Ind. Eng. Chem. Res. 2003, 42, 5229-5238
5229
Modeling and Simulation of Oriented Strandboard Pressing T. E. Fenton, H. M. Budman,* and M. D. Pritzker Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
E. Bernard and G. Broderick Nexfor Technology at Noranda Technology Centre, Pointe-Claire, Que´ bec, Canada H9R 1G5
A two-dimensional model of heat and mass transport within an oriented strandboard (OSB) mat during the hot pressing process is presented. The thickness of the mat changes with time during pressing, and a novel approach to calculating gas flux accounts for the resulting compression of the OSB mat. The gas within the voids is treated as a binary mixture of air and water. Model parameters are obtained from the literature and from direct measurements. The simulation results are compared to measurements taken within panels of two different target thicknesses from an existing industrial manufacturing operation. Introduction Oriented strandboard (OSB) is a panel product made of wood flakes bound with phenol-formaldehyde resin. It is used in a wide array of applications including commercial and residential construction and renovation, furniture, and shelving. OSB can be custom-manufactured to specific requirements in thickness, density, panel size, surface texture, strength, and rigidity. In the manufacturing process, the OSB panel (called a “mat” during pressing), composed of wood, air, bound water, water vapor, and resin, is pressed between two heated platens. Board thickness decreases with time due to compression. When the press opens at the end of the cycle, hot gases are suddenly released from within the mat. Understanding the behavior of OSB during hot pressing requires the consideration of heat and mass transfer within the mat, the effects of compression on densification, other physical parameters affecting strength, and the resin curing kinetics. In one of an important series of papers, Humphrey and Bolton1 presented a two-dimensional model of heat and moisture transfer in particleboard hot pressing. Heat transfer by conduction and convection of water vapor were both considered, and the simulated results appeared to generally follow the trends of their experimental results. However, Humphrey and Bolton did not consider the presence of air within the voids and assumed a constant mat thickness, thus neglecting the effect of mat compression on internal pressure. Tho¨men2 extended this work in a study of mediumdensity fiberboard pressing. This three-dimensional model included mechanical behavior, accounted for air in the mat, and introduced compression of the mat in time. Although the porosity of the mat was allowed to change with time, resulting in compression of the gases, the gas flux expressions do not appear to include the movement of the platen and wood through space. This problem is examined in more detail in the development of the model in the present work. The approach of * To whom correspondence should be addressed. Tel. 519888-4567 ext. 6980. Fax: 519-746-4979. E-mail: hbudman@ cape.uwaterloo.ca.
Tho¨men2 was to describe heat and mass balances on each control volume with difference equations. Several reasons for choosing this method were provided. Nevertheless, it has the disadvantage of being restricted to a specific solution method, whereas formulation in terms of differential equations permits a variety of solution methods to be used. In a study of OSB pressing, Zombori3 developed a partial differential equation formulation for two-dimensional mass and energy transport, including stress relaxation and resin curing. Thickness changes during the simulation due to compression were considered. However, as in Tho¨men’s work,2 the component fluxes were expressed without accounting for the velocity of the platen. In the present work, a general framework for developing the model transport equations for an OSB mat is outlined. Stress relaxation and rheology are not included. Rather, the focus is on correctly defining the fluxes and providing an approach to account for the moving boundary at the mat-platen interface. Thickness and density are explicitly expressed as functions of time. The model includes an overall energy balance and mass balances for the wood, bound water, and both air and water vapor in the board void space. Mass transport is considered to occur by Darcy flow, Fickian diffusion, and the movement of wood and bound water due to compression of the mat between the platens. The model is general and can describe transport in pressing of other mat-formed products by the substitution of the appropriate product-specific parameters. Model simulations are used to predict the evolution of temperature and partial pressure profiles in OSB panels of two different thicknesses. The model is validated by comparing simulated results to data obtained from an existing commercial OSB manufacturing operation. Model Formulation The press consists of two parallel heated platens open to the atmosphere along the four outer edges. The geometry of the press is shown in Figure 1. Since the panels in the press are twice as long as they are wide,
10.1021/ie020656w CCC: $25.00 © 2003 American Chemical Society Published on Web 09/13/2003
5230 Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003
Figure 1. OSB pressing system diagram.
only the width and thickness dimensions (x- and zdirections) are considered in the model in order to reduce computation time. Nevertheless, incorporating the length dimension in a three-dimensional model is straightforward. As a check, a three-dimensional version of the model was tested. It was found to yield substantially identical results at the center of the board when compared to the results from the two-dimensional version. Since measured values are available only at the center of the board, neglecting this third dimension appears to have little consequence in modeling the pressing operation under consideration while dramatically reducing the required computing time. Both platens are maintained at the same temperature and both ends of the press are open to atmosphere. Consequently, temperature, pressure, and moisture profiles will be symmetric across the midplanes. Therefore, only the shaded area of Figure 1 is simulated. Since one of the boundaries (the platen) is moving toward the midplane, the model has the form of a moving boundary problem. The model equations will first be derived in a stationary coordinate system (Cartesian frame) and then converted to a moving frame to maintain a fixed grid. Assumptions. 1. The presence of resin is neglected because it comprises 3% or less of the dry board weight. 2. Radiative heat transfer is negligible both within the mat and between the mat and the platens.1 3. The density of void-free wood substance is constant.4 4. In reality, the density, permeability, and porosity of an OSB mat are functions of temperature, moisture, and pressure and therefore change with time and position. However, in this work, the mat is assumed to densify uniformly during compression; hence, these properties are assumed to be time-dependent but independent of position. This assumption is weak in view of the existing data in the literature. However, the combination of the motion of the platens and density spatial gradients results in a very complex numerical model for solution. Therefore, these gradients are neglected at this point and are left for future investigation. Despite this, the model equations will be formulated in a general manner so that density, permeability, and porosity spatial gradients can be incorporated in future work. 5. The fiber saturation point of various wood species averages ∼30%.4 However, the predicted moisture content of the mat does not exceed 10% anywhere in the mat at any time during the pressing cycle. Therefore, any “condensed” water is assumed to be immediately adsorbed onto the wood so that no free liquid water exists within the OSB. 6. The model presented in this work is general enough to allow for either uniform or nonuniform initial moisture profiles. In the results presented, the initial moisture content of the OSB mat is not uniform and consists of three layers: a top layer of 5% moisture,
which comprises 25% of the mat thickness and is in contact with the top platen; a bottom layer of 5% moisture, which comprises 25% of the mat thickness and is in contact with the bottom platen; and a core layer of 2% moisture, which comprises the remaining 50% of the mat thickness and is sandwiched between the other two layers. 7. Adsorbed water and gaseous water in the void spaces are assumed to coexist at equilibrium throughout the process under local conditions. 8. Compared to other mechanisms, transport of water by the diffusion of adsorbed water is negligible. 9. Perfect heat transfer between the platen and the OSB mat surface is assumed so that the OSB surface reaches the platen temperature immediately. 10. Along the outer edges of the press, which are open to the atmosphere, the partial pressures of both air and water are assumed to be equal to the ambient conditions. Typical pressing room conditions are assumed to be T ) 25 °C, PT ) 97 000 Pa, and RH ) 0.3%. 11. Air and water comprising the gas phase behave ideally. The presence of other gases in the mat is neglected. 12. Transport of gas occurs by a combination of Darcy flow, Fickian diffusion, and movement of the wood due to compression of the OSB mat. 13. The partial pressures and temperatures are initially uniform within the mat. The total internal pressure is initially equal to the total ambient pressure. 14. Following the arguments presented by Tho¨men,2 the heat loss by conduction from the edge of the mat to the surroundings is assumed to be small and is thus neglected. 15. There is symmetry in the temperature and pressure profiles across the midplane between the open edges (x-direction) and the midplane between the plates (z-direction), as described in Figure 1. Mass Balances. In general, the continuity equation for species i is described by
∂Fi/∂t + (∇·ni) ) ri
(1)
The right-hand side of this equation is zero for air and wood since neither one is being generated or consumed within the mat. 1. Wood and Bound Water. The wood itself moves with respect to fixed coordinates because of the compression of the OSB mat. This movement is considered to occur only in the z-direction since the mat does not appreciably elongate during pressing. Thus, with rwood ) 0, the continuity equation for wood is as follows:
∂Fosb ∂vc ∂Fosb + vc ) -Fosb ∂t ∂z ∂z
(2)
Since the bound water moves with the same velocity as the wood, its mass flux is given as MFosbvc, where MFosb represents the mass concentration of bound water. Substitution of this expression into eq 1 yields
M
(
)
(
)
∂Fosb ∂Fosb ∂M ∂M + vc + vc + Fosb + ∂t ∂z ∂t ∂z FosbM
∂vc ) rbw (3) ∂z
It should be noted that rbw * 0 since water is transferred from the bound state to the vapor state and vice versa.
Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 5231
2. Air. The gas fluxes involve Darcy flow, Fickian diffusion, and the velocity of compression of the wood. Since the Darcy velocity is, by definition, a superficial velocity, both the diffusion and compression components of the gas velocities must also be defined in terms of superficial velocities. If only the Darcy flow and diffusion are considered, the flux vector for air would be as follows:
n)
((
)
kx ∂PT ∂wa - FgasDeff -Fa , µ ∂x ∂x kz ∂PT ∂wa -Fa - FgasDeff µ ∂z ∂z
(
))
(4)
The gas fluxes were defined by both Tho¨men2 and Zombori3 in this way. However, this vector does not fully describe the flux. At the mat-platen interface, the total and partial pressure gradients are zero. This is a necessary condition to ensure that there is no flux through the solid platen. Substituting these boundary conditions into eq 4 implies that the flux of air with respect to stationary coordinates is zero at the platen interface. However, this can only be true if the platen is not moving. If the platen is moving, the gas velocity at the boundary must equal the platen velocity in order to ensure zero flux with respect to the platen at the boundary. The component of the air velocity due to compression depends on the local wood velocity vwood. This approach is analogous to the use of a substantial derivative operator, where the local wood velocity is substituted for the local fluid velocity. The total air flux vector is then defined as follows:
na )
((
)
kx ∂PT ∂wa - FgasDeff -Fa , µ ∂x ∂x kz ∂PT ∂wa - FgasDeff + Favc -Fa µ ∂z ∂z
(
))
(
) ( ) (
) ) (
(5)
∂Fa ∂Fa ∂vc ∂ ∂ + Fa + Fa + vc + vc ) ∂t ∂z ∂t ∂z ∂z kx ∂PT kz ∂PT ∂wa ∂ ∂ ∂ Fa + Fa + FgasDeff + ∂x µ ∂x ∂z µ ∂z ∂x ∂x ∂wa ∂ FgasDeff (6) ∂z ∂z
(
(
)
)
3. Water Vapor. The water vapor flux is governed by the same phenomena as air flux and so is defined as follows:
nv )
((
)
kx ∂PT ∂wv - FgasDeff -Fv , µ ∂x ∂x kz ∂PT ∂wv - FgasDeff + Fvvc -Fv µ ∂z ∂z
(
))
(
) (
(7)
However, unlike air, water vapor is generated within the OSB mat due to the vaporization of adsorbed water.
)
∂Fv ∂Fv ∂ ∂ ∂M + vc + vc + + Fv + Fosb ∂t ∂z ∂t ∂z ∂t kx ∂PT ∂vc ∂ ∂ kz ∂PT (Fv + MFosb) ) Fv + F + ∂z ∂x µ ∂x ∂z v µ ∂z ∂wv ∂wv ∂ ∂ FgasDeff + FgasDeff (8) ∂x ∂x ∂z ∂z
(
(
) ( ) (
)
)
Energy Balance. The energy balance equation combines the effects of conductive and convective heat transfer and is written as follows: n
∂ ∂t
(
∑ i)1
n
Fi H ˆ i) + ∇·(
niH ˆ i) ) ∇·(λ∇T) ∑ i)1
(9)
The flux vectors on the left-hand side are the same as those used in the mass balance equations. After some algebra, including the substitution of eqs 2, 6, and 8, the overall energy equation becomes
(
)
(
∂CP,wood ∂T ∂T + vc + + FosbT ∂t ∂z ∂t ∂CP,wood ∂M ∂M vc + vc + Fosb(H ˆ bw - H ˆ v) + ∂z ∂t ∂z ∂H ˆ bw ∂H ˆ bw ∂H ˆa ∂H ˆa + Fa + FosbM + vc + vc ∂t ∂z ∂t ∂z Fa ∂PT ∂H ∂H ˆv ∂H ˆv ˆa Fv ) ∇·(λ∇T) + kx + vc + ∂t ∂z µ ∂x ∂x ˆa ˆv ˆv Fv ∂PT ∂H ∂PT ∂H ∂PT ∂H + kz + k + kz ∂z ∂z µ x ∂x ∂x ∂z ∂z ˆ a ∂wv ∂H ˆv ˆa ∂wa ∂H ∂wa ∂H Fgas Deff + + + Deff ∂x ∂x ∂x ∂x ∂z ∂z ∂wv ∂H ˆv (10) ∂z ∂z
CP,wood Fosb
(
(
( (
The continuity equation for air results from combining eqs 1 and 5 and noting that ra ) 0:
Clearly, rv ) - rbw, so combining eqs 1, 2, 3, and 7 yields the following overall mass balance for water:
)
)
) (
)
( (
)
) )
(
(
)
))
Mat Structure, Properties, and Equations of State. The mat thickness θ(t) changes as a result of the mechanical properties of the OSB mat and the applied forces during compression. However, since this model does not include the mechanical behavior of the mat, θ(t) is treated as an input. Actual thickness was measured on-line during production and fitted to an explicit function of time. Typically, most of the compression of the mat occurs in the first 10 s of the cycle. A curve demonstrating the actual evolution of mat thickness is presented later in the paper. The value of θ(t) is also used for calculating Fosb and , as follows:
Fosb ) (θf/θ(t))Fosb,f
(11)
) 1 - (Fosb/Fwood)
(12)
where the density of void-free wood material, Fwood, is 1460 kg/m3.4 The final value of the mat density Fosb,f is assumed to be the target mean mat density of 690 kg/ m3. The value of Fosb in this simulation varies from ∼440 kg/m3 at the beginning of the cycle to the final value. The calculated porosity thus varies from 0.697 initially to 0.521 at the end of the cycle.
5232 Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 Table 1. Transport Properties property kz kx Deff Dav λ FT µa µv µ φij
) ) ) ) ) ) ) ) ) )
10-15
e(14.444-0.0192Fosb)
59 kz 0.5 2 Dav 2.20 × 10-5(101325/PT) (T + 273.15/273.15)1.75 ((Fosb/1000)(0.217 + 0.4M) + 0.024 )FT 0.001077T + 0.978 1.6863 × 10-5 + (4.06 × 10-8)T 8.801 × 10-6 + (3.69 × 10-8) T ∑i (yiµi/∑jyjφij) (Mj/Mi)-1/2 ) φji-1
Table 2. Component Enthalpy Relations source
property
5 1 3 3 1, 3 1 6 6 6 6
CP,wood ) 103.1 + 3.867(T + 273.15) H ˆ a ) 1003T H ˆ v ) 2.490 × 106 + 2230T - 3.606T2 ∆H ˆ vap ) 2.487 × 106 - 1824T - 4.554T2 H ˆ bw ) 4186T - 0.4 ∆H ˆ vap(1 - M/Mfsp)2 Mfsp ) 0.325-0.1T
After substituting eq 11 into eq 2, the mass balance for wood can be simplified and expressed as follows:
vc )
z dθ θ(t) dt
(13)
1. Ideal gas. The ideal gas law and Dalton’s law of partial pressures are applied in the usual way.
PT ) Pa + Pv
(14)
Fa ) 0.003488Pa/(T + 273.15)
(15)
Fv ) 0.002165Pv/(T + 273.15)
(16)
Fgas ) Fa + Fv
(17)
wa ) Fa/Fgas
(18)
wv ) Fv/Fgas
(19)
2. Transport Properties. The permeability of the mat to gases depends on the direction of flow. Marceau5 measured the thickness direction permeability of finished OSB product from the same process presently being modeled. The values of other transport properties are adopted from published work. The viscosity of the gas mixture is dependent on composition and temperature. Data for the viscosities of pure air and pure water given by Perry et al.6 have been fitted to explicit functions of temperature. The viscosity of the gas mixture is then calculated using a method outlined by Perry et al. The transport properties used in this work are summarized in Table 1. 3. Enthalpies of Components. For an ideal diatomic gas, Cp ) 7R/2. The reference temperature for H ˆ a is arbitrary since H ˆ a appears only inside partial derivatives. For convenience, the reference temperature is chosen to be 0 °C. Since the enthalpy of an ideal gas is independent of pressure, the enthalpy per unit mass of water vapor is set equal to the saturated vapor enthalpy at the same temperature. The enthalpy data for saturated water vapor and heat of vaporization presented by Himmelblau7 have been fitted to explicit functions of temperature. The reference point for these data is free liquid water at 0 °C and the heat capacity of liquid water is ∼4186 J/kg‚°C. The component enthalpy relations are summarized in Table 2. 4. Moisture Equilibrium in Wood. Adsorbed water is assumed to be in local equilibrium with vapor in the voids. The data relating RH to T and M within particleboard used by Humphrey and Bolton1 are assumed to apply to the OSB in this study. For computational
source 10 7 7 11 4
purposes, these data have been fitted according to the following function:
M)
RH (1.1399+0.01235 T)
e
- RHe(-0.1820+0.01655 T)
(20)
The standard definition of RH is used and the vapor pressure of water is calculated using the Antoine equation. The Antoine constants were calculated from saturated steam pressure data, as follows:
RH ) Pv/P*
(21)
P* ) e(23.232-3847.9/(228.74+T))
(22)
After the substitution of these explicit equations of state, only T, Pa, and Pv remain as dynamic state variables. Then, eqs 6, 8, and 10 are solved to simulate the evolution of temperature and pressure in the OSB mat as functions of position and time. Initial Conditions. The initial conditions are chosen by assuming that the OSB is at mechanical equilibrium with the surroundings prior to contact with the hot platens. The temperature of the mat is initially higher than room temperature due to the drying process the wood flakes undergo before mat formation, typically 40 °C. Therefore:
T0 ) 40 °C
(23)
PT,0 ) 97 000 Pa
(24)
It should be recalled that equilibrium between bound water and vapor within the board is also assumed. Since the initial moisture content is not uniform within the mat, neither is Pv. As defined, this model requires an initial condition for Pv, though in reality it is the moisture content of the wood flakes that is specified., as follows:
M0 )
{
0.05 for z g θ/2 and for all x 0.02 for 0 e z < θ/2 and for all x
By specifying the initial moisture content, M0, the corresponding initial value of the partial pressure of water Pv,0 is calculated by rearranging eqs 20 and 21 as follows:
RH0 )
M0e(1.1399+0.01235T0) M0e(-0.1820+0.01655T0) + 1
Pv,0 ) RH(P*)T)T0
(25) (26)
The initial partial pressure of air is then calculated by subtracting Pv, 0 from the total pressure. Boundary Conditions. The symmetry conditions across both of the midplanes lead to the following
Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 5233
conditions:
bra the air, vapor, and energy balance equations become the following:
(∂Pa/∂x)x)0 ) 0
(27)
(∂Pv/∂x)x)0 ) 0
(28)
(∂T/∂x)x)0 ) 0
(29)
(∂Pa/∂z)z)0 ) 0
(30)
(∂Pv/∂z)z)0 ) 0
(31)
(∂T/∂z)z)0 ) 0
(32)
At the mat-platen interface, perfect heat transfer is assumed. The fluxes of the gas species must be zero with respect to the platen, meaning that Darcy and Fickian fluxes are zero at this boundary. These requirements lead to three more boundary conditions:
Tz)θ ) Tp(t)
(33)
(∂Pa/∂z)z)θ ) 0
(34)
(∂Pv/∂z)z)θ ) 0
(35)
Although the model assumes perfect heat transfer from the platen to the mat, the problem would be discontinuous if the initial value of Tz)θ was set to 215 °C while the rest of the mat was defined to be initially at 40 °C. Therefore, to facilitate the numerical solution, Tp(t) was chosen to be initially 40 °C and quickly ramped up, reaching the platen temperature ∼5 s into the simulation run. The remaining three boundary conditions are written for the open end of the press. The partial pressures of air and water at the boundary are assumed to be equal to the partial pressures in the surroundings. Typical pressing room conditions are assumed to be T ) 25 °C, PT ) 97 000 Pa, and RH ) 0.3. The resulting boundary conditions are
(∂T/∂x)x)R ) 0
(36)
(Pa)x)R ) 96 000
(37)
(Pv)x)R ) 1000
(38)
where Pa and Pv are in pascals and (Pv)x)R is determined using the above values of T and RH in eqs 21 and 22. Normalization of Coordinates Rather than solving these equations in fixed spatial coordinates, they are transformed into a dimensionless coordinate system in which the platen positions are constant, as follows:
χ ) x/R
η ) z/θ
τ)t
(39)
Using these new variables, the derivatives become
∂ 1 ∂ f ∂x R ∂χ
∂ 1 ∂ f ∂z θ ∂η
∂ ∂ η dθ ∂ f (40) ∂t ∂τ τ dτ ∂η
The transport eqs 6, 8, and 10 are then written in terms of these transformed variables. After some alge-
Fa
Fv
(
) )
( (
)
∂Fa Fa dθ kx ∂PT 1 ∂ ∂ ) 2 Fa + + + ∂τ ∂τ θ dτ R ∂χ µ ∂χ kz ∂PT ∂wa 1 ∂ 1 ∂ F + 2 FgasDeff + 2∂η a µ ∂η ∂χ θ R ∂χ ∂wa 1 ∂ F D (41) 2 ∂η gas eff ∂η θ
(
(
)
(
(
)
)
)
∂Fv Fv dθ kx ∂PT ∂ ∂M 1 ∂ + + ) 2 + Fosb Fv + ∂τ ∂τ θ dτ ∂τ R ∂χ µ ∂χ kz ∂PT ∂wv 1 ∂ 1 ∂ F + 2 FgasDeff + 2 ∂η v µ ∂η ∂χ θ R ∂χ ∂wv 1 ∂ F D (42) 2 ∂η gas eff ∂η θ
(
)
(
(
)
)
∂M ∂T ˆ bw - H ˆ v) + Fosb(H + ∂τ ∂τ ∂H ˆ bw ∂H ˆa ∂H ˆv 1 ∂ ∂T + Fv ) 2 + FosbM λ + Fa ∂τ ∂τ ∂τ R ∂χ ∂χ ˆ a kz ∂PT ∂H ˆa Fa kx ∂PT ∂H 1 ∂ ∂T + 2 λ + + 2 ∂η 2 ∂η µ R ∂χ ∂χ θ θ ∂η ∂η Deff ∂wa ∂H ˆ v kz ∂PT ∂H ˆv ˆa Fv kx ∂PT ∂H + 2 + + Fgas 2 2 µ R ∂χ ∂χ θ ∂η ∂η R ∂χ ∂χ Deff ∂wa ∂H ˆv ˆ a ∂wv ∂H ˆv ∂wv ∂H + 2 (43) + ∂χ ∂χ ∂η ∂η θ ∂η ∂η
CP,woodFosb
(
(
)
( ) ( )
(
)
( (
( ) ) ))
Experimental Section Simulated results are compared with direct measurements from an existing manufacturing operation. This batch process consists of 11 vertically stacked press cavities. The master panels are approximately 2.4 m wide and 4.8 m long. The ambient conditions are typically 25 °C and 97 000 Pa total pressure. The platen temperatures are held constant between 200 and 215 °C. To collect temperature and pressure data from the mat, a combined temperature and pressure probe was inserted into an OSB mat before press closure, while the mat was still loosely formed. The probe is, in effect, a hollow thermocouple attached to a pressure transducer and was developed by the Alberta Research Council. The probe diameter is 1.8 mm. It has a single hole at the tip. After insertion, this tip lies at η ) 0 (the midplane of the mat) and ∼50 cm from the edge of the mat. The principal disadvantage to using data from a manufacturing process is the lack of precision in probe placement. With a composite mat formed and pressed in a laboratory, multiple temperature and pressure probes can be placed with precision in each mat formed. This is not possible during regular production. During the automated manufacturing operation, the probe must be inserted into the mat very quickly before the press is closed. At the end of the pressing cycle, the probe must be removed quickly after the press opens but before the finished panel is ejected. Consequently, fine positioning of the probe is difficult. This could account for the moderate degree of repeatability. The final
5234 Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003
Figure 2. Predicted evolution of temperature in 11-mm OSB at various positions η along the longitudinal center at χ ) 0.
Figure 3. Predicted evolution of the partial pressure of water at various positions η along the longitudinal center at χ ) 0 in 11mm OSB.
temperatures from 20 measurements on the same day in the 11-mm OSB spanned ∼10 °C. Results Solution Method. The model equations were solved using a commercial package for partial differential equations, FlexPDE version 2.20b, on a Windows 98based computer. Using a Pentium-III 533-MHz processor, the time required to simulate a pressing cycle was approximately 30-60 min for each run, depending on the mat thickness. The simulation was initially carried out on a square grid, 10 nodes along each of the two directions, with the nodes equally spaced. This node density was insufficient to provide numerically stable solutions near the matplaten interface. Therefore, the number of nodes along the mat-platen interface was increased to 100 while along the midplane between the platens the number of nodes was held at 10. FlexPDE automatically smooths the number of nodes between these regions. For this grid, the resulting computation time increased ∼5-fold versus the original 10 × 10 grid. To check the accuracy of the simulation, a material balance for the total water content (both adsorbed and gaseous) of the mat was carried out over the course of the simulation. This balance was satisfied to within 0.18% of the total mass of water. Typical Simulation Results. 1. Temperature Evolution. The intention in the model was to impose a step function from an initial temperature of 40 °C to the final temperature Tp at the plate. However, to alleviate numerical difficulties, the surface temperature was ramped instead according to a steep function that reached the final plate temperature within 2 s. The temperature histories of six points along the longitudinal center (i.e., χ ) 0) of the mat are shown in Figure 2. At η ) 0.6, a rapid temperature increase occurs between approximately t ) 5 s and t ) 20 s. This is followed by an extended period where the temperature increases almost linearly with time. Furthermore, the temperature histories at η ) 0, η ) 0.2, and η ) 0.4 converge at approximately t ) 70 s. This suggests that conduction plays a limited role in heat transfer in the central part of the mat in the latter stages of pressing. Conductive heat flux is proportional to the temperature gradient, which all but disappears during that period. 2. Behavior of Gases. The corresponding evolutions of the partial pressure of water vapor and air are shown in Figures 3 and 4, respectively.
Figure 4. Predicted evolution of the partial pressure of air at various positions η along the longitudinal center at χ ) 0 in 11mm OSB.
Figure 3 shows a rapid increase in the partial pressure of water at the surface of the mat (η ) 1) within the first 6 s. This occurs mainly because the rapid temperature rise at the surface of the mat causes a high rate of desorption and evaporation of bound water. Thermal expansion of the warming gas and the compression of the void spaces due to pressing contribute relatively little to this 100-fold rise in Pv. In layers closer to the midplane of the mat, the rise in Pv is delayed because the temperature does not rise until later in the pressing process. Until the temperature begins to rise at a given point, the partial pressure of water does not rise noticeably, even as water vapor is transported into these areas of the OSB mat. Instead, vapor that enters these cooler regions condenses and adsorbs onto the wood. Therefore, moisture content may temporarily rise, but Pv remains fairly constant until the local temperature rises. Consequently, evaporation and diffusion constantly move water vapor toward the midplane of the mat. Air behaves differently from water vapor in the OSB mat. The most important distinction is that air is neither consumed nor generated by sorption or any other reaction during pressing. The simulated evolution of Pa is shown in Figure 4 at various positions along the longitudinal midplane. Within the first 4.5 s of the simulation, Pa at the surface of the mat drops to half of its initial value. During the same period, Pa at η ) 0.8 increases by 55%. This sets up a diffusive force toward the platen, although on the whole air is clearly being moved toward the center. Since air is not being consumed, the bulk flow of gas
Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 5235
Figure 5. Predicted evolution of moisture content at various positions η along the longitudinal center at χ ) 0 in 11-mm OSB.
away from the platen is carrying air toward the midplane faster than diffusion can act in the opposing direction to equalize the concentration of air. In effect, as water vapor is generated by evaporation and drives toward the midplane, it “carries” air along with it. After this initial drop, Pa at the surface of the mat recovers as diffusion takes over in response to the steep air concentration gradient that has developed. Closer to the midplane of the mat at η ) 0.8, a similar phenomenon is observed later in the process. As air moves from the surface toward the midplane in the first few seconds, Pa increases at points located away from the platen. Once the temperature starts to rise after approximately t ) 4 s, water vapor is generated quickly and the bulk flow of the gas mixture moves both components toward the midplane. The result is a sharp rise in Pv accompanied by a drop in Pa. 3. Moisture Content. The significant movement of moisture within the mat is evident from Figure 5. As the temperature at the mat-platen interface rises during the first 6 s, bound water evaporates and the moisture content M at η ) 1 drops from the initial value of 5 to ∼0.13%. The vapor generated migrates toward the midplane and laterally toward the edge of the mat, condensing as it arrives in cooler parts of the mat. As the temperature begins to rise in layers closer to the midplane, moisture evaporates and moves deeper into the center of the mat. The result is that a “wave” of bound water progresses from the mat-platen interface to the core of the mat during the pressing cycle. The average moisture content of the mat during the simulation was tracked by integrating M across the entire mat. This value decreased steadily, from 3.5% initially to ∼3.14% at the end of the pressing cycle. Comparison of Predicted and Measured Results. 1. 11-mm OSB Mat. The simulated evolution of temperature at the center point of the 11-mm OSB mat follows the measured temperature closely, as shown in Figure 6. The maximum deviation between measured and simulated values was ∼7 °C. The maximum difference between the predicted and measured values of PT at the center of the 11-mm OSB is ∼75 kPa and occurs at t ) 40 s. However, the general trend of the simulated pressure appears to be correct, as shown in Figure 7. Both the simulated and measured data show an immediate and rapid increase from ambient pressure during the first 15-20 s, followed by a more moderate rise during the remainder of the pressing cycle. 2. 29-mm OSB Mat. The model was also used to compare predicted and measured values in the pressing
Figure 6. Comparison of measured and simulated temperatures at the center of 11-mm OSB.
Figure 7. Comparison of measured and simulated pressures at the center of 11-mm OSB.
Figure 8. Comparison of measured and simulated temperatures at the center of 29-mm OSB.
of a 29-mm mat. The same parameter values are used for this simulation as are used in the 11-mm OSB simulation. Unfortunately, the predicted results for the 29-mm mat are less accurate. Shown in Figure 8, the measured temperature at the center point of the mat rises sharply, starting at t ) 75 s. At t ) 250 s, the rate of increase in the measured temperature drops so that from this point onward the temperature increase is small. The simulated temperature, however, behaves differently. Though the temperature rise begins at approximately the same time as in the measured data, the simulated rate of increase is relatively gradual. By the end of the pressing cycle, the two values are approaching each other and may, given more time, converge. However, between t ) 150 s and t ) 250 s, predicted and actual temperatures are at least 35 °C apart.
5236 Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003
Figure 9. Comparison of measured and simulated pressures at the center of 29-mm OSB.
The simulated values of PT are also inaccurate, as seen in Figure 9. In the first 50 s, the simulated pressure rises to ∼180 kPa while the measured value remains well below this. Between t ) 50 s and t ) 170 s, both measured and predicted pressures remain essentially constant, maintaining a prediction error of ∼70 kPa. At that time, the measured pressure begins to climb gradually and eventually approaches, but does not exactly match, the predicted pressure at the end of the cycle. Errors in Predicted Results. While the simulated results for the 29-mm mat are not as accurate as those for the thinner mat, they do not provide enough evidence on their own to invalidate the proposed general transport model. As noted earlier in this work, the precision of the probe placement could not be verified. Furthermore, the values used for the parameters in the model equations require closer scrutiny. Fenton8 examined the effects of changing the values of thermal conductivity, effective diffusivity, permeability, initial mat temperature, and initial moisture content on the simulation results. Uncertainty in the effective diffusivity and permeability were identified as the most likely causes of disagreement between the measured and predicted variables. However, it was clear that neither one of these factors alone could explain the error. Both Deff and kz increase as mat density decreases. However, in solving the model equations, the spatial dependence of density was neglected. This is likely an important error since the local density of OSB near the surface of the finished panel can be more than 30% higher than at the midplane.9 Therefore, the values of Deff and kz in the inner layers of the mat are underestimated, which may explain some of the discrepancy between measured and simulated results. To test this hypothesis, the values of Deff and kz were increased to fit the predicted evolutions of pressure and temperature in the 29-mm OSB mat as closely as possible. The other parameter values used for this modified simulation were the same as in those already presented. The following values of Deff and kz were obtained:
kz,fit ) 3 × 10-15e(14.444-0.0192Fosb)
(44)
Deff,fit ) Dav
(45)
Figures 10 and 11 compare the results from the original simulation already presented to the fitted simulation and measured values. Increases in Deff and
Figure 10. Predicted temperature evolution at the center point of 29-mm OSB using fitted values of Deff and kz.
Figure 11. Predicted total pressure evolution at the center point of 29-mm OSB using fitted values of Deff and kz.
kz change the shapes of both curves dramatically and obviously reduce the magnitude of the error. This suggests that implementing a more complete model that considers a vertical density profile may address some of the accuracy problems encountered in the comparisons of measured and simulated results for the thicker OSB mat. Effect of Wood Velocity Term. In developing the mass and energy balance equations, the component fluxes included a wood velocity term due to compression of the mat between the platens. No other research group has accounted for this flux component. However, these terms are necessary in order to conserve mass in the system. This is best demonstrated using a modified version of the simulation already presented. If the edges of the press were sealed, the total masses of all components would remain constant throughout the pressing cycle. This condition was introduced to the model by modifying the boundary conditions in eqs 37 and 38, making the partial pressure gradients along the press edge equal to zero. Two versions of this modified simulation were executed: one using the transport equations as they are presented in this work and the other using transport equations derived without considering the velocity of the platen in the balance equations. In Figure 12, the total masses of air are shown during these two simulations and are compared to the changing thickness of the mat. It is clear that mass is only conserved when the platen movement is included in the component fluxes. Furthermore, the amount of air not accounted for when the wood velocity is neglected is proportional to the change in the thickness of the mat. This is clear evidence that the failure
Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 5237
rameter values were used for all versions of the model shown in Figures 13 and 14 so that the numerical impact of the wood velocity term can be directly seen. Conclusions
Figure 12. Effect of the compression terms on conservation of mass of air in a simulated sealed 11-mm OSB mat.
Figure 13. Effect of the compression terms on predicted total pressure evolution in 11-mm OSB pressing.
A two-dimensional model for simulating energy and mass transfer within an OSB mat during the pressing process has been presented. The transport model has been derived in a system of stationary spatial coordinates and then solved in a moving system of coordinates in which the platen position is constant. The model predicts the evolution of temperature, partial pressures, and moisture content while the OSB mat is compressed between two heated platens. Air and water vapor movement occur due to several phenomena: the movement of the platens, Darcy flow, and Fickian diffusion. In contrast, other published pressing models have not accounted for the fact that the movement of the platens contributes to the movement of the gases. The effect of this phenomenon has been investigated by comparing results obtained when compression is accounted for versus those obtained for the case where it is neglected. Differences in the predicted values of PT of up to 60 kPa were observed depending on whether this contribution to gas flux is included. The results of simulations for two target panel thicknesses have been presented and compared against measurements taken along the midplanes of OSB mats in an existing batch manufacturing process. Simulated temperature and total pressure results for a 11-mm OSB mat follow the measured values closely in general trend and in magnitude. Simulated results for a 29-mm OSB mat do not agree as well with measured results. It may be possible to account for the deviation between observed and predicted results for the 29-mm mat by incorporating spatial dependence of the mat density and related properties, including permeability and diffusivity. Nomenclature
Figure 14. Effect of the compression terms on predicted total pressure evolution in 29-mm OSB pressing.
to account for wood velocity in the component fluxes implies the convection of those components outside of the system through the platen. Similar results were found for bound and gaseous water. The impact of this error on the simulation results is important. Without these terms the predicted PT at the midplane of the 11-mm OSB mat would be between 40 and 60 kPa lower than when this term is included, as shown in Figure 13. Similarly, the total pressure is underestimated in 29-mm OSB when wood velocity is not considered, as demonstrated in Figure 14. The model that includes the convection terms shows a slightly more rapid increase in the simulated midplane temperatures in both mats, with as much as a 3 °C difference between the model versions. The same pa-
CP,wood ) heat capacity of oven-dry wood substance (J/kg‚ °C) Dav ) binary diffusivity for air and water vapor (m2/s) Deff ) effective diffusivity for air and water vapor within the mat (m2/s) FT ) temperature correction factor for thermal conductivity G ) specific gravity of oven-dry OSB Hi ) enthalpy of component i per unit mass (J/kg) ∆H ˆ vap ) latent heat of vaporization of water (J/kg) kx ) in-plane permeability (m2) kz ) cross-sectional permeability (m2) M ) bound water fraction (kg of bound water/kg of ovendry wood) Mi ) molecular mass of species i (kg/mol) ni ) flux of component i (kg/m2‚s) Pi ) partial pressure of species i (Pa) PT ) total pressure (Pa) P* ) vapor pressure of pure water (Pa) ri ) rate of generation of component i (kg/m3‚s) R ) universal gas constant (J/mol‚K) RH ) relative humidity T ) temperature (°C) t ) time (s) vc ) local velocity vector component due to compression (m/s) vi ) local velocity of species i (m/s) wi ) gas phase mass fraction of species i
5238 Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 x ) fixed coordinate parallel to the platen along the mat width (m) yi ) mole fraction of species i in the gas phase z ) fixed coordinate perpendicular to the platen (m) Greek Letters R ) OSB mat width (m) ) void fraction in the OSB mat η ) moving coordinate perpendicular to the platen θ ) OSB mat thickness (m) λ ) thermal conductivity of the OSB mat (W/m‚°C) µi ) dynamic viscosity of pure component i (Pa‚s) µ ) dynamic viscosity of the gas mixture (Pa‚s) Fbw ) mass concentration of bound water in the mat (kg/ m3) Fi ) mass concentration of species i in the void space (kg/ m3) Fgas ) total mass concentration of gas in the void space (kg/ m3) Fosb ) mass concentration of wood in the OSB mat (kg/m3) Fwood ) density of void-free wood substance (kg/m3) τ ) time coordinate in moving system of coordinates (s) φi, j, φj, i ) mole fraction averaging factor for gas viscosity χ ) moving coordinate parallel to the platen along the width of the mat Subscripts a ) air bw ) bound water f ) final (just prior to the opening of the press) fsp ) at the fiber saturation point l ) free liquid water p ) platen v ) water vapor 0 ) initial (at t ) 0) ∞ ) surroundings
Operators ∇ ) “del” operator ) ∂/∂x i + ∂/∂z k
Literature Cited (1) Humphrey, P. E.; Bolton, A. J. Holzforschung 1989, 43 (3), 199-206. (2) Tho¨men, H. Ph.D. Thesis, Oregon State University, 2000. (3) Zombori, B. G. Ph.D. thesis, Virginia Polytechnic Institute and State University, 2001. (4) Siau, J. F. Transport Processes in Wood; Springer Series in Wood Science; Springer-Verlag: Berlin, 1984; pp 25-29. (5) Marceau, P. Rapport pre´liminaire: perme´abilite´ au gaz des panneaux. Unpublished report, 2001. (6) Perry’s Chemical Engineers’ Handbook, 7th ed.; Perry, R. H., Green, D. W., Maloney, J. O., Eds.; McGraw-Hill: New York, 1984; pp 2-320-2-322. (7) Himmelblau, D. M. In Basic Principles and Calculations in Chemical Engineering, 5th ed.; Amundson, N. R., Ed.; Prentice Hall International Series in the Physical and Chemical Sciences; Prentice Hall: Englewood Cliffs, NJ, 1989; pp 661-666. (8) Fenton, T. E. Master’s Thesis, University of Waterloo, Waterloo, ON, Canada, 2002. (9) Wang, S.; Winistorfer, P. M. For. Prod. J. 2000, 50 (3), 3744. (10) Simpson, W.; TenWolde, A. Physical Properties and Moisture Relations of Wood. In Wood Handbook: Wood as an Engineering Material; USDA Forest Service, Forest Products Laboratory, Madison, WI, 1999; Chapter 3. (11) Stanish, M. A.; Schajer, G. S.; Kayihan, F. AIChE J. 1986, 32 (8), 1301-1311.
Received for review August 23, 2002 Revised manuscript received March 17, 2003 Accepted August 6, 2003 IE020656W
