Determination of effective diffusivity and convective mass transfer coefficient for cylindrical solids via analytical solution and inverse method: Application to the drying of rough rice
Introduction
Thin-layer drying of agricultural products depends not only on the product, but also on the type and condition of drying. A specific drying process can be described by an adequate mathematical model as, for example, the liquid diffusion model which involves the diffusion equation (Luikov, 1968, Crank, 1992, Bird et al., 2001). To solve the diffusion equation, the boundary condition at the external surface of the product must be known. Boundary conditions of the first kind have been used for the description of drying with hot air for several types of grains (Gastón et al., 2002, Doymaz and Pala, 2003, Doymaz, 2005, Mohapatra and Rao, 2005, Hacihafizoglu et al., 2008, Silva et al., 2009). However, boundary conditions of the third kind have been found to be more adequate for the drying with hot air for other agricultural products (Queiroz and Nebra, 2001, Wu et al., 2004, Erdogdu, 2005, Mariani et al., 2008).
Generally, the diffusion equation must be numerically solved for solids with arbitrary geometry and, particularly, with variable thermo-physical parameters (Jia et al., 2001, Gastón et al., 2002, Li et al., 2004, Wu et al., 2004, Carmo and LIMA, 2005, Silva et al., 2008a). Under certain conditions (spherical or cylindrical geometries, infinite slabs, and constant thermo-physical parameters and volume), the diffusion equation has an analytical solution (Luikov, 1968, Crank, 1992). These solutions are used for the description of thin-layer drying for various agricultural products (Lima et al., 2004, Cunningham et al., 2007, Ruiz-López and García-Alvarado, 2007, Hacihafizoglu et al., 2008).
For the determination of thermo-physical parameters, as effective diffusivity and convective mass transfer coefficient, an adequate mathematical model must be tailored to the description of the drying kinetic of a product. Empirical models, generally simple regressions, can be used to determine the thermo-physical parameters (Park et al., 2002, Tello-Panduro et al., 2004, Silva et al., 2008b). However, in the case of the liquid diffusion model an optimization algorithm, based on the inverse method, can be generally used (Mariani et al., 2008, Silva et al., 2008a, Silva et al., 2009, Da Silva et al., 2009). Mariani et al. (2008) proposed an optimization algorithm for the determination of the apparent thermal diffusivity of banana using a numerical solution of the diffusion equation. Da Silva et al. (2009) proposed two algorithms, one deterministic and another stochastic, to determine the effective mass diffusivity of drying of mushrooms, using the analytical solution of the diffusion equation for an infinite slab with boundary conditions of the first kind. Silva et al. (2009), assuming boundary condition of the first kind, proposed an optimizer which scans the domain of the diffusivity to find the minimum of an objective function. The optimizer was coupled to the analytical solution of the diffusion equation for a sphere and applied to the drying kinetic of cowpea. The optimizer needs neither an initial value nor the indication of a search interval for the variable of interest.
This article proposes an optimization algorithm coupled to the analytical solution of the diffusion equation with boundary condition of the third kind. The method aims at the determination of the effective diffusivity and the convective mass transfer coefficient of thin-layer drying for products with cylindrical geometry, and was applied to the drying kinetic of rough rice.
Section snippets
Material and methods
It was assumed in this article that the liquid diffusion model is adequate to describe thin-layer water transport. This model is widely accepted to describe water transport with boundary condition of the first kind (Doymaz and Pala, 2003, Bello et al., 2004, Mohapatra and Rao, 2005, Thakur and Gupta, 2006), as well as with boundary condition of the third kind (Queiroz and Nebra, 2001, Wu et al., 2004).
Boundary condition of the first kind
The experimental data of Section 2.8 were processed by “Prescribed”, software for the optimization of diffusive processes with boundary condition of the first kind, available from Silva et al. (2009). The result is shown in Fig. 2.
Considering the Biot number as infinite, Def = 1.610 × 10−11 m2 s−1 (Def = 5.797 × 10−8 m2 h−1) was obtained with statistical fitting indicators R2 = 0.988760 and χ2 = 1.0373 × 10−2. The statistical indicators can be considered as good, and this result explains why a great quantity of
Conclusions
The use of boundary condition of the first kind for the description of thin-layer drying kinetic of rough rice is an acceptable approximation and produces reasonable results. But, as exposed in Fig. 6, curve (c), there is no instantaneous equilibrium between the external surface and the drying air. Thus, a convective boundary condition is more adequate for a rigorous description of the drying of rough rice, because the resistive effect of the moisture flux at the external surface is taken into
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