Non-Darcy free convection in a thermally stratified porous medium along a vertical plate with variable heat flux

Free convection along an impermeable vertical plate embedded in a thermally stratified, fluid-saturated porous medium is analyzed. The wall heat flux is varied in a power-law form. The non-Darcian effects, such as solid-boundary viscous resistances, high-flow-rate inertia forces, near wall nonuniform porosity distribution and thermal dispersion, have been considered in the present study. Due to the variation of porosity in the near wall region, the stagnant thermal conductivity also varies accordingly. The nonsimilar system of transformed equations is solved with Keller's Box method. It is shown that the thermal stratification effect and the higher value of the exponent m can increase the local Nusselt number. Also the non-Darcian and thermal dispersion effects significantly influence the velocity and temperature profiles and local Nusselt number.     

Unsteady natural convective flow past a moving vertical cylinder with heat and mass transfer

Transient natural convection boundary layer flow of an incompressible viscous fluid past an impulsively moving semi- infinite vertical cylinder is considered. The temperature and concentration of the cylinder surface are taken to be uniform. The unsteady, nonlinear and coupled governing equations of the flow are solved using an implicit finite difference scheme. The finite difference scheme is unconditionally stable and accurate. Numerical results are presented with various sets of parameters for both air and water. Transient effects of velocity, temperature and concentration profiles are analyzed. Local and average skin friction, rates of heat and mass transfer are shown graphically. Received on 1 November 1999     

Combined heat and mass transfer along a vertical moving cylinder with a free stream

The mixed convection flow over a continuous moving vertical slender cylinder under the combined buoyancy effect of thermal and mass diffusion has been studied. Both uniform wall temperature (concentration) and uniform heat (mass) flux cases are included in the analysis. The problem is formulated in such a manner that when the ratio λ(= u _w/(u _w + u _∞), where u _w and u _∞ are the wall and free stream velocities, is zero, the problem reduces to the flow over a stationary cylinder, and when λ = 1 it reduces to the flow over a moving cylinder in an ambient fluid. The partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme. We have also obtained the solution using a perturbation technique with Shanks transformation. This transformation has been used to increase the range of the validity of the solution. For some particular cases closed form solutions are obtained. The surface skin friction, heat transfer and mass transfer increase with the buoyancy forces. The buoyancy forces cause considerable overshoot in the velocity profiles. The Prandtl number and the Schmidt number strongly affect the surface heat transfer and the mass transfer, respectively. The surface skin friction decreases as the relative velocity between the surface and free stream decreases. Received on 17 May 1999     

A note on the free convection boundary layer on a horizontal circular cylinder with constant heat flux

The equations governing the free convection boundary-layer flow on a horizontal circular cylinder on which there is a prescribed surface heat flux are solved using a finite-difference scheme. This numerical solution is then used to compare the accuracy of two proposed series expansions, a Blasius expansion and a Görtler-type expansion. It is shown that the former method is better at estimating temperature profiles while the latter is better at estimating velocity profiles.

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Bemerkung über die freie Konvektionsgrenzschicht an einem horizontalen Kreiszylinder mit gleichförmigem Wärmestrom

Zusammenfassung Die Grenzschichtgleichungen für die freie Konvektion an einem horizontalen Kreiszylinder mit gleichförmigem Wärmestrom durch die Oberfläche wird mit Hilfe eines finiten Differenzverfahrens gelöst. Die numerisch ermittelten Ergebnisse werden nachher für den Vergleich der Genauigkeit von zwei Reihendarstellungen der Lösung der Grenzschichtsgleichungen benützt. Diese Reihen sind vom Blasiusbzw. Görtier-Typ. Es wird bemerkt, daß die Reihendarstellung von Blasius die Temperaturprofile besser beschreibt, während die Reihenentwicklung der Görtlerschen Art für die Geschwindigkeitsprofile eine gute Übereinstimmung mit der exakten Lösung zeigt.

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Nomenclature a radius of the cylinder - g acceleration of gravity - P _r Prandtl number - Q prescribed (constant) heat flux - T temperature of the fluid - t ₀ temperature of the ambient fluid - u velocity in thex-direction - v velocity in they-direction - x co-ordinate measuring distance round the cylinder - y co-ordinate measuring distance normal to the cylinder - G _r Grashof number=g Q a ⁴/v² - coefficient of thermal expansion - x thermal conductivity - v kinematic viscosity - _w skin friction     

Mixed convection boundary layer flow along vertical moving thin needles with variable heat flux

The problem of steady laminar mixed convection boundary layer flow of an incompressible viscous fluid along vertical moving thin needles with variable heat flux for both assisting and opposing flow cases is theoretically considered in this paper. The governing boundary layer equations are first transformed into non-dimensional forms. The curvature effects are incorporated into the analysis whereas the pressure variation in the axial direction has been neglected. These equations are then transformed into similarity equations using the similarity variables, which are solved numerically using an implicit finite-difference scheme known as the Keller-box method. The solutions are obtained for a blunt-nosed needle (m = 0). Numerical calculations are carried out for various values of the dimensionless parameters of the problem, which include the mixed convection parameter λ, the Prandtl number Pr and the parameter a representing the needle size. It is shown from the numerical results that the skin friction coefficient, the surface (wall) temperature and the velocity and temperature profiles are significantly influenced by these parameters. The results are presented in graphical form and are discussed in detail.     

Combined heat and mass transfer by laminar natural convection from a vertical plate with uniform heat flux and concentration

This paper studies combined heat and mass transfer by laminar natural convection from a vertical plate maintained with uniform surface heat flux and species concentration. Very accurate finite-difference solutions of a set of nonsimilarity equations have been obtained for most practical gaseous solutions (Pr?=?0.7, 0.21 ≤ Sc ≤ 2.1) and aqueous solutions (Pr?=?7, 140 ≤ Sc ≤ 1400). Variations of heat and mass transfer rates with buoyancy ratio and Lewis number are presented. Precise correlations have been developed for predicting heat and mass transfer rates of natural convection arising from single (solutal or thermal) buoyancy force and dual buoyancy forces.     

Non-Darcy free convective flow along a vertical cylinder embedded in a porous medium with surface mass flux

The nonsimilar non-Darcian free-convection flow about a vertical cylinder with impermeable surface embedded in a saturated porous medium, where surface temperature of the cylinder varies as x^m, a power function of distance from the leading edge, has been studied by employing the implicit finite-difference method together with the Newton's quasilinearization technique. In the present investigation, effects of the surface mass flux together with the inertial effects on the rate of heat transfer at the surface, on the velocity distribution, and on the temperature distribution are shown graphically.     

Method of singular perturbations in the problem of free convection with constant heat flux on a vertical surface

The process of laminar free convection for medium values of the Grashof number is examined. An asymptotic solution which accounts for the effect of the leading edge and interaction of the boundary layer with the external flow is constructed, A comparison with experimental data shows that the solution obtained is applicable for Ra 10².Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 129–136, March–April, 1977.     

Mixed convection in non-Newtonian fluids along a vertical plate with variable surface heat flux in a porous medium

A nonsimilar boundary layer analysis is presented for the problem of mixed convection in power-law type non-Newtonian fluids along a vertical plate with power-law surface heat flux distribution. The mixed convection regime is divided into two regions, namely, the forced convection dominated regime and the free convection dominated regime. The two solutions are matched. Numerical results are presented for the details of the velocity and temperature fields. A discussion is provided for the effect of viscosity index on the surface heat transfer rate. Received on 13 October 1998     

Effects of MHD free convection and mass transfer on the flow past a vibrating infinite vertical circular cylinder

The effects of MHD free convection and mass transfer are taken into account on the flow past a vibrating infinite isothermal and constant heat flux vertical circular cylinder. The expressions for velocity, temperature, concentration and skin-friction of the fluid are obtained in closed form by using Laplace transform technique. The effects ofPr (Prandtl number),Sc (Schmidt number),Gr (Grashof number,Gr>0 implies cooling andGr<0 heating of the cylinder),Gm (modified Grashof number),M (magnetic field parameter) and variation of time on velocity distribution have been studied graphically. The results presented in this paper agree with the results of Lien and Chen when magnetic parameter approaches zero.

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Effekte der freien MHD Konvektion und der Stoffübertragung auf die Strömung längs eines vibrierenden unendlich langen vertikalen Kreiszylinders

Zusammenfassung Es werden die Effekte der freien MHD Konvektion und der Stoffübertragung auf die Strömung längs eines vibrierenden unendlich langen vertikalen isothermen Kreiszylinders mit konstanter Wärmestromdichte untersucht. Es werden geschlossene Ausdrücke für die Geschwindigkeit, Temperatur, Konzentration und Wandreibung des Fluides mittels der Laplace-Transformation erhalten. Die Effekte der Prandtl-ZahlPr, Schmidt-ZahlSc, Grashof-ZahlGr (Gr>0 bedeutet kühlen,Gr<0 heizen), der modifizierten Grashof-ZahlGm, des ParametersM für das magnetische Feld und das zeitliche Verhalten der Geschwindigkeitsverteilung wurden graphisch untersucht. Die Ergebnisse dieser Untersuchung stimmen mit denen von Lien und Chen überein, wenn der Parameter für das magnetische Feld nahe bei 0 liegt.

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Nomenclature C _p Specific heat at constant temperature - C the species concentration near the circular cylinder - C _w the species concentration of the circular cylinder - C the species concentration of the fluid at infinite - * dimensionless species concentration - D chemical molecular diffusivity - g acceleration due to gravity - Gr Grashof number - Gm modified Grashof number - K thermal conductivity - Pr Prandtl number - r ₀ radius of the circular cylinder - r ₀ dimensionless radius - r, r coordinate and dimensionless coordinate normal to the circular cylinder - Sc Schmidt number - t time - t dimensionless time - T temperature of the fluid near the circular cylinder - T _w temperature of the circular cylinder - T temperature of the fluid at infinite - u velocity of the fluid - u dimensionless velocity of the fluid - U ₀ reference velocity - z, z coordinate and dimensionless coordinate along the circular cylinder - coefficient of volume expansion - * coefficient of thermal expansion with concentration - dimensionless temperature - H ₀ magnetic field intensity - coefficient of viscosity - _e permeability (magnetic) - kinematic viscosity - electric conductivity - density - M Hartmann number - dimensionless skin-friction - frequency - dimensionless frequency     

Calculating mass transfer from vertical wet fabrics using a free convection heat transfer correlation

In this note, the evaporation rate from a vertical wet fabric sheet is calculated using a free convection heat transfer correlation. Chilton–Colburn analogy is used to derive a mass transfer correlation from a heat transfer correlation proposed by Churchill and Chu for free convection from a vertical isothermal plate. The mass transfer rate obtained from this expression has shown excellent agreement with experimental data.     

Parametric excitation of Rayleigh convection in the presence of a variable heat flux on the free surface

The onset of Rayleigh convection in a semi-infinite fluid layer is investigated for a heat flux harmonically modulated along the normal to the surface of the fluid. The problem of the evolution of the velocity and temperature perturbations is solved numerically by means of a finite-difference method. The stability limits and the characteristics of the critical perturbations are determined as functions of the Prandtl numbers. The behavior of the critical Rayleigh number is studied for finite layer depths.Perm'. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, pp. 40–44, November–December, 1996.     

Unsteady convective surface heat flux measurements on cylinder for CFD code validation

The surface convective heat transfer rates to a cylinder have been measured using platinum thin film gauges in a shock tube and the results have been used to validate two numerical codes. The investigations have been carried out at different incident shock Mach numbers and flow Reynolds numbers. The measured and simulated results give an insight into the transient flow fields around the model in the shock tube and are a valuable means of complementing the numerical and experimental techniques used in this study.Received: 27 August 2003, Accepted: 11 November 2003, Published online: 17 February 2004Correspondence to: T. Saito     

Free convection from a vertical permeable circular cone with non-uniform surface heat flux

 In this paper, the problem of laminar free convection from a vertical permeable circular cone maintained with non-uniform surface heat flux is considered. The governing boundary layer equations are reduced non-similar boundary layer equations with surface heat flux proportional to x ⁿ (where x is the distance measured from the leading edge). The solutions of the reduced equations are obtained by using three distinct solution methodologies; namely, (i) perturbation solution for small transpiration parameter, ξ, (ii) asymptotic solution for large ξ, and (iii) the finite difference solutions for all ξ. The solutions are presented in terms of local skin-friction and local Nusselt number for smaller values of Prandtl number and heat flux gradient and are displayed in tabular form as well as graphically. Effects of pertinent parameters on velocity and temperature profiles are also shown graphically. Solutions obtained by finite difference method are also compared with the perturbation solutions for small and large ξ and found to be in excellent agreement. Received on 1 October 1999     

Theoretical solution of simultaneous heat and mass transfer by free convection about a vertical flat plate

In this paper we present a numerical solution for stationary simultaneous heat and mass transfer in the laminar boundary layer on a vertical flat plate. The theory is based on a simple physical model that treats the two buoyancy effects (originating from temperature and concentration differences) as mutually independant and additional forces. Results are given for the case of Pr=0.71, Sc=0.63 and equally directed body forces. Our experimental data for the same conditions show close agreement and these seems to be no need for a more complicated theory.     

Unsteady forced convection heat/mass transfer from a flat plate

Numerical methods are used to investigate the transient, forced convection heat/mass transfer from a finite flat plate to a steady stream of viscous, incompressible fluid. The temperature/concentration inside the plate is considered uniform. The heat/mass balance equations were solved in elliptic cylindrical coordinates by a finite difference implicit ADI method. These solutions span the parameter ranges 10 Re 400 and 0.1 Pr 10. The computations were focused on the influence of the product (aspect ratio) × (volume heat capacity ratio/Henry number) on the heat/mass transfer rate. The occurrence on the plates surface of heat/mass wake phenomena was also studied.     

Unsteady natural convection heat and mass transfer in a saturated porous enclosure

A detailed numerical study has been performed to investigate transient natural convection heat and mass transfer in a porous enclosure. Major dimensionless groups governing the present problem areRa,N,Le, φ andAr. Results are particular presented to illustrate the effects of the combined thermal and solutal buoyancy forces on the temporal evolution of local/average Nusselt and Sherwood numbers. The results show that with the increase in the Rayleigh number, the heat and mass transfer is enhanced as a result of greater buoyancy effect. Additionally, the increase in buoyancy ratioN results in an improvement in the heat and mass transfer rates and in the mean time causes a short time duration for the flow to approach the steady-state condition.     

Oscillatory free convection heat transfer along a semi-infinite vertical plate

Summary The fluctuating free convection flow along a semi-infinite vertical plate is considered when the plate temperature is of the form T _p –T =(T ₀ –T ) where 0 < 1, denotes the frequency of oscillation and the mean temperature T ₀–T is proportional to ⁿ (0 n < 1). Flow and temperature fields have been obtained by means of two asymptotic expansions. For small values of the frequency parameter , a regular expansion is obtained while for large the method of matched asymptotic expansion is used. It is found that the skin friction and the rate of heat transfer obtained from two expansions overlap satisfactorily for a certain value of . For n=1 the flow governing equations to a semisimilar form, and have been solved by finite difference method. The results obtained from the series and the finite difference methods are in good agreement.

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Oszillierender Wärmeübergang an einer halbunendlichen senkrechten Platte bei freier Konvektion

Übersicht Betrachtet wird die fluktuierende freie Konvektionsströmung längs einer halbunendlichen senkrechten Platte, deren Temperatur dem Gesetz T _p –T =(T ₀–T ) [1+ sin {ie1-03}] folgt, wobei 0 < 1 gelte, {ie1-04} die Frequenz ist und der Temperatur-Mittelwert T ₀–T proportional zu ⁿ (0 n < 1) ist. Mit Hilfe zweier asymptotischer Entwicklungen werden die Strömungs- und Temperaturfelder gewonnen. Für kleine Werte des Frequenzparameters wird eine gewöhnliche Entwicklung benutzt, für große die Methode angepaßter asymptotischer Entwicklungen. Es stellt sich heraus, daß die Oberflächenreibung und die Wärmeübergangsrate aus zwei Entwicklungen für ein bestimmtes zufriedenstellend aufeinander fallen. Für n=1 werden die Grundgleichungen zueinander ähnlich und werden nach der Finite-Differenzen-Methode gelöst. Die Ergebnisse nach den Reihenentwicklungen und der Finite-Differenzen-Methode stimmen gut überein.