Spectral method for simulating 3D heat and mass transfer during drying of apple slices
Introduction
Fruits and vegetables are considered as more perishable foods because of high moisture content (Simal et al., 1997). Drying process is one of the well-known methods for preservation of fruits and vegetables. This process prevents occurrence of unpleasant changes such as microbial spoilage and enzymatic reaction by removing water from food products. Moreover, drying by lowering the mass and volume of food products, reduces the cost of packaging, storage and transportation (Goyal et al., 2006, Mujumdar, 2006).
When a moist object is subjected to drying conditions, heat and mass (moisture) transfer happen simultaneously. Heat is transferred by convection from the heated air to the surface of a moist object (food) and by conduction to the interior of food to increase temperatures and to evaporate moisture from the food surface. Moisture transfer is accomplished by diffusion from inside of the food to the surface, and from the food surface to the air by convection due to the heat transfer process (Hernandez et al., 2000, Mujumdar, 2006), although other mechanisms may be involved.
Heat and mass transfer phenomena in a system are described by governing equations of Fourier law and Fick's second law of diffusion. These equations are some simplified descriptions of physical reality of heat and mass transfer represented in mathematical terms (Hussain and Dincer, 2003, Tohidi, 2015).
Mathematical modeling is an important tool in the design and control of drying process especially in food engineering. Many undesirable changes may occur in foods during the drying process or in dried food after drying process which are associated to the temperature and moisture content distribution. Therefore, simulation and prediction of the temperature and moisture distributions in foods as a function of drying time can help us to prevent the undesirable changes in foods during drying process or during preservation (Mishkin et al., 1983, Wang and Brennan, 1993).
The analytical methods and numerical methods (such as the finite difference methods (FDMs), finite element methods (FEMs) and finite volume methods (FVMs)) are the basic mathematical tools that utilized to simulate the model of the heat and mass transfer (Barati and Esfahani, 2011, Lemus-Mondaca et al., 2013, Esfahani et al., 2014, Esfahani et al., 2015, García-Alvarado et al., 2014, Vahidhosseini et al., 2016, Tzempelikos et al., 2015).
Numerical methods mostly compute the approximate solutions of the governing equations through the localization of spatial and temporal variables that can be more realistic and flexible for simulating the aforementioned phenomena. In contrast, analytical methods require the infinite power series in computations that make them deficient and unfavorable with respect to the numerical methods (Zogheib and Tohidi, 2016). Moreover, in analytical methods, symbolic differentiation and integration are time-consuming operations. Therefore, in numerical techniques, alternative tools such as operational matrices of differentiation and also Gauss quadrature rules are replaced instead of direct symbolic differentiations and integration, respectively for speeding up the operations (Shen and Tang, 2006).
In recent years, considerable number of research works have been devoted to numerical simulation of heat and mass transfer phenomena during convective drying of food, such as numerical analysis of coupled heat and mass transfer during drying process in papaya slices and mango with FVM (Villa-Corrales et al., 2010, Lemus-Mondaca et al., 2013), numerical analysis of the transport phenomena occurring during drying process of carrots and mango fruit with FEM (Aversa et al., 2007, Janjai et al., 2008) and numerical simulation of 2D heat and mass transfer during drying of a rectangular object with FDM (Hussain and Dincer, 2003).
Among the numerical methods, the spectral methods are popular and robust tools which have been widely implemented during the recent decades for solving smooth partial differential equations (PDEs) with simple domains.
Numerical methods for solving PDEs can be classified into the local (like FDMs and FEMs) and global categories. In local methods, derivative approximation of an assumed function at any given point depends only on the information from its neighboring. Whereas in global methods, derivative approximation of an assumed function at any given points depends not only on the information from its neighboring points but also on the information from the entire of the computational domain, which force them to achieve a high precision using a small number of discretization nodes (Costa, 2004, Sun et al., 2012).
Spectral methods are global methods and converge exponentially. Spectral methods can provide high accuracy and low computational time and computer memory which make them favorable for solving smooth PDEs such as heat and mass transfer equations. It should be noted that the spectral method becomes less accurate for problems with complex geometries and non-smooth problems, while the FEMs are particularly well suited for solving this problems (Shen et al., 2011).
Spectral methods have been extended rapidly in the past three decades and have been widely implemented in meteorology (Jang and Hong, 2016), computational fluid dynamics (Canuto et al., 2007), quantum mechanics (Graham et al., 2009) and magnetohydrodynamics (Shan et al., 1991). Also, in recent works, researchers have some studies on the implementation of spectral methods for solving radiative heat transfer problem (Kuo et al., 1999, Li et al., 2009, Sun et al., 2012, Zhou and Li, 2017).
According to the authors’ knowledge, there are no results in the literature regarding the application of spectral methods for solving coupled heat and mass transfer equations in food engineering. This partially motivates us to propose such a method for solving the considered systems of coupled heat and mass transfer equations. Moreover, in most of the research works the spectral methods are applied for discretizing spatial variables together with localizing the temporal variable, with low order FDMs, which yields to unbalanced schemes that have high accuracy in spatial variables and low accuracy in time variable (Fakhar Izadi and Dehghan, 2014). Therefore, another motivation of the present study is to propose a spatial-temporal collocation method for the aforementioned processes which is a balanced numerical approach.
The objective of the present study is to extend Jacobi Gauss Lobatto (JGL) spectral collocation method to simulate coupled heat and mass transfer phenomena in three dimensions during convective air drying of apple slices.
Operational matrices of differentiation are implemented for approximating the derivative of both spatial and temporal variables. By using this spectral scheme, the coupled 3D heat and mass transfer together with the initial and boundary conditions will be reduced to the associated system of linear algebraic equations, which can be solved by some robust iterative solvers such as GMRES. Moreover, the experimental data are provided to validate the numerical data for the considered models. They confirm the accuracy of the presented numerical method.
Section snippets
Temperature and moisture measurements
The apple fruits were purchased from local markets in Mashhad, Iran, in June 2015 and stored in the refrigerator. Drying experiments were carried out by a convective air dryer equipped with the control unit to set the temperature of the air. The relative humidity and the air velocity were digitally measured by humidity sensor (Rotronic hygropalm, USA) and air velocimeter (Testo 425, Germany) which were placed in the dryer chamber. Experiments were performed for drying air temperature of 60, 70,
Modeling of external flow and temperature field
Fig. 2 shows the schematic domain of the problem, with its boundary conditions, for the determination of external flow and temperature fields of the drying air around the apple slices. At the left side, inlet velocity is U∞= (0.1 m/s) and inlet temperatures are T∞= (333 K, 343 K, 353 K and 363 K).
In simulations, side walls are considered at U∞ and T∞, and outlet pressure is assumed similar to the outlet condition of flow field. The governing PDEs for the forced convection motion of a drying
Spectral collocation method
In spectral methods the solution is approximated by a finite sum:where are the trial functions and are the expansion coefficients, which should be determined. The choice of trial functions , and test functions (that are used for determining the expansion coefficients) distinguishes the kind of spectral methods. The three main approaches for determination of expansion coefficients are Tau, Galerkin and collocation (pseudospectral) methods (Shen et al.,
External flow analysis
The result of external flow analysis are presented in this section. Fig. 3(A and B) show the velocity and temperature contours around the moist object in XY plane and at the middle of Z plane for an inlet velocity of 0.1 m/s and inlet temperatures of 363 K.
Since the trends of the temperature contours are the same for different drying temperature, only one case is presented. As the results shown, the temperature and velocity contours are seen to be symmetric. It is seen from Fig. 3, that flow
Conclusion
Jacobi spectral collocation method was proposed for simulating distribution of 3D coupled temperature and moisture content inside the apple slices numerically. The presented numerical method needs less computational time with respect to the analytical methods, by making use of the operational matrices in computations. The results of simulations illustrates that the 3D model of the coupled heat and mass transfer provide a better understanding of the transport processes inside the apple slices
Acknowledgments
The author thanks from the editor and reviewers of this paper for their constructive comments and nice suggestions, which helped to improve the paper very much.
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