 . 28 I 4 012 4 a 1 5 5 . 690 6.636 EXPERIMENTAL 1 CM ) 2 . 850 3 . 650 4 650 5 . 900 7 . 200 9 .000 HE II 372 23 4 19.20 466 16 5 13. 13 560.09 607 06 (THEO.-EXP. I/E>P. -O.OBO -O.101 -O.137 -O.184 -O 210 -O.263 THEORETICAL ( CM ) 0.874 1 . 094 1 . 337 I . G05 1 . 897 2.212 ' EXPERIMENTAL I CM I 0.950 1 . 200 1 . 550 1 . 900 2 . 350 2 . 950 RE II 372.23 419.20 466 . 16 513.13 5G0.09 607. OG (THEO.-EXP.I /EXP. -0.080 -0.089 -O.137 -O.155 -O.193 -O.250 THEORETICAL ( CM ) 3 . 495 4 .374 5 . 350 6.420 7 . 506 8 . 048 EXPERIMENTAL ( CM I 3 . 60O 4 . 600 5 . 800 7.400 9.000 I 1 .250 RE II 372.23 419.20 .166 . 16 5 13.13 560.09 607.06 (THEO.-EXP.I /EXP. -O.029 -0.049 -C.078 -O . 132 -O.157 -0.2 13 C^ o THEDRETICAl ( CM I 1 . 747 2.137 2.675 3.210 3 . 793 4.424 EXPERIMENTAL 1 CM ) 2 .050 2 .600 3 . 300 4 . 300 5 . 150 6.400 RE II 372.23 419.20 466. 1 6 513. 13 560.09 607.06 (THEO.-EXP.I /EXP. -O.148 -O. 1 5 9 -0.189 -O.253 -O.263 -O. 309 Table 4-7 A C O M P A R I S O N T H E O R E T I C A L A N D E X P E R I M E N T A L P R E S S U R E D R O P V A L U E S O F ( 4 M M ) WOOD C M I C S V A R • 0 . 0 3 C 9 8 ( f l T T E C I P O R O S I T Y • O . S O O O O ( F I T T E D I S P H E R I C I T Y = 0 5 9 3 0 0 I H E D R E T I C A L ( C M I 2 . 6 8 9 3 . 5 5 2 • 1 . 2 0 1 5 . 0 7 8 5 . 9 - 4 1 E X P E R I M E N T A L 1 C M » 2 . 8 0 0 3 5 5 0 4 . 6 O 0 5 . 5 5 0 6 . 8 0 0 R E II 5 8 2 . 1 8 6 4 7 . 4 0 7 1 2 . 6 3 7 7 7 . 8 5 8 4 3 0 8 ( T H E O . -E:<P 1, 'L.-1» . 0 . 0 3 2 0 . 0 0 0 - 0 . 0 6 9 - 0 . 0 8 5 - O . 1 2 6 T H E O R E T I C A L ( C M I 0 . 9 6 3 1 . 1 8 4 1 . 4 2 7 1 . 6 9 3 1 . 9 8 0 E < P E R I M E N T A L I C M I 1 . C C O 1 . 2 5 0 1 . 5 0 0 1 . 9 0 0 2 . 5 C 0 R E II 5 8 2 . 18 6 4 7 . 4 0 7 1 2 . 6 3 7 7 7 . 8 5 8 1 3 . 0 8 I T H E O . - E X P . I / E X P . - 0 . 0 3 7 - 0 . 0 5 3 - 0 . 0 4 9 - O . 1 0 9 - 0 . 2 0 8 H E I G H T - 0 . 8 1 3 M T H E O R E T I C A L ( C M I 3 . B 5 2 4 . 7 3 6 5 . 7 0 9 6 . 7 7 1 7 . 9 2 2 E X P E R I M E N I A L I C M ) 3 . 5 5 0 4 . 7 5 0 5 . 3 5 0 7 4 0 0 8 . 8 5 0 R E / / 5 8 2 . 18 6 4 7 . 4 0 7 1 2 . 6 3 7 7 7 es 8 4 3 0 8 ( I H C O - E X P 1 / E A P . 0 . 0 8 5 - 0 . 0 0 3 - 0 . 0 2 4 - 0 . 0 8 5 - O . 1 0 5 I cr. T H E O R E T I C A L ( C M I 1 . 9 2 6 2 3 G 8 2 . 8 5 1 3 . 3 8 5 3 . 9 5 1 E X P E R I M E N T A L ( C M ) 1 . 8 ? 0 : . 1 0 0 2 . 7 0 0 3 . 3 5 0 •1 . 3 5 0 R E II 5 8 2 . 1 0 6 4 7 . 4 0 7 1 2 . 6 3 7 7 7 . 8 5 8 4 3 . 0 8 ( I H E O - E X P . I / E X P . 0 . 0 7 0 O . 120 0 . 0 5 7 0 . 0 1 1 - 0 . 0 0 9 H E I G H T = 1 . 0 1 6 M T H E O R E T I C A L ( C M ) 4 . 8 1 5 5 . 9 2 0 7 . I 3 G 0 . 4 6 3 9 . 3 0 2 E X P E R I M E N T A L ( C M ) I 3 0 0 5 . 6 0 0 7 . 1 0 0 8 . 6 0 0 I O . 2 5 0 R E II 5 8 2 . 1 8 6 4 7 . 4 0 7 1 2 . 6 3 7 7 7 . 8 5 8 4 3 . 0 8 ( T H E O - E X P . I / E X P O 1 2 0 0 . 0 5 7 0 . 0 0 5 - O . O I G - 0 . O 3 4 Table 4-8 A C O M P A R I S O N T H E O R E T I C A L A N D E X P E R I M E N T A L P R E S S U R E D R O P V A L U S S O F ( - iMM ] wonn ci 11 p^ ( M E A ' U R T O I T O R O S I T V • 0 . 5 3 4 6 3 I M E A S U R E D I S P H E R I C I T Y » 0 . 5 7 0 7 7 T H E O R E T I C A L ( C M ) 2 . 2 8 2 2 . 7 8 6 • . 3 6 2 3 . 9 9 I 4 . 6 7 3 f X I-' L R I (AE NT A L ( C M ) 2 . O C O 3 . 5 5 0 4 . 6 .10 5 . 5 5 0 6 . B O O R E II 5 8 2 . I B 6 4 7 . 4 0 7 1 2 . 6 3 7 7 7 . 8 5 8 4 3 . 0 8 . ( T H E O . - E X P . I / E • P . - 0 . 1 9 2 - 0 . 2 15 - O . S 6 9 - 0 . 2 8 1 - 0 . 3 13 H E I G H T = 0 . 2 0 3 M T H E O R E T 1 C A L ( C M 1 O . 7 5 4 0 . 9 2 9 1 . 1 2 ' 1 . 3 3 0 1 . 5 5 S E X P E R I M E N T A L 1 C'1 I I . O O O 1 . 2 5 0 1 . 5 0 0 1 ? ' C 0 2 . 5 0 0 RE II 5 8 2 . I B G 4 7 4 0 7 1 2 . 6 3 7 7 7 . 8 5 8 4 3 . 0 8 ( T H E O . - E X P . l / E X P . - 0 . 2 4 6 - 0 . 2 5 7 - 0 . 2 5 3 - O . 3 0 O - 0 . 3 7 7 H E I G H T = 0 . 8 13 M T H E O R E T I C A L ( C M ) 3 . 0 1 6 3 . 7 1 4 4 . 4 8 2 5 . 3 2 1 6 . 2 3 1 C X P E K I M E N T A L I C M I 3 . 5 5 0 4 . 7 5 0 f> . 8 5 0 7 . 4 0 0 8 . 8 5 0 RE II 5 B 2 . 1 8 6 4 7 . 4 0 7 1 2 . 6 3 7 7 7 . 8 5 8 4 3 . 0 8 ( T H E O . - E X P . I / E ' P . - O . t 5 0 - 0 . 2 18 - 0 . 2 3 1 - 0 . 2 8 1 - O . 2 9 6 Ox N3 T H E O R E T I C A L ( C M ) 1 . 5 0 5 1 . 8 5 7 2 . 2 4 1 2 . 6 6 I 3 . 1 1 5 E X P E R I M E N T A L 1 C M l i . B O O 2 . 1 0 O : . 7 0 0 3 . 3 5 0 4 . 3 5 0 RE II 5 B 2 . 18 6 4 7 . 4 0 7 1 2 . 6 3 7 7 7 . B S 8 4 3 . 0 8 ( T H E O . - E X P l / E X P . - 0 . 1 6 2 - 0 . 1 1 6 - O . 1 7 0 - O . 2 0 6 - 0 . 2 8 4 T H E O R E T I C A L ( C M ) 3 . 7 7 0 4 . 6 4 3 5 G 0 3 6 . 6 5 2 7 . 7 B B E A P E R1 ME NT A L ( C M I 4 . 3 0 0 5 . G O O 7 . 1 0 O B . 6 0 0 1 0 . 2 5 0 RE II 5 8 2 . I B 6 4 7 . 4 0 7 1 3 . 6 3 7 7 7 . 8 5 8 4 3 . 0 8 ( T H E O . - E X P . I, E • P . - O 1 2 3 - O . 1 7 1 - 0 2 1 1 - 0 . 2 2 7 - O . 2 4 0 Table 4-9 A C O M P A R I S O N OF T H E U R F t I C A I . A N O E X P E R I M E N T A L P R E S S U R E D R O P V A L U E S O f ( 6 MM ) WOOD C H I P S H E I O U T = 0 6 1 0 M V A R 0 . 0 1 4 9 5 ( F I T T E D ( F I T T E D I P O R O S I T Y ) S P H E R I C I T Y O . 4 7 9 0 0 0 . 6 7 7 0 0 T H E O R E T I C A L ( C M ) 2 . 0 7 1 2 . 5 G 3 3 . 1 0 3 3 . G 9 I 4 . 3 3 1 E X P E R I M E N T A L 1 C M I 2 . O O O 2 . 5 0 O 3 . 0 0 0 3 . 6 5 0 •I . 5 3 0 R E // ao-i. 9 « B 9 5 . 1 7 9 B 5 . 3 6 1 0 7 5 . 5 5 1 I B S . 7 3 ( T H E O . - E X P l . ' E O 0 3 7 O (125 0 0 3 1 0 . 0 1 1 - O . 0 4 4 H E I C U T • 0 . 2 0 3 M T H E O R E T I C A L ( C M ) 0 . 6 9 1 0 . 8 5 4 1 . 0 3 4 1 . 2 3 0 1 . 4 4 4 E X P E R I M E N T A L ( C M I 0 . 5 5 0 0 . 7 5 0 0 . 8 5 0 1 . 1 0 0 1 . 3 5 C R E II 8 0 1 . 9 8 8 9 5 . 1 7 9 8 5 . 3 6 1 0 7 5 . 5 5 1 1 6 5 . 7 3 ( I H E O - E X P I / E X P . 0 . 2 5 7 O 1 3 9 0 . 2 1 6 0 . 1 1 9 O 0 6 9 H E I G H I = 0 . 8 1 3 M T H E O R E T I C A L I C M I 2 . 7 6 5 3 . 4 1 7 4 . 1 3 6 4 . 9 2 2 5 . 7 7 4 E X P E R I M E N T A L I C M I . 2 . G O O 3 . 1 0 0 •1 . 2 0 0 5 . 0 5 0 6 . 5 0 O R E II 8 0 4 . P B 8 9 5 17 9 8 5 . 3 6 1 0 7 5 . 5 5 1 1 6 5 . 7 3 ( T H E O . - E X P I I . 0 0 6 3 0 . 0 O 5 - 0 . 0 1 5 - 0 . 0 2 5 - 0 . 1 1 2 T H E O R E T I C A L ( C M ) 1 . 3 8 2 1 . 7 C 8 2 . 0 6 8 2 4 6 I 2 . 8 R 7 E X P E R I M E N T A L 1 C M I 1 . 2 5 0 1 . 6 5 0 2 . 0 5 0 7 . 5 0 0 3 . 1 IO R E II 8 0 4 . 9 8 8 9 5 . 1 7 9 8 5 . 3 6 1 0 7 5 . 5 5 1 1 6 5 . 7 3 ( T H E O . - E X P I / E X P . O. 1 0 6 O 0 3 5 0 . 0 0 9 - O . O I G - 0 . 0 7 2 H E I G H T -- 1 0 1 6 M T H E O R E T I C A L ( C M ) 3 . 4 5 6 4 . 2 7 1 5 1 7 0 6 . 152 7 . 2 1 8 E X P E R I M E N T A L t C M ) 3 . 2 5 0 4 . 2 5 0 5 . i b O 6 4 5 0 7 . 7 5 0 R E II 8 0 4 9 8 8 9 5 . 1 7 9 0 5 3 6 1 0 7 5 . 5 5 1 1 6 5 . 7 3 ( T H E O . - E x P I / E 0 . 0 6 3 0 . 0 0 5 0 . 0 0 4 - 0 . 0 4 6 - 0 . 0 6 9 Table A-10 A C O M P A R I S O N O r T H E O R E T I C A L A N D E X P E R I M E N T A L P R E S S U R E D R O P V A I U E S O f ( 6 MM ) WOOD C H I P S V A R = 0 0 1 9 1 1 1 M E A S U R E D 1 P O R O S I T Y • 0 . 4 B I 7 6 I M E A S U R E D I S P H E R I C I T Y » 0 . 6 4 2 5 4 T H E O R E T I C A L E X P E R I M E N T A L R E ( T H E O . - E X P l / E X P . ( C M ) ( C M | / / 0 . 7 15 0 . 5 5 0 8 0 4 . 9 8 0 . 3 0 0 0 . S3:- 0 . 7 5 0 8 9 5 . 17 O . 1 7 8 1 . 0 6 ? 0 . 8 5 0 9 8 5 . 3 6 0 . 3 5 8 1 . 2 ' ; 1 , I C O 1 0 7 5 . 5 5 0 . 1 5 6 1 . 4 9 1 1 . 3 5 0 1 1 6 5 . 7 3 0 . 1 0 5 T H E O R E T I C A L E X P E R I M E N T A L RE ( T H E O . - E X P I / E < P . ( C M i i C M I / / I . J 3 0 1 . 2 0 0 8 0 4 . 9 8 0 . 1 4 4 1 . 7 G 7 1 . 6 0 0 8 9 5 . 1 7 0 . 0 7 1 2 . 1 3 8 2 . 0 5 0 9 8 5 . 3 6 0 . 0 4 3 2 . 5 4 3 2 . 5 0 0 1 0 7 5 . 5 5 0 . 0 1 7 2 . 9 8 3 3 . 1 1 0 1 1 6 5 . 7 3 - 0 . 0 4 I H E I ( i l l I * O . G 1 0 M T H E O R E T I C A L ( C M I 2 . I 4 G 2 . 6 5 0 3 2 0 7 3 . B 15 4 . 4 7 4 E X P E R I M E N T A L 1 C M I 2 . OOO 2 . 5 0 0 3 . OOO 3 . 6 5 0 4 5 3 0 R f II 8 0 4 . 9 8 8 9 5 . 1 7 9 B 5 . 3 6 1 0 7 5 . 5 5 1 1 6 5 7 3 ( 1 H E 0 . - E X P l / E X P . O 0 7 3 0 . 0 6 0 0 . 0 6 9 0 . 0 4 5 - 0 . 0 1 2 T H E O R E T I C A L ( C M I 2 . 8 6 1 3 5 3 4 4 . 2 7 6 5 . 0 8 G 5 . 9 G 5 F K P E R I M E N T A L 1 C M | 2 . 6 0 0 3 . 4 0 0 4 2 0 0 5 . 0 5 0 G 5 0 0 R E II 8 0 4 . 9 8 8 9 5 17 9 8 5 . 3 6 1 0 7 5 . 5 5 1 1 6 5 . 7 3 ( T H E O . - E X P 1 / E x P . o too 0 . 0 3 9 0 . 0 1 8 0 . 0 0 7 - 0 . 0 8 2 H E 1 C H T ' 1 . 0 1 6 M T H E O R E T I C A L ( C M ) 3 . 5 7 6 4 . 4 1 7 D . 3 4 4 6 . 3 5 B 7 . 4 5 7 E X P E R I M E N T A L ( C M I 3 . 2 5 0 4.. 2 0 0 5 . I 5 0 G . 4 5 0 7 . 7 5 0 RE II 8 0 4 . 9 8 8 9 5 . 1 7 9 8 5 3 6 1 0 7 5 . 5 5 I 1 6 5 . 7 3 ( T H E O - E X P l / E X P . O . 1 0 O O 0 3 9 0 . 0 3 8 - 0 . 0 1 4 - 0 . 0 3 8 T a b l e 4-11 A COMPARISON OF THEORETCIAL AND EXPERIMENTAL PRESSURE DROP VALUES OF ( MIXED ) WOOD CHIPS VAR = 0.02017 (F ITTED I POROSITY • = 0.48750 I FITTED I SPHERICITY - 0.59300 THEORETICAL ( CM l 2.814 3.463 4.178 4 . 958 5 . 804 EXPER[MENTAL ( CM ) 2 . GOO 3 . lOO 3 . 900 4 . 850 6 . 100 RE II 654.5B 7 2 7.91 801.25 874.59 947.92 (THEO.-EXP,) /ExP. 0.082 0.117 0.07 1 0.022 -0.048 THEORETICAL ( CM ) 0.938 t . 154 1 . 393 1 . 653 1 . 935 EXPERIMENTAL ( CM I O. 850 I . 050 1 . 300 1 . 550 1 850 RE II 654.58 727.91 801.25 874.59 947.92 (THEO.-EXP.I /EXP. O. 103 0.099 0.07 1 0.066 0.046 HE I GUI = 0.813 M THEORETICAL I CM I 3 . 752 4.617 5.570 6.611 7 . 739 EXPERIMENTAL I CM ) 3 . 700 4 . 700 5 950 7 . 200 8.400 RE II 654.58 727.91 801.25 074.59 947.92 (THEO.-EXP. I/EXP. 0.014 -0 018 -0.0G4 -0.0B2 -0.079 CTX U l THEORE1ICAL ( CM ) 1 .876 2 . 309 2 . 705 3 . 3Q5 3 .870 EXPERIMENTAL ( CM ) 1 .650 2 .050 2 550 3. 100 3 . BOO RE II 654.58 727.91 801.25 874.59 947.92 (THEO.-EXP.I/EXP O. 137 O. I2G O.092 0.066 0.018 HEIGHT = I.OIG H THEORETICAL I CM ) 4 . 690 .5.772 6 . 963 8 . 264 9 67.1 EXPERIMENTAL i CM I 4 . 4 50 5 . 70O 7 . 250 8 750 IO.500 R E II 654.58 727.91 801.25 874.59 94 7 92 (THEO.-EXP.I /EXP. 0.054 0.013 -0.040 -0.05G -O.079 A C O M P A R I S O N *Of n IEGRF. 1 C I A L A N D E X P E R I M E N T A L P R C S S U R E D R O P V A L U E S O F ( M I X E D ) W011D C H I P S V A R » O . 3 9 0 3 9 ( M E A o U R C O 1 P O R O S I T Y =• 0.5357a ( M E A S U R E D I S P H E R I C I T Y » 0 . 5 5 1 8 4 T a b l e 4-12 H E I C H I = O . i i 1 0 14 T H E O R E T I C A L E X P E R I M E N T A L R E ( T H E O . - H P . l / E X P . ( C M I I C M I / / 2 . 0 3 8 2 . 6 0 0 6 5 4 . 5 8 -0 .316 2 . 5 1 4 3 . I O C ) 7 2 7 . 9 1 - 0 . 1 8 9 3 . 0 3 7 3 . 9 0 O 8 0 1 . 2 5 - 0 . 2 2 1 3 . 6 0 9 4 8 5 0 8 7 4 5 9 - 0 . 2 5 6 4 . 2 2 9 6 . 1 0 0 9 4 7 . 9 2 - 0 . 3 0 7 T H E O R E T I C A L ( C M ) 0 . 6 7 9 0 . 8 3 8 1 . 0 1 2 1 . 2 0 3 1 . 4 1 0 E X P E R I M E N T A L I C M I 0 . 8 5 0 I . 0 5 0 1 . 3 0 0 1 . 5 5 0 I . 8 5 0 R E II 6 5 4 . 5 8 7 2 7 . 9 I 8 0 1 . 2 5 8 7 4 . 5 9 9 4 7 . 9 2 ( T H E O . - E X P . l / E X P . - 0 . 2 0 1 - 0 . 2 0 2 - 0 . 2 2 1 - O . 2 2 4 - 0 . 2 3 8 H E ] G U T = O . B I 3 M T H E O R E T I C A L ( C M I 2 . 7 1 8 3 . 3 5 2 4 . 0 5 0 4 . 8 1 2 5 6 3 8 E X P E R I M E N T A L ( C M I 3 . 7 0 0 •I . 7 0 0 5 . 9 5 0 7 . 2 0 0 8 . 4 0 0 R E II 6 5 4 . 5 8 7 2 7 . 9 1 8 0 1 . 2 5 8 7 4 . 5 9 9 4 7 . 9 2 ( T H E O . - E X P . l / E X P . - 0 . 2 6 5 - 0 . 2 8 7 -0 .319 - O . 3 3 2 - O 3 2 9 Ox CP-H E I G H T » 0 . 1 0 6 M T H E O R E T I C A L I C M I 1 . 3 5 9 1 . 6 7 6 2 . 0 2 5 2 . 4 0 6 2 . 0 1 9 E X P E R I M E N T A L I C M I 1 . G 5 0 2 . 0 5 0 2 . 5 5 0 3 . 1 0 0 3 . 8 0 0 R E II 6 5 4 . 5 8 7 2 7 . 9 I 8 0 1 . 2 5 8 7 4 . 5 9 9 4 7 . 9 2 ( T H E O . - E X P . l / E X P . - 0 . 1 7 6 - O . 1 8 3 - O . 2 0 6 - O 2 2 4 - O . 2 5 8 H E I C.H T = 1 . 0 1 6 M T H E O R E T I C A L ( C M I 3 . 3 9 7 4 . 1 9 0 5 . 0 6 2 6 . 0 1 5 7 . 0 4 8 E X P E R I M E N T A L I C M I 4 4 5 0 5 . 7 0 0 7 . 2 5 0 8 . 7 5 0 1 0 . 5 0 0 R E // 6 5 4 . 5 8 7 2 7 . 9 1 8 0 1 . 2 5 8 7 4 . 5 9 9 4 7 . 9 2 ( T H E O . - E X P . l / E X P . - 0 . 2 3 7 - 0 . 2 6 5 - O . 3 0 2 - 0 . 3 1 3 - O . 3 2 9 - 67 -c h i p s w i t h f i t t e d s p h e r i c i t y and voidage i s e q u a l to 0.031 which i s s m a l l e r than t h a t w i t h measured s p h e r i c i t y and v o i d a g e which i s e q u a l to 0.265. T h i s may be due i n p a r t to problems a s s o c i a t e d w i t h measurement of the s p h e r i c i t y and voidage parameters. 4.2 Heat Transfer to Spherical Particles Three e x p e r i m e n t s were done at d i f f e r e n t a i r v e l o c i t i e s w i t h the s p h e r i c a l p e l l e t s i n o r d e r to v e r i f y the e x p e r i m e n t a l p r o c e d u r e . The o v e r a l l r e s u l t s are summarized i n T a b l e 4-13. Gas temperature p r o f i l e s are shown as a f u n c t i o n o f time and bed depth i n F i g u r e 4-11 to 4-13. The shape of the t r a n s i e n t temperature wave p a s s i n g though the bed i s s i m i l a r to t h a t shown i n F i g u r e 2-1 f o r the model of A n z e l i u s ( 1 ) . Appendix A, T a b l e 4-37a to 4-37c, l i s t s the raw d a t a f o r a l l t h r e e r u n s . The heat t r a n s f e r c o e f f i c i e n t was e v a l u a t e d u s i n g the model of Chapter 2, where e q u a t i o n 2-17 g i v e s f o r s p h e r i c a l p a r t i c l e s T = t ± + ( T Q - t ± ) <5 (2-17) each T and t ^ b e i n g measured d i r e c t l y i n the experiment. $ i s a f u n c t i o n o f the heat t r a n s f e r c o e f f i c i e n t , and the thermal p r o p e r t i e s o f the s o l i d and g a s . Whereas the thermal p r o p e r t i e s of the a i r are known, the v a l u e s f o r the s o l i d were n o t . T h e r e f o r e two methods were used to e v a l u a t e the heat t r a n s f e r c o e f f i c i e n t . In the f i r s t method, measured v a l u e s of CpS (1795 J/kg K) and a c a l c u l a t e d thermal c o n d u c t i v i t y , k were used. In the second, the heat c a p a c i t y (1773 J/kg K and 1814 J/kg K) as w e l l as the heat t r a n s f e r c o e f f i c i e n t were found by f i t t i n g T a b l e 4-13 A summary of heat t r a n s f e r run c o n d i t i o n s and r e s u l t s f o r s p h e r i c a l p a r t i c l e Sphere A i r . I n l e t A i r Run Diarm? t e r Flow Temp. V e l o c i t y Re* Nu* BI c p No. (mm) (m 3/s) (°C) m/ s W/m2 K J/kg K 3 4.78 0.0109 133.0 0.336 67.6 87 10.15 0.054 1795 2 0.022 139.5 0.6775 83.6 169 12.55 0.066 1795 2 0.033 129.0 1.018 110.31 228 16.56 0.088 179 5 2 0.022 139.5 0.6775 81 .02 169 12.16 0.065 1773 3 0.033 129.0 1.018 103.1 228 15.48 0.082 1814 E q u a t i o n 1-3 0.0109 133.0 0.336 130.0 87 19.52 0.104 1-5 0.022 139.5 0.6775 151.7 169 22.77 0.121 1-5 0.033 129.5 1.018 173.9 228 26.11 0.139 1-6 0.0109 133.0 0.336 21.7 87 3.26 0.0173 1-6 0.022 139.5 0.6775 34.6 169 5.19 0.027 1-6 0.033 129.0 1.018 42.6 228 6.395 0.051 C a l c u l a t e d a t 100°C. Note: A l l the hf were c a l c u l a t e d w i t h f i t t e d k = 2 W/m K because t h e r e was no p r a c t i c a l d i f f e r e n t r e s u l t between u s i n g k ( e s t i m a t e d ) 2.5 W/m K. and k ( f i t t e d ) 2 W/m K. o a. cn o . CO o . cn o o 0.0 20.0 40.0 + (EXP.) AT THE HT.=0.610M O (EXP.) AT THE HT.=1.016M THEORETICAL CURVE AIR VEL0CITY=.336M/S REYNOLDS N0.=87.0 SPHERE DU.=4.78mm INLET GAS TEMPERATURE=133C C0EFFICIENT=67.56W/(K»M,,2) ax 60.0 ( X 1 0 2 ) —I 80.0 —I 100.0 120.0 TIME IN SECONDS Figure 4-11 Temperature profiles for sphere at Re„ = 87 o O O I D ' — s O o w D OS W CU w E-o . CT) C D O . CO + (EXP.) AT THE HT.=0.610M O (EXP.) AT THE HT.=1.016M _ THEORETICAL CURVE AIR VEL0CITY=.678M/S REYNOLDS NO.=169. SPHERE DIA.=4.7Bmm INLET GAS TEMPERATURE=139C C0EFFICIENT=83.6Tr/(K'M*,2) o o o " 0.0 80.0 ~ l 160.0 240.0, ( X 1 0 1 ~~1 320.0 400.0 480.0 TIME IN SECONDS Figure A—12 Temperature profiles for sphere at Rep = 169 o o in " — v o o ~ w Pi D H < W w E-1 00 O a . o . C O ro o o " 0.0 75.0 1 150.0 + (EXP.) AT THE HT.=0.610M O (EXP.) AT THE HT.=1.016M THEORETICAL CURVE AIR VEL0CITY=.883M/S REYNOLDS N0.=228. SPHERE DIA.=4.78mm INLET GAS TEMPERATURE=129C C0EFFICIENT=110.3W/(K»M»*2) i 225.0, ( X 1 0 1 ) 300.0 —1 375.0 450.0 TIME IN SECONDS Figure 4-13 Temperature profiles for sphere at Re p = 228 - 72 -method. The heat transfer coefficient or the coefficient and the heat capacity are found by minimizing the sum of (Experimental T -Theoretical T) . A computer program was written to find the coefficient. The air density (pf) and heat capacity (cp^) were taken from the literature. The diameter and density of the spherical catalyst were measured directly (Table 3-2). Since the heat capacity and thermal conductivity of the pellets were not available, three similar experiments were f i r s t carried out to find out the average value of the solid heat capacity. A known weight of solids was heated up in an oven to 100°C over a period of approximately 24 hours. The solids were then introduced into a vacuum thermoflask containing a measured amount of water. A cork with a themometer was immediately brought to cover the thermoflask and the peak steady state temperature value was measured. The heat capacity of the spheres was calculated by using a simple heat balance equation assuming adiabatic conditions. The value is shown in Table 4-14. The value of the catalyst pellet thermal conductivity was estimated using the weighted average sum of thermal conductivity of i t s chemical components. The details of this calculation are shown in Appendix B. As stated above, the heat transfer coefficient as well as the heat capacity could be evaluated simultaneously by minimizing the sum (Experimental T - Theoretical T) . The comparison of results of the two methods are indicated in Table 4-13. The calculated heat capacities did not agree too closely with the experimental value. A l l further - 73 -Table 4-14 Pr o p e r t i e s of s p h e r i c a l c a t a l y s t Weight % N i c k e l 20 A 1 2 0 3 62 - 68 CaO 4 . 5 - 5 . 3 A f t e r balance with NiO, i t becomes Weight % NiO 25.5 AI2O3 68 CaO 5.5 99% Bulk d e n s i t y 800.95 kg/m3 Surface area 125 - 185 m /g Pore volume 0.25 - 0.35 cm3/g Loss on i g n i t i o n < 10% at 1000°F (538°C) Experimental and measured heat c a p a c i t y 1794.76 (J/kg K) Estimated thermal c o n d u c t i v i t y 3 (W/m K) F i t t e d thermal c o n d u c t i v i t y 2(W/m K) - 74 -d i s c u s s i o n i s based on c o e f f i c i e n t s d e r i v e d u s i n g the e x p e r i m e n t a l heat c a p a c i t y v a l u e s . For the t h r e e p a r t i c l e Reynolds numbers t e s t e d , comparison of the l i t e r a t u r e heat t r a n s f e r c o e f f i c i e n t s and e x p e r i m e n t a l v a l u e s g e n e r a t e d by f i t t i n g the d a t a are l i s t e d i n Ta b l e 4-13. The heat t r a n s f e r c o e f f i c i e n t v a l u e s were found to i n c r e a s e as the Reynolds number was i n c r e a s e d from 87 to 228.2. S i n c e the heat t r a n s f e r c o e f f i c i e n t i s a l s o a f u n c t i o n of the s i z e o f p a r t i c l e s , t h r e e e m p i r i c a l e q u a t i o n s were used f o r the comparison. One of the e q u a t i o n s was develop e d by G.O.G. L o f and R.W. Hawley ( 1 2 ) . The o t h e r two were de v e l o p e d by Colbur n (4) and K u n i i and L e v e n s p i e l (18) r e s p e c t i v e l y . The K u n i i and L e v e n s p i e l e q u a t i o n was f o r p a r t i c l e s of a s i z e s l i g h t l y b i g g e r than 1 mm d i a m e t e r . The p r e d i c t i o n d e v e l o p e d by G.O.G. L o f and R.N. Hawley (12) was based on d a t a f o r b i g g e r p a r t i c l e s , i . e . , from 4.8 mm to 38.1 mm, whereas the C o l b u r n e q u a t i o n was good f o r p a t i c l e s of a s i z e range 2.3 mm to 8.5 mm. The heat t r a n s f e r c o e f f i c i e n t s f o r the 4.77 mm d i a . c a t a l y s t p e l l e t s g e n e r a t e d by f i t t i n g the e x p e r i m e n t a l d a t a were found to be i n between the v a l u e s c a l c u l a t e d by u s i n g the t h r e e e m p i r i c a l e q u a t i o n s f o r the s m a l l e r and l a r g e r p a r t i c l e s . T h e r e f o r e , the e x p e r i m e n t a l v a l u e s can be taken to agree w i t h the l i t e r a t u r e v a l u e s . The N u s s e l t h D P number ( ) equ a l to 16.53, 12.1, and 10.12 w i t h the c o r r e s p o n d i n g e x p e r i m e n t a l Reynolds number 228.2, 169.3 and 87 can be r e a s o n a b l y l o c a t e d between o t h e r d a t a i n F i g u r e 4-37 (page 126)which shows N u s s e l t vs Reynolds number f o r a i r i n packed beds w i t h d i f f e r e n t s i z e s o f - 75 -p a r t i c l e s . Thus, the equipment and procedures were taken to be r e l i a b l e f o r heat t r a n s f e r c o e f f i c i e n t d e t e r m i n a t i o n . The comparison o f the f i t t e d t h e o r e t i c a l and e x p e r i m e n t a l gas temperature v a l u e s are shown i n Ta b l e 4-15 to 4-19. The d i f f e r e n t bed depth temperature h i s t o r y showed r e a s o n a b l e d e v i a t i o n between the t h e o r e t i c a l and e x p e r i m e n t a l v a l u e s . 4.3 Heat Transfer to Wood Chips The c a l c u l a t i o n o f the heat t r a n s f e r c o e f f i c i e n t was s i m i l a r to t h a t f o r the s p h e r i c a l c a t a l y s t , except t h a t e q u a t i o n s f o r the f i x e d bed o f s l a b - s h a p e d p a r t i c l e s were used. The heat t r a n s f e r c o e f f i c i e n t v a l u e s were e v a l u a t e d by m i n i m i z i n g the sum of ( E x p e r i m e n t a l T -T h e o r e t i c a l T) . The t h e o r e t i c a l t emperatures were c a l c u l a t e d by u s i n g e q u a t i o n ( 2 - 3 3 ) . The computer programme w r i t t e n to c a l c u l a t e the heat t r a n s f e r c o e f f i c i e n t s f o r the s l a b - s h a p e d p a r t i c l e s i s shown i n Appendix D-4. Thermal p r o p e r t i e s of the wood c h i p s were r e q u i r e d . S i n c e the wood thermal c o n d u c t i v i t y i s a known f u n c t i o n of m o i s t u r e c o n t e n t , t h r e e samples of wood c h i p s were t e s t e d f o r m o i s t u r e c o n t e n t . The weighed wood c h i p s samples were heated up to 110°C, and the m o i s t u r e c o n t e n t was found from the weight l o s s a f t e r 24 hours (Appendix B - l ) . The heat c a p a c i t y o f Douglas f i r was r e p o r t e d to be 2.70 KJ/kg/°K. However, t h i s v a l u e was exp e c t e d f o r m o i s t Douglas f i r wood. P r i o r t o the experiment, the d r i e d wood c h i p s had absorbed a s m a l l amount of m o i s t u r e , which would be d r i v e n o f f by the hot a i r . T h e r e f o r e , a proper heat c a p a c i t y was found by d e t e r m i n i n g both the heat t r a n s f e r c o e f f i c i e n t and heat c a p a c i t y by m i n i m i z i n g the sum of ( E x p e r i m e n t a l T Table 4-A COMPARISON OF THEORETICAL AND EXPERIMENTAL TEMPERATURE VALUES OF THE SPHERICAL PARTICLES DI A ( M ) TGO ( C ) GO ( K G / ( M " 2 ) / S VO ( M/S I ' 0.00477 139.50 ) - 0.77422 = 0.67750 VAR = 0.053-10366 H ( w/(M-'3) /X ) - 81.025085 K ( W/M/K I • 2.000000 CP ( J/KG ) = 18 14.1 10896 HEIGHT ( M ) • 0.6 10 TI ME EXPERIMENTAL THEORETICAL ( EXP-IHEO 1/E.xP MIN ( C I ( C I 13 250 38.5 39 773 -O 03307 17.250 58.3 61.833 -O.OGOGI 21.250 98.7 98.521 0.00IB1 24.567 121.0 122.317 -0.01080 26.567 128.4 130 679 -O.01775 28.5G7 132.3 135.403 -0.023-15 HE 1GHI ( M ) = 0.013 1 t ME EXPERIMENTAL THEORE I ICAI. ( E »P - t ME'J I / E «'P MIN ( C I I C ) 17 250 37.6 37.021 O.Oi510 2 1.250 47.7 48.117 -0.008'5 2-1.567 G6.I 69 .258 -0.04777 2G.5G7 04.3 85.457 -O.OI3'2 28.567 102.7 101.373 0.01292 32.567 125.5 121.873 0.004;13 38.5G7 134.8 137.615 -0.02008 HE I CUT ( M I = 1.016 TIME EXPERIMENTAL THEORETICAL (EXP-THEO)/£XP MIN ( C ) ( C ) 24.567 40.8 30.720 0.02GIG 26.567 45.9 45.217 0 01488 28.567 54.3 53 991 O.OC570 32.567 83.7 80 132 0.04262 38 5G7 122.8 118 979 0.03112 42.567 131 4 132 377 -0 O0743 Table 4-16 A C O M P A R I S O N O f T H E O R E T I C A L A N D E X P E R I M E N T A L T E M P E R A T U R E V A L U E S O F T H E S P H E R I C A L P A R T I C L E S H E I G H T ( M ) * 0 . 8 1 2 B O I A ( M ) ° 0 . 0 0 4 7 7 T C C ( C ) • 1 2 9 . O O C O ( K G / ( M " 2 ) / S ) • 1 . 0 4 3 2 3 V O ('•• M / S I • O . B 8 2 2 7 V A R 0 . 3 4 1 1 1 8 3 4 H ( W / ( M " 2 ) / K ) K ( W / M / K ) C P ( U / K G ) 1 0 3 . 1 4 7 2 O O O 1 7 7 3 . 0 7 T I M E M I N 9 . 0 0 0 1 3 . 0 0 0 I S O O O 1 8 . 5 3 3 2 0 . 5 3 3 2 4 . 5 3 3 3 1 . 5 3 3 E X P E R I M E N T A L ( C ) 3 3 . I 3 6 . 2 4 0 . 7 G 4 . 6 8 7 . I 1 1 8 . 9 1 2 7 . 2 H E I G U I ( M ) " 1 . 0 1 6 0 T H E O R E T I C A L ( C I 3 2 . 0 6 0 3 5 . 15 I 4 2 . 5 5 4 7 1 . 3 3 9 9 I 3 7 3 1 I B . 9 0 9 1 2 8 . 7 3 4 ( E X P - T H E O I / E X P 0 . 0 3 1 4 2 0 . 0 2 8 9 6 - O . 0 4 5 5 4 - O . 1 0 4 3 2 - 0 . 0 4 9 0 6 - 0 . 0 0 O 0 7 - 0 . 0 1 2 0 6 H E I G H T ( M ) . 0 . 6 0 9 G T I ME M I N 7 . 0 0 0 1 1 . 0 0 0 1 3 . O O O 17 . O O O 2 0 . 5 3 3 3 I . 5 3 3 E X P E R I M E N T A L ( C ) 3 2 . 1 3 8 . 5 5 0 . 6 9 5 . 8 12 1 . 4 1 2 8 . G T H E O R E T I C A L ( C ) 3 2 . 3 6 7 4 4 . 3 6 2 6 2 . 7 5 5 1 0 6 . 0 5 5 1 2 4 . 2 7 5 I 2 B . 9 9 8 ( E X P - T H E O I / E X P - 0 . 0 0 8 3 0 - O . 1 5 2 2 5 - O . 2 4 0 2 2 - O . 1 0 7 0 5 - O 0 2 3 6 8 - 0 . 0 0 3 1 0 T I M E M I N 9 . 0 0 0 I B . 5 3 3 2 0 . 5 3 3 2 2 . 5 3 3 2 4 . 5 3 3 3 I . 5 3 3 E X P E R I M E N T A L ( C ) 3 2 . 0 4 0 . 2 4 8 . 3 6 3 . 0 8 3 . 2 1 2 3 2 T H E O R E T I C A L ( C I 3 2 . O C O 3 9 . 16 I 4 9 . 1 4 5 6 4 . 3 4 4 8 2 . 3 3 4 1 2 3 . 4 2 6 ( E X P - T H E O I / E X P - 0 . O O O O I 0 . 0 2 5 8 4 - 0 . 0 1 7 4 9 - 0 O 2 1 3 3 0 . 0 1 0 4 I - 0 . 0 0 1 8 3 Table 4-17 A COMPARISON OF THEORETICAL AND EXPERIMENTAL TEMPERATURE VALUES OF THE SPHERICAL PARTICLES H E 1 O H T ( M OIA ( M ) TGO ( C ) CO ( K G / ( M " 2 ) / S ) VO ( M/S ) VAR • 0.30545295 H ( W/ (M"2) /K ) • K ( W/M/K ) CP ( J /KG ) 0.0047? 133 OO 0.39755 O.33622 67.560699 2.000000 1794.76538 I T 1 ME MIN 38.233 •16 . 233 52.233 54.233 5B.233 64.233 70.233 EXPERIMENTAL ( C ) 38 . 9 53.0 87 . 8 100. 3 117.1 125 2 129. 1 THEORETICAL ( C ) 34.9G4 55.005 84.SOS 95.003 112.051 126 6 18 131 .620 (EXP-THED)/EXP O.10117 -0.03784 0.03411 0.05281 0.043 12 -0.0033 I -O.01952 HEIGHT ( M ) CO HEIGHT ( M ) - 0.S09G TIME MIN EXPERIMENTAL ( C ) 12.483 35.3 36.233 59.8 40.233 87.2 46.233 119.1 52.233 127.4 64.233 130.8 THEORETICAL ( C ) 32 .000 67.132 9 1.116 1 IB.632 129.854 132.955 (EXP-T1IE0)/EXP 0.09348 -O. 122G I -0.04490 0.00393 -0.01926 -0.01647 TIME MIN 46.233 54.233 58.233 64.233 70.233 78.233 EXPERIMENTAL ( C ) 39.8 45.6 55 . 9 88 . 7 115.7 126.5 THEORETICAL 1 C 1 32 994 4 2.030 53.745 79.856 106.068 126.083 (EXP-THEOl/EXP 0. 17 102 O.07828 O 03854 0.0997 I 0.08325 0.00329 Table 4-18 A COMPARISON OF THEORETICAL AND EXPERIMENTAL TEMPERATURE VALUES HEIGHT OF THE SPHERICAL PARTICLES ( M ) * 0.8 13 DIA ( M ) • 0.0047 7 TGO ( C ) • 139.50 GO ( K G / ( M " 2 ) / S ) • 0.77422 VO ( M/S ) - 0.67750 VAR • 0.08221757 H ( V / ( M " 2 ) / K ) K ( W/M/K ) CP ( O/KG ) 83.584 2 .00 1794.77 HE I CUT ( M ) » 0.610 T I ME MIN 13.250 17.250 21.250 24.5G7 26.567 28.567 EXPERIMENTAL ( C ) 3B .5 58 . 3 98.7 12 1.0 12B.4 133 .3 THEORETICAL ( C ) 39.895 62.98 1 10O.636 124.013 131.847 136.095 (EXP-TH£0)/£XP -0.03624 -0.08030 -0.01962 -0.02490 -0.02684 -0.03868 TIME MIN 17.250 21.250 24.567 26.567 28.567 32.567 3B.567 EXPERIMENTAL ( C ) 37.6 47.7 66 . 1 84 . 3 102 . 7 125.5 134.0 THEORETICAL ( C ) 37.08 I 4 8.806 71.012 87.725 103.800 126.608 138.039 ( EXP-THEQ l/EXP 0.01380 -0.023 19 -0.0743 1 -0.04063 -0.01071 -O 0O883 -0.02395 HEIGHT ( M ) TIME MIN 24.567 26.567 28.567 32.5G7 3B.567 42.567 EXPERIMENTAL ( C ) 40. 8 45 . 9 54 . 3 83.7 122 8 131 .4 THEORETICAL ( C ) 39.994 45.868 55.209 B2.554 12 1.215 133.573 (EXP-THEO)/EXP 0.01975 0.O0O7O -0.01674 0.01370 0.01291 -0.01653 T a b l e 4-19 A COMPARISON OF THEORETICAL AND EXPERIMENTAL TEMPERATURE VALUES HE H.llr ( M ) • O.8128 OF THE SPHERICAL PARTICLES TIME MIN EXPERIMENTAL ( C ) THEORETICAL ( C I { EXP-THEO)/EXP 9 OOO 33 . 1 32 035 0. 032 19 DIA ( M ) « 0.00477 13. .000 36 . 2 34 . 4 14 0. 04935 TGO ( C ) • 129.00 IS .000 40.7 40. 855 -0 . 00380 GO ( K G / ( M " 2 ) / S ) • 1.04323 18 . 533 64.6 68 . 623 -O 06228 VO ( M/S ) » 0.8B227 20 . 533 87 . 1 89 177 -O. 02384 24 .533 118.9 1 ia . 334 0 00476 VAR . 0.22710971 3 1 . 533 127.2 128 742 -o.. 012 12 H ( V / (M ' -2 ) /K ) - 110.310 K ( W/M/K ) . 2.000 HEIGHT ( M ) = 1.0160 CP ( J/KG ) » 1794.77 TIME MIN EXPERIMENTAL ( C ) THEORETICAL ( C ) (EXP-•THEOl/EXP HEIGHT ( M ) • 0.G096 9 .OOO 32 .0 32 .000 -0 OOOOI 18 . 533 . 40. 2 37 . 7 50 0 .06094 TIME EXPERIMENTAL THEORETICAL MIN ( C ) ( C ) ( EXP-THEO)/EXP 20 22 .533 . 533 48 . 3 63 .0 46 61 . 772 . 392 0 0 .03163 .02552 7.OOO 32 . 1 32.24G -0 . 00455 24 . 533 83.2 79 . 536 0 .04404 11.000 38.5 42.645 -0 . 10765 31 .533 123.2 123 .073 0 .00103 13.000 50.6 60.335 -0 . 19238 17.000 95.8 104.879 -0 . 09477 20.533 12 1.4 124.144 -0 . 0226 1 31.533 128.6 128.999 -0 . 00310 - 81 -- T h e o r e t i c a l T) . A heat c a p a c i t y v a l u e e q u a l to 2.516 KJ/kg K was found by f i t t i n g the e x p e r i m e n t a l d a t a of 4 mm wood c h i p s a t Re = 698.3. As expected the f i t t e d v a l u e i s lower than the v a l u e of 2.70 KJ/kg K s i n c e the d r i e d wood c h i p s s h o u l d always have a s m a l l e r heat c a p a c i t y v a l u e than moist wood c h i p s . However, the v a l u e o f 2.515 KJ/kg K was an average heat c a p a c i t y parameter f o r the wood c h i p s d u r i n g the heat t r a n s f e r s t e p . A second v a l u e of 2.6 KJ/kg K was determined by f i t t i n g the e x p e r i m e n t a l data of 4 mm wood c h i p s at Re = 590.2. Si n c e t h e r e was no p r a c t i c a l d i f f e r e n c e between the two v a l u e s , a rounded o f f v a l u e of 2.5 KJ/kg K was chosen as the heat c a p a c i t y parameter f o r subsequent c a l c u l a t i o n . Heat t r a n s f e r experiments were c a r r i e d out over a range of a i r f l o w r a t e s f o r the f o u r d i f f e r e n t wood c h i p s i z e s . Measured and f i t t e d temperature p r o f i l e s and v a l u e s are shown i n F i g u r e 4-14 to 4-26 and T a b l e 4-20 to 4-32 r e s p e c t i v e l y f o r t y p i c a l p o i n t s i n a number of these r u n s . Data f o r the f i r s t few minutes of the runs were n e g l e c t e d i n the f i t t i n g , because of the i n i t i a l d i s t u r b a n c e s to temperatures and f l o w c o n d i t i o n s . T a b u l a t i o n of a l l d a t a p o i n t s , g e n e r a l l y at the two bed depths used i n the f i t t i n g i s found i n T a b l e s 4-37e to 4-37p (Appendix A ) . G e n e r a l l y , the f i t of e x p e r i m e n t a l gas temperature d a t a by the model t r a n s f e r c o e f f i c i e n t would be s i m i l a r to F i g u r e 4-28 and 4-29. The 2 mm wood c h i p a i r temperature p r o f i l e ( w i t h raw data) o b t a i n e d u s i n g the h i g h e r heat t r a n s f e r c o e f f i c i e n t of 17.0 W/m K ( F i g u r e 4-29) rose more q u i c k l y to s t e a d y s t a t e than t h a t o b t a i n e d u s i n g the lower heat t r a n s f e r c o e f f i c i e n t 14.8 W/s K ( F i g u r e 4-28). T h e r e f o r e , the temperature p r o f i l e i s d i c t a t e d by the v a l u e of heat t r a n s f e r o i n o i n . c n o W OS < OS w ,«£ U E-1 CO <: o ° in. o . o "o (EXP.) AT THE HT.=0.406M (EXP.) AT THE HT.=0.610M THEORETICAL CURVE WOOD CHIP THICKNESS=2.44mm REYNOLDS N0.=255.0 00 r o o . 0.0 40.0 80.0 120.0, (X1.0 1 ) 160.0 200.0 240.0 TIME IN SECONDS F i g u r e 4-14 Temperature p r o f i l e s f o r 2.44 mm t h i c k wood c h i p s at R e n = 255 in XT . o rsj. in. cn O o & f 05 E-<d 05 w a. CO O ° . in . o . — o AT THE HT.=0.406M AT THE HT.=0.610M THEORETICAL CURVE WOOD CHIP THlCKNESS=2.44mm REYNOLDS N0.=329.4 C O o . CM 0.0 30.0 60.0 90.0 , ( X 1 0 1 1.20.0 150.0 180.0 TIME IN SECONDS Figure 4-15 Temperature profiles for 2.44 mm thick wood chips at Rep = 329.4 in CJ o w H <: w w o rsi. i n . cn o . CO o ° i n . o + AT THE HT.=0.406M (EXP.) AT THE HT.=0.610M THEORETICAL CURVE WOOD CHIP THICKNESS=2.44mm REYNOLDS N0.=403.1 C O 4> (M 0.0 20.0 40.0 n60.0 '(X10 1 ) TIME IN SECONDS 80.0 100.0 120.0 Figure 4-16 Temperature profiles for 2.44 mm thick wood chips at Rep = 403.1 o in U o 05 ZD H <d DC W S w E -O o i n . o . i n . (EXP.) AT THE HT.=0.406M (EXP.) AT THE HT =0.610M THEORETICAL CURVE WOOD CHIP THICKNESS=2.44mm REYNOLDS N0.=476.1 00 o . CM 0.0 20.0 40.0 60.0 . ( X 1 0 1 80.0 TIME IN SECONDS 100.0 120.0 Figure 4-17 Temperature profi les for 2.44 mm thick wood chips at Re p = 476.1 in o (M . O o 05 H < Oi W CU w E -o in. CD o . in. (EXP.) AT THE HT.=0.406M (EXP.) AT THE HT.=0.610M THEORETICAL CURVE WOOD CHIP THICKNESS=4.03mm REYNOLDS N0.=374.1 00 Ox o . CM 0.0 60.0 "1 120.0 ~I 180.0, ( X 1 0 1 ) 240.0 TIME IN SECONDS 300.0 360.0 Figure 4-18 Temperature profi les for 4.03 mm thick, wood chips at Re„ = 374.1 o m Figure 4-19 Temperature profiles for 4.03 mm thick wood chips at Re,, = 482.9 in o in. cn o . in. o . CXI o o "o (EXP.) AT THE HT.=0.406M (EXP.) AT THE HT.=0.610M THEORETICAL CURVE WOOD CHIP THICKNESS=4.03mm REYNOLDS N0.=590.2 0.0 40.0 80.0 120.0 ( X 1 0 1 ) 160.0 200.0 240.0 TIME IN SECONDS Figure 4-20 Temperature profiles for 4.03 mm thick wood chips at Rep = 590. o i n o i n . CO o K <: OS Cd w E-00 <: o ° i n . o . + o o " O (EXP.) AT THE HT.=0.610M (EXP.) AT THE HT.=0.813M THEORETICAL CURVE WOOD CHIP TfflCKNESS=4.03mm REYNOLDS N0.=698.3 CO o o . 0.0 30.0 60.0 90.0 , ( X 1 0 1 ) 120.0 150.0 180.0 TIME IN SECONDS Figure 4-21 Temperature profi les for 4.03 mm thick wood chips at ReR = 698.3 LO O CM . in. cn u o w 05 D E-1 <3 05 W Cu s w E-1 COo ° in. o . o . CM "O (EXP.) AT THE HT.=0.406M (EXP.) AT THE HT.=0.610M THEORETICAL CURVE WOOD CHIP THICKNESS=7.26mra REYNOLDS N0.=490.7 0.0 40.0 80.0 120.0 C X 1 0 1 ) 160.0 200.0 240.0 TIME IN SECONDS Figure 4-22 Temperature profiles for 7.26 mm thick wood chips at Rep = 490.7 o LO O o O 05 D E-<d 05 W OH w <d O in. CO o . in. (EXP.) AT THE HT.=0.406M (EXP.) AT THE HT.=0.610M THEORETICAL CURVE WOOD CHIP THICKNESS=7.26mm REYNOLDS N0.=832.5 i o . CM 0.0 30.0 60.0 90.0 , ( X 1 0 1 ) 120.0 TIME IN SECONDS 150.0 180.0 Figure 4-23 Temperature profi les for 7.26 mm thick wood chips at Ren = 632.5 o LO o CM . in. cn CJ o 05 D H < 05 W Cu S w 00 O ° in . a. r- AT THE HT.=0.406M (EXP.) AT THE HT.=0.610M THEORETICAL CURVE WOOD CHE? THICKNESS=7.26mm REYNOLDS N0.=774.2 O . CM 0.0 30.0 60.0 ~~I 90.0 , ( X 1 0 1 ) I 120.0 TIME IN SECONDS 150.0 180.0 F i g u r e 4-24 Temperature p r o f i l e s f o r 7.26 mm t h i c k wood c h i p s at Re„ = 774.2 in THE HT.=0.406M (EXP.) AT THE HT.=0.610M THEORETICAL CURVE WOOD CHIP THICKNESS=4.81mm REYNOLDS N0.=399.0 0.0 30.0 60.0 90.0 . ( X 1 0 1 ) 120.0 150.0 180.0 TIME IN SECONDS Figure 4-25 Temperature profiles for 4.81 mm thick wood chips at Rep = 339 IT) o w 05 D <: 05 W w o rg. in . cn o. CO o ° 3-0.0 20.0 HT.=0.40oM (EXP.) AT THE HT.=0.6I0M THEORETICAL CURVE WOOD CHIP TfflOCNESSM.Blmm REYNOLDS N0.=62fl.5 40. 1 60.0 . ( X 1 0 1 80. 100.0 120.0 TIME IN SECONDS Figure 4-26 Temperature profiles for 4.81 mm thick wood chips at Rep = 669.5 o o y—x o O " w 05 D <: 05 w S w CO O o . C D O . to AIR VEL0CTTY=.678M/S WOOD CHIP THICKNESS=7.26mm AT THE HEIGHT=0.81M INLET GAS TEMPERATURE=121C C0EFFICIENT=33.7J/(S»K«M**2) o o " 0.0 45.0 90.0 135.0, ( X 1 0 1 180.0 225.0 TIME IN SECONDS 270.0 F i g u r e 4-27 E x p e r i m e n t a l gas temperature p r o f i l e f o r 7.26 mm wood c h i p s at V f = .678 M/S o o in — N o o """ w D E-W S W 00 o o . o . co ~=F-o . m AIR VEL0CITY=.678M/S WOOD CHIP TfflCKNESS=2.44mm AT THE HEIGHT=0.61M INLET GAS TEMPERATURE=123 C C0EFFICIENT=14.8J/(S»K*M*»2) o o " 0.0 45.0 90.0 135.0, ( X 1 0 1 ) 180.0 TIME IN SECONDS 225.0 270.0 Figure 4-28 Experimental gas temperature profile for 2.44 mm thick wood chips at Vf = .678 M/S o o in /• «. Q CJ ™-o """ W P S D E-» < W CM w E -o o . cn CO o . CO AIR VEL0CITY=.882M/S WOOD CHIP TfflCKNESS=2.44mm AT THE HEIGHT=0.81M INLET GAS TEMPERATURE=131C C0EFFICTENT=17.0J/(S*K*M»*2) o o " 0.0 45.0 90.0 135.0, ( X 1 0 1 ~~l 180.0 TIME IN SECONDS — i 225.0 270.0 Figure 4-29 Experimental gas temperature profi le for 2.44 mm thick wood chips at Vf = .882 M/S in x — ^ m U 2-o W 05 D 05 W W E-1 00 O in . co in . co in . •sr in. + (EXP.) AT THE HT.=0.406M _ THEORETICAL CURVE AIR VEL0CITY=.766M/S REYNOLDS N0.=482.9 WOOD CHIP TfflCKNESS=4.03mm INLET GAS TEMPERATURE=123C FITTED Hf=17.5W/(K»M»»2) 00 0.0 25.0 50.0 ~~1 75.0 , ( X 1 0 1 ) 100.0 125.0 150.0 TIME IN SECONDS F i g u r e 4-30 Temperature p r o f i l e s f o r 4.05 ram t h i c k wood c h i p s w i t h d i f f e r e n t heat t r a n s f e r c o e f f i c i e n t Table 4-20 A C O M - A R I S O N O F T H E O R E T I C A L A N O E X P E R I M E N T A L T E M P E R A T U R E V A L U E S O F 2MM tfOOD C H I P H E I G H T ( M I T H I C K N E S S ( M I T G O ( C ) R E Y N O L D S N O . = O . 0 0 2 * 1 4 1 1 7 . 0 0 = 2 5 5 . O V A R O . 1 0 6 U 0 7 3 H ( v l / S / ( M - ' ? I / K I K ( U / S / M / K 1 C P ( U / K G ) 1 2 . 2 8 7 0 . 0 9 1 0 2 5 1 5 . 8 5 T I M E M I N S . 9 6 7 8 . 4 6 7 9 . 2 8 3 1 0 . 2 8 3 1 2 . 2 8 3 1 4 . 2 8 3 2 2 . 2 8 3 3 4 . 2 8 3 E X P E R I M E N T A L I C I 6 8 3 8 0 . G 8 6 . 2 9 1 6 9 H 9 1 0 3 3 1 1 0 3 1 1 3 . 4 T H E O R E T I C A L ( C I 7 0 . 9 1 2 8 I . 9 0 0 8 7 . 2 4 6 9 2 . 8 1 3 101 7 3 3 1 0 7 . 7 12 I 16 0 9 2 1 1 7 . 0 0 0 ( E X P - T H E O I / E X P - 0 . 0 3 8 2 S - 0 . 0 1 6 13 - 0 . 0 1 2 13 - O 0 1 3 2 5 - 0 . 0 2 8 6 5 - 0 . 0 4 2 7 1 - 0 . 0 5 2 5 1 - 0 . 0 3 1 7 5 H E I G H T ( M I = 0 . 4 0 6 T I M E E X P E R I M E N T A L T H E O R E T I C A L ( E X P - T H E O I / E X P M I N ( C I ( C I 4 . 4 6 7 6 5 . 8 7 1 . 7 5 8 - 0 . C 9 0 5 5 5 . 4 6 7 7 7 . 8 8 0 . 7 3 2 - 0 . 0 3 7 6 9 6 . 4 6 7 8 8 1 8 8 . 2 3 4 - 0 . 0 0 1 5 2 7 . 4 6 7 9 6 . 7 9 4 . 8 7 3 0 . 0 1 8 0 9 8 . 4 6 7 1 0 1 . 9 I 0 O . 1 2 7 0 . 0 1 7 4 0 9 . 2 8 3 1 0 4 . G 1 0 4 . 0 3 3 0 . 0 0 5 12 1 2 . 2 8 3 1 1 0 . G 1 1 1 . 7 0 5 - 0 . 0 0 9 9 9 1 8 . 2 8 3 1 1 4 . 5 1 1 6 . 1 7 7 - 0 . 0 1 4 6 5 Table 4-21 A C O M P A R I S O N OE T I I E O R E I I C A L AND E X P E R I M E N T A L 1 E MI'E RA IIIRE V A L U E S O F ;MM WOOD C H I P T H I C K N E S S ( M l = 0.002-14 T G O ( C I » 123 .00 R E Y N O L D S NO. • 3 2 9 . 4 HEIGHT ( M I = 0 . 6 1 0 V A R O.304 IG771 H ( J / S / 1 M - - 2 l / K I K ( U / S / M / K I CP ( 0 / K G I 14 .790 0 . 0 9 4 0 2 5 1 5 . B 5 T I M E M I N 4 . 2 5 0 5 . 5 8 3 6 . 3 1 7 7 . 3 1 7 8 . 3 1 7 9 . 8 3 3 1 3 . 8 3 3 2 5 . 8 3 3 E X P E R I M E N T A L ( C I 6G . 3 83 . 3 9 0 . 9 98 . 7 104 l 109 . 2 114 .9 120. I T H E O H E T I C A L ( C I 6 2 . 1 5 2 7 6 . 3 8 8 8 3 . 0 5 4 9 I .980 9 9 . 1 9 9 107 .537 118.4 16 122.974 ( E X P - T H E O I / E X P 0 . 0 G 2 5 7 0 . 0 8 2 9 7 0 . 0 6 6 3 1 0 . 0 6 8 0 0 0 . 0 4 7 0 3 0 . 0 1 5 2 3 - 0 . 0 3 0 6 0 - 0 . 0 2 3 9 3 O o HEIGHT I M ) TIME EXPERIMENTAL THEORETICAL ( E X P - T H E O I / E x P MIN I C 1 ( C I 2 . 5 6 7 58 3 6 3 . 5 1 8 - 0 . 0 8 9 5 0 3 . 2 3 3 75 2 7 2 . 4 7 0 0 . 0 3 6 3 0 3 . 7 3 3 8 2 . 5 7 0 . 7 8 4 0 . 0 4 5 0 5 4 . 5 8 3 9 5 . 9 0 8 . 3 5 8 0 . 0 7 8 6 5 5 . 5 8 3 105.7 97 684 0 . 0 7 5 8 4 7 . 3 17 1 1 3 5 108.711 0 . 0 4 220 9 . 8 3 3 1 1 8 . 7 117 .389 0 . 0 1 1 0 5 1 7 . 8 3 3 121.6 122 .819 - 0 . 0 1 0 0 2 T a b l e 4-22 A C O M P A R I S O N Of T H E O R E T I C A L A N D E X P E R I M E N T A L T E M P E R A I U R E V A L U E ! 2 M M W O O D C H I P H E I G H T ( M ) = 0 . 6 1 0 T H I C K N E t S ( M I T G O ( C ) R E Y N O L D S N O . = 0 . 0 0 2 4 4 1 2 5 . 0 0 = 4 0 3 . I V A R O . 1 0 6 9 1 1 3 0 H ( J / S / f M - • 2 ) / K I K ( J / S / M / K ) C P ( J / K G I 1 5 . 8 0 7 0 . 0 9 4 0 2 5 1 5 . 8 5 T I M E M I N 3 . 6 0 0 4 . 4 6 7 5 . 1 3 3 6 . 1 3 3 7 . 4 6 7 9 . 0 5 0 I I . 0 5 0 1 6 . 2 6 7 E X P E R I M E N T A L ( C I 6 5 9 7 9 7 8 8 . 5 9 7 7 1 0 5 . 7 1 1 1 2 1 1 5 . 2 1 2 0 . 6 T H E O R E T I C A L ( C ) 6 6 . 3 4 7 7 7 . 0 7 8 8 4 . 5 9 5 9 4 . 6 4 0 1 0 4 . 6 3 4 1 1 3 . 0 4 0 1 1 9 . 6 0 9 12 1 . 3 1 8 ( E X P - T H E O I / E X P - 0 . 0 0 6 7 8 0 . 0 3 2 9 0 0 . 0 4 4 1 3 0 . 0 3 1 3 2 O 0 1 0 0 9 - 0 . 0 1 6 5 5 - 0 . 0 3 8 2 7 - 0 . 0 3 0 8 3 H E I G H T ( M . ) = 0 . 4 0 6 T I M E M I N E X P E R I M E N T A L I C 1 T H E O R E T I C A L ( C ) ( E X P - • T H E O ) / 2 . 4 3 3 G 6 . 5 7 1 . 6 9 4 - 0 0 7 8 10 3 . 1 0 0 7 8 . 7 8 2 . 4 2 7 - 0 0 4 7 3 6 3 . 6 0 0 8 5 . 3 8 7 . 9 3 8 - 0 . 0 3 0 9 3 4 . 4 6 7 9 8 . B 9 7 . 4 7 6 0 0 1 3 4 0 5 . 1 3 3 1 0 5 . 1 1 0 3 . 3 9 7 0 . . 0 1 6 2 0 7 . 4 6 7 1 1 4 . 7 1 16 . 4 15 - 0 0 1 4 9 5 1 . 0 5 0 1 2 0 . 3 124 . 9 4 5 - 0 O 3 0 G I 6 . 2 G 7 1 2 3 . 9 124 . 8 6 7 - 0 . 0 0 7 8 I Table 4-23 A C O M P A R I S O N O F T H E O R E T I C A L A N D E X P E R I M E N T A L T E M P E R A T U R E V A L U E S O r 2MM WOOD C H I P H E I G H T ( M I T H I C K N E S S ( M 1 T G O ( C I R E Y N O L D S HO. V A R 0 . 3 4 4 5 4 1 5 8 H ( J / S / ( M " 2 I / K I K ( J / S / M / K 1 C P ( U / K G ) 0 . 0 0 2 4 4 13 1 . 0 0 4 7 6 . 7 1 7 . 0 0 0 0 . 0 9 4 0 2 5 1 5 . 8 5 T I M E M I N 2 . 5 8 3 3 . 2 5 0 3 . 9 1 7 4 . 2 5 0 4 . 8 5 0 6 . 2 G 7 7 . 5 1 7 1 7 . 9 3 3 E X P E R I M E N T A L ( C I 6 6 . 8 7 8 . 3 9 0 . 4 9 5 5 1 0 2 . O 1 1 0 . 2 1 1 5 . 2 1 2 6 . a T H E O R E T I C A L ( C I 6 3 . 0 2 9 7 3 . 2 3 3 8 2 . 9 0 7 8 7 . 3 4 3 9 4 . I 19 1 0 9 . G 3 9 1 1 6 . 6 5 5 130.aao ( E X P - T H E O I / E X P 0 . 0 5 6 4 5 0 . 0 6 3 5 1 0 . 0 8 2 8 9 0 . 0 8 5 4 2 0 . 0 7 7 2 6 0 . 0 0 5 0 9 - 0 . 0 1 2 6 3 - 0 . 0 3 2 2 5 O H E I G H T ( M ) ' 0 . 4 0 6 T I M E E X P E R I M E N T A L T H E O R E T I C A L ( E X P - T H E O I / E X P M I N I C I ( C ) 1 . 5 8 3 6 6 . 3 7 3 . 1 3 6 - O . 103 I 1 1 . 9 1 7 7 6 . 1 7 3 . 2 0 4 0 . 0 4 1 8 3 2 . 5 5 0 8 2 . 8 7 9 . 2 9 2 0 . 0 4 2 3 7 2 . 9 1 7 9 7 . 9 8 8 . 6 5 8 0 . 0 9 4 4 0 3.917 1 0 9 . 4 1 0 3 . 1 8 8 0 . 0 5 6 7 8 4 . 8 5 0 1 1 5 . 3 1 1 0 . 2 5 8 0 . 0 4 3 7 3 6 . 5 1 7 1 2 0 . 3 1 2 2 . 0 9 6 - 0 . 0 1 4 9 3 1 3 . 6 6 7 1 2 7 . 4 1 3 1 . 0 0 0 - 0 . 0 2 8 2 6 T a b l e 4-24 A C O M P A R I S O N O F T H E O R E T I C A L A N D E X P E R I M E N T A L T E M P F R A T U R E V A L U E S OF 4 M M WOOD C H I P T H I C K N E S S ( M I T G O ( C ) R E Y N O L D S N O . 0 . 0 0 4 0 3 1 1 1 . 0 0 3 7 4 . I VAR 0 . 3 3 9 3 0 3 0 3 H ( J / S / ( M - ' 2 ) / K ) K ( O / S / M / K I C P ( J / K G ) 1 I . 8 1 0 0 . 0 9 4 0 2 5 1 5 . 8 5 H E I G H T I M ) = O 8 13 T I ME M I N 5 . 5 5 0 7 . 0 5 0 8 . 8 3 3 I 1 . 3 8 3 1 3 . 3 8 3 15 3 8 3 1 9 . 3 8 3 31 . 3 0 3 E X P E R I M E N T A L ( C I 5 1 . 5 6 0 . 8 6 6 . 9 7 7 O 8 2 . 7 B 6 8 9 2 . 2 9 8 , 3 T H E O R E T I C A L 1 C I 4 9 . 5 8 6 5 8 . 0 2 3 G 6 . 7 9 9 7 0 . 1 3 8 8 5 . 3 3 6 9 1 7 8 1 1 0 0 . 3 5 I 1 0 9 . 7 7 7 ( E X P - T H E O I / E X P 0 . 0 2 9 4 1 0 . 0 4 5 6 8 0 . G O 151 - 0 0 1 4 7 8 - 0 . 0 3 1 8 8 - 0 0 5 7 3 8 - 0 . 0 8 8 4 1 - 0 I 1 6 7 5 H E I G H T ( M ) - 0 . 6 1 0 T I M E E X P E R I M E N T A L T H E O R E T I C A L ( E X P - T H E O l / E X P M I N ( C I ( C ) 4 . 1 6 7 5 1 . 4 5 3 . 0 3 2 - 0 . 0 3 1 7 5 6 . 0 5 0 6 6 . 7 6 5 . 7 6 5 0 . 0 1 4 0 2 8 . 8 3 3 7 8 . 9 7 9 . B 0 9 - 0 . 0 1 1 5 3 9 . 8 3 3 8 2 8 8 4 . 1 5 4 - 0 . 0 1 6 3 5 1 1 . 3 8 3 8 7 . 6 8 9 . 7 7 3 - 0 . 0 2 4 8 1 I S . 3 8 3 9 4 . 2 1 0 0 . 3 2 5 - 0 . 0 6 5 0 2 2 5 . 3 8 3 9 9 . G 1 0 9 . 8 4 8 - 0 . 1 0 2 8 9 3 5 . 3 8 3 1 0 1 . 0 1 1 0 . 3 6 0 - 0 . 0 9 2 6 0 Table 4-25 A C O M P A R I S O N UT T H E O R E T I C A L A N D E X P E R I M E N T A L T E M P E R A T U R E V A L U E S O F 4 MM WOOD C H I P H E I G H T ( M I T H I C K N E S S I M I T G O ( C ) R E Y N O L D S N O . = 0 . 0 0 4 0 3 1 2 3 . 0 0 * 4 8 2 . 9 V A R O . 3 7 G 4 3 0 0 S H ( J / S / ( M - - 2 l / K I K ( 0 , ' S / M / K I C P ( J / K G I 1 7 . 4 7 6 0 . 0 9 4 0 2 5 1 5 . 8 5 T I M E M I N 2 . 8 3 3 4 . 1 8 3 5 . 1 6 7 6 . 5 0 0 7 . I G 7 7 . 8 1 7 1 0 . 5 5 O 2 7 . 4 8 3 E X P E R I M E N T A L I C I 5 1 . 7 6 6 G 7 6 . 3 8 7 . I 9 2 . 4 9 6 J 1 0 7 . 4 1 1 9 5 T H E O R E T I C A L ( C I 5 3 0 6 3 6 4 . 5 7 I 7 4 . 5 2 9 8 5 . 0 4 4 8 9 G 9 4 9 3 . 3 7 5 1 0 6 . 9 7 2 1 2 2 . 8 5 7 ( E X P - T H E O I / E X P - 0 . 0 2 6 3 6 0 . 0 3 0 4 7 0 . 0 2 3 2 1 0 . 0 2 3 6 1 0 . 0 2 9 2 8 0 . 0 3 0 J 8 0 . O O 3 9 8 - O . 0 2 8 0 9 H E I G H T ( M ) • 0 . 4 0 6 T I M E E X P E R I M E N T A L T H E O R E T I C A L ( E X P - T H E O I / E X P M I N I C I ( C ) 1 . 8 3 3 5 5 . 4 5 9 . 9 3 7 - 0 . 0 8 1 9 0 2 . 5 0 0 6 9 . 6 6 7 . 8 7 0 0 . 0 2 4 8 6 2 . 8 3 3 7 4 . 1 7 1 . 5 7 6 0 . 0 3 4 0 6 3 . 5 C O 8 6 . 2 7 8 . 5 4 7 0 . 0 8 8 7 B 4 .183 9 5 . 8 8 2 . 0 2 8 0 . 1 4 3 7 6 5 . 1 6 7 1 0 4 . 7 9 2 . 9 1 5 0 . 1 1 2 5 6 7 . 1 6 7 1 1 3 . 9 1 0 4 . 9 2 6 0 . 0 7 6 7 9 1 4 . 5 5 0 1 2 1 . 3 1 2 0 . B 8 3 O . 0 O 3 4 3 Table 4 -26 A C O M P A R I S O N OF T H E O R E T I C A L A N D E X P E R I M E N T A L T E M P E R A T U R E V A L U E S O F 4 M M WOOD C H I P H E I G H T ( M ) = 0 . 6 1 0 1 H I C K N £ S S ( M 1 T G O ( 0 ) R E Y N O L D S N O . 0 . 0 0 4 0 3 1 2 6 . 0 0 5 9 0 . 2 V A R = O . 1 6 6 3 9 8 3 9 H ( J / S / ( M " 2 ) / K ) K ( J / S / M / K I C P ( J / K G ) 2 0 0 0 0 0 . 0 9 - 1 0 2 5 1 5 8 4 9 8 5 4 T I ME M I N 3 . 0 0 0 3 . 6 6 7 4 . 7 3 3 5 . 4 0 0 7 . 0 6 7 8 . 7 5 0 1 7 . 1 3 3 3 2 . 1 0 0 E X P E R I M E N T A L 1 C 1 G G . 7 7 4 . 7 8 1 .9 8 7 . 3 9 B . B 1 0 5 . 5 I IB I 1 2 2 . 1 T H E O R E I I C A L I C ) 6 5 . 5 3 8 7 0 . 5 7 4 8 1 . 8 4 7 8 7 . a a a 9 9 9 10 1 0 9 4 0 7 1 2 4 . 8 2 6 1 2 2 5 5 4 ( E X P - T H E D I / E X P 0 . 0 1 7 4 3 0 . 0 5 5 2 4 0 . 0 0 0 6 5 - O . 0 0 6 7 4 - 0 . 0 1 1 3 2 - 0 . 0 3 7 0 3 - 0 . 0 5 6 9 6 - 0 . 0 0 3 7 2 H E I G H T ( M ) = 0 . 4 0 6 T I M E E X P E R I M E N T A L T H E O R E T I C A L ( E X P - T H E O I / E X P M I N ( C I ( C ) 1 . 6 6 7 5 9 I 6 5 . 2 0 8 - 0 . 1 0 3 3 5 2 . 0 0 0 6 7 . 8 7 0 . 5 4 2 - 0 . 0 4 0 4 4 2 . 6 6 7 7 8 . 1 7 8 . 6 2 0 - 0 . 0 0 6 6 5 3 . 3 3 3 8 8 . 4 8 0 . 8 6 2 0 . 0 8 5 2 7 4 . 4 0 0 9 6 . 4 9 6 . 3 8 4 0 . 0 0 0 1 6 5 . 4 0 O 1 0 4 . 0 1 0 4 . 0 2 1 - 0 . 0 0 0 2 1 7 . 7 5 0 1 1 3 . 3 1 1 5 . 5 1 0 - 0 . 0 1 9 5 1 1 2 . 7 5 0 1 2 0 . 3 1 2 G . 0 0 0 - 0 . 0 4 7 3 8 T a b l e 4-27 A C O M P A R I S O N O F T H E O R E T I C A L A N O E X P E R I M E N T A L T E M P E R A T U R E V A L U E S O F 4MM w o o n C H I P H E I G H T ( M ) • 0 . 8 1 3 T H I C K N E S S ( M ) T G O ( C ) C E V N O L O S N O . • 0 . 0 0 4 0 3 1 2 9 . 0 0 » 6 9 8 . 2 9 V A R 0 . 3 0 6 5 8 7 9 6 H ( J / S / ( M " 2 ) / K ) K ( J / S / M / K ) C P ( J / K G ) 2 3 . 3 9 4 0 . 0 9 4 0 2 5 1 5 . 8 5 T I M E M I N 2 . 6 6 1 3 . 5 6 7 6 . 0 6 7 7 . 0 6 7 8 . 0 6 7 8 . 5 6 7 1 6 . 7 8 3 2 7 . 7 B 3 E X P E R I M E N T A L ( C ) 6 1 . 9 7 2 . 3 8 6 . 2 9 3 8 I O O . O 1 0 6 . 8 12 1 . 4 1 2 6 . 0 T H E O R E T I C A L ( C ) 5 2 . 1 7 6 6 2 . 8 4 0 8 8 . 4 6 7 9 7 . 4 3 1 1 0 4 . 8 8 5 1 0 7 . 8 2 3 1 2 7 . 4 13 1 2 8 . 8 0 3 ( E X P - T H E O I / E X P O . 1 5 7 1 0 0 . 1 3 0 8 5 - 0 . 0 2 6 3 0 - 0 . 0 3 8 7 1 - 0 . 0 4 8 8 5 - 0 . 0 O 9 S 8 - 0 . 0 4 9 5 3 - 0 . 0 2 2 2 5 H E I G H T ( M ) - 0 . 6 1 0 T I M E E X P E R I M E N T A L T H E O R E T I C A L ( E X P - T H E O l / E X P M I N ( C 1 ( C ) 2 . 6 6 7 6 8 . 5 6 7 . 1 9 0 0 . 0 1 9 1 2 2 8 3 3 7 2 . 1 6 9 . 0 S 7 0 . 0 4 2 2 1 3 . 5 6 7 8 2 . 2 7 8 . 7 3 9 0 . 0 4 2 1 1 4 . 0 6 7 8 8 . 0 8 4 . 4 4 2 0 . 0 4 0 4 3 6 . 0 6 7 9 7 . 4 1 0 1 . 5 8 2 - O . 0 4 2 9 3 7 . 0 6 7 1 0 4 . 3 1 0 9 . 3 4 2 - 0 . 0 4 8 3 4 8 . 5 6 7 1 1 4 . 3 1 1 7 . 4 7 6 - 0 . 0 2 7 7 9 2 0 . 7 8 3 1 2 5 . 8 1 2 8 . 7 6 4 - O 0 2 3 5 6 Table 4-28 A C O M P A R I S O N or T H E O R E T I C A L A N D E X P E R I M E N T A L T E M P E R A T U R E V A L U E S OF GMM WOOD C H I P H E I G H T ( M ) • O . C I O T H I C K N E S S ( M I T G O ( C ) R E Y N O L D S N O . = 0 . 0 0 7 2 G I I G . O O = 4 0 O . 7 V A R 0 . 0 5 8 I I 1 3 5 H ( J / S / ( M ' - ; ) / K ) K ( J / S / M / K ) C P I J / K G ) 3 t . 8 2 6 0 . 0 9 - 1 0 2 5 1 5 . 8 5 T I M E M I N 6 . 9 1 7 8 9 17 9 . 9 1 7 1 1 . 4 1 7 1 4 . 4 17 I S . 3 3 3 2 4 . 3 3 3 3 G . 3 3 3 E X P E R 1 M F . N 1 A L 1 C I G 7 . 8 7 8 . G 8 3 . 3 8 9 . 9 9 8 . 6 1 0 2 . 4 1 0 9 . 8 1 1 3 . 3 T H E O R E T I C A L ( C I 6 8 . 9 4 4 8 0 . 3 3 4 8 5 . 2 9 3 9 1 . 7 8 1 1 0 1 . 4 8 3 1 0 5 . 8 2 4 1 1 4 . 0 3 9 1 1 5 . 9 4 1 l £ X P - T I 1 E 0 i / £ x P - 0 . 0 1 6 8 7 - 0 . 0 2 2 0 6 - 0 . 0 2 3 9 3 - 0 . 0 2 0 9 2 - O . 0 2 9 2 3 - 0 . 0 3 3 4 4 - 0 . 0 3 8 6 0 - 0 . 0 2 3 3 1 H E I G H T ( M ) = 0 . 4 0 6 T I M E E X P E R I M E N T A L T H E O R E T I C A L ( E X P - T H E O J / E X P M I N ( C ) I C I 4 . 1 1 7 6 4 . 7 6 0 . 7 0 6 - 0 . 0 6 1 9 2 5 . 2 8 3 7 G . 7 7 7 . 1 3 4 - 0 . 0 0 5 6 6 6 . 1 1 7 8 1 . 5 8 2 . 4 4 7 - 0 . 0 1 1 6 2 6 . 9 17 8 7 . 4 8 7 0 3 0 O . 0 O 1 2 4 7 . 9 17 9 3 . 1 9 2 . 0 5 0 0 . 0 1 1 2 7 9 . 8 G 7 1 0 1 . 2 9 9 7 6 0 0 0 1 1 2 3 t 2 . 9 1 7 1 0 8 2 1 0 7 3 7 9 0 . 0 0 7 5 9 2 0 . 3 3 3 1 1 4 . 1 1 1 4 . 5 0 9 - 0 . 0 0 3 5 0 Table 4-29 4 COMPARISON OF THEORETICAL AND EXPERIMENTAL TEMPERA'IIRE VALUES OF 6MM WOOO CHIP HE I GUI I M I THICKNESS I M I TGO ( C ) REYNOLDS NO. VAR O I9G318B0 H ( J/S/IM-)/K ) K ( O/S/M/K ) CP ( J/KG ) 0.00726 12 1.00 63-2 . S 33.715 0.0940 2515.85 T I ME MIN 4.517 5.517 G . 850 7 .750 8 . 750 I I .067 15.067 25.783 EXPERIMENTAL ( 0 I 69 . 4 77 . 5 87 . 1 92 . 8 97 . 5 105 . 5 112.3 118.0 THEORETICAL I C I 66,805 7 1.422 83.53 1 88.920 94.206 103 . 8 10 113 401 1 19.440 (EXP-THEOl/EXP 0.03738 O 03972 O 04098 0.0418 I O.03379 0.01602 -0.00980 -0.01220 HEIGHT ( M I = 0.406 TIME EXPERIMENTAL THEORETICAL (EXP-THEOI/EXP MIN ( C I ( C ) 2.300 61.4 65.020 -0.05896 3.183 74.8 73.604 0.01598 3.850 82.7 79.415 0.03972 4.517 90.8 84.691 0.06728 5.517 99. t 91 .567 0.07601 6.850 107.1 99.045 0.07521 9.750 115.7 109.830 0.05073 15 .067 120.G I20."468 0.00110 T a b l e 4-30 A C O M P A R I S O N O F T H E O R E T I C A L A N D E X P E R I M E N T A L T E M P E R A T U R E V A L U E S O F 6 M M WOOD C H I P H E I G H T ( M ) - 0 . 6 1 0 T H I C K N E S S ( M I T G O ( C ) R E Y N O L D S N O . 0 . 0 0 7 2 6 1 2 7 . 0 0 7 7 4 . 19 V A R 0 . 0 7 6 0 4 . 1 3 3 H ( U / S / I M - - 2 l / K I K ( J / S / M / K ) C P I J / K G I 4 2 . 8 9 5 0 . 0 9 4 0 2 5 1 5 . 8 5 T I M E M I N 3 5 3 3 4 . 4 3 3 5 . 4 3 3 6 . 4 3 3 7 . 4 3 3 9 . 6 3 3 13 3 0 0 2 1 . 3 0 0 E X P E R I M E N T A L T H E O R E T I C A L ( E X P - T H E O 1 / E f P I C ) ( C I 7 0 . 2 8 0 . G 8 9 4 9 6 . 6 1 0 2 . I 1 1 0 . 4 1 1 7 . 4 1 2 2 . a 6 9 . 0 6 1 7 7 8 0 6 8 6 . 5 0 7 9 4 . 0 7 1 1 0 0 . 4 9 9 1 l t . 0 9 9 1 2 0 B 0 9 1 2 G . 3 B O 0 . 0 1 6 2 3 0 . 0 3 4 6 6 0 . 0 3 2 3 6 0 . 0 2 6 1 8 O . O I 5 G 8 - 0 . 0 0 G 3 3 - 0 . 0 2 9 0 4 - 0 . 0 2 9 1 5 H E I G H T ( M ) • 0 . 4 0 5 T I M E E X P E R I M E N T A L T H E O R E T I C A L ( E X P - T H E O I / E X P M I N I C I I C I 2 . 2 0 0 6 8 . 7 7 3 . 0 3 8 - O . O G 3 1 5 2 . 8 6 7 8 0 . 8 8 0 . 7 8 1 0 . 0 0 0 2 3 3 . 5 3 3 8 9 . 1 8 7 . 6 3 0 0 . 0 1 6 5 0 4 . 4 3 3 9 9 . 3 9 5 . 5 6 3 0 . 0 3 7 6 4 5 . 4 3 3 1 0 6 . 5 1 0 2 . 7 8 0 0 . 0 3 4 9 3 6 . 9 3 3 1 1 3 . 4 1 1 0 B 7 7 O 0 2 2 2 5 9 . 6 3 3 1 1 9 . 8 1 1 9 . 5 8 3 0 . 0 0 1 8 1 1 7 . 3 0 0 1 2 5 . 4 1 2 6 . 3 5 8 - 0 . 0 0 7 6 4 T a b l e 4-31 A C O M P A R I S O N o r T H E O R E T I C A L A N D F. X P t R I MF tl I li L I E M P E K A 1 I J R E V A L U E S O F M I X E D WOOD CI 1 1 P H E I G H T ( M ) T H I C K N E S S ( M I T G O I C ) R E Y N O L D S N O . = O . O O I B t 1 1 6 . 5 0 ' 3 9 9 . 0 V A R » O . 2 0 0 6 9 9 1 2 H ( J / S / ( M " 2 I / K ) K ( J / S / M / K ) C P ( J / K G ) 1 6 . 0 0 0 0 . 0 9 4 0 2 5 I S . 8 5 T I M E M I N 5 . 2 5 0 6 . 2 5 0 8 . 2 5 0 9 . 2 5 0 1 0 . 1 5 0 12 . 1 5 0 14 1 5 0 2 0 . 1 5 0 E X P E R I M E N T A L ( C ) 6 6 J 7 4 1 8 7 . 4 9 2 . 2 9 5 . 7 1 0 1 . 7 1 0 5 4 1 1 0 . 3 T H E O R E T I C A L I C ) 6 3 . 9 4 4 7 2 . 7 9 1 8 2 . 3 17 8 7 . 2 2 2 9 1 . 174 1 0 0 . 4 4 0 1 0 3 . 8 9 5 1 1 2 . 6 2 4 I E X P - T H E O I / E X P O . 0 3 6 9 9 O 0 1 7 6 7 0 . 0 5 8 1 6 0 . 0 5 3 9 9 0 . O 1 7 3 0 0 . 0 1 2 3 3 0 . 0 1 4 2 8 - 0 . 0 2 1 0 7 HEIGHT ( M 1 = 0.4UG T I M E E X P E R I M E N T A L T H E O R E T I C A L I E X P - T H E O I / E X P M I N ( C I ( C ) 3 . 3 8 3 G 7 . 5 G C . 2 8 9 - 0 . 0 1 1 6 9 4 . 0 5 0 75.1 73.447 0 . 0 2 2 0 2 4 . 7 5 0 83.1 7 8 . 4 3 7 0 . 0 5 6 1 I 6 . 2 0 O 9 5 . 2 9 2 . 8 7 4 0 . 0 2 4 4 3 7 . 2 5 0 1 0 0 . 8 9 2 . 7 0 6 0 . 0 8 0 2 9 8 . 2 5 0 1 0 5 . 2 9 G . 9 9 5 0 . 0 7 7 9 9 1 0 . 1 5 0 1 0 9 . 7 1 0 3 . 4 3 8 0 . 0 5 7 0 8 1 6 . 1 5 0 1 1 4 . 6 1 1 3 . 0 6 0 0 . 0 1 3 4 4 Table 4-32 A C O M P A R I S O N O F T H E O R E T I C A L A N O E X P E R I M E N T A L T E M P E R A T U R E V A I I I F S O F WOOD C H I P . H E I G H T ( M I T H I C K N E S S ( H T G O ( C I R E Y N O L D S N O . 0 . 0 0 4 8 1 1 2 8 . 5 0 6 2 9 . 4 V A R 0 . 0 6 2 5 6 7 6 9 H ( J / S / ( M " 2 ) / K K ( J / S / M / K ) C P ( J / K G I 3 3 . 0 7 9 0 . 0 9 4 0 2 5 1 5 . 8 5 T I M E M I N 3 . 3 1 7 4 . 0 0 0 5 . 0 0 0 5 . 7 0 0 6 . 2 5 0 7 . 2 5 0 l O . 7 0 0 1 7 . 6 3 3 E X P E R I M E N T A L I C I 6 7 4 7 S . 5 8 7 . 4 9 5 . O l O O 3 1 0 6 . 9 1 1 8 . 6 1 2 4 . 6 I H E O R E T I C A L ( C I 6 7 . 5 7 7 7 6 . 0 9 0 8 7 . 3 7 0 9 4 . 1 7 0 1 0 0 . 2 5 8 1 0 6 4 3 6 12 1 . 6 3 0 1 2 7 . 2 7 0 ( E X P - T H E O I / E X P - 0 . 0 0 2 6 3 - 0 . 0 0 7 3 1 0 . 0 0 0 3 4 0 . 0 0 B 7 3 0 . 0 0 0 4 2 O 0 0 4 34 - 0 . 0 2 5 5 5 - 0 . 0 2 1 4 3 H E I G H T ( M ) » 0 . 4 0 6 T I M E E X P E R I M E N T A L T H E O R E T I C A L ( E X P - T H E O ) / E x P M I N < C I ( C I 2 . 0 8 3 6 9 . G 7 1 . 0 4 6 - 0 . 0 2 0 7 7 2 . 4 1 7 7 6 . 9 7 6 . 0 8 5 0 . 0 1 0 6 0 2 . 7 5 0 8 3 . 8 8 1 . 0 3 1 0 . 0 3 3 0 4 3 . 3 1 7 9 3 . 5 8 8 . 5 4 3 0 . 0 5 3 0 1 4 . 3 3 3 1 0 4 . 0 9 9 . 7 0 0 0 . 0 4 8 6 7 5 . 7 5 0 1 1 4 . 2 1 1 0 . 6 2 3 0 . 0 3 1 3 2 7 . 2 5 0 1 1 9 . 8 1 1 8 . 4 1 5 0 . 0 1 1 5 6 1 3 . 6 3 3 1 2 6 . 1 1 2 7 . 7 0 2 - O . O I 2 7 I - 112 -c o e f f i c i e n t . F i g u r e 4-30 compares a t y p i c a l e x p e r i m e n t a l temperature p r o f i l e w i t h the t h e o r e t i c a l r e s u l t s o b t a i n e d u s i n g s e v e r a l a r b i t r a r i l y s e l e c t e d heat t r a n s f e r c o e f f i c i e n t v a l u e s . The f i t t e d hf was 17.5 W/m2 K. Meanwhile, f o r hf v a l u e s of 10, 15, 17.5, 20 25 and 35 the c o r r e s p o n d i n g v a r i a n c e s a r e 0.866, 0.458, 0.376, 0.446, 0.564 and 0.939. These numbers i l l u s t r a t e the s e n s i t i v i t y of the c a l c u l a t i o n g i v i n g the f i t t e d v a l u e o f h f . 4.3.1 Effect of Air Velocity A comparison of the a i r temperature vs time p r o f i l e s a t two a i r v e l o c i t i e s f o r 2 mm t h i c k wood c h i p i s shown i n F i g u r e s 4-28 and 4-29. As can be seen, the run a t the h i g h e r a i r v e l o c i t y (.882 m/s) had the h i g h e r heat t r a n s f e r r a t e and the p r o f i l e s approached the s t e a d y s t a t e f a s t e r than i n the case o f the lower a i r v e l o c i t y (0.676 m/s). S i m i l a r e f f e c t s were noted f o r o t h e r p a r t i c l e s i z e s i n the s e r i e s o f runs a t d i f f e r e n t a i r v e l o c i t i e s . The c o n d i t i o n s f o r these r u n s , and the heat t r a n s f e r c o e f f i c i e n t s are g i v e n i n T a b l e 4-33. T h i s t a b l e shows the r e s u l t s of one run f o r 4 mm wood c h i p s a t Re = 374 which was r e p e a t e d to g i v e a measure o f the r e p r o d u c i b i l i t y o f the 2 experiment. The mean heat t r a n s f e r c o e f f i c i e n t was 12.08 ± 0.28 W/m K 2 or 12.1 W/m K ± 2.3%. T h i s i s p r o b a b l y an u n d e r e s t i m a t i o n of the r e a l r e p r o d u c i b i l i t y . As can be seen i n T a b l e 4-33 and F i g u r e 4-37, f o r the same s i z e o f p a r t i c l e s the l a r g e r the Reynolds number the h i g h e r was the heat t r a n s f e r c o e f f i c i e n t . T h i s was expected because of the h i g h e r h e a t t r a n s f e r r a t e under the h i g h e r f l o w r a t e c o n d i t i o n . T h i s e f f e c t was Table 4-33 Summary of heat t r a n s f e r run c o n d i t i o n s and r e s u l t s f o r wood c h i p s Run Chip A i r I n l e t A i r h Rep* Nu* B f I No. T h i c k n e s s Flow* Temp V e l o c i t y * (mm) (m 3/s) °C m/s W/m2 K 2 2.44 0.0193 117 .594 12.3 255 4.037 0.16 2 0.0248 123 .766 14.8 329 4.850 0.192 3 0.0304 125 .938 15.9 403 5.21 0.206 3 0.0360 131 1.11 17.0 476 5.58 0.221 2 4.03 0.0193 111 .594 11.8 374 . 5.67 0.253 1 0.0193 115 .594 12.4 374 5.96 0.266 2 0.0248 123 .766 17.5 483 8.47 0.375 2 0.0304 126 .938 20.0 590 9.61 0.429 2 0.0360 129 1.11 23.4 698 11.24 0.503 2 7.26 0.0193 116 .594 31.8 491 20.04 1.23 2 0.0248 121 .766 33.7 633 21.24 1.301 2 0.0304 127 .938 42.0 774 26.47 1.622 2 4.81 0.0193 116.5 .594 16.0 399 8.21 0.409 2 (mixed) 0.0304 128.5 .938 33.1 630 17.0 0.845 C a l c u l a t e d at 100°C. - 114 -c l e a r l y noted f o r a l l p a r t i c l e s i z e s t e s t e d . A p l o t o f the heat t r a n s f e r c o e f f i c i e n t v e r s u s the Reynolds number f o r the v a r i o u s s i z e s i s shown i n F i g u r e 4-38. The s l o p e s are not the same as w i l l be d i s c u s s e d below. 4.3.2 Effect of Wood Chip Thickness A i r temperature p r o f i l e s a t d i f f e r e n t wood c h i p t h i c k n e s s but e s s e n t i a l l y s i m i l a r v e l o c i t i e s can be compared i n F i g u r e 4-27 and 4-28. As can be seen, t h e r e i s no s i g n i f i c a n t e f f e c t of wood c h i p t h i c k n e s s i n the temperature p r o f i l e s a l t h o u g h the 6 mm wood c h i p i s almost t h r e e times t h i c k e r than the 2 mm wood c h i p . The temperature p r o f i l e f o r the 2 mm wood c h i p seemed to r i s e f a s t e r d u r i n g the t r a n s i e n t s t a g e because i t has a h i g h e r i n l e t gas temperature. V a l u e s of the heat t r a n s f e r c o e f f i c i e n t s and run c o n d i t i o n s a r e g i v e n i n T a b l e 4-33. At a g i v e n v e l o c i t y the heat t r a n s f e r c o e f f i c i e n t i n c r e a s e s w i t h the wood c h i p t h i c k n e s s . For the 2 mm and 4 mm wood c h i p at a g i v e n p a r t i c l e Reynolds number the heat t r a n s f e r c o e f f i c i e n t d e c r e a s e s s l i g h t l y as the t h i c k n e s s i n c r e a s e s . G e n e r a l l y s p e a k i n g , t h e r e would be a h i g h e r heat r e s i s t a n c e f o r the t h i c k e r p a r t i c l e of the same k i n d . In o t h e r words, the heat t r a n s f e r would be slower under t h i s s i t u a t i o n . A d d i t i o n a l l y , the heat t r a n s f e r c o e f f i c i e n t c o n t r o l l e d the heat t r a n s f e r r a t e . Thus the heat t r a n s f e r c o e f f i c i e n t would be s m a l l e r . One f a c t o r which might be of importance f o r the t h i c k e s t wood c h i p s i s the assumption o f o n e - d i m e n s i o n a l heat f l o w i n the model. Such an - 115 -assumption can be a p p l i e d f o r r e a s o n a b l y t h i n s l a b p a r t i c l e s . However, f o r the t h i c k e r p a r t i c l e s the heat b a l a n c e model e q u a t i o n s s h o u l d be r e w r i t t e n a s : k 4+k 4+k 4 X 3 r 2 y 3 r 2 Z 3 r 2 3t cp psTe - (k. 3t x 3r + . k 3t y 3r x=Ll y + k 3t z 3r y=I'2 z z=L ) = h . ( t . - T) f s u r f a c e „ T T 3T / i 3t P-V-C^ - 5 a (k -r— f f P £ 3z x 3r f x + k 3t T y 3r x=L x ' y + k 3t y=L, z 3r z=L„ As can be seen, the r e s u l t i n g t h r e e d i m e n s i o n a l heat t r a n s f e r problem i s f a r more complex than the one d i m e n s i o n a l problem. Undoubtedly, heat t r a n s f e r to the edges of t h i c k e r s l a b s can not be n e g l i g i b l e . However, i n t h i s t h e s i s o n l y the one d i m e n s i o n a l heat f l o w was used i n the s i m u l a t i o n o f the f i x e d bed heat t r a n s f e r . T h i s might a f f e c t the heat t r a n s f e r c o e f f i c i e n t s of the t h i c k e s t p a r t i c l e s . Those t h i c k e r wood c h i p s had r e l a t i v e l y h i g h e r heat t r a n s f e r c o e f f i c i e n t v a l u e s t h a n d i d the t h i n n e r wood c h i p s (2 mm and 4 mm) i n the same range of Reynolds number. T h i s can be observed i n T a b l e 4-33 and F i g u r e 4-37 where o p p o s i t e e f f e c t s appeared r e l a t i v e to what was e x p e c t e d , i . e . , the - 116 -thickness wood chips had a higher heat transfer coef f ic ient , 2 31.08 W/m K, with lower Reynolds number (491) when compared to the 4 mm wood chip which had a lower heat transfer value of 23.4 (W/m2 K.) but with a higher Reynolds number of 698. The heat transfer from the gas to the edges of solids might not be negligible for the thicker wood chips such as the 6 mm wood chips and mixed wood chips. Thus, the heat transfer rate was larger than was expected since only one dimensional heat transfer was assumed in the simulation. In order to match the higher heat transfer rate, the resulting heat transfer coefficient values were larger than they should be. Furthermore, a l l solids were assumed to be uniform slabs. However, the irregular shape of mixed wood chips and 6 mm wood chips would provide high heat transfer rate so that larger fi lm heat transfer coeff icient values were obtained. The effect of part icle size in terms of total volume on the overall heat transfer coeff icient is different from that in terms of thickness. For instance, a larger wood chip particle w i l l have higher overall heat transfer coeff icient than a smaller size part icle with larger thickness under same air velocity or flowrate. This is because the thickness is the major parameter to affect the internal resistance and heat transfer rate. 4.3.3 Effect of Steam-Air Mixture The effect of moist hot gas on the heat transfer to the fixed bed of wood chips particles was studied in two experiments (Appendix A, - 117 -T a b l e 4-7 s to 7-j;) • The p a r t i c l e s s i z e s were 6 ram and 4 mm wood c h i p s . The temperature p r o f i l e p l o t s are shown i n F i g u r e 4-31 to 4-32 which compare the heat t r a n s f e r r a t e w i t h a 30 m o i s t u r e % hot a i r mi x t u r e and dry hot a i r . The d i m e n s i o n l e s s temperature p r o f i l e s were p l o t t e d a g a i n s t time i n t h i s case because the i n l e t d r y gas temperature of 120°C was d i f f e r e n t from the moist gas i n l e t temperature of 115°C. Meanwhile the dry a i r f l o w r a t e was 8.112 x 10 - 1 + kg mole/s or 0.0234 kg/s and the moist a i r f l o w r a t e was 0.027742 kg/s or 1.0546 x 10" 3 kg mole/s. S i n c e the m o i s t u r e was b e i n g condensed d u r i n g the t r a n s i e n t s t a t e , the Reynolds number w i t h s u p e r f i c i a l v e l o c i t y was used f o r the 30 m o i s t u r e % a i r . The m o i s t a i r Reynolds number and v e l o c i t y were 637.4 and 0.98 m/s, and 836.1 and 0.98 m/s f o r the 4 mm and 6 mm wood c h i p s r e s p e c t i v e l y . As r e f e r r e d to the F i g u r e 4-31 and 4-32 the heat t r a n s f e r r a t e was much g r e a t e r f o r the case w i t h moist a i r . F o r the 4 mm wood c h i p s , the time to st e a d y s t a t e f o r the moist a i r i s 600 seconds which i s 400 seconds f a s t e r than t h a t of the dry a i r . S i m i l a r l y , f o r the 6 mm wood c h i p the time to steady s t a t e f o r the moist a i r of 1000 seconds i s a l s o 400 seconds f a s t e r than t h a t of the d r y a i r . There was not o n l y heat t r a n s f e r but a l s o mass t r a n s f e r when the m o i s t hot a i r was i n t r o d u c e d i n t o the f i x e d bed. The m o i s t u r e d r a m a t i c a l l y reduced the i n t e r f a c e heat r e s i s t a n c e of the s o l i d s due to c o n d e n s a t i o n o f water. The i n t e r n a l heat r e s i s t a n c e a l s o would be lowered as the m o i s t u r e p e n e t r a t e d from the s u r f a c e to i n s i d e of the CO o .0 DRY AIR MOIST AIR AT THE HEIGHT=0.61M CO I co WOOD CHE? TmCTCNESS=4.03rnrn DRY AIR REYNOLDS N0.=482.9 MOISTURE PERCENTAGE =30. 0.0 40.0 80.0 120.0, ( X 1 0 1 160.0 TIME IN SECONDS 200.0 240.0 Figure 4-31 Comparison of temperature profiles for dry and moist air for 4.03 mm' thick chips w 05 D <! 05 W CU S w E-CO CO W •-3 :z; o CO Is w \% c n o ' r -o " i n o " o " ^<T) (-o DRY AIR + MOIST AIR AT THE HEIGHT=0.61M WOOD CHE? TfflCKNESS=7.26mm DRY AIR REYNOLDS N0.=632.5 MOISTURE PERCENTAGE =30. I I 1 1 1 I 0.0 40.0 80.0 120.0. ( X 1 0 1 160.0 200.0 240.0 TIME IN SECONDS Figure 4-32 Comparison of temperature profiles for dry and moist air for 7.26 mm thick chips - 120 -s o l i d s . These phenomena not only reduced the thermal resistance but also provide a separate mechanism to introduce heat from the gases to the s o l i d s . There were higher Reynolds numbers for the 30% moisture hot a i r than that of the hot dry a i r but this d i f f e r e n c e was not thought to be the major e f f e c t . The condensation of steam on the wood chips p a r t i c l e s during the unsteady state would undoubtedly cause a large e f f e c t . Figure 4-33 and 4-34 were the plots of temperature p r o f i l e s with experimental moist a i r data and generated by using eq. 2-33 with d i f f e r e n t a r b i t r a r i l y selected heat transfer c o e f f i c i e n t s . Although t h i s model i s not designed for the moist a i r system, and c l e a r l y does not f i t the data well, the experimental temperature p r o f i l e was close to the lower heat transfer value during the e a r l i e r time and close to the higher value near the steady state. This may be because the condensation occurred i n the early stages and there was only a l i t t l e sensible heat from the gas transferred to the s o l i d and i t caused low heat transfer rate conditions. As the process continues, the heat transfer rate would be increased due to the moisture on the s o l i d and i t provided higher heat transfer c o e f f i c i e n t s . This mechanism caused d i f f e r e n t heat transfer c o e f f i c i e n t values during the unsteady state. A model incorporating l a t e n t heat e f f e c t s i s needed to analyze this s i t u a t i o n . 4.3.4 Internal Solid Temperature Two experiments were done to study the i n t e r n a l s o l i d and corresponding gas temperature r e l a t i o n s h i p . The s p e c i a l thermocouple o LO f \ l '—x o o in. co co m_ to in + (EXP.) AT THE HT.=0.610M _ THEORETICAL CURVE MOIST AIR VEL0CITY=.980M/S MOIST REYNOLDS N0.=637.4 WOOD CHIP TfflCKKESS=4.03mm INLET GAS TEMPERATURE=116C Hf=COEFFICIENT W/(K»M»*2) I 0.0 25.0 50.0 75.0 ( X 1 0 1 ) 100.0 125.0 150.0 TIME IN SECONDS F i g u r e 4-33 P l o t of t h e o r e t i c a l and moist a i r e x p e r i m e n t a l temperature p r o f i l e s f o r 4.03 mm t h i c k c h i p s o in CM C J o w 05 E-1 <: as w OH S w E-co o o o in. cn in _ CO a. in 0.0 30.0 60.0 + + (EXP.) AT THE HT.=0.610M _ THEORETICAL CURVE MOIST AIR VEL0CITY=.980M/S MOIST REYNOLDS N0.=836.1 WOOD CHIP THICKNESS=7.26mm INLET GAS TEMPERATURE=116.5C Hf=COEFFICEENT W/(K*M'*2) 90.0 ( X 1 0 1 ) 120.0 150.0 180.0 TIME IN SECONDS Figure 4-34 Plot of t h e o r e t i c a l and moist a i r experimental temperature p r o f i l e s for 7.26 mm thick chips - 123 -m o n i t o r i n g s e t - u p was shown i n F i g u r e 3-13. The r e s u l t i n g temperature p r o f i l e s are shown i n F i g u r e 4-35 and 4-36, and Appendix 4-37q to 4-37r. The gaps between the two temperature v a l u e s were found to g r a d u a l l y i n c r e a s e to maximum v a l u e s , then g r a d u a l l y approach to the s t e a d y s t a t e v a l u e s . The maximum gap v a l u e s were found a p p r o x i m a t e l y to be 7°C and 3°C f o r the 2 mm wood c h i p s experiment at Reynolds number 329 and the 4 mm wood c h i p s experiment a t Reynolds number 374, r e s p e c t i v e l y . Meanwhile, the 2 mm wood c h i p s experiment had an i n l e t gas temperature v a l u e e q u a l to 120°C which was h i g h e r than the i n l e t gas temperature e q u a l to 115°C f o r the 4 mm wood c h i p s e x p e r i m e n t . I f t h i c k n e s s was c o n s i d e r e d to be the o n l y v a r i a b l e , the d e v i a t i o n between the gas and s o l i d temperature would be expected to be b i g g e r f o r the 4 mm wood c h i p s experiment than f o r the 2 mm wood c h i p s c a s e . However, t h e r e are s e v e r a l a d d i t i o n a l f a c t o r s which a f f e c t the f i x e d bed g a s - s o l i d temperature d i f f e r e n c e . They are the f l o w r a t e of f l u i d , f l u i d i n l e t t emperature, heat t r a n s f e r c o e f f i c i e n t and the bed d e p t h . Those major f a c t o r s a f f e c t and c o n t r o l the heat t r a n s f e r r a t e and the heat t r a n s f e r d r i v i n g f o r c e . Thus, the d i f f e r e n c e between the g a s - s o l i d temperature i s a l s o a f f e c t e d by those parameters. 4.4 Correlation of Results The v a l u e s o b t a i n e d f o r the wood c h i p p a r t i c l e Reynolds number, the N u s s e l t number as w e l l as B i o t number are summarized i n T a b l e 4-33. The N u s s e l t number v e r s u s the Reynolds number are p l o t t e d and compared to l i t e r a t u r e v a l u e s as shown i n F i g u r e 4-37. The N u s s e l t number of the wood c h i p s i s found to be below most o t h e r d a t a i n the same range of F i g u r e 4-35 I n t e r n a l wood c h i p and gas temperature p r o f i l e s f o r 2.44 mm t h i c k c h i ps o in rxi F i g u r e 4-36 I n t e r n a l wood c h i p and gas temperature p r o f i l e s f o r 4.03 mm t h i c k c h i p s N3 F i g u r e 4-37 Comparison of wood c h i p and sphere d a t a w i t h l i t e r a t u r e v a l u e s A - 2.44 mm t h i c k c h i p , A- 4.03 mm t h i c k c h i p O- 7.26 mm t h i c k c h i p , 4 - 4.81 mm t h i c k c h i p 4.78 mm d i a . sphere - 127 -Reynolds number. T h i s may be due to the d i f f e r e n c e s i n geometry and hence flow between the p a r t i c l e s which r e s u l t i n lower f i l m heat t r a n s f e r c o e f f i c i e n t s . The heat t r a n s f e r c o e f f i c i e n t v a l u e s of 4 mm and 2 mm wood c h i p s ( T a b l e 4-34) were c o r r e l a t e d as f u n c t i o n s of Reynolds number and wood c h i p t h i c k n e s s . The e m p i r i c a l e q u a t i o n i s : h(W/m2 s) = 0.215 ( R e ) 0 , 7 5 3 ( t h i c k n e s s ( m m ) ) " 0 , 2 1 1 (4-1) T h i s e q u a t i o n shows t h a t the heat t r a n s f e r c o e f f i c i e n t i s p r o p o r t i o n a l to the Reynolds number to the 0.75 power but d e c r e a s e s w i t h i n c r e a s i n g wood c h i p t h i c k n e s s . However the 6 mm c h i p s c l e a r l y do not f i t t h i s e q u a t i o n , as F i g . 4-38 shows an i n c r e a s e i n c o e f f i c i e n t w i t h t h i c k n e s s f o r the t h i c k e r c h i p s . T h i s k i n d of r e l a t i o n s h i p i n d i c a t e s two d i f f e r e n t s i z e e f f e c t s on the heat t r a n s f e r c o e f f i c i e n t . The l a r g e r wood c h i p w i t h l a r g e r e q u i v a l e n t d iameter (Dp) p r o v i d i n g l a r g e r s u r f a c e a r e a w i l l r e s u l t i n a h i g h e r c o e f f i c i e n t . Meanwhile, the t h i c k e r wood c h i p p a r t i c l e c r e a t i n g h i g h e r heat t r a n s f e r r e s i s t a n c e w i l l have lower heat t r a n s f e r c o e f f i c i e n t . The comparison of heat t r a n s f e r c o e f f i c i e n t v a l u e s ( e x p e r i m e n t a l v a l u e s ) g e n e r a t e d by u s i n g the m a t h e m a t i c a l model and the v a l u e s c a l c u l a t e d by u s i n g the e m p i r i c a l e q u a t i o n 4-1 are shown i n T a b l e 4-34. The d e v i a t i o n s are r e a s o n a b l e . To reduce the s c a t t e r of the d a t a , the heat t r a n s f e r c o e f f i c i e n t s might be c o r r e l a t e d by a c c o u n t i n g f o r the t r u e a r e a r a t h e r than the a r e a o f the s i d e s of the c h i p s o n l y . Thus, h? = h f anom/ aT* H e r e % . - « - • > . ! & - ^ - *T - <> - « > < M * ™ : 4 Y L > — Table 4-34 THE EMPIRICAL EQUATION FOR Hf IS : C O E F F I C I E N T ( J / S ( M * « 2 ) / K ) = A * ( R E Y N O L D S N O . * * B ) * ( T H I C K N E SS(mm)* *C ) A * 0 . 2 1 5 3 B=0.7527 C=- .2109 REYNOLDS NO. WOOD CHIP THICKNESS EXPERIMENTAL THEORETICAL ( T H E O . - E X P . ) / E X P . COEFFICIENT COEFFICIENT (mm) ( J / S / ( M * * 2 ) / K ) ( J / S / ( M * * 2 ) / K ) 255 .00 2 .440 12.29 11.55 - 0 . 0 6 0 329 .40 2 .440 14.79 14.01 - 0 . 0 5 3 4 0 3 . 1 0 2 .440 15.81 16.31 0 .032 I 4 7 6 . 7 0 2 .440 17.00 18.50 0 .088 >-N J CO 374 .10 4 . 0 3 0 12.36 13.87 0 .122 I 482 .90 4 .030 17.47 16.81 - 0 . 0 3 8 5 9 0 . 2 0 4 .030 2 0 . 0 0 19.55 - 0 . 0 2 3 698 .29 4 .030 2 3 . 4 0 22 .19 - 0 . 0 5 2 VARIANCE = 0 .0644 DEVI AT 10N(%)= 5 .83 O 2mm WOOD CHIP • 4mm WOOD CHIP + H- 6mm WOOD CHIP • MIXED WOOD CHE? + • • o o • ~ I I I I I ~ 0.0 15.0 30.0 45.0 60.0 75.0 90 ( X 1 0 1 ) REYNOLDS NO. F i g u r e '»-38 P l o t hf versus p a r t i c l e Reynolds number - 130 -the a c t u a l s o l i d / g a s i n t e r f a c e a r e a . The r a t i o of aj/anom of d i f f e r e n t t h i c k n e s s wood c h i p i s shown i n T a b l e 4-35. The h^ v a l u e s were p l o t t e d vs Reynolds number i n F i g u r e 4-39. As can be seen, the T a b l e 4-35 Average a a_ a^ ,a , / \ nom 2, 3 N T t r u e ( n o m T h i c k n e s s (mm) (cm /cm ) (-) 2.44 3.811 5.05 1.33 4.03 2.482 3.407 1.37 7.26 1.435 2.278 1.59 4.81 2.129 3.044 1.43 m o d i f i e d heat t r a n s f e r c o e f f i c i e n t (h-jO d e c r e a s e d w i t h i n c r e a s i n g t h i c k n e s s of the wood c h i p and t h e r e f o r e the s c a t t e r of F i g u r e 4-38 was r e d u c e d . Meanwhile, a s i n g l e e m p i r i c a l e q u a t i o n was developed f o r 2 mm and 4 mm wood c h i p s : h T(W/m 2s) = 0.1670 ( R e ) 0 * 7 5 9 0 ( T h i c k n e s s ( m m ) ) " ° * 2 8 2 2 (4-2) A comparison of e x p e r i m e n t a l h j was shown i n T a b l e 4-36. The d e v i a t i o n i s r e a s o n a b l e , a t about 5.62%. 4.5 Blot Number for Wood Chips The wood c h i p p a r t i c l e B i o t number v a l u e s are g i v e n i n T a b l e 4-33. G e n e r a l l y , the B i o t number i n c r e a s e d w i t h the wood c h i p p a r t i c l e s i z e O 2mm WOOD CHIP • 4mm WOOD CHIP -1- 6mm WOOD CHIP # MIXED WOOD CHIP O O + + + • • 8 • I I I I I 0.0 15.0 30.0 45.0 60.0 75.0 90 ( X 1 0 1 ) REYNOLDS NO. Figure 4-39 Plot of versus particle Reynolds number T a b l e 4-36 THE EMPIRICAL EQUATION FOR Hf(true) IS : COEFFICIENT!; J/S(M«»2)/K)=A*( REYNOLDS NO. * *B )*( THICKNESS (mm ) • *C ) A=0.1670 B=0.7590 C=-2822 REYNOLDS NO. WOOD CHIP THICKNESS EXPERIMENTAL THEORETICAL (THEO.-EXP.)/EXP. COEFFICIENT COEFFICIENT ( m m ) (J/S/(M*»2)/K) (J/S/(M* *2)/K) 255 .00 2 . 440 9 .25 8 . 7 1 -o . 058 329 40 2 . 440 1 1 . 13 10 .58 -0 .049 403 10 2 . 440 1 1 .95 12 . 33 O. 031 476 . . 70 2 . 440 12 .92 14 OO 0. .084 374 10 4 .030 9 .05 10. . 1 1 0. 117 482 . 90 4 .030 12 .77 12 . 27 - o . 039 590. 20 4 .030 14 .60 14 , .29 - o . 021 698 . 29 4 .030 17 08 16. 24 -0 . 049 u> VARIANCE' 0.0441 DEVIATI0N(%)» 5.62 - 133 -such t h a t the 2 mm wood c h i p had the s m a l l e s t B i o t numbers and the 6 mm wood c h i p had the l a r g e s t over the range of f l o w r a t e s t e s t e d . The B i o t modulus i s the r a t i o of i n t e r n a l c o n d u c t i o n r e s i s t a n c e to s u r f a c e c o n v e c t i o n r e s i s t a n c e . A v e r y low v a l u e of the B i o t modulus means t h a t i n t e r n a l - c o n d u c t i o n r e s i s t a n c e i s n e g l i g i b l e i n comparison w i t h s u r f a c e - c o n v e c t i o n r e s i s t a n c e . I f the B i o t number i s l e s s than 0.1, the i n t e r n a l r e s i s t a n c e may be n e g l i g i b l e or w i t h i n 5% e r r o r under c o n s t a n t ambient temperature heat t r a n s f e r . The B i o t number v a l u e s of 2 mm wood c h i p s were r e l a t i v e l y c l o s e to t h i s s i t u a t i o n w i t h v a l u e s of 0.16 to 0.22, whereas the t h i c k e s t c h i p s gave v a l u e s of 1.2 to 1.6 where i n t e r n a l r e s i s t a n c e dominates. 4.6 Comparison with Design Values Used by Blackwell (19,21) In the o r i g i n a l p r o p o s a l f o r the wood c h i p h e a t i n g p r o c e s s B l a c k w e l l et a l . (19,21) used a v a l u e of 57 W/m2 K f o r the f i l m c o e f f i c i e n t , and 24.7 W/m K f o r the o v e r a l l c o e f f i c i e n t which i n c l u d e d the i n t e r n a l r e s i s t a n c e of the 6.35 mm t h i c k wood c h i p . The gas v e l o c i t y was 0.6 m/s at N.T.P., and the p r e s s u r e drop 4.2 cm water/m of bed d e p t h . Average of e x p e r i m e n t a l v a l u e s f o r the c h i p blend (4.8 mm t h i c k ) , and the 7.26 mm t h i c k c h i p s a t 0.82 m/s (100°C) was 32 W/m2 K f o r the f i l m c o e f f i c i e n t , and 15 W/m K f o r the o v e r a l l c o e f f i c i e n t . P r e s s u r e drop at 0.6 m/s f o r the b l e n d was 4.0 cm water/m and 3.1 cm water/m f o r the 7.26 mm t h i c k c h i p s . Thus i t seems th a t the d e s i g n e s t i m a t e s f o r the heat t r a n s f e r c o e f f i c i e n t were o p t i m i s t i c and the p r e s s u r e drop v a l u e s i n e x c e l l e n t agreement w i t h the r e s u l t s of the c u r r e n t e x p e r i m e n t s . - 134 -5. CONCLUSIONS Hot a i r a t about 130°C was passed through 0.2 m d i a x 1 m deep beds o f c o m m e r c i a l l y p r e p a r e d wood c h i p s which had been f i r s t s c r e e n e d f o r t h i c k n e s s . The unsteady s t a t e gas temperature p r o f i l e s were r e c o r d e d . Heat t r a n s f e r c o e f f i c i e n t s between the a i r and the bed of wood c h i p s were c a l c u l a t e d by u s i n g a m a t h e m a t i c a l model of a packed bed of s l a b shaped p a r t i c l e s which was d e v e l o p e d f o r t h i s purpose. The e x p e r i m e n t a l , and m a t h e m a t i c a l t e c h n i q u e s were f i r s t proven on t e s t s w i t h s p h e r i c a l p a r t i c l e s f o r which d a t a were a v a i l a b l e i n the l i t e r a t u r e . The v a l u e s of the heat t r a n s f e r c o e f f i c i e n t s f o r the wood c h i p s were found to i n c r e a s e w i t h Reynolds number d e f i n e d i n terms of an e q u i v a l e n t sphere d i a m e t e r and wood c h i p t h i c k n e s s . The l a t t e r e f f e c t o c c u r r e d because of the o n e - d i m e n s i o n a l m o d e l l i n g assumption i n which heat t r a n s f e r to the wood c h i p edges and ends was n e g l e c t e d . Two e m p i r i c a l e q u a t i o n s f o r the heat t r a n s f e r c o e f f i c i e n t as a f u n c t i o n of Reynolds number and wood c h i p t h i c k n e s s were d e v e l o p e d . A s i n g l e e q u a t i o n t h a t f i t t e d a l l c h i p t h i c k n e s s c o u l d not be found. Measured s o l i d i n t e r n a l temperatures were found to l a g the s u r f a c e temperature by about 5°C d u r i n g the h e a t i n g p r o c e s s . B i o t number ranged from about 0.2 f o r the t h i n n e s t c h i p s to 1.6 f o r the t h i c k e s t c h i p s at the h i g h e s t Reynolds number. The e f f e c t on heat t r a n s f e r r a t e of adding 30% by volume steam to the a i r was s t u d i e d b r i e f l y . Heat t r a n s f e r r a t e s were found to be s i g n i f i c a n t l y f a s t e r f o r m o i s t a i r than f o r d r y a i r but the a c t u a l c o e f f i c i e n t s c o u l d not be e x t r a c t e d from the model, - 135 -presumably because of c o n d e n s a t i o n e f f e c t s . Measured v a l u e s of s p h e r i c i t y and v o i d a g e r e s p e c t i v e l y l e d to d e v i a t i o n s of about ± 7.3% between e x p e r i m e n t a l v a l u e s of the p r e s s u r e drop and p r e d i c t i o n s of the Ergun e q u a t i o n . The t h i c k e s t wood c h i p s had the l e a s t d e v i a t i o n between the t h e o r e t i c a l and e x p e i m e n t a l p r e s s u r e drop v a l u e s . F i t t e d s p h e r i c i t y and v o i d a g e v a l u e s were not markedly d i f f e r e n t from e x p e r i m e n t a l v a l u e s , and p e r m i t t e d p r e d i c t i o n s of p r e s s u r e drop by the Ergun e q u a t i o n w i t h i n - 136 -6. RECOMMENDATIONS 1. Three d i m e n s i o n a l heat t r a n s f e r m o d e l l i n g of the c o n d u c t i o n i n t o the c h i p s i n s t e a d of o n e - d i m e n s i o n a l i s recommended to r e c a l c u l a t e the heat t r a n s f e r c o e f f i c i e n t v a l u e s . 2. To p r o v i d e f u r t h e r d e s i g n i n f o r m a t i o n to the p r o c e s s proposed by B l a c k w e l l e t a l . ( 1 9 ) , f u r t h e r s t u d i e s are n e c e s s a r y : a. Mass t r a n s f e r and heat t r a n s f e r of steam and a i r w i t h m o i s t wood c h i p s s h o u l d be s t u d i e d e x p e r i m e n t a l l y . Mass t r a n s f e r c o e f f i c i e n t s and wet s u r f a c e heat t r a n s f e r c o e f f i c i e n t s s h o u l d be d e t e r m i n e d . b. Small s c a l e p i l o t p l a n t c o n t i n u o u s p r o c e s s o p e r a t i o n i s a l s o r e q u i r e d . c. The r e l a t i o n between the packed bed and moving bed p r e s s u r e drop c a l c u l a t i o n s h o u l d a l s o be s t u d i e d . d. The m a t h e m a t i c a l model o f the f i x e d bed can be extended to i n c l u d e l a t e n t heat e f f e c t s . Thus p r o v i s i o n must be made i n the heat b a l a n c e e q u a t i o n s f o r c o n d e n s a t i o n onto and e v a p o r a t i o n from the wet s u r f a c e of the wood c h i p s . - 137 -REFERENCE 1. A n z e l i u s , A., and Z e i t , F. Ang. Math. U. Mech. j3, 291 (1926 ) . 2. Schumann, T.E.S., J . F r a n k l i n I n s t . _208, 305-16 (192 9 ) . 3. F u r n a s , C C . , Ind. Eng. Chem. 22, 721 (1930 ) . 4. C o l b u r n , A.P., T r a n s . Am. I n s t . Chem. E n g r s . 29, 174-209 (1933). 5. Carslaw, H.S., and J a e g e r , J . C , "C o n d u c t i o n o f Heat i n S o l i d s , " O x f o r d U n i v . P r e s s , London, ( 1 9 4 8 ) . 6. Rosen, J.B., Ind. Eng. Chem. 46_, 1590 (1954 ) . 7. Rosen, J.B., J . Chem. Phys. 20, 387 (1 9 5 2 ) . 8. Thomas H.C, J . Chem. Phys. 1_9, 1213 (1 9 5 1 ) . 9. Neal R. Amundson, Ind. Eng. Chem. 48, 26-43 (195 6 ) . 10. P a u l R. K a s t e n and Neal R. Amundson, Ind. Eng. Chem., 42, 1341 (19 5 0 ) . 11. Bernard W. Gamson, American I n s t . Chem. En g r s . , _39_, 1 (194 3 ) . 12. G.O.G. L o f and R.W. Hawley, Ind. Eng. Chem., 40, 1061 (1948). 13. D a i z o K u n i i and J.M. Smith, A.I.Ch.E.J., _7> 2 9 (1961). 14. Ergun S., Chem. Eng. P r o g r . , _48, 89-94 (1952). 15. John L. C l e a s b y , A.S., C i v i l Eng. Env. Eng. D i v . S p e c i a l l y C onference J u l y 9-11, 1979, San F r a n c i s c o , CA. 16. P e r r y , R.H., C h i l t o n , C H . and K i r k p a t r i c k , S.D., Chem. Eng. Handbook 4 t h e d i t i o n , M c G r a w - H i l l , New York, 1963. 17. R. Byron B i r d , " T r a n s p o r t Phenomena," John W i l e y & Sons I n c . , New York ( 1 9 6 0 ) . 18. K u n i i , D., L e v e n s p i e l 0., " F l u i d i z a t i o n E n g i n e e r i n g , " John W i l e y and Sons, I n c . , New York (1969). 19. B l a c k w e l l , B.R. et a l . , "Use o f Waste Gases to Preheat Wood Chips i n the Pul p and Paper I n d u s t r y - A P r e l i m i n a r y Survey." P r e p a r e d by Beak C o n s u l t a n t s L t d . , Vancouver. DSS C o n t r a c t # SQ79-00135, Dept. Energy Mines and R e s o u r c e s . - 138 -20. H a t t o n , J.V. (ed.) "C h i p Q u a l i t y Monograph," Chapt. 10, No. 5, In Pulp and Paper Technology S e r i e s , TAPPI ( 1 9 7 9 ) . 21. B l a c k w e l l , B.R., L.E. H a r r i s and J.D. Garsho r n " P r e h e a t i n g Wood C h i p s w i t h K r a f t Recovery B o i l e r F l u e Gas," P u l p & Paper Canada 82: 12 (1981), p. T432-7. 22. F u r n a s , C C . , U.S. Bur. Mines B u l l . 361 ( 1 9 3 2 ) . 23. C h i l t o n , T.H., and C o l b u r n . A.P., T r a n s . Am. I n s t . Chem. E n g r e s . , 26, 178 (1931 ) . 24. Thomas, H.C., Ann. N.Y. Acad. S c i . , 49, 161 (1948 ) . 25. K a s t e n , P.R., L a p i d u s L., and Amundson, N.R., J . Phys. Chem., 56, 683 ( 1 9 5 2 ) . 26. M i l n e , W.E., " N u m e r i c a l C a l c u l u s , " pp. 100-30, P r i n c e t o n , P r i n c e t o n U n i v e r s i t y P r e s s ( 1 9 4 9 ) . 27. Kramers, H., and A l b e r d a , G., Chem. Eng. S c i . 2* 173 (1 9 5 3 ) . 28. B r i n k l e y , S.R., J r . , J . A p p l . Phys., 1_8, 582 (1946). 29. B r i n k l e y , S.R., J r . , U.S. Bur. Mines, E x p l o s i v e s and Phys. S c i . D i v i s i o n , Rept. 3172 (1 9 5 1 ) . 30. K u n i i , D. and J.M. Smith, A.I.Ch.E. J . I , N o . l , 29 ( 1 9 6 1 ) . - 139 -N O M E N C L A T U R E C o n s t a n t , depending upon type of ore a. Set of r e a l number 1 a, a , a„ S u r f a c e a r e a per u n i t volume of bed ' nom' T r 1 , , cothto C n_^, C n Lower and upper i n t e g r a l l i m i t s c p f °ps Heat c a p a c i t y of f l u i d ( a i r ) and s o l i d D , D' Diameter of sphere p a r t i c l e i n meter and f e e t P P . i r e s p e c t i v e l y f f r i c t i o n f a c t o r 9 9 GQ, G' Mass f l o w r a t e i n kg/s m and I b /hr f t r e s p e c t i v e l y h^, h^, Heat t r a n s f e r f i l m c o e f f i c i e n t s from gas to s o l i d based on a n o m : L n a i and aj r e s p e c t i v e l y h, h' Heat t r a n s f e r f i l m c o e f f i c i e n t s i n W/m K and B tu/hr °F f t 3 IQ B e s s e l f u n c t i o n of f i r s t k i n d k, ks Thermal c o n d u c t i v i t y of gas, of s o l i d £, L T o t a l bed depth and one h a l f of the c h i p t h i c k n e s s r e s p e c t i v e l y M M o i s t u r e % of wood c h i p p^, p£ F i r s t and second L a p l a c e t r a n s f o r m parameters r e s p e c t i v e l y P, PQ, P^ P r e s s u r e drop AP E, AP,j, Measured and t h e o r e t i c a l p r e s s u r e drop Q Heat f l o w r , r ^ R a d i a l component, d i s t a n c e component Radius o f sphere R L — or — r r x Gas ( a i r ) temperature, gas temperature at bed i n l e t S o l i d t emperature, s o l i d temperature a t time zero tr=R " H t - t i S u p e r f i c i a l v e l o c i t y Second L a p l a c e t r a n s f o r m gas and s o l i d temperature v a r i a b l e Width of wood c h i p F i r s t L a p l a c e t r a n s f o r m gas temperature v a r i a b l e D i m e n s i o n l e s s bed depth (-£•) (1 - e) ^ o r 3(1 - e) k s z V f C p f p f L 2 V f c p f P f R 2 D i m e n s i o n l e s s group f o r Schumann's model D i m e n s i o n l e s s group f o r Schumann's model Bed depth x Thermal d i f f u s i v i t y V f R 2 V f L 2 or — — e£ct e£ct s s Voidage of packed bed L i m i t parameter approach to z e r o l_ . B. Time v a r i a b l e D e n s i t y o f gas, s o l i d V f C p f P f a Y - x D i m e n s i o n l e s s time and h e i g h t groups f o r heat t r a n s f e r f i x e d bed w i t h s p h e r i c a l and s l a b p a r t i c l e s r e s p e c t i v e l y R i g h t hand s i z e complex-plane l i m i t f o r the r o o t s of s i n g u l a r i t i e s I n t e g r a t i o n dummy v a r i a b l e s S p h e r i c i t y /a ? 1 F i r s t L a p l a c e t r a n s f o r m s o l i d temperature v a r i a b l e B i o t number Reynolds number F o u r i e r number N u s s e l t number - 142 -APPENDIX A T a b l e 4-37a Summary of e x p e r i m e n t a l gas temperature d a t a and c o n d i t i o n Sphere Re = 8 7 V e l o c i t y = 0.336 m/s Diameter = 4.78 mm At Height At H e i g h t At Height 0.61 (m) 0.83 (m) 1.016 (m) Time _ _ ( s e c ) (°C) (°C) (°C) 120 27.4 27.1 26.3 329 30 .2 28.7 27.8 569 34.1 32.7 31.7 749 35.3 34.6 34.1 1115 36.5 35.9 36.2 1415 37 .1 36.5 36.8 2174 59.8 38.1 38.0 2294 72.4 38.9 38.4 24.14 87.2 40.3 38.7 253i 101.8 42.7 38.9 2774 119.1 53.0 39.8 2894 123.2 61.5 40.3 3134 127.4 87.8 43.0 3254 128.5 100.3 45.6 3494 129.8 117.1 55.9 3614 130.2 121 .4 64.2 3854 130.8 126.2 88.7 4094 131.3 128.4 109.5 4214 131.5 129.1 115.7 4454 131.8 130.0 123.1 4694 132.1 130.6 126.5 4934 132.5 131.1 128.3 5294 132.8 132.0 130.6 - 143 -Ta b l e 4-37b Sphere Re = 169 V e l o c i t y = 0.6775 ra/s At Height At Heig h t At Height 0.61 (m) 0.83 (m) 1.016 (m) Time _ ( s e c ) (°C) (°C) (°C) 120 27.2 27.3 28.1 180 28.3 27.6 27.5 300 32.1 30.8 29.9 360 33.2 32.1 31.3 555 35.3 34.4 34.3 795 38.5 35.7 35.7 1035 58.3 37.6 36.4 1275 98.7 47.7 37.6 1474 121.0 66.0 40.8 1594 128.4 84.3 45.9 1717.2 132.3 102.7 54.3 1834.2 134.6 117.1 67.3 1954.2 136.2 125.5 83.7 2074.2 136.8 130.3 100.2 2314.2 137.9 134.8 122.8 2554 138.5 136.8 131.4 3421 .2 139.4 138.8 137.9 - 144 -T a b l e 4-37c Sphere Re = 228 V e l o c i t y = 1.018 m/s At Height At H e i g h t At Heig h t 0.6L0 (m) 0.813 (m) 1.016 (m) Time ( s e c ) (°C) (°C) (°C) 180 21.6 22.2 22.2 420 32.1 30.4 28.4 660 38.5 34.6 33.8 780 50.6 36.2 34.8 900 70.7 40.7 35.6 1020 95.8 51.1 37.2 1112 111.4 64.6 40.2 1232 121.4 87.1 48.3 1352 125.3 108.1 63.0 1472 126.8 118.9 83.2 1651 127.8 125.0 110.8 1772 128.3 126.3 119.3 1892 128.6 127.3 123.2 2732 131.2 130.2 129.0 - 1 4 5 -T a b l e 4-37d Wood Chip T h i c k n e s s = 2.44 mm Re = 255 V e l o c i t y = .594 m/s Time ( s e c ) At H e i g h t 0.406 (m) (°C) At H e i g h t 0.610 (m) At Height 0.813 (m) (°C) (°C) 135 41.1 37.8 35.1 195 50.8 43.1 40.2 215 55.1 45.2 43.5 255 62.7 48.8 46.3 268 65.3 50.3 48.2 298 71.9 53.5 52.3 328 77.8 56.9 53.8 358 83.3 60.6 55.4 388 88.1 64.4 57 .0 418 92.3 68.3 58.8 448 96.7 73.2 61.1 508 101.9 80.6 65.1 557 104.6 86.2 68.8 617 107.6 91.6 73.1 737 110.6 98.9 81.0 857 112.3 103.3 87.2 977 113.6 106.2 92.0 1217 115.1 109.4 97.4 1577 116.2 111 .9 102.1 2057 117.1 113.4 105.4 2537 117.7 114.4 107.3 - 146 -Ta b l e 4-37e Wood Chip T h i c k n e s s Re V e l o c i t y = 2.44 mm = 329 = .766 m/s AC Height At H e i g h t At H e i g h t 0.406 (m) 0.610 (m) 0.813 (m) Time — _ _ _ (s e c ) (°C) (°C) (°C) 100 40.9 38.4 37.7 140 46.1 41.9 40.1 202 58.3 48.3 47.1 242 66.0 52.3 49.5 282 75.2 57.8 52.5 322 82.5 62.9 55.4 362 90.3 69.6 59.3 402 95.9 75.2 62.9 462 102.9 83.8 69.3 492 105.7 87.8 72.7 552 110.6 95.6 80.1 612 113.5 101.1 86.2 672 115.6 105.3 91.7 762 117.7 109.8 98.3 822 118.7. 111.9 101.8 942 120.2 114.9 107.1 1143 121.6 117.7 111.9 1373 122.6 119.4 115.1 1613 123.1 120.4 116.9 2613 123.8 121.6 119.1 - 147 -Table 4-37f Wood Chip Thickness = 2.44 mm Re = 403 V e l o c i t y = .938 m/s At Height At Height At Height 0.406 (m) 0.610 (m) 0.813 (m) Time :  (sec) C O (°C) (°C) 86 48.8 43.3 40.1 126 59.3 49.8 45.3 146 66.5 53.7 50.1 165 71.5 56.6 52.2 186 78.7 61.1 57.1 196 83.2 64.3 62.8 216 85.3 65.9 63.3 228 91.2 71.3 65.2 268 98.8 79.7 68.7 288 100.2 84.3 71.0 308 105.1 88.5 73.4 328 107.4 92.3 95.9 368 110.4 97.7 79.9 408 112.7 101.9 83.8 448 114.7 105.7 88.1 483 116.1 108.2 91.1 543 117.8 111.2 95.2 663 120.3 115.2 101.1 773 121.9 117.7 107.2 890 123.2 119.5 111.2 1096 124.5 121.7 114.8 1536 126.2 123.9 118.9 - 148 -T a b l e 4-37g Wood Chip T h i c k n e s s Re V e l o c i t y = 2.44 mm = 476 = 1.11 m/s At H e i g h t At H e i g h t At H e i g h t 0.406 (m) 0.610 (m) 0.813 (m) Time — — — — ( s e c ) (°C) (°C) (°C) 55 47.6 38.8 35.1 95 66.3 49.6 •42.1 115 76.4 55.8 50.2 135 82.8 60.8 57.3 155 91.2 66.8 62.7 175 97.9 73.7 64.0 195 101.6 78.2 65.0 215 106.0 84.6 66.7 235 109.4 90.4 68.7 255 112.1 95.5 70.9 291 115.3 102.0 75.3 321 117.7 106.8 98.8 367 119.4 110.2 84.5 391 120.3 111.8 86.9 411 121.1 113.2 89.4 451 122.3 115.2 93.4 511 124.0 117.9 99.2 571 125.1 119.8 103.4 606 126.1 121.4 106.7 760 127.4 123.8 111.4 1076 129.3 126.8 118.1 - 149 -T a b l e 4-37h Wood C h i p T h i c k n e s s Re V e l o c i t y = 4,03 mm = 374 = .594 m/s At H e i g h t A t H e i g h t At H e i g h t 0.406 (m) 0.610 (m) 0.813 (m) Time ( s e c ) (°C) (°C) (°C) 50 36.4 33.2 31.3 110 43.4 36.6 30.5 170 54.2 42.4 37.7 230 65.3 49.0 43.0 250 69.1 51.4 44.9 303 77.0 57.2 49.2 333 80.9 60.4 51.5 363 87.1 66.7 56.2 423 91.6 72.3 60.8 530 95.6 78.9 66.9 590 97.8 82.8 71.1 683 99.9 87.6 77.0 803 101.6 91.6 82.7 923 102.6 94.2 86.8 1163 103.8 97.3 92.3 1523 104.5 99.6 96.3 1883 104.6 100.7 98.3 2483 104.2 101.1 99.6 4483 114.8 111.1 109.4 - 150 -T a b l e 4-371 Wood Chip T h i c k n e s s = 4.03 mm Re = 483 V e l o c i t y = .760 m/s At He i g h t At H e i g h t At H e i g h t 0.406 (m) 0.610 (m) 0.8L3 (m) Time : : ( s e c ) (°C) (°C) (°C) 60 41.1 35.6 38.2 100 53.1 41.2 42.2 110 55.4 42.3 43.2 130 62.6 45.8 46.0 150 69.6 49.8 48.7 170 74.1 51.7 51.3 190 80.4 55.4 52.9 210 86.2 59.1 55.3 251 95.6 66.6 59.9 290 101.8 72.8 63.7 310 104.7 76.3 . 66.1 350 108.6 81.9 69.9 390 111.4 87.1 74.6 430 113.9 92.4 78.4 469 115.4 96.3 82.1 528 117.2 101.2 87.4 633 119.3 107.4 95.6 753 120.5 111.3 101.7 873 121.3 113.8 105.9 951 121.8 115.0 108.1 1311 123.0 118.1 113.8 1649 123.6 119.5 116.3 1911 123.9 120.1 117.4 2511 124.2 120.9 118.8 - 151 -T a b l e 4-37j Wood Chip T h i c k n e s s = 4.03 mm Re = 590 V e l o c i t y = .938 m/s At Height At H e i g h t At Hei g h t 0.406 (m) 0.610 (m) 0.813 (m) Time : ( s e c ) (°C) (°C) (°C) 20 41.1 39.9 38.4 60 45.1 41.3 39.6 80 52.5 45.9 43.7 100 59.1 50.0 47.1 120 67.8 55.6 51.6 140 74.2 59.8 54.8 160 78.1 62.6 56.9 180 83.6 66.7 59.9 200 88.4 70.8 63.0 220 91.0 74.7 66.0 264 96.0 78.3 68.9 284 99.6 81.9 71.9 324 104.0 87.3 76.7 344 106.1 90.2 79.6 384 109.0 94.5 84.0 424 111.7 98.8 88.8 465 113.3 101.7 92.3 525 115.4 105.5 97.2 585 117.3 108.7 101.7 685 118.5 110.9 104.8 765 120.3 114.1 109.4 1028 122.6 118.1 115.2 2526 126.1 123.2 122.1 - 152 -T a b l e 4-37k Wood Chip T h i c k n e s s = 4.03 mm Re = 698 V e l o c i t y = 1.11 m/s At H e i g h t At H e i g h t At H e i g h t 0.406 (m) 0.610 (m) 0.813 (m) Time _ _ _ (s e c ) (°C) (°C) (°C) 20 41.2 39.0 36.8 40 45.3 40.9 38.1 60 52.8 45.3 42.7 80 58.4 48.7 46.0 100 66.8 53.9 50.6 120 74.9 59.3 55.1 140 80.0 63.0 57.9 160 86.8 68.5 61.8 170 90.8 72.1 64.6 214 100.4 82.2 72.3 244 105.1 88.0 77.3 364 111.6 97.4 86.2 424 115.7 104.3 93.8 484 118.4 109.3 100.0 514 121.3 114.3 106.8 649 123.9 118.9 113.7 676 125.2 121.2 117.2 1007 127.1 124.1 121.4 1247 128.3 125.8 123.8 1667 129.4 127.4 126.0 2567 130.4 128.7 127.7 - 153 -T a b l e 4-371 Wood Chip T h i c k n e s s = 7.28 mm Re = 4 9 1 V e l o c i t y = .594 m/s At He i g h t At H e i g h t At H e i g h t 0.406 (m) 0.610 (m) 0.813 (m) Time ( s e c ) (°C) (°C) (°C) 87 41.8 38.2 36.4 187 55.2 44.9 42.2 247 64.7 • 50.8 46.8 287 70.4 55.4 49.3 317 76.7 58.9 52.4 367 81.5 62.6 54.7 415 87.4 67.8 58.1 475 93.1 73.4 61.9 535 97.6 78.6 65.7 595 101.2 83.3 69.5 685 105.5 89.9 75.4 775 108.2 97.7 80.5 865 110.2 98.6 85.1 980 111.9 100.4 90.4 1220 114.1 107.1 98.0 1460 115.3 109.8 102.9 1700 116.1 111.4 106.8 2180 116.9 113.3 109.5 2660 117.3 114.2 111.2 4660 117.7 115.1 112.9 - 154 -T a b l e 4-37m Wood Chip T h i c k n e s s = 7.28 mm Re = 633 V e l o c i t y = 0.766 m/s At H e i g h t At H e i g h t At H e i g h t 0.406 (m) 0.610 (m) 0.813 (m) Time (sec) (°C) (°C) (°C) 78 47.6 41.7 37.4 98 50.8 43.2 38.2 118 56.2 46.1 40.0 138 61.9 49.2 42.2 171 69.4 53.8 45.4 191 78.8 57.3 47.9 231 82.7 64.1 52.6 271 90.8 69.4 56.6 291 94.2 72.6 58.8 331 99.1 77.5 62.7 371 103.9 82.9 67.2 411 107.1 87.1 71.0 465 110.9 92.8 76.8 525 113.7 97.5 80.1 585 115.7 101.3 87.0 664 117.7 105.5 92.9 784 119.4 109.6 99.6 904 120.6 112.3 104.4 1024 121.3 114.3 108.1 1187 121.9 115.9 111.4 1547 122.7 118.0 115.3 2147 123.3 119.4 117.8 - 155 -T a b l e 4-37n Wood Chip T h i c k n e s s = 7.28 mm Re = 774 V e l o c i t y = 0.928 m/s At He i g h t At H e i g h t At Height 0.406 (m) 0.610 (m) 0.813 (m) Time (s e c ) ( 8 C ) C O C O 69 50.6 42.4 39.0 89 55.1 44.8 40.8 109 61.9 49.1 43.8 132 68.7 53.6 47.1 172 80.8 60.8 53.7 212 89.1 70.2 59.1 232 93.4 74.3 62.3 266 99.3 80.6 67.4 296 103.2 85.2 71.5 326 106.5 89.4 75.4 356 109.2 93.2 79.2 386 111.5 96.6 82.7 416 113.4 99.4 85.9 446 115.0 102.1 88.9 476 116.3 104.3 91.7 518 118.1 107.3 95.6 578 119.8 110.4 100.0 678 121.9 114.4 105.8 798 123.5 117.4 110.7 1038 125.4 120.9 116.6 1278 126.6 122.8 119.7 1398 126.8 123.4 120.7 2358 128.1 125.6 124.2 - 156 -T a b l e 4-37o Wood Chip T h i c k n e s s Re V e l o c i t y = 4.81 mm = 399 = .594 m/s At H e i g h t At H e i g h t At H e i g h t 0.406 (m) 0.610 (m) 0.813 (m) Time ( s e c ) (°C) (°C) (°C) 83 42.9 37.2 36.0 163 57.8 45.3 41.3 203 67.5 51.2 45.2 243 75.1 56.3 48.4 285 83.1 62.4 52.1 315 87.7 66.4 54.6 375 95.2 74.1 59.6 435 100.8 80.9 65.2 495 105.2 87.4 70.2 555 107.9 92.2 74.9 609 109.7 95.7 78.7 729 112.3 101.7 86.9 849 113.7 105.4 92.8 969 114.6 107.7 97.2 1089 115.2 109.3 100.6 1209 115.6 110.3 102.8 1470 116.2 111.8 106.2 1950 116.7 113.2 109.2 4350 117.8 115.1 112.6 - 157 -T a b l e 4-37p Wood Chip T h i c k n e s s Re V e l o c i t y = 4.81 mm = 630 = .938 m/s At H e i g h t At H e i g h t At H e i g h t 0.406 (m) 0.610 (m) 0.813 (m) Time (s e c ) (°C) . (°C) (°C) 78 54.6 42.8 42.2 105 64.4 47.2 45.8 125 69.6 49.9 47.9 145 76.9 54.4 50.9 165 83.8 59.0 54.1 199 93.5 67.4 59.2 240 101.1 75.5 64.6 260 104.8 80.1 67.8 300 109.7 87.4 73.4 345 114.2 97.0 79.8 375 116.8 100.3 85.1 435 119.8 106.9 92.6 495 121.5 111.7 98.9 522 122.5 113.6 101.7 642 124.5 118.6 109.8 762 125.7 121.3 114.8 818 126.1 122.3 116.4 1058 127.2 124.6 120.7 1298 127.8 125.7 122.8 1538 128.2 126.4 124.0 4418 129.1 127.8 126.2 - 158 -T a b l e 4-37q S o l i d i n t e r n a l temperature f o r 4 mm wood c h i p Wood Chip T h i c k n e s s = 4.03 mm Re = 374 V e l o c i t y = .594 m/s S o l i d C o r r e s p o n d i n g I n t e r n a l Temperature Gas Temperature Time (se c ) (°C) (°C) 83 39.1 40.3 143 45.7 47.2 192 51.1 52.8 252 57.2 59.1 312 63.2 65.3 372 69.9 72.2 416 74.2 76.6 476 79.4 81.8 536 84.1 86.4 656 91.7 93.8 776 97.4 99.2 896 101.3 102.7 991 103.6 104.8 1231 107.5 108.7 1471 109.8 110.4 1711 111.4 111.8 2191 113.2 113.6 2671 114.2 114.4 3151 114.8 115.1 3631 115.2 115.4 4111 115.4 115.6 - 159 -T a b l e 4-37r Wood Chip T h i c k n e s s = 2.44 mm Re = 329 V e l o c i t y = 0.766 m/s S o l i d C o r r e s p o n d i n g I n t e r n a l Temperature Gas Temperature Time ( s e c ) (°C) (°C) 113 52.9 53.8 153 58.8 61.3 193 65.1 69.2 233 72.5 77.5 253 74.8 80.2 293 80.8 86.4 333 84.9 90.4 400 90.4 96.0 490 96.4 100.8 550 99.0 103.2 610 101.3 105.2 670 103.3 107.0 810 107.6 110.7 1052 112.1 114.4 1319 115.2 117.1 1799 118.4 119.9 2279 120.2 121.4 2759 121.1 122.3 - 160 -T a b l e 4-37s Gas temperature data f o r 6 mm wood c h i p w i t h moist a i r Wood Chip T h i c k n e s s = 7.28 mm Dry A i r Reynolds Number (100°C) = 633 M o i s t A i r (30% M o i s t u r e ) Reynolds Number (100°C) = 836.1 Dry A i r V e l o c i t y (100°C) = .766 m/s Mo i s t A i r V e l o c i t y (100°C) = .98 m/s At Height At Heig h t At Height 0.406 (m) 0.610 (m) 0.813 (m) Time — . — _ — -( s e c ) ( 4 C ) .CO C O 34 70.4 66.9 65.8 54 76.0 71.8 70.8 74 78.9 74.7 73.8 94 82.5 77.9 76.9 114 85.9 80.6 79.3 134 89.1 80.9 81.3 154 90.9 84.2 82.3 184 94.0 86.5 84.0 266 100.9 90.1 81.4 386 106.9 98.0 91.3 521 110.7 103.0 95.2 881 114.7 110.6 104.1 1121 115.7 112.7 107.9 1471 116.4 114.3 111.1 1951 116.8 115.1 110.9 2431 116.9 115.5 113.7 3551 117.1 115.8 114.2 5471 117.3 116.0 114.6 - 161 -T a b l e 4-37t Gas temperature d a t a f o r 4 mm wood c h i p w i t h moist a i r Wood Chip T h i c k n e s s = 4.03 Dry A i r Reynolds Number (100°C) = 590 M o i s t A i r (30% M o i s t u r e ) Reynolds Number (100°C) = 637.4 Dry A i r V e l o c i t y (100°C) = .766 m/s M o i s t A i r V e l o c i t y (100°C) = .98 m/s At Height At Height At Height 0.406 (m) 0.610 (m) 0.813 (m) Time ( s e c ) (°C) (°C) C O 34 71.7 68.1 67.7 44 77.1 74.2 70.4 64 83.7 80.3 77.6 84 85.8 83.2 80.9 104 89.6 86.6 84.5 124 92.5 88.9 86.9 164 96.8 91.7 89.3 184 98.9 93.1 90.4 255 104.2 96.9 92.7 316 107.1 99.5 94.2 391 109.5 102.2 96.1 631 113.4 108.3 102.1 871 115.1 111.3 106.7 1111 115.9 112.9 109.5 3226 116.6 114.3 111.9 3704 117.1 115.0 113.3 4614 117.4 115.6 114.2 - 162 -APPENDIX B CALCULATION OF WOOD CHIP AND SPHERE CATALYST THERMAL PARAMETERS B-l Wood Chip Thermal Parameters Calculation From the Chemical Engineering Hand Book (16), the thermal conductivity of wood chip i s given as: IF M < 40 k = p(0.1159 + 0.00233 M) + 0.01375 IF M > 40 k = p(0.1159 + 0.00316 M) + 0.01375 where M = moisture % 3 p = g/cm (dry basis) k = Btu/hr f t °F The wood chip moisture content was found to be 6.58%, then the thermal conductivity i s : k = (0.3295) (0.1159 + 0.00233 (6.587.)) + 0.01375 = 0.057 Btu/hr f t °F = 0.057 Btu/hr f t °F x 1.7307 _ V , [ m * — — Btu/hr f t F = 0.09859 W/m K For M = 0, then k = 0.08985 W/m°C Therefore, the average k which was used for c a l c u l a t i o n i s 0.09859 + 0.08985 , ^ - 163 -B—2 Sphere Catalyst Thermal Parameters Calculation wt. % (k) W/m K k x wt % A 1 2 0 3 73.2 3.4614 2.53 CaO 6.4 7.788 0.498 NiO 20.4 0.940 0.192 3.22 (W/m K) A i r (100°F) k(W/m K) = 0.02695 S o l i d ^ = 1.2485 x i 0 ~ 3 m 3/kg = 1.2485 cm 3/g Pore volume = 0.25 - 0.35 cra 3/g Volume % of a i r = 0 t 3 c m * 1 % 1.2485 cra 3/g = 24.03% Volume % of s o l i d = 75.9% The o v e r a l l thermal c o n d u c t i v i t y i s : (3.22) 11^- + 0.0269 2 4 , 0 3 100 100 = 2.5 W/m K. - 164 -APPENDIX C Fixed Beds with Large Spherical Particle Basic equations: i ,z2t 2 t . „ at k S ( ~ 2 + 7 2r"} " C p p s 38 3r *s ks „ = h c(t - T) 3r /r=R P r=R ' „ _ n 3T 3(1 - e) . ,3t. v e c P f p f a l R k s ( a F > r = R z = £x r = sR h R B i V f C P £ P f 0 = n = Y - x x = z/Jc - 165 -R V, C p R D2 f p s V c R *s f a = £ ks e e £ a s — U x ks z x = i x = 3(1 - e ) - — - _ V f C P f p f R 2 Then 3n £ 3n _ _1 TO" V f ' 3z £ 3x _ n _3_x _ 1_ 39 ' 3z £ 3T = 3T / £ N, 3~6~ 3n V, ' (C-4) 3T = 3T , K 3T ,K and 3r = R 3s (C-6) S u b s t i t u t e C4, C5 and C6 i n CI, C2, and C3, i t gives - 166 -- e ,<H>s-l = t " T ' 5 = 1 ( C " 8 ) - » < H > 8 - i = °' 3 = 1 ( c " 9 ) To s o l v e the above e q u a t i o n , double L a p l a c e t r a n s f o r m i s i n t r o d u c e d f i r s t and s u b s t i t u t i o n i s f o l l o w e d . A f t e r the s u b s t i t u t i o n , the d i f f e r e n t i a l e q u a t i o n s may then be i n v e r t e d to g i v e the f i n a l s o l u t i o n . I f the double L a p l a c e t r a n s f o r m i s a p p l i e d i n the form: L(T) = / e t dn = ui r« " P i n L(T) = J 0 e T dn = W r- " P 2 X -L(co) = e co dx = v co ~ P l X L(W) = /" e W dx = V 3 v ^ 2 3v , - C i . , — r + - — = oc(p v - — ) (C-10) 3 s Z S 3s 1 P2 3v — - e . . _ = v _ v ( C - l l ) 3s a ( P 2 V - p ( | ^ ) s = 1 = 0 (C-12) - 167 -We can s o l v e the above e q u a t i o n s by a l g e b r i c s u b s t i t u t i o n , d i f f e r e n t i a l e q u a t i o n i n t e g r a t i o n w i t h ] ^ | s = o = 0 . F i n a l l y , we have, T n - t . a(-9 i) P, t . V • l- + 1 a p 2 + uB(<j>) p p , (C-13) where B(.j>) = * c o t * ~ 1 1 1 O ~ R - ) + -5- <J>cot4> (C-14) B. B. 1 1 (C-15) 4> = / - a p E q u a t i o n C-13 can be i n v e r t e d w i t h r e s p e c t to P 2, i t g i v e s P 1 p CC-16) A l s o e q u a t i o n C-16 can be i n v e r t e d w i t h r e s p e c t to P j , i t g i v e s T - ( T 0 " c i > * + C i (C-17) _t e - x B ( 4 0 where * = L [ — _ J ( c _ 1 8 ) The i n v e r s i o n of e q u a t i o n C-18 has been s o l v e d by Rosen ( 6 , 7 ) , which i s : - 168 --xH HDj + -jp (HD 2 + HD 2) H l " H I T - H C — ( C ' 2 0 ) ( 1 + B J - ) + ( B ^ } 1 l HD ? H 2 = H D — H D 7 — ( C " 2 1 ) ( 1 + ^ + ^ l l _ X(sinh2X + s i n 2 \ ) , m i ~ cosh2X - cos2X 1 ( C 2 2 ) H n _ X(sinh2X - sin2X) ,„ W 2 cosh2X - cos2X ( C _ 2 3 ) X = /3 (C-24) In o r d e r Co e v a l u a t e the i n f i n i t e i n t e g r a l , the a p p r o p r i a t e U n i t s at 3 > 0 or X > 0 i s r e q u i r e d . They have been e v a l u a t e d by Rosen (6,7) and t a b u l a t e d below: - 169 -Function lira 3 > 64 lim 6 < 0.1 lira 8 = 0 _ X(sinh2X + sln2X) (k . 2 1 ~ cosh2X - cos2X " 1 X " 1 (45 } 3 u r > _ X(sinh2X - sln2X) . .2. a H D2 cosh2X - cos2X A (3° 6 H D 1 + ( H D 1 + H D 2 ) H l = HD SDo~T~ (1 + ~ i ) Z + B. B i HD 2 (1 + - V + B i B i s i n ( — — X H 2 ) 2n 2 -— " 3 X 9t Further, If equation C-2 and C-3 combined and the (-r—) or r=R eliminated, i t will give 3x i tr=R is the surface temperature of the solid. As referred to Carslaw and Jaeger (5) the solid temperature within the sphere can be written in terms of its surface temperature. - 170 --2 ct 0 0 t'(8, r) = — — 5 - E (-1)" exp(-a n 2 TT2 8/R2) n TT sin i£i rR , s K n=l /!? exp(a n 2 TT2 A'/R2) t 1 __(X») dA' (C-26) 0 S V—K where t ^ = R = t r = R - t± (C-27) t'(9, r) = t(8, r) - t± (C-28) C-2 Fixed Beds with Slab Shape Particle Basic equations: k 3 " C P ps H ( C " 2 9 ) 9r s x k s ( l 7 >r =L " h f ( t r =L " T ) ' rx= L ( C " 3 0 ) X X X v c o H - ( 1 ~ £ ) ks ( i i ) = e c 0 i i cc - in V f p . P f 3z L * S Q37 ; r =L V P f W Q C J U f X X f Let: z = £x r = sL x r, ._ ks _ 1 hcL B. f 1 - 171 -V f C P f P f O = ; £ z n = Y - x or V £ x = z/£ (1 - e) , M = -—5—- ks L V F C n P c L 2 V T 2 f p s s V f L a = £ Z ks e £ a s Then 1_ 3T 3T , £ N 31 = Tt ( V ~ ) CC-32) I I " Tn ( " £ } + Ix" ( £ } (C-33) - 172 -3 r x = L3s (C-34) S u b s t i t u t e i n C-32, C-33 and C-34 i n C-29 , C-30, and C-31, i t becomes d 2 c , 3 t . ' / = a ( ^ - ) (C-35) 2 2 v 3 n 3s " ^ T i ^ l = t - T, s = 1 (C-36) co ~ P i n _ L ( t ) = / e t dn = io oo " P i n L(T) = / e T dn = W • _ ' " P 2 X -L(OJ) = e in dx = v oo ~ P ? X L(W) = / e W dx = V Take double L a p l a c e t r a n s f o r m of e q u a t i o n s C-35, C-36 and C-37 then -2- t. — = a(p v - — ) (C-38) 3 s z '2 e' — = v - V , s = l (C-39) 3s - 173 -- a ( p 2 V - ^ ) - u ( | ^ ) s = 1 = 0 , B - . I (C-40) Make substitution and solve the differential equations C-38, C-39 and C-40 simultaneously with s_Q = 0. We have, T t , 0 i v (— a - — a) li P l P l V = - - ± — + i i (C-41) P 2P X a p + j j v2 , , cothw <J0 where to = /a (C-42) Take i T ^ v ^ ) and then L~*(pj) of equation C-41, i t becomes, -1 e - B x T - t. + (T. - t.) L (—-) (C-43) l 0 i p 1 W h e r e B " , ^othcu < C ~ ^ to -1 e~ B x The following section is to show how to solve the L (—-—) _1 ~Bx Let $ = L A ( - ) (C-45) P l - 174 -By definition: F(t) = L _ 1 ( f ( s ) ) = J L e S t f ( B ) ds (C-46) Then -B(pj)x Therefore «(B. , x, -) = L - 1 ( - ) i ' a Pl = . J _ rY'+i- V ,e 1 i f Y t i °° i  2 i i V - i - 6 ( p^ > d P l (C- 4 7) To find the limit of y', we may have to find the essential singularities -B( P i)7 P l The essential singularities are pi = 0 and the roots of u± (P x) e' + coth(o± ( P l) - 0 (C-48) where to(p^) = /a p^ a > 0 £' > 0 We find = i a^^ where a^^ is real number and then p 1 = -y±, y^s positive real number. - 175 -Because t h e r e e x i s t an i n f i n i t e number of e s s e n t i a l s i n g u l a r i t i e s the u s u a l method of e v a l u a t i o n of the i n v e r s e t r a n s f o r m by r e s i d u e s i s not p r a c t i c a l . However, s i n c e B(p^) i s a n a l y t i c f o r R(p^) > 0 (Y' > 0) exc e p t at p^ = 0. We can take the path of i n t e g r a t i o n to be along the i m a g i n a r y a x i s w i t h a s m a l l s e m i c i r c l e T o f r a d i u s -»• 0 e x c l u d i n g the o r i g i n , then - n , l i m - i e i . . p.n-xBfp.) * < B f *' 7? = 2 ^ i V 0 I/-!- + / i s . + /r ] f 6 d p i ( C " 4 9 ) i "l To e v a l u a t e the i n t e g r a l around T t r a n s f o r m p o l a r c o - o r d i n a t e s and use B(p^) * 0 as p^ 0, then E.-K) 271 h TY e d p l = 2 ( C - 5 0 ) The f i r s t and second i n t e g r a l s can be combined by making the s u b s t i t u t i o n p^ = - i 3 and p^ = i S r e s p e c t i v e l y and t a k i n g the l i m i t and s u b s t i t u t e e.g., C-50 i n C-49 then: $(B., x, i ) = ± + -Lj. [/" u(x, i g ) + U(x, - 13)] dS (C-51) where - 176 --iSn -xB(-iB) U(x, -13) = - d3 (C-53) U(x, 13) i s Che complex conjugaCe of U(x, - i 3 ) . FurCher, for any complex quanCiCy F, we have F + F = 2R(F), Chen _ i3n - x B ( i 3 ) * < V x « 5 = i + ¥ Q ^~—h ] d3 (c"54) u(p,) Since B(p ) = 1 u(p ) £' + cothu>(p ) For p = 13 and l e t X = (ct3) 1 / 2 ( ^ |) or X2 = Chen we have, B(X) = Hx + i H 2 (C-55) where 2X(|i + HDj) H, = - „ (C-56) 1 + HDX)2 + (HD2)2 (2X)(HD2) H = (C-57) - 177 -HD, - s i "h2X • s l n n ( 2 cosh2X - cos2X Substitute equation C-55 into C-54, i t becomes -xH 4(B., x,J) = I + ^  /Q ^ - g " sin ( S n - xH 2) dB (C-60) Let 3' = — and X = /$r in equation C-60, i t becomes, Finally, the equation becomes: T = t. + ( T Q - t.) *(B., x, IL) (C-62) To evaluate of the i n f i n i t e i n t e g r a l , the appropriate l i m i t s at S' •»• 0 or X •>• 0 is required. They are tabulated below: - 178 -Function lira X < 0.1 Rosen (6,7) lim X 0 X(sinh2X + sin2X) (cosh2X - cos2X) - 1 (4/45) ^ HD„ = sinh2X + sin2X 2 cosh2X - cos2X (4/45) X3 + I X(sinh2X - sin2X) (cosh2X - cos2X) (2/3) X' HD, = sinh2X - sin2X 1 cosh2X - cos2X (2/3) X H l " 2X(|A + HD ) i (|^ + HD X) 2 + (HD 2) 2 i (2X)(HD 2) H2 " (~~ + HD X) 2 + (HD 2) 2 2 3 1  -xH 1 s i n ( - ^ - n - x H 2) ^ - 2x a 31 Further, i f equation C-30 and C-31 combined and the (-|— ) 9r r —L x x eliminated, i t w i l l give. x 3 x i (C-63) - 179 -As referred to Carslaw and Jaeger (5), the solid temperature within the slab can be written in terms of i t s surface temperature. » -a (2n+l) 2 I T 2 0/4L2 (2n+l) Ttr (2n+l) net ( - l ) n f ( V 6) . 1 E e s ... 2 L - ( n=0 a a (2n+l) 2 u 2 X'/4L2 / 0 e S t; = L(X') dX- } (C-64) where t; - t - t. (C-65) x x f ( r x , 6) = t ( r x > 6) - t± (C-66) - 180 -APPENDIX D Appendix D-l C THIS PROGRAME IS WRITTEN FOR THE HEAT TRANSFER FIXEO BEO WITH C INTERNAL HEAT RESISTANCE -SPERICAL PART I C A L S . C NORMANCLATURE C EXPY=EXPERIMENTAL GAS TEMPERATURE (C) C GY=THEORETICAL GAS TEMPERATURE (C) C ZZ=BED HEIGHT IN INCH (IN) C X=TIME IN SECONDS (S) C A(1 ) =HEAT TRANFER COEFFICENT ( W / M * * 2 / K ) C A(2)=THERMAL CONDUCTIVITY OF SOLIO (W/M/K) C A(3)=HEAT CAPACITY OF SOLID ( K J / K G / K ) C K K ' I N I T I A T E POINT C N = FINAL POINT IMPLICIT R E A L * 8 ( A - H , 0 - Z ) DIMENSION A ( 1 0 0 ) , X ( 1 0 0 ) , E X P Y ( 1 0 0 ) . G Y ( 1 0 0 ) , W W ( 1 0 0 ) DIMENSION X X ( 2 0 0 ) , X Y ( 2 0 0 ) . Z Z ( 1 0 0 ) EXTERNAL FUN DATA EXPY/35 . 3 . 5 9 . 8 . 8 7 . 2 . 1 19. 1, 127 . 4 . 1 3 0 . 8 . 3 8 . 9 . 5 3 . 0 . 8 7 . 8 . 100 .3 . 117. 1, 12S.2 , 129. 1 , 3 9 . 8 . 4 5 . 6 . 5 5 . 9 . 8 8 . 7 . 1 1 5 . 7 . 1 2 6 . 5 / DATA Z Z / 2 4 . , 2 4 . , 2 4 . , 2 4 . . 2 4 . . 2 4 . . 3 2 . . 3 2 . , 3 2 . , 3 2 . , 3 2 . . 3 2 . , 3 2 . , 4 0 . . 4 0 . , 4 0 . . 4 0 . , 4 0 . , 4 0 . / DATA X / 7 4 9 . ,2 174. ,24 14. , 2774 . .3 1 34 . ,3854 . , 2294 . , 2 7 7 4 . , 3 1 3 4 . , 3254 . , 3 4 9 4 . , 3 8 5 4 . , 4 2 1 4 . , 2774 . , 3254 . . 3 4 9 4 . . 3854 . .42 14. , 4 6 9 4 . / C C NO. OF DATA N C DATA K K . N / 1 . 1 9 / C C NO. OF UNKOWN PARAMENTERS M C C C INIT IATE VALUE OF A(M) C DATA A / 1 0 3 . 147 .2 . , 1 7 7 3 . 0 7 / DATA A / 6 7 . 5 6 0 7 . 2 . . 1 7 9 4 . 7 6 5 5 / C C 88 CALL S U M ( A , X . E X P Y , S . G Y , N , Z Z . K K . V O . G O , D P , T O ) TG0 = T 0 - 2 7 3 . 15 WRITE(6 ,799) 799 F O R M A T ( / / / / / ) WRITE(6 .998 ) 998 F 0 R M A T ( 1 X , ' A COMPARISON OF THEORETICAL AND EXPERIMENTAL' . 1 ' TEMPERATURE V A L U E S ' , / / . ' OF THE SPHERICAL PART I C A L S ' / ) WRITE(6 .821)DP 821 F O R M A T ( / / / / / / , 1 X , ' DIA ( M ) = ' . F 7 . 5 / ) WRITE(6 ,831)TGO 831 F ORMAT ( 1X , ' TGO (• C ) = ' . F 7 . 2 / ) WRITE(6 .19 1)G0 191 FORMAT(1X , ' GO ( K G / ( M * * 2 ) / S ) = ' . F 7 . 5 / ) • WRITE(6 .915 )V0 915 FORMAT(1X, ' VO ( M/S ) = ' . F 7 . 5 / ) VAR = S WRITE(6 ,86)VAR 86 F O R M A T ( / / , 1 X , 'VAR = ' . F 1 2 . 8 . / / ) - 181 -WRITE(6 .78 )A( 1 ) 78 F O R M A T ( I X , ' H ( W / ( M * * 2 ) / K ) = ' . F 1 0 . 6 / ) WRITE(S, 178)A(2) 178 FORMA T ( IX, ' K ( W/M/K ) = ' . F I 0 . 6 / ) W R I T E ( 6 , 2 7 8 ) A ( 3 ) 278 F O R M A T ( I X , ' CP ( J / K G ) = ' . F 1 3 . 6 / / ) C DO 9 KK = 1 ,M C W R I T E ( G , 7 8 ) K K , A ( K K ) , K K , S A ( K K ) C78 FORMA T( 1X, ' A ( ' , 1 3 . ' ) = ' . F 1 5 . 5 . 5 X , ' O S / D A ' , C 1 ' ( ' . 13. ' ) = ' , F 1 2 . 8 / ) C9 CONTINUE DO 99 I=KK,N I F ( I . E Q . 1)GO TO G1 I F ( ( Z Z ( I ) - Z Z ( I - 1 ) ) . E Q . O . ) G 0 TO 71 61 W W ( I ) = Z Z ( I ) * 2 . 5 4 / 1 0 0 . WRITE(6.81)WW(I) 81 F O R M A T ( / / / , 1X, ' HEIGHT ( M ) = ' . F 6 . 4 . / / ) WRITE(6 .999) 999 F 0 R M A T ( / / , 2 X . ' T I M E ' , 4 X , 'EXPERIMENTAL' , 3 X . 'THEORETICAL ' , 3 X , 1 ' ( E X P - T H E O ) / E X P ' ) WRITE(S ,982) 982 FORMAT(2X, ' M I N ' , 7 X , ' ( C ) ' . 1 0 X . ' ( C ) ' . / ) 7 1 GY(I )=GY( I ) -273 .15 0EV=(EXPY( I ) -GY( I ) ) / E X P Y ( I ) X ( I ) = X ( I ) / 6 0 . W R I T E ( 6 , 9 8 ) X ( I ) . E X P Y ( I ) . G Y ( I ) , D E V 98 FOR MA T( 1 X , F S . 3 , 5 X . F 5 . 1 . 6 X . F 1 0 . 3 , S X , F 1 0 . 5 / ) 99 CONTINUE STOP END C c SUBROUTINE S U M ( A , T I I . E X P Y , S , T G , N , Z Z . K K . V O , G O , R L L , T O ) C NORMANCLATURE C AA=CROSS SECTION AREA OF THE COLUMN ( M * « 2 ) C RLL'DIAMETER OF SPHERE (M) C PS=SOLID DENSITY ( K G / M * * 3 ) C EE-VOIDAGE OF THE BED C RR=ROTAMETER REAOING C V0=SUPERFICAL VELOCITY M/S C 0=VOLUME METRIC FLOWRATE (L /MIN) C TSI= INIT IAL TEMPERATURE (C) C TO=GAS INLET TEMPERATURE (C) C CPF=GAS HEAT CAPACITY ( K J / K G / K ) C PF=GAS DENSITY ( K G / M * * 3 ) C GO=VO*PF C R L = R L L / 2 . RADIUS OF PARTICAL IMPLICIT R E A L * 8 ( A - H , 0 - Z ) EXTERNAL FUN DIMENSION Z ( 1 0 0 ) , T I ( 1 0 0 ) , E X P Y ( 1 0 0 ) , R ( 1 0 0 ) , T G ( 1 0 0 ) DIMENSION RRN( 100) .ZZ( 1 0 0 ) . T S ( 1 0 0 ) . A ( 1 0 0 ) . E X P Y Y ( 1 0 0 ) , T I I( 100) COMMON / B L A / T R . T E . T X S = 0 . DO 77 J=KK.N Z ( J ) = Z Z ( d ) * 2 . 5 4 / 1 0 0 . DATA A A , P S . E E . R R / O . 0 3 2 4 2 9 3 , 1 G 3 0 . , 0 . 4 7 1 4 9 , 4 0 . / RLL=0.0O477 R L = R L L / 2 . D O T0=133.+273.15 - 182 -TSI=32 .+273 .15 T I ( d ) = T I I ( J ) EXPYY(d)=EXPY(J ) + 273 . 15 0=122.9440935+ 13 . 28 1237558*RR V O = Q / A A / 1 0 0 0 . / 6 0 . AVT = DSORT(T0*EXPYY(J ) ) C A 1=0.42188O23980O1 A2 = - .1837520491D-01 A3=0.34 5 3395S220-04 A4 = - . 2368161413D-07 RTSI=25.+273. 15 P F 0 = A 1 + A 2 ' ( R T S I ) + A 3 * ( R T S I * * 2 ) + A 4 * ( R T S I * * 3 ) G 0 = V 0 « P F 0 C 81=0.1O3O3382870O1 82=- .2001190186D-03 B3=0.39710998540-06 C P F = B 1 + B 2 « ( A V T ) + 8 3 « ( A V T * « 2 ) CPF=CPF*1000 . ° C V F = V 0 / ( 2 7 3 . 15 + 25 . )«AVT C RX = 3 . •( 1 . - E E ) / E E * 2 ( J ) / ( R L * * 2 ) * ( A ( 2 ) / ( G O * C P F ) ) R X = 3 . - ( 1 . - E E ) / 1 . * Z ( J ) / ( R L * » 2 ) ' ( A ( 2 ) / ( G 0 * C P F ) ) C RX=125. R E = A ( 2 ) / ( A ( 1 ) * R L ) R R N ( d ) = ( T I ( d ) - Z ( J ) / V F * E E ) / ( A ( 3 ) « P S / A ( 2 ) * ( R L * * 2 ) ) * 2 . / R X C RRN(J )=2 .0 C RE=0.2*RX C W R I T E ( 6 . 9 0 1 ) T I ( J ) , Z ( L ) .RE .RX 901 F O R M A T ( 1 X . ' T I ' . F 1 5 . 5 . ' Z ' , F 1 0 . 5 . ' R E ' . F 1 5 . 5 , ' R X ' . F 1 5 . 5 ) TR=RRN(J) TE = RE TX = RX DS = RE AP=0. BP=80. I F(RX . G T . 4 0 . )BP = 80 . ABSERR=0.OO001 RELERR=0.001 C I F ( R X . L T . 5 0 . ) G 0 TO 9 I F ( R E . L T . 0 . 0 1 ) G 0 TO 2 C SS = ( 3 . / 2 . » R R N ( d ) - 1 . ) / ( 2 . / ( D S 0 R T ( 5 . * R X ) ) ) GO TO 9 2 SS = ( 3 , / 2 . * RRN(J ) - 1 . ) / ( 2 . * ( D S O R T ( R E / R X ) ) ) 3 YY=DERF(SS) YG=( I . + Y Y ) / 2 . ERREST=88. N0FUN=7 FLAG=8. GO TO 10 9 CALL 0 U A N C 8 ( F U N , A P , B P , A B S E R R . R E L E R R , Y . E R R E S T , N O F U N . F L A G ) C YG = Y / 3 . 14 15927+1 . / 2 . C . 10 AP=0. BP=200. - 183 -c C98 2 C 79 C 99 C 87 77 C C C C C C C C C C C C C C C C C C C C c' c CALL Q U A N C 8 ( F U N X , A P . B P . A B S E R R , R E L E R R . Y . E R R E ST ,NOFUN.FLAG) DY = Y / 3 . 14 15927 KL = KK I F ( K K . E Q . 1 )KL=0 T G ( J ) - ( T O - T S I ) * Y G + T S I T S ( J ) = ( T O - T S I ) « Y G + T S I + ( T O - T S I ) « O Y « D S S = ( E X P Y Y ( J ) - T G ( J ) ) * * 2 / ( E X P Y Y ( J ) - 2 7 3 . 1 5 ) / ( F L O A T ( N - K L ) ) + S WRITE(6 .982 )S FORMAT( 1X. 'VAR=' . F 1 5 . 5 ) EX=RE/RX W R I T E ( 6 . 7 9 ) N 0 F U N . F L A G . R X . E X FORMAT( 1 X, 'NOFUN ' . I 4 . 2 X . ' FLAG ' .F7 . 3 , ' RX' .F 10 .5 . ' E X ' . F 9 . 5 / ) W R I T E ( 6 . 9 9 ) R R N ( J ) . Y G . E R R E S T F0RMATC.1X.' RRN = ' . F 1 0 . 5 . 2 X . ' SUM = ' . F 10 . 5 . 2X . F 15 . 8 . / ) W R I T E ( 6 , 8 7 ) T S . T G FORMA T( 1X, ' TS = ' . F 1 5 . 6 . ' TG = ' . F 1 0 . 5 / ) CONTINUE RETURN END SUBROUTINE 0 U A N C 8 ( F U N , A . B , A B S E R R . R E L E R R , R E S U L T , E R R E S T , N O F U N , F L A G ) A u t o m a t i c a d a p t i v e q u a d r a t u r e r o u t i n e b a s e d on N e w t o n - C o t e s 8-pane1 I n p u t : O u t p u t : r u l e . FUN F u n c t i o n FUN(X) to be i n t e g r a t e d A Lower l i m i t of i n t e g r a t i o n B Upper l i m i t of i n t e g r a t i o n ABSERR A b s o l u t e e r r o r t o l e r a n c e RELERR R e l a t i v e e r r o r t o l e r a n c e RESULT I n t e g r a t e d v a l u e ERREST E s t i m a t e d magni tude o f the a c t u a l e r r o r NOFUN Number o f FUN(X) e v a l u a t i o n s r e q u i r e d FLAG I f FLAG i s z e r o then RESULT s a t i s f i e s l e a s t s t r i n g e n t e r r o r t o l e r a n c e . If FLAG i s n o n z e r o bu t XXX.YYY then XXX i s the number o f n o n c o n v e r g e d I n t e r v a l s and YYY i s the f r a c t i o n of t o t a l i n t e r v a l rema1n1ng See F o r s y t h e , M a l c o l m and M o l e r , p p . 102-105 IMPLICIT R E A L * 8 ( A - H . O - Z ) DIMENSION QRIGHT(3 1 ) , F ( 1 6 ) . X ( 1 6 ) . F S A V E ( 8 . 3 0 ) . X S A V E ( 8 . 3 0 ) LEVMIN=1 LEVMAX=30 LEV0UT=6 N0MAX=5OOO NOFIN = N0MAX-8* (LEVMAX-LEVOUT + 2* * (LEVOUT+1)) W0=3956.DO/14175.DO W1 =2 3 5 5 2 . D O / 1 4 1 7 5 . D O W2=-3712.DO/14175.DO W3=41984.DO/14175.DO W4=-18160.DO/14175.DO F LAG = 0 .DO RESULT=O.DO - 184 -C0R11=0.DO E RR E ST =0.DO AREA =0.DO NOFUN=0 I F ( A . E O . B ) RETURN LEV = 0 NIM=1 X0 = A X( 1S)=B 0PREV=O.DO FO=FUN(XO) S T 0 N E = ( B - A ) / 1 S . D 0 X(8)=(X0+X( IS ) ) / 2 . D O X(4 ) = (X0+X(8 ) ) / 2 . 0 0 X( 12) = ( X ( 8 ) + X f l 6 ) ) / 2 . 0 0 X ( 2 ) = ( X 0 + X ( 4 ) ) / 2 . D 0 X ( 6 ) = ( X ( 4 ) + X ( 8 ) ) / 2 . D 0 X( 10) = (X (8 ) + X( 1 2 ) ) / 2 . D O X( 14) = (X( 12)+X( 16) ) / 2 . D 0 DO 25 J = 2 . 16,2 F ( J ) = F U N ( X ( d ) ) 25 CONTINUE N0FUN=9 30 X( 1 ) = (X0+X(2) ) / 2 . D 0 F( 1)=FUN(X( 1 ) ) DO 35 J = 3. 15 . 2 X ( J ) = ( X ( J - 1 )+X(J+ 1 ) ) / 2 . 0 0 F ( J ) = F U N ( X ( J ) ) 35 CONTINUE N0FUN=N0FUN+8 STEP = (X( 1 6 ) - X 0 ) / 16.00 0LEFT = (WO'(F0+F(8 ) ) + W 1 « ( F ( 1)+F(7))+ W2*(F(2)+ F ( 6 ) ) + 1 W 3 * ( F ( 3 ) + F ( 5 ) ) + W 4 « F ( 4 ) ) « S T E P OR IGHT(LEV*1 ) = (WO*(F(8) + F( 16) )+W1*(F(9 )+F(15) )+W2*(F(10)+F(14) )+ 2 W3*(F( 1 1 ) + F (13 ) )+W4 'F (12 ) ) *STEP QN0W = QL E F T + ORIGHT( L E V +1) ODIFF=QNOW-QPREV AREA=AREA+OIFF E5TERR=DABS(ODIFF) /1023 .DO TOLERR=DMAX 1 ( A B S E R R . R E L E R R * D A B S ( A R E A ) ) * S T E P / S T O N E I F ( L E V . L T . L E V M I N ) GO TO 50 I F ( L E V . G E . L E V M A X ) GO TO 62 I F (NOFUN.GT.NOFIN) GO TO 60 I F ( E S T E R R . L E . T O L E R R ) GO TO 70 50 NIM = 2*.NIM LEV=LEV*1 DO 52 1=1,8 F S A V E ( I . L E V ) = F ( 1 + 8 ) XSAVE( I ,LEV)=X(1+8 ) 52 CONTINUE OPREV=OLEFT DO 55 1=1,8 J=- I F (2*d+ 18)=F(d+9) X(2"vJ+18)=X(J + 9) 55 CONTINUE GO TO 30 60 NOFIN = 2 * NOF1N LEVMAX=LEVOUT F L A G = F L A G + ( B - X O ) / ( B - A ) - 185 -GO TO 70 62 FLAG=FLAG+1.DO 70 RESULT=RESULT+QNOW ERREST=ERREST+ESTERR COR 1 1=COR11+QDIFF/1023.DO 72 I F ( N I M . E Q . 2 * ( N I M / 2 ) ) GO TO 75 NIM=NIM/2 LEV=LEV-1 GO TO 72 75 NIM'NIM-M I F ( L E V . L E . O ) GO TO 80 QPREV=QRIGHT(LEV) XO = X(16) FO=F(16) DO 78 1=1,8 F ( 2 * I ) = F S A V E ( I . L E V ) X(2 * I ) = X S A V E ( I . L E V ) 78 CONTINUE GO TO 30 80 RESULT = RESULT + COR 1 1 I F ( E R R E S T . E O . O . D O ) RETURN 82 TEMP=DABS(RESULT)+ERREST I F ( T E M P . N E . D A B S ( R E S U L T ) ) RETURN ERREST=2.DO*ERREST , GO TO 82 ENO C C DOUBLE PRECISION FUNCTION FUN(P) IMPLICIT R E A L - 8 ( A - H , 0 - Z ) COMMON / B L A / R N , E , X I F ( P . E 0 . O . ) G 0 TO 1 I F ( P . L T . O . 1 ) G 0 TO 2 I F ( P . G T . 6 4 . ) G 0 TO 3 R=DSORT(P) RD 1 =R*(DS I N H ( 2 . * R ) + D S I N ( 2 . * R ) ) / ( O C O S H ( 2 . * R ) - D C 0 S ( 2 . *R) ) R02 = R * ( D S I N H ( 2 . * R ) - 0 S I N ( 2 . " R ) ) / ( D C O S H ( 2 . * R ) -DCOS(2 . *R) ) HD1=RD1-1. HD2 = RD2 RH=(1 .+E*HD1)* *2+(E*H0 2 ) * * 2 H1=(HD1+E*(HD1**2+HD2**2) ) /RH H2=HD2/RH F U N = D E X P ( - X * H 1 ) * D S I N ( ( R N ' P - H 2 ) * X ) / P GO TO 5 C 3 R=DSORT(P) HD 1 = R- 1 . HD2 = R RH=( 1 .+E»HD1 )**2 + ( E * H D 2 ) ' * 2 H1 = (HD1 + E * ( H D 1 • * 2 + HD2* * 2 ) ) / R H H2=HD2/RH F U N = 0 E X P ( - X 4 H 1 ) * D S I N ( ( R N • P - H 2 ) * X ) / P GO TO 5 C 2 HD1 = ( 4 . / 4 5 . ) * ( P * * 2 ) H D 2 = ( 2 . / 3 . )*P RH=( 1 .+E*HD1 ) * * 2 + ( E * H D 2 ) * ' 2 - 186 -H1=(HD1*E «(HO 1 * «2+HD2 "2) )/RH H2=HD2/RH FUN=DEXP(-X'H1)*DSIN((RN*P-H2)*X)/P GO TO 5 C 1 H1=0. FUN=OEXP(-X*H1)"(RN*X-2./3.*X) 5 RETURN END C C - 187 -Appendix D-2 C THIS PROGRAM IS WRITTEN FOR THE HEAT TRANSFER FIXED BED WITH C SMALL AND HIGH THERMAL CONDUCTIVITY PARTICALS. C NORMANCLATURE C EXPY=EXPERIMNE IMPLICIT R E A L * 8 ( A - H , 0 - Z ) DIMENSION A ( 1 0 0 ) , X ( 1 0 0 ) , E X P Y ( 1 0 0 ) , G Y ( 1 0 0 ) . W W ( 1 0 0 ) . T S ( 1 0 0 ) 01 MENS ION X X ( 2 0 0 ) , X Y ( 2 0 0 ) , S A ( 1 0 0 ) . D A ( 1 0 0 ) . Z Z ( 1 0 0 ) . Y Y ( 100) EXTERNAL FUN DATA E X P Y / 3 5 . 7 . 4 4 . 1 ,64 . 9 , 92 . 6, 133 .3 , 1 3 6 . 8 . 1 0 0 . . 100 . , 1 5 6 . 9 . 7 1 , 1 .85 . 1 , 9 7 . 1 , 1 1 0 . 9 . 12 1 . 6 . 125 .4 . 128 . 7 . 1 3 7 . 6 . 4 7 . 7 . 6 6 . 1 . 8 4 . 3 , 102 .7 . 125 .5 . 134 .8 . 1 4 0 . 8 . 4 5 . 9 . 5 4 . 3 . 8 3 . 7 . 122 .8 . 1 3 1 . 4 / DATA Z Z / 1 2 . . 1 2 . , 1 2 . , 1 2 . . 1 2 . . 1 2 . . 1 2 . , 1 2 . . 1 16 . , 16 . , 16 . . 16 . , 16 . . 16 . . 16 . , 16 . , 1 24 . , 2 4 . , 2 4 . , 2 4 . , 2 4 . , 2 4 . , 2 4 . . 2 4 . . 1 32 . , 3 2 . . 3 2 . , 3 2 . . 3 2 . . 3 2 . . 3 2 . . 3 2 . / DATA X / 4 5 . . 6 5 . . 8 5 . . 1 0 5 . . 1 4 5 . , 2 0 5 . , 2 7 9 . . 4 3 2 . , 1 4 5. , 6 5 . , 8 5 . , 105. . 145. . 2 0 5 . . 2 7 9 . , 4 3 2 . . 1 1035. . 1 2 7 5 . . 1 4 7 4 . . 1594 . . 1 7 1 4 . , 1 9 5 4 . . 2 3 1 4 . . 1 1474 . , 1594 . , 17 14 . , 1954 . . 23 14 . . 2 5 5 4 . / C C NO. OF DATA N C DATA K K , N / 9 . 16/ C C NO. OF UNKOWN PARAMENTERS M C C C IN IT IATE VALUE OF A(M) C DATA A / 4 0 . , . 1 1 , 2 5 2 5 . 8 5 / C C DATA E P S / 0 . 0 0 1 D 0 / C C 88 CALL S U M ( A , X , E X P Y . S . G Y . T S . N . Z Z . K K . V O . G O , D P , T O ) TG0=T0-273 .15 WRITE(6 .799 ) 799 F O R M A T C / / / / / ) WRITE(6 .998) 998 F 0 R M A T ( 1 X , ' A COMPARISON OF THEORETICAL AND EXPERIMENTAL' . 1 ' TEMPERATURE V A L U E S ' . / / , ' OF THE SPHERICAL P A R T I C A L S ' / ) WRITE(6 ,821)DP 821 F O R M A T ( / / / / / / , IX. ' DIA ( M ) = ' . F 7 . 5 / ) WRITEC6.831)TGO 83 1 FORMATC1X. ' TGO ( C ) WRITE(6 ,19 1)G0 = ' . F7 2 / ) 191 FORMATC1X, ' GO ( KG/ (M WRITE(6 .915 )V0 • * 2 ) / S ) = ' . F7 5 / ) 915 FORMAT(1X . ' VO ( M/S ) VAR = S WRITE(6 ,86)VAR ' . F7 5 / ) 86 F O R M A T ( / / , 1 X , 'VAR = ' . W R I T E ( 6 , 7 8 ) A ( 1 ) F 1 2 . 8 . / / ) 78 FORMAT(1X . ' H ( W / ( M * » WRITE(6 , 178)A(2) 2 ) / K ) = ' , F 1 0 6 / ) 178 F O R M A T ( I X . ' K ( W/M/K ) ' , F10 6 / ) - 188 -V R I T E ( 6 , 2 7 8 ) A ( 3 ) 278 FORMA T ( t X ,' ' CP ( J / K G ) = ' . F 1 3 . 6 / / ) C DO 9 KK=1,M C W R I T E ( 6 . 7 8 ) K K . A ( K K ) . K K . S A ( K K ) C78 FORMAT( I X , ' A ( ' , 1 3 . ' ) = ' , F 1 5 . 5 , 5 X , ' D S / D A ' , C 1 '( ' . 1 3 . ' ) = ' . F 1 2 . 8 / ) C9 CONTINUE 00 99 I=KK,N IF( I . E Q . 1 )G0 TO 6 1 I F ( ( Z Z ( I ) - Z Z ( I - 1 ) ) . E Q . O . )G0 TO 71 W W ( I ) = Z Z ( I ) * 2 . 5 4 / 1 0 0 . 61 WRITE(6.81)WW(I ) 81 F O R M A T ( / / / , IX. ' HEIGHT ( M ) = ' . F 6 . 3 . / / ) WRITE(6 ,999) 999 F O R M A T ( / / . 2 X , ' T I M E ' ,4 X, 'EXPERIMENTAL ' , 3X, 'THEORETICAL ' , 3X , 1 ' ( E X P - T H E O ) / E X P ' ) WRITE(6 .982) 982 FORMAT(2X, ' M I N ' . 7 X . ' ( C ) ' . 1 0 X . ' ( C ) ' , / ) 7 1 GY(I ) =GY( I ) - 273 . 15 TS( I )=TS( I ) - 2 7 3 . 15 OEV=(EXPY(I ) -GY( I ) ) /EXPY( I ) X ( I ) = X ( I ) / 6 0 . W R I T E ( 6 , 9 8 ) X ( I ) . E X P Y ( I ) , G Y ( I ) , D E V C W R I T E ( 6 . 9 8 ) X ( I ) , T S ( I ) , G Y ( I ) . D E V 98 F O R M A T ( 1 X . F 6 . 3 . 5 X . F 5 . 1 . 6 X . F 1 0 . 3 . 6 X . F 1 0 . 5 / ) 99 CONTINUE STOP END C C SUBROUTINE S U M ( A . T I I , E X P Y . S . T G , T S . N . Z Z . K K , VO , G O . R L L . T O ) IMPLICIT R E A L * 8 ( A - H . 0 - Z ) EXTERNAL FUN DIMENSION Z( 100 ) ,T I ( 100) .EXPY( 100) .R( 100) .TG( 100 ) .YY(100) DIMENSION RRN( 100) .ZZ( 1 0 0 ) . T S ( 1 0 0 ) .A( 100) ,EXPYY( 100) .T I I( 100) COMMON / B L A / T X EPS=0.001 S = 0 . DO 77 J=KK.N Z ( J ) = Z Z ( U ) * 2 . 5 4 / 1 0 0 . DATA A A . P S . E E . R R / O . 0 3 2 4 2 9 3 . 3 5 2 . 6 6 7 9 7 , 0 . 5 1 8 3 3 . 1 2 0 . / RLL=0.01045 15 SP=0.4875 R L = R L L / 2 . T0=130.5 + 273. 15 TSI=24 . +273 .15 T I ( J ) = T I I ( J ) EXPYY(J )=EXPY(J )+273 . 15 0=122.9 4 409 3 5+13:28 1237558*RR V 0 = 0 / A A / 1 0 O O . / 6 0 . AVT = DS0RT(TO*EXPYY(J ) ) C A1=0.4218802398D01 A2=- .1837520491D-01 A3=0.3453395622D-04 A4=- .23681614130-07 RTSI=35.+273 . 15 P F O = A 1 + A 2 * ( R T S I ) + A 3 * ( R T S I * * 2 ) + A 4 « ( R T S I * * 3 ) GO=VO*PFO - 189 -c B 1=0. 1030338287001 B2 = - . 2001 1901860-03 8 3=0.39710998540-06 CPF = B 1+B2* (AVT)+B3* (AVT**2 ) CPF=CPF * 1000. C V F = V 0 / ( 2 7 3 . 15 + 2 5 . ) « A V T C V F = V 0 / ( 2 7 3 . 15 + 2 5 . ) * T 0 C R X = A ( l ) * ( 1 . - E E ) / ( G 0 * C P F * R L ) * Z ( d ) R X = A ( 1 ) * ( 1 . - E E ) / ( G 0 * C P F ) * Z ( d ) * 3 . / ( R L * S P ) R R N ( d ) = ( T I ( J ) - Z ( d ) / V F ' E E ) * ( A ( 1 ) *3 . / ( A ( 3 ) * P S * R L * S P ) ) 901 F 0 R M A T ( 1 X , ' Y ' , F 1 5 . 5 . ' Z ' . F 1 0 . 5 . ' Y G ' . F 1 5 . 5 . ' T Z ' , F 1 5 . 5 ) TX=RX ABSERR=0.000001 RELERR=0.001 C AP = 0 . BP=RRN(J) CALL Q U A N C 8 ( F U N , A P , B P . A B S E R R . R E L E R R . Y , E R R E S T , N O F U N , F L A G ) C C XX=2 . *DSQRT(RX*RRN(d) ) DTZ=OBESIO(XX. INO ) * D E X P ( - R X - R R N ( J ) ) * ( T O - T S I ) 0 T Z 1 = D T Z / ( T 0 - T S I ) C 10 YG = Y « ( T O - T S I ) YG1=YG/ (T0 -TS I ) W R I T E ( 6 , 9 0 1 ) R X . R R N ( J ) , Y G 1 , D T Z 1 C C C KL = KK I F ( K K . E Q . 1 ) K L = 0 TG(J)=YG+DTZ+TSI TS(J)=YG+TSI S = ( E X P Y Y ( d ) - T G ( d ) ) * * 2 / ( E X P Y Y ( d ) - 2 7 3 . 15 ) / ( F L O A T ( N - K L ) ) + S C WRITE(6 .982 )S C982 FORMAT( IX, ' VAR = ' . F 1 5 . 5 ) EX=RE/RX C W R I T E ( 6 . 7 9 ) N 0 F U N , F L A G , R X . E X 79 FORMA T( 1X, 'NOFUN ' , I 4 , 2 X , ' FLAG ' , F 7 . 3 . ' RX' . F 1 0 . 5 , ' E X ' , F 9 . 5 / ) C W R I T E ( 6 , 9 9 ) R R N ( d ) . Y G . E R R E S T 99 FORMAT( 1X , ' RRN = ' . F 1 0 . 5 . 2 X . ' SUM = ' , F 1 0 . 5 , 2 X . F 1 5 . 8 , / ) C W R I T E ( 6 , 8 7 ) T S , T G 87 FORMAT(1X, ' TS = ' . F 1 5 . 6 . ' TG = ' . F 1 0 . 5 / ) 77 CONTINUE RETURN END C C C SUBROUTINE Q U A N C 8 ( F U N , A , B , A B S E R R , R E L E R R . R E S U L T . E R R E S T . N O F U N , F L A G ) C C A u t o m a t i c a d a p t i v e q u a d r a t u r e r o u t i n e b a s e d on N e w t o n - C o t e s C 8 - p a n e l r u l e . - 190 -C I n p u t : FUN F u n c t i o n FUN(X) to be I n t e g r a t e d C A Lower l i m i t o f i n t e g r a t i o n C B Upper l i m i t o f i n t e g r a t i o n C ABSERR A b s o l u t e e r r o r t o l e r a n c e C RELERR R e l a t i v e e r r o r t o l e r a n c e C O u t p u t : RESULT I n t e g r a t e d v a l u e C ERREST E s t i m a t e d m a g n i t u d e of the a c t u a l e r r o r C NOFUN Number of FUN(X) e v a l u a t i o n s r e q u i r e d C FLAG If FLAG i s z e r o t h e n RESULT s a t i s f i e s l e a s t C s t r i n g e n t e r r o r t o l e r a n c e . If FLAG i s n o n z e r o C but XXX.YYY then XXX i s the number o f n o n c o n v e r g e d C i n t e r v a l s and YYY i s the f r a c t i o n of t o t a l i n t e r v a l C r e m a i n i n g C C See F o r s y t h e . M a l c o l m and M o l e r , p p . 102-105 C IMPLICIT R E A L * 8 ( A - H . O - Z ) DIMENSION OR I G H T ( 3 1 ) . F ( 1 S ) , X ( 1 S ) . F S A V E ( 8 . 3 0 ) . X S A V E ( 8 . 3 0 ) L EVMIN= 1 LEVMAX=30 LEV0UT=6 NOMA X = 5000 NOFIN = N0MAX-8 * (LEVMAX-LEVOUT + 2* * (LEV0UT+1) ) W0=3956.D0/14 175.D0 W1=23552.DO/14175.DO W2=-3712.DO/14175.DO W3 = 4 1984 .DO/14175 .00 W4 = - 18 1 6 0 . 0 0 / 14 175.DO F LAG = 0 .DO RESULT=0.00 C0R11=0.DO ERREST=0.DO AREA =0.DO N0FUN=0 I F ( A . E O . B ) RETURN LEV = 0 NIM= 1 XO = A X( 16)=B 0PREV=O.DO FO=FUN(XO) S T O N E = ( B - A ) / 1 6 . D O X(8 ) = (X0+X( 16) ) / 2 . D 0 X ( 4 ) = ( X O + X ( 8 ) ) / 2 . D 0 X( 12) = (X (8 ) + X( 1 6 ) ) / 2 . D O X ( 2 ) = ( X O + X ( 4 ) ) / 2 . D O X ( 6 ) = ( X ( 4 ) + X ( 8 ) ) / 2 . D 0 X ( 1 0 ) = ( X ( 8 ) + X ( 1 2 ) ) / 2 . D O X( 14 ) = (X( 12 )+X( 16) ) / 2 . D O DO 25 d = 2 , 1 6 . 2 F ( u ) = F U N ( X ( J ) ) 25 CONTINUE N0FUN=9 30 X ( 1 ) - ( X O + X ( 2 ) ) / 2 . D O F ( 1 ) = F U N ( X ( 1 ) ) DO 35 <J = 3 . 1 5 , 2 X ( d ) = ( X ( d - 1 ) + X ( d + 1 ) ) / 2 . D 0 F ( d ) = F U N ( X ( J ) ) 35 CONTINUE N0FUN=N0FUN+8 - 191 -S T E P = ( X ( 1 6 ) - X O ) / I S . D O QLEFT=(WO*(FO+F(8) )+W1*(F(1)+F(7) )+ W 2 * ( F ( 2 ) + F ( 6 ) ) * 1 W 3 - ( F ( 3 ) + F ( 5 ) ) + W 4 « F ( 4 ) ) ' S T E P OR IGHT(LEV*1 ) = (W0*(F(8 ) + F ( 1 6 ) ) + W 1 « ' ( F ( 9 ) + F( 15))+W2"(F( 10)+F(14))+ 2 W 3 * ( F ( 1 1 ) * F ( 1 3 ) )+W4*F( 1 2 )) "STEP QNOW=QLEFT+QRIGHT(LEV+1) ODIFF=ONOW-QPREV AREA=AREA*OIFF ESTERR = DABS(QDIFF ) / 1023.DO TOLERR =DMAX 1 (ABSERR,RELERR*DABS(AREA) ) * STEP/STONE I F ( L E V . L T . L E V M I N ) GO TO 50 I F ( L E V . G E . L E V M A X ) GO TO G2 IF(NO FUN.GT.NOF IN) GO TO 60 I F ( E S T E R R . L E . T O L E R R ) GO TO 70 50 NIM=2*NIM LEV=LEV+ 1 DO 5 2 1=1,8 FSAVE( I . LEV) =F( 1+8) XSAVE( I .LEV)=X(1+8 ) 52 CONTINUE QPREV=QLEFT DO 55 1=1,8 J = - I F (2- -J+18)=F(J+9) X (2 *J+ 18)=X(J+9) 55 CONTINUE GO TO 30 GO N0FIN=2*N0FIN L EVMAX = LEVOUT F L A G = F L A G + ( B - X O ) / ( B - A ) GO TO 70 62 FLAG = FLAG+ 1.DO 70 RESULT=RESULT+QNOW ERREST=ERREST+ESTERR COR 1 1 =COR 1 1 +00 I F F / 1023.DO 72 I F ( N I M . E Q . 2 " ( N I M / 2 ) ) GO TO 75 NIM=NIM/2 LEV = L E V - 1 GO TO 72 75 NIM=NIM+1 I F ( L E V . L E . O ) GO TO 80 OPREV=ORIGHT(LEV) XO=X(16) F0=F(16) DO 78 1=1,8 F ( 2 * I ) = F S A V E ( I . L E V ) X ( 2 * I ) = X S A V E ( I , L E V ) 7 8 CONTINUE GO TO 30 80 RESULT = RESULT + COR 1 1 I F ( E R R E S T . E O . O . O O ) RETURN 82 TEMP=DA8S(RESULTJ+ERREST I F ( T E M P . N E . D A B S ( R E S U L T ) ) RETURN ERREST = 2 .DO* ERRE ST GO TO 82 END C C - 192 -DOUBLE PRECISION FUNCTION FUN(P) IMPLICIT R E A L * 8 ( A - H , 0 - Z ) COMMON / B L A / X XX=2 . *DSQRT(X*P) F U N = D B E S I O ( X X . I N O ) « D E X P ( - X - P ) I F ( I N D . N E . 0 ) G O TO 5 WRITE(6 ,9 ) IND 9 FORMAT(' IND = ' . 1 3 / ) 5 RETURN END C C - 193 -Appendix D-3 C THIS PROGRAM IS WRITTEN FOR THE HEAT TRANSFER FIXED BEO WITH C INTERNAL HEAT RESISTANCE SLAB-SHAPED P A R T I C A L S . C NORMANCLATURE C EXPY«EXPERIMENTAL GAS TEMPERATURE (C) C GY-THORETICAL GAS TEMPERATURE (C) C ZZ*BED HEIGHT IN INCH (IN) C X ' T I M E IN SECONDS (S) C A(1)=HEAT TRANSFER COEFFICENT ( W / M * * 2 / K ) C A ( 2 ) ' T H E R M A L CONDUCTIVITY OF SOLID (W/M/K) C A ( 3 )=HEAT CAPACITY OF SOILD ( K J / K G / K ) C KK=INITIATE POINT C N=FINAL POINT IMPLICIT R E A L * 8 ( A - H , 0 - Z ) DIMENSION A( 100) ,X( 100) .EXPY( 100) .GY(100) .WW( 100 ) .TS( 100) DIMENSION X X ( 2 0 0 ) , X Y ( 2 0 0 ) . S A ( 1 0 O ) . D A ( 1 0 0 ) . Z Z ( 100) ,YY( 10O) EXTERNAL FUN.FUNX 0ATA E X P Y / 5 4 .4 ,67 . 2 , 7 7 . 7 . 8 7 . 6 . 9 8 . 3 , 109 .3 , 1 1 7 . 3 . 1 2 6 . 3 , 1 6 6 . 5 , 7 8 . 7 , 8 5 . 3 . 9 8 . 8 , 105. 1 . 1 1 4 . 7 . 1 2 0 . 3 , 1 2 3 . 9 , 1 6 5 . 9 . 7 9 . 7 . 8 8 . 5 . 9 7 . 7 . 1 0 5 . 7 , 1 1 1 . 2 . 1 1 5 . 2 , 1 2 0 . 6 . 1 5 8 . 3 . 6 9 . 6 . 7 7 . 8 . B9 . 6 . 9 6 . 6 . 1 0 6 . 0 . 1 16 . 2 , 1 2 2 . 2 / DATA Z Z / 1 2 . , 12. , 12. . 12. , 1 2 . . 12. , 12. . 12. . 1 16. , 1 6 . . 1 6 . . 1 6 . , 1 6 . . 1 6 . , 1 6 . , 1 6 . . 1 24 . . 24 . , 24 . , 24 . , 24 . . 24 . . 24 . , 24 . . 1 3 2 . , 3 2 . , 3 2 . , 3 2 . , 3 2 . , 3 2 . , 3 2 . , 3 2 . / DATA X / 7 2 . , 102 . , 122 . , 142 . . 172 . , 2 14 . . 274 . , 474~. , 1 146. . 186. ,2 16 . , 268 . , 308. , 448 . , 663 . , 976 . . 1 2 1 6 . , 2 6 8 . . 3 0 8 . , 3 6 3 . , 448 . , 543 . , 663 . ,976 . , 1 1 6 5 . , 2 17. , 2 5 7 . , 3 1 7 . , 3 5 7 . . 4 3 7 . , 6 1 5 . . 8 5 5 . / C C NO. OF DATA N C DATA K K . N / 9 . 2 4 / C C NO. OF UNKOWN PARAMENTERS M C DATA M / 3 / C C IN IT IATE VALUE OF A(M) C DATA A / 1 5 . 8 0 7 3 . . 0 9 4 . 2 5 1 5 . 8 5 / C C C C 88 CALL S U M ( A , X . E X P Y . S . G Y , T S , N , Z Z . K K . V O , G O . D P , T O ) T G O - T O - 2 7 3 . 15 WRITE(6 .799 ) 799 F O R M A T ( / / / / / ) WRITE(6 .998 ) 998 F 0 R M A T ( 1 X , ' A COMPARISON OF THEORETICAL AND EXPERIMENTAL ' . 1 ' TEMPERATURE V A L U E S ' , / / , ' 2MM WOOD CHIP ' / ) WRITE(6 .821 )DP 821 F O R M A T ( / / / / / / , 1 X , ' THICKNESS ( M ) = ' , F 7 . 5 / ) WRITE(6 .831 )TGO 831 FORMAT( 1X, ' TGO ( C ) = ' . F 7 . 2 / ) WRITE(6 .19 1) 191 FORMAT( 1X , ' REYNOLDS NO. •= 4 0 3 . 1 ' ) VAR-S - 194 -WRITE(S,86)VAR 86 F O R M A T ( / / , 1X, ' VAR = ' , F 1 2 . 8 . / / ) W R I T E ( 6 . 7 8 ) A ( 1 ) 78 FORMAT( 1X, ' H ( d / S / ( M * * 2 ) / K ) = ' . F 8 . 3 / ) WRITE(6 , 178)A(2) 178 FORMAT(1X , ' K ( d / S / M / K ) = ' . F 8 . 4 / ) W R I T E ( 6 . 2 7 8 ) A ( 3 ) 273 FORMAT( 1X, ' CP ( d /KG ) = ' . F 9 . 2 / / ) C 00 9 KK= 1 ,M C W R I T E ( 6 . 7 8 ) K K , A ( K K ) . K K . S A ( K K ) C78 F O R M A T ( I X , ' A ( ' , 1 3 , ' ) = ' ,F 1 5 . 5 . 5 X , ' D S / D A ' , C 1 ' ( ' , 1 3 . ' ) = ' . F 1 2 . 8 / ) C9 CONTINUE 00 99 I=KK,N I F ( I . E O . 1 ) G 0 TO 6 1 I F ( ( Z Z ( I ) - Z Z ( I - 1 ) ) . E 0 . 0 . ) G 0 TO 71 61 WW(I ) = Z Z ( I ) * 2 . 5 4 / 1 0 0 . WRITE(6.81)WW(I) 81 FORMA T ( / / / , 1X, ' HEIGHT ( M ) = ' . F 6 . 3 . / / ) WRITE(6 ,999 ) 999 F O R M A T ( / / , 2 X , ' T I M E ' . 4 X . ' E X P E R I M E N T A L ' , 3 X , ' T H E O R E T I C A L ' , 3 X . 1 ' ( E X P - T H E O ) / E X P ' ) WRITE(6 .982 ) 982 FORMA T ( 2 X , ' M I N ' , 7 X . ' ( C ) ' . 1 0 X , ' ( C ) ' , / ) 7 1 GY(I )=GY(I ) - 2 7 3 . 15 C TS( I ) =TS(I ) - 2 7 3 . 1 5 D E V = ( E X P Y ( I ) - G Y ( I ) ) / E X P Y ( I ) X ( I ) = X ( I ) / 6 0 . W R I T E ( 6 . 9 8 ) X ( I ) , E X P Y ( I ) , G Y ( I ) , D E V C W R I T E ( 6 . 9 8 ) X ( I ) , T S ( I ) . G Y ( I ) . D E V 98 FORMAT( 1X . F6 . 3 , 5 X , F 5 . 1 , 6 X , F 1 0 . 3 . 6 X . F 1 0 . 5 / ) 99 CONTINUE STOP END C c SUBROUTINE S U M ( A , T I I , E X P Y . S , T G , T S . N . Z Z , K K , V O . G O , R L L , T O ) C AA=CROSS SECTION AREA OF THE COLUMN (M**2) C RLL=THICKNESS OF SLAB (M) C PS=SOLID DENSITY ( K G / M * * 3 ) C EE =V0IDAGE OF THE BED C RR = ROTAMETER READING C VO=SUPERFICAL VELOCITY M/S C 0-VOLUME METRIC FLOWRATE (L /MIN) C TSI= INITAL TEMPERATURE (C) C TO=GAS INLET TEMPERATURE (C) C CPF=GAS HEAT CAPACITY ( K d / K G / K ) C PF=GAS DENSITY ( K G / M * « 3 ) C GO=VO*PF C R L / R L L / S . HALF THICKNESS OF SLAB IMPLICIT R E A L * 8 ( A - H . O - Z ) EXTERNAL FUN.FUNX DIMENSION Z ( 1 0 0 ) . T I ( 1 0 0 ) , E X P Y ( 1 0 0 ) , R ( 1 0 0 ) , T G ( 1 0 0 ) . Y Y ( 1 0 0 ) DIMENSION R R N ( 1 0 0 ) , Z Z ( 1 0 0 ) . T S ( 1 0 0 ) . A ( 1 0 0 ) . E X P Y Y ( 1 0 0 ) . T I I ( 1 0 0 ) COMMON / B L A / T R , T E , T X EPS=0.001 S = 0 . DO 77 d=KK,N Z ( d ) = Z Z ( J ) * 2 . 5 4 / 1 0 0 . - 195 -DATA A A , P S . E E , R R / O . 0 3 2 4 2 9 3 . 3 5 2 . 6 6 7 9 7 . 0 . 5 1 8 3 3 , 1 0 0 . / RLL=0.00244 R L = R L L / 2 . T0 =125.0+273. 15 TSI=24 . +273 .15 T I ( J ) = T I I ( J ) . EXPYY(U)=EXPY(J)+273 . 15 0=122.9440935+13.281237558*RR V O = Q / A A / 1 0 0 0 . / 6 0 . AVT = 0 S Q R T ( T 0 * E X P Y Y ( J ) ) C A 1=0.4218802398001 A2 = - . 18375204910-01 A3=0.3453395622D-04 A4 = - . 2 3 6 8 16 14130-07 RTSI=25.+273 . 15 PF0 = A1+A2* (RTSI )+A3* (RTS I * *2)+A4 * ( R T S I « * 3 ) G0=V0'PF0 C 81=0.1030338287001 B2=- .20011901860-03 83=0.39710998540-06 C P F = B t + B 2 * ( A V T ) + B 3 * ( A V T ' * 2 ) CPF = C P F * 1000. C V F = V 0 / ( 2 7 3 . 1 5 + 2 5 . ) * A V T R X = ( 1 . - E E ) / 1 . * Z ( J ) / ( R L * * 2 ) * ( A ( 2 ) / ( G 0 * C P F ) ) RE = A ( 2 ) / ( A ( 1 ) *RL) R R N ( J ) = ( T I ( J ) - Z ( J ) / V F * E E ) / ( A ( 3 ) * P S / A ( 2 ) * ( R L * * 2 ) ) " 2 . / R X C WRIT E ( 6 , 9 0 1 ) T I ( J ) . Z ( L ) . R E . R X 901 FORMAT( 1 X . ' T I ' , F 1 5 . 5 . ' Z ' . F 1 0 . 5 . ' R E ' . F 1 5 . 5 . ' R X ' . F 1 5 . 5 ) AP=0. BP= 1 .0 Y Y( 1) =0 . 00 777 JK=1 .30 TR=RRN(J) TE = RE TX = RX 0S = RE ABSERR=0.000001 RELERR=0.001 C CALL 0 U A N C 8 ( F U N , A P , B P , A B S E R R , R E L E R R , Y . E R R E S T , N O F U N . F L A G ) YY(JK+1)=YY(JK)+Y C WRITE(6 ,67 1 )YY(JK+1) .Y 671 FORMAT( 1X. 'YY(JK+1) = ' . F 1 0 . 5 . 2 X , 'Y = ' . F 1 0 . 6 / ) IF (DABS(Y /YY(JK+1 ) ) . L E . E P S ) G O TO 10 AP = BP BP=BP+15. 777 CONTINUE C 10 YG = YY(JK+ 1 ) / 3 . 14 15927+1 . / 2 . C WRITE(6, 129)JK 129 F O R M A T ( 1 X , ' J K = ' . 1 4 . / ) C AP = 0 . C - 196 -C CALL Q U A N C 8 ( F U N X , A P , B P . A B S E R R , R E L E R R . Y . E R R E S T , N O F U N , F L A G ) C C DY=Y/3 .1415927 C I F ( Y G . G T . 1 . ) YG=1 . KL = KK I F ( K K . E Q . 1 )KL = 0 T G ( J ) = ( T O - T S I )*YG + TSI C T S ( J ) = ( T O - T S I ) « Y G + T S I + ( T O - T S I ) * O Y * D S S = ( E X P Y Y ( d ) - T G ( J ) ) * « 2 / ( E X P Y Y ( J J - 2 7 3 . 1 5 ) / ( F L O A T ( N - K L ) ) + S C WRITE(5 .982 )S C982 FORMAT( 1X. 'VAR=' . F 15 .5 ) EX=RE/RX C W R I T E ( 6 . 7 9 ) N 0 F U N . F L A G . R X . E X 79 FORMAT( 1X , 'NOFUN ' , I 4 , 2 X . ' FLAG ' . F 7 . 3 . ' R X ' . F 1 0 . 5 . ' E X ' . F 9 . 5 / ) C W R I T E ( 6 . 9 9 ) R R N ( 0 ) . Y G . E R R E S T 99 FORMAT(1X , ' RRN = ' . F 1 0 . 5 . 2 X , ' SUM = ' , F 1 0 . 5 . 2 X , F 1 5 . 8 , / ) C W R I T E ( 6 . 8 7 ) T S . T G 87 FORMA T ( 1X , ' TS = ' . F 1 5 . 6 , . ' TG - ' . F 1 0 . 5 / ) 77 CONTINUE RETURN END C C C SUBROUTINE Q U A N C 8 ( F U N , A . B . A B S E R R . R E L E R R , R E S U L T , E R R E S T . N O F U N , F L A G ) C C A u t o m a t i c a d a p t i v e q u a d r a t u r e r o u t i n e b a s e d on N e w t o n - C o t e s C 8 - p a n e l r u l e . C I n p u t : FUN F u n c t i o n FUN(X) to be I n t e g r a t e d C A Lower l i m i t of i n t e g r a t i o n C B Upper l i m i t of i n t e g r a t i o n C ABSERR A b s o l u t e e r r o r t o l e r a n c e C RELERR R e l a t i v e e r r o r t o l e r a n c e C O u t p u t : RESULT I n t e g r a t e d v a l u e C ERREST E s t i m a t e d magn i tude of the a c t u a l e r r o r C NOFUN Number of FUN(X) e v a l u a t i o n s r e q u i r e d C FLAG If FLAG i s z e r o then RESULT s a t i s f i e s l e a s t C s t r i n g e n t e r r o r t o l e r a n c e . If FLAG i s n o n z e r o C but XXX.YYY then XXX 1s the number o f n o n c o n v e r g e d C i n t e r v a l s and YYY Is the f r a c t i o n o f t o t a l i n t e r v a l C r e m a i n i n g C" C See F o r s y t h e . M a l c o l m and M o l e r . p p . 102-105 C IMPLICIT R E A L * 8 ( A - H . O - Z ) DIMENSION OR I G H T ( 3 1 ) , F ( 1 6 ) . X ( 1 6 ) , F S A V E ( 8 . 3 0 ) . X S A V E ( 8 . 3 0 ) LEVMIN=1 LEVMAX=30 LEV0UT=6 NOMAX=5000 N0FIN=N0MAX-8* (LEVMAX-LEV0UT+2** (LEV0UT+1) ) WO=3956.DO/14175.DO W1=23552.00/14175.DO W2=-3712.DO/14175.DO W3=41984.DO/14175.DO W4=-18160 .00 /14175 .00 F LAG = 0 .DO RESULT=O.DO - 197 -C0R11=0.00 ERREST=0.00 AREA=0.00 N0FUN=O I F ( A . E O . B ) RETURN LEV = 0 NIM= 1 X0 = A X( 16)=B QPREV=0.00 F0=FUN(X0) S T 0 N E = ( B - A ) / 1 6 . D 0 X ( 8 ) = ( X 0 + X ( 1 6 ) ) / 2 . D 0 X ( 4 ) = ( X 0 + X ( 8 ) ) / 2 . D 0 X( 12 ) = (X(8 ) + X( 1G) ) / 2 . D O X(2 ) = (X0+X(4) ) / 2 . 0 0 X ( 6 ) = ( X ( 4 ) + X ( 8 ) ) / 2 . D 0 X( 10) = (X(8)+X( 1 2 ) ) / 2 . 0 O X( 14) = (X( 12)+X( 1G) J / 2 . 0 0 DO 25 0 = 2. 16,2 F ( d ) = F U N ( X ( d ) ) 25 CONTINUE N0FUN=9 30 X ( 1 ) = ( X 0 + X ( 2 ) ) / 2 . D 0 F( 1)=FUN(X(1) ) DO 35 d = 3, 15.2 X(U) = ( X ( J - 1 ) + X ( J + 1 ) ) / 2 . 0 0 F ( J ) = FUN(X(J ) ) 35 CONTINUE N0FUN=N0FUN+8 S T E P = ( X ( 1 6 ) - X 0 ) / 1 6 . D 0 QLEFT=(W0*(F0+F(8 ) )+W1 * ( F ( 1 ) + F ( 7 ) )+W2*(F(2)+F(6) ) + 1 W 3 * ( F ( 3 ) + F ( 5 ) ) + W 4 * F ( 4 ) ) * S T E P QRIGHT(LEV+1) = (WO*(F(8)+F( 16))+W1 * ( F ( 9 )+F( 1 5))+W2*(F( 10 )+ F( 14)) + 2 W 3 * ( F ( 1 1 ) + F ( 1 3 ) ) + W 4 * F ( 1 2 ) ) * S T E P QNOW = QLEFT + QRIGHT(LEV+ t ) OOIFF=ONOW-OPREV AREA=AREA+DIFF ESTERR=OABS(OOIFF) /1023 .DO TOLE RR =DMAX1(ABSERR,RELERR*DABS(AREA ) ) * STEP/STONE I F ( L E V . L T . L E V M I N ) GO TO 50 I F ( L E V . G E . L E V M A X ) GO TO 62 IF (NOFUN.GT.NOFIN) GO TO 60 I F ( E S T E R R . L E . T O L E R R ) GO TO 70 50 NIM=2*NIM LEV=LEV+1 DO 52 1=1.8 F S A V E ( I , L E V ) = F(1 + 8) XSAVE( I .LEV)=X(1+8 ) 52 CONTINUE QPREV=QLEFT DO 55 1=1.8 d=-I F (2*d+18)=F(J+9) X(2*J+18)=X(J+9) 55 CONTINUE GO TO 30 60 N0FIN=2*N0FIN LEVMAX = LEVOUT F L A G = F L A G + ( B - X O ) / ( B - A ) - 198 -GO TO 70 G2 FLAG=FLAG+1.00 70 RESULT=RESULT+QNOW ERREST=ERREST+ESTERR C0R1 1=C0R1 1+00 IFF /102 3 .00 72 I F ( N I M . E 0 . 2 * ( N I M / 2 ) ) GO TO 75 NIM=NIM/2 LEV=LEV- I GO TO 72 75 NIM=NIM+1 I F ( L E V . L E . O ) GO TO 80 QPREV=QRIGHT(LEV) X0=X(16) F0=F(16) DO 78 1=1.8 F ( 2 * I ) = F S A V E ( I . L E V ) X (2 * I )=XSAVE( I . L E V ) 78 CONTINUE GO TO 30 80 RESULT = RESULT + COR 1 1 I F ( E R R E S T . E O . O . D O ) RETURN 82 TEMP=DA8S(RESULT)*ERREST I F ( T E M P . N E . D A B S ( R E S U L T ) ) RETURN ERREST = 2 . 0 0 * ERR E ST GO TO 82 END C C DOUBLE PRECISION FUNCTION FUN(P) IMPLICIT R E A L * 8 ( A - H . O - Z ) COMMON / B L A / R N , E , X I F ( P . E O . O . ) G 0 TO 1 R=DSORT(P) HD1 = ( D S I N H ( 2 . * R ) - 0 S I N ( 2 . * R ) ) / ( D C O S H ( 2 . * R ) - D C O S ( 2 . *R) ) HD2 = (DSINH(2 . *R ) + 0 S I N ( 2 . * R ) ) / ( D C 0 S H ( 2 . * R ) - O C O S ( 2 . * R ) ) R H = ( 2 . * R * E + H D 1 ) « * 2 + ( H D 2 ) * * 2 H 1 = ( 2 . * R ) « ( 2 . * R * E + H D 1 ) / R H H2 = 2 . * R"HD2/RH FUN = OEXP( -X*H1 ) * D S I N ( ( R N * P - H 2 ) * X ) / P GO TO 5 C C 1 H1=0. F U N = ( R N * X - 2 . « X ) 5 RETURN - ENO C C DOUBLE PRECISION FUNCTION FUNX(P) IMPLICIT R E A L * 8 ( A - H . 0 - Z ) COMMON / B L A / R N . E . X I F f P . E Q . O . ) G 0 TO 1 R=DSORT(P) H 0 1 = ( D S I N H ( 2 . * R ) - D S I N ( 2 . • R ) ) / ( D C O S H ( 2 . * R ) - D C 0 S ( 2 . * R ) ) H D 2 = ( D S I N H ( 2 . * R ) + D S I N ( 2 . * R ) ) / ( D C 0 S H ( 2 . * R ) - D C 0 S ( 2 . * R ) ) RH=(2 . *R*E+HD1)* * 2+(HD2)**2 H 1 = ( 2 . * R ) * ( 2 . * R * E + H D 1 ) / R H - 199 -H2 = 2 . *R'HU2/RH F1=DEXP( -X*H1)«DSIN( (RN*P-H2) *X) /P F2=DEXP(-X*H1 )*OCOS((RN*P-H2)*X)/P FUNX=-H1*F1-H2*F2 GO TO 5 C C 1 FUNX=-2. 5 RETURN END - 200 -Appendix D-4 C THIS PROGRAM IS WRITTEN FOR FINDING THE HEAT TRANSFER COEFFICENT C OF WOOD CHIP P A R T I C A L . ( R E F E R TO UBC COMPUTER MANNAL TO USE THIS C PROGRAM ) C NORMANCLATURE C A(1)=EXPERIMENTAL TEMPERATURE (C) C A(2)=BED HEIGHT (IN) C A(3)=TIME IN SECONDS C P=INITIAL HEAT TRANSFER COEFF EI CENT ( W / M « « 2 / K ) IMPLICIT R E A L * 8 ( A - H , 0 - Z ) DIMENSION A ( 2 0 0 . 3 ) , P ( 1 ) ,PLB( 1) .PUB( 1 ) ,PP( 1) EXTERNAL FUN.FUNX.AUX OATA M . N . N V / 1 , 3 , 8 / DATA L O G . N C O N , I T / 1 , 0 , 9 / DATA EPS 1 . E P S 2 / 0 . 0 1 D 0 . . 00100 / OATA P / 5 . / DATA P L B / O . O O / DATA P U B / 9 0 0 0 . D O / DATA A / 9 0 . 9 , 9 4 . . 100 .9 , 104 .3 , 1 0 S . 9 . 1 1 0 . 7 , 112 .8 , 114 .7 , 1 6 8 . 1 . 8 0 . 3 , 8 8 . 9 , 9 3 . 1 , 9 9 . 5 , 1 0 8 . 3 . 11 1 . 3 . 113 .0 , 1 8 4 - 0 . 0 . 1 1 6 . , 1 6 . , 1 6 . . 1 6 . . 1 6 . . 1 6 . , 1 6 . . 1 6 . . 1 2 4 . . 2 4 . . 2 4 . . 2 4 . , 2 4 . . 2 4 . , 2 4 . . 2 4 . , 1 8 4 * 0 . 0 , 1 154. , 184. . 2 6 6 . . 326 . , 386 . .52 1 . .64 1 . , 88 1 . , 1 . 5 6 7 , 1 . 0 6 7 . 2 . 0 6 7 . 3 . 0 6 7 , 5 . 2 6 7 . 10 .517 . 14 .517 , 18 .517 , 1 8 4 * 0 . 0 / 00 7 I=1,N A ( I , 1 ) =A(I , 1 ) + 273. 15 A ( I , 2 ) = A ( I , 2 ) * 2 . 5 4 0 0 / 1 0 0 . 7 CONTINUE C C L=1 C LS= 1 C CALL S U M ( M . A . P P . F F . L , L S ) CALL C L Q F ( M . N , N V . L 0 G , A , P . P L B , P U B . E P S 1 , E P S 2 , IT, I ERR,NCON,AUX) C I F ( I E R R . N E . 0 ) WRITE(6 .10) I ERR 10 F 0 R M A T ( 1 0 X , ' SEARCH FAILED ' . 1 4 , / / ) STOP END C C SUBROUTINE A U X ( M . A . P , F , D , I N D . L . L S ) IMPLICIT R E A L ' S ( A - H . O - Z ) EXTERNAL FUN.FUNX 01 MENS I ON A( 200.M) , P ( M ) , D ( M ) , P P ( 1 ) , D P P ( 1 ) DATA D P P / O . 0 0 0 0 0 1 0 0 / I F ( I N D . E O . 3 ) R E T U R N PP( 1)=P( 1 ) CALL SUM ( M . A . P P . F F . L . L S ) FFF=FF F = F F F - A ( L . 1 ) I F ( I N D . E O . 1 )RETURN I F ( P ( 1 ) . E O . O . O D O ) GO TO 20 PP( 1)=P(1)+DPP( 1 ) CALL S U M ( M A . A . P P . F F . L . L S ) FF 1 =FF D( 1 ) = (FF 1-FFF ) /DPP( 1) RETURN 20 LS=2 RETURN - 201 -END C C SUBROUTINE L I M I T ( P . D U B . I N D . L . C ) IMPLICIT R E A L * 8 ( A - H . O - Z ) DIMENSION P( 1 ) ,DUB( 1) RETURN END SUBROUTINE S U M ( M . A , P P , F F , L , L S ) C AA=CROSS SECTION AREA OF THE COLUME ( M * « 2 ) C RLL=THICKNESS OF SLAB (M) C PS=SOLID DENSITY ( K G / M « « 3 ) C EE =VOIDAGE OF THE BED C RR'ROTAMETER READING C VO'SUPERFICAL VELOCITY M/S C Q=VOLUME METRIC FLOWRATE (L /M IN) C TSI= INIT IAL TEMPERATURE (C) C TO=GAS INLET TEMPERATURE (C) C CPF=GAS HEAT CAPACITY ( K O / K G / K ) C PF=GAS DENSITY ( K G / M - - 3 ) C GO=VO*PF C RL =HALF THICKNESS OF SLAB IMPLICIT R E A L - 8 ( A - H . O - Z ) EXTERNAL FUN.FUNX OIMENSION A ( 2 0 0 , 3 ) , P P ( 1 ) . Y Y ( 1 0 0 ) COMMON / B L A / T R , T E , T X DATA P K . P C P / . 0 9 4 . 2 5 1 5 . 8 5 / DATA R L L / O . 0 0 7 2 6 / OATA A A , P S . E E . R R / O . 0 3 2 4 2 9 3 , 3 5 2 . 6 6 7 9 7 . 0 . 5 1 2 7 4 . 8 0 . / R L = R L L / 2 . 0 0 T0 = 1 16.5 + 273 .15 TSI=24 .+273 .15 0 = 1 2 2 . 9 4 4 0 9 3 5 + 1 3 . 2 8 1 2 3 7 5 5 8 « R R 0=0 -1 .3 V O = 0 / A A / 1 0 0 0 . / 6 0 . AVT=OSQRT(TO*A(L ,1 ) ) C RTSI=25.+273.15 A1=0.4218802398D01 A2 = - . 18375204910-01 A3=0.3453395622D-04 A4 = - . 236816 14130-07 PF = A 1 + A 2 « ( A V T ) + A 3 * ( A V T * * 2) + A4* ( A V T • • 3 ) P F 0 = A 1 + A 2 * ( R T S I ) + A 3 * ( R T S I * * 2 ) + A 4 - ( R T S I * - 3 ) GO=VO-PFO C B 1=0. 103O338287DO1 B2=- .2001190186D-03 B3=0.3971099B54D-06 C P F = B 1 + B 2 - ( A V T ) + B 3 * ( A V T » * 2 ) CPF=CPF*1000 . C GO TO 11 V F = V 0 / ( 2 7 3 . 15 + 25. )*AVT R X = ( 1 . - E E ) / 1 . * A ( L . 2 ) / ( R L - * 2 ) * ( P K / ( G O * C P F ) ) RE = P K / ( P P ( 1 ) *RL) R R N = ( A ( L , 3 ) - A ( L . 2 ) / V F * E E ) / ( P C P - P S / P K • ( R L • • 2 ) ) - 2 . / R X C W R I T E ( 6 . 9 0 1 ) T I ( J ) , Z ( L ) . R E . R X 901 F O R M A T ( 1 X . ' T I ' , F 1 5 . 5 . ' Z ' . F 1 0 . 5 . ' R E ' , F 1 5 . 5 . ' R X ' . F 1 5 . 5 ) EPS=0.001 - 202 -AP=0. BP=1.0 YY(1 )=0 . DO 777 <JK = 1 ,30 TR=RRN TE = RE DS = RE TX = RX ABSERR=0.000001 RELERR=0.0001 CALL 0 U A N C 8 ( F U N , A P , B P , A B S E R R , R E L E R R , Y . E R R E S T , N O F U N , F L A G ) YY(JK+1 )=YY(<JK)+Y I F ( O A B S ( Y / Y Y ( J K + 1 ) ) . L E . E P S ) G 0 TO 10 AP = BP BP=BP+ 15. 777 CONTINUE C 10 YG = YY(JK +1 ) / 3 . 14 15927+1 ./1. C AP=0. C C CALL O U A N C 8 ( F U N X . A P , B P , A B S E R R , R E L E R R , Y . E R R E S T , N O F U N , F L A G ) C C DY = Y / 3 . 14 15927 C I F ( Y G . G T . 1. ) YG=1 . TG=(TO-TSI ) *YG+TSI C TS=(TO-TS I )*YG+TSI + ( T O - T S I ) * D Y * D S FF=TG C W R I T E ( 6 , 9 8 2 ) S C982 FORMAT( 1X, ' V A R ' ' .F 15.5) EX=RE/RX C W R I T E ( 6 , 7 9 ) N 0 F U N . F L A G . R X , E X 79 FORMAT( 1X. 'NOFUN ' . I 4 . 2 X , ' FLAG ' . F 7 . 3 , ' R X ' , F 1 0 . 5 . 1X. ' E X ' ,F 1 0 . 5 / ) C WRITE(6 .99 )RRN.YG.ERREST 99 F ORMAT( 1X, ' RRN = ' . F 1 0 . 5 . 2 X . ' SUM = ' . F 1 0 . 5 , 2 X . F 1 5 . 8 . / ) C W R I T E ( 6 , 8 7 ) T S . T G 87 F O R M A T ( I X , ' TS = ' . F 1 5 . 6 , ' TG = ' . F 1 0 . 5 / ) RETURN END C C C C C C SUBROUTINE 0 U A N C 8 ( F U N , A , B , A B S E R R , R E L E R R , R E S U L T , E R R E S T , N O F U N . F L A G ) C C A u t o m a t i c a d a p t i v e q u a d r a t u r e r o u t i n e b a s e d on N e w t o n - C o t e s C 8 - p a n e l r u l e . C I n p u t : FUN F u n c t i o n FUN(X) to be I n t e g r a t e d C A Lower l i m i t of i n t e g r a t i o n C B Upper l i m i t of I n t e g r a t i o n C ABSERR A b s o l u t e e r r o r t o l e r a n c e C RELERR R e l a t i v e e r r o r t o l e r a n c e C O u t p u t : RESULT I n t e g r a t e d v a l u e C ERREST E s t i m a t e d magn i tude o f the a c t u a l e r r o r C NOFUN Number of FUN(X) e v a l u a t i o n s r e q u i r e d - 203 -C FLAG If FLAG i s z e r o t h e n RESULT s a t i s f i e s l e a s t C s t r i n g e n t e r r o r t o l e r a n c e . If FLAG i s n o n z e r o C but XXX.YYY then XXX i s the number of n o n c o n v e r g e d C i n t e r v a l s and YYY Is the f r a c t i o n of t o t a l i n t e r v a l C r e m a i n i n g C C See F o r s y t h e , M a l c o l m and M o l e r , p p . 102-105 C IMPLICIT REAL-a(A-H.a-Z) DIMENSION QRIGHT(3 1 ) . F ( 16 ) .X( 1 6 ) . F S A V E ( 8 . 3 0 ) . X S A V E ( 8 . 3 0 ) LEVMIN=1 LEVMAX=30 LEV0UT=6 NOMAX=5000 N O F I N = N O M A X - 8 « ( L E V M A X - L E V O U T + 2 * * ( L E V O U T + 1 ) ) WO = 3956 . D O / 1 4 1 75 . DO W1=23552.00/14175.DO W2=-37 1 2 . 0 0 / 14175.DO W3 = 4 1984.00/14 175.DO W4 = - 18160 .00 / 14175.DO FLAG=0.DO RESULT=0.D0 COR 11=0.DO ERREST =0.DO AR EA =0.DO NOFUN=0 I F ( A . E O . B ) RETURN LEV=0 NIM=1 XO = A X( 16)=B 0PREV=O.DO FO=FUN(XO) STONE = ( S - A ) / 1 6 . D O X(8 ) = (X0+X( 16) ) / 2 . D 0 X(4 ) = (X0+X(8 ) ) / 2 . D O X( 12) = (X(8)+X( 1 S ) ) / 2 . D 0 X ( 2 ) = ( X 0 + X ( 4 ) ) / 2 . 0 0 X(6 ) = (X (4 )+X(8 ) ) / 2 . 0 0 X( 10) = (X(8 ) + X( 12) ) / 2 . 0 0 X( 14 ) = (X( 12 )+X( 16) ) / 2 . 0 0 00 25 J = 2. 16.2 F ( J ) = F U N ( X ( J ) ) 25 CONTINUE N0FUN=9 30 X( 1 ) = ( X 0 + X ( 2 ) ) / 2 . 0 0 F( 1)=FUN(X( 1 ) ) DO 35 J = 3. 15,2 X ( J ) = (X (0 -1 )+X(J+ 1 ) ) /2 .DO F ( J ) = FUN(X(J ) ) 35 CONTINUE N0FUN=N0FUN+8 STEP = (X( 1 6 ) - X 0 ) / 1 6 . D 0 0LEFT=(WO*(FO+F(8 ) )+W1* (F (1 )+F(7 ) )+W2* (F (2 )+F(6 ) )+ 1 W 3 * ( F ( 3 ) + F ( 5 ) ) + W 4 * F ( 4 ) ) * S T E P 0RIGHT(LEV+1) = (W0*(F(8 ) + F (16 ) )+W1* (F (9 ) + F (15 ) )+W2*(F( 10)+F( 14)) + 2 W 3 * ( F ( 1 1 ) * F ( 13))+W4*F( 12) ) *STEP ONOW=OLEFT+ORIGHT(LEV+1) ODIFF=ONOW-OPREV AREA=AREA+DIFF - 204 -E5TERR=DABS(ODIFF ) /1023 .DO TOLE RR =DMAX 1 ( A B S E R R . R E L E R R * 0 A B S ( A R E A ) ) * STEP/STONE I F ( L E V . L T . L E V M I N ) GO TO 50 I F ( L E V . G E . L E V M A X ) GO TO G2 IF (NOFUN.GT.NOFIN) GO TO 60 I F ( E S T E R R . L E .TOLERR) GO TO 70 50 NIM=2*NIM LEV=LEV+ 1 DO 52 1=1,8 FSAVE( I . L E V ) = F( 1+8 ) XSAVE( I ,LEV)=X( 1+8) . 52 CONTINUE QPREV=QLEFT DO 55 1=1,8 J = -I F ( 2 * J + 18)=F(J+9) X(2*d+ 18) = X ( J + 9) 55 CONTINUE GO TO 30 60 N0FIN=2*N0FIN L EVMA X = LEVOUT F L A G = F L A G + ( B - X O ) / ( B - A ) GO TO 70 62 FLAG = FLAG+ 1 .00 70 RESULT=RESULT+QNOW ERREST=ERREST+ESTERR COR 1 1 =COR 1 1+ODIFF/1023.DO 72 I F ( N I M . E Q . 2 * ( N I M / 2 ) ) GO TO 75 NIM=NIM/2 LEV=LEV- 1 GO TO 72 75 NIM=NIM+1 I F ( L E V . L E . O ) GO TO 80 OPREV=ORIGHT(LEV) XO=X(16) FO=F(16) DO 78 1=1,8 F ( 2 * I ) = F S A V E ( I . L E V ) X ( 2 * I ) = X S A V E ( I , L E V ) 78 CONTINUE GO TO 30 80 RESULT = RESULT+COR 1 1 I F ( E R R E S T . E O . O . D O ) RETURN 82 TEMP=DABS(RESULT)+ERREST I F ( T E M P . N E . O A B S ( R E S U L T ) ) RETURN ERREST = 2 .DO* ERRE ST GO TO 82 END C C DOUBLE PRECISION FUNCTION FUN(P) IMPLICIT R E A L * 8 ( A - H . 0 - 2 ) COMMON / B L A / R N , E , X I F ( P . E O . O . ) G 0 TO 1 R=DSQRT(P) H D 1 = ( D S I N H ( 2 . * R ) - 0 S I N ( 2 . * R ) ) / ( D C 0 S H ( 2 . * R ) - D C 0 S ( 2 . * R ) ) H D 2 = ( D S I N H ( 2 . * R ) + D S I N ( 2 . * R ) ) / ( D C O S H ( 2 . * R ) - D C 0 S ( 2 . * R ) ) RH=(2 . *R* E + HD 1 ) * *2+(HD2)**2 - 205 -H1=(2.*R)*(2. *R*E+HO 1 )/RH H2 = 2 . *R*HD2/RH FUN=OEXP(-X*H1)*OSIN((RN-P-H2)*X)/P GO TO 5 C C 1 H1=0. FUN=(RN-X-2.*X) 5 RETURN END C C DOUBLE PRECISION FUNCTION FUNX(P) IMPLICIT REAL*8(A-H.O-Z) COMMON /BLA/RN,E,X IF(P.EO.O.)GO TO 1 R=DSORT(P) HD1 = (DSINH(2.*R)-OSIN(2.*R))/(DCOSH(2 . *R )-DCOS(2.*R)) H02 = (0S INH(2 .«R)+OSIN(2 . *R) ) / (OCOSH(2 . *R) -OCOS( 2 . "R)) RH=(2 . *R«E+HD1) * *2+ (HD2) * *2 H1 = ( 2 . « R ) * ( 2 . * R * E+HD1)/RH H2 = 2. *R *HD2/RH F1=OEXP(-X*H1)*DSIN((RN*P-H2)*X)/P F2=0EXP(-X«H1)«DCOS((RN*P-H2)*X) /P FUNX=-H1*r1-H2*F2 GO TO 5 C C 1 FUNX=-2. 5 RETURN END 

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