Buoyancy-induced flow adjacent to a periodically heated and cooled horizontal surface in porous media (original) (raw)
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On natural convection from a vertical plate with a prescribed surface heat flux in porous media
Transport in Porous Media, 1996
This paper presents a theoretical and numerical investigation of the natural convection boundary-layer along a vertical surface, which is embedded in a porous medium, when the surface heat flux varies as (1 + x 2) ~, where # is a constant and x is the distance along the surface. It is shown that for/z >-89 the solution develops from a similarity solution which is valid for small values of x 1 to one which is valid for large values of x. However, when # ~-~ no similarity solutions exist for-1 large values of x and it is found that there are two cases to consider, namely # <-89 and #-~. The wall temperature and the velocity at large distances along the plate are determined for a range of values of/z.
International Journal of Heat and Mass Transfer, 1996
In this paper we investigate the steady, two-dimensional, free convection flow caused by a sinusoidally heated and cooled infinite vertical surface that delimits a semi-infinite porous media. An analytical solution which is valid for small values of the Rayleigh number, Ra, is obtained using a regular perturbation method. A finite-difference technique is used to numerically solve the problem for 0 <~ Ra <~ 150 and for small values of Ra, the results are in very good agreement with the analytical solutions and the streamlines are in the form of a row of counter rotating cells which are situated close to the vertical surface. As the Rayleigh number increases, above a value of about 40, then the cellular flow separates from the plate. At very large values of Ra, a scaling analysis has been performed and the results suggest that the vertical velocity and the local Nusselt number on the plate support better the boundarylayer scalings, than does the mean vertical velocity and the mean Nusselt number along the plate. In the situation in which the flow separates, i.e. for Ra >~ 40, the smallest possible solution domain must be chosen, by using the symmetry of the problem, otherwise it has not been possible to obtain a convergent numerical solution.
Fluid Dynamics Research, 1996
An analysis of the transient, buoyancy-induced flow and heat transfer adjacent to a suddenly heated vertical wall, embedded in a porous medium saturated with a non-Newtonian fluid, is presented. It is shown that the governing equations, under boundary layer assumptions, are of a singular parabolic type and can be solved accurately in a semi-similar, finite domain using a successive relaxation method. The results show that during the initial stage, before effects of the leading edge become influential at a location, heat transfer and flow phenomena in porous media are governed by transient one-dimensional diffusion processes, for both pseudoplastic and dilatant fluids. Results are presented for the transition from this initial stage to a fully two-dimensional transient, which ultimately terminates in a steady convection. Non-Newtonian fluids which are pseudoplastic exhibit a significantly larger change in the heat transfer coefficient during the transition between the initial diffusive and final steady flow conditions, and unlike the free convection in homogeneous media neither dilatants nor pseudoplastics exhibit any undershoot in the heat transfer coefficient. Furthermore, it is shown that the time required to reach steady state increases and the heat transfer coefficient decreases with a decrease in the power law index.
International journal of thermal sciences, 2007
The effect of suction or injection on the free convection boundary layers induced by a heated vertical plate embedded in a saturated porous medium with an exponential decaying heat generation is studied. Similarity solutions are obtained for the governing steady laminar boundary layer equations using Darcy and Boussinesq approximations. The plate is assumed to have a power law temperature distribution. Three distinct cases of uniform lateral mass flux, uniform surface temperature, and uniform heat flux are studied. The effects of suction/injection parameter f w and temperature exponent λ on the flow of heat transfer are studied. Some exact analytical results are obtained for λ = 1, −1/3. Critical values of the suction/injection parameter are obtained for adiabatic surface as a function of the temperature exponent parameter λ.
International Communications in Heat and Mass Transfer, 1997
The unsteady heat transfer process involved in free convection flow along a vertical surface embedded in a porous medium is investigated. An analytical solution has been obtained for the temperature/velocity field for small times in which the transport effects are confined within an inner layer adjacent to the plate. Then, a numerical solution of the full boundary-layer equation is obtained for the whole transient from the initial unsteady state to the final steady state. Detailed results of the effect of the temperature inputs on the transient process are given.
Periodic free convection from a vertical plate in a saturated porous medium, non-equilibrium model
International Journal of Heat and Mass Transfer, 2005
The problem of the free convection from a vertical heated plate in a porous medium is investigated numerically in the present paper. The effect of the sinusoidal plate temperature oscillation on the free convection from the plate is studied using the non-equilibrium model, i.e., porous solid matrix and saturated fluid are not necessary to be at same temperature locally. Non-dimensionalization of the two-dimensional transient laminar boundary layer equations results in three parameters: (1) H, heat transfer coefficient parameter, (2) K r , thermal conductivity ratio parameter, and (3) k, thermal diffusivity ratio. Two additional parameters arise from the plate temperature oscillation condition which are the non-dimensional amplitude (e) and frequency (X). The fully implicit finite difference method is used to solve the system of equations. The numerical results are presented for 0 6 H 6 10, 0 6 K r 6 10, 0.001 6 k 6 10 with the plate temperature oscillation parameters 0 6 X 6 10 and 0 6 e 6 0.5. The results show that the thermal conductivity ratio parameter is the most important parameter. It is found also that increasing the amplitude and the frequency of the oscillating surface temperature will decrease the free convection heat transfer from the plate for any values of the other parameters.
Transport in porous media, 2002
In this paper we analyse how the presence of the thermal capacity of a vertical flat plate of finite thickness, which is embedded in a porous medium affects the transient free convection boundary-layer flow. At the timet = 0, the plate is suddenly loaded internally with a constant heat flux rate q , so that a transient boundary-layer flow is initiated adjacent to the plate. Initially, the transient effects due to the imposition of the uniform heat flux rate at the plate are confined to a thin fluid region near to the surface and are described by a small time solution. These effects continue to penetrate outwards and eventually evolve into a new steady state flow. Analytical solutions have been derived for these transient (small time) and steady state (large time) flow regimes, which are then matched by a numerical solution of the full boundary-layer equations. It has been found that the non-dimensional fluid temperature (or fluid velocity) profiles are reduced when the thermal capacity effects, described by a parameter Q * , are reduced. For small values of Q * , the approach of these profiles to their steady state values is monotonic. However, for large values of Q * , the temperature profiles are observed to locally exceed (pass through a maximum value) the final steady state values at certain distances from the plate. In general, the maxima in the temperature profiles increase in size as Q * increases and the time taken to approach the steady state solutions increases significantly.
Natural convection flow from a porous vertical plate in the presence of heat generation
2010
Abstract: Natural convection flow from a porous vertical plate in presence of heat generation have been presented here. The governing boundary layer equations are first transformed into a non dimensional form and the resulting non linear system of partial differential equations are then solved numerically using finite difference method together with Keller-Box scheme. The numerical results of the surface shear stress in terms of skin friction coefficient and the rate of heat transfer in terms of local Nusselt number, velocity as well as temperature profiles are shown graphically and tabular form for a selection of parameters set of consisting of heat generation parameter Q, Prandtl number Pr.
International Journal of Heat and Mass Transfer, 1985
The Brinkman model is used for the theoretical study of boundary effects in a natural convection porous layer adjacent to a semi-infinite vertica1 plate with a power law variation of wall temperature, i.e. =&xX". It is shown that the dimensionless governing equations based on this model contain two parameters c = (Ra)-'I2 and e = (~u/~)"* where Ra and Dn are the Rayleigh number and the Darcy number based on a reference length and 4 is the porosity. For the limit of &-+ 0, (r + 0 and (F << 6, a perturbation solulion for the problem is obtained based on the method ofmatched asymptotic expansions. The physical problem can then be visualized to be consisting of three layers : an inner viscous sublayer, adjacent to the heated surface, with a thickness of the order of O(u) ; a middle thermal layer of thickness of the order of O(e) ; and the outer potential flow region with thickness of the order of O(l). It is found that the first-order problem of the thermal layer is identical to that based on Darcy's law with slip flow, whose solution was obtained previously. Composite solutions for stream function and temperature, uniformly valid everywhere in the flow field, are constructed from the solutions for the thermal layer and the viscous sublayer. A new parameter Pn,, defined as Pn, = (Ra,f?ax)1~2 with Ru, and Da, denoting the local Rayleigh number and the local Darcy number based on X, is found to be a measure of the boundary effect. It is shown that the viscous effect on the boundary has a drastic effect on the streamwise velocity component near the wall with a lesser effect on heat transfer characteristics. The boundary effect slows down the buoyancy-induced flow with a resulting decrease in heat transfer. The local Nusselt number is found to be of the form NuJ(Ra,) "' = C,-C,Pn, where the values of C, and CZ depending on 1. For an isothermal vertical plate(il = 0), the first-order correction to thelocal Nusselt number is identically zero, i.e. C, = 0. In general, the boundary effect on the local Nusselt number becomes more pronounced as the value of Pn,, Ra, or Da, is increased.