Three dimensional free convection flow and heat transfer along a porous vertical plate

The free convection flow along a vertical porous plate with transverse sinusoidal suction velocity distribution is investigated. Due to this type of suction velocity at the plate the flow becomes three dimensional one. For the asymptotic flow condition, the wall shear stress in the direction of main flow for different values of buoyancy parameter G is obtained. For G=0, the skin friction in the direction of free stream and the rate of heat transfer from the plate to the fluid are given. It is found that these results differ from those obtained by Gersten and Gross.     

Combined forced and free convection MHD row past a semi-infinite vertical plate

Effects of a transversely applied magnetic field on the forced and free convective flow of an electrically conducting fluid past a vertical semi-infinite plate, on taking into account dissipative heat and stress work, have been presented. Without magnetic field, it has been discussed by the authors [1] in an earlier paper. The effects of Gr (Grashof number, Gr>0 cooling of the plate by free convection currents, Gr<0 heating of the plate by free convection currents), Pr (Prandtl number), F (Froude number) and M² (the magnetic field parameter) are discussed. It is observed that reverse type of flow of air exists near the plate when Gr<0.

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Gemischte erzwungene und freie Konvektions-MHD-Strömung an einer halbunendlichen senkrechten Platte

Zusammenfassung Die Wirkung eines transversalen Magnetfeldes auf die erzwungene und freie Konvektion einer elektrisch leitenden Flüssigkeit an einer halbunendlichen senkrechten Platte wurde unter Berücksichtigung der Dissipationswärme und der Kompressionsarbeit mitgeteilt. Das Problem wurde ohne Magnetfeld schon früher [1] behandelt. Diskutiert wurde die Wirkung der Grashof-Zahl Gr (Gr>0 Kühlung der Platte durch freie Konvektion), der Prandtl-Zahl Pr, der Froude-Zahl F und des magnetischen Feldparameters M₂. Bei Gr<0 wird Umkehr strömung in Plattennähe betrachtet.

     

Dissipation effects on MHD free convection flow past a semi-infinite vertical plate

Viscous and Joule dissipation effects are considered on MHD free convection flow past a semi-infinite isothermal vertical plate under a uniform transverse magnetic field. Series solutions in powers of a dissipation number (=gx/c _p) have been employed and the resulting ordinary differential equations have been solved numerically. The velocity and temperature profiles are shown on graphs and the numerical values of ₁(0)/₀(0) (, temperature function) have been tabulated. It is observed that the dissipation effects in the MHD case become more dominant with increasing values of the magnetic field parameter (=M ²/(Gr _x /4)^1/2) and the Prandtl number.     

Natural convection heat transfer along vertical rectangular ducts

Experimental investigations have been reported on steady state natural convection from the outer surface of vertical rectangular and square ducts in air. Seven ducts have been used; three of them have a rectangular cross section and the rest have square cross section. The ducts are heated using internal constant heat flux heating elements. The temperatures along the vertical surface and the peripheral directions of the duct wall are measured. Axial (perimeter averaged) heat transfer coefficients along the side of each duct are obtained for laminar and transition to turbulent regimes of natural convection heat transfer. Axial (perimeter averaged) Nusselt numbers are evaluated and correlated using the modified Rayleigh numbers for laminar and transition regime using the vertical axial distance as a characteristic length. Critical values of the modified Rayleigh numbers are obtained for transition to turbulent. Furthermore, total overall averaged Nusselt numbers are correlated with the modified Rayleigh numbers and the area ratio for the laminar regimes. The local axial (perimeter averaged) heat transfer coefficients are observed to decrease in the laminar region and increase in the transition region. Laminar regimes are obtained at the lower half of the ducts and its chance to appear decreases as the heat flux increases.     

Conjugate free convection along a thin vertical plate with internal nonuniform heat generation in a porous medium

 The steady state heat transfer characteristics of a thin vertical strip with internal heat generation placed in a porous medium is studied in this work. The non-dimensional temperature distribution in the strip is obtained as a function of the intensity and distribution of the internal heat sources. Both the thermally thin as the thick wall approximations are considered in this paper. The mass flow rate of fluid induced by heating the strip decreases as the longitudinal heat conduction effects along the strip decreases. Received on 22 November 2000 / Published online: 29 November 2001     

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.     

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.     

Similarity analysis in magnetohydrodynamics: Hall effects on free convection flow and mass transfer past a semi-infinite vertical flat plate

Heat and mass transfer along a semi-infinite vertical flat plate under the combined buoyancy force effects of thermal and species diffusion is investigated in the presence of a strong non-uniform magnetic field and the Hall currents are taken into account. The induced magnetic field due to the motion of the electrically conducting fluid is negligible. This assumption is valid for a small magnetic Reynolds number. The similarity solutions are obtained using the scale group of transformations. These are the only symmetry transformations admitted by the field equations. The non-linear boundary layer equations with the boundary conditions are transferred to a system of non-linear ordinary differential equations with the appropriate boundary conditions. Furthermore, the similarity equations are solved numerically by using a fourth order Runge-Kutta scheme with the shooting method. Numerical results for the velocity profiles, the temperature profiles and the concentration profiles are presented graphically for various values of the magnetic parameter M in the range of 0-1 with the Hall parameter m taking the values 0.5, 1, 2, and 3.     

黏弹性纳米流体在垂直板上的自然对流与传热分析

基于分数阶Maxwell模型和分数阶Fourier定律构建黏弹性纳米流体在垂直板上的非定常二维边界层自然对流与传热控制方程,利用有限差分和L1算法获得数值稳定解,对不同物理参数下的速度、温度、平均表面摩擦系数和平均Nusselt数的变化趋势进行图形化分析。结果显示,速度和温度边界层均表现出短暂记忆和延迟特性;速度分数导数参数削弱了自然对流,而速度松弛时间的影响却相反;温度分数导数参数削弱了自然对流和热传导,而温度松弛时间的影响却相反。     

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.     

Combined heat and mass transfer by laminar natural convection from a vertical plate

Simultaneous heat and mass transfer in buoyancy-induced laminar boundary-layer flow along a vertical plate is studied for any ratio of the solutal buoyancy force to the thermal buoyancy force by using a new similarity transformation. The effects of the buoyancy ratio and Lewis number on the rates of heat and mass transfer are presented explicitly for most practical gaseous solutions (Pr=0.7, 0.21≤Sc≤2.1) and aqueous solutions (Pr=7, 140≤Sc≤1400). Very accurate correlations of the mass transfer and heat transfer rates are developed for the cases of single and combined buoyancy forces.     

Laminar natural convection heat transfer from vertical plate to power-law fluid

Summary Approximate solutions for laminar natural convection heat transfer between a vertical plate and a power-law fluid with high Prandtl number were obtained using an integral method for cases with various types of boundary conditions. The results were found in good agreement with available experimental evidence.Nomenclature a exponent defined by equations (28) and (29) - A, B, C, D, E constants defined by equations (15) to (19) - C ₁, C ₂, M ₁, M ₂ coefficients for Nusselt number expression defined by (32b), (33b) - f temperature difference, equal to T _s–T - f ⁺ dimensionless temperature difference - g gravitational acceleration - Gr Grashof number defined by (25), (50) and (66), respectively - H heat flux at plate surface - h _x local heat transfer coefficient - K consistency index for Power-law fluid - k thermal conductivity of fluid - K ₁, K ₂ constants defined by (50) and (51) - L height of plate - n flow behavior index for Power-law fluid - P a quantity defined by (54a) - T temperature - T _s plate temperature - T temperature of the bulk of fluid - s constant given by (35) - u velocity component along x-direction - u _x maximum velocity induced by natural convection current, (10) - v velocity component along y-direction - x distance measured along direction parallel to that of gravitational force - x ⁺ dimensionless quantity, defined as x/L - y distance measured away from plate - Nu _x local Nusselt number - Nu _av average Nusselt number - Pr Prandtl number defined by (24) - T temperature difference according to boundary conditions - thermal diffusivity of fluid - coefficient of thermal expression of fluid - boundary layer thickness - ⁺ dimensionless boundary layer thickness - dimensionless velocity profile - dimensionless variable, defined as y/ - dimensionless temperature difference     

Effects of bubble size on heat transfer enhancement by sub-millimeter bubbles for laminar natural convection along a vertical plate

Injection of sub-millimeter bubbles is considered a promising technique for enhancing natural convection heat transfer for liquids. So far, we have experimentally investigated heat transfer characteristics of laminar natural convection flows with sub-millimeter bubbles. However, the effects of the bubble size on the heat transfer have not yet been understood. The purpose of this study is to clarify the effects of the bubble size on the heat transfer enhancement for the laminar natural convection of water along a vertical heated plate with uniform heat flux. Temperature and velocity measurements, in which thermocouples and a particle tracking velocimetry technique are, respectively used, are conducted to investigate heat transfer and flow characteristics for different bubble sizes. Moreover, two-dimensional numerical simulations are performed to comprehensively understand the effects of bubble injection on the flow near the heated plate. The result shows that the ratio of the heat transfer coefficient with sub-millimeter-bubble injection to that without injection ranges from 1.3 to 2.2. The result also shows that for a constant bubble flow rate, the heat transfer coefficient ratio increases with a decrease in the mean bubble diameter. It is expected from our estimation based on both experimental data and simulation results that this increase results from an increase in the advection effect due to bubbles.     

Heat transfer enhancement in a turbulent natural convection boundary layer along a vertical flat plate

An experimental study on heat transfer enhancement for a turbulent natural convection boundary layer in air along a vertical flat plate has been performed by inserting a long flat plate in the spanwise direction (simple heat transfer promoter) and short flat plates aligned in the spanwise direction (split heat transfer promoter) with clearances into the near-wall region of the boundary layer. For a simple heat transfer promoter, the heat transfer coefficients increase by a peak value of approximately 37% in the downstream region of the promoter compared with those in the usual turbulent natural convection boundary layer. It is found from flow visualization and simultaneous measurements of the flow and thermal fields with hot- and cold-wires that such increase of heat transfer coefficients is mainly caused by the deflection of flows toward the outer region of the boundary layer and the invasion of low-temperature fluids from the outer region to the near-wall region with large-scale vortex motions riding out the promoter. However, heat transfer coefficients for a split heat transfer promoter exhibit an increase in peak value of approximately 60% in the downstream region of the promoter. Flow visualization and PIV measurements show that such remarkable heat transfer enhancement is attributed to longitudinal vortices generated by flows passing through the clearances of the promoter in addition to large-scale vortex motions riding out the promoter. Consequently, it is concluded that heat transfer enhancement of the turbulent natural convection boundary layer can be substantially achieved in a wide area of the turbulent natural convection boundary layer by employing multiple column split heat transfer promoters. It may be expected that the heat transfer enhancement in excess of approximately 40% can be accomplished by inserting such promoters.     

A correlation for free convection heat transfer from vertical wavy surfaces

Free convection heat transfer along an isothermal vertical wavy surface was studied experimentally and numerically. A Mach-Zehnder Interferometer was used in the experiment to determine the local heat transfer coefficients. Experiments were done for three different amplitude–wavelength ratios of α = 0.05, 0.1, 0.2 and the Rayleigh numbers ranging from Ra _l = 2.9 × 10⁵ to 5.8 × 10⁵. A finite-volume based code was developed to verify the experimental study and obtain the results for all the amplitude–wavelength ratios between α = 0 to 0.2. It is found that the numerical results agree well with the experimental data. Results indicate that the frequency of the local heat transfer rate is the same as that of the wavy surface. The average heat transfer coefficient decreases as the amplitude–wavelength ratio increases and there is a significant difference between the average heat transfer coefficients of the surface with α = 0.2 and those surfaces with α = 0.05 and 0.1. The experimental data are correlated with a single equation which gives the local Nusselt number along the wavy surface as a function of the amplitude–wavelength ratio and the Rayleigh number.     

Magnetohydrodynamic unsteady free convection past a hot vertical plate

The free convection flow of an electrically conducting liquid from an infinite plate has been studied in the presence of a uniform magnetic field. General expressions for the velocity field, induced magnetic field, skin-friction and temperature distribution have been obtained when the plate is a perfect conductor and its temperature varies with the law t ⁿ e ^at . The results have been presented through some graphs and tables with the magnetic Prandtl number unity as its value.     

Convection regimes and heat transfer characteristics along a continuously moving heated vertical plate

The steady laminar flow and thermal characteristics of a continuously moving vertical sheet of extruded material are studied close to and far downstream from the extrusion slot. The velocity and temperature variations, obtained by a finite volume method, are used to map out the entire forced, mixed and natural convection regimes. The effects of the Prandtl number (Pr) and the buoyancy force parameter (B) on the friction and heat transfer coefficients are investigated. Comparisons with experimental measurements and solutions by others in the pure forced and pure natural convection regions are made. In the mixed convection region, the results are compared with available finite-difference solutions of the boundary layer equations showing excellent agreement. The region close to the extrusion slot is characterized as a non-similar forced-convection dominated region in which Nu_xRe_x^−1/2 drops sharply with increasing Richardson number (Ri_x). This is followed by a self-similar forced-convection dominated region in which Nu_xRe_x^−1/2 levels off with increasing Ri_x until the buoyancy effect sets in. The existence and extent of the latter region depend upon the value of B. A non-similar mixed convection region where increasing buoyancy effect enhances the heat transfer rate follows. Finally, this region is followed downstream by a self-similar natural-convection dominated region in which Nu_xRe_x^−1/2 approaches the pure natural convection asymptote at large Ri_x. Critical values of Ri_x to distinguish the various convection regimes are determined for different Pr and B.     

Heat transfer enhancement for laminar natural convection along a vertical plate due to sub-millimeter-bubble injection

Sub-millimeter-bubble injection is one of the most promising techniques for enhancing heat transfer for the laminar natural convection of liquids. However, flow and heat transfer characteristics for laminar natural convection of water with sub-millimeter bubbles have not yet been fully understood. The purpose of this study is to experimentally clarify the effects of sub-millimeter-bubble injection on the laminar natural convection of water along a heated vertical plate. The use of thermocouples and a particle tracking velocimetry (PTV) technique are applied to temperature and velocity measurements, respectively. The temperature measurement shows that the ratio of the heat transfer coefficient with sub-millimeter-bubble injection to that without injection increases with an increase in the bubble flow rate or a decrease in the wall heat flux and that the ratio ranges from 1.35 to 1.85. Moreover, it is concluded from simultaneous measurement of temperature and velocity that the heat transfer enhancement is directly affected by flow modification due to bubbles rising near the heated vertical plate.     

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