International Journal of Chemical Engineering

Volume 2019, Article ID 9043670, 16 pages

https://doi.org/10.1155/2019/9043670

## A Framework and Numerical Solution of the Drying Process in Porous Media by Using a Continuous Model

^{1}Department of Chemical Process Equipment, School of Chemical Technology, Hanoi University of Science and Technology, Hanoi, Vietnam^{2}Chair of Thermal Process Engineering, Faculty of Process and Systems Engineering, Otto-von-Guericke-University Magdeburg, Magdeburg, Germany

Correspondence should be addressed to Hong Thai Vu; nv.ude.tsuh@gnohuv.iaht

Received 29 October 2018; Revised 11 February 2019; Accepted 4 March 2019; Published 1 April 2019

Academic Editor: Doraiswami Ramkrishna

Copyright © 2019 Hong Thai Vu and Evangelos Tsotsas. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The modelling and numerical simulation of the drying process in porous media are discussed in this work with the objective of presenting the drying problem as the system of governing equations, which is ready to be solved by many of the now widely available control-volume-based numerical tools. By reviewing the connection between the transport equations at the pore level and their up-scaled ones at the continuum level and then by transforming these equations into a format that can be solved by the control volume method, we would like to present an easy-to-use framework for studying the drying process in porous media. In order to take into account the microstructure of porous media in the format of pore-size distribution, the concept of bundle of capillaries is used to derive the needed transport parameters. Some numerical examples are presented to demonstrate the use of the presented formulas.

#### 1. Introduction

The drying process plays an important role in many different industries, for example, in chemicals, pharmaceuticals, and agriculture. The drying process is one of the most complex problems that one finds in process engineering because not only heat and mass transfer takes place simultaneously in the course of the process but also because other phenomena may play a significant role. Although drying processes have been studied experimentally and theoretically for decades, simulating the coupling of heat and mass transfer and other phenomena in drying is still a challenging problem. Many researches were carried out to build suitable models and simulate numerically the drying process in engineering, and among recent works are those of Sekki and Karvinen [1], Antonov et al. [2], Azmir et al. [3], Ramos et al. [4], and Wu et al. [5].

Besides theoretical developments, numerical methods were applied successfully to simulate the drying process of porous media at the macroscopic scale and at the microscopic scale [6]. At the microscopic scale, the drying of porous media can be modelled as a network of pores, and the motion of the liquid-gas interface is modelled at the pore level, for example, in the work of Laurindo and Prat [7], Prat [8], Segura and Toledo [9], Metzger et al. [10], and Hirschmann and Tsotsas [11]. By using this approach [12], which we will refer to as the *discrete approach*, the microscopic structure and therefore the transport properties of the porous medium can be modelled with better accuracy. However, the problem becomes very large, and solving the system of equations of coupled heat-mass transfer becomes in many cases impractical, in particular when dealing with systems of large geometrical dimension. In such cases, the use of the *continuous approach* is more relevant (see for example [13, 14] or [15]). The continuous approach is built on the assumption that the porous medium can be described as a fictitious continuum by using effective coefficients for heat and mass transfer. For many applications, by using the continuous approach, the drying characteristics of porous media can be simulated with a very good accuracy. However, one of the challenges in using the continuous approach is how to determine the transport parameters [16]. Note that since the continuous approach is built on a fictitious continuum, this is also called the continuum approach. A continuous model for the drying process is therefore also called a continuum model.

In developing a drying model at the macroscopic scale, Whitaker [17] used the volume averaging technique to derive a system of macroscopic transport equations from a set of basic transport laws at the microscopic level. In Whitaker’s work [18], a porous medium is assumed to be equivalent to a continuum. A set of conservation equations for mass, energy, and momentum are introduced using average state variables. The continuous model developed by Whitaker is considered as rigorous and the most advanced continuous model today. The theory of Whitaker was later applied to different porous media, for example, by Perré [19], Ouelhazi et al. [20], Boukadida and Nasrallah [21], Boukadida et al. [22], Ferguson [23], Perré and Turner [24], Truscott [25], and Truscott and Turner [26]. Numerical techniques were developed to simulate the drying process using the derived average conservation equations. Among others, Perré and Turner [24] employed the control volume method to solve the problem. The advantage of this numerical method is that it ensures the conservation of mass and enthalpy through the boundaries of elements.

Despite the fact that the derivation of the governing equations of the drying problem at the continuum level can be found elsewhere [18], some effort was made to put these equations into a format that can be solved numerically [24], and there is the need to put all available knowledge in one framework, which is easy to understand and ready to be solved by many of the now-available numerical tools (as example of such tools, see [27, 28] or [29]). Such a framework will not only offer us the tool to solve the drying problem quickly but will also allow us to modify the governing equations in order to reflect the different phenomena that are not yet taken into consideration. In this work, by following the previous foundation laid out by Whitaker [18], Perré and Turner [24], and others, we will revisit the continuous approach starting with the transport equations at the pore scale (microlevel). We will briefly review the set of transport equations at the macro level. After that, we will introduce the transport parameters as a function of the material microstructure, and finally, some numerical solution will be presented and discussed.

#### 2. Governing Equations at the Microscopic Scale (Pore Scale)

We consider here a rigid porous medium with external boundary in which the matrix is made of some solid material and a system of interconnected voids. The voids are also called here “pores”. These pores are connected as a network of pores (voids). At the microscopic level (or pore level), we consider a small part of the porous medium with three phases: solid, liquid, and gas (Figure 1). The solid phase is denoted by , the liquid phase (water) is denoted by , and the gas phase is denoted by . The gas phase has two components (species): air (denoted by ) and vapour (denoted by ). In drying analysis, one of the primary objectives is to compute the distribution of moisture content, temperature, and internal gaseous pressure within the porous medium during the drying process. At the pore level, the (local) moisture content, the temperature, and the gaseous pressure at each point can be determined using suitable laws of physics such as the conservation of mass, the conservation of linear momentum, and the conservation of energy of each phase: solid, liquid, and gas. These conservation laws will be presented in the following, according to the work of Whitaker [18].