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Modeling of Coupled Water Transport and Large Deformation During Dehydration of Apple Tissue

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Abstract

Water loss of fruit during storage has a large impact on fruit quality and shelf life and is essential to fruit drying. Dehydration of fruit tissues is often accompanied by large deformations. One-dimensional water transport and large deformation of cylindrical samples of apple tissue during dehydration were modeled by coupled mass transfer and mechanics and validated by calibrated X-ray CT measurements. Uni-axial compression–relaxation tests were carried out to determine the nonlinear viscoelastic properties of apple tissue. The Mooney–Rivlin and Yeoh hyperelastic potentials with three parameters were effective to reproduce the nonlinear behavior during the loading region. Maxwell model was successful to quantify the viscoelastic behavior of the tissue during stress relaxation. The nonlinear models were superior to linear elastic and viscoelastic models to predict deformation and water loss. The sensitivity of different model parameters using the nonlinear viscoelastic model using Yeoh hyperelastic potentials was studied. The model predictions proved to be more sensitive to water transport parameters than to the mechanical parameters. The large effect of relative humidity and temperature on the deformation of apple tissue was confirmed by this study. The validated model can be employed to better understand postharvest storage and drying processes of apple fruit and thus improve product quality in the cold chain.

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Abbreviations

a w :

Water activity

\( \overline b \) :

Modified left Cauchy–Green tensor

C :

The right Cauchy–Green deformation tensor

\( \overline C \) :

Modified right Cauchy–Green deformation tensor

c p :

Heat capacity of air, J kg−1 °C−1

c ϕ :

Water capacity, kg kg−1 Pa−1

C 1, C 2, C 3 :

Coefficients of Mooney–Rivlin hyperelastic potential, Pa

C 10, C 01, C 11 :

Coefficients of Yeoh hyperelastic potential, Pa

d :

Diameter, m

D :

Stiffness matrix linear elastic model, Pa

D d :

Stiffness matrix in linear viscoelastic model, Pa

D air :

Diffusivity of water in air, m2 s−1

ė :

The strain rate, s−1

E :

Modulus of elasticity, Pa

E 1, E 2, E 3 :

The first, second, and third decay elastic moduli, Pa

E e :

The equilibrum modulus of elasticity, Pa

F :

Deformation gradient tensor

F e :

Elastic part of deformation tensor

F s :

Swelling/shrinkage part of deformation tensor

F T :

Transpose of deformation gradient tensor

G :

Shear modulus, Pa

h :

Mass transfer coefficient, m s−1

h m :

Mass transfer coefficient, kg Wm−2 Pa−1 s−1

h T :

Heat transfer coefficient, Wm−2 °C−1

I :

Second order identity tensor

I g :

Gray scale

\( {\overline I_1} \) :

The first modified strain invariant

\( {\overline I_2} \) :

The second modified strain invariant

J :

Jacobian of the deformation

J e :

Jacobian of the elastic deformation

K :

Hydraulic conductivity, kg s−1 m−1 Pa−1

K b :

Bulk modulus, Pa

Le :

Lewis number

n :

Distance from surface, m

Nu :

Nusselt number

R :

Universal gas constant, 8.314 kJ mol−1

Re :

Reynolds number

RH ∞ :

Relative humidity of the ambient air, %

Pr :

Prandtl number

q i :

The extension of the corresponding springs

Q a :

Second Piola–Kirchhoff stress, Pa

S :

Second Piola–Kirchhoff stress tensor, Pa

S qx :

Viscoelastic stress, Pa

S ISO :

Isochoric, Pa

\( {S^{\infty }}_{\text{ISO}} \) :

Equilibrium isochoric stress, Pa

\( {S^{\infty }}_{\text{Vol}} \) :

Equilibrium volumetric stress, Pa

t :

Time, s

T :

Temperature, K

U :

Deformation, m

V :

Volume, m3

V w :

Molar volume of water, 18 × 10−6 m3 mol−1

X :

Water content, kg kg−1

X ref :

Stress-free water content, kg kg−1

β :

Volumetric shrinkage coefficient

β a :

Nondimensional strain energy factor

\( {\Upsilon_{\alpha }} \) :

Dissipative potential for the viscoelastic contribution

Г a :

Viscoelastic isochoric response, Pa

ε :

Total strain

ε d :

Shrinkage strain

ε e :

Elastic strain

ε x , ε y , ε z :

Elastic strain in the principal directions

ε xy , ε yz , ε xz :

Shear strain

ε 0 :

Nominal initial strain

η :

Specific viscosity, Pa s

k :

Initial bulk modulus, Pa

λ 1, λ 2, λ 3 :

Stretch ratio in the three principal axes

λ s :

Stretch ratio

λ t :

Thermal conductivity, W m−1 °C−1

μ :

Dynamic viscosity of air, kg m−1 s−1

ρ :

Air density, kg m−3

ρ dm :

Dry mass density, kg m−3

ρ sat :

Saturated vapor density, kg m−3

\( \sum\limits_{{i = 1}}^n {{\Upsilon_{\alpha }}} \) :

The dissipative potential

σ :

The Cauchy stress, Pa

σ(t):

The relaxaton stress, Pa

τ a :

Relaxation time, s

ν :

Poisson’s ratio

ϕ :

Water potential, Pa

ϕ ∞ :

Ambient water potential, Pa

χ :

Push-forward operation

\( {\psi^{\infty }}_{\text{VOL}} \) :

Volumetric elastic response

\( {\psi^{\infty }}_{\text{ISO}} \) :

Isochoric elastic response

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Acknowledgements

The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement no. 245288. The authors would also like to acknowledge the Fund for Scientific Research—Flanders (grant no. FWO G.0603.08), the K.U. Leuven (project OT 08/023), and the EC (FP7-226783 project InsideFood) for financial support. Thijs Defraeye is a postdoctoral fellow of the Research Foundation—Flanders (FWO).

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  1. BIOSYST-MeBioS, Katholieke Universiteit Leuven, W. de Croylaan 42, 3001, Leuven, Belgium

    W. A. Aregawi, T. Defraeye, P. Verboven, E. Herremans & B. M. Nicolai

  2. Division of Structural Mechanics, Department of Civil Engineering, Katholieke Universiteit Leuven, Kasteelpark Arenberg 40, 3001, Leuven, Belgium

    G. De Roeck

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  1. W. A. Aregawi
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Correspondence to B. M. Nicolai.

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Aregawi, W.A., Defraeye, T., Verboven, P. et al. Modeling of Coupled Water Transport and Large Deformation During Dehydration of Apple Tissue. Food Bioprocess Technol 6, 1963–1978 (2013). https://doi.org/10.1007/s11947-012-0862-1

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