41
MOISTURE TRANSPORT PHENOMENA AND STRESS DEVELOPMENT IN WOOD by R.L. Zwick March 1986
MOISTURE TRANSPORT PHENOMENA AND
STRESS DEVELOPMENT IN WOOD
by
R.L. Zwick
March 1986
Any enquiries relating to this publication, should be directed to:
Forintek Canada Corp. Western Laboratory 6620 N.W. Marine Drive Vancouver, B.C. V6T 1X2
Forintek Canada Corp.
PROJECT No: F C C-12-] 5-nns
CPS No: j2
c ^
MOISTURE TRANSPORT PHENOMENA AND
STRESS DEVELOPMENT IN WOOD
by
R.L. Zwick
R e s e a r c h S c i e n t i s t
Lumber M a n u f a c t u r i n g
March 1986
This project was financially supported by the Canadian Forestry Service under the Contribution Agreement existing between the
Government of Canada and Forintek Canada Corp.
)
NOTICE
This report i s an i n t e r n a l Forintek document, for release only by permission of Forintek Canada Corp. This d i s t r i b u t i o n does not constitute p u b l i c a t i o n . The report i s not to be copied f o r , or c i r c u l a t e d to, persons or p a r t i e s other than those agreed to by Forintek. Also, t h i s report i s not to be c i t e d , in whole or i n part, unless p r i o r permission i s secured from Forintek Canada Corp.
Neither Forintek Canada Corp., nor i t s members, nor any other persons acting on i t s behalf, make any warranty, express or implied, or assume any l e g a l r e s p o n s i b i l i t y or l i a b i l i t y f o r the completeness of any information, apparatus, product or process d i s c l o s e d , or represent that the use of the disclosed information would not i n f r i n g e upon p r i v a t e l y owned r i g h t s . Any reference i n t h i s report to any s p e c i f i c commercial product, process or service by tradename, trademark, manufacturer or otherwise does not necessarily constitute or imply i t s endorsement by Forintek Canada Corp. or any of i t s members.
SUMMARY
T h i s paper o u t l i n e s the work t o be undertaken f o r r e s e a r c h i n g m o i s t u r e t r a n s p o r t phenomena and s t r e s s development i n wood. The m o i s t u r e t r a n s p o r t model w i l l c o n s i d e r D a r c i a n d i f f u s i o n f o r f r e e water i n wood, and F i c k i a n d i f f u s i o n f o r bound water. The bound water d i f f u s i o n i s assumed t o be d r i v e n by c h e m i c a l p o t e n t i a l . The e q u a t i o n s w i l l be s o l v e d , and e x p e r i m e n t a l l y v e r i f i e d , f o r two di m e n s i o n s .
The s t r e s s model w i l l c o n s i d e r the n o n l i n e a r s t r e s s e s / s t r a i n s d e v e l o p e d , which a r e c h a r a c t e r i s t i c of wood d r y i n g . A v i s c o e l a s t i c model w i l l be e v a l u a t e d f o r t h e c r e e p s t r a i n s . In a d d i t i o n , t h e m o i s t u r e t r a n s p o r t and s t r e s s model w i l l be e x p e r i m e n t a l l y v e r i f i e d on a s e t t i n g of foam or "wood s u b s t i t u t e " . E x p e r i m e n t a l d a t a from foam s h o u l d be e a s i e r t o r e p l i c a t e t h a n from wood.
Heat and mass t r a n s f e r c o e f f i c i e n t s w i l l a l s o be measured f o r t y p i c a l
k i l n c o n d i t i o n s . T h i s w i l l be n e c e s s a r y f o r a p p l i c a t i o n of t h e
m o i s t u r e t r a n s p o r t and s t r e s s development model t o i n d u s t r i a l d r y k i l n
e n vironments.
TABLE OF CONTENTS
Page
LIST OF FIGURES i i i
1.0 OBJECTIVE 1
2.0 INTRODUCTION 1
3.0 STAFF 4
4.0 DIFFUSION THEORY TO MODEL MOISTURE TRANSPORT PHENOMENA 4 4.1 FURTHER DEVELOPMENT OF THE KAYIHAN MODEL 6
4.2 EXPERIMENTAL WORK 7
5.0 STRESS DEVELOPMENT 8
5.1 ELASTIC-PLASTIC MODELS 9
5.2 VISCOELASTIC MODELS 14 5.3 LINEAR ELASTIC MODEL 22 5.4 WOOD PROPERTIES 24 5.5 EXPERIMENTAL WORK REQUIRED FOR THE
STRESS DEVELOPMENT MODEL 24
6.0 OTHER RESEARCH PRESENTLY BEING PERFORMED 25
7.0 HEAT AND MASS TRANSFER COEFFICIENTS 27
8.0 NUMERICAL METHODS 29
9.0 CONCLUSIONS 30
10.0 REFERENCES 31
LIST OF FIGURES
Figure 1 The Forest Industry and Canada
Figure 2 The Forest Industry and B.C.
Figure 3 Minimum Cost of Drying Lumber
Figure 4 Board S l i c e d into Ten Sections f o r Evaluating Stress Development
Figure 5 D e f l e c t i o n of Prongs
Figure 6 Quantitative -Determination of Stress at Each Level
Figure 7 Release and Set S t r a i n
Figures 8-14. Numerical Results from Morgan, et (1982)
Figure 15 Computation of Drying Stresses by Kawai, et a l . (1979)
Figure 16 Maxwell Chain Model
1.0 OBJECTIVE
A p r e v i o u s r e p o r t by Zwick (1985a) d i s c u s s e d t h r e e d i f f e r e n t
approaches t o m a t h e m a t i c a l m o d e l l i n g of m o i s t u r e t r a n s p o r t phenomena
i n wood. The author f o l l o w e d t h e recommendations made i n t h i s r e p o r t
and used d i f f u s i o n t h e o r y f o r t h e computer model. One of t h e
o b j e c t i v e s of the c u r r e n t s t u d y was t o r e f i n e t h e l i t e r a t u r e s e a r c h
i n t o d i f f u s i o n t h e o r y f o r m o i s t u r e t r a n s p o r t phenomena, and d i s c u s s
t h e approach t o be used f o r the computer model.
A second o b j e c t i v e was t o p e r f o r m a d e t a i l e d l i t e r a t u r e s e a r c h on m o d e l l i n g s t r e s s development i n d r y i n g wood. A review of the l i t e r a t u r e w i l l a l l o w a d e c i s i o n t o be made on t h e b e s t approach t o be used f o r s t r e s s m o d e l l i n g . ( T h i s w i l l be d i s c u s s e d i n a subsequent r e p o r t . )
F i n a l l y , a g e n e r a l review of t h e l i t e r a t u r e was performed on the p h y s i c a l c o n s t a n t s t h a t must be e x p e r i m e n t a l l y d e t e r m i n e d f o r m o i s t u r e t r a n s p o r t and s t r e s s development models. S i n c e e x p e r i m e n t a l work can be t h e most time consuming phase of t h e p r o j e c t , t h e t h i r d o b j e c t i v e of t h i s study was t o e s t a b l i s h what experi m e n t s s h o u l d b e g i n immediately.
2.0 INTRODUCTION
F o r i n t e k Canada Corp. i s Canada's n a t i o n a l n o n - p r o f i t r e s e a r c h and
development o r g a n i z a t i o n s e r v i n g the Canadian wood p r o d u c t s i n d u s t r y .
I t i s w e l l known t h a t th e Canadian f o r e s t i n d u s t r y i s undergoing a ,
massive s t r u c t u r a l change. Even though t h i s p r i m a r y i n d u s t r y i s not
as p r o f i t a b l e as i t once was i n t h e m i d - s e v e n t i e s , t h e f o r e s t i n d u s t r y
c o n t i n u e s t o be a major c o n t r i b u t o r t o t h e Canadian b a l a n c e of t r a d e
and t h e B.C. economy ( F i g u r e s 1 & 2 ) .
K i l n d r y i n g p l a y s an i m p o r t a n t r o l e i n the B.C. lumber m a n u f a c t u r i n g
i n d u s t r y . B r i t i s h Columbia produces a t o t a l of 5.5 m i l l i o n m- of
lumber per y e a r ; 70 p e r c e n t from the i n t e r i o r and 30 p e r c e n t from t h e
c o a s t . In t h e B.C. i n t e r i o r , a p p r o x i m a t e l y 72 p e r c e n t of t h e lumber
i s d r i e d , w h i l e on t h e c o a s t , a p p r o x i m a t e l y 17 p e r c e n t i s d r i e d . The
c u r r e n t t r e n d on t h e c o a s t i s t o d r y more lumber.
D r y i n g i s a major c o s t f a c t o r i n t h e p r o d u c t i o n of f i n i s h e d lumber.
A t l e a s t 70 p e r c e n t of t h e t o t a l energy used i n t a k i n g a t r e e from t h e
f o r e s t and c o n v e r t i n g i t t o a m a r k e t a b l e lumber p r o d u c t i s consumed i n
d r y i n g ( P f a f f and Garrahan, 1984). At p r e s e n t , t h e t o t a l energy c o s t
p e r y e a r of o p e r a t i n g lumber d r i e r s i n B.C. i s about $43 m i l l i o n .
In the k i l n d r y i n g p r o c e s s , lumber can be degraded due t o poor d r y i n g
p r a c t i c e s , o p e r a t o r e r r o r , o r d r y i n g w i t h l e s s t h a n optimum
c o n d i t i o n s . Most degrade i n k i l n - d r i e d lumber i s a r e s u l t of wood
s h r i n k a g e e f f e c t s . C o s t s a s s o c i a t e d w i t h t h i s can be about
$30 m i l l i o n a n n u a l l y i n t h e B.C. lumber i n d u s t r y .
FOREST INDUSTRY CONTRIBUTION TO CANADA'S TRADE BALANCE
Net Contribution of Foreign Exchange (Export Dollars) by Sector
$11.4 Forest Products
//////////////////////////. ///////////////^//////////. ////////////////////>//////. //////////////^///////////. //////////////////////////.
$5.4 Metal Ores & Non-Ferous Metals
$3.1 Food,Beverages,Tobacco, Wheat,Grain
$1.1 Fisheries
-$1.0 Petroleum & Natural Gas
-$3.3 Automobiles & Auto Parts
—r— -2 0 -4 2 4
BILLION DOLLARS
6 10 12
F i g , 1 The F o r e s t I n d u s t r y and Canada
B.C. FOREST INDUSTRY AND THE PROVINCIAL ECONOMY - 1982 (Manufacturing Shipments by Value)
Total Shipments: $15.17 Billion
Mining 11.3%
Petroleum & Coal 14.1%
Other 16.8%
Food & Beverages 16.3%
F i g . 2 The F o r e s t I n d u s t r y and B.C.
T r a d i t i o n a l l y , i n d u s t r y i n t e r e s t i n r e d u c i n g c o s t s i n lumber d r y i n g has been l i m i t e d , p r o b a b l y because d r y i n g i s such a p a s s i v e and u n e v e n t f u l o p e r a t i o n . However, i n r e c e n t y e a r s t h e r e has been a renewed i n t e r e s t i n r e d u c i n g d r y i n g c o s t s , and i n r e s e a r c h i n g t h e f i e l d of lumber d r y i n g .
E s t a b l i s h m e n t of optimum d r y i n g p r o c e d u r e s r e q u i r e s a model w h i c h p r e d i c t s how m o i s t u r e m i g r a t e s out of wood d u r i n g k i l n d r y i n g and w h i c h p r e d i c t s s t r e s s e s d e v e l o p i n g i n lumber b e i n g d r i e d . An a c c u r a t e model would be a b l e t o e v a l u a t e e x i s t i n g and e x p e r i m e n t a l k i l n s c h e d u l e s w i t h o u t t h e need f o r a c t u a l k i l n r u n s . A c o m p r e h e n s i v e model would a l s o be e f f e c t i v e i n c o n t r o l l i n g k i l n s t o m i n i m i z e lumber degrade and/or energy use. The u l t i m a t e g o a l would be t o m i n i m i z e d r y i n g c o s t s , as shown i n F i g u r e 3.
F i n a l l y , m o d e l l i n g s t r e s s may e x p l o i t t h e b e n e f i t s of a l t e r n a t e d r y i n g methods such as a p p l y i n g h i g h h u m i d i t y t r e a t m e n t s d u r i n g d r y i n g . There has been some e m p i r i c a l work on i n t e r m i t t e n t l y e x p o s i n g wood t o h i g h h u m i d i t y d u r i n g t h e d r y i n g p r o c e s s . The i n t e n t i o n i s t o r e l i e v e s t r e s s i n t h e wood d e v e l o p e d from s t e e p m o i s t u r e c o n t e n t g r a d i e n t s . E a r l y r e s u l t s show a n o t e d improvement i n q u a l i t y of t h e k i l n - d r i e d wood ( S t o h r , 1 9 8 5 ) . However, i t would be u s e f u l t o q u a n t i t a t i v e l y p r e d i c t t h e f r e q u e n c y and magnitude of t h e s e t r e a t m e n t s .
K i l n d r y i n g lumber has t r a d i t i o n a l l y been a t r i a l - a n d - e r r o r o p e r a t i o n . New f a c i l i t i e s a t F o r i n t e k ' s l a b o r a t o r y a r e now u n r i v a l e d i n Canada f o r t h e a p p l i c a t i o n of modern s c i e n t i f i c p r i n c i p l e s t o d r y k i l n o p e r a t i o n . I n a d d i t i o n , F o r i n t e k i s i n t h e unique p o s i t i o n t o e f f e c t i v e l y t r a n s f e r new t e c h n o l o g y i n t o t h e i n d u s t r y .
3.0 STAFF
R.L. Zwick R e s e a r c h S c i e n t i s t Lumber M a n u f a c t u r i n g Dept.
4.0 DIFFUSION THEORY TO MODEL MOISTURE TRANSPORT PHENOMENA
The m o d e l l i n g of m o i s t u r e t r a n s p o r t phenomena was c l a s s i f e d by Zwick {1985a) i n t o t h r e e a p p r o a c h e s : d i f f u s i o n t h e o r y , i r r e v e r s i b l e thermodynamics and m e c h a n i s t i c m o d e l l i n g . Of t h e t h r e e , Zwick recommended two p o t e n t i a l methods t o m o d e l l i n g m o i s t u r e t r a n s p o r t i n wood; (a) by i r r e v e r s i b l e thermodynamics o r (b) by d i f f u s i o n t h e o r y . Of t h e two, t h e d i f f u s i o n model by K a y i h a n e t a l . (1984) i s t h e more p r o m i s i n g and i s p r o p o s e d f o r t h i s r e s e a r c h .
S e v e r a l r e p o r t s i n a d d i t i o n t o Z w i c k ' s (1985a) have s u p p o r t e d d i f f u s i o n t h e o r y f o r d e s c r i b i n g m o i s t u r e movement i n wood. S i a u (1985) compared f o u r a l t e r n a t i v e e q u a t i o n s t o p r e d i c t wood m o i s t u r e d i f f u s i o n , i n c l u d i n g c h e m i c a l p o t e n t i a l as t h e d r i v i n g f o r c e . However, as o u t l i n e d by Zwick ( 1 9 8 5 a ) , t h e development of t h e c h e m i c a l
MOISTURE CONTENT
F i g . 3 Minimum C o s t o f D r y i n n Lumber
p o t e n t i a l e q u a t i o n (by S i a u ) was inadequate i n comparison t o t h e
K a y i h a n model. None of the f o u r e q u a t i o n s i n v e s t i g a t e d by S i a u c o u l d
c o m p l e t e l y model t h e e x p e r i m e n t a l r e s u l t s . However, S i a u c o n c l u d e d
t h a t c h e m i c a l p o t e n t i a l b e s t p r e d i c t e d f l u x r e v e r s a l (due t o t h e r m a l
d i f f u s i o n ) .
An e x t e n s i v e l i t e r a t u r e review by Rosen (1984) d i s c u s s e d many of the approaches used t o model m o i s t u r e m i g r a t i o n i n wood. Rosen examined d i f f u s i o n t h e o r y f o r m o d e l l i n g h i g h - t e m p e r a t u r e d r y i n g . H i s approach f o r m o d e l l i n g m o i s t u r e m i g r a t i o n above the f i b e r s a t u r a t i o n p o i n t r e l i e d on Darcy's law ( B r a m h a l l , 1971). T h i s c o n c u r s w i t h K a y i h a n ' s model and w i t h S p o l e k ' s (1981b) work. S p o l e k ' s a c c u r a t e m a t h e m a t i c a l d e s c r i p t i o n of a wood t r a c h e i d , and t h e c a p i l l a r y movement of water w i t h i n t h e t r a c h e i d , c o r r e s p o n d s w e l l w i t h e x p e r i m e n t a l r e s u l t s . Rosen extended S p o l e k ' s work t o d e s c r i b e p r e s s u r e development i n s i d e wood d u r i n g h i g h - t e m p e r a t u r e d r y i n g . Rosen's c o n c l u s i o n t h a t d i f f u s i o n below the f i b e r s a t u r a t i o n p o i n t i s d r i v e n by m o i s t u r e c o n t e n t g r a d i e n t s .
H a i s h i (1985) r e c e n t l y i n v e s t i g a t e d the c o n c e n t r a t i o n dependency of d i f f u s i o n c o e f f i c i e n t s f o r F i c k i a n d i f f u s i o n models. He c o n c l u d e d t h a t c o n c e n t r a t i o n g r a d i e n t s a d e q u a t e l y model m o i s t u r e d i f f u s i o n o n l y under moderate and i s o t h e r m a l non-steady s t a t e c o n d i t i o n s .
4.1 FURTHER DEVELOPMENT OF THE KAYIHAN MODEL
Kayihan ^ a l . (1984) s o l v e d t h e c o u p l e d d i f f e r e n t i a l e q u a t i o n s i n one dimension f o r a F i c k i a n d i f f u s i o n model u s i n g c h e m i c a l p o t e n t i a l as the d r i v i n g f o r c e . The n u m e r i c a l s o l u t i o n s were compared t o t h e i r own e x p e r i m e n t a l r e s u l t s and t o S i a u ' s (1985) e x p e r i m e n t s i n n o n i s o t h e r m a l m o i s t u r e d i f f u s i o n . The f l u x e s were det e r m i n e d i n the r a d i a l d i r e c t i o n of t h e wood sample.
The e q u a t i o n s used i n t h e K a y i h a n model a r e b r i e f l y o u t l i n e d below:
C o n s e r v a t i o n of a i r :
C o n s e r v a t i o n o f water:
d -d (n V
+ n + n ) b f
C o n s e r v a t i o n of energy:
d + O h + p h + D h + D h ) '^v v " b b f 'd d t
(n h + n h + n h * + n h - - T l V V b b f f ^2
whe re p d e n s i t y
h e n t h a l p y
d i f f e r e n t i a l e n t h a l p y of bound water
n f l u x
T temperature
k heat c o n d u c t i v i t y
and the s u b s c r i p t s a a i r
b bound water
f l i q u i d ( f r e e ) water
V water vapor
d d r y wood
Kay i h a n a c c o m p l i s h e d the n u m e r i c a l s o l u t i o n t o the i m p l i c i t l y f o r m u l a t e d d i f f e r e n t i a l e q u a t i o n s by t h e f i n i t e d i f f e r e n c e method. The f o r m u l a t i o n i n c l u d e d an " a d a p t i v e mesh t e c h n i q u e " which p o s t u l a t e d a f i n e mesh g r i d t o f o l l o w t h e moving boundary between f r e e water and h y g r o s c o p i c d i f f u s i o n . T h i s t e c h n i q u e a l l o w e d f a s t e r c a l c u l a t i o n t imes w i t h o u t t h e l o s s of a c c u r a c y . F o r i m p l e m e n t a t i o n of t h e model f o r a p r o c e s s c o n t r o l system c a l c u l a t i o n times s h o u l d be m i n i m a l .
4.2 EXPERIMENTAL WORK
The m o i s t u r e t r a n s p o r t model w i l l be extended i n two d i m e n s i o n s .
E x p e r i m e n t a l work w i l l i n v o l v e d e t e r m i n i n g f l u x e s i n t h e t a n g e n t i a l
d i r e c t i o n . In a d d i t i o n , two wood s p e c i e s o f i n t e r e s t t o the Canadian
lumber d r y i n g i n d u s t r y w i l l be used:
1. Lodgepole p i n e ( P i n u s c o n t o r t a )
2. Western hemlock (Tsuga h e t e r o p h y l l a )
N o n i s o t h e r m a l f l u x e s w i l l be e s t a b l i s h e d f o r c o n v e n t i o n a l k i l n
c o n d i t i o n s . N o n - c o n v e n t i o n a l k i l n s c h e d u l e s such as e x t r e m e l y h i g h
temperatures ( > 120°C) w i l l not be examined. A c t u a l e x p e r i m e n t a l
methods and p r o c e d u r e s w i l l be s i m i l a r t o t h o s e of S i a u and Babiuk
(1983). The l e n g t h of time r e q u i r e d t o determi n e m o i s t u r e f l u x e s
d i c t a t e s t h a t t h e sample s i z e s w i l l be f a i r l y s m a l l .
The measurement of the f l u x e s w i l l p r o v i d e e x p e r i m e n t a l v a l u e s f o r the
d i f f u s i o n c o n s t a n t used i n t h e m o i s t u r e t r a n s p o r t e q u a t i o n s .
D e t e r m i n a t i o n of f l u x i s s e n s i t i v e t o the a i r c i r c u l a t i o n i n s i d e
d i f f u s i o n c e l l s (Mackay, 1971). T h e r e f o r e , t h e d e s i g n of t h e
e x p e r i m e n t a l work s h o u l d f o l l o w c l o s e l y t h e work done by S i a u ( 1985)
i n o r d e r t o r e p l i c a t e h i s v a l u e s .
D i f f i c u l t i e s may a r i s e from t h e measurement of some p h y s i c a l v a l u e s
w i t h i n the p i e c e of wood. Temperature measurement u s i n g thermocouples
appears t o g i v e v a l i d r e s u l t s ( K a y i h a n , 1 9 8 5 ) . However, i n t e r i o r
m o i s t u r e c o n t e n t measurements may be d i f f i c u l t . A t p r e s e n t t h e r e are
o n l y a few f e a s i b l e methods t o measure m o i s t u r e c o n t e n t p r o f i l e s
w i t h i n wood. One would be t o shave o f f the wood b l o c k and measure the
m o i s t u r e c o n t e n t of t h e s h a v i n g s by weight. T h i s would be a
d e s t r u c t i v e measure of t h e m o i s t u r e c o n t e n t p r o f i l e s .
A second method i s t o use gamma ray a t t e n u a t i o n , as o u t l i n e d by Plumb
et ( 1 9 8 4 ) . D e s p i t e some l a c k of s u c c e s s w i t h t h i s method ( O l s o n ,
1 9 7 9 ) t h e r e s u l t s r e c e n t l y p r e s e n t e d by Plumb _et a l . appear t o be
en c o u r a g i n g . M o i s t u r e p r o f i l e s were n o n d e s t r u c t i v e l y o b t a i n e d by
gamma a t t e n u a t i o n , w i t h good a c c u r a c y .
A t h i r d method has had some e a r l y e n c o u r a g i n g r e s u l t s but i s s t i l l i n
the e x p e r i m e n t a l s t a g e . Troughton and C l a r k e ( 1 9 8 5 ) o u t l i n e a
pro c e d u r e t o measure m o i s t u r e c o n t e n t w i t h i n f r a r e d r a d i a t i o n . The
b a s i c p r i n c i p l e i s t o expose t h e wood t o a g i v e n q u a n t i t y of heat from
an i n f r a r e d heat s o u r c e and measure the temperature r i s e o v e r a p e r i o d
of t i m e . The A T i s r e l a t e d t o t h e m o i s t u r e c o n t e n t i n t h e wood,
r e g a r d l e s s of whether the water i s i n the f r e e o r bound s t a t e .
A l t h o u g h t h i s i s a s u r f a c e phenomenon, t h e m o i s t u r e c o n t e n t p r o f i l e of
a board c r o s s s e c t i o n can be made. Recent advances i n i n f r a r e d
s e n s i n g and i n f o r m a t i o n d i s p l a y , may a l l o w t h e d e t e r m i n a t i o n of
m o i s t u r e c o n t e n t p r o f i l e s w i t h a c c e p t a b l e r e s o l u t i o n . R e s e a r c h t o
c o r r e l a t e temperature r i s e t o m o i s t u r e c o n t e n t would be n e c e s s a r y .
F i n a l l y , t h e r e are two o t h e r n o n d e s t r u c t i v e methods which may p r o v i d e
i n f o r m a t i o n on t h e m o i s t u r e d i s t r i b u t i o n w i t h i n wood: X-ray computed
tomography (CT) s c a n n i n g , and n u c l e a r magnetic resonance (NMR)
s p e c t r o s c o p y . Both methods have been i n v e s t i g a t e d u s i n g m e d i c a l
s c a n n e r s on wood ( H a t t o r i and Kanagawa, 1 9 8 5 ; Menon et a l . , 1 9 8 5 ) .
The i n f o r m a t i o n from X-ray CT scans showed a s t r o n g dependence on t h e
d e n s i t y of the wood. Scanning w i t h NMR was more s e n s i t i v e t o m o i s t u r e
and c h e m i c a l c o m p o s i t i o n ( i . e . , e x t r a c t i v e s ) .
5 . 0 STRESS DEVELOPMENT
A l t h o u g h l i m i t e d s t u d i e s have been undertaken on s t r e s s development
d u r i n g lumber d r y i n g , t h e a v a i l a b l e l i t e r a t u r e i s e n c o u r a g i n g .
L e s s a r d e t a l . (1982a) have d i s c u s s e d m o d e l l i n g s t r e s s i n lumber f o r
p r o c e s s c o n t r o l a p p l i c a t i o n s . T h i s paper p r o v i d e s an ove r v i e w on
s t r e s s development w i t h e x p e r i m e n t a l work, t h e o r e t i c a l development,
and i m p l e m e n t a t i o n of the r e s u l t s i n t o a p r o c e s s c o n t r o l model. The
s t r e s s development u t i l i z e s an e l a s t i c - p l a s t i c s t r a i n model, w i t h no
c o n s i d e r a t i o n of v i s c o u s c r e e p i n wood.
5.1 ELASTIC-PLASTIC MODELS
The e x p e r i m e n t a l work ( L e s s a r d , 1982b) and t h e t h e o r e t i c a l model,
d i v i d e d a 5 cm-thick board i n t o t e n s e c t i o n s ( F i g u r e 4 ) . The s t r e s s
E i j (CipoTj - fpMj)
^BN - ^SRC' the s t r a i n of j due t o s t r e s s a l o n e
s t r e s s a t l e v e l j u s i n g e l a s t i c model.
s t r a i n o f the board edge boundary.
s h r i n k a g e s t r a i n a t l e v e l j d e t e r m i n e d by m o i s t u r e c o n t e n t at l e v e l j .
= p l a s t i c or memory s t r a i n d etermined a t a p o i n t i n time.
I f the f i r s t s t r e s s e s t i m a t e exceeds the e l a s t i c l i m i t , then a second e s t i m a t e of s t r e s s i s determined w i t h a d i f f e r e n t Young's Modulus:
^ 2 j = E 2 j (^TOTj - ^PMj)
T h i s p r o c e d u r e of d e t e r m i n i n g s t r e s s i n wood was o r i g i n a l l y o u t l i n e d by U g o l e v (1976).
The e x p e r i m e n t a l work r e l i e d on s t r e s s p r o n g s t o q u a n t i t a t i v e l y d e t e r m i n e t h e s t r e s s a t each l e v e l ( F i g u r e s 5 & 6 ) . The r e l e a s e s t r a i n and s e t s t r a i n were r e s p e c t i v e l y measured a f t e r k i l n d r y i n g and oven d r y i n g ( F i g u r e 7), by d e t e r m i n i n g t h e l e n g t h s of t h e t e n t r a n s v e r s e s l i c e s .
The r e s u l t s and c o n c l u s i o n s of L e s s a r d ' s work support the f e a s i b i l i t y of m o d e l l i n g s t r e s s development w i t h i n a board i n c o n j u n c t i o n w i t h t h e t r a n s p o r t of m o i s t u r e . L e s s a r d ' s e l a s t i c - p l a s t i c model was c a p a b l e of p r e d i c t i n g f i n a l s t r e s s e s and time of s t r e s s r e v e r s a l . However, t h e model was i n c o n s i s t e n t i n s e t p r e d i c t i o n , and c r e e p m o d e l l i n g may be n e c e s s a r y t o f u l l y d e s c r i b e the v a r i o u s c h a r a c t e r i s t i c s o f s t r e s s development i n wood.
S t r e s s development i n wood was m o d e l l e d as f o r an e l a s t i c - p l a s t i c m a t e r i a l by Morgan, e t a_l. (1982). The n u m e r i c a l p r o c e d u r e of t h e model was done w i t h f i n i t e element a n a l y s i s , as d e s c r i b e d by Lewis, _et a l . (1984).
The e q u a t i o n s f o r m o i s t u r e t r a n s p o r t i n wood used t h e L u i k o v model (Zwick, 1985a) f o r n o n i s o t h e r m a l d i f f u s i o n . The s t r e s s development was r e l a t e d t o t h e s h r i n k a g e of the wood. The s t r a i n t e n s o r c o n s i s t e d o f :
€= + ^vp + Cm
e q u a t i o n was:
a i j
where ^TOT
w i t h
1
^ i j
^BN
^SRCj
'P„3
F i g u r e 4 Board S l i c e d i n t o Ten S e c t i o n s
F o r E v a l u a t i n g S t r e s s Development
Prong Section
Cantilever beam model for prong
Pure bending stress distribution within the prong
5max
F i g . 5 D e f l e c t i o n o f Prongs
Y
X
Moisture Bias Resultant Stress Stress Stress
F i g . 6 Q u a n t i t a t i v e D e t e r m i n a t i o n o f S t r e s s a t Each L e v e l
Y
Before Slicing _ i
_ 2 3 ^
_4 _5
6 7 _
jrj ~][___j3 ri__ _e
10
1
After Slicing 1 1 1 1 2 _ | 1 3 1 4 ' 1 1 6 1 1 1 6 • 1 7 1 8 1 1 s 1 1 10 1 |-« 1 n L2 ^ •
F i g . 7 R e l e a s e and S e t S t r a i n
where 6 = t o t a l s t r a i n
e„ = e l a s t i c s t r a i n
f = p l a s t i c s t r a i n ^vp
f. = i n i t i a l s t r a i n
P h y s i c a l c o n s t a n t s were o b t a i n e d from wood s c i e n c e l i t e r a t u r e . The n u m e r i c a l a n a l y s i s of t h e s t r e s s development was e x t e n s i v e , but was not compared t o any e x p e r i m e n t a l d a t a . However, the r e s u l t s were d i s p l a y e d i n an i n f o r m a t i v e f a s h i o n ( F i g u r e s 8-14). The p l a s t i c d e f o r m a t i o n i n wood was mod e l l e d t h r o u g h t h e p l a s t i c s t r a i n element of th e s t r a i n t e n s o r . T h i s was s i m i l a r t o t h e approach taken by L e s s a r d , et a l . (1982b).
5.2 VISCOELASTIC MODELS
Kawai e t a l . (1979) i n v e s t i g a t e d t h e m o d e l l i n g o f m o i s t u r e t r a n s p o r t and t h e subsequent s t r e s s development i n wood. The wood was mod e l l e d as a v i s c o e l a s t i c o r t h o t r o p i c m a t e r i a l . The s t r e s s / s t r a i n r e l a t i o n s h i p s were computed by s t r e s s induced from m o i s t u r e c o n t e n t g r a d i e n t s . The m o i s t u r e movement was assumed t o be i n t h e l o n g i t u d i n a l d i r e c t i o n , w h i l e s t r e s s development was i n t h e t r a n s v e r s e ( r a d i a l / t a n g e n t i a l ) p l a n e s .
In s i m i l a r f a s h i o n t o L e s s a r d " s work, the s t r e s s e q u a t i o n was:
<^ij = D i j - f f )
s t r e s s t e n s o r
s t i f f n e s s m a t r i x
t o t a l s t r a i n t e n s o r
= i n e l a s t i c s t r a i n t e n s o r
where ^ i j D. .
e
The s t i f f n e s s m a t r i x was approached d i f f e r e n t l y by Kawai. S i n c e the
s t r e s s e s are mod e l l e d i n two di m e n s i o n s , t h e o r t h o t r o p i c n a t u r e of
wood was c o n s i d e r e d :
^ i i ^11 ^12
• 21 ^22
0 d
0
0
23 J
• 11 = \ -/^T-/^TR^
^22 = 2 ^ / ( 1 -/^T-/*TR>
^ 12 = ^21 = ^11*A*TR = ^22-/*RT
^ 33 = ^RT
Scale: 1mm=100.CX)0N/m2 ^ = Tension
Compression
— Principal stress distributions at 2.5 hours in the timber analyzed.
Scole 1mmrlOO,000 N/m ==Tension
Compression
— Principal stress distributions at 10 hours in the timber analyzed.
Scale 1mm =100,000 N/m? ==Tension
Compression C
— Principal stress distributions at 21.5 hours in tne timber analyzed.
Scale: 1mm =100,000 N/m^ = = Tension
Compression
— Principal stress distributions at 60 hours in the timber analyzed.
F i g u r e s 8-14 N u m e r i c a l R e s u l t s from
Morgan, e t a l (1982)
i Stress
- Centre • Ouarter-section
40 5C 6Z- 70 9C IOC '!0 !2D Timeinrs;
— Surtace Figure 9. — Variation of the major principal stress with time, at the surface, center, and quarter section on section X-X.
TifTie(tir&)
FiguretO— Elastic analysis — variation of the major principal stress with time, at the surface, center, and quarter section on section X-X.
Str«s
_ Centre
-Ouorter-section
PO X 40 5 0 6 0 70 SO 90 100 IC IPO (JO TimelhrsI
-Surfoce
FigureA 1- — Variation of the major principal stress on sec-tion X-X with a faster drying rate.
^ S t r e s s 30DD- lk\.rr' i
200C-
, Stress IkN/ip'!
30001
\
2000r
0
- C»ntre -Ouarter-s«:ton
/ ^0\20 30 40 50 60 70 BO 90 100 nO 120 UOTimeihrsI
Surface 4000\-
Figure\Z. — Variation of the major principal stress on sec-tion X-X when Young's modulus and yield strength vary but Poisson's ratio is constant.
4000
3000
2000-
1000-
0
tooo\
2000
3000:
4000\
stress IkN/mJl
. Centre -Quarter-section
60 70 80 90 100 no 120 130 Time inrg
-Surface
Figure 14. - Variation of the major principal stress on sec-tion X-X when all the strength parameters vary.
where E , E = moduli o f e l a s t i c i t y i n t h e r a d i a l and R T
t a n g e n t i a l d i r e c t i o n s
/^RT' /*TR ^ P o i s s o n ' s r a t i o
G^^ = modulus of r i g i d i t y
The c o m p l e x i t y of the problem i n c r e a s e s by c o n s i d e r i n g t h a t the modulus of e l a s t i c i t y and P o i s s o n ' s r a t i o can be f u n c t i o n s of m o i s t u r e c o n t e n t , g r a i n d i r e c t i o n , temperature, p l u s some o t h e r l e s s i m p o r t a n t p a rameters.
The i n e l a s t i c s t r a i n t e n s o r i n v o l v e s a c r e e p c o m p l i a n c e term d u r i n g d r y i n g (Takemura, 1968):
J = J Q ^ (1 + a • (mc))
where J & J Q = c r e e p and e l a s t i c c o m p l i a n c e s d u r i n g d r y i n g j ' 5, J ' = c r e e p and e l a s t i c c o m p l i a n c e s a t e q u i l i b r i u m
s t a t e o f m o i s t u r e , r e s p e c t i v e l y A (mc) = change i n m o i s t u r e c o n t e n t a = a c o n s t a n t
Takemura showed t h a t the i n e l a s t i c s t r a i n c o u l d be e x p r e s s e d a s :
^ f rt d a
( J - J ) d t 0 0 .
T h i s e q u a t i o n was used by Kawai, &t_ a l . ( 1979), f o r t h e i n e l a s t i c
c o n t r i b u t i o n t o t h e s t r a i n t e n s o r . The g e n e r a l o u t l i n e of computing
the d r y i n g s t r e s s e s i s shown i n F i g u r e 15.
C o n s i d e r a b l e e x p e r i m e n t a l d a t a was c o l l e c t e d t o determine t h e c r e e p
c o m p l i a n c e s under d r y i n g , i n v o l v i n g d i f f e r e n t m o i s t u r e c o n t e n t s and
t h r e e d i f f e r e n t d r y i n g c o n d i t i o n s . A r e g r e s s i o n e q u a t i o n was f i t t e d
t h r o u g h the e x p e r i m e n t a l d a t a , r a t h e r than a t t e m p t i n g t o model t h e
cr e e p c o m p l i a n c e s as a f u n c t i o n of d r y i n g c o n d i t i o n s . T h i s approach
c o u l d p o t e n t i a l l y r e q u i r e c o n s i d e r a b l e e x p e r i m e n t a l d a t a g a t h e r i n g i f
c r e e p m o d e l l i n g f o r a wide range of d r y i n g c o n d i t i o n s was t o be
attempted.
The r e s u l t s from Kawai, e t a l . (1979) were e n c o u r a g i n g . S t r e s s r e v e r s a l w i t h r e s p e c t t o d r y i n g time was p r e d i c t e d by t h e model, i n c l u d i n g the v a r i a t i o n o f s t r e s s t hroughout t h e t h i c k n e s s of t h e p i e c e o f wood. The model was a l s o c a p a b l e o f d e s c r i b i n g t h e d i f f e r e n t s t r e s s e s d e v e l o p e d w i t h i n t h e boards f o r t h e t h r e e d i f f e r e n t d r y i n g c o n d i t i o n s .
A r e c e n t paper by Bazant (1985) d i s c u s s e d some t h e o r e t i c a l a s p e c t s of
c r e e p i n wood. Bazant noted some i n t e r e s t i n g e x p e r i m e n t a l
o b s e r v a t i o n s on a c c e l e r a t e d c r e e p r a t e s as a r e s u l t of d r y i n g . He
r e l a t e d t h e wood c r e e p phenomena t o c r e e p found i n cement and
c o n c l u d e d t h a t the fundamental p r o c e s s o f c r e e p i s i d e n t i c a l i n b o t h
cement and wood.
Start
Solution of the diffusion equation
initial and boundary conditions
Moisture distribution, MC(L.M}
Basic shrinkage,}eB8(L_M)j
Observed shrinkage,{e^s(|.,M)t
Constitutive equation for the drying stress In wood as a viscoelastic and orthotropic material
Theory of moisture diffusion in wood
Partial differential equation
Experimental
Basic shrinkage BS(K/IC)
Observed shrinkage (OS(t)
Memory effect of wood during drying
Correction by the stress equilibrium condition
Drying stresses,|a(L,M)} . & Inelastic strains,<£* (L,M)|
Creep
J(t).J'(t)
Stop
F i g . 15 C o m p u t a t i o n o f D r y i n g S t r e s s e s by K a w a i , e t a l . (1979)
Bazant s t a t e d t h a t c r e e p o c c u r s as a r e s u l t o f m i c r o d i f f u s i o n o f water
i n wood. T h i s would be analagous t o t h e h y g r o s c o p i c movement of water
versus D a r c i a n d i f f u s i o n . The h y g r o s c o p i c movement i n t e r f e r e s w i t h
the c e l l u l o s i c bonds (hydrogen bonds), r e s u l t i n g i n t h e c r e e p
b e h a v i o u r . He a l s o s t a t e d t h a t i t i s t h e f l u x g r a d i e n t upon which t h e
c r e e p depends. I f t h e r e i s no r a t e of change o f m i c r o d i f f u s i o n f l u x
( i e . , s t e a d y s t a t e c o n d i t i o n ) , t h e n c r e e p i s n e g l i g i b l e .
Bazant s u p p o r t e d t h i s argument w i t h e x p e r i m e n t a l o b s e r v a t i o n s e x t r a c t e d from t h e wood s c i e n c e l i t e r a t u r e . He the n p r o v i d e d some e q u a t i o n s f o r the m i c r o - and macro- ( h y g r o s c o p i c and c a p i l l a r y ) d i f f u s i o n of water i n wood, from wich he o b t a i n e d t h e d i v e r g e n c e of the h y g r o s c o p i c m o i s t u r e f l u x , upon which t h e r a t e of c r e e p depends. A Maxwell c h a i n model was used t o t h e o r e t i c a l l y d e s c r i b e t h e e x p e r i m e n t a l r e s u l t s .
The s t r e s s / s t r a i n e q u a t i o n s o u t l i n e d by Bazant a r e :
N
Ev ^ v
u n i a x i a l s t r e s s
u n i a x i a l s t r a i n
p a r t i a l s t r e s s e s
r a t e of s t r e s s , s t r a i n
e l a s t i c modulus i n i n d i v i d u a l Maxwell c h a i n s
v i s c o s i t y i n i n d i v i d u a l Maxwell c h a i n s
s h r i n k a g e s t r a i n r a t e
temperature s t r a i n r a t e
The Maxwell c h a i n model i s i l l u s t r a t e d i n F i g u r e 16.
The c r e e p v i s c o s i t i e s a r e determined from t h e e q u a t i o n :
1
where a e
E V
^ v
Cs
€0,
exp
EyTy RT^ RT
f v = 1
U J U J U J u Hi nv
F i g . 16 M a x w e l l C h a i n Model
where = r e l a x a t i o n time of V^*^ element f o r a g i v e n
r e f e r e n c e temperature and m o i s t u r e c o n t e n t
= r e f e r e n c e temperature
<^^(w) = a m o i s t u r e c o n t e n t f u n c t i o n
R = u n i v e r s a l gas c o n s t a n t
Q = a c t i v a t i o n energy f o r c r e e p c
T = a b s o l u t e temperature
The e q u a t i o n Bazant p r o v i d e s f o r ^ y ( w ) i s :
0v(w) = k(wi - w)
where k = a c o n s t a n t
w = m o i s t u r e c o n t e n t at f i b e r s a t u r a t i o n p o i n t
w = m o i s t u r e c o n t e n t
Bazant attempted t o model t h e c r e e p i n wood f o r a wide v a r i e t y of c o n d i t i o n s , which i n c l u d e : wood under l o n g term s t r e s s ; v a r y i n g or c y c l i c h u m i d i t y c o n d i t i o n s ; and c y c l i c l o a d i n g and u n l o a d i n g w i t h r e c o v e r y of the c r e e p d e f o r m a t i o n . H i s t h e o r y was not extended t o c r e e p phenomena f o r a l l e n v i r o n m e n t a l c o n d i t i o n s . However, f o r k i l n d r y i n g lumber h i s e q u a t i o n s are d e f i n i t i v e on how c r e e p c o u l d be m o d e l l e d .
C a u l f i e l d (1985) approached the t h e o r e t i c a l m o d e l l i n g of c r e e p
b e h a v i o r by k i n e t i c t h e o r y . T h i s work was based upon the l o g a r i t h m i c
r e l a t i o n s h i p between s t r e s s l e v e l and time t o f a i l u r e from d u r a t i o n of
l o a d (DOL) e x p e r i m e n t s . Creep was assumed t o be an a c t i v a t e d energy
p r o c e s s which i n v o l v e s an e x p o n e n t i a l r e l a t i o n s h i p . C a u l f i e l d ' s s t u d y
was on DOL and r a t e of l o a d (ROD r e l a t i o n s h i p s w i t h r e s p e c t t o
c r e e p . He suggested f u t u r e r e s e a r c h be done on a c c e l e r a t e d c r e e p and
a c c e l e r a t e d r u p t u r e , i n c y c l i c h u m i d i t y environments.
5.3 LINEAR ELASTIC MODEL
A r e c e n t paper by Cowin (1985) d i s c u s s e d a new approach t o th e
m o d e l l i n g of s h r i n k a g e i n a porous media. Cowin d e s c r i b e d t h e t h e o r y
as " d r y i n g of l i n e a r e l a s t i c m a t e r i a l s w i t h v o i d s " . The t h e o r y was
based on t h e b u l k d e n s i t y of a porous m a t e r i a l . T h i s b u l k d e n s i t y was
made up of the d e n s i t y of t h e m a t r i x m a t e r i a l , and t h e volume f r a c t i o n
t h a t t h e m a t e r i a l o c c u p i e d .
W i t h the f o l l o w i n g p a r a m e t e r s :
P = b u l k d e n s i t y
7"= d e n s i t y of m a t r i x m a t e r i a l
V= volume f r a c t i o n f i e l d
T h i s r e s u l t i s P =7 • I'
If the subscript R refers to the reference or s t a r t i n g density and i f the material i s i n i t i a l l y s tress and s t r a i n f r e e :
<A(x,t) = V{x,t) -
where
<^(x,t) = change i n volume f r a c t i o n with respect to space and time
</)(x,t) = volume f r a c t i o n
= reference volume f r a c t i o n
The i n f i n i t e s i m a l s t r a i n tensor i s determined from the displacement f i e l d Uj by:
^ i j = 1/2 ( U i , j + Uj^i)
where the comma indicates a p a r t i a l d e r i v a t i v e with respect to the second subscript.
With a balance of l i n e a r momentum:
= + p b i
and balance of e q u i l i b r i a t e d forces:
pk <l) = h i ^ i + g + q
whe re:
T^ . = symmetric st r e s s tensor
b^ = body force vector
h^ = e q u i l i b r a t e d stress vector
k = e q u i l i b r a t e d i n e r t i a
g = i n t r i n s i c e q u i l i b r a t e d body force
q = e x t r i n s i c e q u i l i b r a t e d body force
Cowin then related the stress tensor T ^ j , the e q u i l i b r a t e d s t r e s s vector and the i n t r i n s i c e q u i l i b r a t e d body force g to the s t r a i n Cj^j, change i n volume f r a c t i o n ^ , time rate of change of the volume f r a c t i o n ^ and the gradient of the change i n volume f r a c t i o n <^,j/ for an anisotropic material:
T i j = Cijkn, fkm + Bij0
' i = ^ i j </> , j hi =
where ^ijkm ' ' f = functions of
Cowin assumed that only homogenous shrinking occurred, and that a balance of e q u i l i b r a t e d forces r e s u l t s i n :
q = f <A + w ^ + B i j
where q r e p r e s e n t s l o c a l c e n t e r s of compression or d i l a t i o n due t o
s w e l l i n g o r s h r i n k i n g and w and f are m a t e r i a l c o n s t a n t s .
Cowin's f o r m u l a t i o n does a l l o w f o r d e t e r m i n a t i o n of s t r e s s development at any p o i n t i n time, however, he o n l y p r o v i d e d t h e a s s y m p t o t i c r e s u l t s . In a d d i t i o n , he o n l y c o n s i d e r e d an e l a s t i c m a t e r i a l . F u r t h e r t h e o r e t i c a l development of h i s e q u a t i o n s c o u l d r e s u l t i n m o d e l l i n g e l a s t i c - p l a s t i c or p o s s i b l y v i s c o e l a s t i c s o l i d s w i t h r e s p e c t t o space and time.
O t h e r papers which have been reviewed w i t h r e g a r d t o m o d e l l i n g s t r e s s development i n wood i n c l u d e : Bazant & Chern (1985); M a e g l i n e t a l . (1985); P i e r c e e t a l . (1985a and b ) ; B e l l o & K u b l e r (1975); P r i c e (1985); Cave (1972); L e s s e and K i n g s t o n (1972); and F i s h (1983).
5.4 WOOD PROPERTIES
The s o l i d wood matrix i s composed of a va r i e t y of organic substances. The p r i n c i p l e component i s c e l l u l o s e . However, other polysaccharide polymers, p r i m a r i l y substances known as hemicelluloses, also contribute to the strength of wood. The c e l l u l o s i c chains are bound together with an organic "glue" known as l i g n i n .
The p h y s i c a l properties of wood are dependent upon i t s temperature. The changes i n these properties would have to be considered i f one i s to accurately model the s t r e s s e s / s t r a i n s i n wood. For example, H i l l i s and Rosza (1985) investigated the e f f e c t s of high temperature and chemical e f f e c t s on wood s t a b i l i t y . They found that wood underwent a change in phys i c a l properties at temperatures i n the range of conventional hot a i r k i l n temperatures. Softening points at about 80°C and 100°C occurred, due resp e c t i v e l y to the l i g n i n and hemicelluloses found i n wood. This would promote p l a s t i c deformation of the wood when under s t r e s s . A l i t e r a t u r e search on t h i s t o p i c w i l l form part of future work.
5.5 EXPERIMENTAL WORK REQUIRED FOR THE STRESS DEVELOPMENT MODEL
C h o i c e s of a model are an e l a s t i c , v i s c o e l a s t i c , or e l a s t i c - p l a s t i c
s u b s t a n c e t h a t u t i l i z e s t h e r m o - e l a s t i c t h e o r y . The wood can a l s o be
modelled as a l i n e a r - e l a s t i c m a t e r i a l a c c o r d i n g t o t h e p r o c e d u r e
o u t l i n e d by Cowin (1985).
The r e l a t i v e magnitude of the d i f f e r e n t s t r a i n d e f o r m a t i o n s ( e l a s t i c ,
p l a s t i c and c r e e p ) s h o u l d d i c t a t e what model i s t o be used. On t h e
b a s i s of e x p e r i m e n t a l r e s u l t s from L e s s a r d (1982b) and Kawai e t a l .
(1978), a l l t h r e e s t r a i n e f f e c t s can be t h e same magnitude, but i t
appears t h a t p l a s t i c and c r e e p can be s u b s t i t u t e d f o r each o t h e r .
A l t h o u g h t h e e l a s t i c - p l a s t i c d e f o r m a t i o n i s a s i m p l e t h e o r e t i c a l
model, i t does not model temperature and time e f f e c t s .
R e q u i r e d e x p e r i m e n t a l s t u d i e s a r e d i f f i c u l t t o f i n a l i z e a t t h i s t i m e .
E x p e r i m e n t a l d a t a r e q u i r e d f o r c o n f i r m a t i o n and c a l i b r a t i o n of the
s t r e s s development model w i l l depend on t h e model. However, t h e r e a r e
some e a r l y c o n s i d e r a t i o n s on t h e e x p e r i m e n t a l work worth d i s c u s s i n g ,
r e g a r d l e s s of t h e s t r e s s model chosen.
Hauptmann (1985) has s u g g e s t e d t h a t s i n c e t h e m o n i t o r i n g o f s t r e s s i s
so d i f f i c u l t i n wood, a m a t e r i a l w i t h v o i d s ( i e . , foam) c o u l d be used
as a wood s u b s t i t u t e . T h i s would a l l o w f o r more c o n c l u s i v e
v e r i f i c a t i o n of t h e s t r e s s development model. The m a t r i x m a t e r i a l
s h o u l d be h y g r o s c o p i c (not h y d r o p h i l i c ) which w i l l e x h i b i t s h r i n k a g e
and e x p a n s i o n of t h e body. The e x p e r i m e n t a l b l o c k a l s o s h o u l d be
d e s i g n e d so as t o a l l o w m o i s t u r e d i f f u s i o n i n o n l y one or two
d i m e n s i o n s . Upon v e r i f i c a t i o n o f t h e s t r e s s model, e x p e r i m e n t a l work
can b e g i n w i t h wood.
S t r a i n gauges w i l l be i n s e r t e d i n t o t h e foam m a t e r i a l b e f o r e i t s e t s .
The s e t foam w i l l be s a t u r a t e d w i t h water and t h e n exposed t o
p r e d e t e r m i n e d d r y i n g c o n d i t i o n s . The s t r a i n can then be m o n i t o r e d
d u r i n g t h e d r y i n g p r o c e s s . Foam i s a l s o advantageous because of i t s
i s o t r o p i c p r o p e r t i e s .
An i m p o r t a n t , c l o s e l y a s s o c i a t e d parameter would be the l o c a l , volume-averaged m o i s t u r e c o n t e n t . E x p e r i m e n t a l d e t e r m i n a t i o n of t h i s parameter may be d i f f i c u l t . One method would be t o see i f t h e e l e c t r i c a l r e s i s t a n c e can be c o r r e l a t e d t o t h e l o c a l m o i s t u r e c o n t e n t . Temperature probes can be used t o a c c u r a t e l y measure the l o c a l t e m p e r a t u r e . M i n i a t u r e hygrometers can be used t o measure t h e l o c a l r e l a t i v e h u m i d i t y .
Under i s o t h e r m a l c o n d i t i o n s , vapor p r e s s u r e i s t h e d r i v i n g f o r c e f o r
m o i s t u r e d i f f u s i o n . There have been attempts t o e x p e r i m e n t a l l y
measure l o c a l vapor p r e s s u r e i n wood (Lowry, 1971) but t h e r e can be a
l a r g e amount of e x p e r i m e n t a l e r r o r . Vermass (1978) p r o v i d e s a method
f o r measuring l o c a l vapor p r e s s u r e i n wood by u s i n g embedded c a p i l l a r y
t u b e s . C o m m e r c i a l l y a v a i l a b l e m i c r o e l e c t r o n i c p r e s s u r e s e n s o r s c o u l d
a l s o be used i n s i d e wood.
6.0 OTHER RESEARCH PRESENTLY BEING PERFORMED
An o v e r v i e w of worldwide r e s e a r c h i n lumber d r y i n g i n d i c a t e s a h i g h
l e v e l of i n t e r e s t i n m o i s t u r e t r a n s p o r t and s t r e s s development
phenomena (Rosen, 1985). The r e s e a r c h i n c l u d e s :
A u s t r i a
Measurement of s t r a i n caused by s h r i n k a g e , as a parameter f o r
k i l n c o n t r o l .
F r a n c e
Heat and mass t r a n s f e r d u r i n g wood d r y i n g at h i g h
t e m p e r a t u r e s ( t o 1 8 0 ° C ) , under vacuum and p r e s s u r e up t o two
atmospheres.
T h e o r e t i c a l a n a l y s e s and e x p e r i m e n t a l r e s u l t s on t h e p r e s s u r e
f i e l d s g e n e r a t e d i n t h e c e n t e r of boards b e i n g d r i e d under
c e r t a i n c o n d i t i o n s .
Development of s t r e s s d u r i n g d r y i n g , based on l i n e a r e l a s t i c
phenomena.
D r y i n g s t r e s s e s based on v i s c o e l a s t i c a s p e c t s .
West Germany
I n v e s t i g a t i o n o f t h e development of d r y i n g s t r e s s e s and t h e p o s s i b i l i t y of t h e i r r e d u c t i o n .
Sweden
Computer s i m u l a t i o n of t h e d r y i n g p r o c e s s .
I n v e s t i g a t i o n of m o i s t u r e t r a n s p o r t c o e f f i c i e n t s i n wood.
Development of a d i f f u s i o n model f o r m o i s t u r e t r a n s p o r t
adapted t o p r a c t i c a l p u r p o s e s .
Study of the m e c h a n i c a l p r o p e r t i e s of wood as a f u n c t i o n of d e n s i t y , m o i s t u r e c o n t e n t and t e m p e r a t u r e ; f o r t h e use i n a s i m u l a t i o n model f o r d r y i n g and r e s u l t i n g b u i l d u p of s t r a i n .
Weyerhauser Company, Tacoma, Washington
A d a p t i v e k i l n c o n t r o l , u s i n g i n - k i l n m o i s t u r e measurement and
a wood d r y i n g model (Ka y i h a n model).
U n i v e r s i t y o f M i n n e s o t a
E f f e c t of p r e f r e e z i n g and o t h e r p r e t r e a t m e n t s upon the
p e r p e n d i c u l a r - t o - g r a i n c r e e p of wood d u r i n g d r y i n g .
Energy and e l a s t i c s t r a i n a n a l y s i s f o r t h e d r y i n g of lumber
i n steam heated and d e h u m i d i f i c a t i o n k i l n s .
N o n i s o t h e r m a l m o i s t u r e movement and the c e i l i n g / p a r t i t i o n /
s e p a r a t i o n problem.
U n i v e r s i t y of New Hampshire
Computer a s s i s t e d p r o c e s s c o n t r o l f o r o p t i m a l c o n t r o l t h e o r y .
P r e d i c t i o n o f m o i s t u r e and s t r a i n d i s t r i b u t i o n s d u r i n g d r y i n g
t o produce o p t i m i z e d d r y i n g s c h e d u l e s .
U n i v e r s i t y o f Washington
Fundamental pathways of m o i s t u r e f l o w d u r i n g d r y i n g .
M a t h e m a t i c a l models of m i c r o and macro m o i s t u r e f l o w d u r i n g d r y i n g .
7.0 HEAT AND MASS TRANSFER COEFFICIENTS
L i m i t e d work has been done on heat t r a n s f e r c h a r a c t e r i s t i c s i n s i d e a lumber dry k i l n e nvironment. The work t h a t has appeared has o f t e n been h i g h l y e m p i r i c a l w i t h a n o n t e c h n i c a l approach t o t h e problem (Huber, 1982; Beard ^ a l . , 1983; Horton and Resch, 1976 ; S t e i n h a g e n , 1974; P r i c e and Koch, 1981; G r e e n h i l l , 1936; Lyman, 1963).
C o n v e c t i v e heat t r a n s f e r c o e f f i c i e n t s f o r i d e a l i z e d c o n d i t i o n s have been r e s e a r c h e d t o a l i m i t e d e x t e n t by Plumb e t a l . (1984), and used t o v e r f i y the D i t t u s - B o e l t e r e q u a t i o n ( B u r m e i s t e r , 1983). Spolek (1981a) used a s i m i l a r e q u a t i o n f o r t h e heat t r a s f e r c o e f f i c i e n t i n h i s model, which was s u b s e q u e n t l y used i n t h e work done by Ka y i h a n e t a l . (1984). Meroney and H s i (1975) i n v e s t i g a t e d v o r t e x enhancement methods t o improve heat and mass t r a n s f e r .
I n f o r m a t i o n on t h e i n t e r a c t i o n between mass and heat t r a n s f e r
c o e f f i c i e n t s i s l i m i t e d . The mass t r a n s f e r c o e f f i c i e n t d u r i n g wood
d r y i n g has been i n v e s t i g a t e d by Spolek (1981a). Kayihan et al. (1984)
assumed the mass t r a n s f e r c o e f f i c i e n t was dependent upon the mass
t r a n s f e r r a t e i t s e l f u n l e s s t h e c o n v e c t i v e mass t r a n s f e r was v e r y low
and the mass f l u x i n the wood was h i g h . Under t h o s e c o n d i t i o n s ,
c o r r e l a t i o n s from e x p e r i m e n t a l d a t a were used t o e s t a b l i s h t h e mass
t r a n s f e r c o e f f i c i e n t .
B u r m e i s t e r (1983) has p r o v i d e d an e x c e l l e n t d e s c r i p t i o n o f mass
t r a n s f e r c o e f f i c i e n t s . The approach i s s i m i l a r t o e s t a b l i s h i n g heat
t r a n s f e r c o e f f i c i e n t s , s i n c e t h e d i f f u s i o n e q u a t i o n i s assumed t o be
of the same form as t h e energy e q u a t i o n (under s t e a d y - s t a t e
c o n d i t i o n s ) . C o n s i d e r the f o l l o w i n g s i m p l i f i e d case of heat and mass
d i f f u s i o n :
Energy u a2_T
~8?
D i f f u s i o n u 3 w i
ax Dl2
^1
By'
where u = v e l o c i t y i n x - d i r e c t i o n
T = temperature
Oi = t h e r m a l d i f f u s i v i t y c o n s t a n t
w = mass f r a c t i o n of s p e c i e s 1
Dl2 = d i f f u s i o n c o n s t a n t of s p e c i e s 1 t h r o u g h 2
T h e r e f o r e , the heat t r a n s f e r and the mass t r a n s f e r e q u a t i o n s are o f s i m i l a r form.
^CONV = h ( T c o - T ^ )
mi = hn, ( W „ - WQO)
where ^ ONV = heat t r a n s f e r c o e f f i c i e n t
T^, T^ = temperature i n stream, a t w a l l
m^ = mass t r a n s f e r r a t e
h = mass t r a n s f e r c o e f f i c i e n t m
^co ~ mass f r a c t i o n of s p e c i e s 1 a t t h e w a l l and
i n the stream
h = heat t r a n s f e r c o e f f i c i e n t
These e q u a t i o n s are v a l i d f o r m o i s t u r e s a t u r a t e d porous s u r f a c e s . The d i f f u s i o n e q u a t i o n from B u r m e i s t e r assumes a c o n c e n t r a t i o n g r a d i e n t as the d r i v i n g f o r c e f o r m o i s t u r e t r a n s p o r t .
However, t h i s c o n d i t i o n o c c u r s o n l y i n t h e e a r l y p a r t of t h e d r y i n g
p r o c e s s and o n l y i n v e r y porous woods. The s u r f a c e of the board i s
u s u a l l y not c o m p l e t e l y s a t u r a t e d and t e n d s t o approach t h e d r y b u l b
temperature (Beard et al., 1985). Spolek (1981a) o u t l i n e d a
t h e o r e t i c a l p r o c e d u r e t o determine t h e mass t r a n s f e r c o e f f i c i e n t a t
a d d i t i o n a l s t a g e s of the d r y i n g p r o c e s s and p r o v i d e d some v a l u e s f o r
th e mass t r a n s f e r c o e f f i c i e n t above and below t h e f i b e r s a t u r a t i o n
p o i n t .
The mass t r a n s f e r of a c h e m i c a l s p e c i e s w i t h i n a t u r b u l e n t f l o w f i e l d
i s c h a r a c t e r i z e d by t h e Sherwood ( S h ) , Reynolds (Re) and Schmidt (Sc)
non d i m e n s i o n a l numbers. There are no p u b l i s h e d d a t a on r e l a t i o n s h i p s
between t h e s e numbers w i t h r e s p e c t t o d r y i n g of wood. In a d d i t i o n ,
the t h e o r e t i c a l development of c h e m i c a l - p o t e n t i a l - d r i v e n d i f f u s i o n and
i t s r e l a t i o n s h i p t o t h e no n d i m e n s i o n a l numbers l i s t e d above has not
been d e s c r i b e d .
I n v e s t i g a t i o n w i l l be c o n t i n u e d i n t o the f l o w c h a r a c t e r i s t i c s of a i r
thro u g h a s t i c k e r e d p i l e of lumber, as o u t l i n e d by Zwick (1985b). The
heat and mass t r a n s f e r c o e f f i c i e n t s w i l l be dete r m i n e d and v e r i f i e d .
The i n t e r a c t i o n of s t i c k e r t h i c k n e s s , i . e . , a i r gap; w i t h heat and
mass t r a n s f e r c o e f f i c i e n t s , w i l l be i n v e s t i g a t e d . F i n a l l y , i t w i l l be
v e r i f i e d whether t h e heat and mass t r a n s f e r c o e f f i c i e n t s are
independent of each o t h e r .
8.0 NUMERICAL METHODS
The l i t e r a t u r e on n u m e r i c a l l y s o l v i n g the e q u a t i o n s mentioned i n t h i s paper i s e x t e n s i v e . One problem t o be a d d r e s s e d i s t h e q u e s t i o n of m o d e l l i n g the m o i s t u r e d i f f u s i o n i n two d i m e n s i o n s . Most of the l i t e r a t u r e on n u m e r i c a l l y s o l v i n g c o u p l e d d i f f e r e n t i a l e q u a t i o n s i s i n one dimension, which u s u a l l y reduces the e q u a t i o n t o an o r d i n a r y d i f f e r e n t i a l e q u a t i o n .
Patankar (1980) has o u t l i n e d a l g o r i t h m s t o n u m e r i c a l l y s o l v e s e v e r a l t y p e s of 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 ( p a r a b o l i c , e l l i p t i c , h y p e r b o l i c ) , i n c l u d i n g a n a l y s i s i n t h r e e d i m e n s i o n s . In t h i s s t u d y e i t h e r C a r t e s i a n c o o r d i n a t e s or p o l a r c o o r d i n a t e s ( b o t h i n two dimensions) w i l l be used t o model the wood.
Kay i h a n e t a l . (1984) s o l v e d t h e i r m o i s t u r e t r a n s p o r t model by t h e f i n i t e d i f f e r e n c e t e c h n i q u e , i n one d i m e n s i o n . The s o l u t i o n t o t h e d i f f u s i o n problem r e q u i r e s knowledge of where t h e moving e v a p o r a t i o n f r o n t i s l o c a t e d d u r i n g the d r y i n g p r o c e s s . T h i s can be d e s c r i b e d m a t h e m a t i c a l l y as a moving boundary v a l u e problem. In o r d e r t o c o n s e r v e computation time, the f i n i t e d i f f e r e n c e g r i d s p a c i n g used by K a y i h a n e_t aj.. (1984) was r e l a t i v e l y c o a r s e , except f o r t h e a r e a of the moving e v a p o r a t i o n f r o n t , which had a f i n e r g r i d s p a c i n g . The Kayihan model i s i n t e n d e d f o r a d a p t a t i o n t o a p r o c e s s c o n t r o l system f o r a k i l n .
The n u m e r i c a l s o l u t i o n t o t h e s t r e s s e q u a t i o n s s h o u l d not be
d i f f i c u l t . P r e v i o u s work u s u a l l y r e l i e d on f i n i t e element t e c h n i q u e s
t o e v a l u a t e the s t r e s s / s t r a i n f i e l d s (Johnson, 1984; Lewis e t a l . ,
1984). The f i n i t e element method (FEM) was used i n o r d e r t o
accomodate n o n - c o n v e n t i o n a l shapes and g r i d s p a c i n g s . T h i s n u m e r i c a l
method may be n e c e s s a r y t o use, s i n c e d i m e n s i o n a l changes ( a s s o c i a t e d
w i t h s h r i n k a g e ) s h o u l d be a c c o u n t e d f o r w i t h s t r e s s models f o r wood.
There appears t o be no c o m p u t a t i o n a l advantage, w i t h r e g a r d t o
a c c u r a c y , by u s i n g t h i s t e c h n i q u e o v e r o t h e r methods.
A r e l a t i v e l y new n u m e r i c a l method has been used f o r s o l v i n g 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 . The boundary i n t e g r a l element method (BIEM) i s a t e c h n i q u e e s p e c i a l l y adapted f o r time-dependent problems (Taigbenu and L i g g e t t , 1985), such as d i f f u s i o n i n porous media. S o l u t i o n s u s i n g BIEM appear more a c c u r a t e , and a r e s t a b l e f o r a wider v a r i e t y of c o n d i t i o n s , than when us i n g the FEM. B r e b b i a (1984) p r o v i d e s s o l u t i o n s t o c o u p l e d 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 u s i n g BIEM.
I t i s i n t e n d e d t o extend the m o i s t u r e t r a n s p o r t / s t r e s s model i n t o
p r o c e s s c o n t r o l a p p l i c a t i o n s , where an o b j e c t i v e c o s t f u n c t i o n c o u l d
command the o v e r a l l system. T h i s o b j e c t i v e f u n c t i o n would p r o b a b l y
have t o be s o l v e d by a n o n l i n e a r programming t e c h n i q u e , which i s w e l l
e s t a b l i s h e d i n management s c i e n c e l i t e r a t u r e ( B r a d l e y e t , 1977).
The u l t i m a t e g o a l of optimum p r o c e s s c o n t r o l c o u l d t h e n be attempted.
9.0 CONCLUSIONS
C o n t i n u i n g s t u d i e s s h o u l d i n v o l v e e x t e n d i n g the Kayihan model i n t o two
dime n s i o n s , which w i l l r e s u l t i n b e i n g a b l e t o p r e d i c t m o i s t u r e
c o n t e n t p r o f i l e s i n a board c r o s s s e c t i o n d u r i n g d r y i n g . E x p e r i m e n t a l
work w i l l b e g i n w i t h d e t e r m i n i n g heat and mass t r a n s f e r c o e f f i c i e n t s
of a board, i n an i n d u s t r i a l d r y - k i l n environment. Subsequent
e x p e r i m e n t a l work w i l l i n v e s t i g a t e d i f f u s i o n c o n s t a n t s on wood s p e c i e s
of economic i n t e r e s t t o t h e Canadian f o r e s t i n d u s t r y .
The proposed d r y i n g t h e o r y w i l l go beyond m o d e l l i n g m o i s t u r e t r a n s p o r t
i n wood. T h i s model s h o u l d be c a p a b l e of d e s c r i b i n g d r y i n g o f
no n - h y g r o s c o p i c m a t e r i a l s such as b r i c k , gypsum board, o r cement. I t
c o u l d a l s o be u s e f u l i n the d e s i g n o f d r i e r s f o r o t h e r i n d u s t r i e s .
A l i t e r a t u r e review shows l i m i t e d r e s e a r c h d a t a on t h e m o d e l l i n g of s t r e s s i n wood. No a c c e p t e d model c o r r e c t l y d e s c r i b e s the s t r e s s development i n a boar d . P a r t of t h e reason i s t h a t an adequate m o i s t u r e t r a n s p o r t model has not been u t i l i z e d . In a d d i t i o n , t h e t h e o r e t i c a l r e s u l t s on t h e n o n l i n e a r i t y of wood s t r e s s / s t r a i n r e l a t i o n s h i p s are not m o d e l l i n g e x p e r i m e n t a l r e s u l t s a c c u r a t e l y .
I t i s proposed t o approach t h e s t r e s s model i n a manner s i m i l a r t o t h a t used by Bazant (1985) o r C a u l f i e l d (1985). A model which u t i l i z e s a t h e o r y on c r e e p s h o u l d be a b l e t o v e r i f y e x p e r i m e n t a l r e s u l t s f o r a wide v a r i e t y of d r y i n g c o n d i t i o n s . Only the p h y s i c a l c o n s t a n t s r e l a t i n g t o t h e model would have t o be d e t e r m i n e d . M o d e l l i n g s t r e s s w i t h an e l a s t i c - p l a s t i c model does not appear t o be adequate.
Cowin's f o r m u l a t i o n of t h e s t r e s s e q u a t i o n s may m e r i t an a d d i t i o n a l
e x a m i n a t i o n . The e q u a t i o n s appear sound, y e t p r o v i d e a d i f f e r e n t
approach f o r m o d e l l i n g s h r i n k a g e , s t r e s s and s t r a i n . Cowin's p r e s e n t
model o n l y models a l i n e a r - e l a s t i c m a t e r i a l . C o n s i d e r a b l e time and
e f f o r t may be r e q u i r e d t o determine and q u a n t i f y some of t h e terms
used i n h i s model, f o r t h e case of v i s c o e l a s t i c s t r e s s / s t r a i n
r e l a t i o n s h i p s .
The m o d e l l i n g of s t r e s s development d u r i n g d r y i n g w i l l be u s e f u l f o r
i t s p o t e n t i a l as a p r o c e s s c o n t r o l system i n lumber d r y k i l n s . The
s t r e s s model w i l l be r e q u i r e d f o r the p r e d i c t i o n of degrade d u r i n g
d r y i n g . A l s o , t h e s h r i n k a g e o f wood w i l l be measureable and may
c o n t r i b u t e i n f o r m a t i o n t o the p r o c e s s c o n t r o l system.
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