On steady convection in a porous medium

1972 ◽  
Vol 54 (1) ◽  
pp. 153-161 ◽  
Author(s):  
Enok Palm ◽  
Jan Erik Weber ◽  
Oddmund Kvernvold
Keyword(s):  
Porous Medium ◽  
Nusselt Number ◽  
Rayleigh Number ◽  
Heat Transport ◽  
Abrupt Change ◽  
Darcy’S Law ◽  
Darcy's Law ◽  
Sixth Order ◽  
Steady Convection ◽  
Good Agreement

For convection in a porous medium the dependence of the Nusselt number on the Rayleigh number is examined to sixth order using an expansion for the Rayleigh number proposed by Kuo (1961). The results show very good agreement with experiment. Additionally, the abrupt change which is observed in the heat transport at a supercritical Rayleigh number may be explained by a breakdown of Darcy's law.

Natural convection in porous media

1985 ◽  
Vol 150 ◽  
pp. 89-119 ◽  
Author(s):  
V. Prasad ◽  
F. A. Kulacki ◽  
M. Keyhani
Keyword(s):  
Heat Transfer ◽  
Thermal Conductivity ◽  
Porous Medium ◽  
Rayleigh Number ◽  
Effective Thermal Conductivity ◽  
Darcy’S Law ◽  
Saturated Porous Medium ◽  
Darcy's Law ◽  
Experimental Results ◽  
Rayleigh Numbers

Experimental results on free convection in a vertical annulus filled with a saturated porous medium are reported for height-to-gap ratios of 1.46, 1 and 0.545, and radius ratio of 5.338. In these experiments, the inner and outer walls are maintained at constant temperatures. The use of several fluid–solid combinations indicates a divergence in the Nusselt-number–Rayleigh-number relation, as also reported by previous investigators for horizontal layers and vertical cavities. The reason for this divergence is the use of the stagnant thermal conductivity of the fluid-filled solid matrix. A simple model is presented to obtain an effective thermal conductivity as a function of the convective state, and thereby eliminate the aforementioned divergence. A reasonable agreement between experimentally and theoretically determined Nusselt numbers is then achieved for the present and previous experimental results. It is thus concluded that a unique relationship exists between the Nusselt and Rayleigh numbers unless Darcy's law is inapplicable. The factors that influence the breakdown of Darcian behaviour are characterized and their effects on heat-transfer rates are explained. It is observed that, once the relation between the Nusselt and Rayleigh numbers branches out from that obtained via the mathematical formulation based on Darcy's law, its slope approaches that for a fluid-filled enclosure of the same geometry when the Rayleigh number is large enough. An iterative scheme is also presented for estimation of effective thermal conductivity of a saturated porous medium by using the existing results for overall heat transfer.

Heat transport by parallel-roll convection in a rectangular container

1987 ◽  
Vol 185 ◽  
pp. 205-234 ◽  
Author(s):  
R. W. Walden ◽  
Paul Kolodner ◽  
A. Passner ◽  
C. M. Surko
Keyword(s):  
Rayleigh Number ◽  
Heat Transport ◽  
Critical Rayleigh Number ◽  
Equation Model ◽  
Onset Of Convection ◽  
Infinite Layer ◽  
Lateral Extent ◽  
Prandtl Numbers ◽  
Rayleigh Numbers ◽  
Good Agreement

Heat-transport measurements are reported for thermal convection in a rectangular box of aspect’ ratio 10 x 5. Results are presented for Rayleigh numbers up to 35Rc, Prandtl numbers between 2 and 20, and wavenumbers between 0.6 and 1.0kc, where Rc and kc are the critical Rayleigh number and wavenumber for the onset of convection in a layer of infinite lateral extent. The measurements are in good agreement with a phenomenological model which combines the calculations of Nusselt number, as a function of Rayleigh number and roll wavenumber for two-dimensional convection in an infinite layer, with a nonlinear amplitude-equation model developed to account for sidewell attenuation. The appearance of bimodal convection increases the heat transport above that expected for simple parallel-roll convection.

Natural convective heat transfer in a hemispherical cavity filled with ZnO–H2O nanofluid saturated porous medium

2018 ◽  
Vol 29 (10) ◽  
pp. 1850097 ◽  
Author(s):  
Abderrahmane Baïri ◽  
Najib Laraqi
Keyword(s):  
Heat Transfer ◽  
Porous Medium ◽  
Nusselt Number ◽  
Rayleigh Number ◽  
Convective Heat Transfer ◽  
Convective Heat ◽  
Volume Fraction ◽  
Saturated Porous Medium ◽  
Physical Parameters ◽  
Numerical Work

This three-dimensional (3D) numerical work based on the volume control method quantifies the convective heat transfer occurring in a hemispherical cavity filled with a ZnO–H2O nanofluid saturated porous medium. Its main objective is to improve the cooling of an electronic component contained in this enclosure. The volume fraction of the considered monophasic nanofluid varies between 0% (pure water) and 10%, while the cupola is maintained isothermal at cold temperature. During operation, the active device generates a heat flux leading to high Rayleigh number reaching [Formula: see text] and may be inclined with respect to the horizontal plane at an angle ranging from 0[Formula: see text] to 180[Formula: see text] (horizontal position with cupola facing upwards and downwards, respectively) by steps of 15[Formula: see text]. The natural convective heat transfer represented by the average Nusselt number has been quantified for many configurations obtained by combining the tilt angle, the Rayleigh number, the nanofluid volume fraction and the ratio between the thermal conductivity of the porous medium’s solid matrix and that of the base fluid. This ratio has a significant influence on the free convective heat transfer and ranges from 0 (without porous media) to 70 in this work. The influence of the four physical parameters is analyzed and commented. An empirical correlation between the Nusselt number and these parameters is proposed, allowing determination of the average natural convective heat transfer occurring in the hemispherical cavity.

Thermal Convection in a Rectangular Cavity Filled With a Heat-Generating, Darcy Porous Medium

1987 ◽  
Vol 109 (3) ◽  
pp. 697-703 ◽  
Author(s):  
V. Prasad
Keyword(s):  
Heat Flux ◽  
Porous Medium ◽  
Nusselt Number ◽  
Rayleigh Number ◽  
Aspect Ratio ◽  
Rectangular Cavity ◽  
Stratified Medium ◽  
Maximum Temperature ◽  
Wall Boundary ◽  
Uniform Heat

Two-dimensional, steady natural convection in a rectangular cavity filled with a heat-generating, saturated porous medium has been studied numerically for the case when the vertical walls of the cavity are isothermal and the horizontal walls are either adiabatic or cold. Results are presented in terms of the streamlines and isotherms, the maximum temperature in the cavity, and the local and overall Nusselt numbers. The buoyant flow together with the uniform heat generation produces a highly stratified medium at high Rayleigh numbers. Although the maximum temperature in the cavity θmax invariably increases with the Rayleigh number Ra and aspect ratio A, the rate of increase diminishes with this enhancement in Ra and A. However, the change in the horizontal wall boundary condition from adiabatic to cold reduces θmax. The local heat flux on the bounding walls is a strong function of the Rayleigh number, the aspect ratio, and the wall boundary conditions. The variation in overall Nusselt number is qualitatively similar to that observed in the case of a differentially heated cavity, and the present heat transfer rates are close to that for the cavity heated by applying a uniform heat flux. Several correlations are presented for maximum temperature and overall Nusselt number.

Numerical Results for the Effective Flow and Thermal Properties of Idealized Graphite Foam

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
Christopher T. DeGroot ◽  
Anthony G. Straatman
Keyword(s):  
Heat Transfer ◽  
Nusselt Number ◽  
Thermal Properties ◽  
Effective Properties ◽  
Darcy’S Law ◽  
Dispersion Model ◽  
Fluid Phase ◽  
Reynolds Numbers ◽  
Darcy's Law ◽  
Closed Volume

To simulate the heat transfer performance of devices incorporating high-conductivity porous materials, it is necessary to determine the relevant effective properties to close the volume-averaged momentum and energy equations. In this work, we determine these effective properties by conducting direct simulations in an idealized spherical void phase geometry and use the results to establish closure relations to be employed in a volume-averaged framework. To close the volume-averaged momentum equation, we determine the permeability as defined by Darcy’s law as well as a non-Darcy term, which characterizes the departure from Darcy’s law at higher Reynolds numbers. Results indicate that the non-Darcy term is nonlinearly related to Reynolds number, not only confirming previous evidence regarding such behavior in the weak inertia flow regime, but demonstrating that this is generally true at higher Reynolds numbers as well. The volume-averaged energy equation in the fluid phase is closed by the thermal dispersion conductivity tensor, the convecting velocity, and the interfacial Nusselt number. Overall, it has been found that many existing correlations for the effective thermal properties of graphite foams are oversimplified. In particular, it has been found that the dispersion conductivity is not well characterized using the Péclet number alone, rather the Reynolds and Prandtl numbers must be considered as separate influences. Additionally, the convecting velocity modification, which is not typically considered, was found to be significant, while the interfacial Nusselt number was found to exhibit a nonzero asymptote at low Péclet numbers. Finally, simulations using the closed volume-averaged equations reveal significant differences in heat transfer when employing the present dispersion model in comparison to a simpler dispersion model commonly used for metallic foams, particularly at high Péclet numbers and for thicker foam blocks.

Effects of different thermal modes of obstacles on the natural convective Al2O3-water nanofluidic transport inside a triangular cavity

2022 ◽  
pp. 095440622110614
Author(s):  
Nilankush Acharya
Keyword(s):  
Nusselt Number ◽  
Rayleigh Number ◽  
Heat Transport ◽  
Hartmann Number ◽  
Volume Fraction ◽  
High Heat ◽  
Flow Profile ◽  
Galerkin Finite Element ◽  
Transport Enhancement ◽  
Number Formula

This study investigates the Al2O3-water nanofluidic transport within an isosceles triangular compartment with top vertex downwards. The top wall is maintained isothermally cooled and left as well as right inclined walls are made uniformly heated. Two diamond-shaped obstacles are positioned inside the enclosure. The nanofluidic motion is supposed to be magnetically influenced. This investigation includes a fine analysis of how various thermal modes of obstacles affect the velocity and thermal profiles of the nanofluid. Appropriate similarity conversion leads to having a non-dimensional flow profile and is treated with Galerkin finite element scheme. The grid independency, experimental verification, and comparison assessments are directed to explore the model accuracy. The dynamic parameters like Rayleigh number [Formula: see text], nanoparticle volume fraction [Formula: see text], and Hartmann number [Formula: see text] are varied to perceive the noteworthy changes in isotherms, velocity, streamlines, and Nusselt number. The consequences specify average Nusselt number deteriorates for Hartmann number but escalates for nanoparticle concentration and Rayleigh number. Both heated and adiabatic obstacles exhibit high heat transport, while cold obstacles reveal the lowest magnitude in heat transmission. For Rayleigh number, cold obstacles reveal 34.51% heat transport enhancement, whereas it is 52.72% for heated obstacles compared to cold one. mathematics subject classification: 76W05

Darcy's Law for a Heterogeneous Porous Medium

2001 ◽  
Vol 4 (2) ◽  
pp. 14 ◽  
Author(s):  
F. D. Moura Neto ◽  
S. T. Melo
Keyword(s):  
Porous Medium ◽  
Darcy’S Law ◽  
Darcy's Law ◽  
Heterogeneous Porous Medium

High Rayleigh number convection in a porous medium containing a thin low-permeability layer

2014 ◽  
Vol 756 ◽  
pp. 844-869 ◽  
Author(s):  
Duncan R. Hewitt ◽  
Jerome A. Neufeld ◽  
John R. Lister
Keyword(s):  
Porous Medium ◽  
Rayleigh Number ◽  
Lower Boundary ◽  
Low Permeability ◽  
One Dimensional ◽  
Horizontal Length ◽  
The Relationship ◽  
Geological Formations ◽  
Rayleigh Benard ◽  
Steady Convection

AbstractPorous geological formations are commonly interspersed with thin, roughly horizontal, low-permeability layers. Statistically steady convection at high Rayleigh number $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Ra}$ is investigated numerically in a two-dimensional porous medium that is heated at the lower boundary and cooled at the upper, and contains a thin, horizontal, low-permeability interior layer. In the limit that both the dimensionless thickness $h$ and permeability $\Pi $ of the low-permeability layer are small, the flow is described solely by the impedance of the layer $\Omega = h/\Pi $ and by $\mathit{Ra}$. In the limit $\Omega \to 0$ (i.e. $h \to 0$), the system reduces to a homogeneous Rayleigh–Darcy (porous Rayleigh–Bénard) cell. Two notable features are observed as $\Omega $ is increased: the dominant horizontal length scale of the flow increases; and the heat flux, as measured by the Nusselt number $\mathit{Nu}$, can increase. For larger values of $\Omega $, $\mathit{Nu}$ always decreases. The dependence of the flow on $\mathit{Ra}$ is explored, over the range $2500 \leqslant \mathit{Ra} \leqslant 2\times 10^4$. Simple one-dimensional models are developed to describe some of the observed features of the relationship $\mathit{Nu}(\Omega )$.

Gas Separation Using a Membrane

2014 ◽  
Author(s):  
Nawaf Alkhamis ◽  
Ali Anqi ◽  
Dennis E. Oztekin ◽  
Abdulmohsen Alsaiari ◽  
Alparslan Oztekin
Keyword(s):  
Porous Medium ◽  
Natural Gas ◽  
Gas Separation ◽  
Stokes Equations ◽  
Darcy’S Law ◽  
Circular Cross Section ◽  
Darcy's Law ◽  
Mass Separation ◽  
Functional Surface ◽  
Wide Range

Gas-gas separation, to purify natural gas, is simulated using a membrane supported by a porous medium. Removing acidic gasses from the natural gas is gaining attention recently. Computational fluid dynamics simulations are conducted for asymmetric multi-component fluid flows in a channel. The flow system consists of a circular cross-section channel bounded by a porous layer which supports the membrane wall. The Navier-Stokes equations model the flow in the channel, while the flow in the porous medium is modeled by both the Darcy’s law and the extended Darcy’s law. Mass transport equations, including mass diffusion of mixtures of two gasses (CO2 and CH4), are employed to determine the concentration distribution. The membrane will be modeled as a functional surface; where the flux of each component will be determined based on the local partial pressure of each species, composition, and permeability and selectivity of the membrane. The effect of the porous medium on the membrane performance will be determined for a wide range of Reynolds number. The performance of the system will be measured by maximum mass separation with minimal frictional losses.

Numerical modelling of nonlinear effects in laminar flow through a porous medium

1988 ◽  
Vol 190 ◽  
pp. 393-407 ◽  
Author(s):  
O. Coulaud ◽  
P. Morel ◽  
J. P. Caltagirone
Keyword(s):  
Porous Medium ◽  
Pressure Drop ◽  
Nonlinear Term ◽  
Numerical Solutions ◽  
Darcy’S Law ◽  
Darcy's Law ◽  
Quadratic Term ◽  
Inertial Effects ◽  
Operator Splitting Methods ◽  
Flow Through

This paper deals with the introduction of a nonlinear term into Darcy's equation to describe inertial effects in a porous medium. The method chosen is the numerical resolution of flow equations at a pore scale. The medium is modelled by cylinders of either equal or unequal diameters arranged in a regular pattern with a square or triangular base. For a given flow through this medium the pressure drop is evaluated numerically.The Navier-Stokes equations are discretized by the mixed finite-element method. The numerical solution is based on operator-splitting methods whose purpose is to separate the difficulties due to the nonlinear operator in the equation of motion and the necessity of taking into account the continuity equation. The associated Stokes problems are solved by a mixed formulation proposed by Glowinski & Pironneau.For Reynolds numbers lower than 1, the relationship between the global pressure gradient and the filtration velocity is linear as predicted by Darcy's law. For higher values of the Reynolds number the pressure drop is influenced by inertial effects which can be interpreted by the addition of a quadratic term in Darcy's law.On the one hand this study confirms the presence of a nonlinear term in the motion equation as experimentally predicted by several authors, and on the other hand analyses the fluid behaviour in simple media. In addition to the detailed numerical solutions, an estimation of the hydrodynamical constants in the Forchheimer equation is given in terms of porosity and the geometrical characteristics of the models studied.

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