Porous media approaches to studying simultaneous heat and mass transfer in food processes. I: Problem formulations

Abstract

Heat and mass transfer formulations appearing in the food processing literature are synthesized in a systematic and comprehensive way, under the umbrella of transport in porous media. The entire range of formulations starting from the most fundamental to the semi-empirical are covered. Relationships of different formulations to each other and to the fundamental conservation laws are shown. The important transport mechanism in foods governed by the Darcy’s law is emphasized. Food processing examples of various formulations are provided.

Introduction

A porous medium refers to a solid having void (pore) space that is filled with a fluid (gas or liquid). Generally, many of these pores are interconnected so that transport of mass and heat is possible through the pores, that is generally a faster transport process than through the solid matrix. Porosity refers to volume fraction of void space. A wide variety of materials can be studied as porous media, such as rocks, soils, plant and animal tissues, paper and other packaging materials, stored grains, and packaged foods in cold storage. Concept of porous media is very general—it can have macropores as in an everyday sponge all the way down to nanopores as in a biological membrane.

In food systems, an enormous range of processes can be viewed as involving transport of heat and mass through porous media. Examples include extraction (Schwartzberg & Chao, 1982), drying, frying, microwave heating, meat roasting, rehydration of breakfast cereals (Machado, Oliviera, Gekas, & Singh, 1998), beans (e.g., Hsu, 1983) and dried vegetables (Sanjuan, Simal, Bon, & Mulet, 1999). Illustration of a range of porous media situations and structures in food can be seen in Fig. 1. Methodologies for creating tailor-made porous structures with a wide range of porosities have also been reported (Rassis, Nussinovitch, & Saguy, 1997). Most solid food materials can be treated as hygroscopic and capillary-porous (explained later). Liquid solutions and gels are non-porous. In these materials, transport of water is considered only due to the relatively simple phenomena of molecular diffusion and is not discussed in this article.

Porous media in food systems cover different scales. For the purpose of modeling transport processes in food systems, we can divide a porous media into two general groups, one involving large pores, and the other, small pores (see Table 1 and Fig. 2). In the large pores, fluid flow is mostly outside of the solid. An example of this is in cooling of stacked bulk produce such as oranges and strawberries. The fluid flow in this case is through the empty spaces of these stacked systems and is treated as a Navier–Stokes analog that is a generalization of Darcy flow. The other group consists of situations where the flow is inside the solid (pores are small). Example of this includes moisture transport inside the solid of many food processes such as drying, frying and microwave heating. Here the fluid transport through the pores of the solid is treated in terms of the simplest version of Darcy flow as opposed to its generalization into a Navier–Stokes analog, as mentioned for the other group. More discussion of Fig. 2 will be provided later under “Overview of Problem Formulation”.

Porous and capillary-porous materials can be defined as those having a clearly recognizable pore space (Vanbrakel, 1975). Examples of porous media include silica gel, alumina and zeolites, while those of capillary-porous media include muscle, wood, clay, textiles, packing of sand and some ceramics. The distinction between porous and capillary-porous is based on the presence and size of pores. Porous materials are sometimes defined as those having pore diameter greater than or equal to 10⁻⁷ m and capillary-porous as one having pore diameter of less than 10⁻⁷ m. Bruin and Luyben, 1980, Toei, 1983 treated most food materials as capillary-porous materials that are also adsorptive, in which the capillary suction force and adsorption are the mechanisms of water retention. In capillary-porous or porous materials (these are structured materials), transport of water is a more complex phenomena than in non-porous materials. In addition to molecular diffusion, water transport in porous and capillary-porous materials can be due to Knudsen diffusion (molecular diffusion when the mean free path of molecule is relatively long compared to the pore size), surface diffusion (Jaguste & Bhatia, 1995), capillary flow, and purely hydrodynamic flow.

In non-hygroscopic materials, the pore space is filled with liquid if the material is completely saturated, and with air if it is completely dry. The amount of physically bound water is negligible. Such a material does not shrink during heating. In non-hygroscopic materials, vapor pressure is a function of temperature only. Examples of non-hygroscopic capillary-porous materials are sand, polymer particles and some ceramics. Transport of materials in non-hygroscopic materials does not cause any additional complications as in hygroscopic materials noted below.

In hygroscopic materials, there is a large amount of physically bound water and the material often shrinks during heating. In these materials there is a level of moisture saturation below which the internal vapor pressure is a function of moisture level and temperature and is lower than that of pure water. These relationships are called equilibrium moisture isotherms. Above this moisture saturation, the vapor pressure is a function of temperature only (as expressed by the Clapeyron equation) and is independent of the moisture level. Thus, above certain moisture level, all materials behave non-hygroscopic.

Transport of water in hygroscopic materials can be complex. As water is removed, the unbound water can be eventually in funicular and pendular states (McCabe & Smith, 1976) that are harder to remove. When the unbound water has been removed, considerable bound water is still left. This bound water is removed by progressive vaporization within the solid matrix, followed by diffusion and pressure driven transport of water vapor through the solid.

Modeling transport processes in porous media is conceptually different from continuum based modeling of transport processes. Description of fluid flow and transport in porous media by considering in an exact manner the geometry of the intricate internal solid surface that bound the flow domain is generally intractable (Bear, 1972), although this is being pursued for relatively small dimensions (Keehm, Mukerji, & Nur, 2004). An exact manner here refers to the solution of Navier–Stokes equations to determine the velocity distribution of the fluid in the void space. Even if we could describe and solve such details, solutions are likely to be of little practical value. Thus, the standard continuum treatment cannot be used for porous media.

The approach taken in porous media is still a continuum one, but on a coarser level of averaging as compared to the standard continuum approach that averages at a more microscopic level. All variables and parameters of the continuum approach to porous media are averaged over a representative elementary volume (REV). In this continuum approach, the actual multiphase porous medium is replaced by a fictitious continuum: a structureless substance, to any point of which we assign variables and parameters that are continuous functions of the spatial coordinates of the point and of time (Bear, 1972).

Study of transport in porous media is a very active field. Entire textbooks (Bear, 1972, Kaviany, 1995, Vafai, 2000) and journals (Nassar & Horton, 1997) have been dedicated to this field. However, applications of transport in porous media to food materials have been little, perhaps due to the difficulty in obtaining the many process parameters needed, the complexity of such formulations and the unavailability of software tools to solve the resulting set of equations. Porous media approaches to the study of drying processes have been summarized in Plumb (2000, Chap. 17), where applications to food materials is noted. Although many food researchers are active in studying food structure parameters such as porosity (e.g., Aguilera, 2003, Rahman, 2003) and certainly some of these studies have the intention of relating structure to transport properties, quantitative relationships between structure and transport properties continue to be elusive in the literature.

This paper is organized as follows. The basic transport mechanisms of molecular diffusion, capillarity and Darcy flow are reviewed. Generalization of Darcy’s law for flow through a porous medium to its Navier–Stokes analog is developed. A model that has been used for large pores is presented. For small pores, governing equations are presented in three groups—(1) phenomenological model; (2) mechanistic or first-principle based models; and (3) semi-empirical model. Derivation of the first-principle based model for small pores by combining conservation laws with the fundamental transport mechanisms is presented. Two types of mechanistic models, one that treats evaporation as being distributed throughout the domain while another that treats evaporation as a sharp boundary, are presented. Simplification of the first-principle based model into more commonly used semi-empirical model is shown. Finally, a short discussion is presented on the inclusion of shrinkage or swelling in such models. The companion paper (Datta, 2006) discusses the input parameters to the first-principle based models and the application of one of these models for small pores to convective heating, baking (with and without volume change), frying and microwave heating.