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

Abstract

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).

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Abbreviations

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 :

Domain

∇:

Gradient

0:

Initial

1, 2:

Domain

e:

Element

e1:

Border element

exp:

Experimental

sim:

Simulated

w:

Water

t:

Transpose

References

  1. 1.

    Kotwaliwale N, Bakane P, Verma A (2007) Changes in textural and optical properties of oyster mushroom during hot air drying. J Food Eng 78(4):1207–1211

    Article  Google Scholar 

  2. 2.

    Aguirre L, Frias JM, Barry-Ryan C, Grogan H (2009) Modelling browning and brown spotting of mushrooms (Agaricus bisporus) stored in controlled environmental conditions using image analysis. J Food Eng 91:280–286

    Article  Google Scholar 

  3. 3.

    Devece C, Rodríguez-López JN, Fenoll JT, Catalá JM, De los Reyes E, García-Cánovas F (1999) Enzyme inactivation analysis for industrial blanching applications: comparison of microwave, conventional, and combination heat treatments on mushroom polyphenoloxidase activity. J. Agric. Food Chem 47(11):4506–4511

    Article  Google Scholar 

  4. 4.

    Biekman ESA, Kroese-Hoedeman HI, Schijvens EPHM (1996) Loss of solutes during blanching of mushrooms (Agaricus bisporus) as a result of shrinkage and extraction. J. Food Eng 28(2):139–152

    Article  Google Scholar 

  5. 5.

    Lespinard AR, Goñi SM, Salgado PR, Mascheroni RH (2009) Experimental determination and modeling of size variation, heat transfer and quality indexes during mushroom blanching. J Food Eng 92:8–17

    Article  Google Scholar 

  6. 6.

    Fennema OR (1977) Loss of vitamins in fresh and frozen foods. Food Technology 12:32–38

    Google Scholar 

  7. 7.

    Delgado AE, Sun D-W (2001) Heat and mass transfer models for predicting freezing processes–a review. J Food Eng 47:157–174

    Article  Google Scholar 

  8. 8.

    Cleland DJ, Cleland AC, Earle RL, Byrne SJ (1984) Prediction of rates of freezing, thawing or cooling in solids of arbitrary shape using the finite element method. Int J Refrigeration 7(1):6–13

    Article  Google Scholar 

  9. 9.

    Arce JA, Potluri PL, Schneider KC, Sweat VE, Dutson TR (1983) Modeling Beef carcasss cooling using finite element technique. T ASAE 26:950–954, 960

  10. 10.

    Pham QT (2008) Modelling of freezing processes. In: Evans J (ed) Frozen Food Science and Technology. Blackwell Publishing, Oxford, pp 51–80

    Google Scholar 

  11. 11.

    Santos CA, Carciofi BAM, Dannenhauer CE, Hense H, Laurindo JB (2007) Determination of heat transfer coefficient in cooling-freezing tunnels using experimental time-temperature data. J. Food Process Eng. 30:717–728

    Article  Google Scholar 

  12. 12.

    Sheen S, Hayakawa K (1991) Finite difference simulation for heat conduction with phase change in an irregular food domain with volumetric change. Int J Heat Mass Transfer 34(6):1337–1346

    Article  Google Scholar 

  13. 13.

    Comini G, Del Giudice S, Lewis RW, Zienkiewicz OC (1974) Finite element solution of non-linear heat conduction problems with special reference to phase change. Int J Numer Methods Eng 8:613–624

    MATH  Article  Google Scholar 

  14. 14.

    Mannapperuma JD, Singh RP (1988) Prediction o freezing and thawing times of foods using a numerical method based on enthalpy formulation. J Food Sci 53:626–630

    Article  Google Scholar 

  15. 15.

    Fikiin KA (1996) Generalized numerical modeling of unsteady heat transfer during cooling and freezing using an improved enthalpy method and quasi–one–dimensional formulation. Int J Refrigeration 19(2):132–140

    Article  Google Scholar 

  16. 16.

    Fikiin KA (1998) Some general principles in modeling of unsteady heat transfer in two–phase multi–component aqueous food systems for product quality improvement. In: Nicolaï BM, De Baerdemaeker J (eds) Food quality modeling. Office for Official Publications of the European Communities, Luxembourg, pp 179–186

    Google Scholar 

  17. 17.

    Scheerlinck N, Fikiin KA, Verboven P, De Baerdemaeker J, Nicolaï, BM (1997) Numerical solution of phase change heat transfer problems with moving boundaries using an improved finite element enthalpy method. In: Van Keer R and Brebbia CA (eds) Moving boundaries IV: computational modelling of free and moving boundary problems, pp 75–85

  18. 18.

    Scheerlinck N, Verboven P, Fikiin KA, De Baerdemaeker J, Nicolaï BM (2001) Finite element computation of unsteady phase change heat transfer during freezing or thawing of food using a combined enthalpy and Kirchhoff transform method. T ASAE 44(2):429–438

    Google Scholar 

  19. 19.

    Carslaw HS, Jaeger JC (1959) Conduction of Heat in Solids. University Press, Oxford

    Google Scholar 

  20. 20.

    Comini G, Nonino C, Saro O (1990) Performance of enthalpy–based algorithms for isothermal phase change. Adv Comput Method Heat Trans Phase Change Combust Simulat 3:3–13

    Google Scholar 

  21. 21.

    Goñi SM, Purlis E, Salvadori VO (2007) Three-dimensional reconstruction of irregular foodstuffs. J. Food Eng 82(4):536–547

    Article  Google Scholar 

  22. 22.

    McArdle FJ, Curwen D (1962) Some factors influencing shrinkage of canned mushrooms. Mushroom Sci 5:547

    Google Scholar 

  23. 23.

    Bernas E, Jaworska G, Lisiewska Z (2006) Edible Mushrooms as a source of valuable nutritive constituents. Technol Aliment 5(1):5–20

    Google Scholar 

  24. 24.

    Choi Y, Okos MR (1986) Effects of temperature and composition on the thermal properties of foods. In: Le Maguer M, Jelen P (eds) Food engineering and process applications. Elsevier Applied Science, New York, pp 93–103

    Google Scholar 

  25. 25.

    Mellor JD (1978) Thermophysical properties of foodstuffs. 2. Theoretical aspect. Bull IIR 58:59

    Google Scholar 

  26. 26.

    Miles CA, van Beek G, Veerkamp CH (1983) Calculation of the thermophysical properties of foods. In: Jowitt R et al (eds) Physical properties of foods. Applied Science Publishers, London, pp 269–312

    Google Scholar 

  27. 27.

    Fennema OR, Powrie WD, Marth EH (1973) Low temperatures preservation of foods and living matter. Marcel Dekker Inc., New York

    Google Scholar 

  28. 28.

    Wang H, Zhang S, Chen G (2007) Experimental study on the freezing characteristics of four kinds of vegetables. Food Sci Technol—LEB 40:1112–1116

    Google Scholar 

  29. 29.

    Rahman MS, Sablani SS, Al-Habsi N, Al-Belushi R (2005) State Diagram of Freeze-dried Garlic Powder by Differential Scanning Calorimetry and Cooling Curve Methods. J. Food Sci 70(2):135–141

    Article  Google Scholar 

  30. 30.

    Earle RL (1988) Ingeniería de los Alimentos. (2da Ed.) Ed. Acribia, Zaragoza, España

    Google Scholar 

  31. 31.

    Nesvadba P (2008) Thermal properties and ice crystal development in frozen foods. In: Evans J (ed) Frozen Food Science and Technology. Blackwell Publishing, Oxford, pp 1–25

    Google Scholar 

  32. 32.

    Cleland DJ, Cleland AC, Earle RL, Byrne SJ (1986) Prediction of thawing times for foods of simple shape. Int J Refrigeration 9:220–228

    Article  Google Scholar 

  33. 33.

    Cleland DJ, Cleland AC, Earle RL, Byrne SJ (1987) Prediction of freezing and thawing times for multi-dimensional shapes by numerical methods. Int J Refrigeration 10:32–39

    Article  Google Scholar 

  34. 34.

    Cleland DJ, Cleland AC, Earle RL, Byrne SJ (1987) Experimental data for freezing and thawing of multi-dimensional objects. Int J Refrigeration 10:22–31

    Article  Google Scholar 

  35. 35.

    Chavarria VM, Heldman DR (1984) Measurement of convective heat transfer coefficients during food freezing processes. J. Food Sci 49(3):810–814

    Article  Google Scholar 

  36. 36.

    Califano A, Zaritzky N (1997) Simulation of freezing or thawing heat conduction in irregular two dimensional domains by a boundary fitted grid method. Lebensmittel-Wissenschaft und-Technologie 30:70–76

    Article  Google Scholar 

  37. 37.

    Pan JC, Bhowmik SR (1991) The finite element analysis of transient heat transfer in fresh tomatoes during cooling. Transactions of the ASAE 34:972–976

    Google Scholar 

  38. 38.

    Abdalla H, Singh RP (1985) Simulation of thawing foods using finite element method. J. Food. Process Eng 7(4):273–286

    Google Scholar 

  39. 39.

    Becker BR, Fricke BA (2004) Heat transfer coefficients for forced-air cooling and freezing of selected foods. Internat. J. Refrig. 27:540–551

    Article  Google Scholar 

  40. 40.

    Campañone LA, Salvadori VO, Mascheroni RH (2005) Food freezing with simultaneous surface dehydration: approximate prediction of freezing time. Int. J. Heat Mass Transfer 48(6):1205–1213

    MATH  Article  Google Scholar 

  41. 41.

    Trujillo FJ (2004) A computational fluid dynamic model of heat and moisture transfer during beef chilling, In: School of chemical engineering and industrial chemistry, PhD thesis, University of New South Wales, Sydney, Australia

  42. 42.

    Trujillo FJ, Pham QT (2003) CFD modeling of heat and moisture transfer on a two-dimensional model of a beef leg, In: Twenty first International Congress of Refrigeration, Washington

  43. 43.

    Trujillo FJ, Pham QT (2006) A computational fluid dynamic model of heat and moisture transfer during beef chilling. Int J Refrigeration 29(6):998–1009

    Article  Google Scholar 

  44. 44.

    Harris MB, Carson JK, Willix J, Lovatt SJ (2004) Local surface heat transfer coefficients on a model lamb carcass. J. Food Eng 61(3):421–429

    Article  Google Scholar 

  45. 45.

    Welty JR (1974) Engineering Heat Transfer. John Wiley and Sons, New York

    Google Scholar 

  46. 46.

    Santos MV, Zaritzky N, Califano A, Vampa V (2008) Numerical simulation of the heat transfer in three dimensional geometries, Mecánica Computacional. Vol XXVII No 21 Heat transfer (C) edited by Asociación Argentina de Mecánica Computacional (AMCA), pp. 1705–1718

  47. 47.

    Santos MV, Zaritzky N, Califano A (2010) A control strategy to assure safety conditions in the thermal treatment of meat products using a numerical algorithms. Food Control 21(2):191–197

    Article  Google Scholar 

  48. 48.

    Hossain MdM, Cleland DJ, Cleland AC (1992) Prediction of freezing and thawing times for foods of three-dimensional irregular shape by using a semianalytical geometric factor. Int J Refrigeration 15(4):241–246

    Article  Google Scholar 

Download references

Acknowledgments

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to A. R. Lespinard.

Additional information

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

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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). https://doi.org/10.1007/s00231-011-0831-7

Download citation

Keywords

  • Simulation
  • Freezing
  • Heat transfer
  • Food processing