Transport In Deform Able Food Materials

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Transport in deformable food materials: A poromechanics approach

Ashish Dhall a,1, Ashim K. Datta b,n

a Department of Biological and Environmental Engineering, Cornell University, 175 Riley-Robb Hall, Ithaca, NY 14853, United Statesb Department of Biological and Environmental Engineering, Cornell University, 208 Riley Robb Hall, Ithaca, NY 14853, United States

a r t i c l e i n f o

Article history:

Received 17 February 2011

Received in revised form

26 August 2011Accepted 1 September 2011Available online 16 September 2011

Keywords:

Mathematical modeling

Food processing

Porous media

Solid mechanics

Pressure

Shrinkage

a b s t r a c t

A comprehensive poromechanics-based modeling framework that can be used to model transport and

deformation in food materials under a variety of processing conditions and states (rubbery or glassy)

has been developed. Simplifications to the model equations have been developed, based on driving

forces for deformation (moisture change and gas pressure development) and on the state of food

material for transport. The framework is applied to two completely different food processes (contact

heating of hamburger patties and drying of potatoes). The modeling framework is implemented using

total Lagrangian mesh for solid momentum balance and Eulerian mesh for transport equations, and

validated using experimental data. Transport in liquid phase dominates for both the processes, with

hamburger patty shrinking with moisture loss for all moisture contents, while shrinkage in potato stops

below a critical moisture content.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Factors affecting food safety (presence of pathogens andtoxins) and food quality (porosity, pore size distribution, texture,

and color) are functions of the state (temperature, moisture, and

composition) of the food material and its processing history.

Fundamentals-based understanding of physics of food processing

can help a long way in predicting the state and the history of a

food material and, thus, its safety and quality. The underlying

physics of many food processes, such as drying, rehydration

(soaking), frying, baking, grilling, puffing and cooking, is essen-

tially energy and moisture transport in a deforming porous

medium (Datta, 2007). Although transport in deformable porous

media has been extensively studied for non-food applications

such as geomaterials (soils, rocks, concrete, and ceramics),

biomaterials (plant and animal tissues), gels and polymers, still the

combination of specific characteristics (softness, hygroscopicity andphase transitions) and processing conditions of food materials result

in unique complexities that have rarely been studied.

The general mathematical framework of deformation in satu-

rated and unsaturated porous media (also known as poromecha-

nics) was developed by Biot (1965). The theory was later

extended to include multiphase transport using theory of mix-

tures by various studies (discussed by Schrefler, 2002). An

alternate approach is volume-averaging, i.e., begin with conserva-

tion equations at the microscale and then use averaging or

macroscopization to obtain relationships at the macroscale(Whitaker, 1977). In both approaches, the constitutive relation-

ships can be written either empirically or by invoking second law

of thermodynamics through entropy inequality (nonequilibrium

thermodynamics). Lewis and Shrefler (1998) provide a detailed

review of the similarities and dissimilarities, and the pros and

cons of these poromechanics theories. Although applied exten-

sively to non-food materials, there are no examples of a compre-

hensive poromechanics-based approach in food science literature.

Majority of the existing transport models in food science

literature are either curve fits of lumped empirical data (Ateba

and Mittal, 1994; Ikediala et al., 1996; Bengtsson et al., 1976;

Chau and Snyder, 1988; Fowler and Bejan, 1991) or, in a slightly

improved version, assume purely conductive heat transfer for

energy and purely diffusive transport for moisture (Dincer andYildiz, 1996; Williams and Mittal, 1999; Shilton et al., 2002;

Wang and Singh, 2004; Kondjoyan et al., 2006), solving a transient

conduction (or diffusion) equation using experimentally deter-

mined effective conductivity (or diffusivity). One notable excep-

tion to lumped analysis is the application of Stefan’s moving

boundary approach to track liquid–vapor interface during internal

vaporization (Farkas et al., 1996; Farid and Chen, 1998; Bouchon

and Pyle, 2005). In this type of modeling, the liquid–vapor inter-

face, where all the vaporization occurs, separates completely

saturated and completely dry regions of a food material. Some

examples of detailed description of transport mechanisms based

on a porous media approach are: inclusion of vaporization

Contents lists available at SciVerse ScienceDirect

journal homepage: www.elsevier.com/locate/ces

Chemical Engineering Science

0009-2509/$- see front matter & 2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ces.2011.09.001

n Corresponding author. Tel.: þ1 607 255 2482; fax: þ1 607 255 4080.

E-mail addresses: [email protected] (A. Dhall), [email protected] (A.K. Datta).1 Tel.: þ1 607 255 2871; fax: þ1 607 255 4080.

Chemical Engineering Science 66 (2011) 6482–6497

http://www.elsevier.com/locate/ces


http://dx.doi.org/10.1016/j.ces.2011.09.001

mailto:[email protected]











generated pressure-driven flow during intensive heating pro-

cesses by Ni and Datta (1999), Halder et al. (2007) and

Yamsaengsung and Moreira (2002); nonequilibrium thermody-

namics based hybrid mixture theory approach towards Case-II

diffusion by Singh (2002) and Achanta (1995); and, more recently,

application of Flory–Rehner theory to predict swelling-pressure

driven moisture transport in meat by van der Sman (2007).

Fundamentals-based description of deformation in food mate-

rials is even less frequent than the detailed description of transport itself. Two different approaches are generally followed:

either the experimental shrinkage data is empirically fitted as a

function of moisture content, or the additivity of volumes of

different components is used to predict deformation from moist-

ure loss data (Mayor and Sereno, 2004; Katekawa and Silva,

2006). Modeling of transport in deformable food materials as a

solid mechanics problem and solving the linear momentum

balance for the solid matrix is rare in food, although this approach

is frequently used to study drying of some other materials such as

wood and ceramics. Notable exceptions are study of hygrostress

cracking (Izumi and Hayakawa, 1995), bread baking (Zhang et al.,

2005) and microwave puffing of potatoes (Rakesh, 2010). For

detailed review of drying models that include shrinkage effects,

including pioneering works by Perre and May (2001), Kowalski

(2000) and others, the reader is referred to the review by

Katekawa and Silva (2006).

With this background, the current study is an attempt to

develop a poromechanics-based modeling framework for the

coupled physics of transport and large deformation in food

materials. The macroscale governing equations are based on

extended Biot’s theory of poromechanics (Lewis and Shrefler,

1998). Classical constitutive laws are used in both mass transport

(Darcy and Fick’s laws) and conduction (Fourier’s law) and

deformation (hyperelastic solid) equations.

2. Mathematical model

A mathematical model is developed that describes deforma-tion and transport (energy and moisture) inside a food material

during thermal processing. First, the physics of deformation of the

solid matrix is described, followed by a discussion on special

cases based on the driving mechanism behind deformation. Later,

transport modeling in a deforming food material and special cases

are described.

2.1. Assumptions

(1) Food is treated as a multiphase porous material, in which

all the phases are in continuum. (2) Local thermal equilibrium is

assumed, i.e., temperature is shared by all the phases. Also,

pressure in the liquid water phase is given as the gas pressure

minus the capillary pressure (or the water potential). (3) The solidskeleton is an incompressible hyperelastic material. Solid volume

remains constant during any process. Biological materials exhibit

non-linear stress–strain behavior, often following rubber and

polymers (Ogden, 1972). For rubber, complex stress–strain beha-

viors are accommodated using strain energy density functions.

Neo-Hookean model is used in the present study, since this model

has been found to fit experimental data for rubber-like materials

for large (30–70%) strains with sufficient accuracy.

2.2. Deformation of the solid matrix: model development for a

general case

Macroscopic total stress tensor, r, at any given location in a

food material can be defined as volumetric average of total stress

tensor, r, in the representative elementary volume (REV) around

the location (Lewis and Shrefler, 1998):

r ¼1

V

Z V

r dV ð1Þ

Now, total volume of an REV can be written as a sum of volumes

of the solid and the fluids present in the pores:

V ¼ V s þXi

V i ð2Þ

Therefore, the total stress tensor can also be written as a sum of

averages in the individual phase volumes:

r ¼1

V

Z V s

r dV þX

i

Z V i

r dV

!

¼V sV

1

V s

Z V s

r dV

þX

i

V iV

1

V i

Z V i

r dV

¼ esrs þX

i

eirið Þ ð3Þ

where ei and ri are, respectively, the volume fraction and the

volume-averaged stress of a phase, i. Given that shear stress is

negligible in fluids, stress in a fluid, ri can be approximated as

ri ¼ À piI ð4Þ

Substituting fluid stresses from Eq. (4) in Eq. (3), we obtain

r ¼ esrsÀX

i

ei pið ÞI

¼ ð1ÀfÞrsÀfX

i

ðS i piÞI

¼ ð1ÀfÞ rs þX

i

ðS i piÞI

!ÀX

i

ðS i piÞI ð5Þ

Defining the first term on the right-hand side of Eq. (5) as the

effective stress on the solid skeleton, r0, and the second term as

pore pressure, p f ð ¼ S g p g þS w pwÞ, the well-known effective stress

principle of Terzaghi is recovered:

r ¼r0À p f I ð6Þ

Now, by invoking the quasi-steady state assumption for deforma-

tion (acceleration term equal to zero), the solid momentum

balance leads to divergence-free field of overall stress:

r Á r ¼ 0 ð7Þ

which implies divergence of effective stress is equal to gradient of

pore pressure:

r Á r0 ¼ r p f ð8Þ

In case of two-phase flow, when the pores are occupied by liquid

water and gas (comprising air and water vapor), the pore

pressure, p f , can also be written as p g ÀS w pc . Inserting this

relationship in the solid momentum balance (Eq. (8)), we obtain

r Á r0 ¼ r p g Àr ðS w pc Þ ð9Þ

where the first term on the right-hand side is the gas pressure

gradient, and the second term is a function of the temperature and

moisture content of the food material. Gas pressure gradients are

significant either for processes involving intensive internal vapor-

ization such as microwave heating (Ni et al., 1999) or for processes

involving gas generation reactions such as carbon dioxide in bread

baking (Zhang et al., 2005). For most other processes, such as drying

and rehydration (soaking), gas is at atmospheric pressure and, thus,

the solid momentum balance reduces to

r Á r0 ¼ Àr ðS w pc Þ ð10Þ

In Eq. (10), capillary pressure, pc , has a physical meaning only when

capillary suction is the only attractive force between the solid

A. Dhall, A.K. Datta / Chemical Engineering Science 66 (2011) 6482–6497 6483



surface and the liquid water. In the presence of other attractive

forces like monolayer surface adsorption, multilayer absorption, etc.,

water potential, Cw, is a more appropriate term. Kelvin’s law is

usually applied to relate water potential, Cw (expressed in units of

pressure) to water activity, aw (Lu and Likos, 2004):

Cw ¼RT

vwlnðawÞ ð11Þ

After replacing pc

byÀCw

, Eq. (10) can be used for liquid water in

the presence of multiple attractive forces. On the other hand, some

high moisture food materials (with water activity, aw % 1), which

undergo a change in their capacity to hold water with temperature

rise, require a different approach for estimation of pressure in liquid

water. van der Sman (2007) applied Flory–Rehner theory to estimate

swelling pressure (equal to pore pressure in the absence of gas

phase) for such materials (more in Section 3.1.3).

2.3. Deformation of the solid matrix: special cases

Usual factors that lead to deformation in food materials are

moisture change (examples include drying and rehydration) and

internal pressure generation (examples include puffing and bread

baking). Between the two, deformation due to moisture change isa complex phenomena and is highly dependent on the state of the

food material. The physics of deformation due to gas transport is

relatively easy as the effect of gas pressure can be easily

expressed as a source term in the solid momentum balance (see

Section 2.3.2).

2.3.1. Processes with moisture change as the driving mechanism

Most wet food materials are initially in a soft rubbery state. For

such materials, it is usually observed that total volume change at

equilibrium is equal to volume of moisture lost or gained

(Achanta, 1995). In other words, as long as the material is in a

rubbery state and the drying rate is not too high to cause surface

cracks, the solid matrix remains saturated and the gas phase does

not enter the pores. In such a case, the pore pressure is simply thepressure of liquid water, and Eq. (8) can be written as

r Á r0 ¼ r pw ð12Þ

In a series of papers, moisture transport has been investigated in

detail by Scherer (1989), Smith et al. (1995) for soft and deform-

ing polymer gels, which behave in a similar fashion. Scherer

argued that for a uniform pore size medium with inert liquids in

its pores, effective stress at equilibrium (or during a slow drying

process) is equal to pore pressure:

r0 ¼ pw ð13Þ

As a soft material dries out, two important phenomena happen:

the pores shrink and the bulk modulus of the material increases,

turning a soft, rubbery food into a rigid, glassy state. For uniformmoisture distribution, the volume change is equal to the volume

of water lost. The material will stop shrinking when the liquid–

vapor meniscus moves inside the matrix and, with the increased

bulk modulus, the solid stresses can balance the compressive

capillary pressure, pnc . Until that point, the solid skeleton is too

soft to allow the meniscus to move inside and create compressive

pressure. Assuming the solid skeleton to be elastic, the normal

effective stress (shear stress will be zero at equilibrium as there

are no pressure gradient or external shear load) can be related to

volume change (Scherer, 1989):

dr0 ¼ K dV =V ð14Þ

Inserting the stress–strain relation from Eq. (14) into differential

form of Eq. (13) and integrating from initial stress-free volume,

V 0, to final volume at which shrinkage stops, V n, we obtainZ V n

V 0

K

V dV ¼ À pnc ð15Þ

For a simple material with uniform pore size and a known bulkmodulus-moisture content relationship (hardening of the mate-

rial with moisture loss), an explicit value for critical volume, V n,

can be established from Eq. (15). However, due to the highly

heterogeneous and hygroscopic nature of food material, we can

only say that K and pc are functions of moisture content, M , and

temperature, T . Thus, critical volume, V n, will also be a function of

temperature and moisture at equilibrium:

V n ¼ V nðM ,T Þ ð16Þ

Also, for a general food material with range of pore sizes, the

capillaries will empty at different values of shrinkage. Thus, as

shown in Fig. 1, in food materials, we may observe a gradual

decrease (rather than a sharp change which is expected for

uniform pore size material) in the slope of volume vs. moisturecontent plot to zero. Fortunately, volume vs. moisture content

data is available for many food materials from experiments and

can be used to estimate free strain due to moisture loss, eM , and

other strain measures such as deformation gradient tensor due to

moisture loss, FM. Volume change due to moisture loss can then

be treated as free strain analogous to thermal expansion

(discussed below for both small deformation and large deforma-

tion cases).

Small deformation: For small deformation, volume changes due

to temperature and moisture change, i.e., the moisture and

thermal strains (eM and eT , respectively) are subtracted from the

total strain to get the mechanical strain, em:

em ¼ eÀeM ÀeT ð17Þ

Now, with the effect of liquid (moisture) pressure accounted for

as a free strain, the mechanical strain, em, can be related to the

stress due to mechanical load only, r00, i.e., the effective stress, r0,

minus the pressure of water, pw :

ðr0À pwÞ ¼r00 ¼ D Á em ð18Þ

The solid momentum balance, Eq. (12), can also be written in

terms of r00:

r Á r00 ¼ 0 ð19Þ

Depending on the time scales of the process and deformation, the

food material can be treated as elastic or viscoelastic and the

corresponding stress–strain relationship can be used along with

the solid momentum equation.

Moisture Content

V o l u m e

Critical

Volume, V*Mainly Liquid

Transport

Mainly Vapor

Transport

Liquid + Vapor

Transport

Gradual

Transition

in foods

Ideal

Transition

Fig. 1. Volume change vs. moisture content curve for a typical food material.




Large deformation: For large deformation analysis, a multi-

plicative split (Vujosevic and Lubarda, 2002) in deformation

gradient tensor, F, can be used to separate volume changes due

to moisture and temperature changes from volume change due to

mechanical effects. As shown in Fig. 2, the material is first

assumed to go under stress-free deformations due to moisture

and temperature changes and, then, mechanical stresses act on

this stress-free deformed material. The deformation tensor, F, can

be split as

F ¼ F T FMFel ð20Þ

The dilatation (volume-changing) stress is related to elastic

Jacobian, J el ¼ detðFelÞ, which is obtained as the ratio of total

volume change and volume change due to moisture and tem-

perature effects (details in Section 3.1.2). Thermal Jacobian,

J T ¼ detðF T Þ is often small for food materials and is usually

ignored. Moisture Jacobian, J M , can easily be obtained fromvolume change vs. moisture content relationship (Fig. 1).

2.3.2. Processes with gas pressure as the driving mechanism

For some processes, such as microwave heating or bread-

baking, large internal pressure generation (due to water vapor in

microwave heating and carbon dioxide in baking) can cause

swelling/puffing of the material. In such cases, the gas pressure

gradient term of Eq. (9) (first term on the right-hand side) may

dominate. Swelling due to gas pressure in such cases can be much

larger than shrinkage due to moisture loss, and, therefore, stresses

and strains due to the latter can be ignored. In the absence of

thermal strains, the total strain is approximately equal to the

mechanical strain:

em % e ð21Þ

Also, as the stress due to moisture transport is neglected, the solid

momentum balance (Fig. 9) becomes

r Á r0 ¼ r p g ð22Þ

with effective stress, r0 related to strain, e.

Of course, if deformation due to both phenomena (moisture

change and gas pressure) need to be accounted for, the governing

equation and the constitutive law will take the form

r Á r00 ¼ r p g

r

00

¼ D Á em ð23Þ

2.4. Heat and moisture transport: model development for a general

case

Transport modeling for food processes using the multiphase

porous media approach has been reviewed elsewhere (Datta, 2007).

In this section, only equations relevant to deformable materials are

summarized and the reader should refer to Datta (2007) for details for

rigid materials.

2.4.1. Governing equations

The governing equations for non-isothermal transport of two-

phases (liquid water and gas) in an unsaturated porous medium

are comprised of energy conservation and mass conservation of

gas phase, water vapor and liquid water phase, respectively:

ðreff c p,eff Þ@T

@t þX

ð~ni,G Á r ðc p,iT ÞÞ ¼ r Á ðkeff r T ÞÀl_I ð24Þ

@c g

@t þr Á ð~n g ,GÞ ¼ _I ð25Þ

@ðc g ovÞ

@t þr Á ð~nv,GÞ ¼ _I ð26Þ

@c w@t

þr Á ð~nw,GÞ ¼ À_I ð27Þ

The energy equation is used to solve for temperature and the

mass conservation equations for their respective concentrations.

The gas concentration, c g , is related to pressure by invoking the

ideal gas law. Note that not all four equations are needed for all

processes (Fig. 3). Just as the energy equation is needed only for

non-isothermal processes, the gas phase equation is solved only

in case of significant internal pressure generation when pressure

driven flow and/or deformation due to gas pressure gradients

becomes important. Also, the vapor equation is rarely required as

vapor can be assumed to be at equilibrium with the liquid

moisture (more later).

In a deforming medium, since the solid has a finite velocity,~vs,G, the mass flux of a species, i, with respect to stationary

observer, ~ni,G, (used in Eqs. (24)–(27)) can be written as sum of

flux with respect to solid and flux due to movement of solid:

~ni,G ¼ ~ni,s þc i~vs,G ð28Þ

2.4.2. Mass fluxes

Mass fluxes in an unsaturated porous medium can be attrib-

uted to two primary mechanisms—convection (for both gases and

liquids) and binary diffusion (between vapor and air). Reynolds

number is very low (usually less than one) for transport in food

materials and, therefore, Darcy’s law is applied to determine

convective fluxes. For binary diffusion between vapor and air inthe gas phase, Fick’s law is used:

~n g ,s ¼ Àr g

k g

m g

ðr p g Àr g ~ g Þ ð29Þ

~nv,s ¼ Àrv

k g

m g

ðr p g Àr g ~ g ÞÀ

c 2

r g

!M vM aDbinr xv ð30Þ

~nw,s ¼ Àrw

kw

mw

ðr pwÀrw~ g Þ

¼ Àrw

kw

mw

ðr ð p g À pc ðM ,T ÞÞÀrw~ g Þ

¼ Àrw

kw

mw

r p g À@ pc

@M

r M À@ pc

@T

r T Àrw~ g ð31Þ

T, Mo, σ'= 0

T, M, σ'= 0

To, Mo, σ'= 0

FT

FM

Fel

F

T, M, σ'

Fig. 2. Steps indicating multiplicative split in the deformation tensor, separating

moisture, temperature and mechanical effects.




Dry basis moisture content, M , is defined as

M ¼c wc s

¼c w

ð1ÀfÞrs

ð32Þ

Taking density of solid, rs, as constant, moisture content, M , can

be expressed as M ¼ M ðc w,fÞ (where c w ¼rwew), and Eq. (31) can

be re-written as

~nw,s ¼ Àrw

kw

mw

r p g À@ pc

@M

@M

@c wr c wÀ

@ pc

@M

@M

@fr fÀ

@ pc

@T r T

Àrw

~ g

¼ Àrw

kw

mw

ðr p g Àrw~ g ÞÀDw,c w r c wÀDw,fr fÀDw,T r T ð33Þ

where diffusivity due to moisture gradient, Dw,c w , diffusivity due

to porosity gradient, Dw,f, and diffusivity due to temperature

gradient, Dw ,T , are defined as

Dw,c w ¼ Àrw

kw

mw

@ pc

@M

@M

@c w

Dw,f ¼ Àrw

kw

mw

@ pc

@M

@M

@f

Dw,T ¼ Àrw

kw

mw

@ pc

@T ð34Þ

Eqs. (24)–(27), along with fluxes from Eqs. (29), (30) and (33),

solid velocity, ~vs,G, from solid momentum balance and an explicit

expression for evaporation rate, _I , complete the model develop-

ment. Estimation of evaporation rate, however, is not always easy

(Halder et al., 2010) and an accurate determination of _I is possible

only in some special situations, e.g., when local equilibrium

between liquid water and vapor can be assumed. Details of

estimation of _I are mentioned elsewhere (Halder et al., 2010).

2.4.3. Overall moisture equation

If water vapor can be assumed to be in equilibrium with liquid

water (i.e., time-scale required to achieve equilibrium is smaller

than other relevant time scales for the process), vapor pressure

becomes a function of moisture and temperature (through Clau-

sius–Clapeyron equation and moisture sorption isotherms) and its

conservation equation does not need to be solved. In such cases,

vapor flux (ignoring gravity) can be written as

~nv

,

s¼ À

rv

k g

m g r p

g À

c 2

r g !M

vM

aDbinr

ð pv

ðM ðc w

,

fÞ,T Þ

= p

g Þ

¼ À rv

k g

m g

þc 2

r g

!M vM aDbin

pv

p2 g

!r p g ÀDv,c w r c w

ÀDv,fr fÀDv,T r T ð35Þ

where vapor diffusivity due to moisture gradient, Dv,c w , vapor

diffusivity due to porosity gradient, Dv,f, and diffusivity due to

temperature gradient, Dv,T , are defined as

Dv,c w ¼ Àc 2

r g

!M vM a

Dbin

p g

@ pv

@M

@M

@c w

Dv,

f ¼ À

c 2

r g !

M vM a

Dbin

p g

@ pv

@M

@M

@f

Dv,T ¼ Àc 2

r g

!M vM a

Dbin

p g

@ pv

@T ð36Þ

Now, adding liquid water and water vapor conservation equations

to eliminate evaporation rate, _I , and inserting flux relationships,

we obtain the equation for overall moisture balance

@

@t ðc wÞ þr Á ðc w~vs,GÞ ¼ r Á ðK 1r p g þ Dc w r c w þ Dfr fþDT r T Þ ð37Þ

where

K 1 ¼ Àrw

kw

mw

Àrv

k g

m g

Àc 2

r g

!M vM aDbin pv

p2

g

ð38Þ

Fig. 3. A framework for modeling of transport and deformation in food materials based on the state of the material and its processing conditions.




Dc w ¼ Dw,c w þDv,c w ð39Þ

Df ¼ Dw,fþDv,f ð40Þ

DT ¼ Dw,T þDv,T ð41Þ

are the effective permeability and the effective diffusivities due to

moisture concentration gradient, porosity gradient and tempera-

ture gradient, respectively. In Eq. (37), it is assumed that watervapor can contribute to transport terms but not to accumulation

term (this is because density of vapor is three orders of magnitude

smaller than density of liquid water).

For a majority of food processes, moisture fluxes due to

temperature, porosity and pressure gradients are considered

small as compared to that for moisture gradients (sometimes

without justification). The conditions under which these assump-

tions can be justified are:

Gas pressure is atmospheric ðr p g ¼ 0Þ.

The material is either saturated (f% c w=rw and the porosity

gradient term can be merged with moisture gradient term) or

the material is rigid ðr f¼ 0Þ.

Water activity (in turn, capillary pressure, pc ) is independent of temperature gradient (DT ¼0).

In such cases, the overall moisture balance reduces to the well-

known equation:

@c w@t

þr Á ðc w~vs,GÞ ¼ r Á ðDc w r c wÞ ð42Þ

After ignoring the flux due to solid velocity (again, usually done

without justification), Eq. (42) is extensively used in the food

literature to model drying-like processes. Its great advantage lies

in the fact that rate of evaporation, _I , is not required. Also,

effective diffusivity, Dc w , can be easily estimated by fitting

experimentally observed drying curves. However, the rate of

evaporation may be required to solve Eqs. (24) and (25) (if

pressure gradients are significant).

2.5. Heat and moisture transport: special cases

As discussed in case of deformation, transport models can also

be simplified. Energy and gas phase equations are only required

when temperature and pressure gradients, respectively, are

significant. In the following sections, simplifications based on

the state of a food material, as illustrated in Fig. 3, are discussed.

Two extreme states of a food material are: (1) wet, rubbery state

(above glass-transition temperature); and (2) almost-dry, glassy

state (below glass-transition temperature). In the intermediate

region, near glass transition, moisture transport may exhibit non-

Fickian behavior (Case-II diffusion). Traditional form of Darcy’slaw (which assumes that the flux is proportional to pressure

gradients) breaks down for such regions and needs to be mod-

ified. Various approaches have been explored (especially in the

polymer science literature) to account for non-Fickian or Case II

diffusion. The most fundamental of these approaches is developed

by Cushman and coworkers (Singh, 2002; Achanta, 1995) to

derive modified constitutive equations such as Darcy’s law, Fick’s

law, and solid stress–strain relationship based on nonequilibrium

thermodynamics. The approach Cushman and coworkers

followed, known as Hybrid Mixture Theory, is described in detail

elsewhere (Cushman, 1997), and not discussed further in this

manuscript. We now discuss simplifications in governing equa-

tions of transport based on the state (rubbery or glassy) of a food

material.

2.5.1. Wet-rubbery state: liquid moisture transport as the

dominating mechanism

In the rubbery state, free shrinkage/swelling compensates for

moisture loss/gain which means, at equilibrium, change in

volume of a food material is equal to the volume of water lost/

gained (Section 2.3.1). During rehydration/dehydration of such

materials, the evaporation front stays at the surface of the

material and there is no vapor generation or transport within

the food. So, the evaporation rate,_

I , is equal to zero, there is nogas pressure gradient term in Eq. (37), and the effective diffusiv-

ities reduce to just those of liquid moisture. Therefore, the model

reduces to Eq. (37) for moisture and Eq. (43) for temperature (gas

phase and vapor equations are not required), with solid velocity,~nv,G, from the solid momentum balance:

ðreff c p,eff Þ@T

@t þð~nw,G Á r ðc p,wT ÞÞ ¼ r Á ðkeff r T Þ ð43Þ

For soft materials, shear modulus is very small as compared to

the bulk modulus, which means shear stresses (for an uncon-

strained material) that restrict free swelling/shrinkage are also

small, and volume change at every point in the material can be

approximated by the free volume change, even under large

moisture gradients. Thus, if the only deformation information

required is volume change at every point and estimation of stresses and shear strains is not important, solid momentum

balance can be skipped. Divergence in solid velocity can be

estimated from the solid mass balance (assuming constant and

uniform solid density):

@ðrsesÞ

@t þr Á ðrsesvs,GÞ ¼ 0 ð44Þ

Dses

Dt þesr Á vs,G ¼ 0 ð45Þ

r Á vs,G ¼ À1

es

Dses

Dt ¼

1

1Àew

Dsew

Dt ð46Þ

where Ds=Dt stands for material derivative in the reference frame

of the solid. Divergence of solid velocity, vs,G, from Eq. (46) can

now be inserted in the liquid water and energy equations.

2.5.2. Almost-dry, glassy state: vapor transport as the dominating

mechanism

Food at very low moisture content exists in a rigid-glassy state.

As discussed earlier in deformation analysis, there is no deforma-

tion below a certain moisture content. The material can be

assumed to be rigid and deformation analysis is not required.

Also, the food material can be highly unsaturated at low moisture

contents, which means the permeability of liquid water, kw , can

become very low, while the binary diffusivity of vapor and air,

Dbin, can be very high. In such conditions, the transport can be

dominated by vapor transport terms, i.e., Dw,

c w5

Dv,

c w,

Dw,

T 5

Dv,

T ,and transport in liquid phase can be ignored. From Eq. (27),

ignoring transport terms we get

_I ¼ À@c w@t

ð47Þ

Also, solid velocity terms in all transport equations go to zero and

the diffusivities in Eq. (37) are those of water vapor. The model

(for processes in which transport due to temperature and pres-

sure gradients is small) reduces to Eq. (42) (for moisture) and

Eq. (48) (for temperature)

ðreff c p,eff Þ@T

@t ¼ r Á ðkeff r T Þ þl

@c w@t

ð48Þ

This assumption of neglecting liquid transport terms is, however,

justified only when the material is very dry and may happen only




for a small range of moisture content such as during rehydration

of dry cereals due to high humidity levels.

3. Model implementation and validation

In the following section, the modeling framework developed is

applied to two food processes: single-sided contact heating of a

hamburger patty and hot-air drying of a potato slab ( Fig. 4) to

predict deformation, mass and energy transport kinetics. Ham-

burger patty cooking is selected as an example of single phase

(liquid water only) transport as the patty remains largely rubbery

throughout the cooking the process. On the other hand, potato

drying involves development of air porosity and two-phase

(liquid water and water vapor) transport as the potato undergoestransition from a soft and rubbery to rigid-glassy state during

drying. In each case, the model predictions are validated using

experimental results.

3.1. Contact heating of a hamburger patty

Meat can be processed and cooked in a variety of ways. For the

purpose of this study, single-sided contact heating of hamburger

patties (Fig. 4) bought from a local grocery store (USDA Nutrition

Database, 2010, entry no. 23557, 95% lean and 5% fat) is selected.

A refrigerated hamburger patty of cylindrical shape (diameter

10 cm and height 1.8 cm), initially stored at 5 1C, is heated on a

commercial griddle (HotZoneTM Griddle Model No. GR0215G,

Applica Consumer Products Inc., Miramar, Florida) at a fixed plate

temperature of 140 1C. As temperature rises, water at the surface

of the patty evaporates. Since ground meat is in a rubbery state,

the patty shrinks with loss of moisture, and, at equilibrium (in the

absence of gradients of any temperature and moisture fields) the

shrinkage should be equal to the volume of water lost (Fig. 1).

With further rise in temperature, denaturation of muscle proteins

occurs, which leads to decrease in water holding capacity of the

meat. Since the surface of meat in contact with the griddle gets

No axial displacement

(Heat transfer coefficient)

Hamburger Patty

Simulated geometry

(showing deformation)

Axial

Symmetry

Schematic of the

contact-heating process

Evaporation and

drip losses

Drip loss only

Free surfaces

(Natural convection

heat transfer)

No evaporation

or drip loss

1 . 8 c m

10 cm

Symmetry

Plane

Cross-section of potato slab

(perpendicular to length)

No axial displacement

Insulated for energy & moisture

equations

Free surfaces

Forced convection heat transfer

Surface evaporation moisture loss

Simulated geometry

(showing deformation)

2 cm

x c m

x = 0.4, 0.7, 1.0

Heated Plate

Fig. 4. Schematic of the two processes simulated: (a) single-sided contact heating of hamburger patties, and (b) drying of potato slabs, showing the modeled geometry and

boundary conditions. Input parameters are listed in Tables 1 and 2.

1.00

1.50

2.00

2.50

3.00

20Temperature, °C

M o i s t u r e C o n t e n t , d r y b a s i s

100806040

Fig. 5. Water holding capacity (WHC) Dhall and Datta (submitted for publication)

in terms of moisture content (dry basis, kg water per kg of dry solids) as a function

of temperature showing a large drop in WHC near 60 1C. The error bars are for

standard error for measurements done on three patties.




heated up quickly, the water holding capacity near the surface

may drop faster as compared to the drop in moisture concentra-

tion due to evaporation. This leads to dripping of water from the

patty. The variables of interest for predicting quality and safety

aspects of meat cooking are temperature, moisture content,

shrinkage, evaporation rate and their histories.

3.1.1. Problem details

The patty is simulated as a 2D axisymmetric geometry, as theexchange of heat and mass with the outside environment does

not have angular dependence and only a cross-section of the

cylindrical patty needs to be simulated. The effect of gravity on

mass transfer is ignored as the effect of pressure gradients is

much larger on moisture velocity. Since the patty is in a soft and

rubbery state, evaporation stays on the surface during the entire

cooking process. Even if a rigid glassy region develops at the

heated surface, it is assumed to be small and its effect can be

neglected. Therefore, according to the modeling framework

outlined in Fig. 3, the rubbery state of food can be selected. Also,

as the temperature gradients are significant, the energy equation

needs to be solved along with the moisture transport and solid

momentum balance equations. Since there is no internal gas

pressure generation, vapor and gas equations are not required.

3.1.2. Solid momentum balance

A patty can shrink by 30% or more of its initial volume during

the contact heating process, which necessitates the use of large

deformation analysis for solid deformation. Since the evaporation

front stays at the surface and there is no internal gas pressure

generation, gas pressure gradient term can be ignored for the

solid momentum balance. For large deformation, Lagrangian

measures of stress and strain are used, and the solid momentum

balance (Eq. (19)) is written in Lagrangian coordinates:

r X Á ðS00 Á FTelÞ ¼ 0 ð49Þ

where S00 is the second Piola–Kirchhoff (PK2) stress tensor, and Fel

is the elastic deformation gradient tensor. PK2 stress, S00

, is relatedto Cauchy stress, r00, by the following relationship:

S00¼ J Á FÀ1

el Á r00 Á FÀTel ð50Þ

PK2 stress is energy conjugate to the Green–Lagrange elastic

strain tensor, Eel:

Eel ¼ 12ðFT

elFelÀIÞ ð51Þ

and, thus, S00 and, Eel are related as follows:

S00 ¼@W el@Eel

ð52Þ

Now, we need a constitutive equation for the elastic strain energy

density, W el. Rubbery state means the stress relaxation time

scales are expected to be small (as compared to the time scale

of the cooking process which is in minutes, Deborah number $ 0)

and the solid skeleton can be treated as a hyperelastic material.

Also, the fibers in ground meat are randomly oriented. Therefore,

although meat fibers are anisotropic with different properties

along and across the fibers, the averaged mechanical properties

are isotropic. A modified Neo-Hookean constitutive model is

chosen which accounts for the volume change due to moisture

loss also

W el ¼K

2ð J elÀ1Þ2À

m2

ðI 1À3Þ ð53Þ

where K and m are the bulk modulus and the shear modulus,

respectively. J el is the elastic Jacobian as defined earlier, and I is

the first invariant of the right-Cauchy Green tensor, Cð ¼ FelT

Fel Þ,

for deviatoric part of elastic deformation gradient, i.e., Fel .

Deviatoric part of elastic deformation gradient is related to elastic

deformation gradient, Fel, and its dilatation part, J 1=3

el, as

Fel ¼ J 1=3

elFel ð54Þ

Now, to estimate elastic Jacobian, J el, we need to calculate

Jacobian due to moisture change, J M (Eq. (20)). This is easy, as

under stress-free conditions, a patty shrinks/swells by the amount

of moisture lost/gained. Let V be the REV volume at moisture

volume fraction, ew. Then, change in volume of REV can beequated to change in volume of moisture in REV:

V ÀV 0 ¼ ewV Àew,0V 0 ð55Þ

J M ¼V

V 0¼

1Àew,0

1Àewð56Þ

Similarly, porosity at any time t , fðt Þ, can be determined using

incompressibility of the solid skeleton, equating the initial

volume of solid in an REV to solid volume at time, t :

ð1Àfðt ÞÞV ðt Þ ¼ ð1Àf0ÞV 0 ð57Þ

fðt Þ ¼ 1À1Àf0

V ðt Þ=V 0¼ 1À

1Àf0

J ðt Þð58Þ

Note that while Jacobian due to moisture change, J M , is a statefunction (depending on the moisture content), porosity, fðt Þ, is a

process variable, depending on the actual Jacobian, J (t ).

3.1.3. Moisture and energy transport equations

Moisture flux in case of meat needs to be treated differently

from the discussion in Section 2. Water activity of meat at room

temperature is $ 1, which gives capillary pressure, pc , or water

potential, Cw, close to zero (using Kelvin’s law (Lu and Likos,

2004). Thus, Eq. (31) cannot be used to calculate moisture flux.

Also, with increase in temperature, meat proteins denaturate

leading to a drop in water holding capacity (Tornberg, 2005). As

time scales of temperature rise in the patty during intensive

cooking such as contact-heating are smaller than time scales of

moisture transport, moisture concentration in much of the pattyis more than its water holding capacity at equilibrium.

Liquid water pressure (called swelling pressure) in meat has

been estimated (van der Sman, 2007) by using the Flory–Rehner

theory. Taking the swelling pressure to be zero at equilibrium

moisture volume fraction, and linearizing the Flory–Rehner

expression near equilibrium, it can be shown that the swelling

pressure is proportional to the difference between the actual and

equilibrium moisture concentrations:

pw ¼ C ðc wÀc w,eqðT ÞÞ ð59Þ

where c w ,eq is the equilibrium moisture concentration at a given

temperature and the constant of proportionality, C , though

constant here, can be temperature dependent. Inserting this

expression of liquid pressure, pw , in Darcy’s law (line 1 in

Eq. (31)), we get (ignoring gravity)

~nw,s ¼ ÀðDw,c w r c w þDw,T r T Þ ð60Þ

where the new definitions of diffusivities due to moisture

gradient and temperature gradient are:

Dw,c w ¼rw

kw

mw

C

Dw,T ¼rw

kw

mw

C @c w,eq

@T ð61Þ

Thus, the moisture transport equation reduces to Eq. (62) with

new definitions of diffusivity (Eq. (61)):

@c w

@t þr Á ðc w~vs,GÞ ¼ r Á ðDw,c w r c w þ Dw,T r T Þ ð62Þ




The energy balance equation remains the same as discussed for

rubbery materials in Section 2 (Eq. (43)).

3.1.4. Boundary and initial conditions

Solid momentum balance: Normal displacement of the axisym-

metric boundary and the bottom surface (lying on the griddle) is

set to zero. The other two boundaries are unconstrained and free

to move (Fig. 4).

Liquid water equation: The boundary condition for liquid water

equation consists of two flux terms: evaporation and drip. The

magnitude of the evaporation flux, nw,s,surfe, is simply given by

mass transfer coefficient multiplied by the vapor density differ-

ence between the surface and the boundary:

nw,s,surfe ¼ hmðrv,surf Àrv,ambÞ ð63Þ

Water is lost from the matrix in liquid form (as drip) only when

surface moisture concentration, c w,surf , is more than the water

holding capacity, c w,eq. The drip loss, nw,s,surfd, under such condi-tions is equal to the total moisture flux reaching the surface

subtracted by that taken by surface evaporation, nw,s,surfe:

nw,s,surfd ¼ ~nw,s Á ~N surf Àhmðrv,surf Àrv,ambÞ ð64Þ

Therefore, the total moisture flux at the surface with respect to a

stationary observer is equal to the sum of drip loss, evaporation

loss and flux due to movement of the surface itself:

nw,G,surf ¼ nw,s,surfe þnw,s,surfdþc w~vs,G Á ~N ð65Þ

Energy equation: For energy equation, forced convection heat

transfer boundary condition is applied to get the heat flux at the

surface, qsurf :

qsurf ¼ hðT ambÀT surf ÞÀlnw,

s,

surf ÀX

ð~ni

,

Gc p,

iT Þ Á

~

N surf ð66Þ

In Eq. (66), the first term on the right hand side is the convective

heat transfer coefficient multiplied by the temperature difference,

the second term is the latent heat taken up by surface evapora-

tion, and the third term is energy carried by convection terms

normal to the boundary.

Initial conditions: Initially refrigerated at 5 1C, the composition

of the patty is taken from USDA Nutrition Database (2010) and is

listed in Table 1. Since the weight percentages of the proximates

added up to 100.74, the weight percentages were normalized. The

volume fraction of air in the patties is considered small and, thus,

ignored. From this data, the initial concentrations of water and

solid (protein, fat and ash) can be calculated.

3.1.5. Input parameters and numerical solution

Input parameters used in the hamburger patty cooking simu-

lation are given in Table 1. Bulk modulus and Poisson’s ratio were

estimated considering the patty to be saturated and in a soft,rubbery state throughout the heating duration. In a soft material,

the Poisson’s ratio is expected to be about 0.5. A value of 0.49 was

used to help convergence. For saturated porous materials with

incompressible solid skeleton, the bulk modulus (for small elastic

strains) is given by Hashin (1985)

K ¼1

e f

K f þ4Gs

3ð1Àe f Þ

ð67Þ

Since the bulk modulus of water, K w (2.2 Â 109 Pa) is much greater

than the shear modulus of the solid matrix, Gs (o106 Pa), it

justifies a Poisson ratio close to 0.5, and Eq. (67) reduces to

K ¼K w

ew

ð68Þ

Table 1

Input parameters Dhall (2011) used in the simulations of single-sided contact heating of hamburger patties. Number under source column refer to bibliographic order.

Parameter Value Units Source

2D axisymmetric patty dimensions

Height 1.8 cm Measured

Diameter 10 cm Measured

Patty composition Actual (used) Weight USDA Nutrition Database (2010)

Water 73.28 (72.74) %

Protein 21.41 (21.25) %Fat 5.00 (4.96) %

Ash 1.05 (1.04) %

Initial conditions

Air volume fraction 0 –

Temperature 5 1C Measured

Processing conditions

Ambient temperature 60 1C Measured

Plate temperature 120 1C Measured

Heat transfer coefficient 400 W/m2 K Wang and Singh (2004)

Mass transfer coefficient 0.01 m/s Ni and Datta (1999)

Properties

Water holding capacity Fig. 5 – Measured

Density Choi and Okos (1986)

Water 997.2 kg/m3

Fat 925.6 kg/m3

Protein 1330 kg/m3

Specific heat capacity Choi and Okos (1986)Water 4178 J/kg K

Fat 1984 J/kg K

Protein 2008 J/kg K

Thermal conductivity Choi and Okos (1986)

Water 0.57 W/m K

Fat 0.18 W/m K

Protein 0.18 W/m K

Diffusivity 10À7 m2/s van der Sman (2007)

Bulk modulus K wew

Pa Hashin (1985)

Poisson’s ratio 0.49 – Rubbery state




which is noted in Table 1. Heat transfer coefficient is estimated

from the experimentally measured data of Wang and Singh

(2004). A commercially available finite element software,

COMSOL Multiphysics 3.5a (Comsol Inc, Burlington, MA), was

used to solve the equations. The solid momentum balance is

solved in the total Lagrangian reference frame (i.e., frame moving

with the solid) for the axisymmetric geometry equation in the

structural mechanics module, while convection–conduction and

convection–diffusion equations (in the main COMSOL Multiphy-sics module) were used for energy and moisture transport,

respectively. Deformed mesh equations (again, in the main

COMSOL Multiphysics module) were used to track the material

deformation in the Eulerian reference frame, and move the mesh

accordingly. The transport equations were solved in the Eulerian

reference frame (i.e., frame of the stationary observer) on the

deformed mesh. The computational domain was rectangular,

5 cm Â 1.8 cm, and had an unstructured quadrilateral mesh

consisting of 3864 elements. Linear shape functions were used.

The simulation of 900 s of heating took approximately 4 h of CPU

time for an adaptive timestepping scheme (maximum time step

size of 0.05 s) on a 3.00 GHz dual-core Intel Xeon workstation

with 16 GB RAM. Mesh and timestep convergence were ensured

by checking that any dependent variable (temperature, moisture

content or displacement) did not change by more than 1% of the

total change (at any time at all four vertices of the geometry) by

reducing the timestepsize or mesh-size by half.

3.1.6. Results and discussion

Spatial and temporal distribution of moisture content : Fig. 6

shows a comparison between predicted and experimentally

observed (Dhall, 2011) total moisture loss history of the patty

for 15 min of heating time. Total moisture loss is almost linear

with time, with the patty losing about 17% (26 g for a 155 g patty)

of the initial moisture content in 15 min. The predicted moisture

loss history follows the observed history closely, and the differ-

ence between the two at any time is 5% or less. The cumulative

evaporation and drip losses are also plotted in Fig. 6. Evaporation

loss with time is slightly concave upwards (rate of loss always

increases throughout the heating duration). On the other hand,

cumulative drip loss curve with time is S-shaped and stabilizes

(rate of drip loss goes to zero) at around 5 min as moisture

concentration at the patty surface falls below equilibrium

concentration. Evaporation loss and its rate exceed the drip loss

and the drip loss rate at any time during heating. Contours of

moisture content (dry basis) after every 3 min of heating (starting

at 3 min) are plotted in Fig. 7. It can be seen that the moisture

gradients dominate in the axial direction and end effects are

restricted to a small region near the lateral surface of the patty.

Also, even at the end of heating, the minimum moisture content

(near the griddle plate) is still high (0.891), which means the

surface has not dried up. On the other hand, moisture content

close to the exposed top surface (away from the griddle) rises to

2.731 (from an initial value of 2.6) during the process.

Spatial and temporal distribution of temperature: Fig. 8 shows a

comparison between predicted and experimentally observed

temperature histories at two locations on the central axis of the

patty: (1) at the mid-point between the heated and exposedsurfaces, and (2) on the exposed top surface. With the initial lead

time of about 50 s, temperature at the midpoint follows the

concave downwards curve reaching a value of 56 1C after

15 min. The predicted curve follows the observed one closely,

with the difference between the two at any time being 1 1C or

less. Temperature history at the surface is more interesting. While

the observed history is similar to that of the midpoint, having an

initial lead time followed by a concave downwards curve; the

predicted history shows a quick initial heating period which is

absent in the observed history. The discrepancy between the

predicted and observed histories for the first 300 s of heating can

be attributed to changing ambient conditions of temperature and

relative humidity at the exposed surface during the cooking

process. At the top surface, a fixed ambient air temperature of

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0 3 6 9 12 15

M o i s t u r e l o s s ( f r a c t i o n o f i n i t i a l p a t t y m a s s )

Time (min)

Predicted

Experiment

Evaporation loss

Drip loss

Fig. 6. Cumulative total (evaporationþdrip) moisture loss (predicted and experi-

mentally observed, Dhall and Datta, submitted for publication), evaporation

moisture loss (predicted) and drip loss (predicted) for single-sided contact heating

of hamburger patties. It can be seen that drip loss levels off after 5 min and

evaporation loss dominates for the rest of the heating duration.

3 m i n

1 5 m i

n

Axis of

symmetry

Unheated

Surface

Heated

Surface

Moisture content

(dry basis)

Fig. 7. Contours of moisture content (dry basis) after 3, 6, 9, 12 and 15 min of

single-sided contact heating of hamburger patties showing low moisture at the

heated surface and some accumulation in the center. Moisture gradients can be

seen primarily in the axial direction.




60 1C (Table 1) and zero moisture flux (moisture loss from the top

surface is negligible as compared to the moisture loss from the

bottom surface) were used as boundary conditions which may not

be valid at initial times when patty is just put on the plate.

Therefore, an error in surface temperature prediction may be seen

when the effect of boundary conditions dominates (in the model

the surface gets heated fast due to air temperature of 60 1C).

Similar to moisture content, temperature contours (Fig. 9) show

small end effects. The heated surface reaches around 90 1C early

in the heating process and stabilizes. Temperature at the exposed

surface rises slowly and reaches about 501C after 15 min.

Spatial and temporal distribution of deformation field: Fig. 10

compares the histories of experimentally observed diameter with

the predicted diameter (averaged for diameter at different

heights). The patty diameter reduces to about 91% of the original

value in 15 min, which is as predicted by the simulations. For

reference, the diameter, D(t ), assuming uniform shrinkage

throughout the patty, is computed from the equation:

Dðt Þ

D0 ¼

V ðt Þ

V 0 1=3

ð69Þ

and also plotted. Eq. (69) is obtained by assuming the same

uniform linear shrinkage along the thickness and in the diameter

and relating this to volume shrinkage, using the equation

ðV 0ÀV ðt ÞÞ=V 0 ¼ ðpD20L0ÀpD2ðt ÞLðt ÞÞ=pD2

0L0Þ with Dðt Þ=D0 ¼ Lðt Þ=L0.

As shown in Fig. 10, the diameter assuming uniform shrinkage

is much larger than the predicted or observed diameters at any

time, indicating the non-uniformity in patty shrinkage. Also, this

means that such a simplified relationship as Eq. (69) cannot be

used to predict diameter with solid deformation equations not

solved. Predicted thickness (normalized) and thickness assuming

uniform shrinkage are plotted in Fig. 11. The final value of

thickness is approximately 95% of the initial value, which means

the patty shrinks by less than 1 mm in thickness in 15 min.Predicted values of thickness were not compared to its

observed values because of high variability in patty thickness (it

varied by more than 2 mm at different locations on a single patty)

and also due to variability in shear effects that cause rise of the

bottom surface of the patty near the center. In this simulation, the

bottom surface was considered fixed in the z -direction that could

not be achieved in all the experiments at all times. Some patties

rose by 1–2 mm in the middle, while some others stuck to the

griddle plate. Therefore, uncertainty (more than 2 mm) in height

was more than the total expected change in height ( $ 1 mm) and,

thus, it was meaningless to compare the observed and predicted

thickness values.

Fig. 12, which plots the contours of elastic Jacobian, J elð ¼ J = J M Þ,

at different times, helps us arrive at a very good (albeit, more

involved) method to predict shrinkage. Fig. 12 shows that the

ratio of actual Jacobian, J , to the Jacobian due to moisture change,

0

10

20

30

40

50

60

0 3 6 9 12 15

Time (min)

T e m p e r a t u

r e ( º C )

Center

(Exp.)

Surface

(Exp.)

Surface

(Predicted)

Center

(Predicted)

Center

Surface

Fig. 8. Temperature histories (predicted and experimentally observed, Dhall and

Datta, submitted for publication) at the midpoint and the surface on the central

axis for single-sided contact heating of hamburger patties.

3 m i n

1 5 m i n

Axis of

symmetry

Unheated

Surface

Heated

Surface

Temperature

(°C)

Fig. 9. Temperature contours (in 1C) after 3, 6, 9, 12 and 15 min of single-sided

contact heating of hamburger patties showing constant heated surface tempera-

ture and gradients primarily in the axial direction.

0.89

0.91

0.93

0.95

0.97

0.99

1.01

0 3 6 9 12 15

Time (min)

D i a m e t e r ( n o r m a l i z e d )

Prediction

Experiment

Diameter (assuming uniform

shrinkage throughout the patty)

Fig. 10. Diameter change histories (prediction and experimental observation) for

single-sided contact heating of hamburger patties. Also, diameter calculated

assuming uniform shrinkage throughout the patty (Eq. (69)) is plotted showing

assumption of uniform shrinkage will lead to erroneous results.




J M , lies in the narrow range of 0.98–1.01%. The region near the

heated surface is under tension, while the other cooler regions are

under compression. The narrow range of elastic Jacobian, J el, is

due to the high bulk modulus to shear modulus ratio (Poisson

ratio, n% 0:5). For such cases, if estimation of stresses is not

important, solid momentum balance can be avoided and Jacobian,

J , can be assumed to be equal to the moisture change Jacobian, J M .

In the absence of significant shear strains, the dilatational strains

and, thus, deformation field, can be estimated from Jacobian, J ,

only. Displacements due to this deformation field can now be

calculated and used in the deformed mesh equations to get new

geometry.

3.2. Convective drying of a potato slab

Drying of potato slabs, as described by Wang and Brennan

(1992, 1995), is numerically implemented as a second example.

The potato slabs (Desiree variety) are 45 mm long and 20 mmwide, with thickness varying from 4 to 10 mm. The drying

experiments were carried out by Wang and Brennan at air

temperatures between 40 1C and 70 1C, at a constant absolute

humidity of 16 g (vapor)/kg (dry air). Initially, the potato slab is in

a rubbery state and shrinks with loss of moisture. However,

unlike meat, it becomes rigid towards the end of drying and stops

shrinking with moisture loss, allowing the evaporation front to

move in. As in meat cooking, the variables of interest are

temperature, moisture content, shrinkage, evaporation rate and

their histories.

3.2.1. Problem formulation and modeling details

To reduce computational complexities, a 2D cross-section of

the potatoes (perpendicular to length) is modeled and the end-effects are ignored (Fig. 4). Only half of the width is simulated as

all the physics is symmetric about the center. Initially, the potato

is in a soft and rubbery state, and gradually transitions to a rigid

state. According to the modeling framework outlined in Fig. 3, the

transition state of food can be selected as both rubbery and glassy

states exist at different times and positions during potato drying.

Since this transition occurs at a very low moisture content and

there is no evidence of Case-II diffusion (as discussed in Section

2.5) in potatoes, the traditional constitutive relationship for

moisture flux (Darcy’s law) holds. In this case, the energy balance

(Eq. (24)) is solved along with the moisture balance (Eq. (42)) and

solid momentum balance (Eq. (49)). Assuming equilibrium

between liquid water and water vapor, evaporation rate, _I , is

estimated using Eq. (26). Also, reduction of volume with removal

of moisture stops at M ¼0.3, Jacobian due to moisture change, J M

is written as

J M ¼1Àew,0

1ÀewM 40:3 ð70Þ

J M ¼ J M 9M ¼ 0:3 M r0:3 ð71Þ

For boundary condition of the solid momentum equation, the bottom

and the left edges are treated as a roller (zero normal displacement)

and a symmetry, respectively. The other two edges are free. The

bottom and the left edges are insulated for energy and moisture

transport equations, while surface evaporation and convective heat

and mass transport takes place at the other two edges. Thus, Eqs. (65)

and (66) (with no drip loss) are used as boundary conditions for

moisture and energy transport. Other input parameters used in thesimulation are listed in Table 2. The solution strategy remains the

same, with the simulation of 1000 min of drying taking approxi-

mately 30 min of CPU time for a maximum timestep size of 60 s (784

linear quadrilateral elements) on a 3.00 GHz dual-core Intel Xeon

workstation with 16 GB RAM.

3.2.2. Results and discussion

Figs. 13 and 14 compare model predictions with the experi-

mental observations: (a) temperature history at the top surface

(spatially averaged) for drying a 7 mm thick slab at a drying

temperature of 55 1C; (b) moisture content histories for slabs of

thickness 10 mm, 7 mm and 4 mm at drying temperature of

55 1C; and (c) normalized volume as a function of moisture

content for a 10 mm thick slab at drying temperatures of 701C

0.92

0.94

0.96

0.98

1

0 3 6 9 12 15

Time (min)

H e i g h t ( n o r m a l i z e d )

Prediction

Height (assuming uniform

shrinkage throughout the patty)

Fig. 11. Height change history for single-sided contact heating of hamburgerpatties. Note that height change was too small to be compared with experiments.

Also, height calculated assuming uniform shrinkage throughout the patty is

plotted, showing assumption of uniform shrinkage will lead to erroneous results.

3 m i n

1 5 m i n

Axis of

symmetry

Unheated

Surface

Heated

Surface

Elastic Jacobian

(Jel)

Fig. 12. Elastic Jacobian, J el, (ratio of actual volume to free volume) contours after

3, 6, 9, 12 and 15 min of single-sided contact heating of hamburger patties. It can

be seen that the surface is stretched and the heated interior is compressed by a

maximum of 2% from free volume.




and 40 1C. The surface temperature rises from 32.5 1C to 50 1C in

200 min and stabilizes, reaching about 54 1C after 800 min of

drying. The predicted temperature history closely follows the

observed one. The predicted moisture content histories for three

different values of slab thickness also follow the observed history

very well. The shrinkage of the potato slabs at the two drying

temperature values (70 1C and 40 1C) is a little less than the

volume of moisture lost until moisture content of 0.3, with

volume at 70 1C equal to or more than that at 40 1C. The

simulations capture the trends very well, apart from the small

difference in the observed volumes at the two drying tempera-

tures, which the predictions could not capture. As the experi-

mental error values are not available, it is difficult to conclusively

say if the small difference in volumes at the two temperatures is

real. Nevertheless, the accurate predictions of moisture loss,temperature and shrinkage histories for the drying process serve

to validate the modeling approach followed.

3.3. Importance of solid mechanics analysis

Since the volume change is almost equal to moisture change for

the two food materials studied (above a critical moisture content for

potato), the advantage of solving the solid momentum equation does

not lie in predicting volume change due to moisture content. The real

value of solid mechanics analysis lies in predicting small deviations

from free shrinkage, which lead to stresses and can be important

indicators of food quality, related to cracking, for example. As an

example, Fig. 15 plots the maximum value of elastic Jacobian, J el, as a

function of normalized moisture content for hamburger cooking and

potato drying (10 mm thick slab at 70 1C). The large value of maxð J elÞ

in the case of potato is because of the greater shear modulus for

potato which leads to deviations from free shrinkage. For hamburger

patties, Poisson’s ratio, n, stays close to 0.5 and, thus, much smaller

deviations from free shrinkage are observed. As a potato slab is under

a much larger expansive strains (near the surface as it dries up) as

compared to meat, its surface is more prone to cracking. Thus,

maxð J elÞ can be used as a criteria to predict and avoid drying

situations most prone to cracking. Apart from cracking, other impor-

tant quality parameters, such as porosity development, case hard-

ening (surface drying leading to large increase in shear modulus and

reduced shrinkage), etc., can also be predicted from deformation

analysis.

4. Conclusions

A poromechanics-based approach to mathematically model

the coupled physics of transport and deformation during proces-

sing of food materials is developed. Following this comprehensive

approach, food materials existing in a range of states (glassy to

rubbery) and being processed under a variety of conditions, can

be simulated to predict important food quality and safety para-

meters (spatial and temporal histories of temperature, moisture

and deformation). For deformation, primary driving forces are

identified and their effect on the solid momentum balance is

discussed in detail. The driving forces are: (1) gas pressure, which

causes the food material to swell (gas pressure gradient can be

directly treated as a source term for the solid momentum

Table 2

Input parameters used in the simulations of drying of potato slabs. Number under source column refer to bibliographic order.

Parameter Value Units Source

2D Slab dimensions

Height 4,7,10 mm Wang and Brennan (1992, 1995)

Half width 10 mm Wang and Brennan (1992, 1995)

Initial conditions

Moisture vol. frac.

0.865 (T amb¼551

C) – Wang and Brennan (1992)0.838 (T amb¼40,70 1C) – Wang and Brennan (1995)

Air vol. frac. 0 –

Temperature 32.5 1C Wang and Brennan (1992)

Drying conditions

Temperature 40, 55, 70 1C Wang and Brennan (1992),

Wang and Brennan (1995)

Absolute humidity 0.16 g/kg Wang and Brennan (1992)

Heat transfer coeff. 40 W/m2K Laminar flow

Mass transfer coeff. 0.01 m/s Lewis analogy

Properties

Water activity – – Ratti et al. (1989)

Density Choi and Okos (1986)

Water 998 kg/m3

Air Ideal gas kg/m3

Solid 1592 kg/m3

Specific heat capacity Choi and Okos (1986)

Water 4178 J/kg KSolid 1650 J/kg K

Thermal conductivity Choi and Okos (1986)

Water 0.57 W/m K

Air 0.026 W/m K

Solid 0.21 W/m K

Moisture diffusivity4:49 Â 10À5 exp

À2172

T

m2/s Wang and Brennan (1992)

Binary diffusivity 2:6 Â 10À6e g m2/s Halder et al. (2007)

Bulk modulus

109M 40:3 Pa Estimated for rubbery

106M o0:3 Pa Estimated for glassy

Poisson’s ratio

0:49M 40:3 – Estimated for rubbery

0:3M o0:3 – Estimated for glassy




balance) and (2) moisture change, which can be treated analogous

to thermal expansion/contraction to get the free volume change.

For transport, temperature, moisture, vapor concentration and gas

pressure are the primary variables of interest. As gas does not

enter the pores during processing of wet-rubbery materials, gas

phase equation is not required for such materials. Even if gas is

present, significant pressure generation occurs only for intensive

heating processes such as microwave cooking and processes with

internal generation such as bread baking. Also, solution of vapor

equation is not required unless local equilibrium between vapor

and liquid moisture breaks down. Assuming equilibrium vapor

concentration, liquid water and water vapor flux can be added to

get the total moisture flux relationship, which with further

simplifications takes the form of Fick’s law.

Two different food processes are simulated as implementa-

tions of the modeling framework developed: (1) single-sided

cooking of hamburger patties for which shrinkage is equal to

moisture loss throughout the process and (2) convective drying of

potato slabs for which shrinkage stops under a critical moisture

content. For both the cases, transport of moisture in liquid form

dominates. The difference lies in greater strains experienced by

the potato due to greater shear modulus at low moisture

contents. Accurate predictions of the experimental observations

for two completely different processes show the versatility of the

modeling framework. Being comprehensive and fundamentals-

based, the framework can be widely applicable in food product,

process and equipment design, accounting for both food quality

and safety as design parameters.

Nomenclature

aw water activity

c i concentration of species i, kg mÀ3

c p specific heat capacity, J kgÀ1 KÀ1

c molar density, kmol mÀ3

C constant of proportionality in Eq. (59)

D diameter, m

D stiffness tensor

Dbin effective gas diffusivity, m2 sÀ1

Db effective diffusivity due to gradients of b, m2 sÀ1

Da,b diffusivity of a due to gradients of b, m2 sÀ1

E Green–Lagrange strain tensor

F deformation tensor~ g acceleration due to gravity, kg mÀ3

h heat transfer coefficient, W mÀ2

KÀ1

30

40

50

60

0 200 400 600 800

Time (min)

S u r f a c e t e m p e r a t u r e ( º C )

Predicted

Experiment

(Wang and Brennan, 1992)

0

1

2

3

4

0 200 400 600 800 1000

Time (min)

M o i s t u r e c o n t e n t

( d . b . )

10 mm

7 mm

4 mm

Experiment (Wang

and Brennan, 1992)

Predicted

Fig. 13. (a) Spatially averaged surface temperature and (b) moisture content

histories for drying of potato slabs. Moisture content histories are shown for three

different slab thicknesses of 4, 7 and 10 mm, respectively. Drying temperature is

55 1C and other input parameters are provided in Table 2.

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4

Moisture content (d.b.)

S h r i n k a g e ,

V / V 0

Predicted (40ºC &

70ºC, coincident)

Experiment (40ºC)(Wang and Brennan, 1995)

Experiment (70ºC)

(Wang and Brennan, 1995)

Fig. 14. Volume change vs. moisture content (drying temperatures 40 and 70 1C,

10 mm thickness). Dotted line is for shrinkage equal to moisture loss.

1

1.05

1.1

1.15

1.2

0 0.2 0.4 0.6 0.8 1

M a x e l a s t i c J a c o

b i a n ,

J e l

Moisture content (normalized)

Potato drying

Hamburger

patty cooking

Fig. 15. Maximum value of elastic Jacobian, J el, (ratio of actual volume to freevolume) vs. moisture content (normalized with respect to initial moisture

content) for the two processes simulated showing larger expansive strains for

potato drying as compared to hamburber patty cooking.




hm mass transfer coefficient of vapor, m sÀ1

_I volumetric evaporation rate, kg mÀ3 sÀ1

I identity tensor

J Jacobian

keff effective thermal conductivity, W mÀ2 KÀ1

ki permeability of phase i, m2

K bulk modulus, Pa

K 1 defined by Eq. (41)

L thickness of hamburger pattyM moisture content (dry basis)

M a, M v molecular weight of air and vapor~N normal vector

~ni, j mass flux of species i with respect to j, kg mÀ2 sÀ1

pi pressure of phase or species i, Pa~q heat flux, J mÀ2 sÀ1

R universal gas constant, J kmolÀ1 KÀ1

REV representative elementary volume

S00 Piola–Kirchoff stress tensor, Pa

S i saturation of phase i

t time, s

T temperature

vi, j velocity of species i with respect to j, m sÀ1

vw molar volume of water, m3 molÀ1

V volume, m3

V n critical volume at which shrinkage stops, m3

W strain energy density, Pa

xi mole fraction of species i

Greek symbols

e strain tensor, volume fraction

r density, kg mÀ3

l latent heat of vaporization, J kgÀ1

m shear modulus, Pa

mi dynamic viscosity of a phase, i, Pa s

n Poisson’s ratio

r stress tensor, Pa

r0 effective stress tensor, Par

00 effective stress tensor due to mechanical load only, Pa

f porosity

Cw water potential, Pa

ov, oa mass fraction of vapor and air in relation to total

gas

Subscripts

amb ambient

a, g , s, v, w air, gas, solid, vapor, water

c capillary

eff effective

el elastic

eq equilibrium f fluid

G ground (stationary observer)

i ith phase

m mechanical

M moisture

0 at time t ¼0

surf surface

surfd drip at the surface

surfe evaporation at the surface

T temperature

Superscripts

f Volumetric average of f over an REV

Acknowledgements

This project was supported by National Research Initiative

Grant 2008-35503-18657 from the United States Department of

Agriculture National Institute of Food and Agriculture.

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