Natural convection in a shallow porous cavity—the Brinkman model

The natural convection in a shallow porous rectangular cavity with differentially heated sidewalls is examined using the Brinkman model. The heat transfer rate through the cavity is determined in terms of a Nusselt number, in the limit of vanishingly small aspect ratio. Two types of boundary conditions are considered. Case I deals with a cavity with all rigid boundaries so that the no-slip boundary conditions can be imposed. In case II, the cavity has a free upper surface. The present analysis shows that the Brinkman model and Darcy's law give virtually the same result for the heat transfer rate when the Darcy number, based on the depth of the cavity, is less than the order of 10⁻⁴. We also find that the presence of a free surface can significantly increase the heat transfer rate through the cavity, especially when the permeability of the medium is high.     

Natural convection in fluid–superposed porous layers heated locally from below

A numerical study of two-dimensional natural convection in fluid–superposed porous layers heated locally from below is reported based on the one-domain formulation of the conservation equations. The effects of five dimensionless parameters on overall Nusselt number are investigated: Rayleigh number based on overall layer height, heater-to-cavity length ratio, porous layer-to-cavity height ratio, domain aspect ratio, and Darcy number. Streamline and isotherm patterns indicate that convective motion is restricted to the overlying fluid layer with some penetration into the porous layer. Nusselt numbers increase with a decrease in the heater length and height ratio, and increase with the Darcy number. The size of the heat source does not affect the dependence of the heat transfer coefficient on height ratio and Darcy number. For domains with large aspect ratios, complex flow restructuring is observed with an increase in Rayleigh number. The present results represent an extension of the well studied problem of buoyant convection in fluid–superposed porous layers with a fully heated lower surface.     

Free convection in a shallow cavity with variable properties—2. Porous media

Free convection in a porous shallow cavity with differentially heated end walls has been studied. The governing differential equations are analytically solved by applying the method of asymptotic expansions. The results show that the constant-property solution (Boussinesq approximation) deviates approximately 3% from the variable-property solution, if the properties in the Boussinesq solution are taken as the arithmetic mean between the hot and cold end wall temperature.     

Mixed convection between horizontal plates—I. Entrance effects

Entrance effects in mixed convection between horizontal, differentially heated plates were studied in nitrogen by laser Doppler anemometry in a range 1368 < Ra < 8300 and 15 < Re < 170. Two entrance lengths were deduced from velocity profiles: one for the onset of buoyancy-driven convective instability, and one for the full development of the mixed flow. Explicit expressions for both entrance lengths are given. In addition, unsteady longitudinal convection rolls were observed. These are discussed in terms of an admixture of transverse convection rolls and/or contributions from upstream turbulence. The experimental results show that the critical Ra for the transverse convection roils increases as Re increases.     

Natural convection to horizontal wires — The conduction limit

Free convection measurements were carried out from horizontal platinum wires of 0.0254, 0.0508 and 0.0826 cm in diameter, all 10.2 cm in length, to a 2500 wppm aqueous Carbopol 960 solution. For this highly viscous fluid there was no evidence of fluid motion and the measured fluid temperature distribution and heat transfer rates were in good agreement with the values predicted for pure conduction. Similar results were found for highly concentrated (5000 wppm) aqueous polyacrylamide solutions.     

Coupled FHD–MHD free convection of a hybrid nanoliquid in an inversed T-shaped enclosure occupied by partitioned porous media

The article treats the study on free convection of a hybrid nanoliquid confined within contrariwise T-shaped enclosure saturated by two porous media with different material and structure. Nanocomposite particles of multiwall carbon nanotubes–Fe₃O₄ are dispersed into water. Variable magnetic source located to the bottom wall has been analyzed in terms of heat transport performance and nanofluid motion behavior in the enclosure. The governing equations with boundary conditions formulated using the primitive dimensionless variables have been numerically worked out by the finite element technique. Impacts of the Hartmann number, magnetic number, Rayleigh number, ratio of thermal conductivity, porosity ratio, the Darcy number, and convective heat transfer coefficient at solid–nanofluid interface have been investigated. It has been found that low values of dimensionless convection parameter ratio at the border between solid and nanoliquid phases characterize high values of the nanofluid dimensionless convective heat transfer parameter. The average Nusselt number for the solid state has maximum value for high quantities of ratio of Darcy number and low values of dimensionless convection parameter ratio at the border between solid and nanoliquid phases.     

Fractal analysis of Herschel–Bulkley fluid flow in porous media

The Herschel–Bulkley (HB) fluid is the representative fluid which may be reduced to the power-law fluid, Bingham and Newtonian fluids in appropriate conditions. In this paper, fractal models for velocity and the starting pressure gradient for HB fluid in porous media are derived based on fractal characteristics of porous media and capillary model. The proposed models are expressed as a function of fractal dimensions, porosity, maximum pore size and representative length of porous media. Every parameter in the proposed expressions has clear physical meaning, and the proposed models relate the flow characteristics of HB fluid to the structural parameters of porous media. The variation trends of fractal velocity and starting pressure gradient versus different impact factors are shown, and the analytical expressions reveal the physical principles for flow velocity and starting pressure gradient in porous media.     

Local thermal nonequilibrium conjugate natural convection heat transfer of nanofluids in a cavity partially filled with porous media using Buongiorno’s model

The natural convection heat transfer in a cavity filled with three layers of solid, porous medium, and free fluid is addressed. The porous medium and free fluid layers are filled with a nanofluid. The porous layer is modeled using the local thermal nonequilibrium (LTNE) model, considering the temperature difference between the solid porous matrix and the nanofluid phases. The nanofluid is modeled using the Buongiorno’s model incorporating the thermophoresis and Brownian motion effects. The governing equations are transformed into a set of nondimensional partial differential equations, and then solved using finite element method in a nonuniform grid. The effects of various nondimensional parameters are discussed. The results showed that the Brownian motion and thermophoresis effects result in significant concentration gradients of nanoparticles in the porous and free fluid layers. The increase in Rayleigh (Ra), Darcy (Da), the thermal conductivity ratios for the solid wall and solid porous matrix, i.e., K_r and R_k, enhanced the average Nusselt number. The increase in the convection interaction heat transfer parameter between the solid porous matrix and the nanofluid in the pores (H) increases the average Nusselt number in the solid porous matrix but decreases the average Nusselt number in the nanofluid phase of the porous layer.     

Analysis of temperature gradient bifurcation in porous media – An exact solution

The phenomenon of temperature gradient bifurcation in a porous medium is analyzed by studying the convective heat transfer process within a channel filled with a porous medium, with internal heat generation. A local thermal non-equilibrium (LTNE) model is used to represent the energy transport within the porous medium. Exact solutions are derived for both the fluid and solid temperature distributions for two primary approaches (Models A and B) for the constant wall heat flux boundary condition. The Nusselt number for the fluid at the channel wall is also obtained. The effects of the pertinent parameters such as fluid and solid internal heat generations, Biot number and fluid to solid thermal conductivity ratio are discussed. It is shown that the internal heat generation in the solid phase is significant for the heat transfer characteristics. The validity of the one equation model is investigated by comparing the Nusselt number obtained from the LTNE model with that from the local thermal equilibrium (LTE) model. The results demonstrate the importance of utilizing the LTNE model in the present study. The phenomenon of temperature gradient bifurcation for the fluid and solid phases at the wall for Model A is established and demonstrated. In addition, the temperature distributions for Models A and B are compared. A numerical study for the constant temperature boundary condition was also carried out. It was established that the phenomenon of temperature gradient bifurcation for the fluid and solid phases for the constant temperature boundary condition can occur over a given axial region.     

Free convection in a vertical,porous annulus with constant heat flux on the inner wall—experimental results

Steady-state natural convection in a concentric, tall, vertical annulus filled with saturated, porous media is experimentally studied when the inner wall is heated by applying a constant heat flux and the outer wall is isothermally cooled. Temperature profiles and heat transfer rates are obtained for two sets of aspect and radius ratios, A = 14.4, k = 3.5 and A = 11.08, k = 14. The use of several solid-fluid combinations indicates a divergent behavior by Nusselt vs Rayleigh number curves, as also reported by previous investigators. An analysis of present and previous experimental data shows that the Nusselt number for a given Rayleigh number decreases as the ratio of solid and fluid thermal conductivities increases and vice versa.     

Free convection in a shallow cavity with variable properties—1. Newtonian fluid

The effect of variable properties on free convection in a shallow cavity with differentially heated end walls has been analytically studied. Compared to the results for the linear theory the Boussinesq approximation usually applied is exact, if the fluid properties are taken at the arithmetic mean between the hot and cold temperature at the end walls. For the parameter range considered the deviation between the results for the quadratic theory and the Boussinesq approximation is less than approximately 2.5%. Hence, it has been demonstrated that the arithmetic mean is a reasonable choice. With this reference temperature the Boussinesq approximation leads to sufficiently accurate results.     

Heat transfer in turbulent fluids—I. Pipe flow

The expression for turbulent Prandtl number obtained from the renormalization group procedure is used to describe the process of heat transfer in turbulent pipe flow. The results are in a good agreement with experimental data over the entire range of experimentally accessible Prandtl numbers, 10^-2<σ₀< 10⁶.     

Spectral Fourier–Galerkin benchmark solution for natural convection in an inclined saturated porous medium

This paper presents some novel problems associated with the steady natural convection flow in an inclined square cavity filled with a saturated porous medium. The proposed method is a high-accurate spectral method based on the Fourier–Galerkin technique. The numerical results have demonstrated the advantage for the following reasons. (a) The high-accurate method deals with inclined geometries successfully. (b) The streamlines, isotherms, and the average Nusselt numbers are affected significantly by the inclination of the cavity for high values of Rayleigh number. (c) In contrast with the finite element method a highly accurate and efficient solution with less computational effort is obtained.     

Brinkman–Forchheimer modeling for porous media thermoacoustic system

In this paper, analytical studies have been conducted on the flow and thermal fields of unsteady compressible viscous oscillating flow through channels filled with porous media representing stacks in thermoacoustic systems. The flow in the porous material is described by the Brinkman–Forchheimer–extended Darcy model. Analytical expressions for oscillating velocity, temperature, and energy flux density are obtained after linearizing and solving the governing differential equations with long wave, short stack, and small amplitude oscillation approximations. Experimental work is also conducted to verify the temperature difference obtained across the porous stack ends. To produce the experimental results, a thermoacoustic heat pump is designed and constructed where reticulated vitreous carbon (RVC) is used as the stack material. A very good agreement is obtained between the modeling and the experimental results. The expression of temperature difference across the stack ends obtained in the present study is also compared with the existing thermoacoustic literature. The proposed expression surpasses the existing expression available in the literature. The system of equations developed in the present study is a helpful tool for thermal engineers and physicist to design porous stacks for thermoacoustic devices.     

Natural convection of nanoparticle–water mixture near its density inversion in a rectangular enclosure

Natural convection of mixture of nanoparticles and water near its density maximum in a rectangular enclosure is studied numerically. A non-Boussinesq homogenous model is used in mathematical formulations of governing equations. The finite volume method is used to solve the governing equations. The results are presented graphically in the form of streamlines, isotherms and velocity vectors and are discussed for various nanoparticle volume fractions. It is observed that flow and temperature field is affected significantly in the presence of nanoparticles. The average heat transfer rate considering a non-Boussinesq temperature-dependent density (inversion of density) is lower than considering a Boussinesq temperature-dependent density. The average Nusselt number increases with an increase of nanoparticle volume fraction. It is observed that the density inversion of water leaves strong effects on fluid flow and heat transfer due to the formation of bi-cellular structure. The properties of nanoparticles also affect the fluid flow and heat transfer.     

Thermal dispersion in porous media as a function of the solid–fluid conductivity ratio

Thermal dispersion in porous media is an import phenomenon in combustion and in steam injection systems for Enhanced Oil Recovery methods, among several others engineering applications. In this work, thermal dispersion tensors were calculated within an infinite porous medium formed by a spatially periodic array of longitudinally-displaced elliptic rods. Two different thermal conductivity ratios between the solid and fluid phases were used for analyzing their effect on the thermal dispersion tensor, following a systematic analysis of several porous media modeled by different unit-cell geometry. As such, just one unit-cell, together with periodic boundary conditions for mass, momentum and energy equations, was used to represent the medium. The numerical methodology herein employed is based on the control-volume approach. Turbulence was assumed to exist within the fluid phase and a low Reynolds k–ε closure was used to model it. The flow equations at the pore-scale were numerically solved using the SIMPLE method on a non-orthogonal boundary-fitted coordinate system. Cell-integrated results for the longitudinal dispersion coefficient showed little sensitiveness on porosity, boundary condition type, medium morphology and solid–fluid conductivity ratio, whereas for the transversal direction, all of these parameters modified the numerical value obtained for the dispersion coefficient.     

A three-dimensional enthalpic lattice Boltzmann formulation for convection–diffusion heat transfer problems in heterogeneous media

In this paper, an enthalpic lattice Boltzmann method formulation for 3D unsteady convection–diffusion heat transfer problems is used to overcome discontinuity issues in heterogeneous media. The new formulation is based on the appearance of a source term added to the collision step. The major achievement of the proposed enthalpic LB formulation is avoiding any interface treatments or geometry considerations even when dealing with complex geometries. The performance of the present method is tested for several three-dimensional convection–diffusion problems. Comparisons are made with the control volume method, and numerical results show excellent agreements.     

Mixed convection between horizontal plates—II. Fully developed flow

Fully developed velocity profiles of longitudinal convection rolls in mixed convection between horizontal plates were measured in nitrogen by laser Doppler anemometry for a range 2472 < Ra < 8300 and 15 < Re < 150. It is shown analytically and experimentally that the transverse velocities of the longitudinal convection rolls are independent of the forced flow. The experimentally and numerically obtained w-profiles (Pr = 0.71) are in good agreement with theoretical predictions (Pr → ∞) and other experimental results (Pr = 11.1 and 930) for Rayleigh-Benard convection. A detailed study of the longitudinal velocity modulation Δu[w_max(Ra), Re] is presented. Also, asymmetric roll patterns were found in spite of the small temperature differences used between the horizontal plates.     

Moisture transport in silica gel packed beds—I.Theoretical study

Diffusion mechanisms of moisture within silica gel particles are investigated. It is found that for microporous silica gel surface diffusion is the dominant mechanism of moisture transport, while for macroporous silica gel both Knudsen and surface diffusions are important. A model is proposed for simultaneous heat and mass transfer in a thin packed bed of desiccant particles, which accounts for diffusion of moisture into the particles by both Knudsen and surface diffusions. Using finite difference methods to solve the resulting partial differential equations, predictions are made for the response of thin beds of silica gel particles to a step change in air inlet conditions, and compared to a pseudo-gas-side controlled model commonly used for the design of desiccant dehumidifiers for solar desiccant cooling applications.     

On the onset of thermal convection in rotating nanofluid layer saturating a Darcy–Brinkman porous medium

In this paper, the effect of rotation on the onset of thermal convection in a horizontal layer of nanofluid saturated by a Darcy–Brinkman porous medium is considered. A linear stability analysis based upon normal mode is used to find solution of the fluid layer confined between two free boundaries. The onset criterion for stationary and oscillatory convection is derived analytically and graphically. The effects of the concentration Rayleigh number, Taylor number, Lewis number, Darcy number and modified diffusivity ratio on the stability of the system are investigated. The sufficient conditions for the non-existence of overstability are also derived.