Influence of anisotropy on convection in porous media with nonuniform thermal gradient

Effect of anisotropy on the onset of natural convection heat transfer in a fluid saturated porous horizontal cavity subjected to nonuniform thermal gradients is investigated analytically. The porous layer is heated from the bottom by a constant heat flux while the other surfaces are being insulated. The horizontal boundaries are either rigid/rigid or stress-free/stress-free. The hydrodynamic anisotropy of the porous matrix is considered. The principal directions of the permeability are oriented in a direction that is oblique to the gravity. Based on a parallel flow assumption, closed-form solution for the flow and heat transfer variables, valid for the onset of convection corresponding to vanishingly small wave number, is obtained in terms of the Darcy-Rayleigh number Ra, the Darcy number Da, and the anisotropic parameters K^* and θ. The critical Rayleigh number for the onset of convection arising from sudden heating and cooling at the boundaries is also predicted. The limiting cases Da→∞ (for a viscous pure fluid) and Da→0 (for anisotropic porous media) completed all results. It is demonstrated that effects of anisotropic parameters are strongly significant.     

Double-diffusive natural convection in an anisotropic porous cavity with opposing buoyancy forces: multi-solutions and oscillations

Natural convection by combined heat and mass transfer with opposing horizontal heat and solute gradients has been investigated in an anisotropic porous cavity using the Darcy model. The porous medium is assumed to be both hydrodynamically and thermally anisotropic. The principal directions of the permeability tensor are taken oblique to the gravity vector, while those of thermal and solutal diffusivity coincide with horizontal and vertical coordinate axes. Special attention is given to understand the effect of anisotropic parameters on the existence of unsteady permanent oscillations and multiple steady-state solutions. From the study of analytical solutions, which can be regarded as a verification of the numerical results, simultaneously, it is found that there exists an interval of buoyancy ratio, I_NM, depending on the parametric values, in which multiple solutions exist. For the unsteady case a similar interval, I_NO, for the buoyancy ratio has been observed numerically, in which permanent oscillations exist. Periodicity of the oscillation changes drastically by changing the permeability of the medium. The results indicate that the maximum I_NM and I_NO interval are attained at an orientation angle of θ=45°. The local direction of the flow changes because of the variation in the extent of the thermal and concentration layers, the opposite buoyant mechanism, and anisotropic parameters.     

Forced convection heat transfer inside an anisotropic porous channel with oblique principal axes: Effect of viscous dissipation

An analytical study on laminar and fully developed forced convection heat transfer in a parallel-plate horizontal channel filled with an anisotropic permeability porous medium is performed. The principal axis of the anisotropic porous medium is oriented from 0 to 90 degrees. A constant heat flux is applied on the outer wall of the channel. Both clear (Newtonian) fluid and Darcy viscous dissipations are considered in the energy equation. Directional permeability ratio parameter A¹ is defined to combine both the effect of the dimensionless permeability ratio parameter K¹=(K₁/K₂) and orientation angle φ into one parameter. The effects of the parameter A¹, the Darcy number Da and the modified Brinkman number Br¹ on the heat transfer and fluid flow characteristics in the channels are investigated and presented in graphs. The obtained results show that the parameters A¹, Da and Br¹ have strong effects on the dimensionless normalized velocity and temperature profiles as well as on the Nusselt number. It is found that for a particular value of A¹, called as critical value A_cr¹, the external heat applied to the surface of the channel is balanced by the internal heat generation due to viscous dissipation and the bulk mean temperature approaches the wall temperature. Hence, the Nusselt number approaches infinity for the critical values A_cr¹.     

Transient natural convection of non-Newtonian fluids about a vertical surface embedded in an anisotropic porous medium

An analytical method is carried out to investigate transient free convection boundary layer flow along a vertical surface embedded in an anisotropic porous medium saturated by a non-Newtonian fluid. The porous medium is anisotropic in permeability with its principal axes oriented in a direction that is non-coincident with the gravity force. A step increase in wall temperature or in surface heat flux is considered. On the basis of the modified Darcy power-law model proposed by Pascal [H. Pascal, Rheological behaviour effect of non-Newtonian fluids on steady and unsteady flow through porous media, Int. J. Numer. Anal. Methods in Geomech. 7 (1983) 207–224] and the generalized Darcy’s law described by Bear [J. Bear, Dynamics of fluids in porous media. Dover Publications, Elsevier, New York (1972)], boundary-layer equations are solved exactly by the method of characteristics. Scale analysis is applied to predict the order-of-magnitudes involved in the boundary layer regime. Analytical expressions are obtained for the limiting time required to reach steady-state, the boundary-layer thickness and the local Nusselt number in terms of the modified-Darcy Rayleigh number, the power-law index, the anisotropic permeability ratio, and the orientation angle of the principal axes. It is demonstrated that both the power-law index and the anisotropic properties have a strong influence on the heat transfer rate.     

Natural convection heat transfer in an anisotropic porous cavity heated from side (2nd report,experiment using a Hele‐Shaw cell)

Natural convection heat transfer and flow structure in an anisotropic porous medium of square cavity saturated with a Boussinesq fluid has been studied experimentally using a Hele‐Shaw cell. The permeability ratio defined by K = K_y/K_x was set to three different values: 0.4, 1, and 2.5. The convection patterns at the three permeability ratios are visualized at several different Rayleigh numbers by a pH indicator method. When K is 0.25, the visualized flow is mainly in the vertical direction. On the contrary, for K = 4 the convecting flow is in the horizontal direction. The average heat transfer coefficients are also measured, and the corresponding Nusselt numbers are plotted as a function of K. It is found that the corresponding Nusselt numbers are correlated with (KRa)^1/2. The experimental results of the flow pattern and heat transfer are in good agreement with those obtained by our previous theory. © 2002 Wiley Periodicals, Inc. Heat Trans Asian Res, 31(6): 463–474, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10046     

Numerical simulation of turbulent fluid flow and heat transfer characteristics in heat exchangers fitted with porous media

Numerical simulations have been carried out to investigate the turbulent heat transfer enhancement in the pipe filled with porous media. Two-dimensional axisymmetric numerical simulations using the k–? turbulent model is used to calculate the fluid flow and heat transfer characteristics in a pipe filled with porous media. The parameters studied include the Reynolds number (Re = 5000–15,000), the Darcy number (Da = 10^?1–10^?6), and the porous radius ratio (e = 0.0–1.0). The numerical results show that the flow field can be adjusted and the thickness of boundary layer can be decreased by the inserted porous medium so that the heat transfer can be enhanced in the pipe. The local distributions of the Nusselt number along the flow direction increase with the increase of the Reynolds number and thickness of the porous layer, but increase with the decreasing Darcy number. For a porous radius ratio less than about 0.6, the effect of the Darcy number on the pressure drop is not that significant. The optimum porous radius ratio is around 0.8 for the range of the parameters investigated, which can be used to enhance heat transfer in heat exchangers.     

Influence of periodicity of sinusoidal bottom boundary condition on natural convection in porous enclosure

In this paper, natural convective flow within a rectangular enclosure has been investigated numerically. All the walls of the enclosure are adiabatic except the bottom wall, which is partially heated and cooled by sinusoidal temperature profile. Both situations: medium is hydro-dynamically isotropic and anisotropic are considered. The governing equations are written under assumption of Brinkman-extended non-Darcy model, including material derivative, and then solved by numerically using spectral element method (SEM). Main emphasize is given on effect of periodicity parameter (N) on local heat transfer rate (Nu_x) as well as flow mechanism in the enclosure. The result shows that, as the periodicity is decreased on increasing N, the absolute value of Nu_x at the bottom left corner point increases in both situations. For odd values of N, the local heat transfer profile is symmetric about the line X = 0.5. The entire flow is governed by two type convective cells: (i) rotating clockwise (ii) rotating anticlockwise. Furthermore, increasing of N increases the multiple cellular structures in the form of N + 1. The relative impact of media permeability on Nu_x at (0, 0) shows that enhancement of media permeability along Y-direction is much more effective than the same along X-direction. In contrast to isotropic porous media, the effect of the permeability and permeability orientation angle on the flow is complex and often non intuitive. In particular the present analysis shows that, different periodicity of temperature boundary condition has significant effect on the flow mechanism and consequently on the heat transfer rate for both situations.     

Anisotropy Effects on Heat and Fluid Flow by Unsteady Natural Convection and Radiation in Saturated Porous Media

ABSTRACT

This article deals with a numerical study of fluid flow and heat transfer by unsteady natural convection and thermal radiation in a vertical channel opened at both ends and filled with anisotropic, in both thermal conductivity and permeability, fluid-saturated porous medium. The bounding walls of the channel are gray and kept at a constant hot temperature.

In the present study we suppose the validity of the Darcy law for motion and of the local thermal equilibrium assumption. The radiative transfer equation (RTE) is solved by the finite-volume method (FVM). The numerical results allow us to represent the time–space variations of the different state variables. The sensitivity of the fluid flow and the heat transfer to different controlling parameters, namely, the single scattering albedo ω, the temperature ratio R, the anisotropic thermal conductivity ratio R_c, and the anisotropic permeability ratio R_k, are addressed. Numerical results indicate that the controlling parameters of the problem, namely, ω, R, R_c, and R_k, have significant effects on the flow and thermal field behavior and also on the transient process of heating or cooling of the medium. Effects of such parameters on time variations of the volumetric flow rate q_v and the convected heat flux Q at the channel's outlet are also studied.     

Non—Darcian and Anisotropic Effects on Natural Convection in Horizontal Porous Media Enclosure

Natural convection heat transfer in a horizontal enclosure filled with anisotropic porous media,being isothermally heated at bettom and cooled at top while the vertical walls being adiabatic,is numerically studied by applying the Brinkman model-a modified form of Darcy model giving consideratioin to the viscous effect.The results show that:(1)a larger permeability ratio(K^*) causes a lower flow intensity in the enclosure and a smaller Nusselt number,all Nusselt numbers approach unity in the limit of K^*→∞;a larger thermal conductivity ratio(λ^*) causes a stranger distortion of isotherms in the enclosure and a higher flow velocity near the walls,all the Nusselt numbers approach unity in the limit of λ^*-→0,the permeability and thermal conductivity ratios generally have opposing effects on the Nusselt number.(2) an increasing Darcy number decreases the flow intensity and heat tansfer rates,which is more significant at a lower permeability ratio.In particular,with K^*≤0.25,the Nusselt number for Da=10^-3 would differ from that of Darcy flow up to an amount of 30%,an analysis neglecting the non-Darican effect will inevitably be of considerable error.     

Combined forced and natural convection heat transfer from a horizontal cylinder embedded in a porous medium

This paper reports the results of an experimental study of heat transfer by combined forced and natural convection from a horizontal cylinder embedded in a porous medium composed of randomly packed glass spheres saturated with water. The direction of the flow of water was horizontal and normal to the longitudinal axis of the cylinder. The diameter of the cylinder, D, was 11.45mm and the equivalent diameter of the glass spheres was 3.072mm. It is shown that the condition Gr_k/Re²_D ⩽ 0.5 represents a conservative criterion for segregating heat transfer data that are predominantly governed by forced convection from those in which natural convection effects are significant. A correlation hypothesis for convection heat transfer which is based upon four assumptions, primary among which is that the flow can be (conceptually) regarded as being composed of ‘coarse’ and ‘fine’ components, is presented. This hypothesis is shown to provide a basis for successfully correlating a set of experimental heat transfer data that extends from the Darcy regime into the turbulent regime and spans the intervening Forchheimer and transition regimes. It is suggested that the correlation procedure adopted here may yield useful results if applied to other geometries such as, for example, forced convection heat transfer in ducts packed with porous media.     

Natural convection heat transfer in an anisotropic porous cavity heated from the side: Part 1. Theory

Natural convection heat transfer and flow structure in an anisotropic porous medium in a square cavity saturated with a Boussinesq fluid have been studied analytically and numerically. Based on an asymptotic analysis, three distinctive regimes are found depending on the magnitude of the permeability ratio K. In the vicinity of K = 1 the average Nusselt number and fluid velocity are scaled with (KRa) ^1/2 when either K or the Rayleigh number Ra is varied. In the limit of K → 0 the heat transfer across the cavity approaches the conductive state, and the convecting velocity, which is primarily in the vertical direction, is scaled with KRa. At the other end of the spectrum, namely, K → ∞, the average Nusselt number and the convecting velocity are scaled with Ra and independent of K. The asymptotic results are verified with two‐dimensional numerical calculations. The ranges of K of the respective regimes are also determined based on the numerical results. © 2000 Scripta Technica, Heat Trans Asian Res, 29(5): 373–384, 2000     

Unsteady forced convection heat transfer on a flat plate embedded in the fluid-saturated porous medium with inertia effect and thermal dispersion

For an unsteady forced convection on a flat plate embedded in the fluid-saturated porous medium with inertia effect and thermal dispersion, this paper presents a precise and rigorous method to obtain the entire solutions from one-dimensional transient conduction (ξ=0) to steady forced convection in porous medium (ξ=1) under conditions of uniform wall temperature and uniform heat flux, respectively. It is worth noted that the rate of unsteady heat transfer can be accelerated by the thermal dispersion, which may be regarded as the effect of mixing or agitating, to enhance the heat transfer in porous medium. Additionally, it is found that the time response, from the transient heat conduction to a steady forced convection in Darcy's flow, is τ=1, and is independent of wall heating condition and thermal dispersion strength (φ).     

Non-Darcy effects in buoyancy driven flows in an enclosure filled with vertically layered porous media

Natural convection in an enclosure filled with two layers of porous media is investigated numerically. Constant heat flux is imposed on the left vertical wall and the right wall is assumed to be at a low temperature. The focus of the work is on the validity of the Darcy model when various combinations of fluid Rayleigh number, Darcy number and permeability ratios are considered for fixed values of the modified Rayleigh numbers. It is found that the boundary effects (Brinkman term) have significant importance at higher modified Rayleigh numbers (Rayleigh number based on permeability, Ra_m). Calculations are performed for a modified Rayleigh number up to 10⁵. The results showed that, for the investigated range of parameters, the flow structure and heat transfer could be different than what Darcy model predicts. Two circulations are predicted for Ra_f=1×10⁸, for two different cases, Da=1×10⁻³, K_r=1000 and Da=1×10⁻⁴, K_r=100 (K_r=K₁/K₂). For K_r>1, increasing permeability ratio decreases flow penetration from layer 1 to layer 2 while reverse is true for K_r<1. For low Ra_m (Ra_m?10³) and K_r=1000, the heat transfer is conductive in the right layer, while this is true for the left layer for K_r=0.001. It is possible to obtain no-slip velocity boundary conditions both at the walls and at the interface between the porous layers even for very low permeability.     

Numerical investigation of nonuniform transverse magnetic field effects on the flow and heat transfer of magnetic nanofluid in a sintered porous channel

In this paper, fluid flow and convective heat transfer of a ferrofluid (water and 4 vol% Fe₃O₄) in sintered Aluminum porous channel, which is subjected to a nonuniform transverse magnetic field have been studied. The numerical simulations supposed an ordinary cubic and staggered arrangement organized by uniformly sized particles with a small contact area for the porous media and constant heat flux at the surface of the microchannel. A wire, in which the electric current passes creates a nonuniform magnetic field, which is perpendicular to the flow direction. To do this simulation, the control volume technique and the two‐phase mixture model have been employed. The results show that the obtained local heat transfer coefficient on the channel surface increased with increasing mass flow rate and decreased slightly along the axial direction. Moreover, exerting the above‐mentioned magnetic field increases the Nusselt number that enhances the heat transfer rate while it has no effect on the pressure drop along the channel.     

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.     

Analysis of microchannel heat sinks for electronics cooling

This paper presents an analytical and numerical study on the heat transfer characteristics of forced convection across a microchannel heat sink. Two analytical approaches are used: the porous medium model and the fin approach. In the porous medium approach, the modified Darcy equation for the fluid and the two-equation model for heat transfer between the solid and fluid phases are employed. Firstly, the effects of channel aspect ratio (α_s) and effective thermal conductivity ratio (k?) on the overall Nusselt number of the heat sink are studied in detail. The predictions from the two approaches both show that the overall Nusselt number (Nu) increases as α_s is increased and decreases with increasing k?. However, the results also reveal that there exists significant difference between the two approaches for both the temperature distributions and overall Nusselt numbers, and the discrepancy becomes larger as either α_s or k? is increased. It is suggested that this discrepancy can be attributed to the indispensable assumption of uniform fluid temperature in the direction normal to the coolant flow invoked in the fin approach. The effect of porosity (ε) on the thermal performance of the microchannel is subsequently examined. It is found that whereas the porous medium model predicts the existence of an optimal porosity for the microchannel heat sink, the fin approach predicts that the heat transfer capability of the heat sink increases monotonically with the porosity. The effect of turbulent heat transfer within the microchannel is next studied, and it is found that turbulent heat transfer results in a decreased optimal porosity in comparison with that for the laminar flow. A new concept of microchannel cooling in combination with microheat pipes is proposed, and the enhancement in heat transfer due to the heat pipes is estimated. Finally, two-dimensional numerical calculations are conducted for both constant heat flux and constant wall temperature conditions to check the accuracy of analytical solutions and to examine the effect of different boundary conditions on the overall heat transfer.     

Combined radiation and convection heat transfer in a porous channel bounded by isothermal parallel plates

Combined radiation and convection heat transfer in a porous medium confined between gray isothermal parallel plates is investigated. The medium is absorbing, emitting and scattering. Cases of boundaries at temperatures higher or lower than the medium are considered. In the porous medium, the boundary effect on the fully developed laminar velocity field as proposed by Kaviany is accounted for. For various values of the extinction coefficient, the scattering albedo, the conduction-radiation parameter and the boundary emissivity, Nusselt number, temperature and heat flux distributions are found for the range of values including the extreme limits of the porous medium shape parameter (PMSP), γ=(W²φ/K)^1/2, where W is the channel width, φ the porosity and K the permeability. For the lower limiting value of the PMSP γ, the effect of the porous medium is negligible and the situation approaches that of Poiseuille flow. For this limiting case, results from the present work are compared with those available in the literature. For medium to high values of the PMSP γ, for the purpose of comparison, some results are presented in tabular form. Radiation is found to have a significant effect on various parameters studied. The discrete transfer method was used for the solution of the radiative part of the energy equation. An iterative finite difference scheme was used to solve the energy equation.     

Calculation of turbulent flow and heat transfer in a porous-baffled channel

This study presents the numerical predictions on the turbulent fluid flow and heat transfer characteristics for rectangular channel with porous baffles which are arranged on the bottom and top channel walls in a periodically staggered way. The turbulent governing equations are solved by a control volume-based finite difference method with power-law scheme and the k-ε turbulence model associated with wall function to describe the turbulent structure. The velocity and pressure terms of momentum equations are solved by SIMPLE (semi-implicit method for pressure-linked equation) method.The parameters studied include the entrance Reynolds number Re (1×10⁴-5×10⁴), the baffle height (h=10, 20 and 30 mm) and kind of baffles (solid and porous); whereas the baffle spacing S/H are fixed at 1.0 and the working medium is air. The numerical calculations of the flow field indicate that the flow patterns around the porous- and solid-type baffles are entirely different due to different transport phenomena and it significantly influences the local heat transfer coefficient distributions. Relative to the solid-type baffle channel, the porous-type baffle channel has a lower friction factor due to less channel blockage.Concerning the heat transfer effect, both the solid-type and porous-type baffles walls enhanced the heat transfer relative to the smooth channel. It is further found that at the higher baffle height, the level of heat transfer augmentation is nearly the same for the porous-type baffle, the only difference being the Reynolds number dependence. As expected, the centerline-averaged Nusselt number ratio increases with increasing the baffle height because of the flow acceleration.     

Linear stability of cold water layer saturating an anisotropic porous medium--effect of confinement

The linear stability analysis is applied to a horizontal porous layer saturated with water in the neighborhood of 4 °C. The porous layer considered is two-dimensional and anisotropic in permeability with principal axes arbitrarily oriented. The onset of convection depends on parameters such as the aspect ratio A, the permeability ratio K∗, the orientation angle, θ of the principal axes and the inversion parameter, γ. The relevant linearized equations are solved with the aid of Galerkin and finite element methods. Results for the case of an infinite layer indicate that the presence of a stable layer near the upper boundary for γ<2 changes drastically the critical Rayleigh number and that an asymptotic situation is reached when γ?1. For that asymptotic situation, and with θ=0° or 90°, the incipient flow field consists of primary convective cells near the lower boundary with superposed layers of secondary cells. For 0°<θ<90°, primary and secondary cells coalesce to form obliquely elongated cells.     

Heatline based thermal management for natural convection within right-angled porous triangular enclosures with various thermal conditions of walls

Analysis of natural convection in porous triangles have many important energy related applications in geophysical and solar energy fields. A numerical study on heat distribution and thermal mixing during steady laminar natural convective flow inside a right-angled triangular enclosure filled with porous media subjected to various wall boundary conditions is investigated in this study using Bejan’s heatlines approach. Influence of various thermal boundary conditions and inclination angles (φ) on evaluation of complex heat flow patterns are studied as a function of Darcy numbers (Da) for various regimes of Prandtl (Pr) and Rayleigh (Ra) numbers. Studies illustrate that maximum heat transfer occurs at the top vertex for lower top angle (φ=^°15) at higher Da(Da=10⁻³). As φ increases to ^°45, the maximum heat flux at the top vertex decreases and thermal mixing increases irrespective of Da and Pr. The enhanced convection at higher Da significantly affects the heat flow distribution, which is clearly depicted by high local Nusselt numbers at Da=10⁻³. It is also found that isothermal heating of walls enhances the heat distribution and thermal mixing. Overall, it is shown that heatlines provide suitable guideline on thermal management in porous right-angled triangular enclosures with various heating strategies.