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InterPore Journal, Vol. 1, Issue 2, 2024 https://doi.org/10.69631/ipj.v1i2nr25
Original Research
HYBRID MIXTURE THEORY-BASED
MODELING OF UNSATURATED
TRANSPORT IN A DEFORMING
POROUS FOOD MATRIX DURING
FRYING
Yash Shah , Pawan Singh Takhar*
Department of Food Science and Human Nutrition, University of Illinois at Urbana-Champaign, Urbana, IL, USA
Correspondence to:
Pawan Singh Takhar at
ptakhar@illinois.edu;
Phone: +1 (217) 300-0486
*This author has
previously published as
Pawan P. Singh
How to Cite:
Shah, Y., & Takhar, P.
Hybrid mixture theory-
based modeling of
unsaturated transport in a
deforming porous food
matrix during frying.
InterPore Journal, 1(2),
ipj240824–6.
https://doi.org/10.69631/
ipj.v1i2nr25
RECEIVED:
14 Feb. 2024
ACCEPTED
: 18 June 2024
PUBLISHED
: 24 Aug. 2024
ABSTRACT
Physics-based modeling of deep fat frying is daunting given the intricacies
involved in the transport of different phases (liquid water, gas, and oil) in a
continuously deforming unsaturated porous matrix. To simplify model
development, previous models for frying either ignored volume changes or
used empirical relations. The model developed in this study solved the hybrid
mixture theory-based unsaturated transport equations and mechanistically
accounted for the volume changes of the porous food (potato) matrix. Pore
pressure, the effective pressure on pore walls, was used as the driving force
governing the volume changes. A good agreement was found between the
model predictions and experimental results. The % mean absolute error for
moisture content, oil content, and temperature is 5.57%, 22.42%, and
13.35%, respectively. Evaporation and gas expansion during frying led to high
pressures in the porous matrix with a peak gauge pore pressure of
approximately 19.16 kPa at the center of the sample. The high pressure
restricted the frying oil from penetrating beyond the surface layers. Oil
uptake mainly occurred during the early stages of frying (t<50 s) when the
pressure in the core was low, and towards the end of frying when the matrix
was more susceptible to oil penetration because of decreasing pressure. The
potato cylinder shrunk by 18.55% for a frying time of 300 s. The gauge pore
pressure near the surface became negative, which led to t
he rapid
contraction of the surface layers, and as a result, the porosity near the
surface decreased. The average porosity was predicted to decrease by 5.06%
after 300 s of frying. The evaporation zone expanded with frying time, and its
peak progressively moved towards the core. The insights generated from the
discussed mechanisms will guide the industry in optimizing frying techniques.
KEYWORDS
Unsaturated, Porous, Modeling, Deforming, Pressure, Temperature
@2024 The Authors
This is an open access article published by InterPore under the terms of the Creative
Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND
4.0) (https://creativecommons.org/licenses/by-nc-nd/4.0/).
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1. INTRODUCTION
Deep fat frying is one of the most commonly used food processing operations. It involves completely
immersing a food product in a high boiling point edible fat or oil, typically at temperatures between
150-190 °C, resulting in products that are widely appealing and consumed globally (14, 36). The portfolio
of fried foods is diverse, including products such as chips, French fries, chicken nuggets, chicken patties,
fish sticks, vegetable fritters, donuts, and even fried ice cream. The high frying temperatures and the
presence of multiple phases (liquid water, gas, oil, and solid biopolymers) drive a wide range of chemical
and physical changes, such as protein denaturation, starch gelatinization, retrogradation, glass transition,
moisture evaporation, Maillard reaction, and caramelization (42, 135). These changes play a role in
imparting the food with a unique color (golden brown), texture (crispy, dry crust with a moist core), and
flavor.
Increasing caution has been practiced while consuming fried foods due to their high oil content,
acrylamide accumulation, and toxic compound production during frying (14). Oil consumption has been
linked to obesity and cardiovascular concerns (32, 62), and acrylamide has been identified as a potential
carcinogen (104, 124). Thermo-oxidized compounds like polar and oligomer compounds, trans-fatty
acids, and heterocyclic amines may be present in fried foods, adding to the health and safety-related
concerns of fried foods (99, 133). These growing concerns have renewed the focus on optimizing existing
frying techniques.
Similar to the physicochemical changes that make fried foods desirable, oil uptake and acrylamide
accumulation depend on the complex interplay of various phenomena occurring during frying.
Understanding and predicting the different transport mechanisms during frying could be the first step
in finding ways to reduce oil uptake, or addressing more advanced problems, such as that of acrylamide
accumulation.
There is extensive work available in the food literature on frying, including different types of experimental
(7, 13, 27, 80, 81) and modeling (19, 35, 46, 88, 140) studies. However, given the complexities involved in
frying, there remains significant scope for further improvement and advancement.
Purely experimental approaches can be time-consuming, expensive, and may generate limited insights.
Mathematical models overcome these limitations; however, they require experimental data for
estimating material properties and model validation. Experimental observations can also help refine a
model by including or dropping specific terms in the equations. This strategy will be highlighted in this
study.
Mathematical models for frying range from phenomenological or empirical models (15, 63, 127) to
continuum-scale mechanistic models (18, 67, 68, 93). Further, processes like baking, microwaving, and
frying involve evaporation. Previous models have described frying with either a sharp-moving
vaporization front (59, 60), or with distributed evaporation in the porous matrix (18, 93). Datta (44, 45)
and Halder et al. (67) discuss some of these models.
In recent years, there has been growing interest in the pore-scale modeling of foods (56, 74, 122, 131).
Pore-scale modeling, however, is challenging for naturally occurring porous structures like food because
of their complex pore geometries. The high computational costs limit the applicability of the models to
smaller sample sizes or simplified geometries (74, 144). Additionally, microscale material properties are
needed, which are difficult to measure and are not readily available in the literature. With advances in
modeling techniques and computing power, pore-scale modeling offers promising prospects for future
use. However, its practicality as a predictive tool is currently limited (123).
The current study uses a continuum-scale mechanistic model to study frying. Frying involves
simultaneous mass and energy transfer coupled with microscopic and macroscopic changes in the food
structure. Rapid heat transfer-driven water vaporization, moisture loss, and concomitant oil uptake in a
porous biopolymeric matrix make frying an intricate unsaturated (pores not saturated with liquids but
also containing gas/vapors) transport problem. Biopolymeric matrices like food have a hierarchy of
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length scales and have the potential to shrink or swell, which further adds to the complexity of model
development. A closer look at the multiscale nature of porous foods is provided in Section 2.1.
Continuum-based porous media models for frying can be found in the works of Ni & Datta (93), Halder
et al. (67, 68), Bansal et al. (17, 18), Sandhu and Takhar (108), and Takhar (120). The previous studies
focused on modeling the heat and mass transfer during frying and were able to predict variables like
moisture loss and temperature change with reasonable accuracy. However, there is room for
improvement on three fronts. First, the previous models did not mechanistically account for the volume
changes of the porous matrix during frying. Second, the predicted pressure during frying needs to be
validated using pressure measurements. Third, the understanding of different oil uptake mechanisms is
still incomplete, and there is a need to update previous models based on newer observations available
in the literature.
Though volume change is viewed as a macroscopic quantity, there are accompanying spatial and
temporal changes in porosity. Porosity is the space ‘available’ to the fluid phases to occupy within the
porous structure. Changes in porosity are expected to affect the transport of the fluid phases (140).
Additionally, a shrinking or swelling matrix represents a moving boundary problem, and the effect of this
must be accounted for in the model and solution methodology. Ignoring volume changes may lead to
significant disagreement between model predictions and experimental results (16). For example, Gulati
et al. (65) showed that coupling volume change with the transport problem improved the agreement
between measured and predicted temperatures for microwave drying of potato cubes.
Unlike saturated porous systems, where the change in volume of the matrix can be assumed to be equal
to the loss of the liquid phase from the matrix (47), estimating volume changes for unsaturated systems
is not straightforward. Vapor generation and gas expansion during frying cause the expansion of the
matrix, whereas water loss causes shrinkage. As a result, there is more than one phenomenon causing
volume change. Most previous frying models ignored volume changes of the porous matrix or, in some
cases, empirically accounted (90, 140) for the volume changes. In this study, we accounted for the volume
changes of the porous matrix mechanistically and used pore pressure as the driving force for these
changes.
Pore pressure is the effective pressure applied on the pore walls by the phases occupying the pores.
Many studies in the field of geological sciences have explored the relationship between pore pressure
and deformation (89, 103, 109, 125, 139). However, its use for modeling volume change in porous food
systems remains rare. An example of this can be found in the work of Ditudompo and Takhar (48), who
modeled the transport mechanisms and expansion during the extrusion of starch. They used pore
pressure as the key variable governing the expansion of the extrudate matrix. Further details about
implementing this approach will be provided in Section 2.5.
Unsaturated porous media models can predict the pressure changes during frying. However, in the
previous frying models, a validation of the predicted pressures is unavailable (68, 108). The peak pressure
predicted in some previous models was either much lower or much higher when compared to the data
available in the literature. The measured peak gauge pressure in previous frying studies was of the order
of 10 kPa (110, 135). The peak gauge pressure predicted by the models of Halder et al. (68) and Ni and
Datta (93) was less than 2 kPa. On the other hand, the model developed by Sandhu and Takhar (108)
predicted that the peak gauge pore pressure would be approximately 200 kPa. Therefore, there is a need
for a model which can accurately predict the magnitude of pressure developed during frying. The impact
of pressure development during frying on oil uptake and volume changes will be discussed in Section
4.
1.1. Study Objectives
The study aims to closely highlight the transport mechanisms involved in frying, and by accounting for
volume changes, we present an advancement of the previously published models. The model is validated
using various measured variables detailed below. The focus is on generating insights into the underlying
physics by analyzing the trends of the different model variables. This is further explored through
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sensitivity tests on certain material properties. Additionally, we assess the model’s limitations and present
steps for future improvements.
The specific objectives of the study were: 1) to utilize a hybrid mixture theory (HMT)-based unsaturated
transport model for conventional frying, 2) to mechanistically account for volume changes during frying,
3) to validate the model using measured data on moisture content, oil content, temperature, pressure,
and volume changes, 4) to improve the understanding of transport mechanisms involved in the frying
of a deforming material using simulation data generated with the validated model, and 5) to perform
sensitivity tests on selected material properties
2. MATHEMATICAL MODEL DEVELOPMENT
2.1. Hierarchy of scales in biopolymeric matrices
Biopolymeric matrices, such as those found in foods, exhibit a hierarchy of length scales (130) (Fig. 1).
At the microscale (~1 μm), the solid biopolymers and the vicinal liquid (water) are seen distinctly (23). At
the mesoscale (<103 μm), the water-imbibed biopolymer is seen as an overlaying continuum of vicinal
water and solid biopolymer, and a distinction between the two constituents cannot be made. Bulk phases
like liquid water, oil, and vapors exist at the mesoscale as non-overlaying continua. At the macroscale
(>103 μm), there is a homogenization of all the bulk phases and water-imbibed biopolymers, forming a
macroscale continuum.
Raw potatoes have a regular honeycomb-like arrangement of cells (134), held together by pectic
substances that make up the outer layer (middle lamella) of the cell walls (40). The starch granules are
present inside the cells. The overall tissue structure (parenchyma) in potatoes is made up of the cells and
extracellular spaces occupied by fluid (liquid or gas) phases (40). In terms of length scales, the pores in
Figure 1: Hierarchy of length scales in porous foods.
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the cell membranes are at the microscale, the cell cytoplasm is at the mesoscale, and the tissue is at the
macroscale (20).
Similar naturally occurring hierarchies are seen in soils (24, 57, 87) and other subsurface structures (149).
However, the swelling and shrinking potential, along with the viscoelastic nature of foods, creates a more
complex and interesting problem.
Phenomena occurring at lower scales often show up as mesoscale or macroscale changes (145). For
example, the hydrophilic nature of solid particles causes water imbibition, leading to their swelling and
subsequent deformation of the macroscopic matrix (21). This encourages the use of upscaling
techniques. A range of upscaling techniques is available, and a description, along with the advantages
and disadvantages of some of these techniques, can be found in Battiato et al. (20) and Cushman et al.
(39).
2.2. Hybrid Mixture Theory
Whitaker’s volume averaging method (138) and Hassanizadeh and Gray’s HMT method (69, 70, 71) are
two commonly used upscaling approaches. In the former technique, the mass, momentum, energy, and
entropy balance equations, along with constitutive equations, are volume averaged to higher scales
(mesoscale or macroscale). Though closed-form relations are obtained for the mesoscale counterparts
of the microscale material properties, their experimental determination at the mesoscale is not
straightforward (20, 21).
Hybrid mixture theory presents an alternate approach wherein only the field equations are upscaled. The
constitutive theory is introduced at the higher scales, and the entropy inequality is utilized in the sense
of Coleman and Noll (37). The advantage of this approach is that the material properties manifest at the
macroscale, allowing them to be determined through relatively simpler macroscale experiments (20). A
limitation of this approach is that the behavior of different material components (e.g., elastic solids,
viscous liquids) is not modeled individually. Instead, their combined behavior is represented in a
simplified form, such as viscoelasticity.
The HMT approach was originally developed by Hassanizadeh and Gray (69, 70, 71) for multiphase
systems. While the earlier HMT works assumed a non-swelling porous medium, Achanta et al. (5)
modified and extended the theory for swelling or shrinking systems by accounting for the interaction
between the phases. A further extension was provided by Bennethum and Cushman (20, 22) by adding
another scale (macroscale) to the problem. This requires the field equations to be averaged twice, from
microscale to mesoscale and then from mesoscale to macroscale.
Previous HMT studies assumed the solid phase to be elastic and the liquid phase to be viscous at the
microscale. At the macroscale, these microscale interactions resulted in viscoelastic behavior with short-
memory effects (137). Microscale relaxation processes in a biopolymeric matrix, such as food, impact
liquid transport through the matrix. However, this effect is overlooked when the solid phase is assumed
to be purely elastic. Singh and colleagues (114, 115) extended the HMT approach by assuming the solid
phase as viscoelastic at the microscale for two-scale and three-scale problems, respectively. The resulting
theory was sufficiently general to model both Fickian and non-Fickian fluid transport, including the long-
memory effects of polymers (114).
Many food processing operations, such as baking and frying, involve unsaturated transport and cannot
be accurately modeled using saturated transport models. To address this, Takhar (120) developed HMT-
based two-scale unsaturated transport equations specifically for biopolymeric systems.
This study utilizes Takhar’s (120) HMT-based two-scale unsaturated transport equations for frying. The
two-scale approach simplifies the three-scale approach by merging the micro and mesoscales, assuming
a continuum for adsorbed and bulk fluid phases (145). The resulting scale can interchangeably be called
the microscale or mesoscale. The field equations defined for this scale are upscaled to the macroscale,
where the entropy inequality is invoked. The resulting system of equations is solved for macroscale
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representative elementary volumes (REVs), which require less computational effort and fewer material
property inputs compared to microscale or mesoscale domains, especially when computations are
performed over the entire material.
Hybrid mixture theory-based models have been successfully applied to many food processes. They have
been used to study saturated transport problems such as drying and sorption (1, 2, 96, 116, 117, 119,
121), as well as unsaturated transport problems like frying (18, 108), freezing (145), and extrusion (48).
2.3. Assumptions
Certain assumptions are made to reduce complexities in the model development. The liquid water, gas,
oil, and solid phases are in a continuum. Gravitational forces are considered negligible for driving fluid
flow compared to the other forces. The gas phase is an ideal mixture of water vapors and air. The solid
and liquid phases are assumed incompressible at the microscale (
=
=
= 0), and the gas
phase is assumed compressible (
0). The implication of this assumption will be seen when the mass
balance equations for each phase are presented (Eq. 16, 17, 18 and 19). The overall matrix is still allowed
to shrink or expand. This is possible because =+++, and changes in the
volume fraction of the phases (,where =,,,) will change the density of the mixture even if
individual phase densities () do not vary (30, 53). Further, we assume that there is a local thermal
equilibrium between the phases, which allows us to use a single energy balance equation (common
temperature) for the mixture of phases rather than individual energy balance equations for the phases
(129). The temperature in the matrix can still vary spatially and temporally.
2.4. Field equations
2.4.1 Mass and Momentum Balance Equations
The fluid phases present in the pores are liquid water (), oil (), and gas (). The gas phase is composed
of water vapors () and air (). The two-scale mass balance equation for a fluid phase (where, =
,,) is given by Equation 1 (120):
()
+.(,)
=
, ,=,, (1)
wherein the two-scale mass balance equation for the solid phase (Eq. 2) (120), was used to change the
material time derivatives with respect to the phase (/) to material time derivatives with respect
to the solid phase (/).
+(.) = 0 (2)
The material time derivatives in the equations are also represented by a dot over the variables, for
example,
. represents the gradient operator in Eulerian coordinates. , and , represent the
volume fraction, density, and velocity of phase with respect to the solid phase, respectively. The term
on the RHS of Equation 1,
, is the source/sink term that represents the mass transfer from the
phase to the phase. The source/sink term couples the mass balance equations for different phases and
ensures that mass is conserved. This term is characteristic of multiscale equations and is generally absent
in single-scale equations (17). Evaporation changes liquid water to vapor, whereas condensation does
the opposite. As a result,
=
=
. Since the oil phase does not exchange mass with other
phases,
and
are equal to zero. Similarly, the solid phase does not exchange mass with the fluid
phases ( = 0
).
The generalized Darcy’s law for the liquid phases is given by Equation 3 (120), where , , and,
are the permeability, dynamic viscosity, pressure, and diffusivity of phase , respectively. is the
mixture viscosity of the biopolymeric matrix. The flow of phase is governed by the pressure gradient
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(permeability controlled), concentration gradient (diffusivity controlled), and the time-dependent
resistance of polymers to fluid flow, given by the mixture viscosity term.
,=
B, =, (3)
The gas phase velocity is given by Equation 4,
,=
(4)
and the vapor phase velocity is given by Equation 5,
,=,+,
(5)
where the second term on the RHS is due to binary diffusion of vapors with respect to gas (air+vapor
mixture) and is expressed as Equation 6 (25),
,=
(6)
where is the vapor diffusivity. is the mass fraction of vapors in the air-vapor mixture and is given
by, =
, where and are the density of vapors and air, respectively.
2.4.2 Equations linking different variables and constitutive relations
Some of the variables which appear in the equations above are dependent on each other, and the
equations relating those variables are discussed in this section. The total volume of an REV is occupied
by the solid phase and pores. The porosity () represents the total void space available for fluids to
occupy (pores), and consequently, we can write Equations 7 and 8 as:
+= 1
(7)
++=
(8)
Equation 7 can be used to replace and
in Equation 1 with 1 and , respectively. Water
vapors and air form a gas mixture in the pore volume, and this gives, ==. Dividing Equation
8 by gives Equation 9,
++= 1
(9)
where , and, are the degrees of saturation for the liquid water, gas, and oil phases, respectively.
Further, from Dalton’s Law of partial pressures and the Ideal Gas Law we have Equation 10 and Equation
11:
=+
(10)
=
(11)
Here is the temperature,
is the molecular weight of water vapors, and is the universal gas
constant.
The source/sink term, which determines the rate of evaporation at different spatial points, is given by
Equation 12 (17, 67, 142), where () is the evaporation rate constant, and
is the equilibrium
vapor density.
=(
)
(12)
Increasing is expected to lead to faster evaporation, and for very high magnitudes of , an equilibrium
would exist between the water vapors and the liquid water. Zhang et al. (143) estimated the value to
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be of the order of 1 for evaporation of pure water. The value for is expected to vary depending
on the material and process (18, 142), but there is little information for it available in the literature. We
inversely estimated the value to be about 1.4 and performed a sensitivity test to show the
significance of the value. The results of the sensitivity test are presented and discussed in Section 4.3.
It is interesting to note that, when implementing the transport partial differential equations (PDE),
appears in the source term. Large magnitudes of the source term often led to difficulties in convergence.
The difference between gas pressure and pressure in the liquid water and oil phases is given by water-
air and oil-air capillary pressure, respectively. These relations are expressed as Equations 13 and 14,
where and are the water-air and oil-air capillary pressures, respectively. Further discussion on
capillary pressure is provided in Section 2.8.1.
=
(13)
=
(14)
The pore pressure is the effective pressure on the pore walls and is given by Equation 15 (52):
=++
(15)
Next, by combining Equations 1 to 7, we obtain the equations for liquid water (Eq. 16), oil (Eq. 17), gas
(Eq. 18), and water vapors (Eq. 19). Equations 16 to 19 are solved for ,,, and , respectively.
.()
++
+
1=1
(16)
.()
++
+
1= 0 (17)
+.()
+
1=
(18)
+.()
+
+
1=
(19)
2.4.3 Energy balance equation
Three factors contribute to the efficiency of frying as a heating technique. First, the complete immersion
of food in a bulk heating liquid acts as an energy reservoir, ensuring uniform heating. Second, the high
heat capacity and density of oil make it a superior heating medium compared to hot air or steam (135).
Third, the vigorous bubble generation during frying enhances the turbulence of the surrounding oil,
which increases heat transfer rates. Additionally, the thermal expansion of oil creates buoyancy forces
(105), promoting improved oil circulation as well as uniform mixing and recycling of hot oil. The
combination of these factors result in frying having some of the highest heat transfer rates among
common food processing operations, with the peak magnitude of the convective heat transfer coefficient
during frying being on the order of 10 / (75).
Inside the food, heat is transported by both conductive and convective modes. Additionally, rapid
evaporation of liquid water, particularly near the surface layers, occurs as these layers heat up the fastest.
The two-scale convection-diffusion equation, which was solved for the temperature, , is given by
Equation 20 (23, 94, 120),
+,.=. ()
(20)
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where,
=
,,,
=
,,
=
,,,
Here is the specific heat, is the thermal conductivity, and is the latent heat of vaporization. The
two terms on the left hand side of Equation 20 represent energy storage and heat convection,
respectively. The first term on the right hand side is the heat conduction term, whereas the second term
is the phase change term. Since there are substantial temperature variations during frying, thermo-
physical properties like thermal conductivity, specific heat, density and viscosity are expected to vary
with temperature. The thermo-physical properties used in the current model are summarized in
Table 1.
Table 1: Material properties.
Property Expression or value Reference
Thermal
conductivity
Water
0.869 + 8.949 ×101.587 ×10+
7.975 ×10
132
Vapors
1.317 ×10+ 5.150 ×10+ 3.896 ×
10
1.368 ×10
50
Solid
0.21
67
Oil
0.17
67
Air
2.276 ×10+ 1.155 ×107.902 ×
10+ 4.117 ×107.439 ×
10
132
Gas
+
where
=
,
= 1
=
1 +
.1 +
.
.
=
1 +
.1 +
.
.
25
Specific heat
Water
12010.147 80.407+ 0.30995.3819 ×
10
+ 3.6254 ×10
141
Vapors
13604.7344 90.4303506+
0.277355664.21264496 ×10+
3.18369497 ×109.56147519 ×
10
50
Solid
1650
67
Oil
2223
140
Air
1047.64 0.3726+ 9.4530 ×10
6.0240 ×10
+ 1.2859 ×10
132
Gas
+
where
=
,
= 1
34
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Density
Water
273.15 <<293.15 : 950.704 +
18.9236.037 ×10+ 6.309 ×10
>293.15 : 432.257 + 4.9691.340 ×
10
+ 1.033 ×10
76
Solid
1528
60
Oil
1106.1158 0.6414
55
Air
Ideal gas law
Vapors at
equilibrium
Ideal gas law
Vapors in oil
Ideal gas law
Dynamic
viscosity
(.)
Water
273.15 <<413.15 : 1.380 0.0212+
1.3605 ×104.6454 ×10+
8.9043 ×109.0791 ×10+
3.8457 ×10
413.15 <<553.75 : 4.0124 ×10
2.1075 ×10+ 3.8577 ×10
2.3973 ×10
132
Vapors
1.4202 ×10+ 3.8346 ×10
3.8522 ×10
+ 2.1020 ×10
50
Oil
0.1569 ×10
108
Air
8.3828 ×10+ 8.3572 ×10
7.6943 ×10+ 4.6437 ×10
1.0659 ×10
132
Gas
+
25
Molecular
weight
Vapors
0.018
100
Dry air
0.02897
100
Gas
constants
Ideal gas
constant
= 8.31446
100
Gas constant
for air
=
Gas constant
for vapors
=
Diffusivity
Water
0.9 × 10
..
Based on Ni
(92)
Vapors
(2.775 ×10+ 4.479 ×10
+ 1.656 ×10
)
91, 92, 142
Permeability
()
Gas
=
where
= 3 × 10
()
,
= 1
1.1
61, 67, 93
Water
=
4, 120
Oil
=10
67
Viscoelastic
Parameters
Viscosity of
dashpot in
Maxwell
element
= 1.72 ×10 .
107
Viscoelastic
parameter ()
for spring in
Maxwell
element
0.4:
=
,
(0.1297 +
1.6967)10
> 0.4: =,(0.8823
0.1849)10
Equation
fitted to the
data of
Sandhu &
Takhar (107)
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,
0.33
Estimated
inversely
,
4.0969 ×10
107
Young’s
modulus (E)
118
Bulk modulus of
the spring in the
Maxwell
element
=
3(12)
118
Latent heat of
vaporization
= 2.26 ×10
97
Evaporation
rate constant
= 1.4
Estimated
inversely
Mixture
viscosity
= 0.8 × 10 ×
.
120
Poisson ratio
= 0.492
51
2.5. Equations for volume changes during
frying
Phenomena such as water loss, gas expansion, and rise in
capillary pressure lead to the development of mechanical
stresses within the porous food material. The pore pressure
is used as the variable governing the deformation of the
porous food matrix. The stress due to pore pressure ()
represents the stress exerted on the pore walls by the fluids
occupying the pores, as given by Gray and Schrefler (64) (Eq.
21):
=
(21)
Many fruits and vegetables, including potatoes, have been
shown to exhibit viscoelastic behavior (11). Viscoelastic
materials show both viscous (fluid-like) and elastic (solid-
like) properties when undergoing deformation. Such
materials respond to applied stress in a time-dependent
manner. Here, we use Maxwell’s model (Fig. 2) to study the viscoelastic response of food during frying.
Maxwell’s model is an in-series combination of a spring and a dashpot. The stress acting across the
spring and the dashpot is the same and is equal to the stress applied across the combination. The total
volumetric strain (,) across the combination is given by Equation 22, where , is the strain
across the spring and , is the strain across the dashpot.
,
=
,
+
, (22)
The strains across the spring and dashpot can be calculated as follows:
,=
(23)
,
=
(24)
where, is the volumetric stress, is the bulk modulus of the spring, and is the viscosity of the
dashpot. is calculated using the pore pressure, which acts normally in all directions:
Figure 2: Maxwell’s model for
viscoelasticity.
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=
+
+
3 (25)
The bulk modulus of the spring and viscosity of the dashpot were calculated based on the viscoelastic
model parameters estimated by Sandhu and Takhar (107) as a function of moisture content for fried
potato discs (Table 1) .
2.6. Conversion from Eulerian to Lagrangian coordinates
The pore pressure-driven spatial and temporal changes in the structure of the porous matrix make frying
a moving boundary problem. Solving the equations using an Eulerian framework would require a
continuously deforming mesh, which would increase the computation times (119). An alternative
approach is to transform the equations to Lagrangian coordinates and obtain the solution using a
stationary reference material configuration (119). The initial state of the system (at = 0 ) is chosen as
the reference frame. The details about the conversion from Eulerian to Lagrangian coordinates are
presented in the Appendix (available online here).
=1
() = 1
1
(26)
The gradient operator in Eulerian coordinates, , can be converted to the gradient operator in
Lagrangian coordinates () using Equation 26, and is used to replace in Equations 16 to Equation
19 with .
2.7. Initial and boundary conditions
The initial volume fraction of oil, , in the porous matrix was taken to be 0 as raw potatoes have
negligible oil content. The initial volume fraction of gas in raw potato, , is about 0.05 (113, 148). The
initial mass fraction of liquid water on a dry basis can be represented as Equation 27:
=
(27)
Rearranging, we obtain Equation 28:
=
=
(1
)
(28)
Further simplifying Equation 28 to find an explicit expression for , we obtain Equation 29:
=
(1
)
+ (29)
For = 4.20 g water/g solids (110), using Equation 29, we get = 0.822. The initial porosity can
be determined using, =++. The frying is carried out under atmospheric pressure, and the
initial gas pressure in the matrix is =. The initial temperature inside the matrix is =291.5
(110), and the initial vapor density is estimated as =,
.
The following boundary conditions were used:
=
for Equation 16 (30)
=
(
)
for Equation 17 (31)
=
for Equation 18 (32)
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=
(
)
for Equation 19
(33)
=
(
)+
for Equation 20
(34)
The value of the constants, and heat and mass transfer coefficients that appear in Equations 30 to 34
are summarized in Table 2. The volume fraction of water on the surface of the sample is assumed to be
equal to the volume fraction of water in the bulk oil (Eq. 30). The mass flux of oil is driven by the oil
saturation gradient (Eq. 31). The frying is carried out at atmospheric pressure, thus the gas pressure at
the surface is equal to 1 (Eq. 32). The mass flux of vapors is driven by the vapor density gradient
(Eq. 33). Heat transfer at the surface is driven by convection and radiation, represented by the first and
second terms on the right hand side of Equation 34, respectively. The no-flux boundary conditions are
used for the boundaries lying on the lines of symmetry (Fig. 3A).
Table 2: Supporting expressions required for solving the model.
Property
Expression or value
Reference
Saturated
vapor
pressure =.
+21.2409642 2.711193 ×10+
1.673952 ×10+ 2.433502()
100
Water
activity =
( )
where = 0.867 ×10
, = 0.825, = 0.0575
136
Vapor
pressure at
equilibrium
=
34
Water phase
capillary
pressure
=101325
exp(
)+(1
)
1
2.79 ×10(273)
where
=
, = 0.94,=27.12,= 5.939,= 0.782,=
1.4 (estimated inversely)
26, 78
Figure 3: (A) Schematic diagram of the potato cylinder showing transfer mechanisms and volume change during
frying; (B) meshed 2D axisymmetric geometry.
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Oil phase
capillary
pressure
=
where = 0.024
, =12 ×10 ,=38°
31, 101
Atmospheric
pressure
=101325
100
Vapor
pressure in
oil
= 0.00001
Negligible
value
assumed
Mass
transfer
coefficient
for vapors
= 0.2
(..)
Adapted
from Halder
et al. (67)
Mass
transfer
coefficient
for oil
=
×,
where = 0.82
, =
, =
, , = 5 × 10
,
,= 3.5 × 10
38, 100
Liquid water
volume
fraction in
bulk oil
= 0.01
120
Oil
saturation in
bulk oil
= 0.99
Estimated
using
value
Heat
transfer
coefficient
=144 904 /
Data of Farinu and Baik (2007) were used
58
Temperature
of the oil
at the start of frying was 193°C. Measured temperature profile of the
bulk oil was used in the model.
Measured
Emissivity
= 0.9
84
Stefan-
Boltzmann
Constant
= 5.670373 ×10
100
Material properties
The material properties used in the model are summarized in Table 1 and Table 2. In the next section,
we provide a discussion of some of these properties.
2.8.1 Capillary pressure
Capillary pressure is well understood and extensively researched in fields dealing with unsaturated fluid
flow, such as wood, soil, and geological sciences. However, despite its relevance to unsaturated transport
and capillary pressure-driven liquid movement in many porous foods, there is a lack of capillary pressure
data specifically for food systems. The soft nature of food limits the application of methods that use high
applied pressures or centrifugal forces to drive liquid flow from the porous matrix (111). Furthermore,
foods may undergo microbial contamination or swelling during the equilibration time (128). In the
absence of appropriate capillary pressure data, mathematical models for foods have relied on
simplifications, such as assuming a combined diffusion-like transport for both liquid water and gas
phases (33, 112), or employing a capillary diffusion approach (65, 67). Regardless of which simplification
is used, it is still necessary to estimate a diffusivity value, ideally as a function of moisture content and
temperature. Shah and Takhar (111) reviewed the commonly used mathematical models for unsaturated
food systems and discussed their assumptions and limitations.
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In the absence of capillary pressure data for food systems, we adopted the following equation (Eq. 35)
as developed by Bonneau and Puiggali (26) for pinewood:
=101325
exp(
)+(1
)
12.79 ×10
(273) (35)
Here, the water phase capillary pressure ( ) is given as a function of the saturation of water and
temperature (78). The values of the fitting parameters (, , , and ) are summarized in Table 2.
2.8.2 Permeability and Diffusivity
According to Darcy’s Law (Eq. 3), permeability controls the pressure gradient-driven transport in porous
media. As a result, permeability values are essential to our model solution. The complex structure of food
and the dynamic nature of the changes that occur during processing makes the estimation of
permeability difficult. Datta (43) estimated the permeability for saturated samples of raw potato slices
using a pressure-based method. The flow rate of water moving through a sample under pressure is
measured, and the permeability is estimated using the Darcy’s law. Datta (43) found that the permeability
of potatoes varies between 10 to 10 . An increase in permeability with applied pressure was
observed for some samples. Additionally, swelling was observed in the potato samples during the
experiments, which could affect the estimated permeability values. These factors can explain the
variability in permeability data reported in this study (43) and other permeability estimation studies for
foods.
Dadmohammadi et al. (40) estimated the permeability of saturated raw potato discs using the pressure-
based direct estimation method, an indirect nuclear magnetic resonance (NMR)-based method, and a
phenomenological approach based on approximating the porous media as a bundle of tubes. The latter
method is also referred to as the Carman-Kozeny approach. The permeability of raw potato samples was
found to be of the order of 10 using the three methods.
A range of temperature-dependent changes accompanies the heating of potato samples, including gas
and liquid expansion, cell swelling, cell rounding, cell separation, starch gelatinization, and, eventually,
cell rupture (40). Some of these changes increase permeability, whereas others decrease it. The overall
change in permeability is thus a cumulative result of these phenomena. The gas (air, water vapor, or
from respiration) and liquid (water) present in the extracellular spaces expand upon heating, which is
expected to increase the volume of the pore space, thereby increasing the permeability for extracellular
fluid flow.
The pectin-mediated cell adhesion (41) undergoes changes at higher temperatures, and eventually, there
is cell separation. Cell separation provides more pathways for fluid movement, and as a result, the
permeability increases further. At elevated temperatures between 50 to 100°C, Dadmohammadi et al.
(40) estimated the water permeability to be approximately 10 . Using the pressure-based direct
estimation method, Alam et al. (9) found the water permeability values for potato discs to be between
2 × 10 to 4 × 10 for samples heated to temperatures between 70 to 80°C. In Alam et al.
(9), pressures up to 345 were applied to drive fluid flow.
On the other hand, the swelling of cells and increased turgor pressure due to water uptake can restrict
the extracellular movement of fluids, which is expected to decrease permeability. When cooking times
are longer, cell rupture may occur. The cell contents, including the starch granules, are released into the
extracellular spaces. As a result, the available pore volume and pathways for fluid flow may decrease,
and thus cause a decrease in permeability.
In this study, Equation 36, which appears in the HMT-based derivation of transport equations (4, 120),
was used to calculate the water permeability:
=
(36)
The liquid water diffusivity () expression based on Ni (92) was used and is listed in Table 1. Based on
the above equation, the value used in this study varied spatially and temporally, and ranged from
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approximately 10 to 10 . This is within the range of the experimentally estimated water
permeability values for potatoes (9, 40, 43). Sensitivity testing was performed for and the results are
discussed in Section 4.3.
Very little information is available on oil permeability () in porous foods. A value of 10 (67)
was used in the model, and sensitivity tests were performed to show the effect of the value on the
estimated oil content. Studying oil transport during frying has proven to be complicated. Oil penetration
into the porous matrix is significantly affected by the food’s microstructure and how it evolves with frying
time.
Bouchon et al. (28) showed that only a minor fraction of oil penetrates potato samples during the
immersion phase of frying, whereas significant oil uptake occurs during the cooling phase. Three oil
fractions were identified: structural oil (absorbed during the immersion phase), penetrated surface oil
(penetrated during the cooling phase), and surface oil. The surface oil was determined by washing the
fried potato samples in petroleum ether for 1 s. The total oil content of the fried potato samples
(excluding the surface oil) was determined by the Soxhlet method. Oil dyed with Sudan Red B was added
towards the end of frying as an indicator, and the dye concentration in the extracted oil was determined
by measuring the absorbance at 509.6 . The dye-stained oil represented the penetrated surface oil;
the remaining portion of the extracted oil was the structural oil. It was found that the structural oil
accounted for less than 5% of the final oil content in the fried samples (28). On the other hand, the
penetrated surface oil content increased with frying time and accounted for up to 89% of the total oil
absorbed by the samples. It was concluded that surface damage and porosity increase with frying time,
allowing more oil to penetrate during the cooling phase (28). Therefore, while only a minor fraction of
oil enters deeper in the food during the immersion phase, the frying-driven changes in the food
microstructure have a significant impact on the oil content of the final product. Additionally, pressure
and temperature changes during frying play a critical role in governing the oil uptake (135).
These observations motivated numerous studies focused on elucidating the oil uptake mechanisms,
including those of He et al. (73), Patsioura et al. (98, 99), Touffet et al. (126), Vauvre et al. (133, 134), and
Ziaiifar et al. (147). While there may not be a complete consensus on the relative importance of all
mechanisms, pressure-governed oil transport is considered to be the most important mechanism.
Vaporization, gas expansion, and restriction to the growth of water bubbles by the solid matrix (146)
lead to high pressures in the porous matrix during frying. The high pressures are expected to prevent oil
penetration.
Using an oil dyeing technique, Touffet et al. (126) found that thawing of parfried-frozen potato samples
reduced oil absorption during the immersion stage of frying. It was suggested that the presence of ice
in frozen samples impacted the pressure development. Compared to fresh or thawed samples, the
pressure rise in frozen samples is delayed, allowing the oil to enter the food (99). This highlighted the
significance of pressure-governed oil transport.
However, there is a lack of evidence supporting the diffusive transport of oil during frying. It is common
to find Fick’s Law of diffusion used to model oil transport in the frying literature (15, 35, 42). Such models
define an effective oil diffusivity, which is determined experimentally. The caveat here is that distinct
mechanisms of oil uptake are lumped together as effective diffusion. Fick’s Law-based modeling of
lumped diffusive transport is only analogous to concentration gradient-driven diffusion.
Diffusive transport of oil, in the absence of pressure-driven transport, would make the oil uptake
continuous (99). This, however, is contrary to the discontinuous oil uptake behavior observed
experimentally (73). Additionally, the penetration depth of oil estimated based on diffusive transport
ranged from a few millimeters up to centimeters (98). This conclusion is not entirely accurate, as
experimental studies have shown that oil penetration in potatoes is restricted to just a few layers beneath
the surface (28, 29).
Insights into oil transport in porous materials can also be drawn from petroleum and soil sciences. Oil
spills during crude oil exploration, transportation, and processing contaminate the environment and
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groundwater (79, 82). This poses safety concerns and has led to numerous studies on the movement of
spilled crude oil in soils (3, 72, 102). Oghenejoboh and Puyate (95) conducted experiments to analyze
the movement of six crude oil samples with different physical properties spilled in top (loamy-sand) soil.
During a 25-day period, the oil did not travel beyond a few centimeters from the point of spillage in the
horizontal (longitudinal) direction. In contrast, the oil movement in the vertical direction was significantly
pronounced for all samples. Oghenejoboh and Puyate (95) concluded that the horizontal movement of
crude oil, governed by concentration gradient-driven diffusion, was negligible compared to vertical
movement, which, in addition to concentration gradient, is driven by gravitational forces.
Similar studies for analyzing oil movement in porous food materials are encouraged to resolve the
significance of diffusive transport, although its magnitude is expected to be low. Given the current lack
of evidence supporting diffusive transport compared to pressure-driven flow of oil during frying, we
exclude the diffusion term from the Darcy’s law (Eq. 3) for oil. By doing so, we also highlight the agility
and adaptability of mechanistic models. The importance of experimental studies in complementing and
guiding model development is further emphasized.
Gas flow plays a critical role in unsaturated transport processes because the gas pressure gradient drives
the movement of the liquid phases (water and oil phases in the case of frying). Additionally, evaporation
and subsequent movement of vapor affect heat transfer within the porous matrix. Two properties
required for modeling gas (air and vapors) transport are gas permeability () and vapor diffusivity ().
The following expressions (Eq. 37, 38, and 39)were used for calculating gas permeability (61, 67, 93):
=
(37)
= 3 × 10
(1) (38)
= 1 1.1
(39)
where is the intrinsic gas permeability and is the relative gas permeability. The minimum value
of was restricted to a non-negative, non-zero value of 0.05, estimated inversely. The vapor diffusivity
() expression of Nellis and Klein (91) was used in the model and is provided in Table 1.
3. MATERIALS AND METHODS
3.1. Experimental procedures
Potato cylinders, 44 mm in length and 18 mm in diameter, were cut from fresh whole potatoes using a
stainless-steel fruit corer. The cut samples were then blanched in boiling water for 2 minutes before
frying. The samples were fried in a commercial blend of frying oil at a temperature of 193°C for a
maximum duration of 300 s. The fried samples were immersed in liquid nitrogen for 10 s to arrest the
cooking. The moisture and oil content of the samples were determined using the convection oven
method and the Soxhlet method, respectively. The sample weights were recorded using an electronic
balance1 with a maximum error of ±1.0 . Real-time temperature and pressure measurements were
made at the center of the sample. The temperature measurements were made using three K-type
thermocouples 2 with an accuracy of about ±1.1 ° inserted in three samples. The pressure
measurements were made using a fiber optic pressure sensor3 with an error of ±5 . The sensors
were connected to data loggers, and the extracted data was stored in Microsoft Excel for further analysis.
Fried samples continued to undergo volume changes after removing them from the frying oil. It is thus
vital that volume measurements of the sample are made quickly. To ensure rapid volume measurement,
1 U.S. Solid, Cleveland, Ohio, U.S.A. (model no. USS-DBS8)
2 PerfectPrime, New York, New York, U.S.A.
3 FOP-MH-NS-556A: FISO Technologies Inc., Quebec, Canada
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a liquid displacement technique was used with n-Heptane (=679.5 / at 25°C) as the solvent.
Additional details about the experimental procedures can be found in Shah and Takhar (110).
3.2. Numerical solution
The finite element method (FEM) was used to solve the mathematical model. The FEM approach is a
numerical technique that is capable of solving a wide range of PDEs (77). It can be employed for complex
geometries and can be utilized for coupling equations for various physics modes such as heat transfer,
porous media, structural mechanics, etc. A commercial FEM package4 was used to set up the model,
solve the equations, and post-process the results. Instead of solving for the entire 3D cylindrical
geometry, the model was solved for an axisymmetric rectangular geometry (2D) (Figure 3A). By
assuming symmetry about the horizontal line passing through the center of the cylinder, the model was
solved only for the upper half of the original geometry. This approach significantly reduced the
computational load, making it feasible to conduct additional simulations.
A crucial step in the FEM approach is the discretization of the computational domain. A mapped mesh
(Figure 3B) with 2000 quadrilateral elements, 180 edge elements, and 4 vertex elements was used. A
non-uniform element distribution (element ratio of 3 and 5 in and directions, respectively) was used
to have more elements near the surface exposed to oil. Rapid changes are expected in the surface layers,
and a higher element density is needed to resolve such changes optimally. A mesh independence study
was performed to ensure that the model results do not vary significantly with the number of elements.
The number of elements in and directions were changed by ±10%. Decreasing the number of
elements by 10% in both directions resulted in a coarser mesh with 1620 quadrilateral elements, i.e., a
19% decrease in the total number of elements. Increasing the number of elements by 10% in both
directions resulted in a finer mesh with 2420 quadrilateral elements, i.e., a 21% increase in the total
number of elements. Simulations conducted using the coarser and finer mesh led to an average change
in the value of dependent variables by 1.83% and 1.30%, respectively. Therefore, the mesh with 2000
quadrilateral elements was considered optimum for conducting simulations.
4 COMSOL Multiphysics® v. 6.1: COMSOL, Inc., Burlington, MA, USA
Figure 4: Comparing the predicted and measured ( = 3 replicates) moisture content and
oil content at different frying times. MAE = mean absolute error.
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The backward differentiation formula (BDF) method, which is an implicit method for numerical
integration, was used for time-stepping. The maximum and minimum order for the BDF method was set
to 2 and 1, respectively. The solver was enabled to select the appropriate step size automatically.
MUltifrontal Massively Parallel Solver (MUMPS) (12), which is a direct solver based on Gaussian
elimination, was used for solving the sparse linear equations, which are characteristic of utilizing the
finite element method (77). The automatic Newton method was selected as the nonlinear method in the
software. The solver was allowed to automatically determine a damping factor for each iteration of
Newton’s method. It took 12 to 18 mins for each run to simulate 300 s of frying time on a workstation
with an AMD Ryzen 7 processor with a clock speed of 3.60 GHz and 32 GB RAM.
4. RESULTS AND DISCUSSION
4.1. Results
The model predictions were validated using the experimental results available in the frying literature,
particularly those of Shah and Takhar (110). The percent mean absolute error (%MAE) was calculated,
and a good agreement was found between the model predictions and experimental results (Fig. 4, Fig.
5, Fig. 6). The %MAE for moisture content, oil content, and temperature is 5.57%, 22.42%, and 13.35%,
respectively. The model performance is expected to improve with the availability of more data for
material properties like permeability and capillary pressure. The potential reasons for some of the model
inaccuracies and recommendations for improving model performance are discussed in more detail in
Section 4.4.
The sample temperature, pressure, and volume continued to change after the sample was removed from
the fryer (110). The fried potato cylinders continued to shrink after removal from the fryer as the pressure
in the porous matrix dropped due to water vapor collapse. Shah and Takhar (110) measured volume
change using a modified liquid displacement method which allowed for rapid volume measurements.
However, even with the rapid method, it is not feasible to instantaneously capture the sample volume
after frying. As a result, differences between measured and model-computed volume changes are
expected (Fig. 6).
Spatial profiles were created for model variables like moisture content, oil content, pressure, and
evaporation rate (Fig. 7, Fig. 8, Fig. 9, Fig. 10). The spatial profiles were plotted as a function of the
Lagrangian coordinates. To generate such profiles experimentally would take extensive testing, which
Figure 5: (A) Comparing the predicted and measured ( = 3 replicates) temperature profile at the center of the
sample; (B) pore pressure profile at the center of the sample. MAE = Mean absolute error.
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may be expensive and is not always
feasible. The spatial profiles
provide an intuitive understanding
of the different phenomena
occurring during frying. These
profiles are discussed in detail in
the next section.
4.2. Discussion
During frying, phenomena such as
water loss, oil penetration,
temperature rise, pressure
changes, and volume changes
occur simultaneously, with intricate
coupling occurring between the
different variables. This complexity
makes modeling challenging, but
the insights generated from this
exercise are valuable. To highlight
the coupled nature of these
phenomena, we describe the frying
process as a series of changes beginning with the rapid heat transfer from the hot oil.
There is a steep drop in moisture content during the early stages of frying (Fig. 4, <25 ). The rapid
heat transfer to the moisture-rich potato samples causes a rapid vaporization of the surface water, which
is lost to the bulk oil. Some authors refer to this stage as superficial or surface boiling (59, 135). The
bubbles initially seen around the food are that of the vaporized surface water.
The initial moisture loss from the food is controlled by the external heat transfer. While the steam
bubbles lead to more oil movement, their crowding around the food adds to the heat transfer resistance
(105). The bubble velocity, frequency, flow direction, and oil agitation are important factors affecting the
heat transfer rate (58).
There is rapid oil uptake during the initial frying stages (Fig. 4, <50 ). Injuries to the cellular structure
near the surface are unavoidable during sample extraction from whole potatoes. This, combined with
pores left empty by the escaping water from the surface, allows the entry of oil. During the initial phases,
Figure 6: Comparing the predicted and measured ( = 3 replicates)
volume change of fried samples at different frying times. MAE = Mean
absolute error.
Figure 7: (A) Water phase capillary pressure () and (B) water pressure () as a function of Lagrangian radius.
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when the pressure in the core is low, there is little opposition to the oil seeping into the surface layers.
Thus, a steep increase in oil content is expected. Durán et al. (49) recorded an oil content profile of a
similar nature during the early stages of frying of potatoes.
The core temperature remains almost constant initially (Fig. 5A, <50 ). Heat is transported towards
the core by conduction across the porous matrix and by the convection of the fluid phases. While the
external heat transfer is fast, the internal heat transfer is comparatively slower. Additionally, the
vaporization of water provides a cooling effect, which also prevents the charring of the food (54). As a
result, the temperature at the center does not start increasing immediately. This is confirmed by the
measured temperature profile (Fig. 5A). The pore pressure at the center shows a similar behavior as its
magnitude remains almost unchanged for some time (Fig. 5B, <75 ). Vitrac et al. (135) recorded
similar temperature and pressure trends during the frying of alginate gel cylinders formulated without
starch.
The potato samples undergo shrinkage during frying (Fig. 6), and the volume change is rapid, especially
when the pore pressure in the core is low. The loss of water from the surface layers leads to high
magnitudes of capillary pressure () (Fig. 7A). Consequently, the water pressure (=) in
the surface layers becomes negative (Fig. 7B), and this causes the gauge pore pressure ( =
++) in the surface layers to become negative (Fig. 8A). Negative gauge pore
pressure leads to rapid contraction of the surface layers, and as a result, the porosity near the surface
decreases (Fig. 8B). Comparatively, the porosity of the core does not undergo significant changes (Fig.
8B). The average porosity of the sample decreases sharply during the initial stages of frying (Fig. 8C)
due to the shrinking surface layers. A decrease in porosity near the surface results in the crunchy crust
French fry consumers prefer.
Figure 8: (A) pore pressure (), (B) porosity (), (C) average porosity (), and (D) evaporation rate as a
function of Lagrangian radius (R).
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The evaporation rate radial profile (Fig. 8D) illustrates the existence of an evaporation zone instead of a
commonly assumed sharp boundary or interface. The evaporation rate has a significant value close to
the surface due to the proximity to the frying oil. Evaporation occurs across a region at different rates,
referred to as the evaporation zone here, instead of being isolated to a sharp interface within the sample,
as confirmed experimentally by Lioumbas and Karapantsios (83). Additionally, the evaporation zone
expands with increasing frying time, and the evaporation peak shifts towards the core, which is consistent
with the observations of Lioumbas and Karapantsios (83).
After an initial lag, the temperature in the core of the sample begins to rise (Fig. 5A, >50 ). As the
surface temperature rises, the thermal gradient between the bulk oil and the surface of the sample
decreases, leading to a decrease in the external convective heat transfer rate. Meanwhile, heat flux into
the core is maintained by the thermal gradient between the surface and the core of the sample, causing
the core temperature to increase. The temperature profile then peaks and plateaus (Fig. 5A).
The water loss from the porous matrix continues, albeit at a slower rate, and a moisture gradient develops
across the porous matrix (Fig. 9A). Water from the core, driven by pressure and concentration gradient,
moves towards the surface. The internal mass transfer of water is slower than the water loss from the
surface, resulting in a moisture gradient.
The gas pressure increases as frying progresses (Fig. 9B). The temperature rise brings about vapor
generation and gas expansion in the core of the porous matrix. The gas trapped inside the biopolymeric
matrix exerts pressure on the pore walls. With increasing temperature, the gas exerts greater pressure
on the pore walls. Additionally, the peak of the gas pressure profile shifts towards the core with
increasing frying time (Fig. 9B). This is expected to affect the pore size distribution and pore network
development.
The increase in gas pressure causes the pore pressure to rise (Fig. 5B, >75 and Fig. 8A). The
absolute pore pressure in the model peaks at 120.49 . The peak pressure, as measured by Shah
and Takhar (110), was 134.32 . The model was able to predict the peak pressure during frying
reasonably well with a relative difference of 10.30% between the predicted and measured peak pressure
values.
The increase in pore pressure in the core (Fig. 5B, Fig. 8A) decreases the shrinkage rate of the potato
cylinder (Fig. 6). The negative gauge pore pressure near the surface causes contraction of the matrix,
but the increase in pore pressure in the core during the intermediate stages of frying resists the
contraction. This slows down the overall rate of shrinkage.
Figure 9: Radial profile of (A) moisture content and (B) gas pressure () as a function of Lagrangian radius (R)
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Similarly, after the initial sharp drop in average porosity, its magnitude remains almost constant for a
short time and then continues to decrease but at a slower rate (Fig. 8C). Alam and Takhar (8) estimated
the porosity changes during the frying of thin potato disks (45 mm diameter and 1.65 mm thickness)
using X-ray micro computed tomography (micro-CT) imaging method and also using conventional
laboratory techniques like pycnometry. The sample porosity dropped rapidly until a frying time of 20 s
was reached, similar to the initial sharp drop in porosity predicted by the current model. The porosity
then increased for the remaining frying time (20 – 80 s). The overall percentage change in porosity after
80 s of frying, measured experimentally, was approximately -2.2% (8). The overall change in average
porosity predicted by the current model was approximately -5.06% (Fig. 8C).
After the initial rise in oil content, the profile for oil content plateaus (Fig. 4, =50 to 175 s). The rise
in gas pressure and pore pressure in the core of the porous matrix is expected to restrict oil penetration.
The plateauing of the oil content profile during the middle stages of frying was also seen by Durán et al.
(49) and He et al. (73).
During the immersion phase of frying, a major fraction of the oil remains on the surface while a small
fraction enters the porous matrix (Fig. 10A and B). Furthermore, the oil is only able to penetrate the
surface layers during immersion frying due to the resistance posed by the increasing pressure inside the
porous matrix. This is consistent with the observations of Bouchon and Aguilera (27) and Bouchon et al.
(28), using confocal laser scanning microscopy (CLSM).
After reaching a peak, the pore pressure in the core starts to fall (Fig. 5B, >175 and Fig. 8A). As
water from the core moves outward, water saturation () in the core decreases. The decrease in water
saturation results in an increase in capillary pressure (Fig. 7A) and a subsequent drop in pore pressure
in the core. The pore pressure in the core continues to drop until frying is complete. The partial
breakdown of the cellular structure and the release of vapors from confined cavities further contribute
to this decreasing pressure trend (106).
In the later stages of frying, there is an increase in the oil content (Fig. 4, >175 ). As pore pressure
decreases, the resistance of the food matrix to oil inflow also decreases, which likely contributes to the
rise in oil content during the later stages of frying. The oil phase can penetrate the pore spaces that were
left empty due to water loss.
As frying approaches completion as well as during the subsequent cooling phase, the porous matrix
becomes more susceptible to oil penetration. This could explain the commonly observed increased oil
penetration during the cooling phase compared to the immersion phase (28, 73, 86). Understanding
these mechanisms can guide further research aimed at optimizing frying techniques. Oil uptake occurs
both early in the frying process and in the later stages, including during cooling. By maintaining high
pressures in the core of the porous matrix for extended periods, it may be possible to reduce the oil
content in fried foods.
Figure 10: Oil content (A) 3D spatial distribution and (B) radial plot.
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4.3. Sensitivity testing
Some material properties are not
known with high certainty due to a
lack of data availability in the
literature, typically because of
experimental challenges. Perform-
ing sensitivity tests highlights the
significance of such properties to
the results. Sensitivity tests were
performed for water permeability
(), oil permeability ( ), and
evaporation rate constant ().
The value was changed by
±10% and ±20% from its base value,
and its effect on moisture content,
peak pore pressure, and volume
change was studied (Fig. 11). The
final moisture content was higher
for a lower , and lower for a
higher . A higher magnitude of permeability allows faster water loss from the matrix, and as a result,
the final moisture content is expected to be lower. Though the effect of varying by ±20% on the
moisture content is about ±2%, its effect on pore pressure is more significant (Fig. 11). The peak pore
pressure at the center rises by 16.83% upon lowering , and drops by 16.09% on raising . When
water moves slowly inside the porous matrix for a lower , it results in a higher , and undergoes
evaporation and subsequent vapor expansion, generating a higher . The higher pressure is also
responsible for lowering the overall shrinkage of the potato samples. As a result, the overall shrinkage
reduces for lower values. Conversely, the overall shrinkage increases for higher values.
The oil content of the fried sample increased with an increase in (Fig. 12A). Higher permeability
allows faster oil uptake when the conditions are favorable for oil penetration, i.e., when the pressure
drops in the core. Conversely, lowering only led to a small drop in oil content.
The effect of varying the evaporation rate constant () by ±10% and ±20% on the temperature and
moisture content of the fried samples was studied (Fig. 12B). The temperature and moisture content are
higher for a lower , and are lower for a higher . Increasing leads to faster evaporation, providing
Figure 11: Sensitivity test results for water permeability ().
Figure 12: Sensitivity test results for (A) oil permeability (), and (B) evaporation rate constant ().
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more evaporative cooling and lowering the temperature. Faster evaporation also contributes to more
rapid water loss, as liquid water is converted to vapor and then convected away.
4.4. Limitations of the modeling results
Hybrid-mixture-theory models are based on a rigorously developed porous media transport theory.
These models have been successfully applied to soils (24), food gels (6), swelling polymers used in drug
delivery applications (137), paperboard (10), wood (85) and different food processes (17, 48, 96, 145).
The current study solved the HMT-based unsaturated transport model developed by Takhar (120).
However, the reader is made aware of certain limitations related to both model development and
solution.
The information lost on upscaling has been noted previously (39, 120). Assumptions like the liquid and
solid phase incompressibility, thermodynamic equilibrium between phases, and neglecting gravity-
driven fluid flow were made to simplify the model development. These, however, are not prerequisites
for developing the model.
Solving mechanistic models for porous biopolymeric matrices like foods is challenging due to the
unavailability of appropriate material property data. This is mainly due to experimental difficulties,
especially for properties like capillary pressure and permeability. Foods are generally soft and susceptible
to structural changes when stress is applied. As discussed in Section 2.8.1 and Section 2.8.2, a pressure-
based method is commonly used for estimating capillary pressure and permeability. These techniques
were developed for non-cellular and rigid structures like rocks or soils (66). Attempts made to adapt
these techniques for foods have faced complications (66, 111). For the current model, we adopted values
for material properties based on the little data available in the literature. We have also relied on inverse
estimation for properties like the evaporation rate constant. This approach is expected to have some
impact on the model solution’s accuracy.
The limitations of the experimental properties mentioned are not necessarily weaknesses but rather
opportunities for improvement. There is potential for developing new experimental techniques tailored
specifically to foods. Large-scale collaborative efforts, involving material scientists and food engineers,
are encouraged to address these challenges.
Frying models developed in the past, which may be simplified versions of the currently available models,
played a crucial role in guiding advanced model development. Insights gained from these earlier models
motivated improvements in model setup and computational techniques, as well as stimulated new
experiments. The goal is to continuously update existing models with new data and insights. This work
represents another step in that process, aiming to enhance our understanding of frying.
5. CONCLUSIONS
A multiscale porous media model was developed for frying by utilizing the hybrid mixture theory-based
unsaturated transport equations. While most previous frying models ignored volume changes to simplify
model development, the current model mechanistically predicted the sample deformation using pore
pressure as the driving force governing the swelling and shrinking of the porous matrix. The effect of
the deforming matrix on the transport processes was accounted for by solving the model in a fully
coupled manner. A good agreement was found between model predictions and measured moisture
content, oil content, temperature, pressure, and volume changes. While the pressure predicted by
previous frying models was either too high or too low compared to the measured pressure, the current
model predicted the peak pressure during frying reasonably well. The spatial distribution of different
model variables like moisture content, oil content, pressure, and evaporation rate provided more clarity
on the different mechanisms involved in frying, and such profiles are challenging to generate
experimentally. Internal moisture transport could not keep up with the moisture loss from the surface,
and as a result, a moisture gradient developed across the porous matrix. Vaporization and gas expansion
at elevated temperatures attained during frying caused the pressure to rise, which restricted the oil
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seepage to the peripheral layers of the porous matrix. The evaporation front expanded with frying time,
and the peak of the evaporation front moved towards the core. In the final stages of frying, the pressure
dropped in the core, making the food matrix more susceptible to oil entry, especially during the cooling
phase. Thus, sustaining high pressures in foods for longer times during frying can reduce their oil
content. The importance of the model-generated insights for future process optimization studies was
discussed. The challenges faced during the model solution, particularly those due to the unavailability
of material property data, were highlighted, and further research on devising measurement techniques
for soft food materials was encouraged.
NOMENCLATURE
Symbol
Definition
Mixture viscosity of the biopolymeric matrix (Eq. 3) (.)
Specific heat of phase (Eq. 20),
Diffusivity of phase (Eq. 3),
Modulus of elasticity of the biopolymeric matrix (Eq. 3) ()
Emissivity (Eq. 34) (dimensionless)
Source/sink term (Eq. 1)
Mass transfer coefficient for oil (Eq. 31)
Mass transfer coefficient for water vapors (Eq. 33)
Heat transfer coefficient (Eq. 34)
Permeability of phase (Eq. 3), ()
Intrinsic gas permeability (Eq. 37) ()
Relative gas permeability (Eq. 37) (dimensionless)
Bulk modulus of the spring in the Maxwell element (Eq. 23) ()
Thermal conductivity of phase (Eq. 20),
Mass fraction of liquid water on a dry basis (Eq. 27)
Molecular weight of water vapors (Eq. 11)
Pressure of phase (Eq. 3),
Capillary pressure in phase (Eq. 13), ()
Pore pressure (Eq. 15) ()
Universal gas constant (Eq. 11)
Radial distance ()
Lagrangian radius ()
Saturation of phase in Eulerian coordinates (Eq. 9), (dimensionless)
Temperature (Eq. 11) ()
Velocity of the solid phase (Eq. 2)
,
Velocity of phase with respect to the solid phase (Eq. 1)
Greek
symbols
Definition
Volume fraction of phase (Eq. 1) (dimensionless)
Material time derivative of volume fraction of phase with respect to the solid phase (Eq. 1)
(
)
,
Total volumetric strain (Eq. 22) (dimensionless)
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