Analytical Study of Heat Flux Splitting in Micro-channels Filled with Porous Media

Abstract

The conditions of occurring heat flux splitting (bifurcation) phenomenon in a micro-channel filled with a porous medium including internal heat generations within both the solid and fluid phases under the local thermal non-equilibrium (LTNE) condition is analytically studied in the slip regime. The channel walls are subjected to a constant heat flux. Exact solutions for both the dimensionless temperatures of the two phases and the Nusselt number are obtained. Effects of the pertinent parameters such as heat generation parameter (\(\omega \)), the interphase heat transfer parameter (Vadasz in J Porous Media 15:249–258, 2012) or Biot number (\(\textit{Bi}\)), the fluid-to-solid effective conductivity ratio (k), and the temperature jump coefficient (\(\beta \)) on the dimensionless temperature profile (\(\theta \)) of the two phases as well as the Nusselt number are investigated. Moreover, the validity of one-equation model (the local thermal equilibrium assumption) is analyzed by comparing the Nusselt number obtained by one-equation model (LTE) with that obtained by the two-equation model (LTNE). Results reveal that the conditions at which the heat flux bifurcation (splitting) occurs in the slip regime is the same as those of the no-slip regime. In addition, a kind of heat flux bifurcation in which the solid and fluid phases have the same dimensionless temperature sign is observed in the slip regime, while it was not previously observe in the no-slip regime. It is discussed that the Nusselt number can increase or decrease with respect to \(\omega \) and may have either positive or negative values in both the no-slip and slip regimes. The presence of internal heat generation intensifies the role of \(\beta \) in the Nusselt number reduction. In addition, the accuracy of LTE model increases with increased \(\textit{Bi}\) and with decreased \(\beta \), while it is not a monotonic function of k in the presence of internal heat generation.