A two-dimensional orthotropic model for simulating wood drying processes

ELSEVIER

A two-dimensional orthotropic model for simulating wood drying processes

Ian W. Turner

The School of Mathematics, Queensland University of Technology, Brisbane, Queensland, Australia

Wood is a naturally occurring resource that must be dried before it can be manufactured. Drying is important for a number of reasons that include the protection of the wood against biological damage and the reduction of the moisture content to final equilibrium levels. The complexities involved in modelling this drying process consist of analyzing the heat and mass transfer phenomena that arise in an anisotropic, nonhomogeneous, and hygroscopic porous medium. In this work, a two-dimensional orthotropic mathematical model is formulated and a numerical code based on a structured mesh cell centered control volume approach is implemented in order to allow a more comprehensive numerical investigation of the convective drying of wood to be undertaken. A comparison is made between two different numerical solution techniques; the first numerical method solves the system of equations by treating each equation in an uncoupled form, while the second scheme solves the entire system as a completely coupled set. The most ejlicient numerical algorithm was obtained when the system was solved using the coupled procedure. In order to examine the important differences of the overall kinetics for both low and high temperature drying, simulation results for three different cases of convective drying of wood are presented. These cases include the drying of wood below the boiling point at a relatively low temperature of SOY, a moderate temperature of SOY, and above the boiling point at the high temperature of 120°C. The two-dimensional model highlights the following two very important facts: for an anisotropic medium, where the ratio between longitudinal and transverse permeabilities is of the order of 109 the moisture migration occurs in the longitudinal sense which is the most permeable direction in the wood; and the behavior of the internal gaseous pressure can have a substantial impact on moisture migration.

1. Introduction

Timber is a naturally occurring resource that may be used for a number of different purposes, including building and furniture manufacturing to name but two. It is estimated that of the 1.4 X lo9 m3 of timber produced worldwide each year, approximately 70% is consumed by the United States, the C.I.S. (former Soviet Union), and Europe.’ Furthermore, worldwide trends seem to indicate that significant increases in timber demand will occur in the near future, and as a consequence wood drying technology must be advanced accordingly. Research into innovative techniques that provide decreased drying times while still maintaining end product quality would prove invaluable for the timber industry. Mathematical modelling can become an integral component in the advancement of these technologies. The use of numerical simulators that accu- rately describe the physical mechanisms of the drying

Address reprint requests to Dr. I. W. Turner at the School of Mathemat- ics, Queensland University of Technology, G.P.O. Box 2434, Brisbane,

Q4001, Australia.

Received 29 November 1993; revised 28 April 1995; accepted 6 June

1995

Appl. Math. Modelling 1996, Vol. 20, January 0 1996 by Elsevier Science Inc. 655 Avenue of the Americas, New York, NY 10010

process are extremely powerful tools in providing crucial information that can be used in the optimization of kiln design. The fundamental knowledge gained from analyzing numerous simulation results allows a better understanding of the complex heat and mass transfer phenomena that arise in the wood to be ascertained. This knowledge can then be used to help guide future experimental investigations into new and innovative drying processes such as high temperature convective drying, microwave drying, and vacuum drying.

Untreated wood consists of approximately 75% of its mass as water, and to ensure a suitable and usable end product, most of that moisture content must be removed during processing. Drying of the wood before processing is important for a number of reasons, which include the protection of the wood against biological damage and the reduction of moisture content to final equilibrium levels.

Wood is a complicated hygroscopic cellular porous material comprised of a number of hollow tubular like cells which are bound together with lignin. The cells communicate via small openings in the cell walls called pits. For a complete description of the structure of wood the reader is referred to Siau,’ Spolek,3%4 or Rosen.’ The moisture contained in the wood may exist in a number of different states including, free water in the cell cavities, bound water that is hygroscopically held by the cell walls,

0307-904X/96/$15.00 SSDI 0307-904X(95)00106-T

An orthotropic model for simulating wood drying: I. W. Turner

water vapour in the voids or lumens of the cells, or constitutive water in chemical composition within the cell walls.

Only the free, bound, and vapor phases of the water are removed during the drying process. The basic transport mechanisms that enable moisture movement during the drying of wood are liquid flow due to capillarity, diffusing water vapor, vapor convection in bulk gas flow, and bound liquid diffusion.

The literature is inundated with information relevant to the topic of modelling the wood drying process and some of the more pertinent references are given in the reference list.3-22 It is well noted by all of these authors that because of the complicated structure and the nature of the problem, a very complex heat and mass transfer process arises during wood drying and as such a difficult mathematical formulation results. The difficulties involved in the modelling of timber drying are highlighted by the facts that wood is nonhomogeneous, hygroscopic, strongly anisotropic (with the ratio between longitudinal and transverse permeabilities being in the range of lo2 and 104), and that it shrinks during drying with dimension changes causing internal stresses to occur.

In the past,“.‘6,‘9,23 the emphasis has been on studying an idealized one-dimensional drying model for wood, and already these studies have contributed significantly to the understanding of the physics associated with the overall drying process. Given the advancements made in computer hardware technology in recent times, it is now possible to extend the existing one-dimensional theory and develop a comprehensive and numerically efficient two-dimensional simulation code. The code Wood2D, which was developed for this research, provides a complete description of the moisture content, temperature, and pressure distribution in both the longitudinal and transverse directions of the wood as they evolve in time throughout the drying process. The underlying model can then be used to study in detail the effects that the anisotropic properties of the wood can have on the drying process and assist with the investigation of new drying technologies. The model can also be used to highlight the fact that during drying of wood the liquid movement occurs in the most permeable direction-the longitudinal direction.

Although limited single equation two-dimensional drying models based on simplified diffusion theory and more rigorous two-dimensional finite element models based on the Luikov equations’%“*” exist in the literature, only Perre”*” has successfully implemented a comprehensive two-dimensional model that predicts the moisture content, temperature, and pressure distributions in both the tangen- tial and longitudinal directions of the wood throughout the drying process. Although the underlying physics of Perre’s model and that of the model developed in this work is similar, Wood2D includes some additional terms within its infrastructure which make the final form of the equations slightly more complicated. The main differences between the models are summarized as follows: l The amount of water vapor is not assumed negligible

and is included throughout the entire analysis. The conservation of total liquid becomes inconsistent if the time rate of change of the vapor content is neglected

because, although the magnitude of the vapor content is small in comparison with the liquid content, the change in the vapor content from one time instance to another may be quite considerable. The effect of including these terms leads to additional and more complicated expressions on the left-hand side of the final form of the system of equations.

l The full energy balance equation is treated in the formulation rather than an enthalpy balance. This concept is discussed in detail in the text; however, the main benefits are that a more accurate energy balance arises and the need for treating the evaporation rate source term is eliminated. Furthermore, the implementation of the energy boundary conditions in the finite volume equations becomes very straightforward using this formulation.

l The bound liquid flux incorporates thermomigration effects.

The resulting model consists of a coupled system of three nonlinear partial differential equations which must be solved numerically to study the transient behavior of the solution variables moisture content, temperature, and pressure. Because of the complex nature of these equations, developing a stable numerical solution technique was an onerous task. Great care must be taken in selecting an efficient numerical algorithm and two techniques were investigated in this research-a coupled method and an uncoupled method. Finally, this study will be used to highlight two-dimensional drying phenomena which occur during wood drying at temperatures both below and above the boiling point. In particular, when drying at high temperatures, the board reaches and passes through the boiling point of water and an important overpressure arises. This case exhibits typical two-dimensional heat and mass transfer whereby the heat is supplied to the board in the transverse direction and most of the moisture migrates in the longitudinal direction and leaves the board through the end piece. The model clearly demonstrates the effect that the internal gaseous pressure can have on the overall drying kinetics and highlights the benefits of using high temperature drying processes.

2. Mathematical formulation

2.1 Transport equations

The work consists of the theoretical study of the heat and mass transfer which occurs during the convective drying of a sample of wood placed in a small scale industrial kiln, Figure la. The transverse (z-direction) and longitudinal (y-direction) directions of the wood are considered in the analysis and a two-dimensional mathematical model is constructed. In this study the porous material (wood) is assumed as being a rigid, homogeneous, and anisotropic structure that has some portion of its void space occupied by liquid water. For wood the relevant anisotropic properties are known to be in the mutually perpendicular directions of the y and z axes, and as such, an orthotropic mathematical description is sufficient. The drying configu-

Appl. Math. Modelling, 1996, Vol. 20, January 61

An orthotropic model for simulating wood drying: 1. W. Turner

Figure 1. Development of the mathematical model.

ration is illustrated in Figure lb. Due to symmetry only a quarter of the domain requires computation.

The shaded region, 0 < z <L, and 0 < y Q L, represents the wood being dried by forced convection of hot dry air. The faces z = 0 and y = 0 are exposed to the external drying conditions while the faces z = L, and y = L, are considered as symmetry planes. In order to develop the mathematical model, a continuum approach is adopted and the physics associated with a small averaging volume element of the wood is considered.

Within each individual macroscopic averaging volume, shown as the cross-sectional area contained within the circle in Figure Ic, there is a solid phase (assumed rigid and consists of the wood cell structure together with hygroscopically held bound liquid), a free liquid phase (water) and a gas phase (binary mixture of air and water vapor). This averaging volume V will be associated with every point in space and any shape may be chosen to represent V, provided the dimensions and the orientations are invariant.

In general the averaging volume is selected so that important model parameters that include porosity and density, for example, are measurable across that volume. For the model defined in this work the scale is at the macroscopic level and all quantities will be defined accordingly. The assumptions used in the mathematical formulation are summarized in the following section.

Assumptions

The assumptions used in the formulation of the model are as follows: (9

(ii)

(iii) (iv> (v)

(vi)

(vii)

(viii)

The solid and gas phases are continuous as is the liquid phase above the Fibre Saturation Point (FSP). The gas binary mixture of air and vapor behaves like an ideal gas. Darcy’s law holds for the gas and free liquid phases. Gravity is included for the gas and the liquid phases. Bound liquid is treated as being hygroscopically held within the solid phase. Conduction occurs within each phase and may be treated as constant or may vary with temperature. The phase kinetic energies, the fluid viscous dissipa- tion, and the work done by body forces are negligible. The enthalpies for the three phases are linear func- tions of temperature.

(ix)

(4

(xi>

(xii)

(xiii)

(xiv)

(xv)

(xvi)

(xvii)

Other

The differential heat of sorption (heat of wetting) is included for the treatment of the enthalpy of bound water. Bound water migration is a molecular diffusion process whose flux is proportional to a gradient in the chemical potential. For X> X,, the water vapor pressure varies according to the saturated vapor pressure, P,,,(T). For X< XFsp the relative humidity is defined according to the sorption isotherm data available for the particular species of wood. The capillary pressure is a function of moisture content and temperature. The latent heat of evaporation varies only with temperature. The diffusivity is taken as a function of liquid saturation, temperature, and pressure. The viscosities of the liquid and gas phases depend on temperature. The permeabilities of gas and water can be ex- pressed in terms of relative permeabilities. simplifications are described in the text.

The equations for heat and mass transfer in porous media have been well documented in subsequent research undertaken by the author23-26,28 and the intricate details concerning the development of the continuum equations can be found in those references. In order to study the drying of complicated anisotropic and hygroscopic porous material such as wood, the extension of the existing model to two dimensions, and the inclusion of the effects of bound liquid movement becomes essential. The conservation equations that govern the drying process together with the relevant flux descriptions are summarized in the following paragraphs:

Total mass conservation

+ v’ ( x, &“g + xw &f”wf + xs &b”wb) = o

(la)

Liquid conservation

+ v ’ ( x, &v”gv + xv, Pwf “wf + xs &b”wb ) = ’

(lb)

Air conservation

(lc)

62 Appl. Math. Modelling, 1996, Vol. 20, January


Total energy conservation

+ xw Pwf”wfkvf + xs Pwb”wbkvb)

= V. (K,,VT) + @ (Id)

It should be noted that in order to write equation (Id) in terms of enthalpy alone, the assumption that the vapor and air phases of the gas are ideal was used and since the solid and liquid phases are incompressible, the difference between their internal energies and their enthalpies is assumed to be negligible (i.e., C, = C ). The terms h,, and h,, are defined as the averaged andspecific enthalpies of bound liquid, respectively. The averaged enthalpy of bound water takes into account the integral heat of sorption while the specific enthalpy of bound water accounts for the differential heat of sorption. The terms concerning the heat of sorption are generally negligible outside the region of true absorption. The term - &!?,P, in the energy equation (Id), which arises due to compressive heating effects, is included in this study since it is thought not only to be important from a thermodynamic perspective but it provides an accurate energy balance that may not be satisfied if only an enthalpy balance is considered. Furthermore, the effect that this term may have on overall drying kinetics may become more prominent under extreme drying conditions such as those generated in high temperature convective drying or microwave drying environments when pressure gradients intensify within the medium. A more comprehensive analysis of this postulation will be discussed in future research.

The flux expressions for the gas, free liquid, bound liquid, vapor, and air phases are given as follows:

Gas jlux

xg Pg”g = -yg&yvpg-!g] (2a)

Free liquid jlux

xw Pwf”wf = -ywPw [“(P, -PC) - p,g]

w

(2b)

Bound liquid flux

“kv -+Rln!P+- VT

T

+ ‘Pwb 1

(2c)

Vapor jlux

-itgoB M,P xg Pg,“,, = /g g+v-F]

Air jlux

(24

xg Pga”ga = xg Pg”g - xg Pg”“g” (2e)

The quantities Kg, K,, 6,, 6,rf are second-order tensors that define the gaseous permeability, liquid permeability, bound liquid diffusivity, and effective diffusivity, respectively. For wood, the material axes are in the radial, transverse, and longitudinal directions of an axisymmetric coordinate system. In order to assume that the geometrical axes of the board being modelled are along these natural axes, it is argued that the board is cut along the axis of a perfect cylinder and far away from the main centerline axis of the tree. Under this assumption ;he second-order tensors

are diagonal where, for example, Kg can be written as:

Kg, 0 0 kg = 0 Kg, 0 I I with

0 0 Kgz

(3)

2.2 Equations of state and other constitutiue equations

The equations obtained thus far must be augmented with the following equations of state and constitutive equations in order to close the system:



(i) The binary mixture of gas is an ideal mixture of perfect gases.

P,RT Pp = -

Pg” RT

M P&,” = -

M”

P,,RT P@ = -

M, with P, = P,” + Ppa

and M=M,-(Ma-M”)% g

(ii) The saturated vapor pressure is given by the following relation:

5204.9 25.5058 - 7 Pal

(iii> The relative humidity is defined according to the sorption data available for the wood species being modelled”

F(XW,7 T) = s = exp( u1a’)22x) a”s

where

a, = 17.884 - 0.1423(T)

+ 23.63 x 1O-5(T)2

a2 = 1.0327 - 67.4 x 10-‘(T)

(iv) Enthalpy-temperature relations:

ha, = C,,(T- TR)

h, = C,“(T- Ta) + %,,

&,r =C,,(T-W

L=C,,(T-Ta)

1

/ Cl- @j&b

- (l-+)Pw, 0 Ah, dp

h,,=C,,(T-T,)-Ah,

h,=C,,(T-TR)

(v) Differential heat of sorption:

Ah”” = 0.4h”,,((X,S, -Xwh)/XFsJ2

(vi) The effective diffusivity tensor is assumed to depend on liquid saturation, pressure, and temperature and is given by:

C&W~ TY Pa)

= &S,)D,j 7 + l)l’sl/P

where D, = 2.5 X 10m5 2nd y is the tortuosity factor. The tensor quantity fi(S,> depends on the wood structure.


(vii) The bound water diffusivity tensor is assumed to depend on moisture content and temperature and is an Arrhenius type function*:

(viii) The capillary pressure is given by

(ix)

(x)

pc(XWf~ T) =pco~(T)f*(xvff)

where a(T) is the surface tension and the function f2(Xwf) may depend on the wood species. Relative permeabilities18:

X wf KW 57 x,,= 0 0 1

0 < xv”, 0.95( x,,/x,,P 0.95(1- X,,/X,,)2

< xc, + 0.05

Xwf - xc, X xcr < xvfi 0.05-

- Xbvf X sat - Xc,

0.05 xs=t _ x sat cr

< xs3t + 0.95

where X,,, = 1.33 and XC, is defined according to the species of wood. The fiber saturation point XFsp is given as a function of temperature:

X rsp = 0.598 - O.OOlT

2.3 Boundary conditions

For the configuration shown in Figure lb the boundary conditions for the drying surfaces and the symmetry planes (adiabatic and impervious surfaces) must be specified. In defining the boundary conditions at the drying surfaces of the porous material it is assumed that the driving forces at those boundaries are of the form proposed by18:

Total liquid flux

2.2 = k, CM”

2.2 -x, -x,0 ( X” - X”0) (4a)

Total heat flux = Q( T - TO) (4b)

where x” and T are, respectively, the molar fraction of the gas vapor and temperature at the exchange surface, and x”~ and TO are the molar fraction of the gas vapor and temperature characteristic of the external drying conditions. The quantities k, and Q are the mass transfer and heat transfer coefficients, respectively.

On the exposed surfaces of the material the fluxes are continuous:

Flux of liquid:

( X, &““a” + XW hvf”wf + xs &vb”wb) ’ n

2.2 = k,cM”

2.2 -x, -x,, (X” -x,0) (5a)


X(X” -x,0) Total pressure: Pa = P,

On the symmetry planes the fluxes are zero:

Flux of liquid:

( X, &v”~v + XW &f”wf + xs &b”wb) ’ n = ’

Flux of heat:

( xs P&&v + x, P&&a + XwPwf”wfkf

+ xs &bVwbhwb - Kcffv ‘) ’ n = 0

Flux of air: X, pgyvgV . n = 0

2.4 Initial conditions

(5b) (SC)

(ha)

(6-b) (6c)

Initially the wood has some prescribed moisture content and temperature and the pressure is constant throughout at atmospheric pressure.

At time t=O:

x=x, (7a)

T= T, (%)

P=P, (7c)

2.5 Summary of two-dimensional system

Assuming that a local thermal equilibrium exists between the three phases within the porous structure, and by noting that the vapor pressure Ppv can be determined from the independent variables morsture content and temperature, then the liquid, air, and energy conservation equations (lb-ld) give the two-dimensional system of three nonlinear coupled partial differential equations that govern the drying process as follows:

8X 8T a

x’ at - +a,,%

a ax aT a< =- K az XZ1 az - +KTzljy +KPzl x + Kgrl

I

(*a>

ax aT K,,,-+KK,,-+K ap,

aY aY pY1 aY

ax aT a a<

x2 at - + aT2 at - + a,27

a =-

a2 (

ax aT K

Xz2 az - + K.,.,,~ + KPz2 a5 az + Kg2

1

(8b)

ax aT a a<

x3 at - + aT3; + aP3at

a ax =- K

a2 - + K,,,: + Kp,,z + Kgr3

m3 a2

ax aT KXy3- + KTy3- + K apg

aY aY pY3 aY

The system variables are temperature T, total pressure Pg, and moisture content X. The definition of moisture content adopted for this work is as follows:

i

Xwf + XFsp x= x

x > XFsp

wb xeq < x G XFsp (9)

The equations (8) are highly nonlinear since the capacity coefficients aXi, aTi, api and the kinetic coefficients KXzi,

K,i, Kpzi> K, i, KT,i, Kpyi, K ri on the indepen cr

depend very strongly ent system variab es along with properties f

of the porous media. It is evident from the definition given for the moisture content that step changes will occur in some of the kinetic coefficients at X,,. A smoothing technique must be employed within these coefficients in order to allow the transition of moisture from above to below XFsp to be treated by a stable numerical solution procedure. This problem is overcome by use of an approxi- mation to the heaviside function. For example if a coefficient K, has a particular functional form given by K,, above XFsp and K,, below XFsp, then K, can be com- pactly rewritten as:

Kx=KxfH(X-XFs,)+Kxb[l-H(X-XFs,)]

where H(x) can be approximated by:

1 1 H,(x) = y + -arctan( U) for large n

lr

The boundary conditions (5-6) and initial conditions (7) are summarized as follows:

Exchange surface z = 0

ax aT K - + KTzl

a% xzl az z + KPzl dz + Kgrl

km Mv 2.2 =-

RT, p0 2.2 - x, - xVO (p, -Go)

Kxz2 g + K~z2 g + K~z2 2 + Kgr2

( lOa)

=Q(T-To)

km Mv 2.2

+h,o RT, 2.2 -xv - xv0 (P,” - P,“O)

( lob)

Pp = P, ( 1Oc)



Exchange surface y = 0

K %+K xyl dy

z+K ap, Tyl ay pyl ay

k,Mv 2.2 =-

RT, p. 2.2 -xv -xv,, (P, -PA (lla>

dX C?T K a<

xy2 ay - + KTy2- + KPy2- 8Y JY

=Q(T-To)

k,M” 2.2

+hFo RT, 2.2-x,-n,, (P&.” -P,vo)

Pg=Po

Symmetry plane 2 = L,

K,, z + KTZI; + KP~I aP,

az + Kgrl = 0

ax aT K - + KTz2x + KPz2

% z2 a2 --jg + Kg*2 = 0

K,~~+KT13~+KP.?~+KgrZ=0

Symmetry plane y = L,

K %+K Xyl aY

E+K dPp_ Tyl aY pyl aY

-0

ax aT ap K

Xy2 ay - +KTy2- +K 2’0

aY py2 ay

ax aT K a<

Xy3 ay - +K,,,- +K,,,- =0

aY aY

(lib) (llc)

t 12a)

( 12b)

t 12c)

( 13a)

t 13b)

( 13c)

Initial conditions

q Y, 2, 0) =x,

T( Y, 2, 0) = T,

Pg(Y, 29 0) ‘PO

3. Numerical solution technique

t 14a) ( 14b) t 14c)

The system of equations (8-14) is transformed to nondimensional form using the following scaling:

z z’ = -

Ll y' = +

fft k0 ?j-=-

1 G (YE-

POC,,

T- Tl @=_

To - Tl

together with a number of nondimensional parameter groups.23

The resulting nondimensional system is discretized using the control volume technique2’ and a fully implicit time stepping procedure is used to advance the solution in time. This scheme has the important property of being totally conservative, with mass and energy balances auto- matically satisfied once the solution is sufficiently con- verged.

The basic strategy of this discretization method is to divide the solution domain into a number of strategically placed control volumes and integrate the relevant continuum equations across those control volumes.

Due to the complexities involved with the nonlinear equations, together with the convoluted heat and mass transfer phenomena that arise during extreme drying conditions, great difficulties were met with developing a stable and efficient numerical scheme. Furthermore, the code must overcome other specific problems which arise due to the aspect ratio of the geometry of the wood (in the example considered here the ratio of longitudinal to transverse dimensions is 10) and the strong anisotropic material properties (a factor of between 102-lo4 for the permeabilities). Although the code has many attractive features such as adaptive time step and under-relaxation control incorpo- rated within its infrastructure, it was found that the best results were obtained when the following strategies were implemented: cli

(2)

(3)

Utilization of a cell-centered control volume approach (Figure 2). Application of a refined boundary control volume (BCV) located at the exchange surfaces of the wood. This approach simplifies the implementation of the complicated nonlinear boundary conditions. The values of the moisture content, temperature, and pressure at the center of these BCVs are assumed to be equal to the values of these variables at the actual boundary. In this way interpolation at the boundary becomes unnec- essary. Generation of a nonuniform structured rectangular mesh (Figure 3). This mesh, which is generated according to a geometric progression, provides additional

Figure 2. Control volume approach.



Figure 3. Nonuniform control volume mesh used for all calcu- lations.

3.1 CV-FD equations

refinement at the exchange surfaces in order to satisfy the requirements of item (2) above. The particular mesh used for all computations undertaken in this work contains 210 CVs and has the physical dimensions of 10 X 21 node points in the transverse and longitudinal directions, respectively.

After some mathematical manipulations, the resulting control volume finite difference equations (CV-FDEs) can be summarized as follows:

W,E,N,S

A& = c A”$“,, + b ( 15a) i=l

where CI = (X, 0, P>r and the right-hand side vector b = (Su,, Su2, S,,jT contains any relevant boundary condition information. The (3 X 3) matrices A, and A,, contain the CV-FDE discretization coefficients at the point P and the neighboring F roints Nb, = W, E, N, S, respectively:

A,=

A xl, ATI, API, 42, 42, APZp 43, AT3, AP3, I and

( 15b)

where, for example:

4lN = Kxyl,AZi

A %pij _

6Y” X1,0 St

( 15c)

41p =A&, +-%I, +41s +A& +&I,, and the right-hand side source term S,, is given by

S,, =AxI,Xo’d +ATl,O@“d +ApI,,Po’d

+(Kgrl, -Kg,Iw)AYj (154

+ (contributions from the boundary conditions

that apply at the BCVs)

When these equations are applied to each control volume within the mesh, a square block matrix of dimension

N X M results. This matrix can be visualized as a block pentadiagonal matrix having sub-blocks of dimension [3 X

31, and the main diagonal block consists of a banded matrix having a band width of 11. Due to the nature of the coefficients of the diagonal blocks the matrix is not symmetric; however, it is block symmetric. In order to normal- ize this matrix, the off-diagonal sub-blocks are premulti- plied by the inverse of the diagonal blocks and the diagonal blocks are replaced by I3 identity matrices. This technique becomes particularly useful if an iterative method such as successive over-relaxation (S.O.R.) is used to obtain the solution of the system. However, in this work two numerical solution techniques which use efficient band solvers were implemented. These methods, known as the coupled and uncoupled solution procedures, are discussed in more detail in succeeding paragraphs.

3.2 Treatment of the kinetic coefficients at CV interfaces

An integral component of the CV-FDEs is the calculation of the kinetic coefficients at the control volume surfaces. Accurate evaluation of these coefficients requires the iden- tification of components due to convection and diffusion. The convection component must be treated by using an upstream weighting or upwinding technique. This technique classifies the direction of the flows of liquid and gas by evaluating the relevant fluxes for the cell under consideration and its’ neighboring cells. Then, according to the flow direction, the convection components are evaluated at either the cell center point (P), or the neighboring cell point (N, S, E, W), see Figure 2. Failure to correctly treat the convective terms can cause the drying fronts to appear in misleading positions. The diffusion component is evaluated at the face of the control volume using an appropriate averaging technique.

The two averaging techniques which have been considered in this work are the Weighted average:

( 16a)

and the Harmonic average:

K,Kp (KDiffusion)e= (1 _fe)KE+feKP ( 16b)

where the interpolation factor is defined as the ratio Sz:/Gz, (Figure 2). The Harmonic average is preferred over the weighted average due to the fact that it better represents physical reality across discontinuities that may occur within the material.

3.3 Implementing boundary conditions

Consider the boundary conditions at the West boundary exchange surface. The CV-FDEs are modified as follows:

E,N,S

AA = c A,,,%,, + b ( 17a) i= 1

where all the A, coefficients are set to zero and the S,



terms are adjusted by the boundary flux through the BCV surface as follows:

S”l = S”l - Kmv (f& - %),

(2.2 - P”l& - P,z), AYj (17b)

3.4 The numerical algorithms

Two different numerical techniques were investigated in the search for a robust, efficient solution method. The methods, which are called the uncoupled and coupled solution methods, are briefly discussed in the following paragraphs.

Uncoupled solution method. In this method the equations are solved as individual uncoupled two-dimensional equations. The coupling terms are all transferred to the right-hand side of the discretized equations and an outer loop Gauss-Seidel fixed point iterative procedure is employed to advance the solution in time. In the context of the equations under study here, this implies solving for the moisture content, solving for the temperature, solving for the pressure, and then iterating to take into consideration the coupling effects that exist between the equations. The inner looping procedure determines the solution of each equation by solving the associated point pentadiagonal matrix system via an alternating direction line SOR (AD- LSOR) method. The CV-FDEs are given as follows:

W,E,N,S

A?,= c A,,,X”,, +b, ( 1Sa) i= 1

W,E,NS

A,%= c A”,,@“,,+& ( 18b)

i= 1

W,E,N,S

APP = c A”b,P”b, + b, ( 18c)

i=l

where the bi contain the coupling terms.

Coupled solution method. In this method the coupling between the three equations is retained and the system is solved co_upled together on a line for the triplet (X , Op, P,jT. An efficient block ADLSOR sweeping technique was implemented which solves an 11 banded system of equations (or equivalently a one-dimensional problem) for each line of the solution domain by using a direct band solver based on an LU decomposition. The alternative sweeping is directed from west to east and then from north to south so that the information from the exchange boundary surfaces is quickly propagated throughout the numerical solution.

Inner iteration is necessary to converge the solution sufficiently for a line, while outer iteration is most important to converge the solution due to the nonlinear coefficients. The CV-FDEs for each sweep are summarized as follows:

West-east sweep

Apii’(pk+‘) = yAnh,ii$,T1) + b* where i=l

b * = b + A@(&+ ‘) + A,@’ ( 19a)

North-south sweep

AptiLk+‘) = yA,,iti$,:l) + b* where i= 1

b* = b + ANii(Nkfl) + AsCl’,k’ ( 19b)

After the solution for uck+ ‘) has been computed along a line it can be over-relaxed by application of the following equation at every point along the line:

fib”,d;c; = w ti p (k+ 1) + (1 _ +6”’

and 0 < o < 2 is the relaxation parameter.

(19c)

Although both numerical schemes actually provided convergent solutions, the numerical investigations undertaken here clearly demonstrated that due to the tight coupling of the equations, using the uncoupled method was not very efficient, often taking five to six times longer computation time than the coupled method. This increased processing time can be accounted for by considering the work done by each scheme. For example, to advance the solution in time, the coupled method requires solving a large banded pentadiagonal system for the triplet (X, T, P) at each cell center with iteration necessary due to the nonlinear coefficients; while the uncoupled method solves three smaller individual pentadiagonal systems for X, T, and P but spends a large amount of time iterating due to

1 50

& 120

F

s

a, 90

'j +G .-

r" 60

t3

P

9 30 Q

00 0 20 40 60 80 100

Time (Hours)

Figure 4. Comparison of drying times for the three cases of drying.



no ,rtur. Con L.” L al 20 Hour.

:: : 3: . : :: r ;

f .=

Figure 5. Moisture content surfaces for drying at low temperatures (case I).



Tor,oerslure at 10 Hourm

Figure 6. Temperature surfaces for drying at low temperatures (case 1).


An orthotropic model for simulating wood drying: I. W, Turner

the nonlinear coefficients in each equation as well as complex heat and mass transfer within porous media, the iterating due to the coupling effects of the system. It was best solution procedure to adopt is undoubtedly the cou- concluded from this study that for problems that deal with pled solution technique.

Figure 7. Pressure surfaces for drying at low temperatures (case 1



rk.19lur9 Conbnt sL 20 Hour.

Figure 8. Moisture content surfaces for drying at moderate temperatures (case 2).



. -_.. - *a-

Figure 9. Temperature surfaces for drying at moderate temperatures (case 2).



Pr...ur. e.L 5 Hour*

Pr.*.ur* OL 15 Hour*

Pr...ur* SL 25 Hour.

Figure 10. Pressure surfaces for drying at moderate temperatures (case 2).

3



4. Numerical simulation results

In this section the numerical simulation code Wood2D will be used to study the convective drying of wood at temperatures below and above the boiling point. Three drying cases have been selected to exhibit the two-dimensional heat and mass transport phenomena and the particular drying conditions associated with each case are summarized below.

Case 1 - Low tem-

Dry bulb temperature

Wet bulb temperature

perature drying Case 2 - Moderate

50°C 30°C

temperature drying Case 3 - High

80°C 65°C

temperature drying 120°C 80°C

In all cases a uniformly convected air stream with a velocity of 2 m/s was used for the drying, which provided heat and mass transfer coefficients given by 14 W/m*/K’ and 0.014 m/s, respectively. The dimensions of the wood sample were 5 cm in the transverse and 50 cm in the longitudinal direction which implied a computational domain of 2.5 cm X 25 cm. The wood had an initial moisture content of 150% and an initial temperature of 25°C. The species of wood chosen for the study was spruce and all the necessary correlations (capillary pressure, permeabilities, and diffusivities) are taken directly from the literature’4,‘6,‘8 and are summarized in Appendix A.

The numerical investigation now focuses on the drying kinetics and related moisture content, temperature, and pressure distribution associated with the above drying cases. In particular it will be shown that the two-dimensional code will provide a closer representation of physical reality, as compared with previous one-dimensional models, accentuating the effects of anisotropy.

Figure 4 exhibits the comparison of the drying times for the three test cases. It is evident from the figure that the reduction in drying time can become significant when high-temperature convected air is utilized. For drying below the boiling point (cases 1 and 2) the general shape of the drying curve is similar, displaying a well-defined constant rate period up to 10 hr followed by a significant falling rate period where the bulk of the liquid is removed from the wood. However, above the boiling point (case 3) the curve displays a pronounced shape which differentiates it from the others, showing a rapid moisture extraction rate occurring during the falling rate period of drying for times greater than 5 hr.

In Figures 5-13, three-dimensional surface plots that depict the evolution of the distribution of moisture content, temperature, and pressure throughout the three test drying cases are displayed. Although contour plots are probably a more accurate representation of these distribution, it is felt that the surface plots exhibited here clearly and succinctly demonstrate the phenomena that arise during the wood drying process. These figures show specific snapshots of the distribution at the times 5, 10, 15, 20, 25, 50, 75, and 100 hr for cases 1 and 2; and at 2, 5, 10, 15, 20, 25, 30,

and 50 hr for case 3. Note that in these figures the entire transverse dimension of the wood is displayed in order to enhance the graphical presentations. The main findings from the simulations are summarized as follows.

4.1 Low temperature drying-case 1

A major phenomenon that occurs during low temperature wood drying involves the appearance of an underpressure within the wood. The amplitude of this underpressure will increase as the initial moisture content of the wood approaches the fully saturated value. In this case, liquid extraction by capillarity causes an expansion in the volume of the gaseous phase to arise and the medium tends to become subjected to partial vacuum. This effect can severely inhibit liquid extraction and can cause a steep front to occur in the moisture profile. Examining the graphs for moisture content (Figure 5), temperature (Fig- ure 6), and pressure (Figure 71, indicates that the model evidently exhibits the above phenomenon. These graphs depict that during the constant rate period, times up to 10 hr, the temperature remains constant at the wet bulb temperature of 30°C moisture content reduces significantly in the transverse direction, as a consequence of the low permeability, and the underpressure begins to emerge. Once the falling rate period commences (times greater than 10 hr), the temperature of the wood steadily increases throughout, except at the end piece of the wood which is being cooled due to evaporation. The moisture content along the longitudinal surfaces of the wood has decreased below XFsp and the underpressure, which is now more prominent, causes a slight aspiration of free liquid from the end of the board toward the center.

As drying progresses the moisture distribution displays a distinct uniform parabolic shape which appears to be migrating longitudinally, exiting the wood through the end piece. Eventually the underpressure diminishes and flattens out at the atmospheric value, the moisture content approaches the equilibrium value, and the temperature increases to that of the convected air.

4.2 Moderate temperature drying-case 2

The graphs representing this case are given in Figures 8-10. In general the overall characteristics of the profiles are similar to those observed for case 1. The main difference being that close to the longitudinal surface an overpressure arises (times greater than 20 hr). This effect parallels the rise in temperature in these regions of the board. It is concluded from the figures for this drying case that both an overpressure and an underpressure exist within the wood. Other differences include an increased wet bulb depression in the temperature profiles (times less than 15 hr), and a slightly higher moisture extraction rate. These differences are primarily caused by the characteristics of the drying air. Finally, during the last stages of the drying process (times around 100 hr), the moisture profiles exhibit a complete surface dry out with the moisture content of the wood approaching the equilibrium value and, as expected,


An orthotropic model for simulating wood drying: 1. W, Turner

na,*tw. Conl,nL aL 20 Hour-

Figure 11. Moisture content surfaces for drying at high temperatures (case 3).


TamQsr.hre aL 2 Hour.


Figure 12. Temperature surfaces for drying at high temperatures (case 3).


An orthotropic mode/ for simulating wood drying: I. W. Turner

I+...ur. .t 2 bw.

f .=

.-

.-=

Figure 13. Pressure surfaces for drying at high temperatures (case 3).



the temperature of the board increasing to that of the dry bulb.

4.3 High temperature drying-case 3

From the study undertaken in this work, it appears that high temperature drying has a number of advantages over using low temperatures. In particular, for spruce, it is possible to substantially reduce overall drying times and increase drying rates. This undoubtedly implies faster processing times which provides incentive for future research and development in the area.

During high temperature convective drying, where the wood temperature approaches and passes through the boiling point, rapid internal vaporization causes a very important heat and mass transport process to transpire. Due to the strong anisotropy of the wood a migration of both liquid and gaseous phases occurs in the longitudinal sense which is the most permeable direction.

The results presented in Figures 11-13 distinctly re- flect the occurrence of this phenomenon, again indicating that the model represents reality well. In this particular drying case, only a short constant rate period is evident (time at 2 hr), with the temperature remaining constant throughout at the wet bulb temperature of 80°C and the moisture content sharply decreasing in the transverse direction. The corresponding pressure curve exhibits a slight underpressure due to the expansion of the gaseous volume.

At this stage the profiles are in agreement with those observed with drying cases 1 and 2. After 5 hr of drying, the longitudinal surface temperature begins to increase since the corresponding moisture content along those surfaces has passed through XFsp and is fast approaching the equilibrium value that is determined by the drying air characteristics. It is also evident that liquid is beginning to build up around the end piece of the board. The process has now entered into the falling rate regime. Due to this moisture buildup at the end piece, an evaporative cooling effect keeps the temperature at that position close to the wet bulb. It should be noted that at this stage, both overpressure, due to increased temperature close to the longitudinal surfaces, and underpressure, due to internal capillary extraction, exist within the wood. As the process continues, times lo-30 hr, the temperature increases above 100°C and as a result the internal pressure increases significantly. Moisture continues to migrate longitudinally, con- tinuously feeding the end piece where it finally exits the wood due to evaporation. The temperature is still main- tained at the wet bulb at the end piece.

Eventually (times greater than 30 hr), the board dries throughout and the moisture content distribution reaches the equilibrium value, the temperature increases and plateaus at the dry bulb temperature of 120°C and the internal pressure relaxes back to atmospheric.

5. Conclusions

A comprehensive mathematical model has been developed and a numerical code known as Wood2D has been imple-

mented so that two-dimensional simulations of the convective drying of wood could be investigated. This code has proven to be extremely useful in identifying the convoluted heat and mass transport phenomena that arise throughout the process. In summary the main findings of this research are as follows:

A robust and efficient numerical scheme is obtained using a cell-centered control volume technique. A nonuniform mesh refined around the exchange surfaces of the wood is essential in providing a convergent solution. Due to the tight coupling of the transport equations it was found that the equations should be solved coupled together in order to obtain acceptable computation times. The inclusion of the total gaseous pressure is an essential ingredient in the mathematical formulation for wood drying processes. For the results presented here it was clear that the comportment of the gaseous phase plays an important part in the overall process for drying both below and above the boiling point. In particular the pressure variable allows the effects of underpressure (drying cases 1 and 2) and overpressure (drying case 3) to be thoroughly investigated. The effects of anisotropy, notably for the permeabilities, substantially influence the drying results. It was clearly exhibited that the drying of wood displays classical two-dimensional heat and mass transfer, with heating in the transverse direction causing the liquid and gaseous phases to migrate in the longitudinal direction. This observation highlights the importance of the two-dimensional model.

Acknowledgments

I wish to thank Dr. Julian van Leersum from the School of Mathematics at QUT for many helpful and stimulating discussions regarding this work. I also express gratitude to members of the group in the Centre for Numerical Mod- elling and Process Analysis at the University of Greenwich for allowing me, on a number of occasions, to discuss with them numerous aspects of the control volume method. I found their expertise in this area world class. This work was carried out under a QUT Research and Development Support Scheme and my sabbatical at the University of Greenwich was supported by the QUT Personal Develop- ment Program.

Nomenclature

a Capacity coefficient used in the drying equations

: Molar concentration [mol/m3]

D:r,

Specific heat [J/kg/K] Effective diffusivity tensor [m’/s]

D, Bound liquid diffusivity [m’/s] Internal energy [J/kg] Activation energy of movement of bound water [J/m011


f Gravitational constant [m/s21

7l Intrinsic averaged enthalpy [J/kg] Averaged enthalpy of bound liquid [J/kg]

h AXP

Latent heat of evaporation [J/kg]

H(l) Differential heat of sorption [J/kg] Heaviside function

K eff Effective thermal conductivity [W/m/K] K WO Intrinsic permeability of liquid phase [m*]

KtF Intrinsic permeability of gas phase [m*] K Relative permeability K Kinetic coefficient used in the drying equations

km Mass transfer coefficient [m/s] L Thickness of the material [m] h4 Molar mass [kg/mol]

; Unit outward normal Pressure [Pa]

Q Heat transfer coefficient [W/m*/K] R Universal gas constant [J/mol/K] S Volume saturation T Temperature [K] t Time [s]

t: Velocity [m/s] Averaging volume

x Moisture content dry basis [(kg water>/(kg dry solid)]

X Molar fraction Z Transverse distance [ml

Y Longitudinal distance [m]

Greek symbols

z Surface porosity [m2/m2] Volume fractions

4 Porosity [m”/m3]

Y Tortuousity factor ?P Relative humidity

El. Dynamic viscosity [kg/m/s] 0 Nondimensional temperature

P Intrinsic averaged density [kg/m”] u Surface tension [N/m] 7 Nondimensional time

Subscripts

: Air Bound

C Capillary cr Critical

eq Equilibrium f Free

Fsp Fiber saturation

g Gas

gvs Saturated vapor

B Relative Reference

P Pressure

; Solid Temperature

V Vapor wb Bound liquid wf Free liquid X Moisture content 0 Atmospheric 1 Initial

Appendix A

Wood property Value used in simulations

Porosity Solid density Solid specific heat Capillary pressure

Permeabilities

Effective diffusivity Bound liquid diffusivity

Effective conductivity

r#l = 0.66 p,, = 500 kg/m3 C,, = 1400 J/kg/K P,(Xw,,T)=1.364x105u~T) (Xw,+1.2x10-4)-o~63 Pa (Gas). K =5x IO-l6 m* (Liquid)z%,, =5x10-l7 m* D,rr = K,(Xw,)Dv10-4 m*/s D, = exp( - 9.9 - 4300/ T +9.8X,,) m*/s K,,, = 0.12+0.23X W/m/K

Anisotropy ratios for longitudinal direction

Property Factor

K cl0 K wo D eff

‘% K eff

1,000 1,000

50 2.5 2


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https://www.researchgate.net/publication/225408557_Capillary_Pressure_in_Softwoods?el=1_x_8&enrichId=rgreq-513d4e87fda3eca5f2c2287c66bb272a-XXX&enrichSource=Y292ZXJQYWdlOzIyMzkzMTAxODtBUzoxMDEyNjE0MDc5NDg4MDZAMTQwMTE1NDAwOTQ4Mg==

https://www.researchgate.net/publication/229808748_A_Mathematical_Model_of_Drying_of_Hygroscopic_Porous_Media?el=1_x_8&enrichId=rgreq-513d4e87fda3eca5f2c2287c66bb272a-XXX&enrichSource=Y292ZXJQYWdlOzIyMzkzMTAxODtBUzoxMDEyNjE0MDc5NDg4MDZAMTQwMTE1NDAwOTQ4Mg==



https://www.researchgate.net/publication/37905581_Transport_Process_in_Wood?el=1_x_8&enrichId=rgreq-513d4e87fda3eca5f2c2287c66bb272a-XXX&enrichSource=Y292ZXJQYWdlOzIyMzkzMTAxODtBUzoxMDEyNjE0MDc5NDg4MDZAMTQwMTE1NDAwOTQ4Mg==

https://www.researchgate.net/publication/37905581_Transport_Process_in_Wood?el=1_x_8&enrichId=rgreq-513d4e87fda3eca5f2c2287c66bb272a-XXX&enrichSource=Y292ZXJQYWdlOzIyMzkzMTAxODtBUzoxMDEyNjE0MDc5NDg4MDZAMTQwMTE1NDAwOTQ4Mg==

https://www.researchgate.net/publication/34180903_A_Model_of_Simultaneous_Convective_Diffusive_and_Capillary_Heat_and_Mass_Transport_in_Drying_Wood?el=1_x_8&enrichId=rgreq-513d4e87fda3eca5f2c2287c66bb272a-XXX&enrichSource=Y292ZXJQYWdlOzIyMzkzMTAxODtBUzoxMDEyNjE0MDc5NDg4MDZAMTQwMTE1NDAwOTQ4Mg==




https://www.researchgate.net/publication/291890974_An_application_of_the_finite_element_method_to_the_drying_of_timber?el=1_x_8&enrichId=rgreq-513d4e87fda3eca5f2c2287c66bb272a-XXX&enrichSource=Y292ZXJQYWdlOzIyMzkzMTAxODtBUzoxMDEyNjE0MDc5NDg4MDZAMTQwMTE1NDAwOTQ4Mg==



https://www.researchgate.net/publication/291492857_NUMERICAL_MODELLING_OF_DRYING_INDUCED_STRESSES_IN_POROUS_MATERIALS?el=1_x_8&enrichId=rgreq-513d4e87fda3eca5f2c2287c66bb272a-XXX&enrichSource=Y292ZXJQYWdlOzIyMzkzMTAxODtBUzoxMDEyNjE0MDc5NDg4MDZAMTQwMTE1NDAwOTQ4Mg==



https://www.researchgate.net/publication/279565826_NUMERICAL_MODEL_OF_HEAT_AND_MASS_TRANSPORT_IN_WOOD_DURING_DRYING?el=1_x_8&enrichId=rgreq-513d4e87fda3eca5f2c2287c66bb272a-XXX&enrichSource=Y292ZXJQYWdlOzIyMzkzMTAxODtBUzoxMDEyNjE0MDc5NDg4MDZAMTQwMTE1NDAwOTQ4Mg==



https://www.researchgate.net/publication/300663312_High_Temperature_Convective_Drying_of_Softwood_and_Hardwood_Drying_Kinetics_and_Product_Quality_Interactions?el=1_x_8&enrichId=rgreq-513d4e87fda3eca5f2c2287c66bb272a-XXX&enrichSource=Y292ZXJQYWdlOzIyMzkzMTAxODtBUzoxMDEyNjE0MDc5NDg4MDZAMTQwMTE1NDAwOTQ4Mg==





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A two-dimensional orthotropic model for simulating wood drying processes

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