Thermodynamic and mathematical analysis of modified Luikov's equations for simultaneous heat and mass transfer

Abstract

Interface thermodynamic and mathematical analysis of modified Luikov equations (MLEs) for simultaneous heat and mass transfer in solids were developed. The thermodynamic relations at interface show that air-(wet solid) equilibrium relation may be mathematically approximated by two linear segments. These linear relations conduced to the possibility that MLEs have analytical solution for 1D rectangular, 1D radial cylindrical and 1D radial spherical coordinates system; and, the existence of a topology between solutions. The implications of MLEs thermodynamic and mathematical analysis on lumped equations were discussed. Experimental evolution of moisture and temperature during potato drying were compared with MLEs analytical (for one single linear stage at interface) and numerical (for the 2 linear stages) solutions.

Introduction

Simultaneous heat and mass transfer represents the fundamental phenomena of many engineering problems, such as: the improving of building energy efficiency and indoor air quality [1]; optimization of heat transfer and storage capacity of solar energy on the ground [2]; design and performance optimization of membrane contactors [3]; heat treatments of woods [4]; and, mathematical modeling of drying process [[5], [6], [7], [8], [9], [10], [11], [12]]. The Luikov's Equations (LEs) have been extensively used as reference equations for simultaneous heat and mass transfer in porous solids [1,2,10,[12], [13], [14], [15], [16], [17], [18]]. LEs, originally planted by Luikov [19], written in generalized coordinates system are,

$$C_m\frac{\partial U}{\partial t}=D_{\mathit{w\beta }}{C}_m\nabla \cdot \nabla U+\theta D_{\mathit{w\beta }}\nabla \cdot \nabla T_{\beta }\text{in}{\mathcal{V}}_{\beta }$$

$${\mathit{Cp}}_{\beta }{\rho }_{\beta }\frac{\partial T_{\beta }}{\partial t}=k_{\beta }\nabla \cdot \nabla T_{\beta }+{\mathit{\epsilon \delta \lambda }}_{\mathit{wv}}^0{\rho }_{\mathit{\beta s}}{C}_m\frac{\partial U}{\partial t}\text{in}{\mathcal{V}}_{\beta }$$

$$k_c{C}_m{\rho }_{\mathit{\beta s}}\left(U_i-U\right)=-{\mathbf{n}}_{\mathit{\beta \gamma }}\cdot D_{\mathit{w\beta }}{C}_m\nabla U_i-{\mathbf{n}}_{\mathit{\beta \gamma }}\cdot \theta D_{\mathit{w\beta }}\nabla T_{\mathit{\beta i}}\text{in}{\mathcal{A}}_{\mathit{\beta \gamma }}$$

$$h\left(T_i-T_{\gamma }\right)=-\left(1-\epsilon \right)k_c{C}_m{\rho }_{\gamma }\left(U_i-U_{\gamma }\right){\mathit{\delta \lambda }}_{\mathit{wv}}^0-{\mathbf{n}}_{\mathit{\beta \gamma }}\cdot k_{\beta }\nabla T_{\mathit{\beta i}}\text{in}{\mathcal{A}}_{\mathit{\beta \gamma }}$$

Eqs. (1), (2), (3), (4) are expressed in terms of moisture thermodynamic potential (U) which is related to solid water mass relation (moisture content in dry basis X_β) by the mass capacity (C_m) [2,13,14],

$$X_{\beta }=C_{m}U$$

LEs may be integrated over a finite volume, as example a particle in a fixed bed drying, resulting in averaged equations as function of lumped variables that can be directly applied for process design or simulation. However, the abstract moisture thermodynamic potential (and moisture capacity) limits the application of LEs in engineering. Therefore, García-Alvarado et al., [10] modified the LEs for to be applied in food drying, suggesting the air phase water-to-mass air ratio (moisture content in dry basis X_γ) as moisture potential. These modified Luikov's equations (MLEs) are,

$$\frac{\partial X_{\beta }}{\partial t}=D_{\mathit{w\beta }}\nabla \cdot \nabla X_{\beta }+\theta D_{\mathit{w\beta }}\nabla \cdot \nabla T_{\beta }\text{in}{\mathcal{V}}_{\beta }$$

$${\mathit{Cp}}_{\beta }{\rho }_{\beta }\frac{\partial T_{\beta }}{\partial t}=k_{\beta }\nabla \cdot \nabla T_{\beta }+{\mathit{\epsilon \delta \lambda }}_{\mathit{wv}}^0{\rho }_{\mathit{\beta s}}\frac{\partial X_{\beta }}{\partial t}\text{in}{\mathcal{V}}_{\beta }$$

$$h\left(T_i-T_{\gamma }\right)=-\left(1-\epsilon \right)k_c{\rho }_{\gamma }\left(X_{\mathit{\gamma i}}-X_{\gamma }\right){\mathit{\delta \lambda }}_{\mathit{wv}}^0-{\mathbf{n}}_{\mathit{\beta \gamma }}\cdot k_{\beta }\nabla T_i\text{in}{\mathcal{A}}_{\mathit{\beta \gamma }}$$

$$k_c{\rho }_{\gamma }\left(X_{\mathit{\gamma i}}-X_{\gamma }\right)=-{\mathbf{n}}_{\mathit{\beta \gamma }}\cdot {\rho }_{\mathit{\beta s}}{D}_{\mathit{w\beta }}\nabla X_{\mathit{\beta i}}-{\mathbf{n}}_{\mathit{\beta \gamma }}\cdot \theta D_{\mathit{w\beta }}\nabla T_{\mathit{\beta i}}\text{in}{\mathcal{A}}_{\mathit{\beta \gamma }}$$

In this modified LEs (MLEs), the thermodynamic relations at interface are obtained from Raoult's law, with ideal behavior deviations expressed in water activity (a_w), for water mass relation of air in equilibrium with wet solid phase,

$$X_{\mathit{\gamma i}}=\frac{a_w{p}_w^o/p}{1-a_w{p}_w^o/p}\frac{18}{29}$$

with the extended Antoine Eq. [20] for water vapor pressure as function of temperature,

$$p_w^o=e^{A+B/T_i+Cln\left(T_i\right)+{\mathit{DT}}_i^E}$$

And the extended Henderson Eq. [21] for food sorption isotherms,

$$a_w=1-e^{-k_1{T}_i^{k_2}{X}_{\beta }^{k_3+k_{4}T+k_5{T}^2}}$$

Two main advantages of modified Luikov Equations (MLEs) are: a) the diffusivity (D_βw) is the average water diffusivity in m² ⋅ s⁻¹ reported since Sherwood [22] study on mass transfer during drying; and b), the mass capacity is replaced by well-known (but non-linear) Eqs. (10a), (10b), (10c). Therefore the MLEs represent the unification between original LEs and mass transfer theory commonly used in solids drying [[5], [6], [7], [8], [9], [10], [11], [12],22]. MLEs may be integrated over a finite volume in order to obtain equations in lumped particles moisture and temperature as it will be shown in section 4.

 Much effort has been done to obtain analytical solutions of LEs [1,2,10,[12], [13], [14], [15]] and for understanding their mathematical properties [[16], [17], [18], [19]]. The introduction of non-lineal thermodynamic relations at interface (Eqs. 10a, b and c), complicates the analytical solution of MLEs (Eqs. 6 to 10). Therefore, in this work, an interface thermodynamic and mathematical analysis of MLEs were developed in order to understand the implications and limitations of their mathematical properties including their analytical solution and the application of resulting lumped equations in processes design and simulation. Additionally, the analytical and numerical solutions behaviors of MLEs were compared with respect to experimental moisture and temperature evolution during potato drying. In section 2, thermodynamic implications at interface are discussed and equilibrium models are simplified to linear relation by segments; in section 3, the mathematical properties, including topological transformations among solutions, are deduced from matrix analysis as it has been suggested by Pandey et al. [2]; and, in section 4, comparison of moisture and temperature experimental behavior during potato drying with respect to MLEs are shown.