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A Numerical Modelling Study of Transport Phenomena in
Wood Drying
by
A h m a d H a sh em i E sfah an ian
B.A., Isfahan U niversity of Technology. 1987 M.Sc.. Isfahan U niversity of Technology, 1990
A S ubm itted in P artial Fulfillment of the Requirem ents for the Degree of Do c t o r o f P h i l o s o p h y
in the
D epartm ent of Mechanical Engineering.
We accept this thesis as conforming to the required standard
[Dr. B. Tabarrok). Co-supervisor (D ept, of Mechanical Engineering)
Dr. S. Dost. Co-supervisor (D ept, of Mechanical Engineering)
r. Z. D o n ^ )M ember (D ept, of M echanical Engineering)
M em berY D ept. of Mechanical Engineering)
’6kfS*Member (School of E arth and Ocean Sciences)
iz Oliveira, E xternal E xam iner (Forintek Canada)
© Ah m a d Ha s h e m i Es f a h a n i a n, 1999 U niversity of V ictoria
All rights reserved. This thesis m ay not be reproduced in whole or in p a rt, by photocopy or other m eans, w ithout th e permission of the au th o r.
Supervisors: [Dr. B. Tabarrok] and Dr. S. Dost
A bstract
This thesis presents a num erical sim ulation study for th e heat and mass transfer in- and out-side individual boards of a stack during kiln drying of wood and on th e effect of side gaps betw een th e boards. The objective is to optim ize th e drying process for efhciency and high qu ality products.
\ literatu re survey in th e area is presented. The im portance of th e correct link
between the tra n sp o rt processes in wood, and h eat/ mass transfer and fluid flow in the surrounding drying air is emphasized. O bjectives, m otivations and needs for the present study are also presented. This is followed by a detailed account of the governing equations, description of models, physical properties, and discretization and solution procedures used in the present study.
.\ sam ple stack of planks haa been used to evaluate the perform ance of various tu r bulence models and upw inding schemes of the C FX software developed for predicting the transpo rt p aram eters in air. Given a typical stack set up for drying 105 x 105 mm western hemlock lum ber, th e effects of side gaps on surface coefficients are studied for different air velocities. An optim um gap size for m axim um h e at/m ass transfer is suggested.
T he model developed for heat and mass transfer inside th e wood is validated for a one dim ensional case by com paring the num erical results w ith published results. T he im proved perform ance using a newly proposed relationship for the diffusivity of bound w ater has been dem onstrated. Also a new relationship for th e mass transfer boundary condition a t th e surfaces was proposed to incorporate th e effect of the surface resistance. R elative effects of model unknowns in predicting th e average m oisture content and board center tem peratures are discussed.
I ll
A Fortran program was developed to solve the two-dimensional coupled heat and mass transfer equations inside th e wood during the drying process. T h e model con siders the changes in air tem p eratu re and hum idity due to heat and m ass transfer to and from the boards. T he iterative SOR (Successive Over R elaxation) m ethod was modified to increase accuracy and stability. Predictions for average and local moisure content are in good agreem ent with experim ents. T he effect of side gap on the drying process inside the wood was also exam ined. A plot of stan d ard deviation of each board versus the board average m oisture content is suggested for the judgem ent about the uniform ity of the products. Results indicate th a t w ithout using the ex tra gap size there exists a considerable difference between the m axim um and m inim um final average m oisture content of the boards in each row. T he first and last boards are usually over-dried. By using the previously suggested gap size the m axim um dif ference of the final average m oisture content is almost half the case w ithout the extra gap.
.Average diffusion and surface coefficients are extracted from th e experim ental d a ta of drying a stack w ithout side gap. A software tool has been developed to solve the simple unsteady one-dim ensional diffusion problem. Results are com pared with experim ents. The introduced m ethod can be used to obtain the average diffusion and surface coefficients.
IV
Examiners:
[Dr. B. Tabarrok], Co-supervisor (Dept, of M echanical Engineering)
Dr. S. Dost, Co-supervisor (D ept, of Mechanical Engineering)
Dr. Z. Dong, M ember (D ept, of M echanical Engineering)
Dr. M. D. iber (D ept, of Mechanical Engineering)
D f T ^ T ^ h ^ k M ember (School of E arth and Ocean Sciences)
1
Table o f C ontents
A bstract
ii
List of Tables
viii
List of Figures
ix
N om enclature
1
1
Introduction
9
2
Literature review
15
2.1 G eneral m o d e l s ... 15 2.2 N um erical studies . ... IT 2.2.1 .A p p ro a c h e s ... IT 2.2.2 I n v e s tig a tio n s ... IT 2.3 E xperim ental studies, surface c o e ffic ie n ts ... 26 2.3.1 In v e s tig a tio n s ... 2T3 O bjectives and m otivations
30
3.1 P ractical a p p lic a tio n s ... 33
4
G overning equations
36
4.1 H eat and mass transfer equations in the solid ... 3T 4.1.1 M ovement of w ater v a p o u r ... 39 4.1.2 M ovement of bound w a t e r ... 42 4.1.3 M ovement of free w a te r... 4.5 4.1.4 G eneral form of th e equations for internal n o d e s ... 46 4.1.5 B oundary c o n d itio n s ... 4T 4.1.6 Physical p r o p e r t i e s ... 49 4.2 Governing equations in the fluid p h a s e ... 50 4.2.1 Turbulence m o d e llin g ... 52
T A B L E O F C O N T E N T S vi
4.2.2 D iscretization p ro c e ss ... 57
5 N um erical sim ulation of air flow in stack
60
5.1 Bench m ark case (K ho’s g e o m e t r y ) ... 605.1.1 Solution m ethod ... 61
5.1.2 Effect of th e inlet and ou tlet locations... 62
5.1.3 Mesh arrangem ent ... 63
5.1.4 Prelim inary c o m p u ta tio n s ... 65
5.1.5 Results and d isc u ssio n s... 67
5.2 N um erical com putations of surface coefficients over a com m ercial stack plank geom etry (Li’s c a s e ) ... 72
5.2.1 Prelim inary c o m p u ta tio n s ... 74
5.2.2 Effect of irregularities due to shrinkage and sawing ... 75
5.2.3 Effect of different gap s iz e s ... 77
5.2.4 Effect of Re num ber ... 88
5.2.5 Effect of inlet and wall tem p eratu re/m o istu re c o n t e n t ... 96
5.2.6 Sim ilarities a t different Re n u m b e r s ... 99
6 One dim ensional numerical sim ulation...
102
6.1 In tr o d u c tio n ... 102
6.2 One dim ensional m o d e llin g ... 103
6.2.1 Internal n o d e s ... 105
6.2.2 Surface n o d e ... 107
6.2.3 Node on axis of s y m m e t r y ... 109
6.2.4 ID solution p r o c e d u r e ... 110
6.2.5 B enchm ark c a s e ... 110
6.2.6 Further m odel v e r if ic a tio n ... 115
6.2.7 Sensitivity stud y of th e model unknown p a r a m e te r s ... 116
7 Two dim ensional sim ulation of kiln drying
127
7.1 D iscretization for internal n o d e s ... 1317.2 T reatm en t of boundary n o d e s ... 134
7.3 Changes in drying air condition inside th e stack due to heat and mass transfer ... 137
7.4 Solution procedure for th e two dim ensional c a s e ... 139
7.4.1 Chebyshev acceleration version of SOR m e t h o d ... 139
7.5 Process of obtaining th e model u n k n o w n s... 141
7.6 Validation of th e m o d e l... 144
7.7 Effects of th e optim um gap size on drying c u rv e s... 148
T A B L E OF C O N T E N T S vii
8 Sim ple diffusion case
166
9 Conclusions
171
9.1 Future w o r k ... 175
A D ry wood densities
184
B Com puter program for two-dim ensional drying
186
C C om puter program for one-dim ensional diffusion
226
vm
List o f Tables
5.1 Results of length of separation bubbles and other param eters for Re=S200 67 5.2 C om parison of the results of to tal heat transfer to 7 boards for different
cases a t =4400 (heat transfer rates are for unit length of lum ber) 86 5.3 Com parison of the results of total tim e of drying for different cases at
/2e£)^=4400 ... 88
5.4 Com parison of the results of total heat transfer to 7 boards for different cases at /?eD^=8800 (heat transfer rates are for unit length of lum ber) 95 5.5 Com parison of the results of to tal heat transfer to 7 boards for different
cases a t =13200 (heat transfer rates are for unit length of lum ber) 96
7.1 Drying air schedule used in present work (Li[52]) 141
A .l Wood dry density used for boards of a row in Run No.7 of Li =4400. g=1.5 m m ... 184 A.2 Wood dry density used for boards of a row in Run No. 14 of Li /?e£)^=4400,
g=21.5 m m ... 185 A 3 Wood dry density used for boards of a row in Run N o.10 of Li /?eo^=8800.
g=1.5 m m ... 185 .A.4 Wood dry density used for boards of a row in Run N o.13 of Li Æeo^=8800.
IX
List o f Figures
5.1 Configuration of stack plank investigated by Kho t t al [ 2 9 ] ... 61 5.2 Typical velocity vectors near the leading edge at Reo^ — 8200 . . . . 62 5.3 Typical velocity vectors after the trailing edge at Reo^ = 8200 . . . . 63 5.4 Schem atic view of a typical grid array around the leading edge . . . . 64 5.5 Schem atic view of a typical grid array around the trailing edge . . . . 66 5.6 Dimensionless u velocity at the top of the boards after the leading edge
for th e Kho’s geom etry at Re=8200 68
5.7 Dimensionless u velocity at the top of the boards (D = b o ard s height and w idth = 1 0 0 m m ) ... 68 5.8 Dimensionless u velocity at the top of the boards ... 69 5.9 Dimensionless u velocity at the top of the boards ... TO
5.10 Dimensionless u velocity ID ( = 100mm) after the trailing edge . . . . 71 5.11 Dimensionless u velocity along the centre line after the trailing edge . 71 5.12 Local dimension-less wall shear stress after the leading edge for the
Kho’s geom etry at Re=8200 ... 72 5.13 Local Nu num ber after the leading edge for the K ho’s geom etry at
Re=8200 " . . . 73
5.14 LocaJ Nu num ber after the leading edge for the K ho’s geom etry at R e= 13600 ... 73 5.15 Local Nu num ber after the leading edge for the K ho’s geom etry at
R e = 1 9 1 0 0 " . . . 74
5.16 Schem atic view of a typical grid array around the side g a p ... 75 5.17 C om parison of th e distribution of local Nu num ber for the Li’s geom
etry having vertical side gap (due to shrinkage and sawing) w ith the
case w ithout side gap a t Re=4400 76
5.18 Com parison of th e distrib u tio n of local Nu num ber for the Li's geom etry w ith different boards heights a t Re=4400 ... 77 5.19 C om parison of th e distrib u tio n of local Nu num ber for the Li's geom
L I S T OF F IG U R E S x
5.20 Local Nu num ber on front wall of the first board for th e Li’s geom etry with different side gaps at Re=4400 ... 80 5.21 Local Nu num ber on front wall of the second board for th e Li’s geom
etry w ith different side gaps a t Re=4400 ... 80 5.22 Local Nu num ber on front wall of the third board for th e Li’s geom etry
w ith different side gaps at Re=4400 ... 80 5.23 Local Nu num ber on front wall of th e fourth board for the Li’s geom etry
w ith different side gaps at Re=4400 ... 80 5.24 Local Nu num ber on front wall of the fifth board for the Li’s geom etry
with different side gaps at Re=4400 ... 81 5.25 Local Nu num ber on front wall of the sixth board for the Li’s geom etry
with different side gaps at Re=4400 ... 81 5.26 Local Nu num ber on front wall of the seventh board for the Li's geom
etry with different side gaps at Re=4400 ... 81 5.27 Local Nu num ber on rear wall of the first board for the Li's geom etry
with different side gaps at Re=4400 ... 82 5.28 Local Nu num ber on rear wall of the second board for the Li's geom etry
with different side gaps at Re=4400 ... 82 5.29 Local Nu num ber on rear wall of the third board for the Li's geom etry
with different side gaps at Re=4400 ... 82 5.30 Local Nu num ber on rear wall of the fourth board for the Li's geom etry
with different side gaps at Re=4400 ... 82 5.31 Local Nu num ber on rear wall of the fifth board for the Li’s geom etry
with different side gaps at Re=4400 ... 83 5.32 Local Nu num ber on rear wall of th e sixth board for th e Li’s geom etry
with different side gaps at Re=4400 ... 83 5.33 Local Nu num ber on rear wall of the seventh board for the Li's geom
etry with different side gaps at Re=4400... ... 83 5.34 Velocity vectors inside th e gap of g=41.5 mm at Re=4400 85 5.35 Com parison of the distribution of local Nu num ber for th e Li's geom
etry with different vertical side gaps at Re=8800 ... 89 5.36 Com parison of the distribution of local Nu num ber for the Li's geom
etry w ith different vertical side gaps at R e = 13200 ... 90 5.37 Local Nu num ber on front wall of the first board for the Li's geom etry
with different side gaps at Re=8800 ... 91 5.38 Local Nu num ber on front wall of the second board for th e Li’s geom
etry with different side gaps at Re=8800 ... 91 5.39 Local Nu num ber on front wall of the th ird board for th e Li's geom etry
L I S T OF F IG U R E S XI 5.40 5.41 5.42 5.43 5.44 5.45 5.46 5.47 5.48 5.49 5.50 5.51 5.52 5.53 5.54 5.55 5.56 5.57 5.58 5.59
Local Nu num ber on front wall of the last board for th e Li’s geom etry w ith different side gaps at Re=8800 ... 91 Local Nu num ber on front wall of the first board for th e Li’s geom etry w ith different side gaps a t R e= 13200 ... 92 Local Nu num ber on front wall of the second board for th e Li’s geom e try w ith different side gaps at R e= 13200 ... 92
s geom etry s geom etry s geom etry s geom etry s geom etry s geom etry s geom etry Local N u num ber on front wall of the third board for the Li’s geom etry w ith different side gaps at R e= 13200 ...
Local Nu num ber on front wall of the last board for the Li w ith different side gaps at R e= 13200 ... Local Nu num ber on rear wall of the first board for th e Li w ith different side gaps at Re=8800 ... Local Nu num ber on rear wedl of the second board for the Li w ith different side gaps at Re=8800 ... Local Nu num ber on rear wall of the th ird board for the Li w ith different side gaps at Re=8800 ... Local Nu num ber on rear wall of the fourth board for the Li w ith different side gaps at Re=8800 ... Local Nu num ber on rear wall of the first board for the Li w ith different side gaps at R e= 13200 ... Local Nu num ber on reax wall of the second board for the Li w ith different side gaps at R e= 13200 ... Local Nu num ber on rear wall of the th ird board for the Li w ith different side gaps at R e= 13200 ...
Local Nu num ber on rear wall of the fourth board for the Li’s geom etry w ith different side gaps at R e= 13200 ... 94 M ean tem p eratu re distribution across stack for Re=8800 and g=21.5m m 97 C om parison of the distribution of local Nu num ber for th e Li's geom e try w ith two different inlet and wall tem p eratu res a t Re=4400 . . . 98 C om parison of the distribution of local Nu num ber for th e Li's geom e try with different inlet and wall tem peratures a t R e=4400 ... 98 C om parison of the distribution of local Nu num ber for th e Li’s geom e try w ith g=36.5 and 41.5 m m at Re=4400 ... C om parison of the distribution of loczil Nu num ber for th e Li’s geom e try w ith g=21.5 and 26.5 m m at Re=8800 ... C om parison of th e distribution of local Nu num ber ju s t before optim um gap size (g=36.5 a t Re=4400 and g=21.5 m m a t R e = 8 8 0 0 ) ... C om parison of the distribution of local Nu n u m b er for the optim um gap size (g=41.5 a t Re=4400 and g=26.5 m m a t R e = 8 8 0 0 ) ... 101
s geom etry 92 92 93 93 93 93 94 94 94 99 100 100
L I S T O F F IG U R E S xii
6.1 Schem atic view of different node types in one dim ensional case and th eir control v o lu m e s ... 104 6.2 Configuration of internal node control volum e and th e surrounding nodes 105 6.3 Flow chart of the Fortran program for one dim ensional modelling . . I l l 6.4 T im e dependent average m oisture content of a single board (Stanish
test I), h = o8 W l m y ° K ... 114
6.5 T im e dependent local m oisture content of a single board (Stanish test
I), h ^ 0 8 W / m y ° K ... 114 6.6 T im e dependent tem p eratu re at the board center line (Stanish test 1).
h = 5 8 W l m y ° K ... 115 6.7 T im e dependent average m oisture content of a single board (Stanish
te st 2 ) , /i = 8 7 V F /m V "A '... 116 6.8 Sensitivity of tim e dependent average m oisture content to 5m,n . . . . 119 6.9 Sensitivity of tim e dependent board center tem p eratu re to Smm • • • 119 6.10 Sensitivity of tim e dependent average m oisture content to A% . . . . 120 6.11 Sensitivity of tim e dependent board center te m p e ratu re to/v/j, . . . . 121 6.12 Sensitivity of tim e dependent average m oisture content to . . . . 121 6.13 Sensitivity of tim e dependent board cen ter tem p eratu re to . . . . 122 6.14 Effect of ±50% change in St on tim e dependent average m oisture content 123 6.15 Effect of ±50% change in St on tim e dependent board center tem p eratu re 124 6.16 Effect of ±10% change in St on tim e dependent average m oisture content 124
6.17 Effect of ±10% change in St on tim e dependent board center tem p eratu re 125 6.18 Sensitivity of tim e dependent average m oisture content to a ... 126 6.19 Sensitivity of tim e dependent board center tem p eratu re to o ... 126
7.1 Schem atic view of different node types in two dim ensional case and th eir control v o lu m e s ... 128 7.2 Configuration of internal node control volum e and th e surrounding nodes 128 7.3 Flow chart of the Fortran program for two dim ensional modelling . . 130 7.4 Schem atic view of control volume used for considering the effect of
h e at/m ass transfer between air and b o a r d s ... 138 7.5 T im e dependent stack average m oisture content of th e Li’s geom etry
for g=1.5 m m a t Re=4400 ... 143 7.6 T im e dependent stack average m oisture content of th e Li’s geom etry
for g=21.5 m m a t Re=8800 ... 144 7.7 Average te m p e ratu re a t th e center of th e boards of th e Li's geom etry
for g=1.5 m m a t Re=4400 ... 145 7.8 T im e dependent stack average m oisture content of th e Li’s geom etry
L I S T OF F IG U R E S xiii
7.9 Tim e dependent stack average moisture content of the Li’s geom etry for g=1.5 m m a t R e=8800 ... 146 7.10 D istribution of local m oisture content in horizontal direction of th e
forth board of th e Li’s geom etry for g=21.5 m m at R e = 4 4 0 G ... 147 7.11 D istribution of local m oisture content in vertical direction of the forth
board of th e Li’s geom etry for g=21.5 mm at Re=4400 ... 147 7.12 T im e dependent stack average moisture content of the Li’s geom etry
for different gaps a t R e=4400 ... 148 7.13 Tim e dependent stack average moisture content of the Li’s geom etry
for different gap sizes a t Re=8800 ... 149 7.14 T im e dependent stack average moisture content of the Li’s geom etry
for different gaps a t R e=4400 ... 150 7.15 Tim e dependent stack average m oisture content of the Li’s geom etry
for different gap sizes at Re=8800 ... 150 7.16 D istribution of m oisture content across the fourth board of the Li's ge
om etry at different tim es for Re=4400 and g =1.5m m : a= 6 0 h r (,Vmax=0.374. 'Vmin =0.05), b = e n d of 3rd stage (%mai=0.17, =0.039), c= en d of 4th stage (X m ai=0.169, Xmin=0.072), d = en d of conditioning (Xmaz-=0.163. X m ,n = 0 .1 2 ) ... 151 7.17 D istribution of m oisture content across the fourth board of the Li's ge
om etry at different tim es for Re=4400 and g=41.5m m ; a= 6 0 h r (.Vmar=0.349. =0.036), b = e n d of 3rd stage(.Ymax=0.183 ,Xm,n =0.035). c= en d of 4th stage(Xmax=0.182, =0.052), d = en d of conditioning(.Ym az=0.174. Xmin = 0 .1 0 8 )... 1.52 7.18 Tim e dependent stack average moisture content of the Li’s geom etry
for different gap sizes at R e= 13200 ... 153 7.19 Standard deviation of m oisture content for the Li’s geom etry for g = 1.5mm
at Re=4400 ... 155 7.20 Average deviation of m oisture content for g = 1.5mm at Re=4400 . . . 156 7.21 Average gradient of m oisture content for g = 1.5mm at Re=4400 . . . 156 7.22 Normalized S tan d ard deviation of m oisture content at Re=4400 . . . 157 7.23 Normalized average deviation of m oisture content for g = 1.5mm at
Re=4400 ... 157 7.24 Normalized average garadient of m oisture content for g= 1.5m m at
Re=4400 ... 158 7.25 Tim e dependent S tan d ard deviation of m oisture content for g = 1.5mm
at Re=4400 ... 158 7.26 Standard deviation of m oisture content for the Li’s geom etry for g= 21.5m m
L I S T OF F IG U R E S xiv
7.27 S tan d ard deviation of m oisture content for g= 41.5m m at Re=4400 (initial values th e sam e as those of g=21.5m m c a s e ) ... 160 7.28 S tandard deviation of m oisture content for g=41.5m m a t R e=4400(initial
values the sam e as those of g=1.5m m c a s e ) ... 161 7.29 Total stan d ard deviation of m oisture content for th e Li’s geom etry at
Re=4400 ... 161 7.30 Total stan d ard deviation of m oisture content for th e Li's geom etry at
Re=4400 ... 162 7.31 S tan d ard deviation of m oisture content for the Li’s geom etry for g =1.5m m
at Re=8800 ... 163 7.32 S tan d ard deviation of m oisture content for g=26.5m m at Re=8800 . . 163 7.33 Total stan d ard deviation of m oisture content a t Re=8800 ... 164 7.34 S tan d ard deviation of m oisture content for g = 1.5mm at R e= 13200
(initial values th e sam e as those of Run 13 of L i ) ... 164 7.35 S tan d ard deviation of m oisture content for g=21.5m m a t Re=13200( initial
values the sam e as those of Run 13 of L i ) ... 165 7.36 Total stan d ard deviation of m oisture content a t R e=13200(initial val
ues th e sam e as those of Run 13 of L i ) ... 165
8.1 C om parison of th e distribution of stack average m oisture content as calculated by solving the simple diffusion equation w ith experim ental results of Li [40] for Re=4400 ... 169
XV
A ck now ledgem ent s
First I have to th an k God for providing me guidance, ability, knowledge and patience. T hank you very m uch my father and m other for your full love and support during my whole life, w ithout which I could not reach this position. Many thanks to my wife, Sima, for her patience, love and support. I appreciate your understanding when you had to quit tem porarily your study for staying with me and when I had to stay late at the University.
Posthum ous appreciation is given to Dr. Behrouz T ab arro k ’s role as co-supervisor for his guidance, encouragem ent and support. I also would like to th an k my co supervisor, Dr. Sadik Dost, for all his efforts, guidance and support throughout this research.
I am very gateful for the financial support provided by: the M inistry of C ulture and Higher E ducation of I. R. of Iran, National Science and Engineering Research Council of C anada, and th e University of V ictoria fellowship. I would like to acknowledge Ms. Min Li and Dr. I. H artley for providing the experim ental d a ta which were necessary for com pletion of this work. I would like also to th an k th e external exam iner and other m em bers of th e supervisory com m ittee Drs. L. O liveira. Z. Dong. M. W hale and R. Lueck for th eir generous cooperation and assistance. .\lso instrum ental in the com pletion of this work were the discussions I had with fellow g rad u ate students Ian Spearing, Shafei Zeidan, Anotai Suksangpanom rung, Jon P haroah and M artin Bowers and Drs. Shafei Zeidan and Ibrahim Dincer.
XVI To my parents, brothers and sisters
N om enclature
A coefficient in calculating capillary pressure
Al coefficient in Eqn. 8.1
B coefficient in calculating capillary pressure
Bi coefficient in Eqn. 8.1
Bo coefficient in Eqn. 8.1
B x - B r coefficients in calculating th e flu.x of bound w ater Cu,Cfc coefficients for describing u"*" in turbulent flows
Cp specific heat capacity
Q i, C( 2 coefficients used for e— equation in tu rb u len t flows
D apparent m oisture diffusion coefficient of wood
Dab un-delayed w ater vapour diffusion coefficient Da diffusion coefficient of drying moist air
Dav average diffusion coefficient of m oisture in wood
Db bound w ater diffusion coefficient
Dgff effective w ater vapour diffusion coefficient
Dh hydraulic d iam eter of the duct
Dp coefficient of th e drying m odel related to bound w ater
N O M E N C L A T U R E
Fx coefficient in calculating the flux of free water
G tu rb u len t kinetik energy generation
g side gap size
h convective heat transfer coefficient from surface
ha enthalpy of air
hb en thalpy of bound water
h f enthalpy of free water
hv enthalpy of w ater vapour
hm convective mass transfer coefficient from surface
J vector of to ta l m oisture flux
Ja vector of air mass flux in wood
Jb vector of bound water mass flux
J f vector of free w ater mass flux vector of w ater vapour mass flux
Ki perm eability of u n satu rated wood to liquid w ater flow
K1 3 perm eability of satu rated wood to liquid w ater flow
k kinetik energy of the fluctuation m otion in tu rb u len t flows
rh mass transfer rate
m„ m olar mass of water
N u Nusselt num ber
rijt num ber of stacks per kiln
Pa capillary pressure
Pe Peclet num ber
Pi liquid w ater pressure
P r P ra ü d tl num ber
N O M E N C L A T U R E
Py p artial w ater vapour pressure in wood
Q heat transfer
q beat transfer per unit volume of lum ber
R ideal gas constant
Rh, relative hum idity of the drying air
6 m olar entropy
S satu ration
Sav average surface mass transfer coefficient
S c Schm idt num ber
Serr Summ ation of errors in SOR m ethod
Smin m inim um satu ratio n for free w ater flow
S h Sherwood num ber
S r coefficient of the drying model related to bound water
t tim e
T tem p eratu re
T local tim e average tem perature of air
u'^ dimensionless velocity in turbulent flows
Ua internal energy of dry air inside the wood
Ub internal energy of bound water
Ud internal energy of the dry wood
u f internal energy of free water
Uy internal energy of w ater vapour
V m olar volume
V volume
Vi, Vj velocity com ponents
N O M E N C L A T U R E 4
Xi, Xj C artesian coordinate
X local m oisture content in wood
Xeq equilibrium m oisture content of th e drying air
X p s p m oisture content at the fibre satu ratio n point
Xmax m oisture content if the entire void stru ctu re is filled with w ater
Xmin m inim um m oisture content for liquid flow
vapour mole fraction in hum id air
y"*" dimensionless distance from wall in tu rb u len t flows
N O M E N C L A T U R E
G reek sym b ols
CH a tten u atio n factor of wood stru ctu re in eqns. 5 and 7
Sx distance between two adjacent ponits in x direction
Sy distance between two adjacent ponits in y direction
N t tim e step
N x X direction step size
N y y direction step size
£ viscous dissipation in turbulent flows
É void fraction
Tji viscosity of liquid water
Aa therm al conductivity of air
A therm al conductivity of moist wood
IJ. viscosity of air
y.b chemical potential for bound water
fit tu rb u len t eddy viscosity
residue at each point in the SOR m ethod
p, Pa density of air
Pb density of bound w ater
p u p j density of liquid (free) water
Pd density of d ry wood
Pv density of w ater vapour
Putt to ta l m oisture density in wood
Tut wall shear stress
0 dissipation term in tu rb u len t flows
N O M E N C L A T U R E
u absolute hum idity of the drying air
(I over relaxation factor in the SOR m ethod
S u p erscrip ts
tim e rate
tu rb u len t tim e average values tu rb u len t fluctuation term s tim e step num ber
N O M E N C L A T U R E
S u bscripts
a air
b bound w ater
cell wood cell
d dry wood
db dry-bulb (tem perature)
e f f effective
E S P fiber satu ratio n point
f free w ater
L w liquid w ater
n norm al to the surface
S values at the surface
oo free stream values
('W ) location num bers of control volume cell
V w ater vapour
wb wet-bulb (tem perature)
X com ponent in the x-direction
N O M E N C L A T U R E
A cronyn m s
A D I altern atin g direction implicit
E S P fiber satu ratio n point
S O R successive over relaxation
C hapter 1
Introduction
Drying is aji energy intensive process. This process is used in a variety of applications ranging from food processing to wood drying. For wood drying, th ere is great interest in studying this process because the wood industry is one of th e largest industries in British C olum bia and C anada as well as in other countries.
C anada is a m ajor producer of wood. It is certain th a t th e lum ber industry and wood products will continue to play a key role in C an ad a’s economy over the next century. T he to ta l energy consumption of this in d u stry excluding pulp, paper and printing industries, is approxim ately 1% (27 Pega Joules in 1998) of the total industrial energy consum ption in C anadajl]. Due to its im portance, the C anadian wood in d u stry spends about twenty five million dollars annually on research and developm ent related to wood[l].
M aterial properties of wood are influenced by th e am ount of its m oisture content. Most m echanical properties, such as strength, are b e tte r for dry wood than for green wood, w ith th e exception of toughness and shock resistance. For m ost applications, wood serves best at specific levels of moisture content. T h e m oisture content is the
C H A P T E R 1. IN T R O D U C T I O N 10
am ount of m oisture in wood expressed as a percentage of th e weight of the dry wood substance. Wood is a hygroscopic m aterial, th a t is it gives off or takes in m oisture until it is in balance w ith th e air surrounding it. W hen wood has a tta in e d such a balance, it is said to have reached its equilibrium m oisture content(E M C ). Therefore, wood should be dried ideally to a m oisture content close to th e EMC th a t it will a tta in in service. The values of m oisture content in EMC condition vary between 12 to 18 percent for th e m anufacture of articles for outdoor use, and between 5 and 10 percent for indoor articles[2].
O ne of th e reasons for drying lum ber before shipping to th e processing workshops, is th a t th e tra n sp o rt charges for lumber, moved by rail or truck, are based on the shipping weight. Principally for this reason, m ore than 70 percent of non-coastal lum ber in C anada is d ried [3],
A certain am ount of m oisture content is necessary in wood to prevent decaying caused by destroying organisms. Based on this fact, the lum ber shipped by sea before long trips, e.g. to Jap an , is dried even though in this case th e tran sp o rt charges are based on volume. In addition, wood m ust be relatively dry before it can be satisfactorily painted with oil-based paints, glued or tre a ted with preservatives.
T here are two m ain m ethods for drying wood, k iln d ryin g and air drying. Kiln drying is used alm ost exclusively in preference to air drying, because kiln drying perm its a producer to gear production to dem and and avoids rental of drying yards and, furtherm ore, provides a more uniform product.
T he process of removing m oisture from wood in a kiln can be very costly due to: (a) the cost of supplying equipm ent, labor and energy to accelerate th e process, and (b) losses from dam age caused to the wood during drying.
Researchers in th e field of lum ber drying are m aking efforts in im proving th e drying process, the m ain m otivation being to reduce the m anufacturing costs and degrade
C H A P T E R 1. IN T R O D U C T I O N 11
losses so th a t wood products will continue to be com petitive w ith oth er products such as plastics and m etals.
In order to highlight th e im portance of improving th e energy efficiency of wood drying, it should be noted th a t, in C anada, th e to tal energy which is used for wood and wood products ind u stry comes alm ost exclusively from petroleum products. In 1998. it was about 2 . 2 percent of th e to tal petroleum products consum ed in th e industrial
sector and about 70 percent of this usage is for wood drying[3]. It can be understood th a t even sm all im provem ents in th e energy efficiency of wood drying would bring about significant energy savings and environm ental benefits. This im provem ent would decrease th e energy consum ption per unit of ou tp u t which decreases both pollution and th e release of greenhouse gases such as COg.
Research in lum ber drying is currently concerned with such issues as:
- developing a b e tte r understanding of th e drying process inside th e wood.
- analysing th e num erically predicted m oisture content and te m p e ratu re fields in order to obtain th e internal stresses during drying,
- determ ining energy requirem ents for lum ber drying,
- establishing drying procedures for species of wood not previously processed, to obtain their full potential.
In th e drying process of lum ber of particular species and thickness, the te m p e ra ture and hum idity conditions of th e drying air are set by the k iln schedule. Changes of th e kiln conditions are usually m ade at predeterm ined tim e intervals, or when preselected levels of m oisture content have been reached. T h e schedule used should dry the lum ber in as short a tim e as possible, while m inim izing drying degrade to acceptable levels. Over th e years, kiln-drying schedules have been developed based on experience. Research continues to improve these schedules by taking advantage of
C H A P T E R 1. IN T R O D U C T I O N 12
new technologies and m eeting th e dem ands of th e kiln drying industry. It should be noted th a t there axe m any schedule com binations due to changes in th e velocity, tem p eratu re and relative hum idity of the drying air. Also th ere are different lum ber sizes which can be loaded in different configurations. W ith this wide range of param eters resezirchers have a tte m p te d to improve th e schedules to reach optim um choices. In this direction, a com plete analysis of the com plex process of heat and mass transfer which drives th e m oisture movement during drying is necessary.
T h e forces involved in moving the m oisture are complex. Such com plexities in crease w ith increasing tem p eratu re and m oisture gradient inside th e wood. T h e objec tive of kiln drying is to m ake these forces as high as possible w ithout causing lum ber dam age and decreasing th e quality of lum ber. This is usually done by increasing the kiln te m p e ratu re and by increasing the wet bulb depression (the difference between th e dry- and wet-bulb tem peratures). For some species, there is a practical lim it to th e intensity of tem p eratu re which may cause discolouration, kiln burn or reduced m echanical strength. Besides the dry- and w et-bulb tem p eratu res, there are other param eters which have significant effects on m oisture removing forces, such as: wood stru c tu ra l characteristics, air velocity, and stack configuration. For exam ple, using higher air velocities could result in faster drying. Less obvious factors include the arrangem ent of th e boards in th e kiln. In this stu d y th e effect of th e gaps between th e boards is investigated and its im portance is brought to light. O f course the in tro du ctio n of gaps reduces the num ber of boards th a t can be placed in a kiln. Thus, these are trade-off effects th a t open the way for optim ization.
W hen th e relative hum idity of the drying air is kept very low, in order to provide a high m oisture content gradient inside th e wood, and th e lum ber is still w et, some difficulties such as checking arise. High m oisture content gradients during drying induce surface shrinkage while th e core is unable to shrink. If th e resulting stresses
C H A P T E R 1. IN T R O D U C T I O N 13
are greater th a n the stren g th of th e wood, checking will occur as cracks on th e wood surface. Checks can be avoided by m aintaining a higher relative hum idity for air. particularly in th e early stages of drying. But this will increase th e drying tim e, to tal energy consum ption, an d subsequently th e to tal cost of drying. One of th e questions m ight be: ’’Can the usage of side gap between the boards, w ithout changing th e oth er param eters of the drying schedule, reduce the total tim e and cost required for drying a certain am ount of lum ber?”
Some undesirable conditions such as case hardening can set up high internal stresses th a t cause defects in the dried lum ber during later processes such as sawing. These stresses arise when rapid drying causes shrinkage of the surface while th e core rem ains wet and swollen. F u rth er drying, when the surface is very dry and unyielding, does not facilitate the shrinkage of th e core.
D etailed analysis of th e state of strain and stress at different locations of different boards of th e stack needs a thorough knowledge of th e tem p eratu re and m oisture d istributions in the boards. It will be shown in the following chapters th a t com pre hensive analysis of th e la tte r type has yet to be undertaken. .A.lthough there are some experim ental d a ta for te m p e ratu re and m oisture content at some locations of the boards, an extensive experim ental set of d a ta for these quantities, at different loca tions, is not available for different com binations of stack geom etry and kiln schedules. Experim ental d a ta collection for such a variety of variables is of course very costly and tim e consuming.
The processes of heat and mass transfer in wood drying can be described by th e associated governing differential equations. These equations, however, are nonlinear and coupled, and can only be solved numerically. T he m ain difficulty, however, is not in finding general num erical soloutions for the governing equations bu t th e lack of d a ta on certain physical properties needed to obtain specific solutions.
C H A P T E R L IN T R O D U C T I O N 14
O u tlin e o f T h esis
In this thesis, C h ap ter 2 presents a literature review on the relevant theoretical, num erical and experim ental studies of wood drying. In C h ap ter 3, the objectives and m otivations of th e present work are discussed. T he equations governing th e heat and mass tran sfer and fluid flow are presented for the drying air and th e wood in C h apter 4. Also discussed are the models for m igration and physical properties of different phases of w ater inside the wood. Turbulence models and schem es used for drying air flow are also included in this chapter. C hapter 5 begins w ith studying the perform ance of different turbulence models and schemes in predicting th e air flow characteristics and convective h e at/ mass transfer surface coefficients for a benchm ark case. T hen th e effect of side gap on the heat transfer between the drying a ir and board surfaces of a stack is investigated. A procedure is developed to determ ine the optim um gap size. Based on the results, the optim um gap size is suggested for different air velocities. In C h ap ter 6, the num erical model proposed for the wood in C h ap ter 4
is validated against other num erical and experim ental results by a one-dim ensional sim ulation. Also th e relative effects of some model param eters are brought to light in this C hapter. C h ap ter 7 presents the two dim ensional num erical sim ulation for the drying of a stack under a selected drying schedule. T he effect of air velocity and side gap on th e drying process is investigated as well. In C h ap ter 8, a sim ple
one-dim ensional diffusion case is used to study the m iddle p art of the drying process. Finally in C h ap ter 9, a sum m ary of conclusions is presented and recom m endations for future work are given.
15
C hapter 2
L iterature review
It is evident th a t for an accurate optim um design and control of th e wood drying process, a b e tte r u nderstanding of m oisture movement and accurate prediction of im po rtan t param eters inside th e wood, such as tem p eratu re and m oisture content, are essential. Studies in this field require m athem atical models th a t describe th e process and experim ental findings for th e needed physical properties. Also h e a t/m a ss tra n s fer coefficients between wood surfaces and the drying air are required in boundary conditions for num erical an d theoretical studies.
2.1
G en eral m od els
Research in wood drying s ta rte d w ith modelling th e mass transfer as a sim ple dif fusion process based on P ick ’s law[4]. L ater this type of model was expressed for m oisture concentration, m oisture content and water vapour pressure[5]. T hen a num ber of steady s ta te and un stead y state models for finding th e diffusion coefficient were described. It was indicated by Choong[6] th a t m oisture m ovem ent th ro u g h wood, due
C H A P T E R 2. L I T E R A T U R E R E V I E W 16
to th e difFusion process, involves two mechanisms: (1) vapour movement through cells and
(2) bound w ater movement through the cell walls of the wood.
E xperim ental results also showed th a t the bound water and vapour diffusion coeffi cients in wood vary w ith m oisture content, tem p eratu re and direction[2].
Stamm[7] modelled the wood as a bundle of capillaries of equal length. He indi cated that th e movement of free w ater in the drying wood, above th e fiber satu ratio n point, is a pressure controlled phenom ena. Comstock[8] modeled the flow of free w ater
through wood by using D arcy’s law.
Since th e m oisture content of wood during th e drying process can be below or above the fiber satu ratio n point (E SP), it is necessary to develop a general model to include both diffusion and capillary tran sp o rt. Spolek and Plumb[9] developed an analytical model for the heat and mass transfer in wood drying. A set of coupled partial differential equations for m oisture content and tem p eratu re, as a function of space and tim e, were developed. T h e experim ental results of Plum b et a/[10] showed th a t capillary tran sp o rt was dom inant at m oisture contents above the satu ratio n point.
Some proposed diffusion models for non-isotherm al diffusion and capillary move m ent of water in wood, based on irreversible therm odynam ics, are com pared by S iau [ll].
C H A P T E R 2. L I T E R A T U R E R E V I E W 17
2.2
N u m erica l stu d ies
2 .2 .1
A p p r o a c h e s
T he lite ra tu re survey shows th a t in wood drying there have been two m ain approaches tow ard num erical sim ulation of heat and m oisture transfer. One approach is based on using th e sim ple diffusion equation for the whole process. Some of the studies th a t have used th e diffusion coefficient are: [12-15]. O thers have considered the diffusion tensor [16,17]. In this approach, diffusion coefficients are based on an average value from previous experim ents for one dimensional cases with constant boundary conditions. All of th e investigations in this field have been undertaken for a single board w ith constant boundary conditions over all the boundaries. Due to changes of m oisture content and phases in space and tim e, this approach is not sufficient for prediction of local m oisture content, especially where the geom etry is com plicated and boundary conditions change in space and tim e. .Also th e proposed diffusion coefficients are based on experim ents for boards with low m oisture content (below the fiber satu ratio n point).
The second approach is based on a system of differential equations which takes into consideration th e effect of th e gradient of the therm odynam ic properties of different water phases on th e various m oisture fluxes, by using different forms of driving forces for w ater tra n sp o rt.
2 .2 .2
I n v e s t ig a t io n s
A two-dim ensional, tim e-dependent wood drying model, based on local th ree phase equilibrium , was developed by Kayihan[18]. The model assumes sep arate governing equations for mass conservation o f each phase, and uses th e p artial pressure of each
C H A P T E R 2. L I T E R A T U R E R E V I E W 18
phcLse as one of the m ain param eters. The model predictions show th a t various portions of th e drying curve are sensitive to convective transfer and to individual diffusion coefficients.
Stanish et a/.[19j improved th e above model by introducing some new sub-m odels for each phase and applied it to a one-dimensional case. Results were in very satisfac tory agreem ent with experim ental findings. But the model was complex and required th e solution of three coupled differential equations.
M itchel and Bigbee[20] used th e model of Kayihan[18] for an industrially m oti vated study of western hemlock kiln drying. T he purpose was to evaluate th e relative effect of various wood and kiln param eters on th e final outcom e of the kiln schedule in an effort to understand the variability th a t results in kiln drying. T he changes in the param eters such as dry and wet bulb tem peratures of drying air, air velocity, initial m oisture content and wood density were studied. The authors neither m entioned the m odel unknowns nor com pared their results with experim ental d a ta. T he model was one-dim ensional, and as in th e case of Kayihan[18], the constant bound w ater diffu sion and fully developed form ulation for surface coefficient were used. T he effect of heat and mass transfer to and from the upstream boards on th e conditions of drying air was not considered. The results of this study brought some insight in un d erstan d ing th e significant im pact of variations in the m entioned param eters. However, due to above m entioned shortfalls, the results were neither sufificient nor accu rate enough to draw definitive conclusions.
Josserand et a/. [13] used a sim ple one-dim ensional diffusion equation for an isother m al unsteady case. This model, which assumes th a t the diffusion coefficient is a function of m oisture content, was applied to a two-dimensional case w ith constant boundary conditions. Some researchers, e.g. Simpson and Liu[14], have tried to determ ine th e dependence of th e diffusion coeflficient, in an isotherm al problem , on
C H A P T E R 2. L I T E R A T U R E R E V I E W 19
m oisture content. Such coefficients have been used in three dim ensional problem s for an orthotropic material[17].
T he non-isotherm al m odel for diffusion process was im proved by S iau [Il]. The same problem was num erically solved by .A.vramidis et a/.[2 1] by tak in g w ater chem i
cal potential as th e driving force for diffusion. They derived th e te m p e ratu re gradient coefficient in th e mass balance equation based on th e principles of non-equilibrium therm odynam ics. T he num erical predictions of average m oisture content and te m p e r ature, using finite difference m ethod, were in good agreem ent w ith th e experim ental d a ta of desorption from an initial 20 percent m oisture content (M C). But th e results of the model were not com pared w ith cases with higher MC. In this study, th e con stan t te m p e ratu re and m oisture content were assumed as surface boundary conditions since the air side resistance to h eat/m ass transfer is assum ed to be relatively small.
A two dim ensional tim e dependent Rnite-element model of isotherm al wood d ry
ing, based on w ater poten tial concept, was presented by C loutier et ai. [22]. This study also used a constant surface mass transfer coefficient.
In 1993, Colignan et a /.[23] provided a m ethod of estim atin g th e m acroscopic d ry ing kinetics curve and described th e internal m oisture content profiles of a m aritim e- pine wood. M oisture gradients and th e tim e evolution of the p aram eters were analysed as a function of th e air drying param eters (dry-bulb, wet-bulb and the velocity) and the product param eters (thickness and density).
They assum ed th a t m oisture diffusion occurs only through th e m ain faces of the board and not through th e edges. It was also assum ed th a t th e heat tran sfer has a quasi steady n atu re and, therefore, the energy equation was not solved. T he diffusion process was divided into two stages based on w hether m oisture content of th e wood surface is below th e fiber satu ratio n point (FSP) or above it. T h e au th o rs a tte m p te d to find th e best fit for diffusion coefficients and surface convective coefficients giving
C H A P T E R 2. L I T E R A T U R E R E V I E W 20
the best fit for th e experim ental d a ta a t each stage. This m ethod is in ap pro p riate for expressing tim e dependency of local m oisture content near th e edges, especially for initial stages of drying.
The results of the above stu d y were used as inputs for a model developed for predicting the stresses in the wood which gives rise to flaws called shakes[24]. The model was based on a simplified description of the norm al internal stresses. T he aim was to evaluate th e differences in degradation by the type of drying procedure. The model was based on th e assum ption th a t the change in strain between two subsequent tim es is th e sum of a shrinkage due to local variation of m oisture content, and an elastic strain . T he norm al stress a t each location was then found and com pared with the allowable stress.
The approach in th e above two studies has some lim itations: a) it is one-dim ensional,
b) it neglects the effects of edges on the m oisture content distribution, and
c) it is an isotherm al m odel, does not consider the tem p eratu re effects on the m oisture content.
Liu et û/.[25] developed a m ath em atical model for coupled heat and m ass transfer. The m odel separated th e m oisture flux into th a t of the liquid and th a t of th e vapour phase. T he authors applied the m odel to a one dim ensional tim e dependent case and used the constant surface coefficients based on the boundary layer theory.
H ernandez and Puiggaii[26] sim ulated the drying of a wood sam ple using the model proposed by Stanish et a / .[19]. T hey also studied the drying process using microwaves. T he m ain purpose of this study was to investigate th e drying kinetics (changes of m oisture an d its gradients w ith respect to tem p eratu re over a period of tim e).
C H A P T E R 2. L I T E R A T U R E R E V I E W 21
partial differential equations for one dimensional h eat and mass transfer inside a wood specim en using th e G alerkin finite elem ent m ethod. T he analysis showed th a t the m oisture-tim e and tem p eratu re-tim e curves were strongly affected by th e values of convective mass and heat transfer coefficients. These coefficients were based on fully developed flow, and th e edge and entrance effects were not considered.
T here has been an effort, by Dincer and D ost[12], in proposing an analytical model to determ ine the m oisture diffusivities in geom etrical solid objects (namely, infinite slab, infinite cylinder, sphere) during the drying process. Com parison with the available d a ta showed the validity of the model for different ranges of Biot num bers (Bi), nam ely th e ratio of internal resistance to external resistance during h e a t/ mass transfer. The authors applied their model to a slab of wood[15].
The profiles of the surface and centre tem p eratu re and also average m oisture content in the drying of a single pinus radiata board were predicted as a function of tim e by Pang et a /.[28]. In this work, the heat and mass transfer inside the wood along the air stream direction was not taken into account. The governing equations were solved in the direction perpendicular to the air stream . Since the num erical results of previous studies were not satisfactory[29-31], the direct experim ental d a ta of surface coeflBcients were applied to the equations. The same process was repeated for different locations of stream wise direction considering separate boundary conditions for each location. T hey reported th a t variations have been noted in the external mass transfer coefficients in the stream wise direction which leads to differential drying of the board. T he effect of reversing the air flow periodically was studied to determ ine the reduction of these differences in the extent of drying across each board, and some practical suggestions were given.
For the following reasons, the model and suggestions of [28] cannot be extended to study an array of boards. Firstly, the h e at/m ass transfer of upstream boards will
C H A P T E R 2. L I T E R A T U R E R E V I E W 22
affect th e te m p e ratu re and relative hum idity of air dow nstream . This problem was not addressed since the [29] study considered only one board. In o th er words, the surface m a ss/h e a t transfer coefficients defined were based on entrance values of con c e n tra tio n /te m p e ra tu re instead of the mean values. Secondly, due to th e com plexity of flow and having a lim ited num ber of boards in a row, th e distrib u tio n of local surface coefficients for each board is different from th a t for the other boards in the sam e row. T hirdly, in the experim ents, m ass/h eat transfer occurred from one board and th e o th er boards did not take part in the process. Fourthly, th e mass transfer of naphtaline, which waa used in tests would provide the boundary condition of sur face concent ra tio n /te m p e r at ure across the test section. B ut in the real process this changes with tim e and space. Finally, the m ass/h eat transfer for side edges of the boards were not considered neither in the experim ents nor in com putations. This may suggest a lower drying rate and in consequence a higher predicted m oisture content th an th e actu al values during drying.
Recently, Turner[32] form ulated a two-dimensional m ath em atical m odel for an o rtho tro p ic single wood board and developed a num erical code based on the finite volume approach. He expressed the governing equations in the longitudinal and stream wise directions. T he effect of air stream tem p eratu re was studied while the surface coefficients were assumed to be independent of tim e and space. .Although the heat and m ass transfer across the wood thickness is much m ore significant th an those in the stream wise direction, they were neglected. In solving the model equations, it was assum ed th a t th e board is a perfectly sym m etrical body and the external transfer coefficients over th e end are th e sam e as those over the flat surfaces.
A simplified one dim ensional description of drying kinetics was ob tain ed for the drying behaviour of an array of pinus radiata by Pang et a/. [33]. Mass and heat bal ances over a control volume in th e stream wise direction were coupled w ith simplified
C H A P T E R 2. L I T E R A T U R E R E V I E W 23
characteristic drying curves. It was assumed th a t the drying rate curves for a specific m aterial will rem ain sim ilar, irrespective of th e external condition, and it contains three m ain periods: initial constant drying rate, and two falling periods w ith differ ent slopes[34]. This assum ption would not hold for locations near the edges of the boards especially when there is a large gap betw een boards. T he stu d y determ ined the changes in air condition (hum idity and tem p eratu re) and local average m oisture content through th e stack using th e above simplified one dim ensional approach. The influence of air flow reversals under different strategies were investigated in order to make some suggestions for decreasing m oisture content differences along th e stack.
Gong and Plumb[35,36] proposed a theoretical model for the effect of heterogenous and anisotropic n atu re of wood on heat and mass transfer process. T hey im plem ented their model using a finite difference program capable of predicting the tw o-dim ensional d istribution of m oisture content and tem p eratu re in wood. T hey conducted experi ments using a 7-ray a tte n u atio n system and a near-infrared reflectance analyzer. The
internal and surface m oisture content were m easured by the m entioned instrum ents. T heir m ain concern was the difference between the transfer process in radial and ta n gential directions, and th a t the process in any other direction could be determ ined based on these two ones. In the model, they ignored the contribution of th e gradient of tem p eratu re in th e mass transfer equation and the gradient of m oisture content in the energy equation. Also in th e energy equation, the changes in th e enthalpy and internal energy of each phase were not considered separately. Instead, a te m p e ratu re and m oisture content dependent relationship for moist wood specific heat was em ployed (sim ilar to th e works of Pang et a /.[28] and A vram idis et a / .[27]). .Also heat of vaporization and ra te of changing the liquid to vapor at any point were used for the m entioned param eters.
C H A P T E R 2. L I T E R A T U R E R E V I E W 24
models using a comm on set of drying d ata for spruce lum ber. Experim ental d a ta of the initial values of relevant param eters were supplied to the authors from 16 coun tries, for com parison of predicted average m oisture contents, and m oisture content near th e surfaces and in th e core. The m ajority of th e sim ulation results were not in good agreem ent w ith th e test runs. Some explanations were given for poor agree m ents such as: uncertain coefficients for the models considered as the most im p o rtant reason; nature of simplifications; and the solution m ethods used for heat and mass transfer equations. The review em phasized th a t ■‘there is no universal model available for wood drying, but ra th e r models which are best suited for specific problems due to perform ance or ease of use” . From the comparison of different sim ulations with exper im ental d a ta presented in [37], it can be observed th a t th e following programs perform b e tte r th a n others: VVOODRY-S originated from Sutherland et. a /.[38]; S.A.LIN.A, re lated to Salin’s thesis[39]; SAHA developed by Ranta-M aunus[40]; and PR O FIL based on Vogel's work[41].
The model by Salin was developed following th e earlier work by Stanish et a /.[19]. It considers different phases of m oisture but neglects th e bulk flow of water vapor and air. The model considers th e drying of each wood board as an isotherm al one dim ensional process. T he tre a tm e n t of bound w ater diffusion as a function of m oisture content and te m p e ratu re is the m ost im portant m odification of this model com pared to its original version. Solution is by the finite difference m ethod, and the convective surface coefficients are velocity dependent.
T he original SAHA program is a one-dimensional isotropic m odel which only considers the diffusion m echanism for mass transfer driven by a gradient in m oisture content. T he value of th e diffusion coefficient was determ ined by experim ents over a range of m oisture contents, tem p eratu re and density for heartw ood and sapwood. H eat transfer is governed by conduction. For th e b oundary condition a t the surface.
