Forced convection heat transfer to an elastic fluid of constant viscosity flowing through a channel filled with a Brinkman-Darcy porous medium

An analysis is presented of fully developed flow and heat transfer in a channel confined by two parallel walls subjected to uniform heat flux in a highly porous medium saturated with an elastic fluid of constant viscosity. The Brinkman-extended Darcy model is used for studying the effect of the boundary viscous frictional drag on the heat transfer characteristics. The approximate integral method is employed to obtain a solution.

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Wärmeübergang bei Zwangskonvektion an ein elastisches Fluid konstanter Zähigkeit, das durch einen, von einem porösem Medium mit Brinkman-Darcy Charakteristik ausgefüllten Kanal fließt

Zusammenfassung Die Untersuchung bezieht sich auf die voll ausgebildete Strömung und den Wärmeübergang in einem von zwei parallelen Wänden begrenzten Kanal bei konstantem Wärmefluß am Rande. Der Kanalraum wird von einem hochporösem Medium ausgefüllt, das mit einem elastischen Fluid konstanter Zähigkeit getränkt ist. Der Einfluß des Reibungswiderstandes an den Kanalgrenzen auf das Wärmeübertragungsverhalten wird mit Hilfe des von Brinkman erweiterten Darcy-Transportgesetzes ermittelt. Die Lösung erfolgte unter Anwendung eines näherungsweisen gültigen Integralverfahrens.

     

Thermal nonequilibrium,non-darcian forced convection in a channel filled with a fluid saturated porous medium—A perturbation solution

This paper concentrates on the analysis of the thermal nonequilibrium effects during forced convection in a parallel-plate channel filled with a fluid saturated porous medium. The flow in a channel is described by the Brinkman-Forchheimer-extended Darcy equation and the thermal nonequilibrium effects are accounted for by utilizing the two energy equations model. Applying the perturbation technique, an analytical solution of the problem is obtained. It is established that the temperature difference between the fluid and solid phases for the steady fully developed flow is proportional to the ratio of the flow velocity to the mean velocity. This results in a local thermal equilibrium at the walls of the channel if the Brinkman term which allows for the no-slip boundary condition at the walls is included into the momentum equation.     

On forced convection in channels partially filled with porous substrates

 This paper numerically simulates the forced convection flow in the developing region of a parallel-plate channel partially filled with two porous substrates of equal thickness deposited at the inner walls of the channel. The major objective of the present work is to investigate the impact of several operating and design parameters on the thermal performance of the channel under consideration. The physical problem is simulated by using Darcy–Brinkman–Forchheimer model. For a prescribed amount of porous material, the current investigation discusses the comparison between inserting this entire amount at one side of the channel and inserting half of this amount at each side of the channel. Received on 25 May 2000 / Published online: 29 November 2001     

An approximate solution for free convection flows in a thermally stratified porous medium

A simple and sufficiently accurate method for finding the rate of heat transfer for buoyancy-induced flows over bodies of arbitrary shape embedded in a thermally stratified porous medium is given.     

Ice-water convection in an inclined rectangular cavity filled with a porous medium

This paper reports on the results of a numerical study on the equilibrium state of the convection of water in the presence of ice in an inclined rectangular cavity filled with a porous medium. One side of the cavity is maintained at a temperature higher than the fusion temperature while the opposite side is cooled to a temperature lower than the fusion temperature. The two remaining sides are insulated. Results are analysed in terms of the density inversion parameter, the tilt angle, and the cooling temperature. It appears that the phenomenon of density inversion plays an important role in the equilibrium of an ice-water system when the heating temperature is below 20°. In a vertical cavity, the density inversion causes the formation of two counterrotating vortices leading to a water volume which is wider at the bottom than at the top. When the cavity is inclined, there exist two branches of solutions which exhibit the bottom heating and the side heating characteristics, respectively (the Bénard and side heating branches). Due to the inversion of density, the solution on the Bénard branch may fail to converge to a steady state at small tilt angles and exhibits an oscillating behavior. On the side heating branch, a maximum heat transfer rate is obtained at a tilt angle of about 70° but the water volume was found to depend very weakly on the inclination of the cavity. Under the effect of subcooling, the interplay between conduction in the solid phase and convection in the liquid leads to an equilibrium ice-water interface which is most distorted at some intermediate cooling temperature.     

Effects of viscous dissipation and boundary conditions on forced convection in a channel occupied by a saturated porous medium

Forced convection with viscous dissipation in a parallel plate channel filled by a saturated porous medium is investigated numerically. Three different viscous dissipation models are examined. Two different sets of wall conditions are considered: isothermal and isoflux. Analytical expressions are also presented for the asymptotic temperature profile and the asymptotic Nusselt number. With isothermal walls, the Brinkman number significantly influences the developing Nusselt number but not the asymptotic one. At constant wall heat flux, both the developing and the asymptotic Nusselt numbers are affected by the value of the Brinkman number. The Nusselt number is sensitive to the porous medium shape factor under all conditions considered.     

Entropy generation analysis of thermally developing forced convection in fluid-saturated porous medium

Entropy generation for thermally developing forced convection in a porous medium bounded by two isothermal parallel plates is investigated analytically on the basis of the Darcy flow model where the viscous dissipation effects had also been taken into account. A parametric study showed that decreasing the group parameter and the Peclet number increases the entropy generation while for the Brinkman number the converse is true. Heatline visualization technique is applied with an emphasis on the Br 〈 0 case where there is somewhere that heat transfer changes direction at some streamwise location to the wall instead of its original direction, i.e., from the wall.     

Conjugate natural convection between horizontal concentric cylinders filled with a porous medium

A conjugate problem of natural convection between two horizontal concentric, isothermal cylinders filled with a fluid-saturated porous medium is studied. The flow field and energy equations are solved under the conditions of equality in temperature but with jump in heat flux at the fluid-solid interface. Numerical results by a finite-difference technique are presented for a large variation in the parameters entering the problem. A comparison of the solution with non-conjugate problem is also given.Das konjugierte Problem der freien Konvektion zwischen zwei horizontalen, konzentrischen, isothermen Zylindern, die mit einem fluid-gesättigten porösen Medium gefüllt sind, wurde hier untersucht.Das Strömungsfeld und die Energiegleichung sind unter den Bedingungen gelöst worden, daß die Temperatur gleich ist, aber ein Wärmeflußsprung am Übergang von der Flüssigkeits- zur Festphase stattfindet. Die numerischen Ergebnisse werden mit dem Finite-Differenzen-Verfahren für viele Parametervariationen dargestellt. Zudem wurde ein Lösungsvergleich mit einem nicht konjugierten Problem gegeben.     

An analysis of species separation in a thermogravitational column filled with a porous medium

In the past, the analysis of species separation in a thermogravitational column filled with porous media has been based on strong dependency of thermal and molecular diffusion to dispersion. In this work, we suggest an alternative and show that the dispersion effect is negligible for the conditions in a packed hermogravitational column and that compositional dependency of the thermal diffusion should be accounted for.     

Radiation effects on mixed convection about a cone embedded in a porous medium filled with a nanofluid

The problem of steady, laminar, mixed convection boundary-layer flow over a vertical cone embedded in a porous medium saturated with a nanofluid is studied, in the presence of thermal radiation. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis with Rosseland diffusion approximation. The cone surface is maintained at a constant temperature and a constant nanoparticle volume fraction. The resulting governing equations are non-dimensionalized and transformed into a non-similar form and then solved by Keller box method. A comparison is made with the available results in the literature, and our results are in very good agreement with the known results. A parametric study of the physical parameters is made and a representative set of numerical results for the local Nusselt and Sherwood numbers are presented graphically. Also, the salient features of the results are analyzed and discussed.     

Natural convection flow with solidification between two vertical plates filled with a porous medium

Quasi-steady solidification between two vertical flat plates filled with a saturated porous medium has been investigated. The medium is homogeneous and isotropic. The convection flow of liquid takes place in the porous medium in the variable space between the two walls. One of the vertical walls is set to a temperature lower than the solidification temperature of the medium and therefore a frozen crust is formed on this wall. The second wall has a high temperature then the fusion temperature of the medium. The problem has been simplified by assuming laminar flow and the Brinkman and the Oberbeck–Bousinesq’s approximations. The results are presented in terms of the velocity for different properties of the porous medium. Various velocities are displayed in dependence of the Rayleigh and Darcy numbers. The study indicates that asymmetric boundary conditions have an important effect on the temperature and flow field. In addition, the growth of the thickness of the frozen layer with time has been derived from a simple analytical solution of the interface energy equation.     

Influence of inclination angle on buoyancy-driven convection in triangular enclosure filled with a fluid-saturated porous medium

The present investigation deals with the numerical analysis of steady-state laminar buoyancy-driven convection in an inclined triangular enclosure filled with fluid saturated porous media using the Darcy law equation. One wall of the enclosure is isothermally heated and the other is cooled, while the remaining wall is adiabatic. The effect of inclination angle on natural convection is investigated by varying the angle of inclination (φ) between 0° and 360°. The governing transformed equations are solved numerically using a finite-difference method. Obtained results are shown in the form of streamlines, isotherms, mean Nusselt numbers and dimensionless stream function for different values of the Rayleigh number Ra in the range 100 ≤ Ra ≤ 1,000. It is found that the values of the maximum and minimum mean Nusselt number are reached for φ = 330° and φ = 210° , respectively. However, the lowest flow strength is formed at φ = 240° for all values of Ra.     

Non-Darcian effects on conjugate natural convection between horizontal concentric cylinders filled with a porous medium

An analysis is presented of steady conjugate free convection between two horizontal concentric cylinders filled with a fluid-saturated porous medium; the innermost cylinder surface is maintained at a high temperature and the outermost cylinder surface at a lower one. The velocity-pressure-gradient relation is taken to be nonlinear, with departure from the linear Darcy situation measured by a parameter F₀. The investigation is based on the numerical solution, by a finite-difference method, of the full momentum and energy equations. The streamline and isotherm patterns as well as the local and mean Nusselt numbers are plotted for several physical parameters to show some of the flow and heat transfer characteristics. It is found that all parameters play an important role in the flow and heat transfer characteristics. The model can be applied to a variety of engineering problems.     

Forced convection from an isolated heat source in a channel with porous medium

A numerical study is made of flow and heat transfer characteristics of forced convection in a channel that is partially filled with a porous medium. The flow geometry models convective cooling process in a printed circuit board system with a porous insert.The channel walls are assumed to be adiabatic. Comprehensive numerical solutions are acquired to the governing Navier-Stokes equations, using the Brinkman-Forchheimer-extended Darcy model for the regions of porous media. Details of flow and thermal fields are examined over ranges of the principal parameters; i.e., the Reynolds number Re, the Darcy number Da (≡K/H²), the thickness of the porous substrate S, and the ratio of thermal conductivities R_k (≡k_eff/k). Two types of the location of the porous block are considered. The maximum temperature at the heat source and the associated pressure drop are presented for varying Re, Da, S, and R_k. The results illustrate that as S increases or Da decreases, the fluid flow rate increases. Also, as R_k increases for fixed Da, heat transfer rates are augmented. Explicit influences of Re on the flow and heat transport characteristics are also scrutinized. Assessment is made of the utility of using a porous insert by cross comparing the gain in heat transport against the increase in pressure drop.     

Combined free and forced convection in a porous medium between two vertical walls with viscous dissipation

Viscous dissipation effects in the problem of a fully-developed combined free and forced convection flow between two symmetrically and asymmetrically heated vertical parallel walls filled with a porous medium is analyzed. The equation of motion contains the modified Rayleigh number for a porous medium and the small-order viscous dissipation parameter. Particular attention is given to the solutions near the critical Rayleigh numbers at which infinite flow rates are predicted. Information concerning the multiplicity of solutions at critical Rayleigh numbers is also deduced from perturbation solutions of the governing equation.     

Exact analytic solutions for free convection boundary layers on a heated vertical plate with lateral mass flux embedded in a saturated porous medium

 The problem of the self-similar boundary flow of a “Darcy-Boussinesq fluid” on a vertical plate with temperature distribution T _w(x) = T _∞+A·x ^λ and lateral mass flux v _w(x) = a·x ^(λ−1)/2, embedded in a saturated porous medium is revisited. For the parameter values λ = 1,−1/3 and −1/2 exact analytic solutions are written down and the characteristics of the corresponding boundary layers are discussed as functions of the suction/ injection parameter in detail. The results are compared with the numerical findings of previous authors. Received on 8 March 1999     

TEMPERATURE PROFILES OF LOCAL THERMAL NONEQUILIBRIUM FOR THERMAL DEVELOPING FORCED CONVECTION IN POROUS MEDIUM PARALLEL PLATE CHANNEL

Based on the two-energy equation model, taking into account viscous dissipation due to the interaction between solid skeleton and pore fluid flow, temperature expressions of the solid skeleton and pore fluid flow are obtained analytically for the thermally developing forced convection in a saturated porous medium parallel plate channel, with walls being at constant temperature. It is proved that the temperatures of the two phases for the local thermal nonequilibrium approach to the temperature derived from the one-energy equation model for the local thermal equilibrium when the heat exchange coefficient goes to infinite. The temperature profiles are shown in figures for different dimensionless parameters and the effects of the parameters on the local thermal nonequilibrium are revealed by parameter study.