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000 Hot-pressing process modeling for medium density

fiberboard (MDF) Technical Report 1 .

Noberto Nigro and Mario StortiCentro Internacional de Metodos Computacionales en Ingenierıa

http://minerva.unl.edu.ar/gtm-eng.html

October 28, 2018

$Id: hotpress.tex,v 1.6 1999/06/17 15:39:19 mstorti Exp $

Abstract

In this paper we present a numerical solution for the mathematicalmodeling of the hot-pressing process applied to medium density fiber-board. We have started our development from the paper of Carvalho& Costa [2] and the model presented by Humphrey in his PhD. thesis[4] with some modifications and extensions in order to take into ac-count mainly the convective effects on the phase change term and alsoa conservative numerical treatment of the resulting system of partialdifferential equations.

Contents

1 Hot-pressing mathematical model 2

1.1 Multiphase model . . . . . . . . . . . . . . . . . . . . . . 21.2 Energy balance . . . . . . . . . . . . . . . . . . . . . . . 31.3 Steam mass balance . . . . . . . . . . . . . . . . . . . . 31.4 Gas mixture mass balance . . . . . . . . . . . . . . . . . 4

2 Summary of equations and boundary conditions 5

3 Numerical method 5

4 Physical and transport properties 7

4.1 Thermal conductivity . . . . . . . . . . . . . . . . . . . 74.1.1 Density correction . . . . . . . . . . . . . . . . . 74.1.2 Moisture correction . . . . . . . . . . . . . . . . . 74.1.3 Temperature correction . . . . . . . . . . . . . . 74.1.4 Heat flux direction correction . . . . . . . . . . . 8

4.2 Permeability . . . . . . . . . . . . . . . . . . . . . . . . 84.2.1 Variation of vertical permeability with board

material density . . . . . . . . . . . . . . . . . . 9

1

4.2.2 Horizontal permeability . . . . . . . . . . . . . . 104.3 Vapour Density . . . . . . . . . . . . . . . . . . . . . . . 104.4 Saturated vapour pressure . . . . . . . . . . . . . . . . . 104.5 Latent heat of evaporation and heat of wetting . . . . . 104.6 Specific heat of mattress material . . . . . . . . . . . . . 114.7 Porosity . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

5 Numerical results 11

5.1 Original numerical experiment . . . . . . . . . . . . . . 135.2 Further numerical experiments . . . . . . . . . . . . . . 18

6 Appendix. Derivation of the averaged energy balance

equation 19

1 Hot-pressing mathematical model

Following the Carvalho & Costa paper [2] the following dimensionlessnumbers and reference values appear:

Foz =1

ρs(1)

Fomz = ǫ (2)

ρv0 =MMw

R(3)

Npz =1

ǫ(4)

Pez = 1 (5)

Pemz =1

ǫ(6)

ρ0 =MMa

R(7)

ρv0 =MMw

R(8)

where ρs is the solid material density, ǫ represents the material poros-ity, MMw,MMa are the water and air molecular weight in Kg/Kmoland R is the particular gas constant for air (R = 8314KJ/(KmolK)).Foz,Fomz are the thermal and mass Fourier number, ρ0, ρv0 are refer-ence values for air and vapor density, Npz is the number of voids in aunit volume, Pez ,Pemz are the thermal and mass Peclet values repre-senting the relative importance of convective effects in the transportequations. In order to match our dimensional form with the dimen-sionless set of equations presented in the Carvalho & Costa paper [2]we have adopted a unit value for T0 − T∞, τ , Lx,y,z.

1.1 Multiphase model

In order to avoid modeling the material down to the scale of the mi-crostructure (the fibers in this case), non homogeneous materials aresolved via “averaged equations” so that the intrincated microstructure

2

results in a continuum with averaged properties. This averaged equa-tions and properties can be deduced in a rigorous way through thetheory of mixtures and averaging operators[14].

We will not enter in the details of all the derivations, but will derivethe averaged energy balance equation.

1.2 Energy balance

ρsCp∂T

∂t= ∇ · (k∇T − ρvvg(CpvT + λ)) − m(λ+Ql) (9)

The main difference between this equation and that presented byboth Carvalho & Costa [2] and Humphrey [4] is in the addition of thewater evaporation heat in the convection term instead of consideringthe phase change effect only on the temporal term. This term shouldbe included because both phases, the solid material and the vapor arein relative motion and we think that its influence is not negligible in ahigh temperature process.

1.3 Steam mass balance

Carvalho & Costa [2] have proposed the following steam mass conser-vation equation:

m =MMw

Rǫ∇ ·

[

−Dv∇(Pv

T) +

1

ǫvg

Pv

T

]

(10)

Considering that the steam is treated as an ideal gas

Pv

T= MMw

−1Rρv (11)

so, the resulting mass balance for the steam may be written as:

m = ∇ · [−ǫDv∇ρv + vgρv] (12)

We prefer to adopt this kind of expression instead of 10 because itis written in a conservative form that is more agreeable for a numeri-cal treatment. The left hand side term represent the mass interfacialtransport and those in the right hand side take into account the masdiffusion and the mass convection. However, it should be noted thatthis last expression does not have a temporal term like in every con-sistent balance equation. For example, if no evaporation is consideredthe 12 is only valid for a static situation that it is not the general case.Then, we rewrite the steam mass balance as:

ǫ∂ρv∂t

= ∇ [ǫDv∇ρv − vgρv] + m (13)

This is another difference between our model and that proposed byCarvalho & Costa [2].

3

1.4 Gas mixture mass balance

Finally, because the gas phase is composed by two main constituents,the steam and the air we may use an additional equation for the masstransport of the whole gas phase.

∂P

∂t= −1

ǫ∇ ·

(

−Kg

µ

P

T∇P

)

T +m

ǫMMa

TR+P

T

∂T

∂t(14)

Again, assuming the ideal gas law as state equation for this phase,

∂

∂t

(

P

T

)

= −1

ǫ∇ ·

(

−Kg

µρg∇P

)

+mR

ǫMMa

(15)

finally we arrive to the following expression:

ǫ∂ρg∂t

= −∇ · (ρgvg) + m (16)

In order to close the system of equations we need to introduce arelationship between m and ∂Pv

∂t . Here, we have proposed the followingstrategy to avoid the definition of this relationship. We consider thesteam mass balance above presented (13) and we have proposed anadditional equation for the total water content H of the following form:

ρsH = ǫρv + ρL (17)

where ρL is the bound water density (= bound water per unit volume).The bound water only may be transfered to the gas phase (solid tosteam) assuming that no liquid phase is considered. So,

m = −∂ρL∂t

(18)

and then

ρs∂H

∂t= ǫ

∂ρv∂t

+∂ρL∂t

(19)

= ∇ · [ǫDv∇ρv − vgρv] (20)

Air mass balance equation arises substracting (13) from (16)

ǫ∂ρa∂t

= ∇ · [−ǫDv∇ρv − vgρa] (21)

Due to the fact that the mean macroscopic diffusive fluxes should benull

Dv∇ρv +Da∇ρa = 0 (22)

the air mass balance is transformed in:

ǫ∂ρa∂t

= ∇ · [ǫDa∇ρa − vgρa] (23)

expression very similar to (13) but here valid for the air. As it isobvious for the air transport there is no evaporation.

4

2 Summary of equations and boundary

conditions

In order to clarify the mathematical model that is finally used forthe simulation of hot-pressing process we present the following briefsummary of partial differential equations.

ρsCp∂T

∂t= ∇ · (k∇T − ρvvg(CpvT + λ)) − m(λ+Ql) (24)

ρs∂H

∂t= ∇ · [ǫDv∇ρv − vgρv] (25)

ǫ∂ρa∂t

= ∇ · [ǫDa∇ρa − vgρa] (26)

The boundary conditions are the following:

• At the press platen:

T = Tplaten (27)

vg = 0,∂ρa∂z

= 0,∂ρv∂z

= 0 (28)

• At the center line (r = 0):

∂T

∂r= 0,

∂H

∂r= 0,

∂ρa∂r

= 0 (29)

• At the mid plane (z = 0):

∂T

∂z= 0,

∂H

∂z= 0,

∂ρa∂z

= 0 (30)

• At the exit boundary (r = Rext):

∂T

∂r= 0, null diffusive heat flux (31)

Pv = Pv,atm (32)

Pa = Pa,atm (33)

3 Numerical method

The above system of equations contains three main unknowns, the tem-perature, the moisture content and the air density representing the de-pendent variables of the problem also called the state variable. In thiswork we have used as independent variables the time and three spatialcoordinates. Due to the physical and geometrical inherent complex-ity of this problem this may be computed only by numerical methods.For the spatial discretization we have employed finite elements withmultilinear elements for all the unknowns. Due to the high convectiveeffects we need to stabilize the formulation by some special technique

5

like SUPG (“Streamline Upwind - Petrov Galerkin”, otherwise spuriousoscillations are frequently encountered. Once the spatial discretizationis performed the partial differential system of equations is transformedinto an ordinary differential system of equations like:

U = R(U) (34)

where U is the state vector containing the three unknowns in eachnode of the whole mesh. So, the system dimension is 3N where Nis the number of nodes in the mesh. The numerical procedure is asfollow: Knowing the state vector at the current time (tn), i.e. Uj(t

n) =[Tj ,Hj , ρa,j](t

n) where j represent an specific node in the mesh. To getthe residual, right hand side of equation 34 the following steps shouldbe done:

• From the gas state equation (T, ρa) → Pa,

• From soption isotherms (T,H) → HR → Pv,

• From vapor state equation (T, ρv) → Pv,

• Compute some state dependent coefficients from additional con-stitutive laws, like → D∗,Kg, kx,y, Cp, and so on ...

• Compute gradients of T , P , so on... at some specific Gauss pointsin each element of the mesh.

• Compute each element residual contribution and gather in aglobal vector

Once the whole residual vector at t = tn is computed the unknownsvariables at the next time step is updated from the following equation:

Un+1 = Un +∆tR(Un) (35)

This kind of scheme, called explicit integration in time is very simpleto be implemented but it has two major drawbacks, one is the limi-tation of the time step to ensure numerical stability and the other isthe bad convergence rate for ill-conditioned system of equations. Inthis application the last disadvantage is very restrictive because thecharacteristic times of each equation are very different. To circumventthis drawback we have implemented an implicit numerical scheme like:

Un+1 −∆tR(Un+1) = Un (36)

where the residue is computed at the new state variable U(tn+1) in-stead of using the current value U(tn). The non-linearities and the timedependency of the state vector makes this implementation more diffi-cult and time consuming but more stable. Due to the non-linearitiesa Newton-Raphson method is highly recomended being the jacobiancomputation one of the more difficult problems to be done specially forthis application where the coefficients of the balance equations have astrong dependency with the state vector. In order to get a robust codewe adopt a numerical computation of the jacobians,

J =∂R

∂U(37)

6

using a finite difference method like:

J ≈ Jnum =R(U + δU)−R(U)

δU(38)

If we performe this strategy in an elementwise form the cost involved isO(N). In our application and despite of the ill-condition of the problemwe have found a good convergence at each time step, in average 4iterations per time step to reduce 10 orders of magnitude in the globalresidue.

4 Physical and transport properties

4.1 Thermal conductivity

Following Humphrey [4] the influence of board density, moisture con-tent and temperature on the thermal conductivity is considered onceat each time. The effect of each variable is defined as a factor uponthe main definition:

κz = 1.172× 10−2 + 1.319−4ρs (39)

where κz is the thermal conductivity in the pressing direction inW/m/K and ρs is the oven dry density of the material in Kg/m3.Considering the basal or reference value as:

κz = 0.09W/m/K for ρs = 650Kg/m3 (40)

4.1.1 Density correction

Fκ,ρ = (ρs − 586)× 1.46× 10−3 + 1 (41)

4.1.2 Moisture correction

From Kollmann and Malmquist (1956) and Humphrey (1982)

Fκ,H = (H − 12)× 9.77× 10−3 + 1 (42)

with H the moisture content of the board material in %

4.1.3 Temperature correction

From Kuhlmann (1962) and Humphrey (1982)

Fκ,T = (T − 20)× 1.077× 103 + 1 (43)

with T in Celsius degrees.

7

4.1.4 Heat flux direction correction

By far the greater part of conductive heat translation takes place in thevertical plane. However the energy lost from the matress is largely theresult of radial vapour migration from the centre towards the atmo-sphere. The associated horizontal relative humidity gradient lead to ahorizontal temperature gradient. Even though this gradient is alwayslower than the vertical one its influence should be taken into accountif multidimensional analysis is required. Following the few Ward andSkaar laboratory measurements (1963) they observed that at a firstglance a factor of approximately 1.5 may be a good initial guess beforedoing some extra measurements. Then

κxy = 1.5κz (44)

4.2 Permeability

The evaporation and condensation of water changes the vapour densityand consecuently its partial pressure in the voids within the composite.A vertical pressure gradient leads to the flow of water vapour from thepress platens towards the central plane of the board. At the sametime an horizontal vapour flow is set up in response to the pressuregradient established in the same direction. The relation between thepressure gradient and the flow features may be assigned to the materialpermeability. Permeability is a measure of the ease with which a fluidmay flow through a porous medium under the influence of a givenpressure gradient. Different mechanisms may be involved in this flow,a viscous laminar flow, a turbulent flow and a slip or Knudsen flow.In this study only the first type is included with the assumption thatDarcy law is obeyed. This hypotesis should be verified experimentally.However for our preliminar work we have assumed it. This may bewritten as:

∇p = − µg

Kgvg (45)

where ∇p is the driving force and vg is the flow variable.Kg

µg

is

called the superficial permeability with µg the dynamic viscosity ofthe gas phase and Kg the specific permeability. When Darcy law isapplied to gaseous flow the compressibility of the fluid should be takeninto account. Humphrey suggested to use a correction factor for thesuperficial permeability as:

Kg

µg

P

P(46)

where P is the mean pressure value along the flow path and P isthe local total pressure.

8

Relative to the viscosity the kinetic theory of gases suggests that atnormal pressure the viscosity is independent of pressure and it variesas the square root of the absolute temperature,

µ ∝√T 103 < p < 106 (47)

Corrections with the absolute temperature are often considered bySutherland law:

µ =B T 3/2

T + C(48)

with B and C characteristic constants of the gas or vapour with µ inKgm−1 s−1. Values for B and C are available from Keenan and Keyes(1966). For this application the following expression was adopted:

µ = 1.112× 10−5 × (T + 273.15)1.5

(T + 3211.0)(49)

with T in C.

4.2.1 Variation of vertical permeability with board ma-

terial density

Even though we consider that the press is closed and the density profileis set up and fixed it is included here some conclusions from Humphreythesis with results obtained by Denisov (1975) for 19 mm boards andothers from Sokunbi (1978). The data to be fitted are the following:

Table I: Vertical permeability density correction data

Mean density Kg/m3 Mean vertical permeability m2 × 1015

425 64

475 40

525 24

575 16

625 11

675 7

725 5

775 3

825 2

875 2

9

4.2.2 Horizontal permeability

Sokunbi measures included in figure 2.7 of Humphrey thesis shows therelation between the board thickness in mm with horizontal permeabil-ity. For approximately 15mm board thickness and ρs ≈ 650Kg/m3 wehave that the horizontal permeability is 100 the vertical value. How-ever, if using 586Kg/m3 for the oven dry material instead we have afactor of 59 that agrees with that reported by Carvalho-Costa .

4.3 Vapour Density

For the pressure range likely to occur during hot pressing (between103 and 3× 105) a linear relationship between saturated vapour pres-sure and vapour density may be assumed. Fitting experimental dataHumphrey had proposed the following expression:

ρv = Psat × 6.0× 10−8 ×HR (50)

with ρv in Kg/m3, Psat in N/m2 and the relative humidity HR in%. By definition

HR = Pv/Psat (51)

and applying ideal gas law for gaseous phase

Pv

ρv= R/MwT (52)

Taking R = 8314J/Kmol/K and Mw = 18Kg/Kmol with T ≈400K we may get the 6.0× 10−8 factor in 50

4.4 Saturated vapour pressure

Following the Kirchoff expression with data presented by Keenan andKeys (1966) we include here the following equation:

log10 Psat = 10.745− (2141.0/(T + 273.15)) (53)

with Psat in N/m2 and T in C.

4.5 Latent heat of evaporation and heat of wetting

Using Clausius-Clapeyron equation in differential form and after somesimplification the latent heat of vaporization of free water may be writ-ten as:

λ = 2.511× 106 − (2.48× 103 × T ) (54)

10

with λ in J/Kg and T in C.For the differential heat of sorption we follow Humphrey that used

Bramhall (1979) expresion:

Ql = 1.176× 106 × exp(−0.15H) (55)

with Ql in J/Kg and H in %.

4.6 Specific heat of mattress material

It is computed by adding the specific heat of dry wood and that ofwater according to the material porosity and assuming full saturation.The specific heat of dry mattress material is taken as 1357K/Kg/Kand the specific heat of water has been taken to be 4190J/Kg/K.From Siau (1984) the expression for specific heat of moist wood is:

Cp = 41800.268+ 0.0011(T − 273.15) +H

1 +H(56)

4.7 Porosity

According to Humphrey the volume of voids within the region may becomputed by

ǫ =Vvoids

V= (1− ρ

ρs) (57)

where ǫ is the porosity, ρ is the density of the region and ρs is thedry density of the board material.

In Carvalho-Costa paper they included the expression from Suzukiand Kato (1989):

ǫ = 1− ρs1/ρf + yr/ρr

1 + yr(58)

where ρr is the cure resin density, ρf is the oven dry fiber densityand yr is the resin content (resin weight/board weight). In its paperCarvalho-Costa had used yr = 8.5% with ρf = 900Kg/m3 and ρr =1100Kg/m3.

5 Numerical results

In this section we present some results obtained from Humphrey PhD.thesis. This numerical experiment allows the validation of the mathe-matical model and its numerical implementation for future applicationsto hot-pressing process simulation and control. This experiment con-sists of an round fiberboard of 15 mm of thickness and 0.2828 m ofradius that according to its axisymmetrical geometry needs as spatial

11

0 0.05 0.1 0.15 0.2 0.25

−0.05

0

0.05

0.1

r [m]

z x

10 [m

]

Ejemplo Tesis Humphrey ; malla FEM utilizada

Figure 1: Finite element mesh

coordinates only the radius r and the axial coordinate z assuming novariation in the circumferencial coordinate. The axisymmetrical do-main is discretized in 20×20 elements in each direction with a gradingtowards the press platen and the external radius as may be visualizedin figure 1. In order to follow the same assumptions as Prof. Humphreywe have fixed the air to a very low value (ρa ≈ 10−6) because he hadassumed that the press was close with no air inside the fiberboard.

We have adopted the following boundary conditions:

• thermally adiabatic at r = 0, at z = 0 and at r = Rext = 0.2828m

• temperature at z = 15mm/2 fixed to follow the time evolutionof press platen temperature Tpress = T (t)

• moisture transfer isolated at z = 0,z = 15mm/2 and r = 0

• We experienced with two possibilites for the vapour pressureboundary condition at the external radius r = Rext = 0.2828m:

pv(r = Rext) = HRatmPsat(Tatm) (59)

For the press platen temperature we have applied a ramp from 30

at t = 0 to 160 at t = 72 seconds with a least square fitting fromHumphrey data. As initial conditions we have assumed a uniformtemperature of T (t = 0) = 30 in the whole solid material with auniform moisture content of H = 11%.

In the next section we include the results obtained by using themodel above cited to the original problem presented by Humphrey.Next we present some other experiments showing some phenomenathat deserve more attention for future studies.

12

0 1 2 3 4 5 6 7 8

x 10−3

20

40

60

80

100

120

140

160

z [m]

T [C

]

Ejemplo Tesis Humphrey ; T=T(z,t) para r = 0

1 s

10 s

50 s

100 s

200 s

300 s

400 s

Figure 2: Axial temperature profile at r = 0

5.1 Original numerical experiment

Figure 2 shows the temperature distribution in time and with theaxial coordinate at r = 0 (centerline). We can note the penetra-tion in the axial direction of the temperature profile in time fort = 1, 10, 50, 100, 200, 300 and 400 seconds. Figure 3 shows the samekind of plot for moisture content.

The following figures show similar plots but at different locations,

• 4 : temperature at external radius

• 5 : moisture content at external radius

• 6 : temperature at axial centerline (z = 0)

• 7 : moisture content at axial centerline (z = 0)

• 8 : moisture content at press platen

The radial temperature profile at press platen is not shown becauseits distribution is fixed as input data.

Figure 9 shows several isotherms at t = 200 seconds distributed inthe r, z plane.

Figure 10 shows a three-dimensional view for moisture content rep-resented by the third coordinate axis at t = 200 seconds as a functionof r, z.

These results are in good agreement with those presented byHumphrey in his thesis. However the cited author did not presenthis results at some locations that in our opinion should be treatedwith some care, for example at the external radius.

13

0 1 2 3 4 5 6 7 8

x 10−3

0

2

4

6

8

10

12

14

16

18

H [%

]

z [m]

Ejemplo Tesis Humphrey ; H=H(z,t) para r = 0

400 s 300 s

200 s 100 s

50 s

10 s

1 s

Figure 3: Axial moisture content profile at r = 0

0 1 2 3 4 5 6 7 8

x 10−3

20

40

60

80

100

120

140

160

180

200

z [m]

T [C

]

Ejemplo Tesis Humphrey ; T=T(z,t) para r = Rext

400 s 300 s

200 s

100 s

50 s

10 s

1 s

Figure 4: Axial temperature profile at r = R = 0.2828m

14

0 1 2 3 4 5 6 7 8

x 10−3

0

2

4

6

8

10

12

Ejemplo Tesis Humphrey ; H=H(z,t) para r = Rext

z [m]

H [%

]

1 s

10 s

50 s

100 s

200 s

300 s 400 s

Figure 5: Axial moisture content profile at r = R = 0.2828m

0 0.05 0.1 0.15 0.2 0.25 0.3 0.3520

30

40

50

60

70

80

90

100

110

r [m]

T [C

]

Ejemplo Tesis Humphrey ; T=T(r,t) para z = 0

1 s = 10 s = 50 s

100 s

200 s

300 s 400 s

Figure 6: Radial temperature profile at z = 0

15

0 0.05 0.1 0.15 0.2 0.25 0.30

2

4

6

8

10

12

14

16

r [m]

H [%

]

Ejemplo Tesis Humphrey ; H=H(r,t) para z = 0

1 s = 10 s = 50 s = 100 s

200 s

300 s

400 s

Figure 7: Radial moisture content profile at z = 0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

1

2

3

4

5

6

7

8

r [m]

H [%

]

Ejemplo Tesis Humphrey ; H=H(r,t) en la prensa

1 s

10 s

50 s

100 s

200 s

300 s

400 s

Figure 8: Radial moisture content profile at press platen

16

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

1

2

3

4

5

6

7

8x 10

−3

r [m]

z [m

]

isocurvas de T = T(r,z) para t=200 s

Figure 9: Isotherms at t=200 seconds

00.05

0.10.15

0.20.25

0.30.35

0

2

4

6

8

x 10−3

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

r [m]

H = H(r,z) para t=200 s

z [m]

Figure 10: 3D view for moisture content at t=200 seconds

17

5.2 Further numerical experiments

Humphrey only include results for moisture and temperature in thevertical and radial directions at both central planes, r = 0 and z = 0respectively. No mention about the vertical distribution at r = R =0.2828m or about the moisture content at press platen. Moreover, hehad used a uniform mesh of 10 × 10 elements without showing whathappen at the last annuli of elements corresponding to the externalradius.

Our results present some overshooting in the temperature profilevery close to the radial exit contour and at the first moment we thoughabout a spurious numerical problem, but it is due to large variationsof the magnitude of vapour pressure and density at the boundary. Intypical runs, vapour pressure varies from near 2 atm in the centerof the board to 0.01 atm at the external radius. We think that thisproblem will be fixed if we solve for the air density also, but then a veryfine grid will be necessary at the exit boundary, since large variationsof the vapour molar fraction are expected. (see 11). Molar fractionvaries from nearly 1 at the interior of the mat to a 2% at the externalatmosphere. This variation is produced in a thin layer of width δproportional to the diffusivity of vapour in air which is very small.

mat

Total pressure

Vapour partial pressureair

δ

Figure 11: CAPTION

18

6 Appendix. Derivation of the averaged

energy balance equation

The microscopic energy balance equation in the gas phase is

∂

∂t(ρh) +∇ · (ρvh) = −∇ · (k∇T ) (60)

For the other phases (solid and bound water) a similar expressionholds, but neglecting the advective term. Applying the volume av-erage operator[14], we arrive to the following equation averaged on thegas phase:

∂

∂t(ǫg 〈ρh〉g) +∇ · (ǫg 〈ρhv〉g) = ∇ · (ǫg 〈k∇T 〉g) +Qg (61)

ǫg is the volumetric fraction of phase g (i.e. gas) and 〈X〉g is theaverage of quantity X on the volume occupied by phase g

〈X〉g =1

Ωg

∫

Ωg

X dΩ (62)

The term Qg is the total enthalpy flux through the solid-gas interfaceΓ

Qg =

∫

Γ

(ρh)g(v −w) · n dΓ (63)

where (X)g is the value of property X on the g side of the interfaceand w is the velocity of the interface. Assuming that hg is constanton all Ωg (for a certain volume control) then

Qg = 〈h〉g∫

Γ

(ρ)g(v −w) · n dΓ (64)

= 〈h〉g m (65)

where m is the rate of mass of water being evaporated. A commondrawback of averaged equaitons is that, when products of variableslike ρh appear in the microscopic equation, the average of the product〈ρh〉g is obtained in the averaged equation. Now, it is not true that

〈ρh〉g = 〈ρ〉g 〈h〉g (66)

so that the averaged equation contains more unknowns than the orig-inal equation. A common assumption is to assume that no correlationexists between variables and so (66) is approximmately valid.

Then, applying the volume average operator over the gas, solid, andbound water phases and assuming no correlations between variables weobtain the following averaged equaitons for the three phases:

∂

∂t(ǫgρghg) +∇ · (ǫgρgvghg) = ∇ · (ǫg 〈k∇T 〉g)− mhg gas phase(67)

∂

∂t(ǫsρshs) = ∇ · (ǫs 〈k∇T 〉s) solid phase (68)

∂

∂t(ǫlρlhl) = m hl liquid phase (69)

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Now, summation of these three equations gives:

∂

∂t(ǫgρghg + ǫsρshs + ǫlρlhl) +∇ · (ǫgvgρghg) =

∇(ǫg 〈k∇T 〉g + ǫs 〈k∇T 〉s) + m (hl − hg) (70)

Now,

ǫg 〈k∇T 〉g + ǫs 〈k∇T 〉s = keff∇〈T 〉 (71)

where keff is the average conductivity of the solid+water+gas mixture.The gas is assumed as an ideal mixture, so that the enthalpy is thesum of the enthalpy of its constituents, and neglect the contributionof the air constituent so that

ρghg = ρaha + ρvhv ≈ ρvhv (72)

Taking a reference state for the entalphy at a point on the adsorbedstate

hv = Cpv(T − Tref) + λ+Ql (73)

We also neglect the entalphy of the gas phase with respect to thesolid+water phases and put

ǫsρshs + ǫlρlhl = (ρCp)eff (T − Tref) (74)

where ρeff , and Cpeff are averaged properties for the moist board, asa function of temperature and moisture content. Finally, the averagedequation is

∂

∂t[(ρCp)eff (T − Tref)] +∇ · [ǫgρvvg (Cpv(T − Tref) + λ+Ql)] =

∇ · (keffT )− m(λ+Ql) (75)

References

[1] G. Bramhall. Mathematical model for lumber drying. i. principlesinvolved. Wood Sciences and Technology, 12(1):14–21, 1979.

[2] L.M.H. Carvalho and C.A.V. Costa. Modeling and simulationof the hot-pressing process in the production of medium densityfiberboard (mdf). Chem. Eng. Comm., 170 1–21, 1998.

[3] O. B. Denisov, P.P. Anisov, and P.E. Zuban. Untersuchung derpermeabilitat von spanfliesen. Holztechnologie, 16(1):10–14, 1975.

[4] P. Humphrey. Physical aspects of wood particleboard manufacture.PhD thesis, Univ. of Wales, U.K., 1982.

[5] P. Humphrey and A. Bolton. The hot pressing of dry-formedwood-based composites. part ii. a simulation model for heat andmoisture transfer, and typical results. Holzforschung, 43:199–206,1989.

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[6] J. Keenan and F. Keyes. Thermodynamic properties of steam.John Wiley and Sons., London, 1966.

[7] F. Kollmann and Malmquist L. Uber die warmeleizahl von holzund holzwerkstoffen. Holz Roh-Werkstoff, 14(16):201–204, 1956.

[8] G. Kuhlmann. Untersuchung der termischen eigenshaften von holzund spanplatten in abhangigkeit von feuchtogkeit und temperaturim hygroskopischen bereich. Holz Roh-Werkstoff, 20(7):259–270,1962.

[9] P. Kuts and N. Grinchik. Two-phase filtration equations andsorption isotherms as applied for studying kinetics and dynamicsof drying process. In A. Mujumdar, editor, Proc. Drying of solids.

Recent International developments, 1986.

[10] J. Liu, S. Avramidis, and S. Ellis. Simulation of heat and moisturetransfer in wood during drying under constant ambient conditions.Holzforschung, 48:236–240, 1994.

[11] O.K. Sokunbi. Aspects of particleboard permeability. Master’sthesis, University of Wales, U.K., 1978.

[12] M. Suzuki and T. Kato. Influence of dependent variables on theproperties of mdf. Mokuzai Gakkaishi, 35(1):8, 1989.

[13] R. Ward and C. Skaar. Specific heat and conductivity of parti-cleboard as a function of temperature. For. Prod. J., 13(1):32,1963.

[14] S. Whitaker. Heat and mass transfer in granular porous media.In A.S. Mujumdar, editor, Advances in Drying, volume 1, pages23–61. McGraw Hill, Hemisphere P.C., Washington, 1980.

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