Porous media approaches to studying simultaneous heat...

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Journal of Food Engineering 80 (2007) 80–95

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

A.K. Datta *

Department of Biological and Environmental Engineering, Cornell University, 208 Riley-Robb Hall, Ithaca, NY 14853-5701, USA

Received 15 December 2005; received in revised form 27 April 2006; accepted 2 May 2006Available online 24 July 2006

Abstract

Heat and mass transfer formulations appearing in the food processing literature are synthesized in a systematic and comprehensiveway, 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. Theimportant transport mechanism in foods governed by the Darcy’s law is emphasized. Food processing examples of various formulationsare provided.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Darcy; Capillary; Interface; Evaporation; Model

1. 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 transportof mass and heat is possible through the pores, that is gen-erally a faster transport process than through the solidmatrix. 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 andother packaging materials, stored grains, and packagedfoods in cold storage. Concept of porous media is very gen-eral—it can have macropores as in an everyday sponge allthe way down to nanopores as in a biological membrane.

In food systems, an enormous range of processes can beviewed as involving transport of heat and mass throughporous media. Examples include extraction (Schwartzberg& Chao, 1982), drying, frying, microwave heating, meatroasting, rehydration of breakfast cereals (Machado, Olivi-era, Gekas, & Singh, 1998), beans (e.g., Hsu, 1983) and

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dried vegetables (Sanjuan, Simal, Bon, & Mulet, 1999).Illustration of a range of porous media situations andstructures in food can be seen in Fig. 1. Methodologiesfor creating tailor-made porous structures with a widerange of porosities have also been reported (Rassis, Nussi-novitch, & Saguy, 1997). Most solid food materials can betreated as hygroscopic and capillary-porous (explainedlater). Liquid solutions and gels are non-porous. In thesematerials, transport of water is considered only due tothe relatively simple phenomena of molecular diffusionand is not discussed in this article.

Porous media in food systems cover different scales. Forthe purpose of modeling transport processes in food sys-tems, we can divide a porous media into two generalgroups, one involving large pores, and the other, smallpores (see Table 1 and Fig. 2). In the large pores, fluid flowis mostly outside of the solid. An example of this is in cool-ing of stacked bulk produce such as oranges and strawber-ries. The fluid flow in this case is through the empty spacesof these stacked systems and is treated as a Navier–Stokesanalog that is a generalization of Darcy flow. The othergroup consists of situations where the flow is inside thesolid (pores are small). Example of this includes moisture

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Nomenclature

Aspec specific area, m2

Aw water activity, dimensionlessc mass concentration, kg/m3

c constant in Eq. (19)cp specific heat, J/kg KC molar concentration, mol/m3

D mass diffusivity, m2/sDc capillary diffusivity due to concentration gradi-

ent, m2/sDT capillary diffusivity due to temperature gradient,

m2/sF0 microwave energy flux at surface, W/m

2

g gravity, m/s2

h enthalpy, J/kgh heat transfer coefficient, W/m2 Kh capillary head, mhm mass transfer coefficient, m/s_I rate of evaporation, kg/m3 sk permeability, m2

k thermal conductivity, W/m Kk constant in Eq. (19)Kij coefficients, primarily empirical, various unitsL length, mM molecular weight, kgM moisture content, kg of water/m3

Md moisture content, dry basis, kg of water/kg ofdry solids

Mw moisture content, wet basis, kg of water/kg oftotal

n mass flux, kg/m2 sp pressure, partial pressure, PaP pressure, Pa_q heat source, W/m3

R gas constant, 8.315 kJ/kmol Ks distance, mSw water saturation = DVw//DV, dimensionlessSg gas saturation = DVg//DV, dimensionlesst time, sT temperature, �C or Ku velocity, m/s

x distance, mX location of interface, mV total volume, m3

Greek letters

dmicro microwave penetration depth, mdinfra infrared penetration depth, mk latent heat, kJ/kgl viscosity, kg/m s/ porosity, dimensionlessq density, kg/m3

qapp apparent density, kg/m3 of total volume

s tortuosity, dimensionless

Subscripts

a airc capillaryd dry basise effectiveeff effectiveg gasgr gas, relativei initialir irreducibleinfra infraredI, II regionsl liquidmicro microwaveo oils solidv vaporvs saturated vaporva vapor in airw water, wet basiswr water, relativea aircap capillarydiff diffusionpress pressure

A.K. Datta / Journal of Food Engineering 80 (2007) 80–95 81

transport inside the solid of many food processes such asdrying, frying and microwave heating. Here the fluid trans-port through the pores of the solid is treated in terms of thesimplest version of Darcy flow as opposed to its generaliza-tion into a Navier–Stokes analog, as mentioned for theother group. More discussion of Fig. 2 will be providedlater under ‘‘Overview of Problem Formulation’’.

1.1. Porous and capillary-porous materials

Porous and capillary-porous materials can be defined asthose having a clearly recognizable pore space (Vanbrakel,

1975). Examples of porous media include silica gel, aluminaand zeolites, while those of capillary-porous media includemuscle, wood, clay, textiles, packing of sand and someceramics. The distinction between porous and capillary-porous is based on the presence and size of pores. Porousmaterials are sometimes defined as those having pore diam-eter greater than or equal to 10�7 m and capillary-porous asone having pore diameter of less than 10�7 m. Bruin andLuyben (1980) and Toei (1983) treated most food materi-als as capillary-porous materials that are also adsorptive,in which the capillary suction force and adsorption arethe mechanisms of water retention. In capillary-porous or

Fig. 1. Examples of food systems that have been modeled as porous media, illustrating range of porous structure. On the left is chicory roots in bulkcourtesy of Hoang et al. (2003), in the middle is micrograph (SEM) of bread and on the right is micrograph (Environmental SEM) of raw potato.

Table 1Table showing classes of porous media problems in relation to food and their relationship to this article

Porous media characteristics Food applications Equations to solve

Large pores, applied pressure (Section 5) Cooling of stacked produce such as potato orstrawberries in a cold room

Navier–Stokes analog of Darcy equation; energyequation; species equations

Small pores, pressure mostly frominternal evaporation (Section 6)

Most food processes involving significant heatingsuch as drying, frying, microwave heating, etc.

Darcy equation replacing momentum equation;energy equation; species equations for eachphase

Small pores, capillarity only, nosignificant internal evaporation (Section 7)

Lower temperature processes where heating is notintense such as rehydration, storage

Darcy equation only for capillary pressure; speciesequation for liquid phase

Small pores, capillarity plus other modes;small amount of evaporation (Section 8)

Lower temperature processes where heating is notintense (same as above)

Semi-empirical formulation with effective diffusivityfor a combined species equation; energy equationwith empirical evaporation term

Food systems

Darcy flowNavier stokesequivalent ofDarcy flow

Non-isothermalformulation

Isothermalformulation

smallpores

largepores

strong evaporation

weakevaporation

Capillary pressureformulation

Capillary diffusivityformulation

Distributed evaporation

Evaporation with a sharp interface

Semi-empiricalformulations

Semi-empiricalformulations

Phenomenologicalformulation (Luikov)

Fig. 2. Schematic showing the various porous media formulations and their interrelations in the context of food processes.

82 A.K. Datta / Journal of Food Engineering 80 (2007) 80–95

porous materials (these are structured materials), transportof water is a more complex phenomena than in non-porousmaterials. In addition to molecular diffusion, water trans-

port in porous and capillary-porous materials can be dueto Knudsen diffusion (molecular diffusion when the meanfree path of molecule is relatively long compared to the pore


size), surface diffusion (Jaguste & Bhatia, 1995), capillaryflow, and purely hydrodynamic flow.

1.2. Hygroscopic vs. non-hygroscopic materials

In non-hygroscopic materials, the pore space is filledwith liquid if the material is completely saturated, and withair if it is completely dry. The amount of physically boundwater is negligible. Such a material does not shrink duringheating. In non-hygroscopic materials, vapor pressure is afunction of temperature only. Examples of non-hygroscopiccapillary-porous materials are sand, polymer particles andsome ceramics. Transport of materials in non-hygroscopicmaterials does not cause any additional complications asin hygroscopic materials noted below.

In hygroscopic materials, there is a large amount ofphysically bound water and the material often shrinks dur-ing heating. In these materials there is a level of moisturesaturation below which the internal vapor pressure is afunction of moisture level and temperature and is lowerthan that of pure water. These relationships are called equi-librium moisture isotherms. Above this moisture satura-tion, the vapor pressure is a function of temperature only(as expressed by the Clapeyron equation) and is indepen-dent of the moisture level. Thus, above certain moisturelevel, all materials behave non-hygroscopic.

Transport of water in hygroscopic materials can be com-plex. As water is removed, the unbound water can be even-tually in funicular and pendular states (McCabe & Smith,1976) that are harder to remove. When the unbound waterhas been removed, considerable bound water is still left.This bound water is removed by progressive vaporizationwithin the solid matrix, followed by diffusion and pressuredriven transport of water vapor through the solid.

1.3. Modeling porous media as a continuum

Modeling transport processes in porous media is con-ceptually different from continuum based modeling oftransport processes. Description of fluid flow and transportin porous media by considering in an exact manner thegeometry of the intricate internal solid surface that boundthe flow domain is generally intractable (Bear, 1972),although this is being pursued for relatively small dimen-sions (Keehm, Mukerji, & Nur, 2004). An exact mannerhere refers to the solution of Navier–Stokes equations todetermine the velocity distribution of the fluid in the voidspace. 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 por-ous media.

The approach taken in porous media is still a continuumone, but on a coarser level of averaging as compared to thestandard continuum approach that averages at a moremicroscopic level. All variables and parameters of the con-tinuum approach to porous media are averaged over a rep-

resentative elementary volume (REV). In this continuumapproach, the actual multiphase porous medium isreplaced by a fictitious continuum: a structureless sub-stance, to any point of which we assign variables andparameters that are continuous functions of the spatialcoordinates of the point and of time (Bear, 1972).

Study of transport in porous media is a very activefield. Entire textbooks (Bear, 1972; Kaviany, 1995; Vafai,2000) and journals (Nassar & Horton, 1997) have beendedicated to this field. However, applications of transportin porous media to food materials have been little, perhapsdue to the difficulty in obtaining the many process param-eters needed, the complexity of such formulations and theunavailability of software tools to solve the resulting set ofequations. Porous media approaches to the study of dry-ing processes have been summarized in Plumb (2000,Chap. 17), where applications to food materials is noted.Although many food researchers are active in studyingfood structure parameters such as porosity (e.g., Aguilera,2003 & Rahman, 2003) and certainly some of these stud-ies have the intention of relating structure to transportproperties, quantitative relationships between structureand transport properties continue to be elusive in theliterature.

1.4. Organization of this paper and its companion

This paper is organized as follows. The basic transportmechanisms of molecular diffusion, capillarity and Darcyflow are reviewed. Generalization of Darcy’s law for flowthrough a porous medium to its Navier–Stokes analog isdeveloped. A model that has been used for large pores ispresented. For small pores, governing equations are pre-sented in three groups—(1) phenomenological model; (2)mechanistic or first-principle based models; and (3) semi-empirical model. Derivation of the first-principle basedmodel for small pores by combining conservation lawswith the fundamental transport mechanisms is presented.Two types of mechanistic models, one that treats evapora-tion as being distributed throughout the domain whileanother that treats evaporation as a sharp boundary, arepresented. Simplification of the first-principle based modelinto more commonly used semi-empirical model is shown.Finally, a short discussion is presented on the inclusion ofshrinkage or swelling in such models. The companionpaper (Datta, 2006) discusses the input parameters to thefirst-principle based models and the application of one ofthese models for small pores to convective heating, baking(with and without volume change), frying and microwaveheating.

2. Various transport mechanisms in porous media

Transport in a porous media can be due to several differ-ent mechanisms. Primarily three mechanisms are consid-ered—molecular diffusion (for gases), capillary diffusion


(for liquids), and convection (pressure driven or Darcyflow). These three mechanisms are discussed in this sectionby treating the food as a porous medium and consideringthe transport of water, vapor, air and other componentsinside the food.

2.1. Molecular diffusion

Gases such as water vapor and air in the porous mediacan move by molecular diffusion if the pores are largeenough. Molecular diffusion of a gas species (e.g., vapor)in a gas mixture (e.g., vapor and air) is described by Fick’slaw

ndiffg ¼ �Dgocgos

ð1Þ

where g denotes a gas species, ndiffg is the mass flux of gasdue to diffusion, Dg is the molecular diffusivity of the gasin the porous medium, cg is the gas concentration and sis the distance. The molecular diffusivity of the gas in bulkcan be related to the molecular diffusivity of the gas insidethe porous medium using the tortuosity, s, and porosity, /

Dg ¼Ds

/ ð2Þ

where D is the gas diffusivity in the bulk. Mixture of vaporand air, when present, is treated as an ideal gas.

2.2. Darcy flow of gases due to gas pressure

Transport of gases in a porous material due to pressurecan be described by Darcy’s law, written as:

npressg ¼ �qgkglg

oPos

ð3Þ

where npressg is the mass flux of gas, P is total pressure in thegas phase, and qg and lg are the density and viscosity of thegas, respectively. The permeability in the gas phase is givenby kg = kkgr where k is called the intrinsic permeability andkgr is the relative permeability of the gas phase. A simpleinterpretation of intrinsic permeability can be given as fol-lows. As a first approximation, a porous medium can beconsidered as a bundle of tubes of varying diameter embed-ded in the solid matrix. By considering steady and fullydeveloped pipe flow through these tubes, we can show(e.g., Datta, 2002) that the coefficient determining flow inEq. (3) (called the hydraulic conductivity when the fluidis water instead of a gas) can be written as

qgkglg|ffl{zffl}

hydraulic conductivity

¼qglg|{z}

fluid property

1

8s

Xi

Dbir2i|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

matrix property

ð4Þ

where Dbi is the volume fraction of pores in the i-th classhaving radius ri. Eq. (4) shows that, as expected, the coef-ficient in Eq. (3) depends on both fluid and matrix proper-

ties. The relevant fluid properties are density q andviscosity l. The relevant solid matrix properties are poresize distribution, shape of pores, porosity, and tortuosity,s. Tortuosity can be defined as the ratio between the actualpath traveled by a fluid element between two points dividedby the straight line path between the same two points. Thesole effect of matrix property can be included in intrinsicpermeability or permeability k such that

k ¼ 18s

Xi

Dbir2i ð5Þ

The units of permeability, k, can be seen from Eq. (5) to bem2. Permeability is also expressed in the units of Darcy.The Darcy is a non-SI metric unit of permeability, equalto cm2 cP/(atm s) where cP denotes the centipoise. By usingthe value of atmospheric pressure, it follows that 1Darcy = 9.8692 · 10�13 m2 which may be convenientlyremembered as Darcy � 1 lm2.

Like Fourier’s law of heat conduction and Fick’s law ofmass diffusion, Darcy’s law (Eq. (3)) is an empirical rela-tionship. The volumetric flux, npressg =qg, having the unitsof m3/m2 s or m/s, is also the average velocity or the super-ficial velocity through the porous medium. Thus,

u ¼ � kglg

oPos

ð6Þ

In transport terminology, u represents the velocity thatcontributes to convection. Note that the average velocityin the pores is higher, given by u//.

2.3. Darcy flow of liquid due to gas and capillary pressures

Capillary flow is due to the difference between the rela-tive attraction of the molecules of the liquid for each otherand for those of the solid. A familiar example of this is therise of water in an open tube of small cross-section. As theradius becomes very small, capillary rise increases signifi-cantly. Capillarity and other forces are the reason, forexample, that the food tissue does not get completelydrained by gravity or lose all its water from evaporation.In a porous solid food, the liquid will be attracted or heldmore tightly when there is less of it, i.e., at lower concentra-tions. Conversely, the liquid will be held less tightly whenthere is more of it. Due to the differences in capillary attrac-tion, flow of liquid can occur from locations in the solidhaving more water to locations having less water, i.e., fromhigher concentration to lower concentration of water. Thisis referred to as unsaturated flow and is extremely impor-tant in food processing, for example, in drying of foodmaterials.

Transport of liquids in a porous material due to pressureis also described by Darcy’s law. The discussion surround-ing Eqs. (4)–(6) are also applicable here. However, the pres-sure in the liquid phase is not the same as that in the gasphase. The capillary attraction referred to in the previousparagraph results in a negative pressure on the liquid (for


example, water is held by this pressure). Thus, the mass fluxof liquid can be written as

npress; capl ¼ �qlklll

oðP � pcÞos

ð7Þ

¼ �qlklll

oPos|fflfflfflfflfflffl{zfflfflfflfflfflffl}

mass flux due to gas pressure

þ qlklll

opcos|fflfflfflffl{zfflfflfflffl}

mass flux due to capillarity

ð8Þ

Here npress; capl is the mass flux of liquid, P is total pressure inthe gas phase, pc is the capillary pressure of the liquid andkl is the permeability in the liquid phase, given by kl = kklr,where klr is the relative permeability in the liquid phase.The negative pressure pc due to capillary and other attrac-tive forces is a function of the concentration of the liquid(such as water content), cl, and temperature, T, for a par-ticular material. When water is the fluid of interest, thisrelationship is a moisture characteristic curve that is ob-tained from experiments. Although little or no data isavailable for food materials, moisture characteristic curvehas been obtained for other materials such as soil and per-haps we can learn from these (see discussion in companionpaper Datta, 2006). Since pc depends on cl and T, we canwrite Eq. (8) as

npress; capl ¼ �qlklll

oPosþ ql

klll

opcocl

oclos|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

capillary fluxdue to conc: grad:

þ qlklll

opcoT

oTos|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}

capillary fluxdue to temp: grad:

ð9Þ

¼ �qlklll

oPos

�Dcoclos|fflfflfflffl{zfflfflfflffl}

capillary fluxdue to conc: grad:

�DToTos|fflfflfflffl{zfflfflfflffl}

capillary fluxdue to temp: grad:

ð10Þ

where

Dc ¼ �qlklll

opcocl

ð11Þ

DT ¼ �qlklll

opcoT

ð12Þ

Eq. (10) can also be interpreted as

npress; capl ¼ ql u|{z}velocity due to

gas pressure

�Dcoclos|fflfflfflffl{zfflfflfflffl}

capillary diffusionconc: effect

�DToTos|fflfflfflffl{zfflfflfflffl}

capillary diffusiontemp: effect

ð13Þ

The quantities Dc and DT in the above equations aretermed diffusivity in analogy to Eq. (1) describing Fick’slaw. It is important to note that, although the term ‘‘diffu-sivity’’ is being used here for movement of a liquid (such aswater) through a solid, the mechanism of flow is primarilycapillarity, not molecular diffusion. The capillary diffusivitydue to gradient in concentration, Dc, is often the more sig-nificant component of the two (Dc and DT). Capillary dif-fusivity Dc reduces as the concentration cl of liquid reduces,

the exact variation being quite material specific and canonly be obtained from experiment. Although very little di-rect measurement of capillary diffusivity in foods is avail-able, data supporting decrease of capillary diffusivity asmoisture content decreases are quite common in food liter-ature (Saravacos, 2001).

2.3.1. Special case: Porous media is nearly saturated

When the solid is nearly saturated with liquid, as in verywet food where pores are completely filled with water, thecapillary pressure can be very small and Eq. (8) can beapproximated as

npressl ¼ �qlklll

oPos

ð14Þ

2.3.2. Special case: Porous media is highly unsaturated

When the solid is highly unsaturated (like in many dry-ing applications), capillary pressure can be so large that it isthe dominant force compared to the gas pressure. Undersuch conditions, Darcy’s law has only the negative capil-lary pressure due to the binding of water from capillaryand other attractive forces. Eq. (8) can be approximated as

ncapl ¼ qlklll

opcos

ð15Þ

or, alternatively, using Eq. (10),

ncapl ¼ �Dcoclos� DT

oTos

ð16Þ

For such a solid, capillarity is the primary mode of trans-port for the liquid. In contrast to the positive pressures (ap-plied or gravity) of liquid in a saturated solid, the differencein the negative pressures (due to differences in liquid con-centration, cl or temperature, T) drives the flow in anunsaturated solid.

2.4. Reynolds number in porous media

In an unconsolidated porous medium such as storedgrain, often the mean grain diameter is used as the lengthdimension. Using such a dimension, Darcy’s law is consid-ered valid (Bear, 1972) as long as the Reynolds numberbased on average grain diameter does not exceed somevalue between 1 and 10:

Re < 1 � 10 for Darcy’s law to be valid ð17ÞIn the wide range of food applications, Re can cover a sig-nificant range. For example, for deep bed forced air dryingof grains in a bin, Re based on grain diameter (�0.004 m)can be around 20 for flow rates in the range 0.035–0.045 m3/s (Istadi & Sitompul, 2002). On the other hand,in microwave heating of a vegetable tissue such as a potato(Ni, 1997), where the pore structure is not at all well devel-oped, Reynolds number (Re ¼ uav

ffiffiffikp

q=l) for the flow gen-erated by the internal vapor formation is of the order of10�5.


3. Obtaining velocity: Navier–Stokes analog of Darcy

equation

The Darcy’s law for porous media, given by Eq. (3) orits equivalent,

rP ¼ � lk

u ð18Þ

is solved to obtain the velocity, u, in the porous mediumthat represents convection. Depending on the physical sit-uation, a more general form of Eq. (18) that has the formof the Navier–Stokes equation, is solved, the origin ofwhich is now explained (Ingham & Pop, 1998). Addition-ally, because of the similarity with the Navier–Stokes equa-tion, this more general form is also used at times in manyCFD solvers to obtain velocity in a porous medium evenfor scenarios where the physical situation does not requireit. Eq. (18) has been generalized over the years. Dupuit’s(also known as Forscheimer and Ward’s) modification ofthe Darcy’s law leads to

rP ¼ � lk

u� ck1=2

qu2|fflfflfflffl{zfflfflfflffl}additional

ð19Þ

where c is an empirical constant that is obtained fromexperiment. The combined effect of c and k can be obtainedexperimentally, e.g., in Hoang, Verboven, Baelmans, andNicolai (2003), as discussed below. The additional termrepresents the non-linearity in the pressure drop-velocityrelationship and is assumed due to the form force imposedon a fluid by any solid surface obstructing the flow path.

Brinkman’s modification leads to

rð/P 0Þ ¼ � lk

/u0 � ck1=2

q/2ju0ju0 þ ler2u0|fflfflffl{zfflfflffl}viscous analogy

ð20Þ

The last term represents the effect of viscous shear stressanalogy with the Navier–Stokes equation. Note that P 0

and u 0 are force and velocity, respectively, per unit fluidarea, so that u = u 0/ and P = P 0/. The term le is not justfluid viscosity, but an effective viscosity that is a function offluid viscosity and geometry of the medium. For example,in Khaled and Vafai (2003), le is defined as le = ls, wheres is tortuosity of the porous medium. By including the iner-tia and transient effects, the equation becomes

qDu0

Dt¼ �rð/P 0Þ � l

k/u0 � c

k1=2q/2ju0ju0 þ ler2u0 ð21Þ

qou0

ot|{z}Transient

þ u0ru0|fflffl{zfflffl}Inertia

2664

3775 ¼ �rð/P 0Þ|fflfflfflffl{zfflfflfflffl}

Pressure

� lk

/u0 � ck1=2

q/2ju0ju0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}Darcy and Forscheimer resistance

þler2u0|fflfflffl{zfflfflffl}Viscous

ð22Þ

The inertial term u 0$u 0 was shown (Vafai & Tien, 1981) tobe negligible except for highly permeable media. This sim-plification leads to Eq. (23) for the velocity of fluid in theporous medium that is the Navier–Stokes analog of Darcyflow

q/

ouot¼ �rP � l

ku� c

k1=2qjujuþ le

/r2u ð23Þ

Here the variables u 0 and P 0 have been changed back to uand P, respectively, that are the usual continuum basedquantities, based on total area instead of pore area. Eq.(23) can be readily implemented in a software that solvesthe Navier–Stokes equation. For example, it can be imple-mented in the software FIDAP (Fluent, Inc., Lebanon,NH, USA) directly and in the software CFX (Ansys,Inc., Canonsburg, PA, USA) we are given a choice to in-clude the above equation as

q/

ouot¼ �rP � Ruþ le

/r2u ð24Þ

where R(u) is a resistance term that includes the Darcy andits non-linear effects, and is obtained from experiment (e.g.,see Ergun, 1952 and Hoang et al., 2003).

4. An overview of problem formulations

The food applications in the literature can be groupedinto various porous media formulations, as shown inFig. 2. In the larger pores, Navier–Stokes analog ofDarcy’s law (Eq. (23)) is used to obtain the velocity inthe convective term of the transport equation whereas insmaller pores, the linear Darcy equation (Eq. (3) or itsequivalent) is used to obtain the velocity. In the latter case,due to its simplicity, Darcy’s law is substituted directly forthe velocity in the convective term. For the small pores,problem formulation can be further divided into strongevaporation where evaporation produces significantlyincreased gas pressure and contributes to the Darcy flow,whereas weak evaporation does not lead to gas pressuredriven Darcy flow. The strong evaporation problems havebeen treated using either a distributed evaporation formu-lation where the evaporation is distributed throughout thematrix or they have been treated using an evaporationfront where the evaporation occurs at a sharp front andthe matrix is divided into two parts, one where evaporationhas already occurred and the material is dried and the otherbeing the rest of the material. For the weak evaporation sit-uation, vapor transport is ignored and liquid transport bycapillarity can be formulated in two equivalent ways, oneusing capillary pressure and the other using a capillary dif-fusivity. Finally, many empirical formulations exist in foodand their relationship to the more rigorous problem formu-lations are discussed.

5. Problem formulation in systems with large pores

Examples of problem formulation in large pores arecooling of stacked bulk produce such as spherical productsand potato (Xu & Burfoot, 1999; Xu, Burfoot, & Huxtable,2002), chicory roots (Hoang et al., 2003) and Pears (Nahor,Hoang, Verboven, Baelmans, & Nicolai, 2005). TheNavier–Stokes analog of the Darcy equation (Eq. (23) orits more general form given by Eq. (22)), together with spe-cies transport and energy equations, are used in these stud-ies. Internal gradient in temperature and moisture in the


solid phase was considered in Xu and Burfoot (1999) andXu et al. (2002) and ignored in Hoang et al. (2003). Asan example of an application, consider the work of Hoanget al. (2003) for cooling of chicory roots in bulk, as picturedon the left of Fig. 1. The authors considered the tempera-tures of the solid (chicory root) phase and the air phaseat a spatial location to be different and thus used twoenergy equations for the solid phase temperature, Ts, andthe air phase temperature, Ta (Hoang et al., 2003).

qao

otð/haÞ þ qa

o

oxið/uihaÞ|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

convective transport in air

¼ ooxi

/kaoT aoxi

� �|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}

conduction in air

þ hAspecðT s � T aÞ|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}convective exchange between air and product

ð25Þ

(a)

(b)

Fig. 3. Typical result showing air temperature (a) and product temper-ature (b) from the work of Hoang et al. (2003).

qso

otðð1� /ÞhsÞ ¼

o

oxið1� /Þks

oT soxi

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

conduction in product

þhAspecðT a � T sÞ

þ ð1� /Þ _q|fflfflfflfflffl{zfflfflfflfflffl}respirative heat generation

þ khmAspecðqvs � qvaÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}evaporative heat loss

ð26Þ

Here h stands for enthalpy, Aspec is specific area (m2/m3)

and the subscript i stands for the dimensions in the carte-sian coordinate system. The velocities in the above equa-tions are obtained by solving Eq. (24). Diffusion of vaporin air was considered using an additional species equation.The exchange of energy between the product and air phasesis considered through a heat transfer coefficient, leading toa source term in one equation and a sink term of equalmagnitude in the other equation. For moisture transport,diffusion within the product is ignored and moisture lossfrom the surface is calculated using a mass transfer coeffi-cient and the difference between vapor pressure at the sur-face and that far from the surface being the driving force.This rate of moisture loss from the product is used as asource term for moisture in the air equation. Representa-tive result from this study is shown in Fig. 3.

6. Problem formulation in systems with small pores: Strongevaporation

This formulation covers the vast majority of food pro-cessing situations when our interest is transport within aplant or animal tissue or a structured food material. Excel-lent reviews of heat and moisture transfer in porous mediain the context of foods have appeared in the past (Bruin &Luyben, 1980; Fortes & Okos, 1980). For our discussion, aporous medium can be conceptually viewed as shown inFig. 4, with the various phases and the associated modesof transport. This representation would be used in themechanistic models described below.

6.1. Phenomenological models

One of the well known theories for porous media trans-port is the work of Luikov (1975). In this theory, the mac-roscopic heat and mass transfer governing equations arederived based on the phenomenological theory of non-equilibrium thermodynamics. Non-equilibrium thermody-namics is a term coined several decades ago for studies con-cerned with time-dependent thermodynamic systems, suchas multiphase transport being discussed in this paper.The word phenomenological (as opposed to mechanistic,used in the next section) refers to type of model where anobserved phenomena is taken as the starting point andmodels are formulated to extract the essential features ofthe phenomena. By contrast, mechanistic models oftenhave fundamental laws as the starting point. In the workof Luikov (1975), by choosing temperature, moisture

Fig. 4. Schematic of an unsaturated porous medium showing the phasesof water, vapor, air and the solid matrix, together with the associatedmodes of transport.


content and gas pressure as primary variables, the finalequations were written as

oTot¼ K11r2T þ K12r2M þ K13r2P

oMot¼ K21r2T þ K22r2M þ K23r2P

oPot¼ K31r2T þ K32r2M þ K33r2P ð27Þ

This model includes convective flow of gas as well as cap-illary flow of liquid. Three simple parallel final equationsare very favorable to the analytical solution in some simplecases. However, the model has several disadvantages. Mostimportantly, physical interpretation of the parameters isnot clear. This is because all flux expressions are basedon phenomenological relationships. Another importantdrawback is the use of a constant phase conversion factorthat is a ratio of water transport in the vapor phase towater transport in the liquid phase. Although phase con-version factor provides solution with some simplicity, itsassumed value really makes the solution semi-empirical.Furthermore, gas diffusion is not described in the equationsas diffusive transport and liquid bulk flow is not describedas convective transport, at least in any explicit manner. Re-cent works used the theory to calculate the relative effectsof various parameters on the solution. It is fair to say thatthis theory is used less than the mechanistic models pre-sented below.

6.2. Mechanistic models: Distributed evaporation

formulation

In contrast to a phenomenological model (e.g., Luikov,1975), mechanistic models of heat and mass transfer equa-tions for porous media have been developed, perhaps the

most elaborate work being that of Whitaker (1977). InWhitaker (1977), starting from conservation equationsfor heat and mass for each phase (solid, liquid, gas plusvapor), different phases are volume averaged. Althoughthe final equations look like nothing but simple continuityand flux equations, the rigorous study for the transitionfrom the individual phase at the ‘microscopic’ level to rep-resentative average volume at the ‘macroscopic’ level pro-vides the fundamental and convincing basis. Thus therelationship between the porous medium and its fictitiousequivalent continuum is clearly shown in this mechanisticformulation. The big advantage of this (and other) mecha-nistic model is that the physics of the model is better under-stood, the assumptions are very clear and the parametersare well defined.

Mechanistic formulations are generally quite similar toeach other. One such mechanistic formulation is discussedhere in detail from the work of Ni, Datta, and Torrance(1999) that has been used to study conventional heating,microwave heating, baking (with and without volumechange) and deep frying (see also Yamsaengsung & More-ira, 2002). The rate laws for the transport are first devel-oped following the schematic in Fig. 4. The total flux ofvapor, ~nv, and that of air, ~na, are composed of convectiveor Darcy flow (Bear, 1972) and diffusion (Bird, Stewart,& Lightfoot, 1960), respectively:

~nv ¼ �qvkglgrP

|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}flux of vapor

due to pressure

�C2gqg

MaMvDeff ; grðpv=P Þ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}flux of vapor due to

binary diffusion in air

ð28Þ

~na ¼ �qakglgrP �

C2gqg

MaMvDeff ; grððP � pvÞ=P Þ ð29Þ

The total liquid flux is written from Eq. (10) as

~nw ¼ �qwkwlwrP � Dwqw/rSw � DTrT ð30Þ

Note that pv/P and (P � pv)/P in the vapor and air equa-tions above are the mole fractions of vapor and air, respec-tively. The permeabilities in the above equations are givenby kg = kkgr and kw = kwr, respectively, where kgr is the rel-ative permeability of gas and kwr is the relative permeabil-ity of water. The water saturation, Sw, is defined in terms ofthe fraction of the pore volume occupied by the water ascw = qw/Sw. Note that / here is apparent porosity that rep-resents combined volume fraction of liquid water and gasesas compared to the total volume. The conservation equa-tions for vapor, liquid water, air, and energy in the porousmedium are written, respectively, as

ocvotþr � ð~nvÞ ¼ _I ð31Þ

ocwotþr � ð~nwÞ ¼ �_I ð32Þ

ocaotþr � ð~naÞ ¼ 0 ð33Þ


ðqcpÞeffoTotþr�ð~nvhvþ~nahaþ~nwhwÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

convective energy transport

¼r�ðkeffrT Þ|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl}conduction

� k_I|{z}evaporation

þ _q|{z}heat source

ð34Þ

where _I is the spatially distributed volumetric rate of evap-oration (units of kg/m3 s) that can also vary with time. Thequantities hv = cpvDT, ha = cpaDT and hw = cpwDT areenthalpies of vapor, air and water, respectively, with cpsas specific heats of the respective phases. The effectiveproperties are defined as

ðqcpÞeff ¼ qscpsð1� /Þ þ qwcpw/Sw þ qgcpg/ð1� SwÞ ð35Þkeff ¼ ksð1� /Þ þ kw/Sw þ kg/ð1� SwÞ ð36Þ

The set of equations are often solved by transforming theconcentrations in terms of saturations. Concentrationsare related to Sw by cv = pv(1 � Sw)/Mv/RT,ca = (P � pv)(1 � Sw)/Ma/RT and cw = qw/Sw. After thetransformation to saturation, the unknowns in Eqs. (31)–(34) are P, Sw, T, _I and pv, which together is one more un-known than the number of equations.

6.2.1. Closure: Equilibrium vs. non-equilibrium description

of evaporation

The closure to the set of equations (Eqs. (31)–(34)) isprovided by using the additional equation describingsolid-water vapor equilibrium relation for the particularfood material

pv ¼ pvðSw; T Þ ð37Þthat is obtained from experiment (see companion paper byDatta, 2006). Almost all published work on evaporation inporous media have used Eq. (37). However, in a recentimplementation of Eqs. (31)–(34) for bread baking process(Zhang, Datta, & Mukherjee, 2005), it is noted that the ac-tual vapor pressure in a fairly rapid process such as bakingis likely to be lower than that given by Eq. (37). See alsodiscussion in Zhang and Datta (2004). This lower vaporpressure is due to the fact that the time to reach equilib-rium, that is normally assumed in experimentation leadingto Eq. (37), is much longer than in a rapid process such asbaking. While this needs further study and work on simplersystems has been reported (see Fang & Ward, 1999), a non-equilibrium approach to evaporation has been used in oneporous media model (Bixler, 1985), where the evaporationrate is given by

_I ¼ Cðpv;eqm � pvÞ ð38Þ

where C is an empirical constant and pv,eqm is the equilib-rium vapor pressure given by Eq. (37). A large value ofC would force the system to be close to equilibrium, i.e.,pv � pv,eqm. This description (Eq. (38)) of evaporation isan alternative to Eq. (37). Either Eq. (37) or Eq. (38) canbe used for closure to the set of governing transport equa-tions (Eqs. (31)–(34)). Further work investigating this non-equilibrium formulation for evaporation is underway inauthor’s research group.

The above two sets of general equations (Eq. (27) or(31)–(34)) have set the starting point for most literature

studies on modeling of drying in porous media. The mech-anistic model (Eqs. (31)–(34)) has been developed also inother materials processing context, such as sandstone ora generic porous media (Wei, Davis, Davis, & Gordon,1985), wood (Turner & Mujumdar, 1996) and concrete(Constant, Moyne, & Perre, 1996). Other researchers havedeveloped heat and mass transfer equations for porousmedia starting from conservation equations and mechanis-tic flux models (Chen & Pei, 1989; Ilic & Turner, 1989; Nas-rallah & Perre, 1988; Stanish, Schajer, & Kayihan, 1986).Their equations have similarity with Whitaker’s (1977)and will also be referred to as mechanistic models hereafter.Although they do not always discuss the phase averagingexplicitly, many of the steps and assumptions in Whitaker’smodel are implied in these formulations. Representativeresults from this model are deferred to the companionpaper (Datta, 2006).

6.3. Mechanistic models: Sharp interface formulation

In this type of formulation for problems involving evap-oration, a sharp interface is assumed where all the evapora-tion takes place. The interface divides the entire region intotwo parts—one containing water (or ice) while the othercontaining no water or the dry region. These formulationsare also called front-tracking or interface tracking or mov-ing boundary problem where the location of the sharpinterface moves with time and is tracked by using a bound-ary condition of heat or mass balance at the interface. Asharp boundary may be a reasonable approximation whosevalidity would depend on the specific application. Thisapproximation has been used to model several processesincluding drying, freeze drying and frying (Farid, 2002).

Many variations of sharp interface formulations exist inthe literature, having various degrees of simplification. Anexample of sharp interface formulation is provided herefrom the work of Farkas, Singh, and Rumsey (1996) fora frying problem where the 1D computational domain isdivided into two regions, core (the inner portion or theregion I) and the crust (the outer portion in contact withoil or region II). The heat and mass transfer in the coreare described by, respectively,

ðqcpÞeffoTotþ �Dw

ocwox

� �cpw

oTox|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

energy convected by liquid

¼ kIIeffo2Tox2|fflfflffl{zfflfflffl}

conduction in solid and liquid

ð39Þ

ocwot¼ Dw

o2cwox2

ð40Þ

where the subscript eff stands for effective properties. Thediffusivity, Dw, in the above equations is, strictly speaking,capillary diffusivity of water due to water concentrationgradient. In practice, however, capillary diffusivity data isnot available and it is estimated from available effective dif-fusivity data. From these two equations, temperature, Tand concentration of water, cw, in the core region are ob-tained. In the crust, energy transport is due to conduction


and vapor transport, and the vapor transport is entirelydue to pressure driven flow. The heat and mass transferequations are given by, respectively,

ðqcpÞeffoTotþ �qv

kvlv

opvox

� �cpv

oTox|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

energy convected by vapor

¼ kIeffo2Tox2|fflfflffl{zfflfflffl}

conduction in solid and vapor

ð41Þ

o

ox�qv

kvlv

opvox

� �¼ 0 ð42Þ

From these two equations, temperature, T, and partialpressure of vapor, pv, in the crust region are obtained. Atthe crust-core interface, X, all the evaporation is assumedto be concentrated. Thus, the movement of this interface(that divides the entire region into the crust and the core)is related to the rate of evaporation. Energy balance at thisinterface relates the heat fluxes to the rate of evaporation:

�kIeffoTox|fflfflfflffl{zfflfflfflffl}

heat flux in core

� �kIIeffoToxþ qvkv

lv

opvoxðhw � hvÞ

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

heat flux in crust

¼ dXdtð1� /Þqs hIs � hIIs

� �þ /qvðhv � hwÞ

� |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

energy change due to interface movement

ð43Þ

Eq. (43) provides the location of the interface, X, whichcompletes the solution for the two region problem. It isimportant to note how the simplifications in the modeldeveloped here was guided by improved physical under-standing from preliminary experimentation.

6.4. Making various assumptions in mechanistic models

Heat and mass transfer in porous media are complex.Several assumptions are often necessary to make the prob-lem tractable. It is often difficult to make some assump-tions a-priori, i.e., before the physics of the problem isunderstood in detail. To get around this, sometimes thebest way to decide if a certain assumption can be made isto perform sensitivity analysis with respect to that assump-tion, i.e., range of values for a particular parameter, or thepresence or absence of a term in the governing equation.

In a food with large water content, capillary diffusion ofliquid water can be the dominant mechanism for moisturetransport and molecular diffusion of vapor may not con-tribute as much. In such a situation, it may be possible toignore molecular diffusion. If significant internal evapora-tion and pressure generation take place, as in microwaveheating, pressure driven or Darcy flow becomes quiteimportant (Ni et al., 1999). In a somewhat dried material,when liquid phase is not continuous, transport is primarilyin the vapor phase following the mechanisms of pressuredriven flow and molecular diffusion (Perre, Nasrallah, &Arnaud, 1986) and it may be possible to ignore capillarydiffusion. Thus, different mechanisms can be dominant atdifferent stages of processing and separate models can bedeveloped for the stages.

Transport mechanisms such as Soret effect (mass fluxdue to temperature gradient) and Dufour effect (heat fluxdue to concentration gradient) are generally consideredsmall (Itaya, Kobayashi, & Hayakawa, 1995) in compari-son with the mechanisms mentioned in Section 2, and aregenerally not used. Property data to consider Soret orDufour effects are also generally unavailable. Gravity isignored in most studies. Perhaps this can be justified fromthe fact that the food material is generally unsaturatedwhere capillary forces are much stronger than gravity.

An important assumption made in almost all multiphaseporous media studies is that the solid, liquid and gas(vapor + air) at any location are in thermal equilibrium,so there is only one temperature at any given location. Thisassumption has been justified and used for even very rapidheating as in microwaves (Ni, 1997). Additionally, byassuming local phase equilibrium, isotherm relating vaporpressure above a solid to its moisture and temperature isused for the closure of the set of governing equations.

Properties are generally considered isotropic due to thelack of detailed information. Most foods experience shrink-age (or swelling) during processing that is coupled withheat and mass transfer. To avoid the computational com-plexities of a coupled model, shrinkage (see section onshrinkage later) is generally ignored except in studies wherethe goal is to compute the stress development and cracking(e.g., Itaya et al., 1995) as opposed to moisture and heattransport only.

7. Problem formulation in systems with small pores: Weakevaporation

In this section we discuss simplifications of formulationspresented in the previous section, for the special case whenevaporation is small and contributes to insignificantamount of gas pressure driven flow. It is possible todevelop a formulation having multiphase transport thatuses a simplified version of the liquid, vapor, air and energyequations (Eqs. (31)–(34)), but without the term containingpressure driven flow. This formulation would be simplerand may describe well processes where pressure driven flowis not as significant, such as in lower temperature drying.Such a model may have been used in the literature, but isnot very common, and will not be elaborated here. Instead,this section considers situations when capillarity is the pri-mary mode of species transport and the process is nearlyisothermal so that the energy equation can be ignored.

7.1. Capillary pressure formulation

For most of the food processing situations, flow isunsaturated. If pressure development from evaporation issmall and no other applied pressures are present, capillarypressure dominates and the driving force for transport isthe spatial variation in capillary pressure (due to variationin degree of saturation). In unsaturated flow, when evapo-ration and flow in vapor form are insignificant and spatial

Fig. 5. Capillary pressure and moisture content in the bread and meatregions of a sandwich during storage (Geedipalli & Datta, 2002).


variations in temperature can be ignored, Eqs. (31)–(34)simplifies to only the water transport equation (Eq. (32)).

ocwotþr � �qw

kwlwrðP � pcÞ

� �¼ �_I ð44Þ

which can be written following Eq. (15) for pc > >P (capil-lary pressure is large) and _I ¼ 0 (no significant evaporationat lower temperature)

ocwotþr � qw

kwlwrpc

� �¼ 0 ð45Þ

where pc is the capillary pressure. In this form of the trans-port equation for a porous medium, both water content cwand capillary pressure, pc, are present. This equation is alsoknown as the Richard’s equation. Note that gravity hasbeen ignored here, which is a reasonable assumption infairly unsaturated flows, as in the case of water movementin drying. In porous media literature, instead of capillarypressure, capillary head, h, is often used. Using this termi-nology, Eq. (45) can be rewritten as

ocwotþr � qw

kwqwglwrh

� �¼ 0 ð46Þ

By transforming to the single variable h, this equation canbe written as

ocwoh

� �ohotþr � qw

kwqwglw

rh� �

¼ 0 ð47Þ

where ocw/oh is the specific moisture capacity. Inputparameters, mainly the relationship between capillary pres-sure and moisture content has been discussed in the com-panion paper (Datta, 2006).

This formulation has been used only recently in foodcontext. Washing of solutes in a packed bed of tea leaf inteabag infusion is studied using this porous medium formu-lation (Lian & Astill, 2002). Rehydration of tea leaves wasstudied (Weerts, Lian, & Martin, 2003a; Weerts, Lian, &Martin, 2003b) that included the effect of anisotropy ofthe microstructure and temperature. Study of rehydrationin other materials using similar formulation has also beenproposed by Saguy, Marabi, and Wallach (2005). Moisturetransport in multidomain foods such as a sandwich (breadand meat being the two domains) was studied using thisformulation (Geedipalli & Datta, 2002). Capillary pressureformulation for the sandwich problem is conceptually morenatural in comparison to effective diffusivity formulationdiscussed in the following section because of multiple mate-rials (domains) involved that have different capillary pres-sure variation with moisture content. Here, moisture isnot the driving force, rather water activity (that relates tocapillary pressure) is the driving force. Thus, at equilib-rium, the water activity (or capillary pressure) will be thesame throughout the two domains, rather than the mois-ture, as can be seen in Fig. 5. This physics is harder toimplement using an effective diffusivity formulation(described next section), although, moisture transport ina composite food, similar to the sandwich problem above,

has been studied using an effective diffusivity formulation(Guillard, Broyart, Bonazzi, Guilbert, & Gontard, 2003).A variation of the capillary pressure formulation, theLucas Washburn equation, has been used to model the cap-illary rise in rehydration of food (Lee, Farid, & Nguang,2006).

7.2. Capillary diffusivity formulation

Using discussion in Section 7, the capillary pressure for-mulation just discussed can be written as an equivalent for-mulation in terms of a capillary diffusivity of liquid waterthat resembles the commonly used diffusion equation.Thus, Eq. (45) can be written following Eq. (16) as

ocwot�r � ðDwrcwÞ ¼ 0 ð48Þ


where Dw stands for the capillary diffusivity of water givenby

Dw ¼ �q2wgkwlw

ohocw

ð49Þ

and DT is ignored. Eq. (48) is the familiar form of the dif-fusion equation describing moisture transport. Note, how-ever, that this equation is derived for liquid transport bycapillarity. Its extension to situations where combined li-quid and vapor transport is present, as is commonly used,is far from obvious, as is explained in the next section.

8. Problem formulations in small pores: Simplified and

semi-empirical

The simplified formulations in the previous section stillshowed clear relationship to the original conservationequations (Eqs. (31)–(34)) for multiple phases. In practice,formulations have been used, not all of which can beclearly traced back to the fundamental conservation equa-tions. These will be referred to as empirical formulationsand discussed in this section.

8.1. Effective diffusivity formulation

The combined process described by Eqs. (31) and (32)may be conceptually written as (Zhang & Datta, 2004):

oMot¼ r � ðDeffrMÞ ð50Þ

where M (=cv + cw) now represents total moisture. Follow-ing analogy to Eqs. (31) and (32), M is in terms of mass ofwater (including vapor) per unit volume of the food mate-rial, however, if the volume stays fixed (no shrinkage orswelling), M can also be replaced by mass of water (includ-ing vapor) per unit mass of dry solids, i.e., moisture contenton dry basis. Generally, the mass content of vapor wouldbe negligible.

The diffusivity, Deff, represents some sort of effective dif-fusivity for the combined transport of liquid and vapor.Since Eq. (50) for the two combined phases cannot bederived from fundamental laws governing species massconservation that are based on well known transport mech-anisms, the effective diffusivity in this equation is somewhatof an empirical factor. The effective diffusivity formulationlumps the different transport mechanisms such as capillaryflow and pressure driven flow, as well as all other possiblemechanisms. It is not easy to relate Deff to capillary andvapor diffusivity. In almost all studies, the value of effectivediffusivity Deff is actually obtained from fitting experimen-tal data to the solutions of the same equation (Eq. (48)). Allpossible food processes, including microwave heating,microwave drying, and combined microwave and conven-tional drying have been modeled using an effective diffusiv-ity value Deff (see, e.g., Shivhare, Raghavan, & Bosisio,1994). In microwave heating, however, the internal absorp-tion of energy and internal evaporation can be quite signif-

icant, leading to pressure driven flow which is not capturedin an effective diffusivity, Deff.

8.2. When temperatures are needed

Use of Eq. (50) does not require the knowledge of tem-peratures, i.e., solution of the energy equation. Thus, iftemperatures are not of interest, Eq. (50) is used to describemoisture transport. If temperatures are of interest, Eq. (34)can be rewritten as

ðqcpÞeffoTot¼ rðkeffrT Þ � k_I ð51Þ

Here convective energy transport through the porous med-ium has been ignored. In the application of microwave hear-ing, for example, it was shown (Ni, 1997) that the convectiveenergy transport is not particularly significant. Internalheat generation term has also been ignored from the aboveequation, one reasoning being that strong internal evapora-tion can produce strong pressure driven flow that is beingignored. This formulation (Eqs. (50) and (51)) is commonin food literature. Although the evaporation term, _I , in thisequation stands for distributed evaporation throughout thedomain, it is obtained or estimated in a number of ways inpractice that do not make it clear whether the evaporation isat the surface or distributed throughout:

8.2.1. Obtained from experiment

The term _I is found experimentally, from weight lossmeasurements with time. This measured value is substi-tuted in Eq. (51).

8.2.2. Obtained using a surface mass transfer coefficient

In some studies, rate of moisture transport from the sur-face is estimated using a mass transfer coefficient as

nA ¼ hmAðpv; surface � pv;1Þ ð52Þ

and this amount is distributed volumetrically, to obtain thevolumetric rate of evaporation as

_I ¼ nAV

ð53Þ

where V is the volume of the sample. Sometimes, the evap-oration term, _I , is treated in ways that are questionable(e.g., see discussion in Zhang & Datta, 2004).

9. An example of models that cannot be justified from

physical considerations

In the final section on models of simultaneous heat andmass transfer, the author felt compelled to provide exampleof a model that could not be justified from physical consid-erations. Many heat and mass transfer studies in food haveused the following two equations for total moisture andtemperature, respectively:

oMot¼ o

oxDeff

oMox

� �ð54Þ


qcpoTot¼ o

oxkeff

oTox

� �þ k oM

otð55Þ

In Eq. (54), the term oM/ot represents the transient term inthe diffusion equation that stands for the rate of change ofmoisture at a location due to diffusion. However, in Eq.(55), the term oM/ot stands for evaporation rate! It wasshown in Zhang and Datta (2004) that this formulationdoes not satisfy the mass conservation and the use of thismodel is not recommended.

10. Presence of shrinkage or swelling

Heat and mass transfer in a porous medium and shrink-age (or swelling) of the medium are often coupled. Themost common example of this is the shrinking of a foodmaterial being dried (Mayor & Sereno, 2004). Althoughthe deformation (shrinkage or swelling) and transport arecoupled in any real process, depending on the situation,they can be modeled in a number of ways—(1) transportproblem is solved with a prescribed deformation, i.e., cou-pling is one way; (2) transport problem is solved on a fixedmatrix and the resulting information is used to do themechanics problem, again a one-way coupling; (3) two-way coupling where the deformation strongly affects thetransport and vice versa. We will discuss the items 1 and3 below. Item 2 is often studied in the context of predictingcracking during drying of food materials (e.g., Haghighi &Segerlind, 1988; Itaya et al., 1995). Here transport is con-sidered to be not affected and therefore will be consideredoutside the scope of this manuscript.

10.1. Simple models: One-way coupling

This relates to item 1 in the previous paragraph, whendeformation is prescribed. Simple models to consider theinfluence of shrinkage to the heat and mass transfer gener-ally use pre-determined relationships of dimensionalchanges with moisture and use it to modify the geometrywith time in the simulation. For example in Ketelaars,Kaasschieter, Coumans, and Kerkhof (1994), a shrinkagecoefficient that is the ratio of density at any moisture con-tent to the initial density has been used in the moisture dif-fusion equation. The relationship

LL0¼

qw þ qapp;sMdqw þ qapp;sMd;0

ð56Þ

has been used (Hawlader, Ho, & Qing, 1999) to calculatethe new length L at an average moisture content (dry basis)of Md, where subscript 0 denotes the initial condition.Shrinkage at any spatial location was calculated using theshrinkage velocity u(x)

uðxÞ ¼ uðLÞ xL

�nð57Þ

with n = 1 for a linear shrinkage model and n = 2 for non-linear shrinkage. Accuracy of the prediction depends lar-gely on that of the assumed shrinkage model. Such simple

models should be readily implementable in commercialsoftware that can simulate transport processes with a mov-ing boundary.

10.2. More complex models: Two-way coupling

This refers to the two-way coupling mentioned earlier.More complex models that include the deformation effectduring shrinkage or swelling have been reported that areof consequence to food applications (Achanta & Okos,1996; Hasatani & Itaya, 1996; Singh, Cushman, & Maier,2003; Zhang & Datta, 2004). A number of generalized for-mulations have been reported in the recent years (e.g.,Schrefler, 2002; Singh et al., 2003). Schrefler (2002, 2004)presents a rigorous general thermodynamic formulationwhere the applications are presented primarily in the areaof soil mechanics for small deformation of the solid phaseand using elastoplastic constitutive laws. Singh et al. (2003)proposed a multiscale theory for flow through viscoelasticbiopolymers. Their later work (e.g., Singh, Maier, Cush-man, Haghighi, & Corvalan, 2004; Singh, Maier, Cush-man, & Campanella, 2004) applied the theory to moisturetransport during drying and imbibition of soybeans. InZhang and Datta (2004), a multiphase porous mediummodel was fully coupled with deformation, i.e., effect oftransport on deformation as well as the effect of deforma-tion on transport are both included. Results for this model,as applied to bread baking, is discussed in the companionpaper (Datta, 2006).

11. Summary

Relationship between various models used to studysimultaneous heat and mass transfer in food processes isclearly shown here, starting from the most elaborate multi-phase porous medium model that includes evaporation andgoing down in complexity to the simplest equation of iso-thermal diffusion. The fundamental transport modes ofmolecular diffusion, capillary diffusion and pressure drivenDarcy flow are clearly shown in the detailed models. Therelationship of a simple model to the detailed models, suchas when using effective diffusivity, is discussed. A widelyused model that is inconsistent, i.e., does not satisfy conser-vation equations, is also noted. Two types of heat and masstransfer formulations are covered, one involving distrib-uted evaporation and the other involving a sharp movinginterface where evaporation occurs. Knowledge of theinterrelationship between models should improve our fun-damental understanding of food processes and complementexperimental work toward better optimization of the cur-rent processes and inventing new ones.

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Porous media approaches to studying simultaneous heat and mass transfer in food processes. I: Problem formulationsIntroductionPorous and capillary-porous materialsHygroscopic vs. non-hygroscopic materialsModeling porous media as a continuumOrganization of this paper and its companion

Various transport mechanisms in porous mediaMolecular diffusionDarcy flow of gases due to gas pressureDarcy flow of liquid due to gas and capillary pressuresSpecial case: Porous media is nearly saturatedSpecial case: Porous media is highly unsaturated

Reynolds number in porous media

Obtaining velocity: Navier-Stokes analog of Darcy equationAn overview of problem formulationsProblem formulation in systems with large poresProblem formulation in systems with small pores: Strong evaporationPhenomenological modelsMechanistic models: Distributed evaporation formulationClosure: Equilibrium vs. non-equilibrium description�of evaporation

Mechanistic models: Sharp interface formulationMaking various assumptions in mechanistic models

Problem formulation in systems with small pores: Weak evaporationCapillary pressure formulationCapillary diffusivity formulation

Problem formulations in small pores: Simplified and �semi-empiricalEffective diffusivity formulationWhen temperatures are neededObtained from experimentObtained using a surface mass transfer coefficient

An example of models that cannot be justified from physical considerationsPresence of shrinkage or swellingSimple models: One-way couplingMore complex models: Two-way coupling

SummaryReferences

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