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

Arman Fathi-Kelestani,Meysam Nazari,Yasser Mahmoudi

doi:10.1007/s10973-020-09843-0

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.

Highlights

Convective heat transfer in porous media has been a subject of intense studies due to its wide range of application in the industries such as oil recovery, geothermal engineering, thermal insulation, carbon storage, heat transfer augmentation, solid matrix or micro-porous heat exchangers and porous radiant burners [1, 2]
The current work presents an analytical solution to investigate the effects of pulsating flow on the velocity, temperature distributions and Nusselt number in a channel filled with a porous medium under local thermal non-equilibrium (LTNE) condition and considering internal heat generation in the fluid and solid phases
We present the validation of the unsteady velocity field in comparison with the analytical solutions of Siegel and Perlmutter [23] presented for pulsating flow in a channel without porous medium

Summary

A A defined function of time

A1,A2 Constant coefficients bm,n Time-dependent coefficient B A defined function of time. M Viscosity ratio, eff∕ m Counter Nu Nusselt number n Counter p Pressure ( Pa) Pr Prandtl number Ps A defined dimensionless parameter, scps ∕ks,eff qw Heat flux (W m−2) r1 , r2 Roots of a characteristics equation obtainied from Eq (88) Re Reynolds number, umDH∕ Rm,n Time-dependent coefficient Sf Internal heat generation within fluid phase,. ( ks,eff T ∕qE H ) u Longitudinal velocity (m s−2) um A characteristic velocity (m s−1) U Dimensionless velocity V1 , V2 Time-dependent coefficients w∗ Womersly number x Axial coordinate (m) X Dimensionless x coordinate y Vertical coordinate (m) Y Dimensionless y coordi√nate Z Constant parameter, 1∕ MDa. Dimensionless frequency Coefficient of pressure amplitude n , m Eigenvalues of the unsteady velocity and temperature equations Dimensionless temperature defined as:. F Density of the fluid (kg m−3) s Density of the solid (kg m−3) ∗ Dimensionless period of oscillation ∅n Eigenfunctions of the unsteady velocity equation Γm Eigenfunctions of the unsteady energy equation

Introduction

Results and discussion

Conclusions