Numerical simulation of mushrooms during freezing using the FEM and an enthalpy: Kirchhoff formulation


The shelf life of mushrooms is very limited since they are susceptible to physical and microbial attack; therefore they are usually blanched and immediately frozen for commercial purposes. The aim of this work was to develop a numerical model using the finite element technique to predict freezing times of mushrooms considering the actual shape of the product. The original heat transfer equation was reformulated using a combined enthalpy-Kirchhoff formulation, therefore an own computational program using Matlab 6.5 (MathWorks, Natick, Massachusetts) was developed, considering the difficulties encountered when simulating this non-linear problem in commercial softwares. Digital images were used to generate the irregular contour and the domain discretization. The numerical predictions agreed with the experimental time–temperature curves during freezing of mushrooms (maximum absolute error <3.2°C) obtaining accurate results and minimum computer processing times. The codes were then applied to determine required processing times for different operating conditions (external fluid temperatures and surface heat transfer coefficients).

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CG :

Global capacitance matrix

Cp :

Specific heat J (kg °C)−1

Cp ap :

Apparent specific heat J (kg °C)−1

D :

Button (cap) diameter (m)

e :

Thickness (m)

E :

Kirchhoff function (W m−1)

FG :

Global force vector

h :

Surface heat transfer coefficient (W (m2 °C)−1)

H :

Volumetric enthalpy (J m−3)

k :

Thermal conductivity (W (m °C)−1)

L :

Length of mushrooms (m)

KG :

Global conductance matrix

MG :

Global convective matrix

N :

Vector containing the shape functions

N j :

Shape function j

t :

Time (s)

T :

Temperature, vector of nodal temperatures (°C)

T ext :

External fluid temperature (°C)

T f :

Initial freezing temperature (°C)

T ref :

Reference temperature (°C)

x :

Mass fraction

v :

Velocity (m s−1)

Δt :

Time increment (s)

δΩ :

Surface of the domain

ε :

Residual (Wm−3)

ρ :

Density (kg m−3)

Ω :






1, 2:





Border element










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The authors acknowledge the financial support provided by the Consejo Nacional de Investigaciones Científicas y Tecnológicas (CONICET), Agencia Nacional de Promoción Científica y Tecnológica (ANPCYT), and Universidad Nacional de La Plata, Argentina. We also acknowledge the support given by Dr. R. Mascheroni, Dr. A. Califano and Dr. N. Zaritzky.

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Correspondence to A. R. Lespinard.

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M. V. Santos and A. R. Lespinard contributed equally to this work.

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Santos, M.V., Lespinard, A.R. Numerical simulation of mushrooms during freezing using the FEM and an enthalpy: Kirchhoff formulation. Heat Mass Transfer 47, 1671 (2011).

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  • Simulation
  • Freezing
  • Heat transfer
  • Food processing