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A mathematicalphysical model to describe heat and mass transfer in the mat during MDF hotpressing is proposed. Different transport mechanisms involved in this process are represented by a set of three coupled differential equations based on the principles of mass and energy conservation. A moisture diffusion term is included in the vapor conservation equation. This new model improves the physical approach to better understand the heat and mass transfer involved in this process. The unknown variables in the set of equations, including mat temperature, gas pressure, and moisture content are linked to physical properties such as mat density, thermal conductivity, diffusion coefficient and permeability. The final system will be solved by finite element method.
Key words: medium density fiberboard, hotpressing process, heat and mass transfer, physical properties, wood based composites
The hotpressing operation plays an important role in the medium density fiberboard (MDF) manufacturing process for many reasons. This process step affects the quality of the end product due to the mechanisms of heat and mass transfer involved during mat compression. An appropriate hot pressing is the key to reducing the risk of board blister or blown board. Hot pressing parameters also determine the vertical density profile (VDP). On the other hand, hot pressing has a significant impact on production costs due to the high energy consumption and pressingtime required. Understanding the heat and mass transfer processes involved is therefore fundamental to optimize this process. Numerical modeling is one approach to reach this goal and optimize hotpressing.
In MDF hot pressing many physical, mechanical and chemical processes are involved: heat and mass transfer in the mat, viscoelastic deformation of the wood fibers and resin cure. Nowadays, many models have been reported to help predicting the influence of the important parameters involved in the process and affecting the final product. An extensive literature on hotpressing models can be found.
Kayihan and Johnson (1983) were the first to develop a onedimensional model for hotpressing based on Luikov’s (1975) model. Harless et al. (1987) presented a onedimensional model to predict the density profile without considering the mechanisms of convection and diffusion during the process. Humphrey and Bolton (1989) proposed a cylindrical geometry for particleboard taking into account conduction, convection and phase change. Their contribution was essential for the description of the hotpressing process and more specifically, heat and mass transfer in wood based composites.
Suo and Bowyer (1994) also proposed an approach similar to Harless et al. (1987) to describe the development of the density profile in particleboard. Suo and Bowyer (1994) account temperature, moisture content, and the stressstrain behavior developed during hotpressing for each one of the layers of the mat.
The other models of hotpressing in panels are based upon models already developed for wood drying. For example, the models proposed by Carvalho and Costa (1998) and Zombori (2001) are based on Stanish’s (1986) model as well as the work of García (2002) which is based on Turner and Perré (1995) model. In brief, we can mention that the absence of free water and the presence of resin in the fibers, orientation and geometry of the fibers, flakes or particles, and mechanisms of transport presented in each one of the respective processes are the main differences between the models developed for hotpressing of woodbased panels compared to wood drying models.
Other models proposed for wood drying such as those by Defo (1999) and Benrabah (2002) have not yet been adapted to the modeling of hotpressing. The principal reason can be attributed to the parameters considered in both models such as the ratio of vapor diffusion for example, which must be determined for woodbased panels.
Only the Mansilla (1999) and Godbille (2002) models have been validated under production conditions found in particleboard plants. In both cases, modifications were applied to the equations of transport and to the physical parameters involved in order to improve the predictions of the models. The use of constant gas permeability is one of them.
Nigro and Storti (2001) have modified the Carvalho and Costa (1998) model and Humprey (1989) model by adding a water evaporation heat term. The Haselein (1998) model was based on Humphrey (1989) model. It simulates all the pressing cycle in panels but the presence of resin is neglected. Haselein (1998) has explained the effects of some variables such as press closing time on the formation of the density profile. Haselein (1998) work was based on Ren’s (1991) model to characterize the rheological coefficients of the material during mat compression. On the other hand, mat compression behavior was studied statistically by Dai and Steiner (1994).
Most of the models of hotpressing of woodbased panels have been developed for a batch press. However, Thoemen (2000) developed a model of continuous pressing based on the model developed by Humphrey (1989). It takes into account the rheological behavior of the mat and consequently the development of internal stresses based on the approach proposed by Ren (1991). Thoemen (2000) work does not consider the kinetic of adhesive polymerization, in contrast to the model presented by Carvalho and Costa (2001).
As we have seen previously, an important evolution of the models of hotpressing of woodbased composites occurred over the last 20 years. However, the following limitations of the models developed so far can be identified:
The type of numerical method used to solve the models equations changes from one investigator to the other. Computing time has not yet been seriously considered;
Many of the physical parameters involved in the models have not been determined or validated in specific conditions, especially at high temperatures;
There is a lack of knowledge for the appropriate description of the heat and mass transfer boundary conditions;
Most of the models were validated in laboratory conditions which are not representative of industrial conditions;
There is a need for a fundamental characterization of the resins used in terms of thermal and mechanical behavior during hotpressing;
Transport mechanisms have been neglected in several models such as the diffusion term in the mat during hot pressing.
Therefore, the objective of this study is to present a physical model of heat and mass transfer during MDF hotpressing. The proposed model includes a diffusion term. This model will be the basis of a model of continuous hot pressing and subsequently solved by the finite element method.
There are three governing equations in this model: one for the conservation and flow of gas, one for the conservation and flow of water vapor and one for the conservation and flow of energy. The three dependent variables are pressure (P), moisture content (M) and temperature (T).
The mass or density conservation equation accounts for each mass unit that is created, flows or accumulates in the volume element (Figure 41).
The terms IN and OUT express the flow produced through the volume element shown in Figure 41. Mathematically it can be written as where is the flux vector of component .
The term ACCUMULATION indicates the rate of mass or density increase in the system. Mathematically it can be written as .
The “SOURCE” term refers to the production or consumption of products involved in the system during a specific process. It can be represented by .
We can express the mass conservation equation according to the absence (Equation 41) or presence (Equation 42) of a source term. Without a source term this system can be described as:
With a source term this system may be described as (Figure 41):
In this volume element (Figure 41), the conservation equation can be expressed as:
In our volume element, we consider the gas as a mixture of air and water vapor following the ideal binary gas law. i.e.
where is the density of air (kg m^{3}), is the density of vapor (kg m^{3}), and is the density of gas (kg m^{3}) .
Each one of these components fills the void spaces and the wood cells of the mat. Then, the gas pressure will be considered as the sum of air and water vapor pressure or in other words the gas density will be the result of the sum of air and vapor densities. Some other components can be found such as gaseous products from the resin but they can reasonably be neglected.
According to the ideal binary gas law, the air density can be defined as:
where is the molecular weight of air (kg mol^{1}), is the total pressure (Pa), is the vapor pressure (Pa), is the ideal gas constant (J mol^{1} K^{1}) defined at the end in the section on constant values (see appendix 41), and is the temperature (K).
On the other hand, the vapor density can be defined as:
where is the molecular weight of the vapor (kg mol^{1}) and:
where is the saturated vapor pressure and is the relative vapor pressure defined as the ratio of the partial vapor pressure in the air to the saturated vapor pressure expressed as a fraction (Siau 1995). It can be expressed as (Siau 1995, García 2002):
where and are positive constants.
During the hotpressing process, air and water vapor pressure are transported mainly in the thickness direction by gas bulk flow and diffusion due to total pressure and moisture content gradients (Figure 42).
The gas flux can be described by Darcy’s law in response to a total pressure gradient produced during the process. We know that the flow is turbulent when the Reynolds number is higher than 2300. On the other hand, slip flow exists when the pathway diameter is close to the diameter of the fluid molecules. Both cases do not correspond to gas flow in a fiberboard mat during the hotpressing process (Zombori 2001, Kamke and Wolcott 1991). As a consequence, laminar viscous flow can be assumed and Darcy’s law can be written as:
BULK FLOW LAMINAR
where is the superficial permeability (m^{2}) and is the gas viscosity (Pa s). The gas permeability tensor (m^{3} s kg^{1}) is defined as:
During hotpressing the rate of evaporation can be seen as a source term. Then, by definition:
Therefore, we obtain:
where is the oven dry density of wood (kg m^{3}) , and the moisture content expressed as a fraction.
Similarly to equation (43), we can write:
The above equation expresses that the source term for the gas, is equal to the sum of the variation of the total gas mass per unit time plus the divergence of the gas flow.
The flow of water vapor can also be represented by Darcy’s law, assuming viscous laminar flow. However, contrarily to García (2002) we have introduced a new term. As recognized by Thoemen (2001) bound water is present in the cell walls whereas water vapor is found in the cell lumens and mat voids. Boundwater diffusion was not considered in Thoemen (2001) model. García (2002) neglected not only boundwater diffusion but also water vapor diffusion in the vapor conservation equation. García (2002) and Thoemen (2001) also recognized that both types of diffusion are present during hotpressing but at the same time they consider that the contribution of this phenomenon is insignificant compared to bulk flow. In spite of the minor contribution of both types of diffusion, these authors are in agreement that bound water diffusion is involved during hotpressing. In the current model we consider that the diffusion process is not negligible in hotpressing and is produced simultaneously with bulk flow. The diffusion process can be expressed by Fick’s law (Siau, 1995). On the other hand, diffusion measurements at high temperatures (above 100°C) and data which describe each mechanism separately have not yet been reported in the literature (Thoemen, 2001). We can therefore consider it in the transport equation as follows:
with:
where is the diffusion tensor (kg m^{1} s^{1}), is the water vapor diffusion coefficient (m^{2 }s^{1}), and C is the concentration of the diffusing substance (kg m^{3}).
As for the gas, the source term for vapor can be expressed as:
Similarly to equation (413), we can obtain:
The above equation shows that the rate of evaporation and the other source terms for the vapor are equal to the sum of the variation of the total vapor mass per unit time plus the divergence of the vapor flow.
Our system includes the presence of dry wood , bound water in the cell wall, air which fills the void spaces between the fibers, and water vapor which fills a portion of the cells lumen and void spaces. The resin is present to allow the adhesion between fibers. Free water is not considered because the initial moisture content of fibers before hotpressing is assumed to be below fiber saturation point. However, this assumption can be discussed because in particular cases free water can be generated by condensation during hot pressing.
Then, the total density of the system (m_{t}) can be expressed as:
where:
According to thermodynamic relationships (Siau, 1995), each one of these components has an associated internal energy (J kg^{1}) which can be calculated from the difference between the total enthalpy or heat content and the entropy from:
where is the temperature of the system (K).
Then, the total energy can be expressed as:
and thus the rate of change of internal energy is obtained as:
The internal energies can be calculated as:
and its enthalpy can be expressed as:
The development of equations (423), (424), (425), (427) and (428) can be found in García (2002). Equation (426) was deduced from a relationship between the enthalpies of vaporization and vapor as:
Resin energy is included in the energy conservation equation as a source term as shown in equation (435).
The flow of energy occurs by conduction and convection phenomena. Conductive transfer occurs from the hot press platens, and convective transfer is due to the movement of the air and water vapor mixture from panel faces to the center and edges. The effect of radiation is neglected because it is insignificant compared with the conduction and convection flows (Bowen 1970). Heat convection can be expressed on the basis of Darcy’s law while heat conduction is described by Fourier’s law. Considering conduction and convection, the energy flux can be written as follows:
where the enthalpy of the gas is defined as:
and is the thermal conductivity tensor (W m^{1} K^{1}) which depends on the thermal conductivity value during hotpressing.
Heat is generated from mat compression and polymerization of the resin during hotpressing. The heat generated from mat compression was calculated by Bowen (1970) and found to represent about 2 % of the total energy supplied during hotpressing and therefore can be neglected. Mansilla (1999) has estimated that the heat generated by resin polymerization contributes for about 13 % of the total energy involved during the process. Bowen (1970) estimated this value to be about 22 %. The heat generated is strongly related to the type of resin used. This term produces a faster increase of the temperature in the mat if the polymerization reaction of the adhesive increases due to its exothermic curing reaction. This heat source can be considered in a source term . The adhesive cure kinetics is described by an Arrheniustype equation as proposed by Zombori (2001).
This equation is based on the parameters , , , and where is the reaction constant (s^{1}), is the activation energy (J mol^{1}), is the extent of the reaction, is the order of the reaction and the temperature (K). Thus this equation establishes a relationship between temperature and the extent of the polymerization of the thermosetting resin as described in equation (433). The enthalpy relationship can be obtained from differential scanning calorimetry (DSC) data where is the resin average enthalpy (J kg^{1}).
Thus, the source term can be described as:
Similarly to equation (417), we can write:
The above equation shows that the sum of the rate of change of total energy plus the divergence of the heat flux by conduction and convection is equal to the source term.
From equations (413), (417) and (435), we can write the complete system in matrix form as:
System (436) written as a function of unknown variables P, M and T and with the addition of diffusion term can be expressed as:
This set of coupled differential equations is the final system to describe the heat and mass transfer in the mat during the MDF hotpressing process.
We therefore have a system (Equation 436) of three equations (Equations 437, 438 and 439) with three unknowns: for total pressure (Pa), for temperature (K) and for fractional moisture content. This model incorporates the diffusion and therefore improves the physical approach to better understand heat and mass transfer during MDF hotpressing.
Properties such as gas permeability, diffusion coefficient, porosity, viscosity, thermal conductivity, hygroscopicity, and rheological coefficients are necessary for modeling the hotpressing process. The parameters used in the current work are presented below.
The superficial gas permeability, (m^{2}), determines the rate of gas bulk flow. The diffusion coefficient (m^{2 }s^{1}) determines the molecular mass flow resulting from a gradient of concentration of the diffusing substance.
Gas permeability was determined by Garcia (2001) for MDF panels as a function of panel density. The superficial gas permeability equation obtained is:
where is the gas superficial permeability (m^{2}) and is the panel density (kg m^{3}). Equation (440) is valid for panel densities ranging from 400 to 1150 (kg m^{3}).
Little information is available for the diffusion coefficients in wood based composites especially at high temperatures values. Rosenkilde and Söderström (1997) determined diffusion coefficients for MDF samples at 56 ºC while Ganev et al. (2003) determined the same coefficient at 20 ºC. However, both temperature conditions are very far from the high temperatures involved during hotpressing. For that, an extrapolation was done from diffusion coefficients for solid wood (Siau, 1995) due to unavailable data.
The porosity or fractional void volume ( ) may be calculated from the oven dry cellwall specific gravity ( ) and the fractional moisture content (M) (Siau 1995).
where is the specific gravity of wood defined as the ratio of the oven dry mass density to the normal density of water .
The gas viscosity (Pa s) was adapted from Humphrey and Bolton (1989). The general form of the dynamic viscosity equation is:
where is the absolute temperature, and and are constants characteristics for the air and and are characteristic constants for the vapor. Then, we can write:
The thermal conductivity (W m^{1} K^{1}) determines the rate of conduction heat transfer during hotpressing and it depends mainly on moisture content and mat density.
An equation for thermal conductivity was adapted from one developed for particleboard by Godbille (2002). The final equation is:
where is the thermal conductivity (W m^{1} K^{1}), is the moisture content (fractional), and is the panel density (kg m^{3}). Equation (445) is valid for densities ranging between 450 and 800 (kg m^{3}).
The Hailwood and Horrobin (1946) sorption model is the most frequently used. However, this model gives somewhat incorrect values at temperatures above 150 °C which deviate significantly from experimental results at all relative humidity levels (Thoemen 2003).
A sorption isotherm model was adapted from the literature by García (2002). From this model, the resulting relative vapor pressure equation is:
where are constants defined in appendix and h is the vapor relative pressure (h=HR/100).
An equation for equilibrium moisture content (EMC) adapted from Malmquist (1959) can be used because it was found to give the best EMC values at high temperatures among seven EMC models compared by Vidal and Cloutier (2005). The resulting EMC equation is:
where is the fractional moisture content (kg kg^{1}), are extrapolated parameters in function of , is the relative vapor pressure, and is the temperature .
A model of heat and mass transfer in the medium density fiberboard mat (MDF) during hot pressing is proposed in this paper. The model unknowns are temperature, pressure and moisture content. The model is based on the conservation of gas, water vapor and energy. It includes a term to take account of moisture diffusion. Material properties were taken from the literature. This model will be used as the basis of a finite element model of heat and mass transfer in medium density fiberboard mat in batch pressing of laboratory panels and continuous pressing of industrial panels.
M_{a} 
28.951E3 
(kg mol^{1}) 

R 
8.31439 
(kg mol^{1}) 

M_{v} 
18.015E3 
(kg mol^{1}) 

Saturated vapor pressure 
C_{1}=24.912; 
C_{2}=5006.91 
(K) 
From García (2002) 

Dry wood energy 
C_{3}=103.1 
(J kg^{1}K^{1}); 
C_{4}=3.867 
(J kg^{1}) 
From García (2002) 
Vapor energy 
C_{5} =34.083 
(J mol^{1} K^{1}); 
C_{6}=240990 
(J kg^{1}) 
From García (2002) 
Air Energy 
C_{7}=29.161 
(J mol^{1} K^{1}); 
C_{8}=298290 
(J kg^{1}) 
From García (2002) 
Air viscosity 
C_{9}=1.37E6; 
C_{10}=358.9 
From Thoemen (2000) 

Vapor viscosity 
C_{11}=1.12E5; 
C_{12}=3211 
From Thoemen (2000) 

Relative vapor pressure 
C_{13}=0.186575; 
C_{14}=752.02 
(K); 
C_{15}=1163.31 
(K); 
C_{16}=12.7441; 
C_{17} =1.09603; 
C_{18}=2.36069; 
C_{19}=1.84447 
From García (2002) 
Equilibrium Moisture Content (from Malmquist, 1959)
where for equations
: Temperature
Gas permeability tensor
Diffusion tensor
Thermal conductivity tensor
The authors are grateful to the Natural Sciences and Engineering Research Council of Canada (NSERC) for supporting this project under Discovery grant no 12195402. They also extend their gratitude to Pablo García for his interest to enhance the quality and scientific integrity of this work.
© Marcia Vidal Bastias, 2006