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Numerical Simulation of the Two-Dimensional Heat Diffusion in the Cold Substrate and Performance Analysis of a Thermoelectric Air Cooler Using The Lattice Boltzmann Method

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Abstract

This article presents numerical simulations of the two-dimensional temperature distribution in the cold substrate and a performance analysis of a thermoelectric cooler with the Lattice Boltzmann Method. A detailed and concise procedure for the derivation of the source term of Lattice Boltzmann method for a thermal diffusion problem is presented. Numerical simulations are performed using the Bhatnagar-Gross-Krook collision operator with two velocity schemes, namely D2Q4 and D2Q9. The numerical validation is performed by comparisons with an approximate analytical solution and a finite difference method solution. Later, performance parameters of the thermoelectric cooler, based on thermal resistances, are computed from the obtained temperature distribution. The results show that the Lattice Boltzmann Method is capable of simulating the addressed thermal diffusion problem, with very small relative errors (maximum errors of 0.09%) for the temperature distribution. Excellent agreement is observed for the performance parameters, ensuring the robustness of the method. Furthermore, the procedure for the solution of the differential equation can be easily applied to solve other problems.

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Abbreviations

A:

Area, m2

ci :

Discrete velocities in the LBM formulation, lattice length (lattice time)−1

cs :

Speed of sound in the LBM formulation, lattice length (lattice time)−1

d:

Thickness of the cold substrate, m

Di :

Differential operator

fi :

Particle distribution function in the LBM formulation

k:

Thermal conductivity of the cold substrate, Wm−1 K−1

kt :

Thermal conductance of the pellet, W/K

I:

Electrical current, A

Np :

Number of pellets

Q:

Heat transfer rate, W

Q0 :

Heat source rate input, W

R:

Electrical resistance of the pellet, Ohm

Rb :

Heat sink base resistance, KW−1

Rb,sp :

Spreading resistance, KW−1

Rc,sp :

Resistance between the heat source and the cold substrate, KW−1

Rcons :

Constriction resistance, KW−1

Rconv :

Convection resistance, KW−1

t:

Time, s

T:

Temperature, K

wi :

Weight coefficients in the LBM formulation

x:

Physical position, m or position in LBM formulation, Lattice length

α:

Thermal diffusivity, m2s−1 or artificial diffusion coefficient in LBM formulation (in Lattice units)

αt :

Seebeck coefficient, VK−1

δ:

Dirac delta function

Δ:

Difference operator

ω:

Relaxation frequency

∇:

Gradient operator

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Acknowledgements

This work was supported by CNPq and FAPESP.

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Authors and Affiliations

  1. Institute of Science and Technology, Federal University of the Jequitinhonha and Mucurí Valleys, Diamantina, Minas Gerais, Brazil

    Matheus dos Santos Guzella

  2. Heat Transfer Research Group, Department of Mechanical Engineering, Engineering School of São Carlos, University of São Paulo, EESC-USP, São Carlos, São Paulo, Brazil

    Matheus dos Santos Guzella, Gustavo Ribeiro dos Santos & Luben Cabezas-Gómez

  3. Department of Mechanical Engineering, College of Engineering, The University of Texas At San Antonio, San Antonio, TX, 78249, USA

    Antonio Campo

  4. Department of Thermal and Fluid Sciences, Federal University of São João del Rei, UFSJ, São João del-Rei, Minas Gerais, Brazil

    Luiz Gustavo Monteiro Guimarães

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  1. Matheus dos Santos Guzella
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Correspondence to Matheus dos Santos Guzella.

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dos Santos Guzella, M., dos Santos, G.R., Cabezas-Gómez, L. et al. Numerical Simulation of the Two-Dimensional Heat Diffusion in the Cold Substrate and Performance Analysis of a Thermoelectric Air Cooler Using The Lattice Boltzmann Method. Int. J. Appl. Comput. Math 7, 130 (2021). https://doi.org/10.1007/s40819-021-01073-8

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