Penetrative convection in fluid layers with internal heat sources

Summary The stability problem for penetrative convection in a fluid layer that is heated internally is analyzed using the methods of linear instability theory and unconditional nonlinear energy theory. Critical Rayleigh numbers in the case of a constant heat source are determined numerically from both theories. Although it is not known whether exchange of stabilities holds for this problem, a comparison between the linear and nonlinear results suggests that for top temperatures close to 4°C stationary convection is predominant when the heat source is not too large. The nonlinear results delimit a band of Rayleigh numbers where possible subcritical instabilities could arise.     

Conjugate natural convection in enclosures with external and internal heat sources

Several special issues should be paid attentions on conjugate heat transfer in enclosures with external and inner heat sources, including velocity scale, diffusion coefficient on fluid–solid interface, temperature scale for independent multiple heat sources, coupling of velocity and pressure for solid body immersed in fluid. That boosts the present study to analyze and develop convenient practices of solving the conjugate heat transfer firstly, and then applies it to examine the conjugate natural convection and conduction in enclosures with external and inner heat sources. Results are produced in terms of the isotherms, streamlines and Nusselt numbers with wide ranges of external and internal Rayleigh numbers under different thermal boundary conditions. The analysis mainly reveals that the practices of solving the conjugate heat transfer perform well, and the sole temperature scale is flexible than combined temperature scale for arbitrary strengths of external and internal heat sources.     

Onset of bouyancy-driven convection in superposed reacting fluid and porous layers

A linear stability analysis is applied to the onset of bouyancy-driven convection in a horizontal layer of reacting fluid overlying a porous region saturated with the same fluid. The fluid is assumed to be undergoing zero-order exothermic reactions in both regions. At the interface between fluid and porous layers the boundary conditions proposed by Nield [1] are employed; these include the empirical slip condition suggested by Beavers and Joseph [2]. Predictions for the onset of convection and critical wavenumbers are obtained from the analysis by the collocation method and solution of the resulting generalized eigenvalue problem. The effect of variable fluid/porous layer depth ratio, Frank–Kamenetskii number or thermal boundary conditions on the onset of fluid motions is studied, and ignition values of the Frank–Kamenetskii number for the system are calculated.     

Anisotropy effects on non–Darcy natural convection from concentrated heat sources in porous media

Summary In this investigation, closed-form exact similarity solutions are obtained for non-Darcian natural convection arising above a line heat source or a point source embedded in an unbounded porous medium saturated by an incompressible fluid. The porous medium is assumed to be hydrodynamically anisotropic with its principal axes oriented in a direction that is oblique to the gravity vector. In the formulation of the problem, use is made of the Forchheimer model to take into account the inertia effects which are expected to be significant for high permeabilities and Rayleigh numbers. Scale analysis is applied to predict the orders of magnitudes involved in the boundary-layer regime for which the conditions of validity are presented. Effects of both anisotropy parameters (K* and θ) and Forchheimer number Fs are observed to be strongly significant. It is demonstrated that a minimum (maximum) intensity of the thermal convective plume above concentrated heat sources can be obtained if the porous matrix is oriented with its principal axis with higher permeability parallel (perpendicular) to the vertical direction. Moreover, it is shown that for a fixed Rayleigh number an increase of the inertia effects tends to reduce the convective flow induced upward.     

Natural convection heat transfer of a viscous fluid in a vertical porous channel

Approximate analytic solutions to second-order nonlinear systems arising in natural convection flow and heat transfer in vertical porous channels are obtained via the Galerkin–Legendre Spectral Method. Furthermore, existence, uniqueness, and concavity results are established using Green’s functions and degree theory. We find that an increase in either the Darcy number or the quadratic density temperature variation results in an increase in the velocity and the temperature of a Newtonian fluid. Finally, parametric zones for the occurrence of reverse flow are considered, and the resulting influences on the obtained approximate solutions are analyzed.     

Thermal convection in a fluid layer sandwiched between two porous layers of different permeabilities

Summary A linear stability analysis is carried out for the convective instability problem in a horizontal fluid layer sandwiched between two porous layers of different permeabilities. The velocity boundary condition is a general one and is via-media between the free and rigid boundary conditions. The thermal condition at the porous-fluid interface is assumed to be neither constant heat flux nor constant temperature, but a condition leading to a third-type of boundary condition. The principle of exchange of stability is valid for the problem and the critical eigenvalue is obtained for the general boundary condition using the single-term Rayleigh-Ritz technique. The results of several works are recovered as limiting cases of the present study.     

Influence of a magnetic field on natural convection in a shallow porous enclosure saturated with a binary fluid

Summary An analytical and numerical investigation is conducted to study the effect of an electromagnetic field on natural convection in a horizontal shallow porous cavity filled with an electrically conducting binary mixture. A uniform heat flux is applied to the horizontal walls of the layer while the vertical walls are adiabatic. The solutal buoyancy forces are assumed to be induced either by the imposition of constant fluxes of mass on the horizontal boundaries (double diffusive convection) or by temperature gradients (Soret induced convection). Governing parameters of the problem under study are the thermal Rayleigh number, R _T , Hartmann number, Ha, Lewis number, Le, buoyancy ratio, ϕ, aspect ratio, A, and normalized porosity of the porous medium, ɛ. An analytical solution, valid for a shallow layer (A ≫ 1) is derived on the basis of the parallel flow approximation. In the range of the governing parameters considered in this study, a good agreement is found between the analytical predictions and the numerical results obtained by solving the full governing equations.     

Effects of throughflow and internal heat generation on the onset of convection in a fluid layer

Summary The throughflow and internal heat generation effects on the onset of convection in an infinite horizontal fluid layer are investigated. The boundaries are considered to be rigid (however permeable) and perfectly conducting. The resulting eigenvalue problem is solved by using the Galerkin method, and the effects of various parameters in the stability results are analyzed. The results indicate that the stability of the system is significantly affected by both throughflow and internal heat generation in the fluid layer. The Prandtl number comes into play due to the presence of throughflow and it has a profound effect on the stability of the system. It is found that, in the presence of internal heating, throughflow in one direction supresses convection while throughflow in the other direction encourages it.     

Optimum relationships in a plane multilayer wall with internal heat release

We consider the possibility of reducing the heat flows through a flat multilayer wall with internal heat release.     

Effect of variable viscosity on free convection over a non-isothermal axisymmetric body in a porous medium with internal heat generation

Summary. A similarity solution is proposed for the analysis of the steady free convection boundary layers over a non-isothermal axisymmetric body embedded in a fluid saturated porous medium. The effect of temperature dependent viscosity on heat transfer rates in the presence of internal heat generation is investigated. The linearized version of the Arrhenius law for the temperature dependent viscosity is considered. It is shown that the heat transferred is more for a less viscous fluid.     

Natural convection of the earth liquid core in the presence of internal heat sources

Results of natural-convection heat transfer in the earth liquid core in the presence of internal sources are presented. Consideration is given to boundary conditions of the I, II, and III kind for the temperature. The effect of internal heat sources on the hydrodynamics and heat transfer of the core is studied.     

Boundary element method for natural convection in non-Newtonian fluid saturated square porous cavity

The main purpose of this work is to present the use of the Boundary Element Method (BEM) in the analysis of the natural convection in the square porous cavity saturated by the non-Newtonian fluid. The results of hydrodynamic and heat transfer evaluations are reported for the configuration in which the enclosure is heated from a side wall while the horizontal walls are insulated. The flow in the porous medium is modelled using the modified Brinkman extended Darcy model taking into account the non-Darcy viscous effects. The governing equations are transformed by the velocity–vorticity variables formulation enabling the computation scheme to be partitioned into kinematic and kinetic parts. To analyse the effects of the available non-Newtonian viscosity and to evaluate the presented approach, the power law model for shear thinning fluids (n<1), for shear thickening fluids (n>1) and in the limit for the Newtonian fluids (n=1) is considered. Numerical model is tested also for the Carreau model adequate for many non-Newtonian fluids. Solutions for the flow and temperature fields and Nusselt numbers are obtained in terms of a modified Rayleigh number Ra^*, Darcy number Da, and the non-Newtonian model parameters. The agreement between the results obtained with finite difference method is very good indicating that BEM can be efficiently used for solving transport phenomena in saturated porous medium.     

Convection in superposed fluid and porous layers

This paper reports an analytical study of the stability and natural convection in a system consisting of a horizontal fluid layer over a layer of saturated porous medium. Neumann thermal boundary conditions are applied to the horizontal walls of the enclosure while the vertical walls are impermeable and adiabatic. At the interface between the fluid and the porous layers the empirical slip condition, suggested by Beavers and Joseph, is employed. An analytical solution is obtained using a parallel flow approximation, for constant-flux thermal boundary conditions, for which the onset of supercritical cellular convection occurs at a vanishingly small wavenumber and can thus be predicted by the present theory. The critical Rayleigh number, Ra _c , and Nusselt number, Nu, are found to depend on the depth ratio, η, the Darcy number, Da, the thermal conductivity ratio, γ and the slip parameter α. Results are presented for a wide range of each of the governing parameters. The results are compared with limiting cases of the problem and are found to be in agreement.     

Nonlinear double-diffusive convection with local heat and solute sources

Effects of local heat and solute sources on nonlinear double diffusive convection has been investigated. These sources introduce three new parameters. Dependence of the horizontal cells' size, critical solute Rayleigh number, and heat and solute fluxes on these new parameters is determined. In particular, it is found that local sources can significantly reduce the horizontal cells' size. For the case where the strength of the local solute source is insignificant, the preferred horizontal flow cross-section pattern is found to be squares. On the other hand, if the strength of the local solute source is significant, then hexagons become the preferred pattern when flow amplitude ϵ exceeds a critical value. However, both squares and hexagons can be stable for an even higher value of ϵ. The convective motion is upward at the center of the hexagonal cells if a solute sink acts on the flow. When a solute source acts on the flow, downward motion at the cells' center will occur. Furthermore, the local solute source is also found to be able to provide finite amplitude instability.     

Nonstationary regime of heat exchangers with internal heat sources

Analytic relations are obtained for determining the stability of thermoelectric devices and heat exchangers with heat sources whose strength is a linear function of temperature.     

Unconditional nonlinear stability for convection in a porous medium with vertical throughflow

Summary Linear and nonlinear stability analyses of vertical throughflow in a fluid saturated porous layer, which is modelled using a cubic Forchheimer model, are studied. To ensure unconditional nonlinear results are obtainable, and to avoid the loss of key terms, a weighted functional is used in the energy analysis. The linear instability and nonlinear stability thresholds show considerable agreement when the vertical throughflow is small, although there is substantial deterioration of this agreement as the vertical throughflow increases.     

Thermal convection in sections of multilayer cryogenic heat insulation

An experimental investigation has been made of heat transfer during natural convection in multilayer heat insulating structures consisting of anisotropic permeable porous materials.Translated from Inzhenerno-fizicheskii Zhurnal, Vol. 61, No. 2, pp. 239–243, August, 1991.     

Investigation of natural convection in cylindrical liquid layers

An experimental study has been made of heat transfer in cylindrical gaps under natural convection, using water and 96% ethanol. A comparison has been made of the convection coefficient in horizontal and vertical tubes, and the conditions for generating convection have been defined more accurately.     

Heating of a cylindrical body with internal heat sources and a variable heat transfer coefficient

An approximate method is described for solution of the problem of the temperature distribution In an infinite cylinder when the coefficient of external heat transfer is a function of time. A constant power heat source acts in the body during the entire period of heating.