Theory on Thermal Instability of Binary Gas Mixtures in Porous Media

1976 ◽  
Vol 98 (1) ◽  
pp. 35-41 ◽  
Author(s):  
M. L. Lawson ◽  
Wen-Jei Yang ◽  
S. Bunditkul
Keyword(s):  
Fluid Layer ◽  
Perturbation Technique ◽  
Critical Rayleigh Number ◽  
Lower Boundary ◽  
Horizontal Layer ◽  
Linear Perturbation ◽  
Onset Of Convection ◽  
Hurwitz Stability ◽  
Binary Gas Mixture ◽  
The Galerkin Method

A theory is developed which predicts the instability of a horizontal layer of porous medium saturated with a binary gas mixture. The lower boundary of the system is maintained at a higher temperature and the upper one at low temperature. The transport equations and coefficients are developed on the basis of kinetic theory. A linear perturbation technique is employed to reduce the governing equations for momentum, heat, and mass transfer to eigenvalue differential equations which are solved by the Finlayson method, the combination of the Galerkin method and the Routh-Hurwitz stability criterion. Only neutral stationary stability is found to occur in the system. Its criterion can be predicted by a simple algebraic equation. Both the critical Rayleigh and wave numbers for the onset of convection are governed by five independent dimensionless parameters, two of which are most influential. The critical Rayleigh number may be lower or greater than that for pure fluid layer depending upon whether thermal diffusion induces the heavier component of the mixture to move toward the cold or hot boundary, respectively. The theory compares well with the experimental results.

The Rayleigh—Jeffreys problem with boundary slab of finite conductivity

1968 ◽  
Vol 32 (2) ◽  
pp. 393-398 ◽  
Author(s):  
D. A. Nield
Keyword(s):  
Finite Thickness ◽  
Critical Rayleigh Number ◽  
Horizontal Layer ◽  
Wave Number ◽  
Linear Perturbation ◽  
Finite Conductivity ◽  
Solid Layer ◽  
Onset Of Convection ◽  
Infinite Conductivity ◽  
Bounded Below

Linear perturbation analysis is applied to the problem of the onset of convection in a horizontal layer of fluid heated uniformly from below, when the fluid is bounded below by a rigid plate of inlinite conductivity and above by a solid layer of finite conductivity and finite thickness. The critical Rayleigh number and wave-number are found for various thickness ratios and thermal conductivity ratios. Both numbers are reduced by the presence of a boundary of finite (rather than infinite) conductivity in qualitative agreement with the observation of Koschmieder (1966).

Natural Convection in an Inclined Fluid Layer With a Transverse Magnetic Field: Analogy With a Porous Medium

1995 ◽  
Vol 117 (1) ◽  
pp. 121-129 ◽  
Author(s):  
P. Vasseur ◽  
M. Hasnaoui ◽  
E. Bilgen ◽  
L. Robillard
Keyword(s):  
Magnetic Field ◽  
Transverse Magnetic Field ◽  
Hartmann Number ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Integral Form ◽  
Transverse Magnetic ◽  
Horizontal Layer ◽  
Governing Equations ◽  
Onset Of Convection

In this paper the effect of a transverse magnetic field on buoyancy-driven convection in an inclined two-dimensional cavity is studied analytically and numerically. A constant heat flux is applied for heating and cooling the two opposing walls while the other two walls are insulated. The governing equations are solved analytically, in the limit of a thin layer, using a parallel flow approximation and an integral form of the energy equation. Solutions for the flow fields, temperature distributions, and Nusselt numbers are obtained explicitly in terms of the Rayleigh and Hartmann numbers and the angle of inclination of the cavity. In the high Hartmann number limit it is demonstrated that the resulting solution is equivalent to that obtained for a porous layer on the basis of Darcy’s model. In the low Hartmann number limit the solution for a fluid layer in the absence of a magnetic force is recovered. In the case of a horizontal layer heated from below the critical Rayleigh number for the onset of convection is derived in term of the Hartmann number. A good agreement is found between the analytical predictions and the numerical simulation of the full governing equations.

Throughflow effects on convective instability in superposed fluid and porous layers

1991 ◽  
Vol 231 ◽  
pp. 113-133 ◽  
Author(s):  
Falin Chen
Keyword(s):  
Porous Layer ◽  
Convective Instability ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Vertical Direction ◽  
Single Layer ◽  
Layer Depth ◽  
Onset Of Convection ◽  
Precise Control ◽  
Porous Layers

We implement a linear stability analysis of the convective instability in superposed horizontal fluid and porous layers with throughflow in the vertical direction. It is found that in such a physical configuration both stabilizing and destabilizing factors due to vertical throughflow can be enhanced so that a more precise control of the buoyantly driven instability in either a fluid or a porous layer is possible. For ζ = 0.1 (ζ, the depth ratio, defined as the ratio of the fluid-layer depth to the porous-layer depth), the onset of convection occurs in both fluid and porous layers, the relation between the critical Rayleigh number Rcm and the throughflow strength γm is linear and the Prandtl-number (Prm) effect is insignificant. For ζ ≥ 0.2, the onset of convection is largely confined to the fluid layer, and the relation becomes Rcm ∼ γ2m for most of the cases considered except for Prm = 0.1 with large positive γm where the relation Rcm ∼ γ3m holds. The destabilizing mechanisms proposed by Nield (1987 a, b) due to throughflow are confirmed by the numerical results if considered from the viewpoint of the whole system. Nevertheless, from the viewpoint of each single layer, a different explanation can be obtained.

Experimental investigation of convective stability in a superposed fluid and porous layer when heated from below

1989 ◽  
Vol 207 ◽  
pp. 311-321 ◽  
Author(s):  
Falin Chen ◽  
C. F. Chen
Keyword(s):  
Porous Layer ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Linear Stability Theory ◽  
Convective Stability ◽  
Onset Of Convection ◽  
Temperature Distributions ◽  
Critical Wavelength ◽  
Crystal Film ◽  
Flux Curve

Experiments have been carried out in a horizontal superposed fluid and porous layer contained in a test box 24 cm × 12 cm × 4 cm high. The porous layer consisted of 3 mm diameter glass beads, and the fluids used were water, 60% and 90% glycerin-water solutions, and 100% glycerin. The depth ratio ď, which is the ratio of the thickness of the fluid layer to that of the porous layer, varied from 0 to 1.0. Fluids of increasingly higher viscosity were used for cases with larger ď in order to keep the temperature difference across the tank within reasonable limits. The top and bottom walls were kept at different constant temperatures. Onset of convection was detected by a change of slope in the heat flux curve. The size of the convection cells was inferred from temperature measurements made with embedded thermocouples and from temperature distributions at the top of the layer by use of liquid crystal film. The experimental results showed (i) a precipitous decrease in the critical Rayleigh number as the depth of the fluid layer was increased from zero, and (ii) an eightfold decrease in the critical wavelength between ď = 0.1 and 0.2. Both of these results were predicted by the linear stability theory reported earlier (Chen & Chen 1988).

The Onset of Convection in a Layer of Cellular Porous Material: Effect of Temperature-Dependent Conductivity Arising From Radiative Transfer

2010 ◽  
Vol 132 (7) ◽  
Author(s):  
D. A. Nield ◽  
A. V. Kuznetsov
Keyword(s):  
Rayleigh Number ◽  
Porous Material ◽  
Effective Conductivity ◽  
Critical Rayleigh Number ◽  
Horizontal Layer ◽  
Wave Number ◽  
Effect Of Temperature ◽  
Onset Of Convection ◽  
Cellular Foams ◽  
Variation Parameter

The onset of convection in a horizontal layer of a cellular porous material heated from below is investigated. The problem is formulated as a combined conductive-convective-radiative problem in which radiative heat transfer is treated as a diffusion process. The problem is relevant to cellular foams formed from plastics, ceramics, and metals. It is shown that the variation of conductivity with temperature above that of the cold boundary leads to an increase in the critical Rayleigh number (based on the conductivity of the fluid at that boundary temperature) and an increase in the critical wave number. On the other hand, the critical Rayleigh number based on the conductivity at the mean temperature decreases with increase in the thermal variation parameter if the radiative contribution to the effective conductivity is sufficiently large compared with the nonradiative component.

scholarly journals Finite amplitude thermal convection with spatially modulated boundary temperatures

1995 ◽  
Vol 449 (1937) ◽  
pp. 459-478 ◽  
Keyword(s):  
Stability Analysis ◽  
Thermal Convection ◽  
Fluid Layer ◽  
Perturbation Technique ◽  
Lower Boundary ◽  
Finite Amplitude ◽  
Resonant Wavelength ◽  
Two Dimensional ◽  
Dimensional Solution ◽  
One Dimensional

Finite amplitude thermal convection in a fluid layer between two horizontal walls with different fixed mean temperatures is considered when spatially modulated temperatures with amplitudes L 1 * and L u * are prescribed at the lower and upper walls, respectively. The nonlinear steady problem is solved by a perturbation technique, and the preferred mode of convection is determined by a stability analysis. In the case of a resonant wavelength excitation, regular or non-regular multi-modal pattern convection can be preferred for some ranges of L 1 * and L u *, provided the wave vectors for such patterns are contained in a certain subset of the wave vectors representing a linear combination of modulated upper and lower boundary temperatures. In the case of non-resonant wavelength excitation, a three (two) dimensional solution in the form of multi-modal (rolls) pattern convection can be preferred, even if the boundary modulations are one (two) or two (one) dimensional, provided the wavelengths of the modulations are not too small. Heat transported by convection can be enhanced by boundary modulations in some ranges of L 1 * and L u *.

scholarly journals Effects of modulation on Rayleigh-Benard convection. Part I

2004 ◽  
Vol 2004 (19) ◽  
pp. 991-1001 ◽  
Author(s):  
B. S. Bhadauria ◽  
Lokenath Debnath
Keyword(s):  
Rayleigh Number ◽  
Linear Stability ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Horizontal Layer ◽  
Time Dependent ◽  
Steady Temperature ◽  
Prandtl Numbers ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

The linear stability of a horizontal layer of fluid heated from below and above is considered. In addition to a steady temperature difference between the walls of the fluid layer, a time-dependent periodic perturbation is applied to the wall temperatures. Only infinitesimal disturbances are considered. Numerical results for the critical Rayleigh number are obtained at various Prandtl numbers and for various values of the frequency. Some comparisons have been made with the known results.

Neutral Instability and Optimum Convective Mode in a Fluid Layer with PCM Particles

2005 ◽  
Vol 127 (12) ◽  
pp. 1289-1295 ◽  
Author(s):  
Chuanshan Dai ◽  
Hideo Inaba
Keyword(s):  
Rayleigh Number ◽  
Fluid Layer ◽  
Critical Rayleigh Number ◽  
Gaussian Function ◽  
Critical Conditions ◽  
Onset Of Convection ◽  
Apparent Specific Heat ◽  
Change Material ◽  
Raphson Iteration ◽  
Convective Mode

Linear stability analysis is performed to determine the critical Rayleigh number for the onset of convection in a fluid layer with phase-change-material particles. Sine and Gaussian functions are used for describing the large variation of apparent specific heat in a narrow phase changing temperature range. The critical conditions are numerically obtained using the fourth order Runge-Kutta-Gill finite difference method with Newton-Raphson iteration. The critical eigenfunctions of temperature and velocity perturbations are obtained. The results show that the critical Rayleigh number decreases monotonically with the amplitude of Sine or Gaussian function. There is a minimum critical Rayleigh number while the phase angle is between π∕2 and π, which corresponds to the optimum experimental convective mode.

scholarly journals Thermosolutal Natural Convection in Partially Porous Domains

2012 ◽  
Vol 134 (3) ◽  
Author(s):  
Dominique Gobin ◽  
Benoît Goyeau
Keyword(s):  
Mass Transfer ◽  
Porous Medium ◽  
Natural Convection ◽  
Heat And Mass Transfer ◽  
Fluid Layer ◽  
Horizontal Layer ◽  
Clear Fluid ◽  
Onset Of Convection ◽  
The Stability ◽  
Double Diffusive

In many industrial processes or natural phenomena, coupled heat and mass transfer and fluid flow take place in configurations combining a clear fluid and a porous medium. Since the pioneering work by Beavers and Joseph (1967), the modeling of such systems has been a controversial issue, essentially due to the description of the interface between the fluid and the porous domains. The validity of the so-called one-domain approach—more intuitive and numerically simpler to implement—compared to a two-domain description where the interface is explicitly accounted for, is now clearly assessed. This paper reports recent developments and the current state of the art on this topic, concerning the numerical simulation of such flows as well as the stability studies. The continuity of the conservation equations between a fluid and a porous medium are examined and the conditions for a correct handling of the discontinuity of the macroscopic properties are analyzed. A particular class of problems dealing with thermal and double diffusive natural convection mechanisms in partially porous enclosures is presented, and it is shown that this configuration exhibits specific features in terms of the heat and mass transfer characteristics, depending on the properties of the porous domain. Concerning the stability analysis in a horizontal layer where a fluid layer lies on top of a porous medium, it is shown that the onset of convection is strongly influenced by the presence of the porous medium. The case of double diffusive convection is presented in detail.

Surface tension and buoyancy effects in cellular convection

1964 ◽  
Vol 19 (3) ◽  
pp. 341-352 ◽  
Author(s):  
D. A. Nield
Keyword(s):  
Surface Tension ◽  
Order Differential Equation ◽  
Boundary Surface ◽  
Lower Boundary ◽  
Horizontal Layer ◽  
Linear Perturbation ◽  
Perturbation Techniques ◽  
Fourier Series Method ◽  
Coupled Cells ◽  
Tightly Coupled

The cells observed by Bénard (1901) when a horizontal layer of fluid is heated from below were explained by Rayleigh (1916) in terms of buoyancy, and by Pearson (1958) in terms of surface tension. These rival theories are now combined. Linear perturbation techniques are used to derive a sixth-order differential equation subject to six boundary conditions. A Fourier series method has been used to obtain the eigenvalue equation for the case where the lower boundary surface is a rigid conductor and the upper free surface is subject to a general thermal condition. Numerical results are presented. It was found that the two agencies causing instability reinforce one another and are tightly coupled. Cells formed by surface tension are approximately the same size as those formed by buoyancy. Bénard's experiments are briefly discussed.

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