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
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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
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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.
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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.
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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.
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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
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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.
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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
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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
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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  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.
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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.
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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.
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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
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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.
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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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