Abstract
In order to avoid insufficient sterilization caused by uneven heating and nutrition loss generated by excessive heating, researchers and scientists have focused on the topics of controllable heat transfer and predictable temperature fields in food processing. In this paper, the temperature distribution of power-law liquid food under the combination impacts of the thermal radiation and natural convection is studied by the numerical technique. Three kinds of open symmetrical heating containers commonly used are paid attention to, and the height and cross-sectional area of containers are set to equal for the convenience of comparing the effect of the container geometry on the temperature fields. The governing equations are treated in a dimensionless way as well as the boundary conditions. The finite element method is used to obtain the solutions, which has been proved robust and accurate. It has been displayed that the temperatures in the cylindrical container (i.e., the noodle bowl) are the highest, which proves its best heating effect. The liquids of egg yolk and egg white are adopted as special cases of non-Newtonian power-law fluids. The temperature fields of egg yolk and egg white are less affected by the generalized Grashof number than the thermal radiation number NR. The technique of adjusting the generalized Grashof number and the thermal radiation to control the heat transfer behavior applied in the present research could be extended to related food engineering applications.
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Abbreviations
- c p :
-
Fluid specific heat (J kg−1 K−1)
- g :
-
Gravity (m s−2)
- Gr:
-
Generalized Grashof number (\({\text{Gr}} = {{T_{0} L^{n + 1} g\rho \beta } \mathord{\left/ {\vphantom {{T_{0} L^{n + 1} g\rho \beta } {(\mu U_{0}^{n} }}} \right. \kern-0pt} {(\mu U_{0}^{n} }})\))
- H :
-
Height of the container (m)
- k :
-
Thermal conductivity (W m−1 K−1)
- k*:
-
The mean absorption coefficient
- n :
-
Power-law index
- N R :
-
Thermal radiation number (\(N_{\text{R}} = 3kk^{*} /(16\sigma^{*} T_{1}^{3} )\))
- p :
-
Pressure (Pa)
- Pr:
-
Generalized Prandtl number (\(\Pr = {{c_{\text{p}} \mu L^{1 - n} } \mathord{\left/ {\vphantom {{c_{\text{p}} \mu L^{1 - n} } {(k}}} \right. \kern-0pt} {(k}}U_{0}^{1 - n} )\))
- q r :
-
Heat flux (W m−2)
- Re:
-
Reynolds number (\(\text{Re} = \rho H^{n} /(\mu U_{0}^{n - 2} )\))
- T :
-
Temperature (K)
- u, v:
-
Velocities along x and y, respectively (m s−1)
- x, y :
-
Cartesian coordinates along the bottom and normal to it, respectively (m)
- \(\varOmega\) :
-
Calculation domain
- \(\varGamma\) :
-
Boundary of \(\varOmega\)
- \(\beta\) :
-
Thermal expansion coefficient (K−1)
- \(\rho\) :
-
Fluid density (kg m−3)
- \(\tau\) :
-
Shear stress (N)
- \(\mu\) :
-
Consistency index (Pa sn)
- \(\sigma^{*}\) :
-
The Stefan–Boltzmann constant
- 0:
-
Initial state of the container
- 1:
-
Initial state of the fluid
References
Tattiyakul J, Rao MA, Datta AK. Heat transfer to three canned fluids of different thermo-rheological behaviour under intermittent agitation. Food Bioprod Process. 2002;80(1):20–7.
Kelder JDH, Ptasinski KJ, Kerkhof PJAM. Power-law foods in continuous coiled sterilisers. Chem Eng Sci. 2002;57(21):4605–15.
Zhang BB, Yue MA, Wang D, et al. Effect of ultra high pressure (UHP) and ultra high temperature(UHT) sterilization treatments on the quality of watermelon beverage. Food Sci. 2014;35(17):72–6.
Hakeem AKA, Saranya S, Ganga B. Comparative study on Newtonian/non-Newtonian base fluids with magnetic/non-magnetic nanoparticles over a flat plate with uniform heat flux. J Mol Liq. 2017;230:445–52.
Estellé P, Mahian O, Maré T, et al. Natural convection of CNT water-based nanofluids in a differentially heated square cavity. J Therm Anal Calorim. 2017;128(3):1765–70.
Hinojosa JF, Cabanillas RE, Alvarez G, et al. Nusselt number for the natural convection and surface thermal radiation in a square tilted open cavity. Int Commun Heat Mass Transf. 2005;32(9):1184–92.
Hinojosa JF, Buentello D, Xaman J, et al. The effect of surface thermal radiation on entropy generation in an open cavity with natural convection. Int Commun Heat Mass Transf. 2017;81:164–74.
Gonzalez MM, Jesus HP, Estrada CA. Numerical study of heat transfer by natural convection and surface thermal radiation in an open cavity receiver. Sol Energy. 2012;86(4):1118–28.
Miroshnichenko IV, Sheremet MA. Radiation effect on conjugate turbulent natural convection in a cavity with a discrete heater. Appl Math Comput. 2018;321:358–71.
Martyushev SG, Sheremet MA. Conjugate natural convection combined with surface thermal radiation in an air filled cavity with internal heat source. Int J Therm Sci. 2014;76:51–67.
Bahiraei M, Hosseinalipour SM. Thermal dispersion model compared with Euler–Lagrange approach in simulation of convective heat transfer for nanoparticle suspensions. J Dispers Sci Technol. 2013;34(12):1778–89.
Sheikholeslamia M, Roknib HB. Numerical simulation for impact of Coulomb force on nanofluid heat transfer in a porous enclosure in presence of thermal radiation. Int J Heat Mass Transf. 2018;118:823–31.
Sheikholeslami M. Numerical approach for MHD Fe3O4–ethylene nanofluid transportation inside a permeable medium using innovative computer method. Comput Methods Appl Mech Eng. 2019;344:306–18.
Sheikholeslami M. New computational approach for exergy and entropy analysis of nanofluid under the impact of Lorentz force through a porous media. Comput Methods Appl Mech Eng. 2019;344:319–33.
Belabid J, Belhouideg S, Allali K, et al. Numerical simulation for impact of copper/water nanofluid on thermo-convective instabilities in a horizontal porous annulus. J Therm Anal Calorim. 2019;138:1515–25.
Sheremet MA, Pop I. Marangoni natural convection in a cubical cavity filled with a nanofluid. J Therm Anal Calorim. 2019;135:357–69.
Memon RA, Solangi MA, Baloch A. Analysis of stresses and power consumption of mixing flow in cylindrical container. Punjab Univ J Math. 2011;43:47–67.
Varma MN, Kannan A. CFD studies on natural convective heating of canned food in conical and cylindrical containers. J Food Eng. 2006;77(4):1024–36.
Anese M, De Bonis MV, Mirolo G, et al. Effect of low frequency, high power pool ultrasonics on viscosity of fluid food: modeling and experimental validation. J Food Eng. 2013;119(3):627–32.
Delouei AA, Nazari M, Kayhani MH, et al. Immersed boundaryl-thermal lattice boltzmann methods for non-Newtonian flows over a heated cylinder: a comparative study. Commun Comput Phys. 2015;18(02):489–515.
Jahanbakhshi A, Nadooshan AA, Bayareh M. Magnetic field effects on natural convection flow of a non-Newtonian fluid in an L-shaped enclosure. J Therm Anal Calorim. 2018;133:1407–16.
Ishak A, Nazar R, Pop I. Boundary layer flow and heat transfer over an unsteady stretching vertical surface. Meccanica. 2009;44(4):369–75.
Sheikholeslami M, Domiri Ganji D, Younus Javed M, et al. Effect of thermal radiation on magnetohydrodynamics nanofluid flow and heat transfer by means of two phase model. J Magn Magn Mater. 2015;374:36–43.
Eid MR, Alsaedi A, Muhammad T, et al. Comprehensive analysis of heat transfer of gold-blood nanofluid (Sisko-model) with thermal radiation. Results Phys. 2017;7:4388–93.
Ziabakhsh Z, Domairry G. Analytic solution of natural convection flow of a non-Newtonian fluid between two vertical flat plates using homotopy analysis method. Commun Nonlinear Sci Numer Simul. 2009;14(5):1868–80.
Khodabandeh E, Bahiraei M, Mashayekhi R, et al. Thermal performance of Ag-water nanofluid in tube equipped with novel conical strip inserts using two-phase method: geometry effects and particle migration considerations. Powder Technol. 2018;338:87–100.
Dogonchi AS, Chamkha AJ, Ganji DD. A numerical investigation of magneto-hydrodynamic natural convection of Cu–water nanofluid in a wavy cavity using CVFEM. J Therm Anal Calorim. 2019;135(4):2599–611.
Ko S, Suli E. Finite element approximation of steady flows of generalized Newtonian fluids with concentration-dependent power-law index. Math Comput. 2018;88(317):1061–90.
Prieto JL. Stochastic particle level set simulations of buoyancy-driven droplets in non-Newtonian fluids. J Nonnewton Fluid Mech. 2015;226:16–31.
Madhu M, Kishan N. Finite element analysis of heat and mass transfer by MHD mixed convection stagnation-point flow of a non-Newtonian power-law nanofluid towards a stretching surface with radiation. J Egypt Math Soc. 2016;24(3):458–70.
Li B, Zheng L, Lin P, et al. A mixed analytical/numerical method for velocity and heat transfer of laminar power-law fluids. Numer Math Theory Methods Appl. 2016;9(03):315–36.
Abel MS, Mahesha N. Heat transfer in MHD viscoelastic fluid flow over a stretching sheet with variable thermal conductivity, non-uniform heat source and radiation. Appl Math Model. 2008;32(10):1965–83.
Lin P, Liu C. Simulations of singularity dynamics in liquid crystal flows: a C0 finite element approach. J Comput Phys. 2006;215(1):348–62.
Turan O, Sachdeva A, Poole RJ, et al. Laminar natural convection of power-law fluids in a square enclosure with differentially heated sidewalls subjected to constant wall heat flux. J Nonnewton Fluid Mech. 2012;166(17–18):1049–63.
Tatsuo N, Toru T, Mitsuhiro S, et al. Numerical analysis of natural convection in a rectangular enclosure horizontally divided into fluid and porous regions. Int J Heat Mass Transf. 1986;29(6):889–98.
Acknowledgements
The work is supported by the Fundamental Research Funds for the Central Universities and the National Natural Science Foundation of China (No. 11402188).
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Huang, Z., Li, B., Si, X. et al. Natural convection and radiation heat transfer of power-law fluid food in symmetrical open containers. J Therm Anal Calorim 144, 1287–1298 (2021). https://doi.org/10.1007/s10973-020-09616-9
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DOI: https://doi.org/10.1007/s10973-020-09616-9