## THERMAL SCIENCE

International Scientific Journal

### A NEW HYBRID ALGORITHM FOR SOLVING TRANSIENT COMBINED CONDUCTION RADIATION HEAT TRANSFER PROBLEMS

**ABSTRACT**

A new algorithm based on the lattice Boltzmann method (LBM) and the Control Volume Finite Element Method (CVFEM) is proposed as an hybrid solver for two dimensional transient conduction and radiation heat transfer problems in an optically emitting, absorbing and scattering medium. The LBM was used to solve the energy equation and the CVFEM was used to compute the radiative information. The advantages of the proposed methodology is to avoid problems that confronted when previous techniques are used to predict radiative heat transfer, essentially, in complex geometries and when there is scattering and/or non-black boundaries surfaces. This method combination, which is applied for the first time to solve this unsteady combined mode of heat transfer, has been found to accurately predict the effects of various thermo-physical parameters such as the scattering albedo, the conduction-radiation parameter and the extinction coefficient on temperature distribution. The results of the LBM-CVFEM combination were found to be in excellent agreement with the LBM-CDM (Collapsed Dimension Method)this proposed numerical approach include, among others, simple implementation on a computer, accurate CPU time, and capability of stable simulation.

**KEYWORDS**

PAPER SUBMITTED: 2010-07-22

PAPER REVISED: 2010-12-30

PAPER ACCEPTED: 2011-02-24

**THERMAL SCIENCE** YEAR

**2011**, VOLUME

**15**, ISSUE

**3**, PAGES [649 - 662]

- S. Succi, The Lattice Boltzmann Method for Fluid Dynamics and Beyond, Oxford University Press, (2001)
- Higuera, F.J., et al., Lattice gas dynamics with enhanced collisions, Europhys. Lett, 9 (1989), pp. 345-349
- Mohamed, A. A., et al, Lattice Boltzmann Simulation of Natural Convection in an Open ended Cavity, International Journal of Thermal Sciences, 48, (2009), 10, pp. 1870-1875
- R. Benzi, R., et al, The lattice Boltzmann equation: theory and applications, Phys. Rep, 222 (1992), pp. 145-197
- Semma, E., et al., Investigation of flows in solidification by using the lattice Boltzmann method. International Journal of Thermal Science, 47, (2008), 3, pp. 201-208.
- S. Chen, G.D. Doolen, Lattice Boltzamann method for fluid flows, Ann. Rev. Fluid Mech, 30 (1998), pp. 329-364
- He, X., et al, A novel thermal model for the lattice Boltzmann method in incompressible limit, J. Comput. Phys, 146 (1998), pp. 282-300.
- El Ganaoui, M., Djebali, R., Aptitude of a lattice Boltzmann method for evaluating transitional thresholds for low Prandtl number flows in enclosures, Comptes Rendus Mécanique, 338, (2010),2, pp. 85-96
- Wolf-Gladrow, D.A., Lattice Gas Cellular Automata and Lattice Boltzmann Models: An Introduction, Springer-Verlag, Berlin-Heidelberg, (2000).
- Shan, X, Simulation of Rayleigh-Benard convection using a lattice Boltzmann method, Phys. Rev. E, 55 (1977), pp. 2780-2788.
- Mezrhab, A. et al, Hybrid lattice-Boltzmann finite-difference simulation of convective flows, Comput. Fluids, 33 (4) (2005), pp. 623-641.
- El Ganaoui, M, Semma, E., A lattice Boltzmann coupled to finite volumes method for solving phase change problems, Thermal Science, 13, (2009), pp.205-216.
- Ho, J.R. et al, Lattice Boltzmann scheme for hyperbolic heat conduction equation, Numer. Heat Transfer B, 41 (2002), pp. 591-607.
- Mishra, S.C., et al, Application of the lattice Boltzmann method for solving the energy equation of a 2-D transient conduction-radiation problem, International Journal of Heat and Mass Transfer 48 (2005), pp. 3648-3659.
- Meng, FL., et al, Radiative heat transfer by the discrete transfer method using an unstructured mesh. In: HTD-Vol.244, Radiative heat transfer: theory and applications. New York: ASME (1993), pp. 55-66.
- Chai, J.C., et al, Evaluation of spatial differencing practices for the discrete-ordinates method. J Thermophys Heat Transfer, 8 (1994),1 , pp. 140-144.
- Baek, S.W., Kim, M. Y., Analysis of radiative heating of a rocket plume base with the finite-volume method, Int J Heat Mass Transfer 40 (1997), 7, pp. 1501-1508
- Rousse, D.R., Numerical prediction of two-dimensional conduction, convection, and radiation heat transfer. I- formulation, Int J Therm Sci, 39 (2000), pp. 315-331
- Rousse, D.R. et al, Une fonction d'interpolation produisant des coefficients positifs pour le rayonnement, Cong Français Thermique SFT, 15-17 mai (2000)
- Rousse, D.R. et al, Numerical prediction of two-dimensional conduction, convection, and radiation heat transfer, II- validation, Int J Therm Sci, 39 (2000), pp. 332-353