Density Maximum Effect on Double-diffusive Natural Convection in a Porous Cavity with Variable Wall Temperature

The effect of density maximum of water on double-diffusive natural convection in a two-dimensioned cavity filled with a water saturated isotropic porous medium is studied numerically. The horizontal walls of the cavity are insulated. The opposing vertical walls are kept at different temperatures θ _h (linearly varies with height) and θ _c (θ _c ≤ θ _h ). The concentration levels at cold wall and hot wall are, respectively, c ₁ and c ₂ with c ₁ > c ₂. Brinkman-Forchheimer extended Darcy model is used to investigate the average heat and mass transfer rates. The non-dimensional equations for momentum, energy, and concentration are solved by finite volume method with power law scheme for convection and diffusion terms. The results are presented in the form of streamlines, isotherms, and isoconcentration lines for various values of Grashof numbers, Schmidt number, porosity, and Darcy numbers. It is observed that the density maximum of water has profound effect on the thermosolutal convection. The effects of different parameters on the velocity, temperature, and species concentrations are also shown graphically.     

Explicit Conditions for Local Thermal Equilibrium in Porous Media Heat Conduction

Based on the traditional formulation of heat transfer in porous media it is demonstrated that Local Thermal Equilibrium (Lotheq) applies generally for any boundary conditions that are a combination of constant temperature and insulation. The resulting consequences are being analysed and discussed. Among these consequences it is shown that the linear relationship between the average temperature difference of the two phases and the heat transferred over the fluid-solid interface is inappropriate for use in connection with conditions of Lack of Local Thermal Equilibrium (La Lotheq).     

Validation of Thermal Equilibrium Assumption in Transient Forced Convection Flow in Porous Channel

The validity of the local thermal equilibrium assumption in the transient forced convection channel flow is investigated analytically. Closed form expressions are presented for the temperatures of the fluid and solid domains and for the criterion which insures the validity of the local thermal equilibrium assumption. It is found that four dimensionless parameters control the local thermal equilibrium assumption. These parameters are the porosity , the volumetric Biot number Bi, the dimensionless channel length _max and the solid to fluid total thermal capacity ratio C _R. The qualitative and quantitative aspects of the effects of these four parameters on the channel thermal equilibrium relaxation time are investigated.     

The Onset of Convection in an Anisotropic Porous Layer Using a Thermal Non-Equilibrium Model

The stability of a horizontal fluid saturated anisotropic porous layer heated from below and cooled from above is examined analytically when the solid and fluid phases are not in local thermal equilibrium. Darcy model with anisotropic permeability is employed to describe the flow and a two-field model is used for energy equation each representing the solid and fluid phases separately. The linear stability theory is implemented to compute the critical Rayleigh number and the corresponding wavenumber for the onset of convective motion. The effect of thermal non-equilibrium and anisotropy in both mechanical and thermal properties of the porous medium on the onset of convection is discussed. Besides, asymptotic analysis for both very small and large values of the interphase heat transfer coefficient is also presented. An excellent agreement is found between the exact and asymptotic solutions. Some known results, which correspond to thermal equilibrium and isotropic porous medium, are recovered in limiting cases.     

Natural Convection in a Nanofluid Saturated Rotating Porous Layer with Thermal Non-Equilibrium Model

In the present article, we study the effect of local thermal non-equilibrium on the linear and non-linear thermal instability in a nanofluid saturated rotating porous layer. The Darcy Model has been used for the porous medium, while the nanofluid layer incorporates the effect of Brownian motion along with thermophoresis. A three-temperature model is been used for the effect of local thermal non-equilibrium among the particle, fluid, and solid–matrix phases. The linear stability analysis is based on normal mode technique, while for nonlinear analysis a minimal representation of the truncated Fourier series analysis involving only two terms has been used.     

Conjugate Natural Convection in a Porous Enclosure with Non-Uniform Heat Generation

Effects of a conductive wall on natural convection in a square porous enclosure having internal heating at a rate proportional to a power of temperature difference is studied numerically in this article. The horizontal heating is considered, where the vertical walls heated isothermally at different temperatures while the horizontal walls are kept adiabatic. The Darcy model is used in the mathematical formulation for the porous layer and finite difference method is applied to solve the dimensionless governing equations. The governing parameters considered are the Rayleigh number (0 ???Ra ???1000), the internal heating and the local exponent parameters (0 ????? ???5), (1 ????? ???3), the wall to porous thermal conductivity ratio (0.44 ???Kr ???9.9) and the ratio of wall thickness to its width (0.02 ???D ???0.5). The results are presented to show the effect of these parameters on the fluid flow and heat transfer characteristics. It is found a strong internal heating can generate significant maximum fluid temperature more than the conductive solid wall. Increasing value thermal conductivity ratio and/or decreasing the thickness of solid wall can increase the maximum fluid temperature. It is also found that at very low Rayleigh number, the heat transfer across the porous enclosure remain stable for any values of the thermal conductivity ratio.     

Validation of the Local Thermal Equilibrium Assumption in Forced Convection of Non-Newtonian Fluids through Porous Channels

In this paper, we assess the validity of the local thermal equilibrium assumption in the non-Newtonian forced convection flow through channels filled with porous media. For this purpose, the problem is solved numerically using local thermal non-equilibrium and non-Darcian models. Numerical solutions obtained over broad ranges of representative dimensionless parameters are utilized to map conditions at which the local thermal equilibrium assumption can or cannot be employed. The circumstances of a higher modified Peclet number, a lower modified Biot number, a lower fluid-to-solid thermal conductivity ratio, a lower power-law fluid index, and a lower microscopic and macroscopic frictional flow resistance coefficients, are identified as unfavorable circumstances for the local thermal equilibrium (LTE) condition to hold. Quantitative LTE validity maps that reflect the proportional effect of each parameter as related to others are presented.     

Thermal Analysis of Natural Convection Porous Fins

This work introduces a simple method of analysis to study the performance of porous fins in a natural convection environment. The method is based on using energy balance and Darcy’s model to formulate the heat transfer equation. The thermal performance of porous fins is then studied for three types of fins: long fin, finite-length fin with insulated tip and a finite-length fin with tip exposed to a known convection coefficient. It is found from the analysis that the effect of different design and operating parameters such as: Ra number, Da number, thermal conductivity ratio, Kr and length thickness ratio on the temperature distribution along the fin is grouped into one newly defined parameter called S_H. The effect of the variation of S_H on the porous fin thermal performance is established. The effect of varying the fin length and thermal conductivity ratio on the heat transfer rate from the fin is investigated and compared with that for a solid fin at certain conditions. It is found that the heat transfer rate from porous fin could exceed that of a solid fin. It is also found that increasing the fin length and effective thermal conductivity enhances the heat transfer from the fin up certain limit, where a further increase in these parameters adds no improvement to the fin performance. On Leave from Jordan University of Science and Technology, Irbid-Jordan     

Onset of Buoyancy-Driven Convection in Porous Media Saturated with Cold Water Cooled from Above

When porous media saturated with initially stagnant cold water around the density maximum temperature are cooled from above, convection may be induced in an unstable lower layer. In this study, the onset of buoyancy-driven convection during time-dependent cooling is investigated using the propagation theory, which transforms disturbance equations similarly, and also considering the density inversion effect. The critical Darcy–Rayleigh number Ra _D,c is found as a function of the dimensionless density maximum temperature θ _max. For Ra _D > Ra _D,c the dimensionless critical time τ _c to mark the onset of instability is presented as a function of Ra _D and θ _max. These critical conditions are compared with previous theoretical results.     

Local Thermal Non-Equilibrium and Heterogeneity Effects on the Onset of Convection in a Layered Porous Medium

The effect of local thermal non-equilibrium on the onset of convection in a porous medium consisting of two horizontal layers is studied analytically. Linear stability theory is applied. Variations of permeability, fluid conductivity, solid conductivity, interphase heat transfer coefficient and porosity are considered. It is found that heterogeneity of permeability and fluid conductivity have a major effect, heterogeneity of interphase heat transfer coefficient and porosity have a lesser effect, while heterogeneity of solid conductivity is relatively unimportant.     

The Effect of Thermal Expansion on Porous Media Convection. Part 2: Thermal Convection Solution

The impact of thermal expansion and the corresponding non-Boussinesq effects on porous media convection are considered. The results show that the non-Boussinesq effects decouple from the rest, and solving the Boussinesq system separately is needed even when non-Boussinesq effects are being investigated. The thermal expansion is shown to have a lasting impact on the post-transient convection only for values of Rayleigh number substantially beyond the convection threshold, where the amplitude of convection is not small. In the neighbourhood of the convection threshold the thermal expansion has only a transient impact on the solution. It is also evident from the results that the neglect of the time derivative term in the extended Darcy equation might introduce a significant error when oscillatory effects are present.     

Steady and Unsteady Heat Transfer in a Channel Partially Filled with Porous Media Under Thermal Non-Equilibrium Condition

Steady and pulsatile flow and heat transfer in a channel lined with two porous layers subject to constant wall heat flux under local thermal non-equilibrium (LTNE) condition is numerically investigated. To do this, a physical boundary condition in the interface of porous media and clear region of the channel is derived. The objective of this work is, first, to assess the effects of local solid-to-fluid heat transfer (a criterion indicating on departure from local thermal equilibrium (LTE) condition), solid-to-fluid thermal conductivity ratio and porous layer thickness on convective heat transfer in steady condition inside a channel partially filled with porous media; second, to examine the impact of pulsatile flow on heat transfer in the same channel. The effects of LTNE condition and thermal conductivity ratio in pulsatile flow are also briefly discussed. It is observed that Nusselt number inside the channel increases when the problem is tending to LTE condition. Therefore, careless consideration of LTE may lead to overestimation of heat transfer. Solid-to-fluid thermal conductivity ratio is also shown to enhance heat transfer in constant porous media thickness. It is also revealed that an increase in the amplitude of pulsation may result in enhancement of Nusselt number, while Nusselt number has a minimum in a certain frequency for each value of amplitude.     

Mixed Convection Boundary-Layer Flow in a Porous Medium Filled with Water Close to its Maximum Density

The steady mixed convection boundary-layer flow over a vertical impermeable surface in a porous medium saturated with water close to its maximum density is considered for uniform wall temperature and outer flow. The problem can be reduced to similarity form and the resulting equations are examined in terms of a mixed convection parameter λ and a parameter δ which measures the difference between the ambient temperature and the temperature at the maximum density. Both assisting (λ > 0) and opposing flows (λ < 0) are considered. A value δ₀ is found for which there are dual solutions for a range λ_c < λ < 0 of λ (the value of λ_c dependent on δ) and single solutions for all λ ≥ 0. Another value of δ₁ of δ, with δ₁ > δ₀, is found for which there are dual solutions for a range 0 < λ < λ_c of positive values of λ, with solutions for all λ≤ 0. There is also a range δ₀ <  δ < δ₁ where there are solutions only for a finite range of λ, with critical points at both positive and negative values of λ, thus putting a finite limit on the range of existence of solutions.     

Thermal Capacity Effect on Transient Free Convection Adjacent to a Vertical Surface in a Porous Medium

In this paper we analyse how the presence of the thermal capacity of a vertical flat plate of finite thickness, which is embedded in a porous medium affects the transient free convection boundary-layer flow. At the time t = 0, the plate is suddenly loaded internally with a constant heat flux rate q, so that a transient boundary-layer flow is initiated adjacent to the plate. Initially, the transient effects due to the imposition of the uniform heat flux rate at the plate are confined to a thin fluid region near to the surface and are described by a small time solution. These effects continue to penetrate outwards and eventually evolve into a new steady state flow. Analytical solutions have been derived for these transient (small time) and steady state (large time) flow regimes, which are then matched by a numerical solution of the full boundary-layer equations. It has been found that the non-dimensional fluid temperature (or fluid velocity) profiles are reduced when the thermal capacity effects, described by a parameter Q ^*, are reduced. For small values of Q ^*, the approach of these profiles to their steady state values is monotonic. However, for large values of Q ^*, the temperature profiles are observed to locally exceed (pass through a maximum value) the final steady state values at certain distances from the plate. In general, the maxima in the temperature profiles increase in size as Q ^* increases and the time taken to approach the steady state solutions increases significantly.     

A Two-Equation Analysis of Convection Heat Transfer in Porous Media

This paper presents a two-equation analysis on the convection heat transfer in porous media based on the modeling developed by Carbonell and Whitaker (1984). The porous system under consideration is bounded by two parallel walls and heated uniformly from one side surface. The Darcy flow is imposed and the fully developed heat transfer is assumed. General solutions, which take into account the additional convective and conductive terms, are obtained for the temperature fields and the Nusselt number. The detailed studies are presented for the porous systems characterized by consolidated and unconsolidated circular unit cells. The results show that, for the consolidated unit cell case, a prediction without the additional convective term overestimates the heat transfer, while for the unconsolidated unit cell case, this effect is negligible. The additional conductive terms are also examined and found to act conventionally as part of the conductive terms.     

Thermal Non-equilibrium Heat Transfer and Entropy Generation due to Natural Convection in a Cylindrical Enclosure with a Truncated Conical,Heat-Generating Porous Bed

Natural convection in enclosures driven by heat-generating porous media has diverse applications in fields like geothermal, chemical, thermal and nuclear energy. The present article focuses on heat transfer and entropy generation characteristics of a heat-generating porous bed, placed centrally within a fluid-filled cylindrical enclosure. Pressure drop and heat transfer in the porous bed are modelled using the Darcy–Brinkmann–Forchheimer approximation and the local thermal non-equilibrium model, respectively. Energy flux vectors have been utilised for visualising convective energy transfer within the enclosure. The study of a wide range of Rayleigh number (\(10^{7}\)–\(10^{11}\)) and Darcy number (\(10^{-6}\)–\(10^{-10}\)) reveals that heat transfer in the porous region can be classified into conduction-dominated and convection-dominated regimes. This is supplemented with an entropy generation analysis in order to identify and characterise the irreversibilities associated with the phenomenon. It is observed that entropy generation characteristics of the enclosure closely follow the above-mentioned regime demarcation. Numerical computations for the present study have been conducted using ANSYS FLUENT 14.5. The solid energy equation is solved as a user-defined scalar equation, while data related to energy flux vectors and entropy generation are obtained using user-defined functions.     

Interaction of Transverse Heterogeneity and Thermal Development of Forced Convection in a Porous Medium

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 whose permeability and thermal conductivity vary in the transverse direction. It was found that there is a significant interaction between heterogeneity and thermal development.     

Forced Convection Heat Transfer on a Flat Plate Embedded in Porous Media for Power-Law Fluids

The effect of the presence of an isotropic solid matrix on the forced convection heat transfer rate from a flat plate to power-law non- Newtonian fluid-saturated porous medium, has been investigated. Numerical results are presented for the distribution of velocity and temperature profiles within the boundary layer. The effects of the flow index, first-order and second-order resistance on the velocity, and temperature profiles are discussed. The missing wall values of the velocity and thermal functions are tabulated.     

Finite Element Modeling of Liquid Deuterium Flow and Heat Transfer in a Cold-neutron Source

A finite element simulation of flow and heat transfer in the moderator cell of a cold-neutron source (CNS), in which liquid deuterium subject to internal heat generation is flowing, is reported. The numerical scheme consists of a stabilized equal-order method. A time-accurate approach is adopted to resolve the large-scale eddies of the flow, with a Smagorinsky's model for the subgrid-scale effects. The thermal coupling follows a staggered strategy, with SUPG-type upwinding. A specific wall-law is developed that accounts for the correct partition of the heat deposited at the wall by radiation between the liquid deuterium and the helium gas flowing at the outer side of the wall. The average flow and thermal structure are presented. The turbulent fluctuations are both illustrated in physical space and decomposed into spectral components. The wavenumber spectrum suggests that adequate resolution of the large-scale eddies has been attained with just 200,000 nodes, while a DNS analysis would have required at least 1010 nodes. Usefulness of the approach in the design process of the CNS is highlighted.     

The effect of local thermal non-equilibrium on conduction in porous channels with a uniform heat source

We examine the effect of local thermal non-equilibrium on the steady state heat conduction in a porous layer in the presence of internal heat generation. A uniform source of heat is present in either the fluid or the solid phase. A two-temperature model is assumed and analytical solutions are presented for the resulting steady-state temperature profiles in a uniform porous slab. Attention is then focussed on deriving simple conditions which guarantee local thermal equilibrium.