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Ind. Eng. Chem. Res. 2006, 45, 1974-1988
Simulation and Optimization of the Continuous Oriented Strand Board Pressing Process Clement W. B. Lee, Hector M. Budman, and Mark D. Pritzker* Department of Chemical Engineering, UniVersity of Waterloo, 200 UniVersity AVenue West, Waterloo, Ontario, Canada N2L 3G1
This study has the following two objectives: (1) development of a 3-dimensional steady-state model for heat and mass transfer during continuous pressing of oriented strand board (OSB) and comparison with data from an industrial operation and (2) optimization of operating conditions for continuous pressing. An extensive sensitivity analysis indicates that the dependency of the permeability on the board density is crucial for better prediction of the temperature and pressure values measured in the mat during pressing and may explain discrepancies between data and numerical predictions reported in previous studies. Once the model is calibrated to experimental data, a genetic algorithm is employed to optimize the overall hot-pressing of OSB by determining conditions that maximize the operating profit, while ensuring that delamination is prevented and sufficient resin curing is achieved. This algorithm is shown to determine operating conditions that result in a higher profit than that obtained by the current practice of the industrial operation. 1. Introduction Oriented strand board (OSB) is a composite panel made of wood flakes bound by phenol-formaldehyde resin. Plywood has been replaced by OSB for many applications in the past 20 years because of its high cost of manufacturing. Advancements in manufacturing techniques have further reduced the cost of manufacturing OSB. Since the most expensive part of the process is the hot-pressing stage, its optimization will likely provide the greatest advantage in cost savings. In the past few years, some commercial mills have been able to lower costs by 16%.1 A motive for the current study is to search for greater cost savings by closely examining the continuous hot-pressing cycle and by optimizing the operating conditions. To understand the hot-pressing cycle, energy and mass transfer have to be evaluated through the development of an adequate model. Previous models have been formulated but have not predicted temperature and pressure profiles that are entirely accurate for the full pressing cycle.2-7 An objective of the present work is to develop a model to describe the fast compression that the boards undergo during continuous pressing as compared to the slower compression during batch pressing. The major constituents of the board include resin, water, air, and wood. The properties of each constituent change as a function of the three state variables considered in this study, i.e., partial pressure of air, partial pressure of water vapor, and temperature. These material property functions have been selected from a number of sources in the literature. As will be shown, an extensive parameter-sensitivity analysis indicates that the panel temperature and pressure during pressing are highly sensitive to changes in its permeability. However, the dependence of permeability on density as described in the literature is typically based on the properties of finished boards rather than on boards undergoing consolidation compression. In this work, a different approach for determining the dependence of permeability on density suitable for the conditions that the board undergoes during continuous pressing is discussed in detail. Once the pressing cycle is modeled, an optimization scheme is employed. Optimization of the pressing process has not been * To whom correspondence should be sent. Tel.: (519) 888-4567, ext. 2542. Fax: (519) 746-4979. E-mail:
[email protected].
Figure 1. Cross-sectional view of continuous OSB press showing typical dimensions. Direction of motion of the panel through the unit is normal to the plane of the page.
previously reported in the literature. In the present work, a genetic algorithm is employed for optimization. The genetic algorithm has a clear advantage over other methods such as gradient techniques since it can progress through local cost function minima and maxima in search of a global optimum. Local maxima and minima are likely to exist in this system because of its discontinuous nature. 2. Model Formulation The model in the present work is based on that previously presented by Fenton et al.,7 but with some modifications. First, the earlier batch model is adapted for the case of continuous pressing operation. Also, more attention is focused on the dependence of the permeability on the board void fraction, and more realistic boundary conditions at the mat-platen interface are considered compared to those used in the previous work. Finally, effects of resin curing are included in the model. As will be shown, resin curing may have a significant effect on the system in the later stages of pressing. 2.1. Geometry of System. The coordinates describing the geometry of the system are shown in Figure 1. The distances along the mat width and height are denoted as x and z, respectively. The position along the moving direction of the board through the machine (i.e., normal to the plane of the page) is defined as y. Assuming that both the geometry and energy/ mass transfer at a given position do not vary with time, continuous pressing is modeled in three dimensions at steady state with respect to a stationary frame of coordinates (i.e., Eulerian formulation). The system contains a moving boundary
10.1021/ie058051y CCC: $33.50 © 2006 American Chemical Society Published on Web 02/10/2006
Ind. Eng. Chem. Res., Vol. 45, No. 6, 2006 1975
corresponding to the height of the mat as it moves through the press and is compressed. For this analysis, we have transformed x and z to dimensionless forms χ and η,
z x χ) , η) R θ(y)
(1)
where R is the half-width of the panel and θ(y) is the mat thickness at position y. The second transformation converts the moving boundary to a stationary one at η ) 1 but leads to some modification of the continuity and transport equations to be presented. More details concerning these modifications are given in our previous study.7 The height of the panel as it is conveyed through the machine is controlled by the spacing betwen the heated belts. For the industrial plant of interest, the variation of the height with distance y is described by the following empirical function:
θ ) 0.003 318 exp(-0.285y/Vls) + 0.004 932
(2)
Since the system is assumed to be at steady state with respect to a stationary frame, the time elapsed for a particle on the board to travel a distance y in the machine direction is given by the ratio of y divided by the linear velocity Vls of the board. 2.2. Governing Equations. Strictly speaking, equations for the rheological behavior of the mat should be solved simultaneously with the energy and mass transport equations to account for the development of a vertical density profile (VDP) during pressing and the consequent variation of material properties in the η direction. To assess the effect of such a variation on the evolution of the core temperature and pressure during pressing, we compared the simulations obtained when a VDP typical of finished panels is assumed to apply throughout a pressing cycle to the situation where a uniform profile with the same average density is used. These simulations showed almost identical temperature and pressure profiles in the two cases, particularly near the discharge end of the machine. Some difference was observed near the entry, but this quickly disappeared further along the machine direction. In view of this result and the extra computational difficulty of incorporating rheological equations into the model, we assumed the density varies in the y direction but is uniform with respect to χ and η. It should be noted that material properties such as diffusivity and thermal conductivity still vary in the χ and η directions to some extent because of their dependence on the local temperature and moisture content. The model is described by four transport equations over the entire geometrical domain of the system: three for the material balances of wood, air, and water and a fourth equation for an overall energy balance. In the coordinate system transformed according to eq 1, the wood continuity equation becomes
Vls
due to the vaporization of bound water in the wood into the voids between the wood flakes. Then, an overall water balance can be obtained by combining the balance for bound water and water vapor as follows:
∂Fosb Fosb ∂Vc )∂y θ ∂η
(3)
FVVls
∂FV FV ∂Vc ∂ ∂M + Vls + + VlsFosb ) ∂y ∂y θ ∂η ∂y κχ ∂P ∂wV 1 ∂ 1 ∂ FV + 2 FgasDeff + 2 ∂χ µ ∂χ ∂χ ∂χ R R κη ∂P ∂wV 1 ∂ 1 ∂ FV + 2 + FgasDeff (5) 2 ∂η µ ∂η ∂η ∂η θ θ
(
(
)
)
(
(
)
)
The final transport equation is the energy balance as shown below:
(
) ( )
( )
∂H ˆ wood ∂M + FwoodVls(H ˆ bw - H ˆ v) + ∂y ∂y ∂H ˆ bw ∂H ˆ air ∂H ˆv FwoodVlsM + FairVls + FvVls ) ∂y ∂y ∂y κx ∂P ∂wair 1 ∂ 1 ∂ H ˆ air + 2 H ˆ F FgasDeff + 2 ∂χ air µ ∂χ ∂χ air R R ∂χ ∂wv 1 ∂ 1 ∂ κx ∂P F FgasDeff H ˆv + 2 H ˆ + 2 ∂χ v µ ∂χ ∂χ v R R ∂χ κz ∂P ∂wair 1 ∂ 1 ∂ F + 2 FgasDeff + H ˆ H ˆ 2 ∂η air µ ∂η air ∂η air θ θ ∂η κz ∂P ∂wv 1 ∂ 1 ∂ H ˆ + 2 H ˆ + F FgasDeff 2 ∂η v µ ∂η v ∂η v θ θ ∂η
FwoodVls
(
(
(
(
) ) ) )
( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
1 ∂ ∂T 1 ∂ ∂T λ + 2 λ + R2 ∂χ ∂χ θ ∂η ∂η ∂X η dθ ∂X (6) ∆H ˆ rxnwresinFwood Vls - Vls ∂y θ dy ∂η The last term on the right side of eq 6 includes the contribution from the enthalpy of resin curing on the energy balance. As indicated from the left sides of eqs 3-6, bulk flow in the direction of travel of the mat through the machine is included, with the air, vapor, and energy being convected with the solid phase at the line speed velocity Vls. The equations do not contain terms for Darcy, diffusive, and conductive flows in this direction since the pressure, concentration, and temperature gradients in the y-direction are never large enough (even at the entry end of the machine) for these modes of transport to be significant compared to convective flow from the movement of the mat. These transport equations are solved along with the rate equation for resin curing, which in the transformed coordinate system can be written as
Vls
∂X ) krxn(1 - X)a ∂y
(7)
The continuity of air is represented by the following equation:
∂Fair Fair ∂Vc ∂ + ) FairVls + Vls ∂y ∂y θ ∂η κχ ∂P ∂wair 1 ∂ 1 ∂ Fair + 2 FgasDeff + 2 ∂χ µ ∂χ ∂χ ∂χ R R κη ∂P ∂wair 1 ∂ 1 ∂ Fair + 2 FgasDeff (4) 2 ∂η µ ∂η ∂η ∂η θ θ
(
(
)
)
(
(
)
)
The water vapor continuity equation is very similar to the air continuity equation except for the addition of a generation term
where krxn and a are the rate constant and the reaction order, respectively. 2.3. Material Properties. The properties in eqs 3-7, excluding permeability, are defined by functions of the state variables in Table 1. The empirical wood moisture-humidity relationship in Table 1 was obtained in an earlier study7 by fitting to previously used data2. These data were used since they more completely cover the temperature range encountered during pressing than do some more recent experiments and correlations.16 An important aspect of this study is to examine the effect of the board permeability-density relationship on the
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Table 1. Property Functions (Excluding Permeability) Used in This Study property density of gas component i
mass fraction of gas component i
equation
source (ref)
Pi(MWi)
Fi ) wi )
R(T + 273.15) Fi n
∑F
i
i)1
mat density and porosity
z F θ osb,f
mass concentration of wood in board
Fosb )
void fraction of mat
)1-
Fosb Fcell-walls
8
moisture-humidity relationships
relative humidity
RH )
Pv P*
9
1704.5 T + 231.73
log10P* ) 10.141 -
RH ea+bT - RH eg+hT a ) 1.1399 b ) 0.01235 g ) -0.1820 h ) 0.01655 M)
moisture
2,7
mass transport properties
273.15 (101P325)(T +273.15 )
1.75
binary diffusivity of air-water gas mixture
DAB ) (2.2 × 10-5)
effective diffusivity of air-H2O gas mixture
Deff ) 0.52DAB
dynamic viscosity of pure gas i
T + 273.15 n µi ) µ0,i 273.15 air: µ0,air ) 1.730 7 × 10-5, n ) 0.65 H2O: µ0,v ) 8.869 × 10-6, n ) 1.1
6
(
2
µ)
∑ i)1
dynamic viscosity of air-H2O gas mixture through porous media
)
( )
9, 11
yiµi
2
∑y φ
j ij
j)1
φij )
10
(
) [ () ( ) ]
MWi 1 1+ MWj x8
-0.5
1+
µi µj
-0.5
MWj MWi
12
0.25 2
energy transport properties
[
Fosb (0.217 + 0.4M) + 0.024 1000
]
thermal conductivity of OSB mat
λ ) FT
temperature correction factor specific enthalpy of wood specific enthalpy of bulk liquid water specific enthalpy of air
specific enthalpy of water vapor
FT ) 0.001 077T + 0.978 H ˆ wood ) 103.1(T + 273.15) + 1.933 5(T + 273.15)2 H ˆ w ) 4 185T H ˆ air ) 968.6(T + 273.15) + (3.388 × 10-2)(T + 273.15)2 + (5.517 × 10-5)(T + 273.15)3 - (1.694 × 10-8)(T + 273.15)4 ∆H ˆ vap ) (2.469 × 106) + (1.565 × 103)(T + 273.15) 5.690(T + 273.15)2 from steam tables
specific enthalpy of bound water
H ˆ bw ) H ˆ w - 0.4∆H ˆ vap 1 -
fiber saturation point
Mfsp ) 0.325 - 0.001T
heat of vaporization of water
(
M Mfsp
)
2
8 2 13 14 14 14 14 10 8
resin curing kinetics
(
Eact
rate constant
krxn ) 0.25 exp -
reaction order
Eact ) 12 423 J mol-1 a ) 0.587
R(T + 273.15)
)
15 15
Ind. Eng. Chem. Res., Vol. 45, No. 6, 2006 1977
Figure 2. Typical belt temperature along the direction y of the continuous press.
computed temperature and pressure profiles. Comparison of the use of various functions reported in the literature to an approach adopted in this study is presented in Section 3. 2.4. Boundary Conditions. In a previous model,7 the platen boundary was assumed to be impermeable to gases. However, in actual practice, screens are placed between the OSB mat and the platen to allow for gas release and prevent pressure buildup that could result in delamination of the board as it is discharged from the press. Zombori et al.6 formulated a mixed boundary condition for the platen surface to accommodate gas release from the mat-platen boundary. A similar boundary condition is also used in the present work, i.e.,
[
]
κη,surf 1 (P - P∞) + FgasDeff,surf(wair - wair,∞) Fair θ µ (8) 1 κη,surf (P - P∞) + FgasDeff,surf(wv - wv,∞) (9) nv|η)1 ) Fv θ µ
nair|η)1 )
[
]
where nair|η)1 and nv|η)1 represent the mass fluxes of air and water vapor, respectively, leaving the mat-platen interface. At the outer edges of the mat (i.e., χ ) -R, +R), the partial pressures of air and water vapor are assumed to be equal to that in the ambient gas phase surrounding the pressing unit. The remaining mass-transfer boundaries are defined by symmetry with respect to the χ-y plane midway between the platen surfaces and the η-y plane midway between the edges of the mat. The boundary conditions for energy transfer are quite simple as well. The mat-platen surface is considered to be a perfect heat-transfer boundary, with temperature varying along the length of the machine in a preset manner. The edges of the mat are assumed to be perfectly insulating, since any heat losses from the edges of the mat are negligible because of their small surface area. The remaining boundaries for energy transfer are once again constrained by symmetry conditions.
3. Results and Discussion The governing equations given in Section 2 were solved using the software package FlexPDE (Version 3.10) on a Pentium IV 2.4 GHz processor. This package uses a modified NewtonRaphson method to solve the set of equations according to a finite element grid. The grid in the η-χ plane contained 525 nodes as the best compromise between computational accuracy and time. The mesh network in the η-χ plane was automatically generated by the software. A relative error of 10-3 for each state variable was chosen as the convergence criterion for the model simulations, again as a compromise between computational accuracy and time. Also, the use of a smaller relative error caused an oscillation in the spatial dependence of some of the computed values. The process optimization was carried out by combining the MatLab genetic algorithm routine with the FlexPDE code for solving the continuous pressing model. This was accomplished by a user-written MatLab code serving as the interface between FlexPDE and the genetic algorithm routine to exchange files and data from one to the other. 3.1. Comparison of the Model Predictions to Industrial Plant Data. The initial objective is to calibrate the model for continuous OSB pressing to experimental data obtained from an industrial operation. The length of the press in this operation is 50 m, while the mat width (2R) is 3.08 m. The gap between the heated platens along the y direction varies according to the expression given in eq 2. The platens consist of eight heated zones along the machine direction with lengths ranging from ∼5 to 6.6 m. The temperature profile of the platens along the machine length is shown in Figure 2. The ambient conditions surrounding the unit are 40 °C, 93 840 Pa, and 11% relative humidity. The press produces finished OSB panels with a thickness of ∼11 mm at a line speed of 0.5 m s-1. Industrial data were collected using a temperature and pressure probe inserted into the center of the board (i.e., η ) 0 and χ ) 0) as it moved along the y direction during pressing.
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Figure 3. Measured pressure profiles at the board center (η ) 0, χ ) 0) along the y-direction of an industrial continuous hot press to produce 11 mm OSB boards at a line speed of 0.5 m s-1.
Figure 4. Measured temperature profiles at the board center (η ) 0, χ ) 0) along the y-direction of an industrial continuous hot press to produce 11 mm OSB boards at a line speed of 0.5 m s-1.
Data were recorded onto a memory chip in the probe to simultaneously monitor the evolution of mat thickness, belt temperature, core temperature, and core pressure in the y direction. Typical pressure and temperature profiles yielding a final board thickness of 11 mm at a line speed of 0.5 m s-1 through the pressing machine are shown in Figures 3 and 4, respectively. The curves from five different runs are shown in each of the plots to illustrate the good repeatability of the plant data. The average temperature and pressure profiles over these five runs were used for comparison with the model.
Initially, simulations were conducted on the basis of the same physical parameters used by Fenton et al.7 The relation for the dependence of board permeability on porosity was taken from the work of Marceau.17 The permeability κχ in the χ direction along the board width is much larger than κη in the η direction. As in previous studies,2,7 the following relationship is used:
κχ ) 59κη
(10)
Initial comparisons between model predictions and the experi-
Ind. Eng. Chem. Res., Vol. 45, No. 6, 2006 1979
Figure 5. Comparison of measured and predicted pressure profiles at the board center (η ) 0, χ ) 0) along the y-direction of an industrial continuous hot press using various permeability-porosity relationships reported in the literature. Final board thickness is 11 mm and line speed is 0.5 m s-1.
Figure 6. Comparison of measured and predicted temperature profiles at the board center (η ) 0, χ ) 0) along the y-direction of an industrial continuous hot press using various permeability-porosity relationships reported in the literature. Final board thickness is 11 mm and line speed is 0.5 m s-1.
mental data without any fitting are presented in Figures 5 and 6. Different model predictions are shown depending on whether the permeability-porosity relationship reported by Zombori et al.,6 Marceau,17 or von Haas et al.18 is used. Regardless of the relationship used, significant discrepancies between the predicted and measured profiles are apparent. First, the model-predicted pressure curve shows very little buildup in the y-direction compared to what is observed. Also, the pressure is significantly underpredicted toward the end of the pressure cycle by ∼120 kPa. Finally, the computed temperature profile shows a maximum underprediction error of 20 °C and has different curvature than the measured profile. The objective was then to improve the fitting by adjusting the minimum number of parameters. For this purpose, we conducted an extensive sensitivity analysis on the physical parameters for which an exact value for the specific type of wood used is not available in the literature. Changes with respect
to the original nominal values of the diffusivity and conductivity specified in Table 1 and the permeability were considered in turn. Figures 7 and 8 show the effect of changes in thermal conductivity of (20% on the center pressure and temperature evolution as a function of position along the length of the press. It is clear that changes in the conductivity have a negligible effect on the predicted core pressure values. In addition, the core temperature as the board is discharged would be predicted properly by increasing the base-case conductivity by ∼20%, but the profile over the entire length of the press is not predicted well. Although not included here, (20% changes in the diffusivity have little effect and cannot explain the differences between the numerical predictions and the experimental data. The property that may be most responsible for the discrepancy between the measured and predicted data is the board permeability. Several factors such as board density, porosity, and flake alignment have been reported to significantly affect the perme-
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Figure 7. Effect of conductivity on computed pressure profiles at the board center (η ) 0, χ ) 0) along the y-direction of an industrial continuous hot press. The number corresponding to each predicted curve represents the factor by which the base conductivity defined in Table 1 has been multiplied.
Figure 8. Effect of conductivity on computed temperature profiles at the board center (η ) 0, χ ) 0) along the y-direction of an industrial continuous hot press. The number corresponding to each predicted curve represents the factor by which the base conductivity defined in Table 1 has been multiplied.
ability of OSB boards. Although the latter has been shown to play an important role,19 flake deposition was not explicitly addressed in this work and so flake-alignment effects are not considered. On the other hand, the effects of porosity and density on permeability were explicitly addressed. Much work has been done in relating permeability to board density or porosity.18-26 However, this approach has generally involved measuring the permeability of a finished board at room temperature once it has been removed from a press. Previous models on transport processes occurring during pressing have been based on the permeability values for finished boards.5-7 Large discrepancies between experimental data for temperature and pressure within
the panels and numerical predictions were obtained using this approach, similar to those shown in Figures 5 and 6. This may not be too surprising given that the mat is subjected to higher pressure and undergoes large changes in thickness and temperature during pressing so that the permeability would span a much wider range of values than that obtained in finished boards. Unfortunately, no studies have been reported on the variation in permeability of wood products as they undergo hightemperature compression and resin curing. Thus, the approach taken is to carry out a least-squares fitting of the model to the measured pressure and temperature profiles in Figures 5 and 6 using the permeability κη as an adjustable parameter and eq 10
Ind. Eng. Chem. Res., Vol. 45, No. 6, 2006 1981
Figure 9. Comparison of the variation of board permeability κη along the y-direction of an industrial continuous hot press using different functions reported in the literature and the one obtained in the present study. Also included is the variation of board density with position.
to relate κχ to κη. Since the board thickness is also known at each position y, the density corresponding to each permeability value can be determined. This correlation between permeability and density is then stored as a look-up table used by the FlexPDE software for interpolation purposes in subsequent model runs. An important aspect of the variation of mat permeability with position is the behavior once the board reaches its final thickness at a distance ∼5 m from the entry point. Winistorfer et al.27 and Wang and Winistorfer28 obtained direct evidence that mat density also continues to slowly change after the platens have reached their final positions. Another important factor affecting the permeability during this portion of the process is resin curing. Analysis of our results shows that the average extent of curing over the η-direction at χ ) 0 when the OSB is discharged from the machine is 0.56. However, relatively little curing occurs over the portion of the panel within 5 m of the entry point, since it has been in the press for only a short time and the temperature has not risen significantly over most of its thickness. Consequently, most curing occurs after the panel has been compressed to essentially its final thickness and is further than 5 m from the entry point. Resin curing likely causes progressive obstruction of channels across the board, leading to a gradual reduction in permeability. Thus, over this region, the permeability should be affected more strongly by the extent X of resin curing than by porosity. Accordingly, we have assumed the permeability decrease during this portion of the process depends only on X according to a linear relationship. The variation of permeability κη obtained by this fitting and the corresponding board density with position y is shown in Figure 9. Also included are plots based on permeability-density relationships previously reported in the literature. Comparisons of the computed center-board pressure and temperature profiles obtained using the new calibrated permeabilities to the measured profiles originally shown in Figures 5 and 6 are presented in Figures 10 and 11. Much better agreement between the
computed and measured profiles is obtained than with the other approaches shown in Figures 5 and 6. The agreement between the model-fit and measured pressures in Figure 10 is very good, whereas the agreement between the corresponding temperature profiles in Figure 11 is not as good. It may be worth noting from a comparison of the five sets of pressure and temperature measurements shown in Figures 3 and 4 that the temperature data show considerably more variability than do the pressure data from run to run. The variation in temperature increased above 10 °C at some positions, which matches the deviation between the measured and model temperatures shown in Figure 11. Thus, the discrepancy evident in Figure 11 may not be due entirely to model error, and a significant portion may be due to the variability in the temperature measurements. It is clear from Figure 9 that the calibrated permeability depends more strongly on density throughout OSB pressing than do the values based on the correlations from previous studies. Second, the variation in permeability values is, as expected, much larger than the ones reported in the literature, because the latter were based on finished boards. Third, we initially attempted to calibrate the permeability assuming it varies smoothly according to a single function for the entire pressing process, but this did not yield the best results. More satisfactory results were obtained by relaxing this requirement and removing constraints on how the permeability changes as the curvature of the measured pressure and temperature profiles changes. Another important feature of the calibrated permeability curve is the gradual decline in the permeability after the board reaches its final thickness at a distance ∼5 m from the entry point, although it is not evident on the scale of the plot in Figure 9. To determine whether this small change in permeability is significant for the behavior of the system, a calculation was carried out under the conditions where the permeability was allowed to decrease according to the calibrated function shown in Figure 9 over the first 5 m into the press, but the small decrease after 5 m was ignored. The resulting profile of the
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Figure 10. Comparison of measured pressure profile at the board center (η ) 0, χ ) 0) along the y-direction of an industrial continuous hot press with the computed profile based on the permeability-density relationship obtained in this study.
Figure 11. Comparison of measured temperature profile at the board center (η ) 0, χ ) 0) along the y-direction of an industrial continuous hot press with the computed profile based on the permeability-density relationship obtained in this study.
pressure at the center of the board appears as the curve labeled “no attenuation” in Figure 12. A large deviation between the measured and predicted pressure is observed, with a significant underprediction at the end of the cycle compared to the measured values. This clearly demonstrates the sensitivity of the pressure to permeability during this stage when the permeability drops to very low values. Presumably, the permeability reaches such low values at this stage of pressing since most of the heretofore connected pathways for gas transport are now pinched off from one another. Evidence exists from previous work that the permeability may undergo distinct changes during pressing. Although mat permeability measurements while panels are being pressed have not been reported to our knowledge, related investigations on the evolution of VDP and horizontal density distributions (HDD) have indicated that the response of the mat to hot pressing goes through different stages. From in situ density, temperature, and
pressure measurements, Wang and Winistorfer28 proposed that VDP formation proceeds through five successive stages before and after the platens reach their final positions. These stages are related to changes in the mat compression behavior as the temperature rise propagates inward from the platens and moisture is transferred from the surface to the core and out the panel edges. Higher temperature and/or moisture content cause the wood plasticity to rise and its resistance to compressive stress to decrease, leading to densification. This process is further complicated by resin curing and bond formation that are affected not only by temperature and moisture gradients within the mat but also by the fact that the various layers can be all in compression states, all in springback states, or in mixed states at different times during a complete press cycle. Monte Carlo simulations of mat formation by Zombori et al.29 indicate that reduction of void space during pressing also proceeds in a stage-
Ind. Eng. Chem. Res., Vol. 45, No. 6, 2006 1983
Figure 12. Effect of board permeability during the final stage of pressing (i.e., once the board has reached its final height) on the computed pressure profile at the board center (η ) 0, χ ) 0) along the y-direction of an industrial continuous hot press.
like manner, with most voids between flakes being eliminated before the lumen volume within flakes is affected. Our fitting results in Figure 9 indicate that the average panel density changes smoothly during the process, unlike the permeability. However, this difference in behavior may arise since the permeability is strongly affected not only by the amount of void space but also by the configuration of this void space such as its detailed structure, distribution within the panel, and connectivity. Another factor affecting the permeability that is often overlooked is the presence of fines and their distribution within a panel. A panel with a given overall density may have a wide range of permeabilities depending on these other characteristics of its void structure. The permeability may undergo a number of changes as the various stages during pressing, resin curing, and bond formation proceed. Finally, as mentioned previously, it has not been possible in previous studies to measure OSB permeability during pressing; it is only possible after panels have been removed from the presses and usually after cooling to room temperature. It is likely that some relaxation of the permeability occurs once a panel is removed from a press. Although Bourbie and Zinszner,30 Doyen,31 Bryant et al.,32 and Cade et al.33 did not study the consolidation of wood products, their work on the permeability of other porous composite materials during compression may be relevant. They showed that the permeability can change sharply during such processes and that its dependency on porosity cannot be described in terms of a single function. Obviously, permeability measurements obtained during continuous pressing are needed to confirm the behavior shown in Figure 9. Comparisons between simulated and measured data were made for a separate pressing run obtained at a line speed of 0.6 m/s. These experimental data were not used during the previous calibration, and consequently, this comparison provides a test of the predictive accuracy of the model with the new permeability dependence. The profiles for the center board pressure and temperature as a function of position y are presented in Figures 13 and 14 and show good agreement in both shape and magnitude. 3.2. Model Application: Genetic Algorithm For Process Optimization. In this work, the optimization of the continuous
pressing process was carried out by applying a genetic algorithm to the model presented in the previous section. The advantage of genetic algorithms over other optimization techniques such as gradient-based methods is that they are suitable for problems that may have many local maxima and minima. Also, gradient methods are suitable for optimization of objective functions that are continuous with respect to the decision variables. In the current work, the generation of a dense and almost continuous mapping of the objective function with respect to the decision variables would require a large number of lengthy runs of the model. Consequently, it is particularly desirable to base the optimum on a smaller number of runs, a situation for which a genetic algorithm is especially convenient. To understand the algorithm, a few key terms should be outlined. Each parameter is encoded into a binary string called a gene. The binary string can contain several 0’s and 1’s that represent a discretized parameter. Several process parameters and settings are combined together for the manufacturing of OSB. For example, this may correspond to a specific combination of line speed, a set of temperature values along the pressing machine, etc. Such a set of parameters encoded into a string of genes is called an indiVidual. A group of these individuals is called a population that can be formed through genetic operators. The genetic operators include mutation and crossoVer operations. A mutation involves flipping 0’s with 1’s and 1’s with 0’s in the genetic code. A crossover involves an exchange of a string from one individual with a string from another individual. Fitness governs which individuals participate in a crossover and is determined by an objective function that corresponds to profit in the case of this study. Therefore, a fitter individual represents a set of process parameters and settings that would yield a higher profit for OSB pressing. The genetic algorithm for optimization of OSB pressing proceeds as follows: 1. Design variables are encoded according to physical bounds and an assumed number of bits. 2. Initial population of 16 individuals is generated randomly. 3. SelectionsThe mating population is selected from the entire population. The probability of an individual being selected
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Figure 13. Comparison of measured and predicted pressure profiles at the board center (η ) 0, χ ) 0) along the y-direction for a line speed of 0.6 m s-1 using the permeability-porosity relationship obtained in this study. Final board thickness is 11 mm.
Figure 14. Comparison of measured and predicted temperature profiles at the board center (η ) 0, χ ) 0) along the y-direction for a line speed of 0.6 m s-1 using the permeability-porosity relationship obtained in this study. Final board thickness is 11 mm.
is related to its fitness. If the fitness is nonpositive, the best overall individual replaces the negative individual. 4. Mating population is randomized to determine the pairs that will mate. 5. Individuals are encoded into binary strings. 6. ReproductionsThe mates combine to form offspring through crossovers. The crossover point is randomized. 7. Mutations1% of the genes in the population are mutated. 8. Each individual is decoded. 9. Individuals are evaluated by the objective function to give a fitness value. 10. Maximum, minimum, and mean fitness values are displayed. 11. If termination criteria are met, the algorithm stops. The termination criteria include the following: the generation number reaches 10, the relative change in maximum fitness is
