Modeling of Coupled Moisture and Heat Transfer During Wood Drying

Transcription

1 th International IUFRO Wood Drying Conference - 3 Modeling of Coupled Moisture and Heat Transfer During Wood Drying P. Horáček Mendel University of Agriculture and Forestry Brno, Faculty of Forestry and Wood Technology, CZ-13 Brno, Czech Republic, horacek@mendelu.cz ABSTRACT Development of an accurate mathematical model of sufficient generality to evaluate drying processes in materials having diverse moisture transport properties is proposed. The process of wood drying is interpreted as simultaneous heat and moisture transfer with local thermodynamic equilibrium at each point within the timber. The model is a mathematical one treating wood as a continuous and homogenous medium not trying to explain the several parallel physical mechanisms of moisture transport actually taking place in the internal structure of wood. The transfer of moisture from a wooden surface and the corresponding mass transfer coefficient in relation to the heat transfer coefficient is considered. The Finite Element Method is used for numerical approach in the paper. The predicted values by the theoretical model are compared with experimental data taken under actual drying conditions to demonstrate the efficiency of the predictive model. INTRODUCTION In hygroscopic porous material like wood, mathematical models describing moisture and heat movements may be used to facilitate experimental testing and to explain the physical mechanisms underlying such mass transfer processes. A number of theoretical models have been suggested in wood drying (e.g. Chen and Pei 199) but they are not completely applicative to simulate real drying processes as seen in industry. The process of wood drying can be interpreted as simultaneous heat and moisture transfer with local thermodynamic equilibrium at each point within the timber. Drying of wood is in its nature a unsteady-state non-isothermal diffusion of heat and moisture, where temperature gradients may counteract with the moisture gradient. For a theoretical description of these phenomena, thermodynamic models seem to be most suitable. Although unsteady-state non-isothermal experiments have been conducted (e.g. Avramidis et al. 199), we are convinced that practical solutions of mathematical models that couple moisture and heat transfer with inhomogeneous material, combined with an experimental verification is still lacking (Irudayaraj et al. 199). A considerable volume of research has been carried out regarding modeling moisture and heat transfer in materials like polymers, wood, or agricultural products (e.g. Parrauffe and Mujumdar 19, Salin 1991, Kamke and Vanek 199). For wood, model developments have been based on either a mechanistic approach with the transfer phenomena derived from Fick s and Fourier s laws, or on the principles of thermodynamics and entropy production. These models may be divided into three categories: (a) diffusion models (Rosen 197), (b) models based on transport properties (Plumb et al. 195, Stanish et al. 19) and (c) models based on both the transport properties and the physiological properties of wood related to drying (Pang 199, 1997). The modeling of moisture fluxes under unsteadystate non-isothermal conditions has been noticeably absent in literature (Avramidis et al. 199), although the theory behind it was proposed twenty years ago (Siau 193). Heat and moisture transfer should be considering as coupled processes; the thermally induced mass transfer, Soret effect (Avramidis et al. 199, Siau 19) and the heat flux resulting from moisture diffusion, the Duffour effect (Siau 199), should be taken into account. Luikov (19) and in much details Whitaker (1977) developed a unique approach that describes the simultaneously heat and moisture transfer in drying 37

2 th International IUFRO Wood Drying Conference - 3 processes, based on irreversible thermodynamic processes. The difficulty with the mathematical formulation is the number of combined transfer mechanisms, the interdependencies among these mechanisms, and the different variables controlling them. MATERIAL AND METHOD Diffusion equations are used to describe moisture and heat transport phenomena in wood. Moisture content and temperature gradient are set as driving forces, since all the other possible factors related to moisture content are applicable only in the hygroscopic region. Thus, the model is purely mathematic treating wood as a continuous and homogenous medium, without explaining those physical mechanisms associated with the moisture transport that take place simultaneously inside wood. The three-dimensional transfer of heat and moisture is generally described as follows: dt(w)=div(d grad(w)) dt(t)=1/cρ div(λ grad(t)), where ρ is wood density, c specific heat, T temperature, t time, λ thermal conductivity and D moisture diffusion coefficient. Non-isothermal diffusion is analysed by using a thermodynamic model with the gradients of both water potential and temperature considered. The proposed equations are inspired by Siau and Avramidis (199), who found with these equations the best agreement between the experimental fluxes and thermodynamic model. The differentiated unsteady-state equations for twodimensional cross-section are w dw T w dw T w Dx Dx Dy Dy x + x dt x + = y + y dt y t T T Eb w T λx λy cρ x x + = y y +, 1, c t t where dw ϕ w Eb = dt RT ϕ T refers to the Soret effect (thermodiffusion) based on slope of the sorption isotherms w/ φ and activation energy for water diffusion E b. In general, these isotherms are specific to wood specimen, since differential heat of sorption for different wood differs noticeably (Babiak 199). To solve the equations, the initial conditions and the boundary conditions have to be considered. Especially boundary conditions affect the correct description of the real physical process going on at the body boundary in the case of unsteady/state heat and mass transport. At the boundary we can describe the concentration as a function of Newman s boundary conditions (3. order) (Newman 1931), details are described by Crank (1975). Boundary conditions are obtained from heat and moisture transfer between wood surfaces and external air w D x T λ x x= L x= L = h = h T c ( w( L, t) w ) ( T ( L, t) T ). where h c is mass transfer coefficient, h T heat transfer coefficient. Determination of moisture diffusion coefficient is based on models derived by Siau (1995) and Skaar 19), thermal conductivity by MacLean (191), specific heat of wood by Skaar (19), and density of wood by Kollmann (1951). In contrast to regularly applied assumption by Crank (1975), the initial values of temperature and moisture don t have to be uniformly distributed within the specimen. The numerical value used for heat and mass transfer coefficient in simulation of wood drying has been discussed in Soderstrom and Salin (1993), Avramidis et al. (199), Siau (1995) and Pang (199) together with correction coefficient introduced by Plumb et al. (195) and Siau (1995). An analytical solution to the problem of coupled heat and moisture transfer in wood is difficult to find. Therefore, the Finite Element Method (FEM) was used as the numerical approach in the paper. A software tool for the solution of systems of partial differential equations FlexPDE was applied. FlexPDE uses a Galerkin finite element model (Zienkiewicz 1977), with quadratic or cubic basis functions involving nodal values of system variables only. This model assumes that the dependent variables are continuous over the problem domain, but does not require or impose continuity of derivatives of the dependent variables. RESULTS The results presented methodological approach are the model prediction with data obtained in industryscale experiments. Model is applied to single boards as well as to stacked lumber piles, simulated there are moisture and heat distribution within -D domain. The predicted values by the theoretical model are compared with experimental data taken under actual drying 373

3 th International IUFRO Wood Drying Conference - 3 conditions to demonstrate the efficiency of the predictive model. Beech (Fagus sp.) and walnut (Juglans sp.) wood species was selected and samples in a shape of rectangular prisms of butt are cut from sawn timber (Fig.1). Samples were classified for wood densities. Samples are then kiln-dried according to standard schedules with a continuous monitoring of the moisture content. During the drying experiments, subsequent samples were removed and measured for moisture distribution. These data was compared with the theoretical results calculated on two cross-sections, A x7mm and B x1mm (Fig.1). with similarly results. From the mathematical model proposed, moisturecontent and temperature distributions as well as profiles are calculated and the results are presented in Figs.-9. 9mm 7mm mm mm 1mm FIGURE 1. Drying samples of butt used in experiments and cross-section simulated by the model. To estimate moisture diffusion of wood experiments are performed in a set-up with controlled air parameters, i.e. temperature, relative humidity (RH), which corresponds to the equilibrium moisture content (EMC) at a constant air flow. The drying conditions summarize Tab.1, Fig. and Fig.3. temperature ( C) / RH (%) / EMC (%) Temperature Relative humidity EMC time (hours) FIGURE. Example of industrial kiln drying conditions used in drying-schedule of beech wood species; 1 - heating, - drying, 3 - conditioning. Table 1: Drying conditions used for numerical simulation. Wood species Fagus sp. Juglans sp. Wood density 55 kg.m -3 kg.m -3 Initial temperature C Initial moisture content 1.5 % 15. % Equilibrium moisture content.5-1,5 % Dry bulb drying temperature C Relative humidity 3- % Air velocity 3 m.s -1 The derivation is based on thermodynamics potentials and coefficients used are considered a-priori as variables. In that way we can substitute the usual form of Luikov equations (19), where particularly the coefficients for coupled mass and heat transfer are difficult to obtain. Instead of coupling coefficients in Luikov equations determining the flux of moisture and heat caused by a thermal gradient, respectively by moisture gradient, there is used the numerical solution of non-isothermal unsteady-state moisture diffusion 37

4 th International IUFRO Wood Drying Conference - 3 temperature ( C) / RH (%) / EMC (%) Temperature Relative humidity EMC time (hours) FIGURE 3. Example of industrial kiln drying conditions used in drying-schedule of walnut wood species The predicted pattern of moisture and temperature at center, quarter and surface points (cross-section B) against the time showing both drying and heating dynamics, and moisture content and temperature difference between the surface and the interior of the wood is presented in Fig. and Fig.5. temperature ( C) dimesionless distance (-) FIGURE 5. Simulated temperature profiles in a beech (Fagus sp.) wood sample for transerverse direction at different drying periods (in seconds) dimesionless distance (-) The Fig. displays continual pattern changes of moisture distribution within the body with sharply decreasing effect of the temperature gradient from the very beginning of the drying process up to the time about 3s. Due to specimen dimensions, the Soret effect reveled in our case on contrary to the heat flux resulting from moisture diffusion that was omitted. Example of model simulations indicate that at given conditions, the external wood surface dries very rapidly with an accompanying rise of the internal temperature. A wet-line drying front appears since a rate of moisture transport within the body is slower than from wood surface. The front therefore travels inward through the wood as drying proceeds. FIGURE. Simulated moisture content profiles in a beech (Fagus sp.) wood sample for transverse direction at different drying periods (in seconds). 375

5 th International IUFRO Wood Drying Conference middle point quarter quaver surface average MC FIGURE. Curves of calculated average moisture content, and moisture at middle, quarter, quaver and surface points of the cross-section B against time for a modeled domain at beech wood species (Fagus sp.) showing the drying dynamics and moisture difference between the surface and the interior of the wood. The drying curves in distinct points within crosssection (B) together with the average moisture content over the whole body are clearly depicted in Fig.. Especially during the initial stages of drying, the simulated process behaves as strongly unsteady-state one with slow convergence to more steady-state form. In the beginning of drying process, the important influence of quick temperature elevation within the cross-section there is manifested by the increase of moisture content up to time of 3s when the temperature is distributed equally within body (Fig.5). The general shape of pattern of selected variables against time changes reasonably before/after completing the heating phase experimental MC modeled MC error FIGURE 7. The differences between simulated by the model and experimentally measured average moisture content of beech wood specimens (Fagus sp.). Fig.7 shows prediction of the average moisture content simulated by diffusion model together with experimentally measured values according to drying schedule depicted in Fig.. The maximum difference between values doesn t exceed 1% of MC with maximum disparity at the beginning of drying. error of the model (%MC) middle point quarter quaver surface average MC FIGURE. Curves of calculated average moisture content, and moisture at middle, quarter, quaver and surface points of the cross-section B against time for a modeled domain at walnut wood species (Juglans sp.) showing the drying dynamics and moisture difference between the surface and the interior of the wood. 37

6 th International IUFRO Wood Drying Conference - 3 The results proved that predicted values are in close relation with experimental data and evaluation of drying schedules currently used in the practice is therefore possible by the means of proposed model. Similar results we obtained both for the other cross-section (A) and walnut specimens (Fig. and Fig. 9) experimental MC modeled MC FIGURE 9. The differences between simulated by the model and experimentally measured average moisture content of walnut wood specimens (Juglans sp.). According to results obtained, the optimisation of currently used drying-schedules with cost reduction through fast drying and reduction of defects seams to be feasible. The benefits of the paper can be summarized as follows: (1) a validation of the proposed model for drying conditions in industry-scale drying chamber, () optimization of existing drying schedules, and (3) possible development of new schedules for other species under specified conditions. CONCLUSION A simple model based on unsteady-state nonisothermal moisture diffusion is presented that can be used to predict heat and moisture transfer in the hygroscopic moisture content and below C temperature ranges. For any wood described by wood density in this paper, the model predicts a distribution of both moisture and temperature fields in the body during error error of the model (%MC) the drying process. Examination of results indicates that an application of nonlinear finite element analysis of coupled problem is possible and in the case of arbitrary initial distribution of moisture and temperature within the body. The numerical solution of unsteady-state nonisothermal diffusion provides the reasonable results. Comparison of predicted values with experimental data taken both from literature and experiments is still needed to predict the moisture content and temperature fields during drying and/or the phenomenon of thermal diffusion during the initial stages of drying (desorption). With this model, the following goals can be achieved: (1) theoretical description of heat and moisture transfer by means of coupled unsteady-state non-isothermal diffusion of moisture and heat in wood through applying nonlinear finite-element analysis, () prediction of heat and moisture transfer for the complete moisture and temperature range, for single boards as well as stacked lumber. ACKNOWLEDGMENTS This work was supported by the Research Project of Ministry of Education, Youth and Sport of the Czech Republic No. MSM3. REFERENCES Avramidis S., Hatzikiriakos S.G., Siau J.F. (199): An irreversible thermodynamics model for unsteadystate nonisothermal moisture diffusion in wood. Wood Science and Technology : Babiak M. (199): Wood water system. TU Zvolen. 3 p. Chen P., Pei D.C.T. (199): A mathematical model of drying processes. Int. J. Heat Mass Transfer 3(): Crank J. (1975): Mathematics of diffusion. Cranendon Press, New York - Oxford. Irudayaraj J., Haghighi K., Stroshine R.L. (199): Nonlinera finite element analysis of coupled heat and mass transfer problems with application to timber drying. Drying technology (): Kamke P.A., Vanek M. (199): Comparison of wood drying models. Proc. th IUFRO Wood Drying Conference, Rotorua, New Zeland: 1-1. Kollmann F. (1951): Technologie des Holzes und der Holzwerkstoffe. 1. Band, Springer Vrelag Berlin, 5p. Luikov A.V. (19): Heat and mass transfer in capillary-porous bodies. Pergamon Press, New York, 53 p. MacLean J.D. (191): Thermal conductivity of wood. Heating, piping, and air conditioning 13: Newman A.B. (1931): The drying of porous solids: Diffusion and surface emission equations. Trans.Am.Inst.Chem.Engr. 7:

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