Modeling unsteady heat and mass transfer with prediction of mechanical stresses in wood drying
Introduction
Wood drying is a complex thermal process commonly done in the industry and is known for its high energy consumption combined with the selection of optimum operating conditions in order to obtain a quality product (i.e. free high residual stresses, cracks, deformations significant or collapse). In addition to air velocity and its relative humidity, the moisture content and the temperature are critical parameters of drying and the transport modeling of these variables would allow to develop more efficient processes and with lower energy costs, Cloutier and Fortin [1]. The various complexities inherent in wood require strong simplifications to model and simulate the drying of wood. In general terms, it implies accepting the hypothesis of the continuous medium, eliminating heterogeneity, considering the anisotropy only relevant in the main directions, assuming it as non-deformable for the effects of transport phenomena and thermodynamic equilibrium for heat and mass balances, Pang [2] and Salinas et al. [3].
Heat and mass transport models in wood generally follow two approaches. The first one is based on the classical concepts of diffusion, incorporating in the conservation equations coefficients that depends on the wood properties, Luikov [4]. Some authors who proposed models based on this approach are [2,[5], [6], [7]]. In particular, a model that incorporates free water and bound water conditions during the drying of tropical wood is reported by [8]. A second approach is based on phenomenological multiphase transport inside the wood, Whitaker [9]. Perré and Degiovanni [10], Perré et al. [11], Turner and Ferguson [12,13] and Turner [14] have described simulations of heat and mass transfer in wood according to the approach of Whitaker [9]. The mathematical models to predict the stress-strains of the wood associated with the drying process was subsequently developed by Ferguson [5].
The first mathematical model to predict the relations between stress and strains of wood associated with the drying process was developed by Ferguson [5]. The stress-strain in the wood drying requires a coupling between the constitutive transport equations with the equilibrium equations of solid mechanics. This procedure becomes complex due to the anatomical and physical characteristics of the wood, Pang [2], Ferguson [5], Moutee et al. [15] and Turner and Ferguson [12]. According to Pang [2], the evolution of drying stress is the result of five partial deformation components: (i) elastic strain; (ii); free shrinkage; (iii) mechanical sorption strain; (iv) thermal strain; and (v) sustained load over time (creep). In this context, the stresses developed in the wood during the drying process have a defined behavior [2,3,5]. In the initial stage of the drying process, the wood begins to dry the surface, which is the region that first reaches the fiber saturation point (FSP) resulting in a contraction of the surface layers. Meanwhile, the adjacent inner layers prevent their free contraction inducing a state of stresses on the surface and contraction inside the wood. As drying progresses, the process undergoes a transition from the stresses state in the layers located towards the center of the wood, which lose moisture below the FSP. In this transition, the mechanical stresses are reversed resulting in a state of compression on the surface and tension in the inner layers of the wood, Salinas et al. [3].
The mathematical models that characterize the behavior of the tensions during the drying process are constituted by a system of non-linear partial differential equations and a numerical approach is required to obtain the solution. The Finite Element Method (FEM) has been widely used to solve solid mechanic problems. Svensson and Mãrtensson [16], Pang [2], Svensson and Mãrtensson [17] y Moutee et al. [15] have implemented a solution for the stress-strain problem in wood drying using FEM. On the other hand, the Control Volume Finite Element Method (CVFEM) has also been used to predict the residual stresses in wood drying process, Ferguson [5] and Salinas et al. [3]. In this context, Ferguson [5] compared FEM versus CVFEM and reported that the prediction of drying stresses is more efficient with CVFEM. According to the author, this result is due to the fact that CVFEM mixes the topological flexibility of the Finite Elements with the efficiency of the Control Volume in the conservation of the transported properties.
According to Pang [2], in order to simplify the models, thermal and creep strains are usually ignored due to the time scale of the drying process. In the first hours of drying, the wood is heated and the thermal expansion due to temperature gradients slightly influences the transitional state of stresses. However, throughout the drying process, the effect of temperature variation significantly influences the physical and mechanical properties of wood [18]. In this way, when the moisture content reaches values below FSP, the stresses state depends on the variation of the wood properties, which changes with the temperature, coupled to the moisture content distribution. Therefore, the mathematical models reported in the literature assume a total deformation as a result of shrinkage and mechano sorptive effects induced by moisture variations, which are added to the mechanical response of the wood as a result of its elasticity. The determination of drying stresses has been accomplished by a simplified one-dimensional approach [16,17,[19], [20], [21]], two-dimensional models proposed by Perré and Turner [22,23] and Salinas et al. [24], and three-dimensional models developed by Ormarsson et al. [[25], [26], [27]] and by Thibeault et al. [28].
The main contribution of this paper was to develop a new two-dimensional model to predict normal, tangential and torsional stresses in wood drying based on a coupled transient diffusion heat and mass transfer model. The novelties implemented in the physical-mathematical modeling of wood drying in this study include: (1) the strategy of imposing local mass transfer coefficients on each one of the wood surfaces that varied parabolically in space; (2) non-orthotropic mass transfer diffusion coefficient of moisture content in wood found by solving an inverse problem with a method developed by our group [3,3) the effect of moisture content on thermal conductivity and specific heat of Pinus radiata, that results in an increased coupling between heat and mass transfer by diffusion of water in wood. The numerical solution of the mathematical model was accurately accomplished by the Control Volume Finite Element Method, implemented in an in-house program coded in Fortran [29]. The results include the description of the evolution of the moisture content, temperature and stress distributions in wood drying by hot air. Experimental results of low temperature convective drying of Pinus radiata wood in a chamber with air flow of 1.6 m/s at 44/36 °C/°C, obtained by Salinas et al. [3] which reports all the thermophysical and mechanical properties necessary for the implementation of the model, allow validating the predicted results of all the dependent variables.
Section snippets
Physical model and experiments
Fig. 1a and Fig. 1b depict the experimental setup which considers a piece of solid wood (Pinus radiata) exposed to a drying environment and schematizes the procedure applied to each sample in order to obtain the slats for which dimensions and weights are measured, respectively. According to these figures, five samples of solid wood were located inside a climate chamber Heraeus Votsch with initials moisture content (M) and temperature of 158% on a dry basis and 20 °C, respectively. The drying
Physical description
Fig. 3 illustrates the domain adopted from sample wood transversal plane. This domain is identified with two convection edges and two other symmetry edges. Air flow edges are characterized by a variable convective coefficient h along the axes X and Y for tangential and radial directions, respectively. The distribution of convective effects is described according to Fig. 4 in which it is indicated a function S(x) or S(y) that describes the behavior of h along the corresponding surface. This
Results and discussion
Fig. 6 compares the experimental and simulated moisture contents curves. This figure depicts the simulated curves corresponding to the surface tangential, surface radial and center of wood, according to the positions indicated in Fig. 4. According to Fig. 6, the simulation of average moisture content presents a curvature well correlated with the experimental data. It is worth noting that the numerical average values of the drying curve are calculated by integrating the variable function of
Conclusions
For low temperature convective drying of radiata pine boards, a new modeling of simultaneous heat and mass transfer phenomenon in wood drying with prediction of the mechanical stresses was performed. This allowed to indicate the following conclusions:
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The simultaneous evolution of the coupled diffusion of heat and mass with the prediction of drying stresses in wood was accurately simulated and the efficiency of the model proposed in this work was validated with experimental data of Pinus radiata
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors acknowledge Agencia Nacional de Investigación y Desarrollo, ANID-Chile for support received in the Fondo Nacional de Desarrollo Científico y Tecnológico, FONDECYT, project N°3170214.
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