Pulsating flow in a channel filled with a porous medium under local thermal non-equilibrium condition: an exact solution

Abstract

The present work investigates analytically the problem of forced convection heat transfer of a pulsating flow, in a channel filled with a porous medium under local thermal non-equilibrium condition. Internal heat generation is considered in the porous medium, and the channel walls are subjected to constant heat flux boundary condition. Exact solutions are obtained for velocity, Nusselt number and temperature distributions of the fluid and solid phases in the porous medium. The influence of pertinent parameters, including Biot number, Darcy number, fluid-to-solid effective thermal conductivity ratio and Prandtl number are discussed. The applied pressure gradient is considered in a sinusoidal waveform. The effect of dimensionless frequency and coefficient of the pressure amplitude on the system's velocity and temperature fields are discussed. The general shape of the unsteady velocity for different times is found to be very similar to the steady data. Results show that the amplitudes of the unsteady temperatures for the fluid and solid phases decrease with the increase in Biot number or thermal conductivity ratio. For large Biot numbers, dimensionless temperatures of the solid and fluid phases are similar and are close to their steady counterparts. Results for the Nusselt number indicate that increasing Biot number or thermal conductivity ratio decreases the amplitude of Nusselt number. Increase in the internal heat generation in the solid phase does not have a significant influence on the ratio of amplitude-to-mean value of the Nusselt number, while internal heat generation in the fluid phase enhances this ratio. Keywords Porous media • Pulsating flow • Convective heat transfer • Local thermal non-equilibrium • Analytical solution List of symbols a n , a j Time-dependent coefficient a sf Interfacial area per unit volume of porous media (m −1) A A defined function of time A 1 ,A 2 Constant coefficients b m,n Time-dependent coefficient B A defined function of time B 1 ,B 2 Constant coefficients Bi Biot number, Bi = a sf h sf H 2 k s,eff c pf Fluid specific heat (J kg −1 K −1) c ps Solid specific heat (J kg −1 K −1) C Constant parameter, C =

Experimental and Pore-Scale Analysis of Flow and Thermal Fields in a Packed Bed Channel

“…The present experiment and that of Nie et al [27] both were conducted for steady heat transfer between spheres and passing air flow. Nie et al [27] also considered transient data, however, based on theoretical studies the average Nu is not a time dependent parameter [21].…”

“…𝑐 𝜇 = 0.09, 𝜎 𝐾 = 1, 𝜎 𝜖 = 1.3, 𝑐 𝜖1 = 1.44 and 𝑐 𝜖2 = 1.92 (21) Turbulence shear production is as [34] 𝑃 𝐾 = 𝜇 𝑡 . (∇𝑢 ⃗ + (∇𝑢 ⃗ ) T ) (22) Boundary conditions used to solve the governing equations are the same as the experiment, which are describes in the following.…”

“…In the volume-averaged method for packed bed energy storage systems, the local thermal non-equilibrium (LTNE) model is deployed to calculate temperatures of the fluid and particle phases in the packed bed by solving different energy equations for each phase in the bed [18][19][20][21]. However, for the LTNE model, the internal heat transfer coefficient between the spheres and fluid has to be known a priori.…”

Effects of Neumann and Robin boundaries on the thermal instability

“…The former case assures a rapid and higher amount of heat transferred between phases while the latter corresponds to a case in which the saturated fluid holds higher thermal conductance than the solid skeleton. A small H or γ can reverse the approach of the LTE towards a non-equilibrium one [10][11][12][13].…”

“…However, the feature of the LTE is not always true as we can encounter in some cases a disparity in the thermal conductivity or/and a weakness in the heat transfer coefficient, which can yield a relaxation of the local thermal equilibrium regime. From a practical standpoint, the local thermal non-equilibrium (LTNE) model has a vital effect on the cooling process of countless engineering devices [10,17]. On the other hand, the approach of the LTNE can highlight difficulties in the modelling of thermal boundaries, especially in the case of isoflux [18][19][20].…”