Unsaturated fluid transport in swelling poroviscoelastic biopolymers

Chemical Engineering Science 109 (2014) 98–110

Contents lists available at ScienceDirect

Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces

Unsaturated fluid transport in swelling poroviscoelastic biopolymers Pawan S. Takhar 1,n Food Science and Human Nutrition, 1304 W Pennsylvania Ave., University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

H I G H L I G H T S







Continuum thermodynamics based theory yielded general unsaturated transport laws. Inclusion of poroviscoelasticity resulted in temporally non-local transport laws. The equations can describe both Darcian and non-Darcian modes of fluid transport. Transport and stress relations include hydrophilic and phobic interaction terms.

art ic l e i nf o

a b s t r a c t

Article history: Received 27 July 2013 Received in revised form 28 December 2013 Accepted 15 January 2014 Available online 31 January 2014

The hybrid mixture theory was used to obtain the two-scale unsaturated transport and thermomechanical equations for biopolymers. The two-scale laws of conservation of mass, momentum, energy and entropy were utilized, the constitutive theory was formulated and the entropy inequality was exploited to obtain various equilibrium, near-equilibrium and non-equilibrium relations. The system was treated as poroviscoelastic with the viscoelastic biopolymers interacting with the viscous water and oil phases at pore-scale via hydrophilic and hydrophobic forces. The gas phase exchanged mass with the liquid water due to evaporation/condensation away from equilibrium. The exploitation of entropy inequality resulted in temporally non-local generalized Darcy's laws for the liquid phases, near-equilibrium swelling and capillary pressure relations, generalized stress relations, near-equilibrium Gibbs free energy relation and the rate of evaporation relation. The generalized Darcy's law relations include novel integral terms with long-memory effects. These can describe the effect of biopolymer–fluid interaction on both Darcian and non-Darcian modes of fluid transport depending upon the state of the biopolymers (glassy, rubbery or glass-transition). The resulting transport laws for various phases include the cross-effect terms in the form of volume fraction gradients. The unsaturated generalized Darcy's law relations were validated by making comparison to the experimental data on moisture transport, heat penetration and pressure development during frying of potato cylinders. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Multiphase flow Hybrid mixture theory Darcy's law Non-Darcian flow Poroviscoelastic Temporally non-local

1. Introduction Although, significant progress has been made in developing continuum-thermodynamics based transport and thermomechanical relations for saturated biopolymeric systems, the work remains scarce for unsaturated transport. In numerous food and bioprocessing applications, the fluid and heat transport is of unsaturated type that involves the movement of both liquid and gas (mixture of dry air and water vapors) through the biopolymeric matrix. Examples include drying of food and fiber, frying, baking, bubble transport in expanding biopolymers exiting an extruder, puffing and biofiltration. In many cases, the fluid

n

Tel.: þ 1 217 300 0486. E-mail address: [email protected] 1 Author has previously published as Pawan P. Singh.

0009-2509/$ - see front matter & 2014 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ces.2014.01.016

transport is coupled with heat flow. The liquid may change to vapor form due to heat, sudden pressure changes, water vapor pressure differences between the liquid and air phase, etc. During transport, the surrounding biopolymeric matrix may also undergo physicochemical changes such as glass-transition, gelatinization, browning reactions and color changes. The porous media equations developed for soils may not be adequate to describe the biopolymeric transport problems because the matrix is composed of long chain viscoelastic biopolymers in the latter as opposed to small-sized elastic soil particles in the former. In addition, in numerous bioprocessing applications such as frying and starch expansion, the external temperature may exceed the boiling temperature of water causing rapid changes in the physicochemical structure of biopolymers in few seconds. In biological systems, a macroscale elementary volume may be comprised of hydrophilic (e.g. polysaccharides), hydrophobic (e.g. lipids, cholesterol and triglycerides) or amphiphilic (e.g. proteins)

P.S. Takhar / Chemical Engineering Science 109 (2014) 98–110

molecules. Thus, different components in an elementary volume may interact with both water and oil. E.g. proteins and minerals in the core of fried meat products have affinity for water and the lipids have affinity for oil. Some other examples include the interaction of hydrophobic gases with lipid bilayers in cells (Truskey et al., 2004) and unfolding of hydrophobic portions of proteins by interaction with fat (Morin et al., 2004). Thus, for modeling transport in complex biological systems, both hydrophilic and hydrophobic interaction terms need to be included in the transport equations. For understanding the physics of fluid and heat transfer, and their interaction with biopolymers (related to quality attributes), the continuum thermodynamics based multiscale modeling can be utilized. In this communication, hybrid mixture theory (HMT) is utilized, which involves using the mathematical filtering theorems to average the microscale field equations to obtain equations at higher scales (Hassanizadeh and Gray, 1979). At macroscale, the constitutive equations are formulated by exploiting the entropy inequality in the sense of Coleman and Noll (1963). During upscaling (averaging), some microscale information is lost. However, the effect of microscale transport mechanisms and thermomechanical processes on macroscale behavior is incorporated. One significant advantage of upscaling first and then formulating the constitutive theory is that the material coefficients in derived relations appear at the macroscale. The macroscale material coefficients are easier to measure via experiments or obtain from other experimental studies using conventional methods. Upscaled equations are solved in macro and mesoscale representative elementary volumes (REV), which saves the computational time significantly in comparison to time needed for solving the microscale equations. HMT was initially used to explain the thermomechanical behavior of swelling and shrinking porous media like clays and food gels (Hassanizadeh and Gray, 1979; Achanta et al., 1994; Bennethum and Cushman, 1996a). In these studies, the solid and fluid phases were modeled as elastic and viscous, respectively, at the microscale, which resulted in a viscoelastic system with a short memory at macroscale (governed by the first-order time derivative of strain tensor). Such systems are equivalent to the viscous systems. These theories ignored the microscale relaxation processes of the polymeric matrix, which arise due to conformational changes in the flexible thread-like polymer chains (Ferry, 1980). Most solid biopolymers exhibit viscoelastic effects with a long memory when subjected to a wide range of thermomechanical processes (Slade and Levine, 1991). Singh et al. (2003a–c) applied HMT to saturated systems by treating the solid phase as viscoelastic and obtained integro-differential fluid transport equation. The integral term included fluid volume fraction gradient obtained via logical arguments and chain-rule. Kleinfelter et al. (2007) used HMT for developing generalized Darcy's law for poroelastic unsaturated soils. The capillary pressure relations were obtained at mesoscale between the vicinal water and vapor phase, and at macroscale between the bulk water and vapor phase. Kleinfelter et al. (2007) note that very few theoretical efforts have been applied to unsaturated systems, despite the availability of abundant experimental literature. Muraleetharan and Wei (1999) developed unsaturated transport equations for porous soils using mixture theoretic approach by treating the solid phase as elastic. In the constitutive theory, the dependent variables for each phase were considered as functions of volume fraction of only that phase. Thus, the resulting transport equations for a given phase did not include the cross-effect terms due to the volume fraction gradients of other phases, which are physically expected in unsaturated systems. Zhu et al. (2010) studied unsaturated transport in biopolymers by ignoring the mass exchange between liquid and vapor phases, assuming

99

constant gas phase pressure, and making an a priori assumption in constitutive theory that dependent variables are functions of time derivatives of volume fractions. These assumptions limit the applicability of the resulting equations to only a small class of unsaturated problems where phase change is not involved and pressure could be considered constant. Making the assumption in constitutive theory that dependent variables are functions of time derivative of volume fraction instead of strain tensor, further limits the applicability of resulting equations in studying viscoelastic biomaterials. To my knowledge, the theoretical study of Gray and Hassanizadeh (1991b) is most complete and makes physically valid assumption in regard to the unsaturated transport processes in soils. Gray and Hassanizadeh (1991b) used HMT by treating the solid phase as elastic and the liquid phase as viscous, and included the interfacial effects. The resulting transport equations included cross-effect terms via porosity and saturation gradients. The hydration stress was not included as the liquid phase was not considered a function of the solid phase strain tensor. This paper discusses unsaturated transport and thermomechanical relations for biopolymers obtained using HMT. The system will be considered as poroviscoelastic to represent the interaction of viscoelastic biopolymers with fluid phases in pores. It will be demonstrated that the integral term of Singh et al. (2003c) can also be obtained using the continuum mechanics based relations instead of using the logical arguments. The model will involve the interaction of both hydrophilic and hydrophobic biopolymers with the water and oil phases, respectively. In the past HMT based studies for soils and polymers, only the interaction of hydrophilic solids with the water phase was included and the oil phase was treated as non-interacting (Bennethum and Cushman, 1996a; Singh et al., 2003b,c). After obtaining the general theory, three simplified cases will be discussed. The solution of generalized Darcy's law relations will be compared to unsaturated transport during high temperature frying. 1.1. System description and notation The four phases in the system are denoted as—solid (s), water (w), oil (o), and gas (g). The Greek letters α and β used as superscripts in the variable names are the general representation of phases. For example, at macroscale ɛ α denotes the volume fraction of the phase α, with α being s, w, o, and g or either of these. α The variables with two superscripts, such as β b e denote source/ sink terms representing transfer of a quantity from the phase β to the phase α. The equations are presented in the indicial notation.

2. Constitutive theory Two-scale laws of mass, momentum, energy and entropy are presented in Achanta et al. (1994) and Singh et al. (2003b). The number of variables in the balance laws is more than the number equations (Bennethum and Cushman, 1996b). This is expected because these balance laws are general and thus applicable to all kinds of materials. The system is closed by formulating the constitutive theory through axioms of constitution (Eringen, 1980) and imposing restrictions via entropy inequality (Coleman and Noll, 1963). In this step, the information on material behavior is utilized to relate dependent and independent variables. Following is the list of independent variables: ðmÞ

α

ɛw ; ɛ o ; ɛ g ; ρα ; vα;s ; T; T ;l ; EsKL ; dkl ; dkl ; dkl ; g αl ; h ; l w

o

where m ¼ 0–p; and α ¼ s; w; o; g

g

ð1Þ

Here ɛ was not considered as independent because it is related to the volume fraction of other phases via equation, ∑α ¼ s;w;o;g ɛ α ¼ 1. s

100

P.S. Takhar / Chemical Engineering Science 109 (2014) 98–110

The ɛα account for the solid–fluid interaction at moderate to high fluid contents. The vα;s is equal to vαl  vsl , which denotes the l velocity of the phase α relative to the solid phase. Various phases are assumed to be in thermal equilibrium inside the macroscale REV, due to which a single temperature value T can be used for each. The thermal gradient T ;l incorporates heat conduction. To model the system as poroviscoelastic, the fluid phase is considered α as viscous (dkl terms) and the solid polymers are considered as

time dependent manner. Similarly, in the vicinity of hydrophobic polymers, the oil phase is also expected to flow in a time dependent manner, which is included through various order time

generalized Kelvin–Voigt viscoelastic (EsKL terms, with m ¼ 0  p representing the order of material time derivative). The equations derived for a general Kelvin–Voigt viscoelastic system denoted using various order material time derivatives of strain (EsKL) can be transformed to equivalent integral form using the method discussed in Eringen (1980) and as shown by Singh et al. (2003c) for saturated transport. Since, the strain tensor is in Lagrangian coordinates its various order material time derivatives do not lead to higher order Eulerian spatial gradient terms. Following is the list of dependent (or constitutive) variables:

Physical pressure : pα  ðρα Þ2

ðmÞ

α

α

b ; ɛ_ α ; t αkl ; Aα ; ηα ; qαl ; β b e ; β Tb l ; β ϕ α

where α ¼ s; w; o; g

ð2Þ

which are functions of the independent variables listed in (1). All the macroscale variables in the list (2) had a microscale counterpart before upscaling, except the volume fraction, ɛ α . The variable ɛ α appeared at the macroscale during upscaling. Therefore, it does not have a corresponding equation. This is the closure issue. The system is closed by following (Bowen, 1982), which involves treating ɛ_ α as dependent variables. The axiom of equipresence requires that initially all dependent variables should be considered as functions of the same list of independent variables (Eringen, 1980). It can be shown that the Helmholtz free energies are functions of the subsets of the list (Hassanizadeh, 1986). Additionally, from physical knowledge of the system it is known that the Helmholtz free energy of a given phase would depend upon density of that phase only, and the ðmÞ

strain rates in solid polymers (EsKL ) would not affect the Helmholtz free energy of the gas phase. Therefore, to save derivation steps, the dependence of the Helmholtz free energies on subsets of the list (1) is postulated: s

ðmÞ

As ¼ As ðρs ; EsKL ; E_ KL ; E€ KL ; …EsKL ; TÞ s

s

ðmÞ

Aw ¼ Aw ðɛ w ; ɛo ; ɛ g ; ρw ; EsKL ; E_ KL ; E€ KL ; …; EsKL ; TÞ; s

ðmÞ

s s Ao ¼ Ao ðɛ w ; ɛ o ; ɛg ; ρo ; EsKL ; E_ KL ; E€ KL ; …EsKL ; TÞ;

Ag ¼ Ag ðɛw ; ɛ o ; ɛg ; ρg ; TÞ;

where m ¼ 0  p:

ð3Þ

ðm ¼ 0Þ For the solid phase, the ɛ α (α ¼ w; o; g) are related to and ρs via the solid phase mass balance equation. Therefore, ɛ α are not independent of EsKL and ρs, and thus not included as independent variables for As above. Inclusion of ɛ α for the fluid phases allows incorporating the effect of concentration gradients on fluid transport. In a multiphase system, since the change in volume fraction of single phase will affect the volume fraction of other phases, the cross-effects are expected. These cross-effects are included through three independent volume fraction terms in (3). In Muraleetharan and Wei (1999) these effects were not taken EsKL

ðmÞ EsKL

ðmÞ

derivatives of strain (EsKL ) in the Ao equation. To my knowledge, this effect has not been included in the previous porous media studies. Next, some macroscale quantities are defined, which will be used while describing the results:

s Terzaghi stress : t se kl  ρ

∂Aα ; ∂ρα

where α ¼ s; w; o; g

∂As s s x x ∂EsKL k;K l;L

w Hydration stress : t sw kl  ρ

∂Aw s s x x ∂EsKL k;K l;L

o Lipophilic or hydrophobic stress : t so kl  ρ

∂Ao s s x x ∂EsKL k;K l;L

ð4Þ

Bennethum and Weinstein (2004) defined pα as physical (or classical) pressure because it appears in the physical stress equation of various phases as shown in later sections. Terzaghi stress is the elastic stress in the solid phase. Hydration stress results from physicochemical forces between the hydrophilic biopolymers and the adsorbed water. The hydrophobic stress results from physicochemical forces between the hydrophobic biopolymers and the oil phase. It has not been defined in the previous multiscale studies. The hydration stresses have been measured to have a significant magnitude in foods at low fluid contents when a few molecular layers of water are present (Wolfe, 1987). The importance of hydrophobic stress causing uptake of oil during frying of foods has been experimentally observed by Pinthus and Saguy (1994). Hydrophobic stress would also be important for expansion of bread crumbs during extrusion in the presence of oil and shortenings (Ganjyal, 2009). In the next step, the material time derivative of Aα in (3) is computed and substituted in the entropy inequality as shown in the previous studies (Achanta et al., 1994; Bennethum and Cushman, 1996b; Singh et al., 2003b). Exploiting the entropy inequality in the sense of Coleman and Noll (1963) imposes restrictions on the system to ensure that it follows second law of thermodynamics. Past work in soil physics (Gray and Hassanizadeh, 1994), and food (Achanta et al., 1997; Singh et al., 2004b; Takhar, 2011) and chemical engineering (Kim et al., 1996) has shown that thermodynamically viable resulting equations are also experimentally viable. This step results in a large number of non-equilibrium, equilibrium and near-equilibrium relations. Non-equilibrium relations hold both at and away from the thermodynamic equilibrium; equilibrium relations hold only at equilibrium; and near-equilibrium relations hold at and near equilibrium. A subset of these developed relations can be selected for solving various problems. Pertinent equations needed to describe our system are discussed in the next section.

3. Results

w

into account. Inclusion of for A allows generalization of the hydration stress of soils based studies for its application to biopolymers. In soil based studies, only the zeroth order time derivative of strain (EsKL) was used to include hydration stress (Bennethum and Cushman, 1996b; Bennethum et al., 1996, 2000; Murad and Cushman, 2000). For a poroviscoelastic system, various ðmÞ

order time derivatives of strain (EsKL , m ¼ 0  p) are expected to affect the flow of water in the vicinity of hydrophilic polymers in a

The definition of thermodynamic equilibrium depends upon the knowledge of the physical system. In an unsaturated transport process, it is expected that at equilibrium, the volume fraction (ɛα ) of various phases would become constant; the rate of deformation α tensor (dkl ) and velocity of various phases with respect to the solid α;s phase (vl ) would become zero; the thermal gradients (T ;l ) would α α vanish and the phase change variables (β b e , β Tb l , etc.) representing evaporation (or condensation) would also become zero.

P.S. Takhar / Chemical Engineering Science 109 (2014) 98–110

3.1. Equilibrium relations At equilibrium, the net entropy generation rate, Λ, attains a minima and the following variables become zero: ðmÞ

w o g ɛ_ w ; ɛ_ o ; ɛ_ g ; vα;s ; T ;l ; EsKL ; dkl ; dkl ; dkl ; l

where m ¼ 0 p; and α ¼ s; w; o; g ð5Þ

If Λ ¼ CðνÞv, one would obtain ∂Λ=∂ν ¼ CðνÞ þ νðð∂CðνÞÞ=∂νÞ. At equilibrium, ν ¼ 0, which will make the second term in the previous equation to drop. Therefore, for Λ to attain a minima, CðνÞ must also be zero. Thus, at equilibrium, in the entropy inequality the coefficients of variables becoming zero, must also become zero. This results in the following equilibrium relations: ps  pw þ

ps  po þ

ps  pg þ

∑

α ¼ w;o;g

α

∂A

ɛ α ρα

∑

∂A ¼ 0; ∂ɛ g

ð8Þ

ɛw sw ɛo so t þ t ɛ s kl ɛ s kl

¼ 0;

ðmÞ

ɛ α qαl

ð9Þ

m ¼ 1p

pg ɛ g;l ɛ g ρg

ðmÞ

EsKL and T ;l . To obtain the thermoviscous effects, two-term linearα ization is performed around dkl and T ;l . The material coefficients in the resulting linearized equations are functions of the independent variables in the list (1), around which linearization was not performed. For example, various α material coefficients such as thermal conductivity (kkl ) and viscoesm lastic properties (GKLMN ) in equations listed below are functions of variables such as fluid concentration (ɛα ), temperature (T) and strain-rate (EsKL ). To avoid the equations to appear unwieldy, the parametric dependence of the material properties on these variables is not shown in the results presented below. The following are the resulting near-equilibrium relations: pw  ps 

po  ps  where α ¼ w; o; g

ð12Þ

∑

β ¼ s;w;o;g βaα

β ¼ s;w;o;g βag

β bg Tl

ð16Þ

ɛ α ρα

∂Aα ¼ N ow ɛ_ w þ Noo ɛ_ o þ Nog ɛ_ g ∂ɛ o

ð17Þ

ɛ α ρα

∂Aα ¼ N gw ɛ_ w þ Ngo ɛ_ o þ N gg ɛ_ g ∂ɛ g

ð18Þ

ɛα ρα

∑

∑

α ¼ w;o;g

α ¼ w;o;g

ð11Þ

¼ 0;

∂Ag β ∑ ɛ  ∑ β ;l β ¼ w;o;g ∂ɛ

∂Aα w g o _ w þN w _ ¼ Nw wɛ o ɛ_ þ N g ɛ ∂ɛ w

∑

ð10Þ

α ðmÞ p ∂Aα β α α ∂A ɛ  ∑ ɛ ρ Es  ∑ β ;l ðmÞ KL;l β ¼ w;o;g ∂ɛ m¼0 s ∂EKL where α ¼ w; o;

pα ɛ α;l  ɛα ρα

relations. Singh et al. (2003a) clarified that linearization can be performed around single or multiple variables. In some cases multiple variable linearization should be preferred to obtain explicit expressions for cross-effects. Gray and Hassanizadeh (1991b) performed single-term linearization around porosity and saturation (related to volume fractions). Since, the volume fractions of various phases cannot vary independently, a multivariable linearization is performed around ɛ_ α for the entropyinequality terms involving pressure. To obtain expressions for the thermoviscoelastic effects, two-term linearization is used around

ðmÞ

α

∂EsKL

t αkl ¼  pα δkl ; α ¼ s;w;o;g

ð7Þ

ɛ α ρα

α ¼ w;o;g

ɛ α ρα

∂Aα ¼ 0; ∂ɛo

∑

t skl ¼  ps δkl þ t se kl þ ∑

ð6Þ

ɛ α ρα

α ¼ w;o;g

β ¼ s;w;o

∂Aα ¼ 0; ∂ɛw

∑

β bα Tl

pg  ps 

α ¼ w;o;g

t skl ¼  ps δkl þ t se kl þ

¼ 0;

ð19Þ

ð13Þ ð14Þ

ðmÞ

s

p

s

¼ 0;

ðmÞ s p ɛ w sw ɛ o so t þ t þ ∑ Gm Es xs xs þ H sklm T ;m ɛ s kl ɛs kl m ¼ 1 KLMN MN k;K l;L

s s qsl ¼ kkl T ;k þ ∑ J m KLM xl;K ELM

where α; β ¼ s; w; o; g and α a β;

ð15Þ

where, the Gibbs free energy (Gα ) is given by Gα ¼ Aα þ pα =ρα . Bennethum and Weinstein (2004) defined pα as physical pressure because it is related to the physical stress at equilibrium via (9) and (11). In absence of volume change due to fluid adsorption or gas expansion (∂Aα =∂ɛβ ¼ 0), (6)–(8) imply that the physical pressure in various phases will be equal at equilibrium (ps ¼ pw ¼ po ¼ pg ). Eq. (9) generalizes the Terzaghi's stress principle for elastic soils (first two terms on RHS) to the stress in biomaterials composed of hydrophilic and hydrophobic polymers (see (4) for hydrophilic and hydration stress expressions). Eq. (15) states that at equilibrium Gibbs free energy (or chemical potential) of various phases is equal. The near-equilibrium form of this equation will result in the rate of evaporation term in the next section. Near-equilibrium form of (10)–(15) will be presented in the next section. 3.2. Near-equilibrium relations As discussed in Hassanizadeh and Gray (1980), Achanta et al. (1994), Bennethum and Cushman (1996b) and Singh et al. (2003a), the coefficients of variables becoming zero at equilibrium (list (5)) can be linearized around these variables to satisfy entropy inequality. The linearization procedure results in near-equilibrium

ð20Þ

m¼1

α

Gα  Gβ ¼ 0;

101

t αkl ¼  pα δkl þ ναklmn dmn þ H αklm T ;m ; α

α

qαl ¼ kkl T ;k þ Gαlmn dmn ; pα ɛα;l  ɛ α ρα

where α ¼ w; o; g

where α ¼ w; o; g

ð22Þ

p ∂Aα β ∂Aα ðmÞ ɛ  ∑ ɛ α ρα ðmÞ EsKL;l  ∑ β ;l ∂ɛ β ¼ w;o;g m¼0 ∂EsKL

∑

β ¼ s;w;o;g βaα

β bα Tl

where α ¼ w; o pg ɛg;l  ɛ g ρg

∂Ag β ɛ  ∑ Tb βlg ¼ Rgkl vg;s ; k β ;l β ¼ w;o;g ∂ɛ β ¼ s;w;o;g ∑ α

e Gβ  Gα ¼ ξαβ b

ð21Þ

where α; β ¼ s; w; o; g and α a β;

¼ Rαkl vα;s ; k ð23Þ ð24Þ ð25Þ

Eqs. (16)–(18) provide the relation between various pressures away from equilibrium. The swelling potential for the liquid phases (πw, πo) and expansion potential for the gas phase (πg) are defined similar to the wettability potential of Gray and Hassanizadeh (1991a) as follows: πw 

πo 

∂Aα ∂ɛw

ð26Þ

∂Aα ∂ɛo

ð27Þ

∑

ɛ α ρα

∑

ɛ α ρα

α ¼ w;o;g

α ¼ w;o;g

102

P.S. Takhar / Chemical Engineering Science 109 (2014) 98–110

πg 

∑

α ¼ w;o;g

ɛ α ρα

∂Aα ∂ɛ g

ð28Þ

Eqs. (26) and (27) extend the wettability potential definition of Gray and Hassanizadeh (1991a) and swelling potential definition of Bennethum and Weinstein (2004) to unsaturated systems with multiphases. πw and πo can be termed as hydrophilic and hydrophobic swelling potentials, respectively. Using (26)–(28) and (16)– (18), the following expressions for the swelling and expansion potentials can be obtained: _ w þ Nw _ o þ Nw _g π w ¼ pw  ps  ½N w wɛ oɛ g ɛ

ð29Þ

π o ¼ po ps  ½Now ɛ_ w þ N oo ɛ_ o þ N og ɛ_ g 

ð30Þ

π g ¼ pg  ps  ½N gw ɛ_ w þ Ngo ɛ_ o þ Ngg ɛ_ g 

ð31Þ

Eqs. (29)–(31) indicate that away from equilibrium the swelling/ expansion potential is governed by not only the difference in physical pressure between the fluid and solid phases, but also the rate of change of volume fraction of various phases. If the fluid phase is entrapped in the pores, the ɛ_ α terms would become zero to result in the expected relation that the swelling potential is equal to the difference in physical pressure between a fluid phase and the solid phase. In Achanta and Cushman (1994), an expression for the expansion potential due to gas phase was presented, but its physical significance could not be found. Here it is noted that the expansion potential due to gas phase is important for applications involving evaporation at high temperatures such as expansion during extrusion, frying and baking. If condensation occurs in some application, this potential will become negative leading to collapse of the surrounding matrix. Eqs. (19)–(22) are two-scale thermoviscoelasticity and thermoviscous relations. Eq. (19) extends the equilibrium stress relation ðmÞ

to near-equilibrium. It includes the viscoelasticity term (EsMN xsk;K xsl;L ) in gradient form. In a subsequent section, this term will be transformed into an integral form. The last term of (19) describes the effect of temperature gradient on solid phase stress in anisotropic materials. For isotropic materials, this term will become zero as expected due to vanishing of third-order tensors. Eqs. (23) and (24) are generalized expressions for calculating the velocity of a phase relative to the solid phase. In a subsequent section, generalized Darcy's law will be developed using these equations. The N α ; ναklmn ; K and Rαkl are the material coefficients that appear during the linearization procedure. Eq. (25) is the nearequilibrium phase-change equation, where ξα is the material coefficient. Eq. (25) states that the rate of mass exchanged between two phases is directly proportional to the difference in Gibbs free energy (or chemical potential) between them when the system is away from equilibrium. 3.3. Capillary pressure Achanta and Cushman (1994) defined capillary pressure of water phase as the difference between πg and πw and found a relation that was consistent with derivation of Hassanizadeh and Gray (1993). When oil phase is also present and has affinity for the hydrophobic polymers, the following definitions can be used: p

cw

g

 π π

w

ð32Þ

pco  π g π o

ð33Þ

Combining these definitions with Eqs. (29)–(31) results in ¼ p  p  ½N w ɛ_ þ N o ɛ_ þ N g ɛ_ 

ð34Þ

pco ¼ pg  po  ½Nw0 ɛ_ w þ N o0 ɛ_ o þ Ng0 ɛ_ g 

ð35Þ

p

cw

g

w

w

o

g

g o where the coefficients, N α  N gα  N w α , and N α0  N α  N α (see (29)–(31)). Away from equilibrium, (34) and (35) provide viable expressions for the capillary pressure for water and oil phases, which depend upon the rate of change of volume fraction of various phases along with pressures. These equations extend the dynamic capillary pressure relations of Hassanizadeh et al. (2002) to systems composed of multiphases. In Hassanizadeh et al. (2002), since only water and air phases were present, the time rate terms included only saturation term. When three fluids are present, the rate of change of volume fraction of various phases will be interdependent. Thus, all fluid phases will affect the dynamic capillary pressure as shown by (34) and (35).

3.4. Total stress and heat flux In a multiphase system, the total stress and heat flux can be defined as (Singh et al., 2003b) t kl ¼ ql ¼

∑

ɛ α t αkl

ð36Þ

∑

ɛ α qαl

ð37Þ

α ¼ s;w;o;g

α ¼ s;w;o;g

where p ¼ ∑α ¼ s;w;o;g ɛ α pα . Substituting stress and heat flux relations from (19) to (22) in (36) and (37) results in the following expressions for the total stress and total heat flux in a multiphase system: p

s

ðmÞ

s w sw o so s s s s s t kl ¼  pδkl þɛ s t se ∑ Gm KLMN E MN xk;K xl;L þ ɛ H klm T ;m kl þ ɛ t kl þ ɛ t kl þ ɛ m¼1

þ

ql ¼

∑

α ¼ w;o;g

∑

α ¼ s;w;o;g

ɛ

α

α ðναklmn dmn þ H αklm T ;m Þ;

ðmÞ

s

p

α

ð38Þ

s s ɛ α kkl T ;k þ ∑ ɛs J m KLM xl;K E LM þ m¼1 ðmÞ term (EsMN )

α

∑

α ¼ w;o;g

ɛ α Gαlmn dmn :

ð39Þ

The viscoelastic in total stress equation, (38), can be transformed from gradient to integral form by using the Laplace transformation procedure discussed in Singh et al. (2003c). This procedure requires the assumptions that the material is free from strain initially and that the stress relaxation function is not an explicit functions of strain. Using the Laplace transformation method of Singh et al. (2003c), (38) converts to the following form involving convolution: Z t  ∂Es w sw o so s t kl ¼  pδkl þɛ s t se GsKLMN ðt  τÞ MN dτ xsk;K xsl;L kl þ ɛ t kl þ ɛ t kl þ ɛ ∂τ 0 þɛ s H sklm T ;m þ

∑

α ¼ w;o;g

α

ɛ α ðναklmn dmn þ H αklm T ;m Þ;

ð40Þ

where GsKLMN is the stress relaxation function for the anisotropic solid phase. The integral form of stress equation has the advantage that the stress relaxation function is easier to measure with static or dynamic mechanical testing (Ferry, 1980). The stress and heat flux equations can be simplified for isotropic materials. Under isotropy, the third order tensors become zero and various material coefficients can be written as (Segel and Handelman, 1977) α

α

kkl ¼ k δkl ;

where α ¼ s; w; o; g

GsKLMN ¼ Gs1 δKL δMN þ Gs2 ðδKM δLN þ δKN δLM Þ where ναklmn ¼ λα δkl δmn þ μα ðδkm δln þ δkn δlm Þ;

α ¼ w; o; g:

ð41Þ

Using (41), the total stress and heat flux equations reduce to the following form for isotropic poroviscoelastic materials: Z t ∂Es w sw o so s Gs1 ðt  τÞ MM dτ t kl ¼  pδkl þɛ s t se kl þ ɛ t kl þ ɛ t kl þ ɛ ∂τ 0

P.S. Takhar / Chemical Engineering Science 109 (2014) 98–110

Z þ2

t 0

Gs2 ðt  τÞ

 ∂EsKL α α dτ xsk;K xsl;L þ ∑ ɛ α ðλα dmm δkl þ 2μα dkl Þ; ∂τ α ¼ w;o;g ð42Þ

ql ¼

∑

α ¼ s;w;o;g

α

ɛ α k T ;l

ð43Þ

Here the memory functions Gs1 and Gs2 are the bulk and shear relaxation moduli, respectively. As noted in Section 3.2, viscoelastic properties (Gs1 and Gs2) are functions of fluid concentration (ɛα ),

can be used to simplify M αKL and Rαkl M αKL ðtÞ ¼ M α ðtÞδKL Rαkl ¼ Rα δkl :

3.5. Generalized Darcy's law equations The inertial term is expected to be smaller in magnitude in low velocity flow in a porous matrix in comparison to the pressure and α concentration gradient terms. Substitute t αkl from (21) and β Tb l from (23) into the momentum balance equation of Achanta et al. (1994), neglect inertia term (ɛ w ρw ðDw vw l =DtÞ) and simplify to obtain the gradient form of generalized Darcy's law for the water and oil phases: α

Rαkl vα;s ¼  ɛ α pα;l þ ðɛα ναklmn dmn Þ;k k þ ðɛ α H αklm T ;m Þ;k  ɛα ρα þ ɛ α ρα g αl ;

p ∂Aα β ∂Aα ðmÞ ɛ ;l ɛ α ρα ∑ ðmÞ EsKL ;l β β ¼ w;o;g ∂ɛ m¼0 ∂EsKL

∑

where α ¼ w; o:

ð44Þ ðmÞ EsKL

By performing Laplace transformation of the term, rearranging and taking inverse Laplace transformation as described in Singh et al. (2003c), the following integro-differential form of generalized Darcy's law is obtained: α

Rαkl vα;s ¼  ɛ α pα;l þ ðɛα ναklmn dmn Þ;k þ ðɛ α H αklm T ;m Þ;k  ɛ α ρα k  ɛα ρα

Z 0

t

M αKL ðt  τÞ

∂EsKL;l dτ þ ɛ α ρα g αl ; ∂τ

∂Aα β ɛ β ;l β ¼ w;o;g ∂ɛ ∑

where α ¼ w; o:

α

For gas phase, the generalized Darcy's law equation is

 ɛ α ρα

0

M α ðt  τÞ

g

∂EsMM;l dτ þ ɛ α ρα g αl ; ∂τ

ð46Þ

Next, various simplified cases of the generalized Darcy's law relations followed by the solution of equations for an unsaturated transport problem are presented. 3.6. Case I: generalized Darcy's law for porous isotropic materials For flow in isotropic materials, the third order tensors in (45) and (46) will vanish. Additionally, (41) and the following relations

where α ¼ w; o;

ð48Þ

g

∂Ag β ɛ þ ɛg ρg g gl : β ;l β ¼ w;o;g ∂ɛ ∑

ð49Þ The first term on the right hand side of (48) is the classical pressure gradient term of Darcy's law; the second and third terms denote the Brinkman correction of importance in high velocity flows; and the fourth term incorporates the cross-effects caused by the concentration gradient of various fluids on flow of water and oil phases. The integral term is temporally non-local and is novel in context of Darcy's law, and the last term incorporates the effect of body or gravity forces on fluid flow. Following (Singh et al., 2003c), the memory function can be represented as M α ðtÞ ¼ Bc GðtÞ

ð50Þ

where Bc is the material coefficient relating the viscoelastic relaxation to fluid flow, and G(t) is the stress relaxation function measured using uniaxial or shear testing. Bc can be estimated inversely during fluid transport simulations (Singh et al., 2004a) or by analyzing the asymptotic behavior of the ratio of diffusivity to the stress relaxation function of materials in glassy and rubbery states (Takhar, 2011). The integral term of (48) describes the effect of viscoelastic relaxation in the poroviscoelastic matrix on flow of water and oil phases. Due to its non-local form, it is able to incorporate longmemory effects of strain in the biopolymers on the flow of fluids. In most HMT based studies for soils and biopolymers (Achanta et al., 1994; Bennethum and Cushman, 1996a; Singh et al., 2003a), short-memory effects were included that described the effect of viscous resistance of polymers on fluid flow. Singh et al. (2003c) obtained a long-memory integral form for saturated systems with the volume fraction gradient showing up in the final equation instead of the strain tensor gradient. From continuum mechanics it is known that EsMM of (48) is equal to the first invariant of strain (Eringen, 1980): ð51Þ IsE

In addition, is related to the ratio of infinitesimal volume change of the solid phase due to swelling to its Lagrangian volume: s

∂A β ɛ þ ɛg ρg g gl β ;l β ¼ w;o;g ∂ɛ

t

¼  ɛg pg;l þ ðɛg λg dmm Þ;l þ ð2ɛ g λg dkl Þ;k ɛ g ρg Rg vg;s l

I sE ¼

g

∑

Z

∂Aα β ɛ β ;l β ¼ w;o;g ∂ɛ ∑

For gas phase:

g

¼  ɛ g pg;l þ ðɛ g νgklmn dmn Þ;k þ ðɛ g H gklm T ;m Þ;k  ɛ g ρg

α

Rα vα;s ¼  ɛα pα;l þ ðɛ α λα dmm Þ;l þð2ɛ α λα dkl Þ;k  ɛ α ρα l

I sE ¼ EsMM ð45Þ

Rgkl vg;s k

ð47Þ

Using (41) and (47), the following generalized Darcy's law relations for the isotropic porous materials is obtained. For α ¼ w; o:

ðmÞ

temperature (T) and strain-rate (EsKL ) but this dependence is not shown in (43) and thereafter to avoid the equations to appear unwieldy. For the fluid phases, λα and μα are the dilational and shear viscosities, respectively. For a single-scale homogeneous material only the integral term shows up in the stress equation (Christensen, 1982). As shown in Eq. (42), in a poroviscoelastic material composed of multiphases, the total stress involves numerous terms such as pore pressure, Terzaghi stress, hydrophilic stress, hydrophobic stress, viscoelastic stress and viscous stress. Eq. (42) can be used for applications such as stress-crack initiation, expansion, crust formation, sheeting and drying. In the Fourier's law relation, (43), the thermal conductivity α of mixture is given by k ¼ ∑α ¼ s;w;o;g ɛ α k , where the linearity is assumed.

103

s

dv  dV : s dV

ð52Þ

For a saturated system (ɛ w ¼ 1  ɛs ) without oil phase, IsE can be written as s

I sE ¼ EsMM ¼

s

dv  dV 1 1  ɛs ɛw ¼ s 1 ¼ ¼ s s ɛ ɛs ɛ dV

ð53Þ

Thus, EsMM;l of (48) can be written as ðɛw =ɛ s Þ;l which reduces the integral term to volume fraction gradient based form obtained by Singh et al. (2003c). The change to volume fraction gradient based form makes it easier to obtain the solution, but the equation can be utilized only for saturated systems. For an unsaturated system,

104

P.S. Takhar / Chemical Engineering Science 109 (2014) 98–110

the above equation can be written as s

I sE ¼ EsMM ¼

s

dv  dV 1 1 ɛ ϕ ¼ s 1 ¼ s ¼ s s ɛ ɛ ɛ dV s

ð54Þ

Thus, for the unsaturated system either EsMM;l of (48) or ðϕ=ɛ s Þ;l will need to be used to obtain the solution. Using arguments similar to Singh et al. (2003c) it can be shown that in the glassy and rubbery states, when the relaxation time of biopolymers becomes very large and very small, respectively, the integral term will predict Fickian (or Darcian) type of fluid flow. In the vicinity of glasstransition, when the relaxation time of the biopolymers is of the order of the experimental time, the integral term retains its form, thus predicting non-Fickian (or non-Darcian) mode of fluid transport. Similar observations have been made in the experimental studies in polymer science (Kim et al., 1996). The reader is referred to Singh et al. (2003c) for additional details on various modes of fluid transport. For α ¼ o in (48), the generalized Darcy's law can be used to study time dependent diffusion of lipids in lipophilic domains for applications such as chocolate blooming (Galdamez et al., 2009) and oil diffusion in polypropylene membranes (Ponsard-Fillette et al., 2005). In Darcy's law for the gas phase, (49), in addition to the classical pressure gradient term (the first term on RHS) and Brinkman correction terms (second and third terms on RHS), the cross-effect terms (fourth term on RHS) show up for unsaturated transport. The cross-effect terms couple the gas flow equation to the fluid flow equations via volume fraction gradients. Various cross-effect coefficients, ∂Aα =∂ɛβ , need to be determined by devising appropriate experiments. 3.7. Case II: isotropic biomaterials with negligible Brinkman, cross-effect and gravity terms

and for the gas phase, take α = g and drop the last term of (55). By taking Rα in the above equations to the right hand side " # Z t ∂EsMM;l Kα α α α α α α α α dτ ; vα;s ¼  ɛ p þ ɛ D ɛ þ ɛ ρ M ðt τÞ ;l ;l l μα ∂τ 0 ð56Þ

where K α  μα =Rα

ð57Þ

is the coefficient of permeability, Dα  ρα =Rα ð∂Aα =∂ɛα Þ

3.8. Case III: isotropic biopolymers with a short-memory If the isotropic biopolymers are assumed to have a shortmemory, the second and higher-order time derivatives of strain tensor in (44) can be neglected. The first-order strain tensor tensor can be retained to describe swelling/shrinking behavior (Thomas and Windle, 1982; Achanta et al., 1994). The integral term of (56) can be replaced by the following term (Singh et al., 2003c): ɛα ρα

α ∂Aα _ s αK Bα ɛ_ α;l s E KL;l ¼ ɛ α _ μ ∂E

ð61Þ

KL

where Bα is the mixture viscosity (Achanta et al., 1994; Singh et al., 2003a). Thus, the generalized Darcy's law relation (56) becomes  α  K Kα ¼  α ɛ α pα;l þ ɛα Dα ɛ α;l þ ɛ α α Bα ɛ_ α;l ; where α ¼ w; o: vα;s ð62Þ l μ μ If both concentration and pressure-gradient driven flow are present, (62) can be solved. Examples of such problems would involve rapid drying, rapid evaporation during expansion of biopolymers and frying of foods. Solution of this equation would generalize the previous work of (Achanta et al., 1997; Singh et al., 2004a, 2004b; Takhar, 2011) with food biopolymers, where only the diffusive flux term was solved. 3.9. Numerical simulation of unsaturated transport equations and comparison to experimental data

If the fluid velocity is not high, the Brinkman correction term in (48) can be neglected. This is expected to occur in low Reynolds number flow in biomaterials. By performing additional simplification by neglecting the cross-effect and gravity terms, the generalized Darcy's law relation (48) reduces to Z t α ∂EsMM;l α α α α α ∂A α α α Rα vα;s dτ; ¼  ɛ p ɛ ρ ɛ  ɛ ρ M ðt  τÞ ;l ;l l ∂ɛα ∂τ 0 where α ¼ w; o; ð55Þ

where α ¼ w; o;

It is straightforward to deduce that performing a similar simplification for the gas phase would result in Darcy and diffusive flow terms. The diffusive term would apply to the problems where the diffusion of gas phase dissolved in the biopolymeric mixture occurs.

Frying of foods involves both concentration and pressuregradient driven flow (Halder et al., 2007). The unsaturated transport equations developed above can be tested using experimental data on frying. The frying experiments were conducted in Takhar lab at University of Illinois. Potatoes of Russet variety were obtained from a local grocery store and cut into cylindrical shapes of 0.04 m length and 0.018 m diameter. The cylinders were immersed in high temperature oil at 190 1C. The cylinders contained a high amount of moisture (5.45 g water/g solids). Their initial temperature before immersion in oil was 24 1C. After the cylinders were immersed in the oil, the oil temperature reduced to 174 1C within 10 s and stayed at 17571 1C throughout the process. The frying process was performed for different time amounts ranging from 20 s to 300 s. At the end of the process the moisture and oil content of potato cylinders were determined using a calibrated moisture meter and Soxhlet method (AOAC, 1996). Thermocouples were inserted in the center of potato cylinders and in the oil to record temperature versus time.

ð58Þ

is the coefficient of diffusivity, and Mα  M α =Rα

ð59Þ

is the non-Fickian memory term. Following Achanta et al. (1997), ρα ð∂Aα =∂ɛα Þ represents coefficient of elasticity (E) of the biopolymeric mixture. Therefore, eliminating Rα between (57) and (60) K α ¼ Dα μα =E

ð60Þ

Therefore, experimentally determined diffusivity can be converted to permeability using (60), if E of the biopolymeric mixture is known.

Fig. 1. Unsaturated transport in potato cylinders heated in high temperature oil.

P.S. Takhar / Chemical Engineering Science 109 (2014) 98–110

3.9.1. Problem description Fig. 1 depicts the transport mechanisms inside a single potato cylinder. When the potatoes are immersed in hot oil, heat penetration from the oil to potatoes causes evaporation of water near the surface. The evaporation front moves toward the center with time. The driving forces for moisture movement are both concentration gradient (diffusion) and pressure gradient (Darcian flow). At sites of evaporation, high gas pressure is developed. The gas pressure gradient causes vapor movement toward the surface. Rate of evaporation is a near-equilibrium process. Sites of vapor formation serve as sinks of heat in macroscale REV. In this section, the Darcy's law relations listed under Case-III will be used by coupling them with the mass balance equations. The two-scale heat transport relation published in the literature will be utilized (Bear and Bachmat, 1991). 3.9.2. Model equations The two-scale mass balance equation for the phase α is (Achanta et al., 1994) Dα ðɛα ρα Þ α þ ɛ α ρα vαl;l ¼ ∑ β e^ ; Dt β ¼ w;o;g βaα

where α ¼ w; o; g

ð63Þ

where the material time derivative is taken with respect to the phase α. The material time derivative of a variable ψ can be converted to the one with respect to the solid phase s using the relation: Dα ψ Ds ψ ¼ þ vα;s ψ ;l l Dt Dt

ð64Þ

The solid phase mass balance equation is (Singh et al., 2003b)

which can be obtained by simplifying Gibbs free energy based relation, (25), as shown by Fang and Ward (1999) for pure water ρveq ¼ ρveq =ðRv TÞ

ð65Þ

Using (64) to obtain material time derivative in (63) with respect to solid phase, and using (65) to eliminate vsl;l from (63) and simplifying, results in Ds ðɛα ρα Þ ɛ_ s α þ ðɛ α ρα vα;s Þ;l  ɛ α ρα s ¼ ∑ β e^ ; l Dt ɛ β ¼ w;o;g βaα

where α ¼ w; o; g

ð66Þ

where the velocity of a phase is given by the generalized Darcy's law Eq. (62). The solid phase volume fraction, ɛs can be obtained from its relation with the porosity, ɛs ¼ 1 ϕ. For problems involving evag poration/condensation, w e^ represents mass lost by the water phase, which is equal to the mass gained by the gas phase. In the problem considered here, oil and solid phases are not considered to exchange α mass with other phases. Thus, β e^ ¼ 0 for the solid and oil phases. The rate of evaporation was calculated using the following relation,

ð67Þ v

¼ aw ps , and R is the gas constant for water vapors. The where vapor concentration ρv was calculated by taking α ¼ v in (66) and using the following relations: ρveq

Dv v ðω Þ;l ωv g;s v;s vl ¼ vl þvv;g l

vv;g ¼ l

ð68Þ

where the first row of (68) is the binary-diffusion equation for the mixture of vapors and air (Bird et al., 2006), and the second equation adds the velocity of gas mixture with respect to solid phase to the vapor diffusive velocity with respect to the gas mixture to result in vapor velocity with respect to the solid phase. To calculate heat-transfer in the porous biomaterial, the following two-scale convection-diffusion equation with phase-change effects was used (Bear and Bachmat, 1991): ðρC p Þeq

∂T g þ ðρC p Þf vfl ;s T ;l  ðkT ;l Þ;l ¼  λw e^ ∂t

ð69Þ

where various quantities for the macroscale mixture of different phases are given by (Bennethum and Cushman, 1999) ðρC p Þeq ¼

∑

α ¼ s;w;o;g

ðρC p Þf ¼ vfl ;s ¼ k¼

Ds ɛ s þ ɛ s vsl;l ¼ 0 Dt

105

∑

ɛ α ρα C αp

α ¼ w;o;g

∑

α ¼ w;o;g

∑

α ¼ s;w;o;g

ɛ α ρα C αp

ɛ α vα;s l α

ɛα k :

ð70Þ

On LHS of (69), the first term represents energy storage, the second term represents energy convected by the moving fluids and the third term represents conduction. The RHS term represents a heat sink within the REV due evaporation (or a heat source due to condensation). Eqs. (63)–(70) coupled with the Darcy's law relations (62) form a system that can be solved for studying unsaturated transport problems for biopolymeric systems. Various initial and boundary conditions, material properties and supporting equations solved for transport in the potato cylinder are listed in Tables 1 and 2. These tables are self-explanatory. Some points worth noting are—the second column in these tables lists the equation numbers where a property or initial/boundary condition is needed; detailed methodology of quantities listed as measured is described in Maneerote (2009), Lalam et al. (2013) and Sandhu et al.

Table 1 Initial and boundary conditions. The second column lists the equation numbers where an initial/boundary condition is needed. Initial/boundary condition Equation I.Cs. Fluid transport Gas transport

Quantity

Source

(66) with α ¼ w; o ɛ w i ¼ 0:8, ɛ oi ¼ 0 (66) with α ¼ g pgi ¼ patm

Vapor diffusion

(66) with α ¼ v

Energy transport

(69)

B.Cs. Fluid transport Oil transport

(66) with α ¼ w (66) with α ¼ o

Vapor diffusion Energy transport

(66) with α ¼ v (69)

ɛw i measured using moisture meter From potato composition

ρvi ¼ 0:022 kg=m3 T i ¼ 297 K

Using P s  T relation and ideal gas law

ew s ¼ 0:01

Measured as per Sandhu et al. (2013)

Measured using thermocouples as per Lalam et al. (2013)

Q mo ¼ hmo ðɛ o  ɛ oo Þ with hmo ¼ 2:0  10  6  4  10  6 m=s Calculated using Sh ¼ 0:82Re0:5 Sc1=3 ; Green and Perry (2008) Calculated as per Halder et al. (2007) ρv Q mv ¼ hmv ðρv  ρvo Þ, with hmv ¼ 1:1 1:6 m=s Farinu and Baik (2007) Q ¼ hðT  T o Þ þ εsðT 4  T 4 Þ, where h follows profile of o

Farinu and Baik (2007)

External medium

s ¼ 5:670373  10  8 W=m2 K4 , ε ¼ 0:9 ɛ oo ¼ 1, ρvo ¼ 0, T o ¼ T oil

Stefan–Boltzman constant Charm (1978)

106

P.S. Takhar / Chemical Engineering Science 109 (2014) 98–110

Table 2 Supporting relations and properties. Second column lists the equation numbers where a supporting relation or property is needed. Diffusivity equation was extrapolated to the frying temperature range by adjusting the pre-exponential factor inversely. Relation/property Moisture diffusivity

Equation Quantity (62)

Source 



 31580 RT exp ðð  0:0025T þ 1:22Þ MwÞ ðm2 =sÞ

Hassini et al. (2007)

Dw ¼ 3:6  10  5 exp

Oil diffusivity

(62)

1:22  10  8 expð  2:8 þ 2M o Þ ðm2 =sÞ

Halder et al. (2007)

Vapor diffusivity

(68)

Dv ¼  2:775  10  6 þ 4:479  10  8 T þ

Nellis and Klein (2008)

1:656  10  10 T 2 ðm2 =sÞ Permeability

(62)

Coefficient of elasticity

(60)

Capillary pressure (pcw)

(52)

Capillary pressure (pco)

(60)

Mixture viscosity

(62)

Evaporation rate constant Porosity

(67)

K α ¼ Dα μα =E ðm2 Þ, with α ¼ w; o E ¼  6422:4M w þ 8179:4 ðPaÞ, if M w o 0:8 ¼ 2050:2M w þ 1699:4 ðPaÞ, if M w 40:8 pcw ¼ pg  pw ¼ 8:4  104 sw 0:63 ðPaÞ pco ¼ pg  po ¼ 2γ cos θ=r ðPaÞ with γ ¼ 0:024 N=m, θ ¼ 38o ,

Eq. (60) Measured as per (Dit-u dompo et al., 2013) Spolek and Plumb (1981) Bradford and Leij (1996), Pinthus and Saguy (1994)

r ¼ 12  10  6 m

(65)

Bα ¼ 0:8  10  6 E=E ðPa  sÞ ξ ¼ 0:3 s  1 Using ϕ_  ð1  ϕÞ



 1 ∂f ðɛ w Þ w ɛ_ ¼ 0 f ðɛ w Þ ∂ɛ w

where, f ðɛw Þ ¼ 0:1568ρw ɛ w =½ρs ð1  ϕÞ þ 0:3714

Following Achanta et al. (1997)'s method As per Halder et al. (2007)'s method 1 Ds js , js Dt ϕ ¼ 1 ɛ s and js ¼ f ðɛ w Þ in (65) Estimated from ϕ versus ɛ w data Substituting vsl;l ¼

(2013); and the heat transfer coefficient varies as a function of frying time. For heat transfer coefficient, the profile of Farinu and Baik (2007) was used in which, h ¼ 150 W=m2 K at t¼0 s, attains a peak value of 860 W/m2 K at t¼70 s, and reduces to 500 W/m2 K toward the end of frying. 3.9.3. Solution methodology The solution was obtained by implementing Eqs. (62)–(70) in a commercial finite elements package (Comsol Multiphysics, Ver 4.3, Burlington, MA, U.S.A.). For water, oil and vapor, the variables ɛ w , ɛ o and ρv were treated as dependent, respectively. For the gas phase, Pg was calculated. T was calculated using the heat transfer equation (69). The axisymmetric one-dimensional finite element geometry was used to model transport in the radial direction of potato cylinders. A mapped type mesh was used with 40 quadrilateral elements with an element-ratio (ratio of center to surface element length) of 2. The solution was obtained using the PARDISO method with backward Euler method for time-stepping. The timesteps were selected automatically by the solver with maximum step-size of 0.1 s. The time-dependent process was simulated up to 300 s. The solution was obtained over an Apple Macbook Pro with a 4-core, 2.7 GHz Intel i7 Processor and 16 GB RAM. Each simulation run was completed in 40–45 s. A relative tolerance of 0.01 and an absolute tolerance value of 0.001 was selected. 3.9.4. Validation of numerical simulations The calculated ɛ w and ɛ v values were converted to moisture content on mass basis (M) as a function of ðr; tÞ. The average moisture content was calculated on volume-basis M avg ðtÞ ¼ R R MðtÞ dv= dv. Fig. 2 shows comparison of experimental and predicted average moisture content values. The experimental moisture content values were calculated by taking the average of data collected in three frying trials. The experimental moisture content data shows greater variation initially due to inaccuracies introduced while pulling out of potato samples from the fryer when the moisture content was varying rapidly. Fig. 2 shows that the model predicted experimental moisture content with sufficient accuracy (R2 ¼ 0:8; CV ¼ 0:1). Fig. 3 shows the comparison of calculated and experimental center temperature (R2 ¼ 0:9; CV ¼

Fig. 2. Comparison of experimental and predicted average moisture content.

Fig. 3. Comparison of experimental and predicted temperature at the center of potato cylinder.

P.S. Takhar / Chemical Engineering Science 109 (2014) 98–110

0:1). The diffusivity values of Hassini et al. (2007) were extrapolated from the drying temperature range of 40–80 1C to the frying temperature range of 170–180 1C. The heat and mass transfer coefficient values in the frying temperature range are also not known precisely (Halder et al., 2007). Despite of these uncertainties, the model predicted average moisture content and center temperature with reasonable accuracy. With a better estimate of diffusivity, and heat and mass transfer coefficient values, the accuracy of model predictions is expected to further improve. In Fig. 3, the simulated surface temperature is lower than the experimental oil temperature because of cooling caused by water evaporation at the surface of potato cylinders during frying. Similar observations were made in an experimental frying study about the surface temperature of potato discs (Sandhu et al., 2013).

Fig. 4. Oil distribution as a function of radial distance from the center.

107

Several interesting observations can be made about the underlying transport mechanisms as discussed in the next section.

3.9.5. Simulation runs Using hot-stage attached to a light microscope, it has been shown that potato starch granules are resistant to oil uptake during frying (Aguilera et al., 2001). The same has been shown using confocal laser scanning microscopy during frying of potato products (Pedreschi et al., 2008) and dye uptake experiments with chicken nuggets (Lalam et al., 2013). Moreira et al. (1997) showed that in tortilla chips 20% oil uptake occurs during frying and the rest during cooling. However, the reasons for the resistance of food matrix to oil uptake during frying are still not fully understood. The simulation runs are used in this section to discuss the possible reasons. Simulation results in Fig. 4 show that oil penetrated only the surface layers of potato cylinders. A large part of the potato remained devoid of oil. To obtain insight into the reasons for oil penetration only in the surface layers, the pressure development inside the cylinder is discussed (Fig. 5a–d). Fig. 5a shows that due to evaporation, positive gas pressure is developed inside the potato cylinder. The gas pressure increases throughout the cylinder up to 20 s and then start decreasing at the center. The gradient of gas pressure (dp=dr) near the surface layers remains negative even toward the end of frying although its magnitude starts decreasing. This gas pressure development under the surface due to evaporation and its negative gradient would resist oil penetration caused by diffusive flux due to large pool of oil outside the potato cylinder. Potato cylinder's potential to uptake oil increases during the frying stage itself. This can be described using pore pressure

Fig. 5. Gas (a), capillary (b), water (c) and pore (d) pressure distribution across the radial cross-section of the potato cylinder.

108

P.S. Takhar / Chemical Engineering Science 109 (2014) 98–110

Fig. 6. Pore pressure at the center of potato cylinder as a function of frying time.

development (average pressure exerted by various fluids against the solid walls, ppore ¼ sw pw þ sg pg þso po (Ehlers and Bluhm, 2002)) (Fig. 5). Fig. 5b shows that capillary pressure due to water phase (pcw) becomes significantly larger than the atmospheric pressure during frying. This is expected because dry layers formed near the surface during frying will cause the hydrophilic matrix to adsorb water. Since, pw ¼ pg  pcw , capillary pressure greater than the atmospheric pressure would make pw negative (Fig. 5c). A negative pw made ppore negative (Fig. 5d). Negative water pressure is also observed in soils in the unsaturated region due to affinity of solid matrix to water (Mercury et al., 2003). After 90 s, the pore pressure inside large part of the cylinder becomes negative and its gradient (dppore =dr) becomes positive. Therefore, more oil can penetrate the potato cylinder if sufficient time is allowed, or smaller sized potatoes are used or subsequently during the cooling stage. The pore pressure is plotted as a function of time at the center of the cylinder (Fig. 6). Pore pressure becomes larger than the atmospheric pressure initially and then starts decreasing and becomes negative. After the peak negative pressure is attained, pressure starts increasing slowly but remains negative throughout the frying. A similar pattern has been observed in pore pressure measurement for various frying conditions with potatoes and chicken nuggets (Sandhu et al., 2013; Lalam et al., 2013). In the experimental work, the reason for negative pore pressure was not known. Most past studies have focussed on capillary forces in the oil phase being the primary reason behind oil uptake during frying (Ziaiifar et al., 2008). Here it is worth noting that the capillary forces in the water phase are equally important, because they make the overall pore pressure negative, thus increasing a food matrix's potential to suck oil after the gas pressure lowers. In summary, to optimize the unsaturated transport processes, the developed fluid transport equations can be solved to obtain information about fluid penetration, pressure magnitudes and gradients, rate of evaporation, etc.

4. Conclusions The continuum thermodynamics based hybrid mixture theory was used to obtain the unsaturated transport equations for poroviscoelastic biopolymers. The solid phase was treated as viscoelastic and the fluid phases were considered as viscous at

the microscale. The viscoelastic solid interacted with the viscous fluids to result in a viscoelastic matrix with long memory at the macroscale. The hydrophilic and hydrophobic (or lipophilic) interaction of the biopolymers with the water and oil phases was incorporated. The liquid water exchanged mass with the water vapors in gas phase. The equations for generalized Darcy's laws for various phases, near-equilibrium swelling and capillary pressure relations, generalized stress relations, near-equilibrium Gibbs free energy based rate of evaporation relation were obtained. The generalized Darcy's law relations for the water and oil phases included cross-effects resulting from the presence of other phases in the unsaturated system. The Darcy's law relations included novel integral terms incorporating the effect of viscoelastic relaxation on fluid transport. These equations can be used for predicting both Darcian and non-Darcian fluid transport during processing of food and biopolymeric materials in a wide range of temperatures and fluid contents. The resulting Gibbs-free energy relations showed that the evaporation (or condensation) of water away from equilibrium is governed by the difference in free energies between water and gas phases. Using the developed equations, one can predict water, vapor and oil distribution in pores, the interaction between flowing fluids and biopolymers, and thermomechanical changes in the biopolymeric matrix. Solving these equations would provide insights into the physical mechanisms at micro and macro scales, and would allow one to modify the process and product parameters to obtain desirable product characteristics. The solution of equations predicted experimental data on moisture content and temperature profile during high temperature frying of potato cylinders with reasonable accuracy despite of extrapolation of the experimental parameters by 100 1C. The solution of equations also predicted the trend in pore pressure profiles in agreement with the experimental data of Lalam et al. (2013) and Sandhu et al. (2013). The simulations showed that pore pressure becomes negative during frying due to high magnitude of water capillary pressure. Nomenclature Latin symbols Aα Bα CV CðνÞ Cp Dα dvs dVs α

dkl β bα e EsKL EsMM;l s E_ KL ðmÞ s EKL

E E f ðɛ w Þ g αl Gα

Helmholtz free energy of the phase α (J/kg) mixture viscosity of the biopolymeric matrix (Pa s) coefficient of variation (dimensionless) a general representation of coefficients in entropy inequality specific heat (J/(kg1 K)) coefficient of diffusivity of the phase α (m2/s) Eulerian volume of the solid phase matrix after expansion (m3) Lagrangian volume of the solid phase before expansion (m3) rate of deformation tensor for the phase α (s  1) net rate of mass transfer from the phase β to the phase α (kg/(m3 s)) Lagrangian strain tensor of the solid phase (dimensionless) gradient of the trace of Lagrangian strain tensor (m  1) first order material time derivative of EsKL (s  1) mth order material time derivative of EsKL (s  m) coefficient of elasticity of the biopolymeric matrix (Pa) average coefficient of elasticity (Pa) function relating Jacobian(js) to ɛ w (dimensionless) gravitational or body force acting on the phase α (m/s2) Gibbs free energy of the phase α (J/kg)

P.S. Takhar / Chemical Engineering Science 109 (2014) 98–110

Gs1 Gs2 γ h hmo hmv

bulk relaxation modulus of the viscoelastic solid (Pa) shear relaxation modulus of the viscoelastic solid (Pa) surface tension (N/m) surface heat transfer coefficient (W/(m21 K)) surface mass transfer coefficient for the oil phase (m/s) surface mass transfer coefficient for the vapor phase (m/ s) js solid phase Jacobian (dimensionless) α macroscale thermal conductivity of the α phase (W/ k (m1 K)) k macroscale thermal conductivity of the mixture of various phases (W/(m1 K)) permeability of the α phase (m2) Kα M α ðtÞ memory function of the poroviscoelastic material in Darcy's law for phase α (J/kg) M moisture content on mass dry basis (g/g solids) Mα M α =Rα , memory function in Darcy's law Eq. (56) (m5/ (kg s)) material coefficient of swelling and expansion potential N αβ relations (29)–(31) (Pa s) Nβ material coefficient of near-equilibrium capillary pressure relation (34) (Pa s) N β0 material coefficient of near- equilibrium capillary pressure relation (35) (Pa s) α net external body source of energy for the phase α (W/ h kg) pα physical pressure in the phase α (Pa) pcw capillary pressure due to water phase (Pa) pco capillary pressure due to oil phase (Pa) qαl heat flux for the α phase (W/m2) surface oil flux (m/s) Qmo Qmv surface vapor flux (m/s) r average pore radius (m) material coefficient for the Darcy's laws (Pa s/m2) Rαkl α material coefficient for the isotropic form of Darcy's law R relations (Pa s/m2) R2 coefficient of determination [dimensionless] REV representative elementary volume (m3) β bα net heat transfer from the phase β to the phase α (W/ Q m3 ) t Time [s] sα degree of saturation by the phase α (dimensionless) s Stefan–Boltzman constant (W/(m21 K4)) α t kl stress tensor for the α phase (Pa) Terzaghi stress (Pa) tse kl tsw hydration stress (Pa) kl tso lipophilic or hydrophobic stress (Pa) kl T temperature (1K) β bα net momentum transfer from the phase β to the phase α Tl (N/m3) vαl velocity of the α phase (m/s) xαk Eulerian coordinate in the α phase (m) Lagrangian coordinate in the α phase (m) X αK Greek symbols δkl ɛα ε ηα Λ Λα

Kronecker delta function in Eulerian coordinates (dimensionless) volume fraction of the α phase (dimensionless) coefficient of emissivity (see Table 1) (dimensionless) entropy of the α phase (J/(kg1 K)) net entropy production in the system (mixture of various phases) at macroscale (W/(m31 K)) net entropy production in the α phase at macroscale (W/(m31,K))

λα λ LHS RHS μα ναklmn ν ωv πα

β bα

ϕ

ϕ ρα ρveq ξα ξ

109

dilational viscosity of the α phase in the porous matrix (Pa s) latent heat of vaporization in (69) (J/kg) left hand side right hand side shear viscosity of the α phase in the porous matrix (Pa s) material coefficient in Eq. (21) (Pa s) a general representation of variables becoming zero at equilibrium, see Eq. (5) ¼ ρv =ρg , mass fraction of the vapor phase (dimensionless) swelling pressure in the α phase [Pa] entropy transfer from the β phase to the α phase (W/m31 K) porosity of the biopolymeric matrix (dimensionless) density of the phase α (kg/m3) density of vapor phase at equilibrium (kg/m3) near-equilibrium material coefficient for the exchange of mass between two phases (J m3 s/kg2) mass exchange material coefficient for (67) (s  1)

Subscripts atm i k, l K, L eq

atmospheric initial indices for Eulerian coordinates indices for Lagrangian coordinates denotes “equilibrium” in (67), and “equivalent” in (69) and (70)

Superscripts α β m s w o g

general representation of a phase general representation of a phase order of the material time derivative of EsKL that varies from 0 to p solid phase water phase oil phase gas phase

Special symbols Dα =Dt material time derivative with respect to velocity of the α phase particle (s  1) α;s velocity of the α phase relative to the solid phase vl ( ¼ vαl vsl ) (m/s) Dot Ds =Dt, material time derivative with respect to velocity () of the solid phase particle

Acknowledgements This work was supported in part by the National Science Foundation award #0756762 and USDA-NIFA award #200935503-05279. Thanks to Jaspreet Sandhu and Harkirat Bansal for providing help with the experimental work on frying. References Achanta, S., Cushman, J.H., 1994. Nonequilibrium swelling-pressure and capillarypressure relations for colloidal systems. J. Colloid Interface Sci. 168 (1), 266–268. Achanta, S., Cushman, J.H., Okos, M.R., 1994. On multicomponent, multiphase thermomechanics with interfaces. Int. J. Eng. Sci. 32 (11), 1717–1738. Achanta, S., Okos, M.R., Cushman, J.H., Kessler, D.P., 1997. Moisture transport in shrinking gels during saturated drying. AICHE J. 43 (8), 2112–2122.

110

P.S. Takhar / Chemical Engineering Science 109 (2014) 98–110

Aguilera, Jose Miguel, Cadoche, Laura, Lopez, Carlos, Gutierrez, G., 2001. Microstructural changes of potato cells and starch granules heated in oil. Food Res. Int. 34 (10), 939–947, ISSN 0963-9969. AOAC, 1996. Official Methods of Analysis of AOAC International, 16th edition. Gaithersburg, Md: Association of Official Analytical Chemists. Method No. 991.36. Bear, J., Bachmat, Y., 1991. Introduction to Modeling of Transport Phenomena in Porous Media. Kluwer Academic Publisher, Dordrecht, The Netherlands Bennethum, L.S., Cushman, J.H., 1996a. Multiscale, hybrid mixture theory for swelling systems. 1. Balance laws. Int. J. Eng. Sci. 34 (2), 125–145. Bennethum, L.S., Cushman, J.H., 1996b. Multiscale hybrid mixture theory for swelling systems. 2: Constitutive theory. Int. J. Eng. Sci. 34 (2), 147–169. Bennethum, L.S., Weinstein, T., 2004. Three pressures in porous media. Transp. Porous Media 54, 1–34. Bennethum, L.S., Cushman, J.H., 1999. Coupled solvent and heat transport of a mixture of swelling porous particles and fluids: single time-scale problem. Transp. Porous Media 36 (2), 211–244. Bennethum, L.S., Cushman, J.H., Murad, M.A., 1996. Clarifying mixture theory and the macroscale chemical potential for porous media. Int. J. Eng. Sci. 34 (14), 1611–1621. Bennethum, L.S., Murad, M.A., Cushman, J.H., 2000. Macroscale thermodynamics and the chemical potential for swelling porous media. Transp. Porous Media 39 (2), 187–225. Bird, R.B., Stewart, W.E., Lightfoot, N.L., 2006. 2nd editionTransport Phenomena, vol. 2. John Wiley and Sons, New York. Bowen, R.M., 1982. Compressible porous-media models by use of the theory of mixtures. Int. J. Eng. Sci. 20 (6), 697–735. Bradford, S.A., Leij, F.J., 1996. Predicting two and three-fluid capillary pressuresaturation relationships of porous media with fractional wettability. Water Resour. Res. 32 (2), 251–259. Charm, S.E., 1978. Fundamentals of Food Engineering, 3rd edition The Avi Publishing Co., Westport, CT Christensen, R.M., 1982. Theory of Viscoelasticity. Academic Press, NY Coleman, B.D., Noll, W., 1963. The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Ration. Mech. Anal. 13, 167–178. Dit-u dompo, S., Takhar, P.S., Ganjyal, G., Hanna, M., 2013. The effect of temperature and moisture on mechanical properties of extruded cornstarch. J. Texture Stud. 44, 225–237. Ehlers, W., Bluhm, J., 2002. Porous Media: Theory, Experiments and Numerical Applications. Springer-Verlag, Berlin Eringen, A.C., 1980. Mechanics of Continua. R.E. Krieger Pub. Co., Huntington, New York, U.S.A Fang, G., Ward, C.A., 1999. Examination of the statistical rate theory expression for liquid evaporation rates. Phys. Rev. E 59 (1), 441–453. Farinu, A., Baik, O., 2007. Heat transfer coefficients during deep fat frying of sweet potato: effects of product size and oil temperature. Food Res. Int. 40, 989–994. Ferry, J., 1980. Viscoelastic Properties of Polymers. John Wiley and Sons Inc., NY Galdamez, J.R., Szlachetka, Kristin, Larry Duda, J., Ziegler, Gregory R., 2009. Oil migration in chocolate: a case of non-Fickian diffusion. J. Food Eng. 92 (3), 261–268. Ganjyal, G., 2009. Personal Communication. MGP Ingredients, Atchison, KS Gray, W.G., Hassanizadeh, S.M., 1991a. Paradoxes and realities in unsaturated flow theory. Water Resour. Res. 27 (8), 1847–1854. Gray, W.G., Hassanizadeh, S.M., 1991b. Unsaturated flow theory including interfacial phenomena. Water Resour. Res. 27 (8), 1855–1863. Gray, W.G., Hassanizadeh, S.M., 1994. Paradoxes and realities in unsaturated flow theory – reply. Water Resour. Res. 30 (5), 1625–1626. Green, D.W., Perry, R.H., 2008. Perry's Chemical Engineers Handbook. The McGrawHill Companies, NY Halder, A., Dhall, A., Datta, A.K., 2007. An improved, easily implementable, porous media based model for deep-fat frying. Part I. Model development and input parameters. Food Bioprod. Process. 85 (C3), 209–219. Hassanizadeh, S.M., 1986. Derivation of basic equations of mass-transport in porous-media. I. Macroscopic balance laws. Adv. Water Resour. 9 (4), 196–206. Hassanizadeh, S.M., Gray, W.G., 1979. General conservation equations for multiphase systems. I. Averaging procedure. Adv. Water Resour. 2, 131–144. Hassanizadeh, S.M., Gray, W.G., 1980. General conservation equations for multiphase systems. III. Constitutive theory for porous media flow. Adv. Water Resour. 3, 25–40. Hassanizadeh, S.M., Gray, W.G., 1993. Thermodynamic basis of capillary-pressure in porous-media. Water Resour. Res. 29 (10), 3389–3405.

Hassanizadeh, S.M., Celia, M.A., Dahle, H.K., 2002. Dynamic effect in the capillary pressure-saturation relationship and its impacts on unsaturated flow. Vadose Zone J. 1, 38–57. Hassini, L., Azzouz, S., Peczalski, R., Belghith, A., 2007. Estimation of potato moisture diffusivity from convective drying kinetics with correction for shrinkage. J. Food Eng. 79, 47–56. Kim, D.J., Caruthers, J.M., Peppas, N.A., 1996. Experimental verification of a predictive model of penetrant transport in glassy polymers. Chem. Eng. Sci. 51 (21), 4827–4841. Kleinfelter, N., Park, M., Cushman, J.H., 2007. Mixture theory and unsaturated flow in swelling soils. Transp. Porous Media 68, 69–89. Lalam, S., Sandhu, J., Takhar, P.S., Thompson, L., Alvarado, C., 2013. Experimental study on transport mechanisms during deep fat frying of chicken nuggets. LWT-Food Sci. Technol. 50 (1), 110–119. Maneerote, J., 2009. Transport Phenomena and Frying Conditions (thesis). Mercury, L., Azaroual, M., Zeyen, H., Tardy, Y., 2003. Thermodynamic properties of solutions in metastable systems under negative or positive pressures. Geochim. Cosmochim. Acta 67 (10), 1769–1785. Moreira, R.G., Sun, X.Z., Chen, Y.H., 1997. Factors affecting oil uptake in tortilla chips in deep-fat frying. J. Food Eng. 31 (4), 485–498. Morin, L.A., Temelli, F., McMullen, L., 2004. Interactions between meat proteins and barley (hordeum spp.) [beta]-glucan within a reduced-fat breakfast sausage system. Meat Sci. 68 (3), 419–430. Murad, M.A., Cushman, J.H., 2000. Thermomechanical theories for swelling porous media with microstructure. Int. J. Eng. Sci. 38 (5), 517–564. Muraleetharan, K.K., Wei, C., 1999. Dynamic behavior of unsaturated porous media: governing equations using the theory of mixtures with interfaces. Int. J. Numer. Anal. Methods Geomech. 23, 1579–1608. Nellis, G., Klein, S., 2008. Heat Transfer, 1st edition Cambridge University Press, Cambridge, UK Pedreschi, F., Cocio, C., Moyano, P., Troncoso, E., 2008. Oil distribution in potato slices during frying. J. Food Eng. 87 (2), 200–212. Pinthus, E.J., Saguy, I.S., 1994. Initial interfacial tension oil uptake by deep-fat fried foods. J. Food Sci. 59 (4), 804–807. Ponsard-Fillette, M., Barres, C., Cassagnau, P., 2005. Viscoelastic study of oil diffusion in molten PP and EPDM copolymer. Polymer 46 (23), 10256–10268. Sandhu, J., Bansal, H., Takhar, P.S., 2013. Experimental measurement of physical pressure in foods during frying. J. Food Eng. 115 (2), 272–277. Segel, Lee A., Handelman, G.H., 1977. Mathematics Applied to Continuum Mechanics. Macmillan, New York Singh, P.P., Cushman, J.H., Bennethum, L.S., Maier, D.E., 2003a. Thermomechanics of swelling biopolymeric systems. Transp. Porous Media 53 (1), 1–24. Singh, P.P., Cushman, J.H., Maier, D.E., 2003b. Three scale thermomechanical theory for swelling biopolymeric systems. Chem. Eng. Sci. 58, 4017–4035. Singh, P.P., Cushman, J.H., Maier, D.E., 2003c. Multiscale fluid transport theory for swelling biopolymers. Chem. Eng. Sci. 58 (11), 2409–2419. Singh, P.P., Maier, D.E., Cushman, J.H., Campanella, O., 2004a. Effect of viscoelastic relaxation on moisture transport in foods. Part II. Sorption and drying of soybeans. J. Math. Biol. 49 (1), 20–35. Singh, P.P., Maier, D.E., Cushman, J.H., Haghighi, K., Corvalan, C., 2004b. Effect of viscoelastic relaxation on moisture transport in foods. Part I. Solution of general transport equation. J. Math. Biol. 49 (1), 1–19. Slade, L., Levine, H., 1991. A Food Polymer Science Approach to Structure–Property Relationships in Aqueous Food Systems: Non-Equilibrium Behavior of Carbohydrate–Water Systems. Spolek, G.A., Plumb, O.A., 1981. Capillary pressure in softwoods. Wood Sci. Technol. 15, 189–199. Takhar, P.S., 2011. Hybrid mixture theory based moisture transport and stress development in corn kernels during drying: coupled fluid transport and stress equations. J. Food Eng. 105 (4), 663–670. Thomas, N.L., Windle, A.H., 1982. A theory of case-II diffusion. Polymer 23 (4), 529–542. Truskey, G.A., Yuan, F., Katz, D., 2004. Transport Phenomena in Biological Systems. Pearson Education Inc., New Jersey, USA Wolfe, J., 1987. Lateral stresses in membranes at low water potential. Aust. J. Plant Physiol. 14 (3), 311–318. Zhu, H., Dhall, A., Mukherjee, S., Datta, A.K., 2010. A model for flow and deformation in unsaturated swelling porous media. Transp. Porous Media 84, 335–369. Ziaiifar, A.M., Achir, N., Courtois, F., Trezzani, I., Trystram, G., 2008. Review of mechanisms, conditions, and factors involved in the oil uptake phenomenon during the deep-fat frying process. Int. J. Food Sci. Technol. 43 (8), 1410–1423.