Assessment of the local thermal non-equilibrium condition in developing forced convection flows through fluid-saturated porous tubes

The present investigation is concerned with the numerical simulation of forced convection heat transfer flows in a tube filled with a fluid-saturated porous medium. Steady state incompressible flows with isothermal tube walls are considered along with a uniform inlet approach velocity and temperature conditions. In addition, the generalized form of the momentum equation is considered by accounting for the solid boundary and the Forchheimer quadratic inertial effects without invoking the boundary layer approximations. Moreover, the energy transport is simulated using the two-equation model, which accounts separately for the local fluid and solid temperatures. The numerical solution is obtained through the application of the finite volume method. The validity of the local thermal equilibrium (LTE) was tested over a wide domain of the employed dimensionless parameters, namely; the Peclet number, Biot-like number, effective fluid-to-solid thermal conductivity ratio, Reynolds number, Forchheimer dimensionless coefficient and Darcy number. The validity of the LTE condition was examined for the full tube length and upon excluding the first tube diameter length.     

The effect of local thermal non-equilibrium on forced convection boundary layer flow from a heated surface in porous media

A steady two-dimensional forced convective thermal boundary layer flow in a porous medium is studied. It is assumed that the solid matrix and fluid phase which comprise the porous medium are subject to local thermal non-equilibrium conditions, and therefore two heat transport equations are adopted, one for each phase. When the basic flow velocity is sufficiently high, the thermal fields may be described accurately using the boundary layer approximation, and the resulting parabolic system is analysed both analytically and numerically. Local thermal non-equilibrium effects are found to be at their strongest near the leading edge, but these decrease with distance from the leading edge and local thermal equilibrium is attained at large distances.     

The linear stability of a developing thermal front in a porous medium: The effect of local thermal non-equilibrium

In this paper we analyze the stability of the developing thermal boundary layer which is induced by a step-change in the temperature of the lower horizontal boundary of a uniformly cold semi-infinite porous medium. Particular attention is paid to the influence of local thermal non-equilibrium between the fluid and solid phases and how this alters the stability criterion compared with corresponding criterion when the phases are in local thermal equilibrium. A full linear stability analysis is developed without approximation, and this yields a parabolic system of equations for the evolving disturbances. Criteria for the onset of convection are derived as a function of the three available nondimensional parameters, the inter-phase heat transfer coefficient, H, the porosity-scaled conductivity ratio, γ, and the diffusivity ratio, α.     

A note on the modeling of local thermal non-equilibrium in a structured porous medium

The traditional two-temperature thermal energy equations modeling local thermal equilibrium in a saturated porous medium are modified to model the case of convection when the medium has transverse structure.     

Forced convection in porous microchannels with viscous dissipation in local thermal non-equilibrium conditions

Fully developed, steady-state forced convection, in parallel-plate microchannels, filled with a porous medium saturated with rarefied gases at high temperatures, in local thermal non-equilibrium (LTNE) condition, is investigated for the first-order slip-flow regime (0 ≤ Kn ≤ 0.1). Both velocity and temperature jumps at the walls are accounted for. An analytic solution is proposed for the Darcy–extended Brinkman flow model with assigned uniform heat flux at the microchannel walls and viscous heat dissipation in the fluid phase. The solution for NTLE includes the shear work done by the slipping effects. A closed-form expression of the Nusselt number is derived. A validation analysis with respect to the case of channels filled with saturated porous medium is accomplished. The results show that the internal dissipation increases as the velocity slip increases. In addition, the heat dissipation strongly affects the fluid temperature profiles. The increases in velocity slip and temperature jump lead to decreases of temperature gradients in the fluid and solid along the sections. The heat transfer at channel walls is enhanced due to an increase in the bulk heat transfer.     

Thermally developing forced convection in a channel occupied by a porous medium saturated by a non-Newtonian fluid

The classical Graetz methodology is applied to investigate the thermal development of forced convection in a parallel plate channel filled by a saturated porous medium, with walls held at constant temperature, for the case of a non-Newtonian fluid of power-law type. A Brinkman-Forchheimer model is used for the momentum equation. The analysis for the case of small modified Darcy number leads to expressions for the local Nusselt number and average Nusselt number as functions of the dimensionless longitudinal coordinate, the power-law index, a modified Darcy number, and a modified Reynolds-Forchheimer number (with the last three parameters being involved via a boundary-layer thickness).     

Effect of rotation on the onset of thermal convection in a sparsely packed porous layer using a thermal non-equilibrium model

Linear and nonlinear stability of a rotating fluid-saturated sparsely packed porous layer heated from below and cooled from above is studied when the fluid and solid phases are not in local thermal equilibrium. The extended Darcy–Brinkman model that includes the time derivative and Coriolis terms is employed as a momentum equation. A two-field model that represents the fluid and solid phase temperature fields separately is used for energy equation. The onset criterion for both stationary and oscillatory convection is derived analytically. It is found that small inter-phase heat transfer coefficient has significant effect on the stability of the system. There is a competition between the processes of rotation and thermal diffusion that causes the convection to set in through oscillatory mode rather than stationary. The rotation inhibits the onset of convection in both stationary and oscillatory mode. The Darcy number stabilizes the system towards the oscillatory mode, while it has dual effect on stationary convection. Besides, the effect of porosity modified conductivity ratio, Darcy–Prandtl number and the ratio of diffusivities on the stability of the system is investigated. The nonlinear theory is based on the truncated representation of Fourier series method. The effect of thermal non-equilibrium on heat transfer is brought out. The transient behavior of the Nusselt number is investigated by using the Runge–Kutta method. Some of the convection systems previously reported in the literature is shown to be special cases of the system presented in this study.     

Combined free and forced convection flow through a porous medium

Local similarity solutions of the forced and free convection flow past a porous medium bounded by a semi-infinite vertical porous plate are obtained numerically.     

Laminar forced convection in a heat generating bi-disperse porous medium channel

Thermal management of heat generating electronics using the Bi-Disperse Porous Medium (BDPM) approach is investigated. The BDPM channel comprises heat generating micro-porous square blocks separated by macro-pore gaps. Laminar forced convection cooling fluid of Pr = 0.7 saturates both the micro- and macro-pores. Bi-dispersion effect is induced by varying the porous block permeability Da_I and external permeability Da_E through variation in number of blocks N². For fixed Re, when 10^?5 ? Da_I ? 10^?2, the heat transfer Nu is enhanced four times (from ～200 to ～800) while the pressure drop Δp1 reduces almost eightfold. For Da_I < 10^?5, Nu decreases quickly to reach a minimum at the Mono-Disperse Porous Medium (MDPM) limit (Da_I → 0). Compared to N² = 1 case, Nu for BDPM configuration is high when N² ? 1, i.e., the micro-porous blocks are many and well distributed. The pumping power increase is very small for the entire range of N². Distributing heat generating electronics using the BDPM approach is shown to provide a viable method of thermo-hydraulic performance enhancement χ.     

Effects of a porous medium on forced convection of a reciprocating curved channel

Enhancement of heat transfer rates of a reciprocating curved channel partially installed by a porous medium is investigated numerically. The distribution of heat transfer rates on the heat surface of the reciprocating curved channel is rather non-uniform that easily causes a thermal damage to destroy the channel. A method of using the porous medium to enhance heat transfer rates of the channel is then developed to solve the thermal damage. The arbitrary Lagrangian–Eulerian method is firstly modified for treating a moving boundary problem of the porous medium. Main parameters of Reynolds numbers, porosities, frequencies and amplitudes are examined. The results show that the enhancements of heat transfer rates of most porous medium situations are achieved. However, heat transfer rates of a few porous medium situations are unexpectedly inferior to those of without porous medium situations.     

A thermal resistance analysis on forced convection with viscous dissipation in a porous medium using entransy dissipation concept

The analytical solution of a two-equation model presented in an earlier study is examined. Heat transfer characterization is classified into two regimes which are dominated by fluid conduction or solid conduction and interstitial heat exchange, respectively by using the entransy dissipation concept. The computed pattern of variation of thermal resistance with shape factor S at a fixed Brinkman number for a low ratio of the fluid to solid effective thermal conductivities implies the occurrence of temperature gradient bifurcation as S decreases. Therefore, the thermal diffusion term in the fluid phase in the two-equation model is not negligible for both regimes.     

Stability of a fluid-saturated porous medium heated from below by forced convection

A linear stability analysis determining the onset of convection in a bounded rectangular cavity containing a fluid-saturated porous medium is performed for insulated sidewalls, isothermal top wall, and bottom wall heated by forced convection. The nature of the bottom wall heating necessarily involves the Biot number, Bi. Numerical calculations of the critical Rayleigh number, R_c made over the range of Biot numbers 10⁻⁴?Bi?10⁴ for cavity aspect ratios 0?(a,b)?5 cover all effective bottom heating conditions from the constant heat flux global limit, R_c=27.096 found as Bi→0 to the isothermal global limit, R_c=4π² found as Bi→∞. Marginal stability boundaries, preferred cellular modes and disturbance temperature contours are displayed graphically.     

Periodic free convection from a vertical plate in a saturated porous medium, non-equilibrium model

The problem of the free convection from a vertical heated plate in a porous medium is investigated numerically in the present paper. The effect of the sinusoidal plate temperature oscillation on the free convection from the plate is studied using the non-equilibrium model, i.e., porous solid matrix and saturated fluid are not necessary to be at same temperature locally. Non-dimensionalization of the two-dimensional transient laminar boundary layer equations results in three parameters: (1) H, heat transfer coefficient parameter, (2) K_r, thermal conductivity ratio parameter, and (3) λ, thermal diffusivity ratio. Two additional parameters arise from the plate temperature oscillation condition which are the non-dimensional amplitude (ε) and frequency (Ω). The fully implicit finite difference method is used to solve the system of equations. The numerical results are presented for 0 ? H ? 10, 0 ? K_r ? 10, 0.001 ? λ ? 10 with the plate temperature oscillation parameters 0 ? Ω ? 10 and 0 ? ε ? 0.5. The results show that the thermal conductivity ratio parameter is the most important parameter. It is found also that increasing the amplitude and the frequency of the oscillating surface temperature will decrease the free convection heat transfer from the plate for any values of the other parameters.     

Simulation of laminar impinging jet on a porous medium with a thermal non-equilibrium model

This work shows numerical simulations of an impinging jet on a flat plate covered with a layer of a porous material. Macroscopic equations for mass and momentum are obtained based on the volume-average concept. Two macroscopic models are employed for analyzing energy transport, namely the one-energy equation model, based on the Local Thermal Equilibrium assumption (LTE), and the two-energy equation closure, where distinct transport equations for the fluid and the porous matrix follow the Local Non-Thermal Equilibrium hypothesis (LNTE). The numerical technique employed for discretizing the governing equations was the finite volume method with a boundary-fitted non-orthogonal coordinate system. The SIMPLE algorithm was used to handle the pressure–velocity coupling. Parameters such as porosity, porous layer thickness, material permeability and thermal conductivity ratio were varied in order to analyze their effects on flow and heat transport. Results indicate that for low porosities, low permeabilities, thin porous layers and for high thermal conductivity ratios, a different distribution of local Nusselt number at the wall is calculated depending on the energy model applied. The use of the LNTE model indicates that it is advantageous to use a layer of highly conducting and highly porous material attached to the hot wall.     

Natural convection in a heat generating hydrodynamically and thermally anisotropic non-Darcy porous medium

Natural convection in a two-dimensional square cavity containing hydrodynamically and thermally anisotropic porous medium with internal heat generation is analyzed numerically by generalized non-Darcy approach. The properties considered for the study are permeability ratio (K¹), inclination of the principal axes (θ), ratio of Forchheimer constants (F¹) and thermal conductivity ratio (k¹). Results are presented in terms of isotherms, streamlines and maximum temperature in the cavity to understand the flow physics. It is observed that the anisotropic properties have significant influence on the flow behaviour and heat transfer. A correlation for maximum temperature in the cavity for a wide range of parameters (10⁷ ? Ra ? 10⁸, 10^?6 ? Da ? 10^?3, 0° ? θ ? 90°, 1 ? F¹ ? 100, 0.1 ? K¹ ? 10 and 0.1 ? k¹ ? 10) is developed.     

Thermomechanical response of porous biological tissue based on local thermal non-equilibrium

Abstract

Understanding of heat transfer and related thermomechanical interaction in biological tissue is very important to clinical applications. It is quite natural to treat the living tissue as a porous medium, such as the living tissue in the presence of blood. Based on a non-equilibrium heat transfer model, the thermomechanical response of porous biological tissue exposed to an instantaneous thermal shock is investigated in this work. The governing equations are established based on local thermal non-equilibrium model in the context of the generalized thermoelastic theory and solved by time-domain finite-element method. The effect of porosity coefficient on the thermal-mechanical response of the porous tissue is studied and illustrated graphically. Comparisons are made between the proposed results and those from the local thermal equilibrium models to reveal the difference of these two models in terms of thermoelastic response.     

Transient forced convection heat transfer from a circular cylinder embedded in a porous medium

Studies of the transient heat transfer past a circular cylinder in a steady-state viscous flow are presented for some fluid saturated fibrous porous media. Numerical results have been obtained according to the Darcy-Brinkman model by means of the finite element method. Analysis of the influence of the Darcy and Peclet numbers on the mean Nusselt number exhibits the successive conduction, transition and convection regimes. The duration necessary to reach the steady-state convection heat transfer appears as a function of the Peclet and Darcy numbers.     

The effect of local thermal non-equilibrium on impulsive conduction in porous media

We examine the effect of local thermal non-equilibrium on the evolution of the stagnant temperature field in a semi-infinite porous medium. When local thermal equilibrium pertains, the temperature field which is induced by a step change in the temperature of a plane boundary is given by the classical conduction solution involving the complementary error function. When thermal local equilibrium does not apply, then conduction takes place more rapidly in one phase than in the other, although local thermal equilibrium is always approached as time increases. This note examines the evolution of the temperature field in each phase in detail using numerical methods, and the numerical solutions are supplemented by asymptotic solutions valid for both small and large times.