SIMULATION OF WOOD DRYING STRESSES USING CVFEM

Transcription

1 Ltin Americn Appied Reserch 4:2- (2) SIMULATION OF WOOD DRYING STRESSES USING CVFEM C. H. SALINAS, C. A. CHAVEZ, Y. GATICA nd R.A. ANANIAS Deprtmento de Ingenierí Mecánic, Universidd de Bío-Bío, Av. Coo 22, Concepción, CHILE. Mster s Progrm (c), Ciencis y Tecnoogí de Mder, Universidd de Bío-Bío, Av. Coo 22, Concepción, CHILE. crchvez@umnos.ubiobio.c Doctor Progrm (c), Ciencis e Industris de Mder, Universidd de Bío-Bío, Av. Coo 22, Concepción, CHILE. ygtic@ubiobio.c Deprtmento de Ingenierí en Mders, Universidd de Bío-Bío, Av. Coo 22, Concepción, CHILE. nnis@ubiobio.c ABSTRACT The drying of soid wood nd ssocited stresses were simuted by ppying the Contro Voume Finite Eement Method (CVFEM) to trnsvers section of soid wood on the rdi/tngenti pne. The trnsport of moisture content nd stresses produced by its grdients ssocited with the phenomen of shringe nd mechnic sorption were modeed simutneousy. In prticur, we used CVFEM progrm (Fortrn 9) tht ows integrting differenti eqution of non-iner trnsient diffusion, defining tringur finite eements with iner interpotion of the independent vribe within itsef. The mode ws vidted by compring the experiment nd nytic resuts vibe in the speciized iterture. Finy, we showed the origin resuts of the simution ppied to the drying of spen wood (Popuus tremuoides) t three drying tempertures. Keywords: Simution, drying, wood, stress, CVFEM I. INTRODUCTION The present study focuses on het nd mss trnsfer couped with strin/stress probem during drying process in terms of modeing nd simuting the drying of soid wood. A coection of reted wors cn be found in Turner nd Mujumdr (997) nd n updted review of these methods is given in Hernndez nd Quinto (25). In prticur, Coutier nd Fortin (994) deveops numeric mode tht predicts the drying curve using the wter potenti mode. In the present wors, this mode is dopted to simute the trnsport of moisture content within the wood s described in Sins et. (24). The effects of het nd mss trnsport cuse strin/ stress within the wood. Modeing this phenomenon is compex process due to the effects tht the drying process produces on the wood. These ed to stresses tht cuse permnent nd trnsitory deformtions due to vritions of moisture contents. The modes proposed for wood focus miny on deformtions cused by the trnsport of energy (temperture) nd mss (moisture content). Some wors (Perre et., 99; Chen et., 997; Png, 2; Png, 27) propose one-dimension modes for determining the deformtions resuting from het nd mss trnsport; notby, deformtion by shringe nd mechnic sorption. Liewise, in two-dimension (Turner nd Ferguson, 995; Lin nd Coutier, 996; Ferguson, 998; Kng nd Lee, 24) nd three-dimension (Ormrsson et., 2) modes hve been proposed for deformtion. Numericy, we use the Contro Voume Finite Eement Method (CVFEM) to sove the trnsport nd deformtion equtions induced during the drying process. In gener terms, the method CVFEM consists of Finite Voume tht is mde up with Finite Eements. This mode offers dvntges reted miny to its intrinsic quity of conservtivey given by Finite Voume Method nd the topoogic verstiity bestowed by Finite Eements Method (Big nd Ptnr, 98). Thus, we consider iner orthotropic vritions of the properties nd independent vribes within the Finite Eement, considering the discrete vribe centered on the Contro Voume. The numeric pproch eds to the formution of iner gebric equtions systems tht re soved through itertive nd direct methods (Guss Side with SOR nd Guss Eimintion, Lpidus nd Pinder, 982). The im of the present wor is concerning with simution of the drying/stress probem, foowing systemtic vritions of geometric nd physic prmeters for the nysis of stbiity nd consistency of the gorithms deveoped. Moreover, we vidte the resuts obtined by compring them with the experiment, numeric nd nytic dt vibe in the iterture. II PHYSICAL MODEL We study physic mode of the wood strin/stress probem during drying process. We consider the nonuniform trnsitory effects induced by the vrition in the moisture content (M); tht is: stress (s ij ), strin (e ij ), nd dispcements (u i = (u,v)). As shown in Fig., we consider trnsvers twodimension section of wood on the rdi-tngenti pne. The properties re given in Tbe (Coutier et., 992). The dimensions of this piece of wood re: wide L=.45 (m) nd thicness H=.45 (m). The initi nd contour conditions re: ) for the probem of moisture content trnsport, initi moisture 2

2 C. H. SALINAS, C. A. CHAVEZ, Y. GATICA, R. A. ANANIAS content of M=M ini, Neumnn-type contour conditions of no-fow on the symmetry xes (x=l/2 nd y=), nd surfce convection of x= nd y=h/2; nd b) for the strin/stress probem, non-deformed initi stte (s ij =e ij =u i =), Dirichet-type contour conditions on the symmetry xes (u= on x=l/2 nd v= on y=), nd free contour conditions on the surfces x= nd y=h/2. III MATHEMATICAL MODEL The mthemtic mode considers tht the oc vrition in the concentrtion of moisture content, equivent to the fow divergence, cn be written ccording to the mode proposed in (Coutier nd Fortin, 994). Tht is: C + q m = () t where C is the concentrtion of moisture content (g wter /g wood-moist ) nd q m is the fow of moisture content (g wter /m 2 wood-moists). Assuming sm vritions in temperture nd wood wter in equiibrium, the fow cn be described in function of conductivity nd wter potenti s: qm = K( M, T, P) ψ (2) In prticur, s impemented in Sins et. (24), the trnsport is described indirecty through the wter potenti ψ (J/g), considering wood to be twodimension orthotropic medium on the pne x, y. This cn be expressed s : H/2 D h A h Y Tngenti WOOD X Rdi L/2 v = q c = Figure. Digrm of the physic probem: Trnsvers section of wood on the rdi-tngenti pne with non-uniform trnsient vritions in moisture content. Tbe : Wood properties Drying Temp. (ºC) Convec. Coef. (g 2 /m 2 s J) 4.4e e - 9.6e - Specific grvity Esticity modue (MP) 25 Shringe Coef. (/ºC). Creep Coef. (/P) -. e -7 Poisson rtio ().25 Initi MC (%) 5 Equiibrium MC (%) Becuse the coefficients c ψ, nd yy re now dependent on the vribe trnsported nd, in the cse of conductivity, so on direction, these configure non-iner, orthotropic mss trnsport. C Symmetry B u = q c = ψ ( c ) ψ ψψ = + yy () t x x where, yy conductivity (g 2 wter/m wood-moist sj) nd Gmρm M c = cpcity ψ ψ (g2 wter/jm wood-moist). The physic trnsport prmeters to be determined experimenty re conductivity nd yy in the min directions X nd Y, respectivey, nd the vrition of moisture content in retion to the potenti ( M ψ ). Aterntivey, Eq. cn be expressed bsed on the moisture content (M). M M ( cmm) = + yy (4) t x x ψ G where =, yy = yy nd mρ m c M =. M ψ M The mthemtic mode for two-dimension deformtion resuting from the mechnic equiibrium ppied to ech point in the bsence of body force cn be written s Zieniewicz nd Tyor (2). σ τ xy + = x (5) τ yx σ yy + = x In terms of the grdient, this resuts in: σ = to i =, 2 (6) i where =, two-dimension grdient x σ = ( σ, τ xy) stress in x σ 2 = ( τyx, σ yy ) stress in y. The non-uniform distribution of moisture content produces stress induced by free deformtion nd stress sustined over time. The modeing of these stresses cn be done through n impicit function of five prmeters tht define n initi deformtion ε : shringe α, mechno sorptive creep coefficient m, stress σ, vrition of concentrtion ΔC nd, in the cse of pne deformtion, the Poisson rtio ν. Tht is: ( α+ mσ ) ( α+ mσ ) ΔC stress pne (7) ε = βδ C = ( +ν )( α+ m σ ) ( +ν)( α+ mσ ) ΔC strin pne Tbe 2: Vribes nd properties. c Γ Eqution ψ M Gmρm M ψ u i ψ Gmρ m ; x ; Potenti yy i yy ui Dij x ψ Moisture content Dispcement 24

3 Ltin Americn Appied Reserch 4:2- (2) Thus, the stresses re determined bsed on the foowing expression: σ = D( ε ε ) + σ (8) where σ represents the vues of the initi stresses nd D is the esticity mtrix of mteri. In order to impement the Initi Deformtion Method, the forementioned probem is integrted over time in discreet terms (Turner, 996). For this, ΔC is considered to be the vrition in moisture content between time t n = ndt nd time tn+ = ( n+ ) dt, where dt is the interv of time considered; tht is: Δ C = Cn+ Cn nd, iewise, Δ σ = σn+ σ n nd Δ ε = ε n+ ε n. Thus, we cn pose the foowing expression tht describes the stress mentioned in time t n+ s toty expicit function of the vribes nd/or prmeters in time t n. Tht is: σ ( ) n+ = D εn+ εn+ + σ with εn+ = εn + β ΔC (9) The mthemtic mode for the cse of moisture content trnsport nd stress (Eqs., 4, 6) cn be presented in the generic form by non-iner, second-order eqution of trnsient diffusion in which the independent vribe is denominted s (see specific vribes nd properties in Tbe 2). Tht is: c ( J) S with + = J = Γ () t Fujo where is the trnsported vribe, t is time (s), c is cpcity, Γ is the diffusion coefficient nd S = S S is the source. Considering voume (domin) with contour Ω=Ω +Ω, we hve the foowing initi nd contour conditions: to t Inici vue = = t Ω Derichet. Γ t Ω Newmn n IV NUMERICAL MODEL Upon integrting Eq., in domin, ccording to Green s Theorem, the divergence integr in domin becomes n integr in the contour Ω. Tht is: ( ) c d ( Γ ) ds = Sd t n () Ω where n is the norm extern unitry vector to Ω. is the grdient equ to, x Now, considering tht domin mde up of by n Finite Voumes (FV) ( with ( =, n) ) of contour Ω nd tht, simiry, ech FV consists of n prti contributions of Finite Eements (FE) of contour Ω with ( =, n) (see Fig. 2), we cn write: n n = with = EF nd = = Ω = n = Ω Figure 2. Finite Voume mde up of n FE. When considering the term of oc vrition ( ρ ) nd the source term S in the centroid of the FV t s predominnt nd evuting the tempor term through finite difference t (t=m Δt) with (t=(m-) Δt) we get: m m ( c) d = (( c) ( c ) ) (2) t Δt ( S-S m Sd = ) () m The diffusion term n is integrted on the contour of the FE equ to Ω with ( =, n) for time t=m Δt (omitted for simpicity). Thus, nd considering tht ech FV is mde up of prti contributions of n FE, s shown in Fig., the prti diffusive contributions of the FE to the FV centered on the oc nodes i (i=, 2, nd ) cn be written s: n ds = n ds+ n ds (4) ( ) ( ) ( ) Ω Ω Ω c CD CD CD c Simir expressions cn be written for the contributions centered on 2 nd. The subscripts, b nd c represent the contour segments G, bg nd cg, respectivey (see Fig. 2). For the bove integrtion, we define iner vrition within the FE, tht is: ( x, y) = Ax+ By+ c (5) where A, B nd C re constnts defined bsed on the nod vues, 2 nd equ to: y2 y y y y y2 A = D D D x x2 x x x2 x B = D D D xy 2 xy 2 xy xy xy2 x2y C = D D D D = ( x x2)( y2 y) ( x2 x)( y y2). According to Eq. 4, the diffusive contribution of the contour defined by the segment G is given by: c G b 25

4 C. H. SALINAS, C. A. CHAVEZ, Y. GATICA, R. A. ANANIAS CD =, ( n, ) x ny ds (6) x Ω nd, in considertion of Eq. 5, the grdient of is equ to: =, = ( A, B). (7) x The diffusive fows J x nd J y in the orthotropic directions x nd y, when considering diffusion Γ x nd Γ y, respectivey, re given by: J = Γx, Γ y = ( Jx, Jy) (8) x With the grdient nd norm constnt in the contour G, the resuting integr is equ to: CD = J n + J n d (9) ( ) x x y y where d is segment ength G. The norm unitry vectors n is defined s: yg y xg + x n = ( nx, ny) =,. (2) d d According to these st definitions, we get: CD = CD + CD CD, (2) where ( y )( ) ( )( ) 2 y yg y x x2 x xg CD = + D D ( y )( ) ( )( ) y yg y x x x xg CD = + 2 D D ( y )( ) ( )( ) y2 yg y x2 x x xg CD = + D D D = x x y y x x y y. ( )( ) ( )( ) Liewise, we obtin expressions for the contributions of the contour segments CD c. Thus, the contribution of the eement to the FV centered on node : ( ) ( ) 2 2 ( CD CDc ) CD = CD + CD + CD + CD c c (22) Finy, when considering the trnsient contributions nd those of the source term given by (2) nd (), in ddition to the diffusive terms given by (22), the discreet two-dimension trnsient diffusion eqution posed impicity for t t = mδt in generic voume, is given by: n CL + CD = CS ( =, n) (2) = With ρ CL = + S Δt nd ρ m CS = ( ( ) + S ). Δt Thus, for ech vue, we obtin n gebric eqution formuted for the verge vue of in ech subdomin ( ), configuring system of n x n equ- tions of the form [A]{}={B}. V RESULTS The resuts from the three simutions (moisture content trnsport in spen wood, het diffusion nd therm stresses in stee pte, nd moisture content trnsport nd drying stress in spen wood) re presented herein. The first two simutions re done in order to vidte the computer codes for modeing, respectivey, the trnsport of moisture content nd stresses. The third simution shows n ppiction of this in the context of the wood drying process, which motivted this study. A. Trnsport of moisture content in wood Beow re shown resuts of trnsient moisture content trnsport on two-dimension section of the spen wood on the rdi-tngenti pne. The drying probem is modeing s discussed in the item III, it is iustrted in Fig. nd its properties re given in Tbe. Figure shows chrcteristic mesh used in modeing tht presents exponenti-type refinement towrds the surfces with convection. This type of mesh is optim for cpturing the rge moisture content grdients in these regions. In Fig. 4, the consistency nd convergence re nyzed in retion to the mesh type (uniform, cosine, nd ogrithmic eement distribution) for moisture content vues in the CD yer. We note tht the distributions tht concentrte the eement towrds the convection regions best pproch the convergent soution Figure. Mesh with x ogrithmic refinement. M(%) Uniform 2x2 Uniform x Uniform 4x4 Log x Log 4x4 Cos x Figure 4. Moisture Content v/s Mesh nysis. 26

5 Ltin Americn Appied Reserch 4:2- (2) Figure 5 shows detis of the moisture content distribution simuted for T equ to 2, 5 nd 5 (ºC). We cn see the dvnce of the drying front: greter concentrtion of moisture content isoines refect greter grdients. The orthotropic behvior cn be observed in the c of por symmetry of the moisture content Experiment 2ºC Experiment 5ºC Experiment 5ºC CVFEM 2ºC CVFEM 5ºC CVFEM 5ºC M (%) ) 2 ºC b) 5 ºC. c) 5 ºC. Figure 5. Drying fronts v/s time M(%) M(%) M(%) Figure 6. Drying curves with dt=. (h). Finy, Fig. 6 shows the resuts of drying curves obtined by the present mode with those experiment resuts obtined by Coutier et. (992). The good greement observed shows tht this cptures the essence of the physic phenomenon under study. B. Het diffusion nd therm stresses. The trnsient two-dimension simution of het trnsport nd the ssocited stresses ws done for stee pte. The probem is shown in Fig. 7 nd the properties re given in Tbe. The pte dimensions re: ength L=.6 (m) nd height H=.2 (m). The contour condition effects considered re: restricted dispcement in the Y direction of yer AD (v= in y=), restricting in the X direction in the yer BC (u= in x=l), nd freedom in the yers CD nd DA. Ntury, point B is restricted in X nd Y ((u,v)=(,) in B). This probem hs n nytic soution (Boey nd Weiner, 96). The non-uniform distribution of tempertures produces stresses induced by dittion, s it does when sustined over time (creep). The modeing of these stresses cn be done in mnner equivent to tht of stresses presented for the vrition of moisture content in wood. In prticur, for n initi deformtion ε in function of: therm dittion α, creep coefficient m, stress σ, temperture vrition ΔT, nd for pne strin, the Poisson rtio ν. The probem ws soved with the present gorithm nd some resuts re presented in Fig. 9-. H D C ( ) 2 ( 5 t 625 y H e ) for t> Y Tyt (,) = for t = u = v = A B X ( uv, ) = (,) L Figure 7. Two-dimension stee pte probem Tbe : Properties of the therm pte. Esticity Modue 5 MP Coefficient of Expnsion. (/ºC) Coefficient of Creep 5-9 (/P) Poisson Rtio.25 27

6 C. H. SALINAS, C. A. CHAVEZ, Y. GATICA, R. A. ANANIAS Figure 8 shows mppings of the stte of strin nd stress for t= (h). These mps reve the consistency in terms of the imposition of contour conditions (Fig. 8). The effects of the restrictions to the free deformtion dded to the creep phenomenon in terms of its quittive consistency cn be pprecited in Fig. 8b. The nysis of convergence nd consistency reted to the size contrsted with the nytic soution, s shown in Fig. 9. As cn be seen, trnsitory convergent resuts cn be obtined with uniform meshes of 5x2. Finy, Fig. shows the resuts of the distribution of norm stresses in the direction x (σ ) of the BC yer, contrsting with the nytic resuts. This greement refects pproprite modeing of the stress/strin probem under study. C. Moisture content nd drying stresses. Beow re shown resuts of computer code for moisture content couped with strin/stresses probem in piece of spen wood during the drying process. For this, we determine, in ech time of integrtion, the distribution of moisture content tht ows ter ccution of the free deformtions ( ε ) tht motivte drying stresses. Figure shows spti distributions of the ccution prmeters for the strin/stress probem for the permnent stte (4 h) nd drying temperture equ to 5ºC. In Fig. nd b, re we cn pprecite the impementtion of the contour conditions nd s the gretest dispcements concentrte on the restriction of degree of freedom. Figure c nd d show in deti how the norm stresses concentrted in A, B, nd C (see Fig. ), focusing on tension in A nd C (surfce) nd compression in B (center). On the other hnd, the sher stresses re shown in Fig. e concentrte on the digon sighty dispced towrds point D (surfce). E+6 2E+6 E+6 -E+6 Anytic CVFEM dt =. CVFEM dt = u: ) Dispcement u s: -.5E+6 -E E+6.5E+6 2E+6 2.5E X(m) b) Norm stress σ Figure 8. Stress nd dispcement: t=, dt=. (h). E+6-2E Figure. Norm stress σ : BC yer, dt=.(h) ) Dispcement u (Rdi). u(m).4e-4.2e-4.e-4 2.8E-4 2.6E-4 2.4E-4 2.2E-4 2.E-4.8E-4.6E-4.4E-4.2E-4.E-4 8.E-5 6.E-5 4.E-5 2.E-5.E+ 2.5E+6 5 v(m) 2E+6.5E+6 E+6 5 Anytic CVFEM Mesh x5 CVFEM Mesh 26x CVFEM Mesh 52x2 5.E+ -2.E-5-4.E-5-6.E-5-8.E-5 -.E-4 -.2E-4 -.4E-4 -.6E-4 -.8E-4-2.E-4-2.2E-4-2.4E-4-2.6E-4-2.8E-4 -.E-4 -.2E-4 -.4E Figure 9. Norm stress σ : Point B, dt=. (h) b) Dispcement v (Tngenti). 28

7 Ltin Americn Appied Reserch 4:2- (2) 5.E+5.E+ -5.E+5 -.E+6 -.5E+6-2.E+6-2.5E E+5.E+ -5.E+5 -.E+6 -.5E+6-2.E+6-2.5E E+5-2.E+5 -.E+5.E+..5.E c) Stress σ. d) Stress σ s (P) 4.E+5 2.E+5.E+ -2.E+5-4.E+5-6.E+5-8.E+5 -.E+6 -.2E+6 -.4E+6 -.6E+6 -.9E+6-2.2E+6 syy (P) 4.E+5 2.E+5.E+ -2.E+5-4.E+5-6.E+5-8.E+5 -.E+6 -.2E+6 -.4E+6 -.6E+6 -.8E+6-2.E+6-2.2E+6-2.4E+6 sxy (P) 5.E+4.E+ -2.7E+4-5.E+4-7.E+4 -.E+5 -.5E+5-2.E+5-2.5E+5 -.E+5 -.5E+5-4.E+5 e) Stress τ xy. Figure. Strin nd dispcement: T=5ºC, t=4 (h). Figure 2 show the trnsitory norm stress σ t centerine X=L/4, spit in two drying stge ccording to fiber sturtion point (FSP): Figure 2, show º Stge, upper FSP, where the center hve compressive stresses, induced by tensie effort in the surfce. And Fig. 2, show 2º Stge, under FSP, where the center becme deveoped tensie stresses, s rection to contrction. In these pictures we cn see how the cross section of timber is subjected to tensie nd compressive during drying process, which re highy vribes. Figure shows the trnsitory evoutions of the stresses in the center nd the surfce of the domin of ccution, points B nd C of Fig. for three drying temperture (2, 5 y 5 ºC). There, the dynmic of the vrition of the intensities of the norm stresses cn be pprecited. Bsicy, t the onset of drying on the surfce nd in the center, we observe mred tension nd sight compression, respectivey. This contrsts with the fin stte of stresses t the end of the drying, in which the norm stresses re inverted. The tter is one of the reevnt chrcteristics of the simuted physic probem. Furthermore, we cn see tht the residu stress re very simir for three cses studied, but there re strong differences in the stress trnsitory evoution, speciy, how the stresses re inverted. Of course, t initi drying the surfce stress hve one oscitory performnce becuse the critic drying condition. VI CONCLUSIONS The resuts presented in Fig. -6 ed us to concude tht the two-dimension simution of moisture content trnsport modeed ccording to wter potenti ws effective. The resuts given in Fig. 7- show tht n effective simution ws so done for the stress/strin phenomenon cused by free deformtions nd the ssocited creep phenomenon. The resuts of the ppiction shown in Fig. - indicte consistent quittive resuts, in terms of the simution of the phenomenon of moisture content trnsport correted with the stresses ssocited with the moisture content grdients during the trnsient process of wood drying. Furthermore, re not high differences in residu stresses for drying temperture studied. Nevertheess there re hrd differences in how the trnsitory stress re performed, speciy when re compred stresses in the center versus surfce..e+6 5.E+5.E+ -5.E+5.E+ -5.E+5 -.E+6 -.5E+6 h 2 h 9 h h 45 h 56 h 6 h 6 h 67 h ) º Stge. 67 h 7 h 82 h 8 h 5 h b) 2º Stge. Figure 2. Norm stress σ t centerine. 29

8 C. H. SALINAS, C. A. CHAVEZ, Y. GATICA, R. A. ANANIAS 2E+6 E+6 -E+6-2E+6 -E E+6 E+6 -E+6-2E+6 -E+6 ) 2 ºC. S Center S Surfce MC Averge -4E E E+6 -.5E+6-2E+6 b) 5 ºC. S Center S Surfce MC Averge S Center S Surfce MC Avernge 5 5 c) Ti 5 (h) ºC. Figure. Trnsitory stress σ Mesh Log x. REFERENCES Big, B.R. nd S.V. Ptnr, A contro voumen finito eement method for two dimension incompressibe Fuid fow nd het trnsfer, Num. Het Trnsfer, 6, (98). Boey, B.A. nd J.H. Weiner, Theory of Therm Stresses, John Wiey nd Sons, Inc., New Yor (96). Chen, G., R.B. Keey nd J.F.C. Wer, The drying stress nd chec deveopment on high-temperture in sesoning of spwood, Pinus rdite bords. Hoz s Roh-und Werstoff, 55, (997). Coutier, A. nd Y. Fortin. Wood drying modeing bsed on the wter potenti concept: Effect of the hysteresis MC (%) MC (%) MC (%) in the M-ψ retionship, Drying Tech., 2, (994). Coutier, A., Y. Fortin nd G. Dhtt, A wood drying finite eement mode bsed on the wter potenti concept, Drying Technoogy,, 5-8 (992). Ferguson, W.J., The contro voume finite eement numeric soution technique ppied to creep in softwoods, Int. J. Soid Structures, 5, 25-8 (998). Hernndez, R. nd P. Quinto, Secdo en Medios Porosos: Un Revisión s Teorís Actumente en Uso, Cinétic, 9, 6-7 (25). Kng, W. nd J.H. Lee, Simpe nytic methods to predict one-n two-dimension drying stresses nd deformtions in umber, Wood Sci. Tech., 8, (24). Lpidus, L. nd G.F. Pinder, Numeric soution of prti differenti equtions in science nd engineering, John Wiey & Sons, Inc. (982). Lin, J. nd A. Coutier, Finite eement modeing of the viscoestic behviour of word during drying, 5 th IU- FRO Interntion wood drying conference, 7-22 (996). Ormrsson, S., D. Cown nd O. Dhbom, Finite eement simutions de moisture reted distortion in minted timber products of norwy spruce nd rdit pine, 8 th IUFRO Interntion wood drying conference (2). Png, S., Modeing of stress deveopment during drying nd reief during steming in Pinus rdite umber, Drying Technoogy, 8, (2). Png, S., Mthemtic modeing of in drying of softwood timber: Mode deveopment, vidtion nd prctic ppiction, Drying Technoogy, 25, 42-4 (27). Perre, P., M. Moser nd M. Mrtin, Advnces in trnsport phenomen during convective drying with superheted stem nd moist ir, Int. Journ of Het nd Mss Trnsfer, 6, (99). Sins, C., R. Annis nd M. Aver, Simución de secdo convencion de Mder, Mders Cienci y Tecnoogí, 6, -8 (24). Turner, I. nd A.S. Mujumdr, Mthemtic modeing nd numeric techniques in drying technoogy, Mrce Deer Inc., New Yor (997). Turner, I.W. nd W.J. Ferguson, Unstructured mesh cecentered contro voume method for simuting het nd mss trnsfer in porous medi: Appiction to softwood drying, prt I nd II, App. Mth. Modeing, 9, (995). Turner, I.W., A two dimension orthotropic mode for simuting wood drying process, Appied Mthemtic Modeing, 2, 6-8 (996). Zieniewicz, O.C. nd R.L. Tyor, The Finite Eement Method, Fifth edition, pubished by Butterworth- Heinemnn (2). Received: November 2, 29. Accepted: Jnury, 2. Recommended by Subject Editor Edurdo Dvorin.