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A Finite Element Analysis of Coupling Between Water Absorption and Swelling of Foodstuffs During Soaking

Abstract

Most foodstuffs are viscoelastic in nature and they experience volume change when soaked. Moisture transport affects the volume change of foodstuffs in a soaking process and vice versa. Studies of soaking of foodstuffs that involve rigorous two way coupling between simultaneous moisture transport and large viscoelastic deformation of the material are missing, to the best of the authors’ knowledge. In this article, these two important phenomena, i.e., swelling and moisture transport during a soaking process of foodstuffs are coupled non-empirically based on a fundamental mathematical model developed by Zhu et al. (Transp Porous Media 84:335–369, 2010). A finite element analysis is performed to study boiling of plane sheet pasta at 100°C. This coupling of the transport and deformation problems is carried out using a Newton scheme based on a Lagrangian description of the spatial domains. In this study, the dependence of relaxation modulus of pasta on the moisture content is investigated, starting from data in the literature, and it is found from the results that pasta can be considered as a hydro-rheologically simple viscoelastic material as analogous to a thermo-rheologically simple material. A novel theoretically determined expression of the diffusion coefficient is also used to completely separate the viscoelastic effect from diffusion. Sorption curves are calculated and the predictions agree well with experimental results obtained by Cafieri et al. (J Cereal Sci 48:857–862, 2008). Thickness change of plane sheet pasta and the stress field at different times are also calculated. The numerical model presented here can be successfully used to predict simultaneous moisture migration and swelling of viscoelastic food materials during a soaking process.

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Abbreviations

a M :

Time–moisture shift factor

a w :

Water activity

A l :

Free energy of the liquid phase

B :

Material coefficient related to the bulk modulus

B c :

Material constant

C :

Right Cauchy-Green strain tensor of the solid

\({{\bar{\bf C}}}\) :

Right Cauchy-Green strain tensor associated with \({{\bar {\bf F}}}\)

D :

Diffusion coefficient

E :

The Lagrangian strain tensor

\({{\bar E}(s)}\) :

Laplace transform of the Young’s modulus

F :

Deformation gradient of the solid

\({{\bar {\bf F}}}\) :

Multiplicative decomposition of the deformation gradient

g(t):

Normalized relaxation function

I :

Identity tensor

J :

Determinant of the deformation gradient

k l :

Permeability

k i :

Intrinsic permeability

k r :

Relative permeability

K :

Relaxation modulus

K i :

Coefficients of the Prony series

\({{\bar K}(s)}\) :

Laplace transform of the bulk modulus

L :

Total length of the domain

M :

Moisture content on a dry basis

M 0 :

Reference moisture content on a dry basis

M l :

Material coefficient

\({{\bar M}}\) :

Initial moisture content

n :

The unit normal vector

n k :

Number of terms in the Prony series

p α :

Pressure of the α phase

p c :

Capillary pressure

r i :

Radius of the pores in the ith class

R :

The gas constant

s :

The complex parameter in Laplace transform

S :

The second Piola-Kirchhoff stress in the solid phase

S a :

Degree of air saturation

S l :

Degree of water saturation

S′:

The effective second Piola-Kirchhoff stress

t :

Time

t :

The total stress

t a :

The stress in the air phase

t l :

The stress in the water phase

t s :

The Cauchy stress of the solid phase

t′:

The effective Cauchy stress

T :

Temperature

u s :

Displacement of the solid phase

U :

Volumetric part of the energy function

v l :

Molar volume of the liquid

v s :

Velocity of the solid phase

V l :

Volume of the liquid phase

V P :

Volume of the pores

W :

Energy function

\({{\bar W}}\) :

Deviatoric part of the energy function

x :

Position vector in the deformed configuration

X :

Position vector in the undeformed configuration

β :

Constant for the initial condition

Δβ i :

Volume fraction of pores having radius r i

\({\epsilon^{\alpha}}\) :

Volume fraction of the α phase

κ :

Initial bulk modulus

μ :

Viscosity of the liquid phase

\({\bar{\nu}}\) :

Laplace transform of the Poisson’s ratio

ξ :

Coordinate of the parent element

ξ M :

“Shifted time”

π l :

Swelling pressure due to interaction of the solid phase and the bulk fluid

ρ α :

Density of the α phase

τ :

Tortuosity

τ i :

The relaxation times in the prony series

\({\phi}\) :

The porosity

ω :

Weighting function

a:

Air phase

l:

Liquid phase

s:

Solid phase

(·)0 :

Initial value of (·)

α :

α phase

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Correspondence to Subrata Mukherjee.

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Zhu, H., Mukherjee, S. & Dhall, A. A Finite Element Analysis of Coupling Between Water Absorption and Swelling of Foodstuffs During Soaking. Transp Porous Med 88, 399–419 (2011). https://doi.org/10.1007/s11242-011-9746-5

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  • DOI: https://doi.org/10.1007/s11242-011-9746-5

Keywords

  • Soaking pasta
  • Swelling porous material
  • Hydro-rheologically simple
  • Finite element method
  • Non-empirical diffusion coefficient