A Mathematical Model of Wood Behaviour

Copyright © IF.\(: :\ ulolllatic \kasun:I1lt'1l1 alld Contro l in \\"ood\\"orking I lldu:-.lr\". Br;lIi:-.I;t\;t. Czechos lmakja, I'lHli

A MATHEMATICAL MODEL OF WOOD BEHAVIOUR B. A. Todorov*, Chr. K. Shec htov* and S. G. Georgieva** " l liglll'!" Fornl· Tl'rillli((l/ IlI.Ililllll', SII/ia, HlIlgaria ""FlIntillln' (11111 Dnigll IlI.Ililllll' , SII/ia, fill/garill

Abstract. The mathematical model of wood behaviour in kiln drying process assoslates two submodels: two - dimensional heat - and moisture transfer model and one - dimensional tensely state model, which takes into account elastic and plastic properties of wood. One could solve three kinds of tasks with the model of behaviour: first, to investizate existing kiln drying schedules to recieve information of heat - and moisture fields distribution and tensions in tirneduring the process;second,to generate new schedules with constrained conditions - actual tensions to be less then maximum admissible and third, to control the process with the last constrains by microcomputer system. Keywords. ~'Iodels; process control; numerical analysis; !>'ood processing; stess control . INTRODUCTIO:-J

:IAT!!P'ATICAL IIODEL

Drying of wood is a typical unsteady state process of heat- and moisture transfer, carried out by the potentials of the transfer: temperature and chemical potential. The uneven distribution of the temnerature and moisture content as well as th~ir al teration in the process of drying cause elastic and plastic deformations, responsible for the dried wood quality. The comp l ex phenomena necessitate the creation and the application of various dryin~ methods and algorythns to control tile process. Liukov and :Iic ilailov (1963 ) and Liuko v (1974, 1978) introduce a system of differential equations of parabolic type to des cribe heat-and mass exchange processes in capillary coloidal bodies to which wood is referred as ..'ell. Tholaas, Lewis and :- Ior~an (1980) solve the differential equations system by Liukov usinR finite difference elements ~ethod. According t o the authors there is a reasonable agreeme nt between the obtained numerical results and the experinen tal ones , taken fron \eyll,ertll . Siau (19 7 1) . ;\engert (1976, 1977), \uan Bui , [hoong and l\'Jdcl (1980) create a nodel of r.lOisturc trans fer in the wood, using rich's second law . Ho !>'ever, they e:,anine the process wi thout addit ional effects or phase transision ancl trans fer under the tenperature gradient. Lapshin (1966) and Jgole\' (1971) i\'esti gate flat stressed condition of wood as an elastic bo dv in the initial period of drying. !Juring this' period estimated deformations and stresses are near to those , experimentaly determined. Sinniak (19 75) creates algoryth r: for determi ning an one- dimensional tensely state of the wood as an elastic- plastic body for both periods of the process . She obtains analytical presentations for the resultant stresses at three types parabolic functions of moistu re content distribution, hut not as a solution of the heat - and moisture transfer equations . 1-19

The mathematical p.lOdel of wood behavior in convective kiln drying associates the di fferencial equations of heat - and moisture transfer sug~ested by Liukov and ~Iichailov (1967) and Sinniak ' s one- dimensiona l ten sely state model. The equations of heat- and moisture transfer have the following presentation: f r 7 at 0 ~lj (1 ) ar a'rt + - c- ar all 'S'T

0

il'rv

+

[)8

?

'1 ~ t

(2 )

with convective boundary conditions

where t is wood temperature; 1 - time; a -termal diffusion coefficeint; f - phase transition criterion; C - heat capacity of <;ross !>'ooel; IJ - moist'Jre content ; D - mois ture diffusion coefficient; r - heat of ? 0 vaporisation; '1- - Laplass operator; at heat transfer coefficient; ta - air temperature; ts - gross wood surface tenperature; ~ - thermal conductivity coeficient of gross wood; '1 - \a~la operator; lie - equilibrium moisture content of the air; Vs - surface moisture content of gross wood; Po - density of d ried woo d . The one- Jimensional tensely state model treats infinite flat plate with thikness 2R and sYMetrically distribution of the moisture content with respect to the Y

B. :\ . Todorm . Chr. 1\. . Shcc htm and S. (; . Gcorg ic\'a

150

ax is , Fig . 1 .

thl' t)"l' c of func t iont is one and the same: ~1

~Itb+ [ [' +C 4 (t-60)] (U .- Uf ) x J X

x

( 9)

YIUI

~ xb

- basic value of the modules experi mentally de termin e d at 60·C: :.Ixb ~ E , E~ , Kb b

-~'"=x:-::-.""o'---t-:x=-.nR-

Th e basic v alu es of th e modu l es are shown in TABLE 1 in kgf/cm2 and coefficients Cl"" ,C 4 fr om Eq . (9) .

x

TABLE 1

lR

Fi g . 1.

Infinit e flat plate and moisture cont e nt distributi on .

During the first period of th e cess, when mo istur e content in laye rs of the pl a t e is gr ea t e r fibr e saturated point - U t he f stersses are determined by th e equations: a

d r y i ng prothe mi dd l e then t he result a nt following (5)

X

where E ~ EeK I(Ee + K ). Ee - mo du l of x x'X x x ' x elastic defor mat ions; Kx - modul of r es idual deformations; Ex - result an t defo r ma tions. The resultant defo rmation s are de ter mined by th e elastic a nd the equilibrium defo r ma tions:

Basic Values of De formation ~Io ­ Jules anJ EquatIon Co e ffIcIents for Thr ee Wood Sp ec I es

Def liood :-Iod Spec . e Eb K b

Eb

Birch Oak Beec h Birc h Oak Beec h Birch Oak Beech

Basic Valu es

Coeff iceints Cl

- 5 . 62 - 5.00

650 900 1000 2350 2000

- 18 . 70 - 34 .4 0 - 4.38 - 51 . 20 - 5.13 -1 4 . 50 -1 4 . 1 0

HiOO

510 620 615

C2 - 0 . 047 - 0 . 280 +0 .1 57 - 0 . 08 1 - 0 .51 5 +0.656 - 0.038 +0 . 100 +0 . 1 38

C4 - 1 23 0 . 420 2.1 70 - 1 73 - 146 0 . 414 - 1 34 0 . 650 - 4 28 14.600 - 280 2.750 - 73.5 0 . 390 - 124 3 . 250 96 . 5 3 .520 C

J

-

NUNERICAL SO LUTIO N OF THE NODEL Th e so lution of Eq . (1 ) and (2) is obtained by finite differences me tho d w'i th the Crank ~icho l son 's scheme on rectangular two - dimensiona l mesh o r grid with I x J nodes si tu ated on I of the sec tion . The approxi mation o f Eq . (1) and (2) has the followin g pr esenta tion:

(6 )

n l - ~t ,n)+ - 4 t n1J ..+l ) +O . ~"F 0 (n t.l+J l ' +t.nI - l'J +t n1]+ .. 1 +t·. .. 1J IJ where

a

Cl 0)

is a shrinkage coeffic i en t .

Du ri ng the second pe ri od of th e process , when the mo istur e content i n the mi ddle laye rs of the pl a t e is l ess then th e f i br e saturated point th e r esu lt ant st re sses a

X

( K a _am)/(K /E - 1 )

x x

X

X

X

'

!In. +.l - 1J

~

{.n 0 -c '(V n + l vn +l Un +l V n+1 ' l'J'+ . ~ . o l· +lJ· + 1. . 1. l'+ " l+ IJJ 1J+

(~)

wh e r e am ar e th e max i mum values of the x stresses o bta i ned durin g th e firs t per i od . The unknown value of th e equilibrium defor mation is determined b)" th e condi tion of equality of strain and compression st r esse s at s vmetr ic al distributio;l of t:~L' :-10i!': ,I, :' content: R

faxdx o

o

(8)

The defo r r..ation !·1Oc'.~1 1es: :.: ' Eex and ': x for different "'ooJ species ha\'e d ifferent va lues, depend on the t empe r a tur e and t he mo isture content bu t for th a t t)lree 110lb l es

o

0

where Fo ~ aVT/V-; Fo ' DVT/V -; VT - step in time; V - step in space . The indexes abo\'e in Eq . (10) and (11) shO\,' t and II in t he kno,,'n n-th and unkno wn n+l t h moment of the ti me . The indexes below - i and i - show th e componen t s of t and il to th e ~espective nodes' of th e gr i d . The ma trix equations, obta ined as a r esu lt of the approximation are solved b)" the consecutive uppe r r elaxa ti on method , Potter (1973): ",P Fo D+ 1 t ( tl+ 1j +tl:i j + ti j 2(1 +?Fo)

.-\

~[atheJl1atical ~[oclel

ISI

of \\'oocl Beh,I\'iour

computed

(~
from the following equation:

E •

a(Uf-Uj)--r

o.

E •

J J

(19 ) (20)

E.

J J

For the second period of the process, when Uo
(21)) where

The relaxation parameter is determined by Chebishov's polinomials: (~o =1; j=ll (1-11 2/2); for p;;'l w p + l

2

m

11 (1-1l 2wp 14). m

S7

The vectors V.. and W.. are formed by the IJ 1J known values of the co~ponents t and U in the known n-th moment of time:

v .. IJ

1-2Fo tn. + Fo (t n .+tn .+ 1+2Fo IJ 2(1+2Fo) 1+1J l-lJ

(14 ) +t n.. l+t n.. 1+"'IUn) .. IJ+ IJIJ l-2Fo' n Fo' n n w.. - - - U .. + [ lJi+1J·+lJi-lJ·+ IJ 1+2Fo' IJ 2(1+2Fo') +\jn. l+U n . ,+8 (t n .+t n .+tn. +tn. _ IJ + IJ 1+1J l-lJ IJ +1 IJ-l

J-l K.E.(UJ-U.) + Kj+lE j + l (UJ-U i + l )] ~ [ J J J j=l K.-E. Kj +l-E j +1 (22) J J

t.

t.

n J-l E .om E. 1 0 . 1 r .....L..L + J + J + j=l K. -E. K +l-E +1 j j J J

J-1 K.. E. ~

(~

+

2 j = 1 K.-E. J J

K E j +, j + 1)

(23)

(24)

Kj + 1E j + 1

U - moisture content at the central node. J The resultant deformations and stresses are computed by the equations: (25)

E •

J

-i

-4t

n

ij

)]

o.

(15)

In the parameter of relaxation Il m is the maximum own value of the matrixes 2FoI (1+2Fo)<1,

(16 )

2Fo'l (1+ZFo' )<1.

(17 )

The boundary conditions anproximation in finite differences Rohsenov's approach (1973) is followed. The numerical solution of Eq. (1) and (2 ) gives a possibility for deeper investigation of the tensely state of the wood and optimizing the process of kiln drying. SinCe the one- dimensional stressed condition, values of tJj and UJj are necessary only or shortly tj and Uj . Then the defromations and the stresses are presented on the space grid as vectors with components E., Ee, K· e J J J E j' E j' 0 j' determined in nodes with nunber j .

The unknown equilibrium defor mation value is calculated by numerical integration of Ea.. (8). For the first drying period, when Uo;;'U f

(1 S )

where X is a number of the node to which the resultant deformations and stesses are

J

=

E.(K.E.-om)/(K.-E.)

JJJJ

JJ

(26)

Eq. (12) ... ( 2 6) represent the numerical SQlution of the model which reflects two basic interrelated phenomena: heat- and moisture transfer and a tensely state of moist wood. RESULTS AND DISCUSSION The adequacy of the heat- and moisture transfer l~oJel has been cO ;1 f ~ lrmed by experimental investigations with beech woo d , ;-o do,QV (198.+ ) but however, no quantiti1tiv~ assessment of the stresed condition has been made by experinental data. One could solve three types of tasks with the model of behaviour. 1. Analysis of temperature and moisture content distribution and evolution in wood and the related deformations and stresses at given control parameters from existing drying schedules: tepmerature, relative humidity and velocity of the environment. 2. A Generation of new drying schedules with restricting condition: the resultant stresses should be less then the maximum admissible for a given wood species. 3. An autonatic control with a restricting condition from item 2 above by microcomputer system. On Fig. 2 are shown results from an analysis of normative drying schedule of beech wood with thickness of 60 mm used in Bulgaria, recieved as a solution of the model. nuring the first period the computed stresses on the surface inconsiderably exceed

B ..\. T"dor()\, ChI. 1\. . Shcchl()\ a lld S, ( ;, (;coq("icla

152

0,98

0,45

100

0,40

0,84

~ ~ I::::,

0,75

70

&:

0::. 056

I

u

~

"b


020 ' ~ 0,42

V1 V1
20

tI)

10

§ 025 "-

::::J ......

~

.c

~ 0

1:: 0 15

~


::.

~ Cl

~

~

30

, /

0,05

----- ..... ----c::

--- ---

S

-40

20

40

80 Time 'C , h

60

100

120

140

160

Theoretical results from an investigation of &ll~arian ' s normative schedu l e for ~eeech material Kith thickness of GO nm , recieved as a solution of the Kood behaviou mode l : 1 - relative hunidity of the environent ; 2 - temperature of the enviroment; 3 - naxinun adnissible strain stresses at the surface ; J - computed strain stresses at the surface ; 5 - conputed conpression stresses in the middle layers; 6 - maximum admissihle conpression stresses in the middle layers; 7 average ];1O i sturc content of the nateri:ll .

the maximum admissible \'alues at each transition from higher to 10Kcr r e lativ e humidity . TlVo peaks of stresses appear on t:le surface exceeding about 1 0 to 1 , 5 .1 0 Pa at an avera ge are the differences bet~een the stresses in the central layer . Quick reduction of the resul tant stresses in compress ion h'i t~l naximum admissible ones in ~bserveJ at avera ge moisture content 0 . 15 k~/k~. Ohviously, a defin it e reserve exist in this period for an acceleration of the process. On Fig. 3 are sho~n r esu lts fron ~cneratior of nel, dryin~ schedule for tcec:1 h'o03 ,,:i th thich.ness of 60 ,l n , recei\'c,; as a solution of the nodel "'i th a restrict ion

"'here BI and B, are constants le ss t:;en 1 onax _ J:la ximun admissible \'alues of stre J

5

~

0

Fig. 2.

I '\

6

0

Vi

-30 O{JO

\

"

.!::

-20

, ,...

, .,.,. 7'.". 0,65

045' .... ... '

c:: 6

0

,, ,

0,56

I

-70 0,14

0,00

40

I

"'J

,--

V1

~ 028

~ 0,10 ::.

~

...

50

lJ)

, ,..-,

.

~2

60

0,70

~

..i

80

-- 0;35

..... 0;30 :r: c::

66°C ---

64°C

90

sses in I,j node , function of the tenperatu re and the moisture content calculated at the same node .

For beech ~ooJ the naxinurn admissible stresses are estinated and approcsimated in folo"in :; eO,uations:

ab (t ) a,

,)

=

ab

+

Cl (t - t o )

(29)

= 11.10" Pa ; C = - 0 .2; C,= - 2.7');

l

c_= n. (lO J ,)

In accordance "ith the results obtained, restriction (2 7 ) is satisfied during the "hole period of dry i n~ . The stean tTeatnent is usually made at average mois ture content of "oDd bc t" een 0.:: and 0 . 19 kg/kg in p rac ti,:e. T!le solution resul ts of the nodel S:1O'" that just in this in t en'al of the a,' e ra~e moisture content , the relative huni Jity of the air is ~reatest . This r::caLS ~~)1"':

of steJ.r·,

model predicts tile r.l~;>l:: :10: : l~: l : tre3t:-~c: , t, ~.' .l:iL\ is ap!ll ie . . l i:l

practice ::is "ell. This fact is an indirect

,\ \Ltthclllatical \i ()(lc l

()r

\\'()()d Ikh;l\-i"ur

0,45

90 80

0{34

0\0. 40 ~

70

~

;,; 035 ::t

I::)

60

070

Q:

& 50

- 0;30

~c

I

l{)

's;:> 40

80;5

b 30

I

,

,..",""-------

I

, I

I

~

I~

20

Qj

'- 10

~

V)

0)5

e~ 0)0

0,28

III

----~----....

- 10

c

0.14

-20

0,00

-40

- 30 0.00

Fig . .).

I ...............

I

I

:

I

/"-

li

L -_ _ u~

o

I ... _- .... 1

- ..... _-- ----

"

6

__~__~__~__~__~__~__~~__~__~__~__~__~__~__~__~__~~~ ~ ~ W ~ W ~ 140 1W 180 Time T, h

A neK relative ~unidity schedule gene rati on of th e env ir onen t a t the sane temp erature increasenent frol~ fi~ . 2 , thickness of th e ma t er i a l, re c ieved as a solution of KOOt! hehav i ou r nodel Kith cons trai ned condition fron Eo . (27): 1 - relative hunidity ("I f the enviror>en t; 2 - teT'lperature of the envi r 0nen t; 3 - maxinuJ'l a dniss i hle stresses at the surface ; 4 - compu t ed st r esse s a t t he su rf ace ; 5 - co mputed st re sses in the niJJ l e la ye r s ; 0 - nax i num adm issible stresses in the n i Jd l e layers ; 7 - avera~e nois tur e content of th e ma terial .

assess nen t of thy \'al id i. t\' of th e pto de l and re gu lari ty of th e basic ass~lnp ti on on ,,'hich it is creat ed . Th e application of t~te nodel as a ba sic algorythm of control requires a consideration of th e uneven dry i ng of planks anJ boa r ds in the fi gu r e , Khich is also obse rv ed i n practice. It i s r ecol:lmendeJ t hat during d r y in g proc ess . The con tr ol infornation of no isture con t ent shou l d be ob t a in ed from thr ee control strai ght li nes in threesec t o rs of the figure: sectors Kith nax i J'l um, average anJ minimum i n t es it y o f d ryin g . CO\CLUSIO\ The model of beha\'iour of moist hOOJ in con vec tive ki l n d ryi ng p r ocess is a fir st step to study the complicated and interrelated phenome na of heat - and T:lO i s t ure transf e r and a tensely state of h'ood . In spite o f restricted c ha racter o f t he noJe l it can ana lyse ex i s ti g d r yinR schedu les, to genera te neK ones Kith restriction (2 7) or s e rv e as a basic algorythn for !'liC r OCol~p uter con trol of the process . Another tKO- dinensio nal tensel y state moJel could be join t o the heat- and mo i s t ure submode l. AMC- P

7·'

I

0~~-~-~------------~5----------+-~------------~~~

<:t

0,05

I

.~

I

REFERf\C[S Sapshin , J . ( 1 9()()) . In\'estigation of gross KOOt! t e nsely s t a t e a t the be InnIn of ryinr, nro cess .• 1. D. lesls , ;'os 0"'. Liukov , A. V. (IS~S ) . Handbook of Hea t- and \! ass Transfer . Ene r gy , ' Ios ko,,'. Liukov , ,-'I . \' ., J . c\ . \ii cha ilov (1963) . Tl1eo r y of Ifea t- anJ :,Iass Transfe r. Go sene r go l:d a t, :"osko"-. Po tt er , n. (l9~3 ) . Computa tional Ph ys ic s . Kil ey & Sons , \eK Yo rk . Rohseno"-, 11. ~I. , J . P. Har tnet ( 1 973) . lianclbook o f Hea t Transfe r. >icGra,," Hill , \e'," York. Shechto\", Ch r. I\. ( 1 9 7 3) . Un tersuchung de r feuchtigkeits und Jer temperaturvertei lung in einem ho l: s ta be l "eherend seines trocknun gs pro: esses . For . Hochsch. Kiss. band 1 9 , ser. :Iech . Tec h. des Hol: es ., 1' .

S3 - 90 .

Sinn~ak ,

A. \ . (lg - S) . On compu tin g t otal t ens i ons of "ooJ as an elastic - plastIC body Ju rin g d ryin g . Lesnoy J., \0 4, p . 15-1 - 159 Thomas , H. R., R. K. Le"is and K. ~organ ( 1 930) . An app li ca ti on of fin ite elemen t me thod t o th e drying of ti mb er. Kood an d Fibr e , Vo l . 1 1 , \0 -1 , pp.23 7~~ .) .

B ..-\. Todor()\·. Chr. K. Shecht()\ and S. G. Georgin<\

Todorov, B. A. (1994). ~athernatical rnodeling of unsteady state heat- and moisture transfer in capillary- porous colloid bodies. Eng. - Phys. J., Vol. 47, \0 ~, pp. 651-653. Ugolev, B. \. (1976). Deformations and Tensions of Wood During DrYIng. Lesn. Prom. , ~!osko",.

Wengert, E.~. (1977). Predicting average moisture content of ,,"ood in a changing enviroment. ~ood and Fihre, Vol. 7, \0 4, pp. 264-273. Vengert, E.~!. (1977). Some considerations in model ing and rneaap,uring moisture flo,," durig drying in ,,"ood. Wood Science, Vol. 10, No I, pp. 23-36. j'liley, A. T., E. T. Choon.r; (1975). An analysis of free- ,,"ater flo,," durin drying in soft,,"oocis. Wood Science, Vol. 7, \0 4, pp. 310-318. Xuan, B., E. T. Choong and W. G. Rudd (1980). Numerical methods for solving The equation for diffusion through ,,"ood during drying. Wood Science, Vol. 13, No 2, pp. 117-121.