Buoyancy-affected laminar film condensation

The flow and heat transfer in a laminar condensate flim on an isothermal vertical plate is modelled mathematically. The strict Boussinesq approximation is adopted to account for buoyancy due to local temperature variations within the film. A similarity transformation reduces the governing boundary-layer type equations to a coupled set of ordinary differential equations and the resulting three-parameter twopoint boundary value problem is solved numerically for Prandtl numbers,Pr, ranging from 0.001 to 1000 and Jakob numbers,Ja, between 0.0001 and 1.5. The principal effects of the favourable buoyancy are to reduce the thickness of the condensate film and increase the film velocity at the smooth liquid-vapour interface, whereas the friction and heat transfer at the plate are enhanced. In accordance with the classical Nusselt theory, it is found that the temperature varies nearly linearly across the film. The computed similarity profiles for velocity reveal, however, substantial departures from the parabolic distribution assumed in the simplified Nusselt analysis.     

Laminar mixed convection adjacent to vertical, continuously stretching sheets

The analysis of laminar mixed convection in boundary layers adjacent to a vertical, continuously stretching sheet has been presented. The velocity and temperature of the sheet were assumed to vary in a power-law form, that is, u _w (x)=Bx ^m and T _w (x)−T _∞=Ax ⁿ . In the presence of buoyancy force effects, similarity solutions were reported for the following two cases: (a) n=0 and m=0.5, which corresponds to an isothermal sheet moving with a velocity of the form u _w =Bx ^0.5 and (b) n=1 and m=1, which corresponds to a continuous, linearly stretching sheet with a linear surface temperature distribution, i.e. T _w −T _∞=Ax. Formulation of the present problem shows that the heat transfer characteristics depends on four governing parameters, namely, the velocity exponent parameter m, the temperature exponent parameter n, the buoyancy force parameter G ^*, and Prandtl number of the fluid. Numerical solutions were generated from a finite difference method. Results for the local Nusselt number, the local friction coefficient, and temperature profiles are presented for different governing parameters. Effects of buoyancy force and Prandtl number on the flow and heat transfer characteristics are thoroughly examined. Received on 17 July 1997     

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     

Radiation-conduction interaction on mixed convection from a horizontal circular cylinder

A mixed convection flow of an optically dense viscous incompressible fluid along a horizontal circular cylinder has been studied with the effect of radiation when the surface temperature is uniform. Using appropriate transformations, the boundary layer equations governing the flow are reduced to local nonsimilarity form. Solutions of the governing equations are obtained employing the implicit finite difference method. Effects of varying the pertinent parameters, such as, the Planck number, R _w the surface temperature parameter, θ_w and the buoyancy parameter, α on the local skin-friction and local heat transfer coefficients are shown graphically as well as in tabular form against the curvature parameter ξ, while taking Prandtl number Pr = 1.0. It is found that an increase of R _d,θ_w or α leads to increases in the values of the local skin-friction and the local rate of heat transfer coefficients. At the stagnation point asymptotic solutions for large value of α are also obtained and the effect of the other pertinent parameters on the formation of the flow separation are studied. Received on 28 July 1998     

Numerical investigation of flow and combined natural-forced convection from an isothermal square cylinder in cross flow

Unsteady flow and heat transfer from a horizontal isothermal square cylinder is studied numerically using a three-dimensional computational model to investigate the influence of buoyancy on the forced flow and heat transfer characteristics. The numerical model is based on a horizontal square cylinder subjected to laminar fluid flow in an unconfined channel. The governing equations in 3D form are solved using a fractional step method based on the finite difference discretization in addition to a Crank–Nicholson scheme employed to the convective and the viscous terms. Two working fluids–air (Pr = 0.7) and water (Pr = 7)–are considered, and the flow and heat transfer simulations were carried out for the Reynolds and Richardson numbers in the intervals 55 ≤ Re ≤ 250 and 0 ≤ Ri ≤ 2, respectively. The flow characteristics such as time-averaged drag/lift, rms drag/rms lift coefficients as well as Strouhal number were computed. The heat transfer from the cylinder is assessed by mean Nusselt number (and rms Nusselt number) over the total heated cylinder walls. As the buoyancy increases, the mass and the velocity of the fluid flowing underneath the cylinder increases. The fluid is injected into the near wake region with an upward motion which significantly alters the flow field in the downstream as well as upstream regions. The effects of Reynolds, Richardson and Prandtl numbers on the flow field and temperature distributions are discussed in detail. It is shown that the flow and heat transfer characteristics are influenced more for air than water. To fill the void in the literature, useful empirical correlations of practical importance are derived for pure forced and pure natural as well as mixed convection. The mixed convection correlations, in terms of the ratio of pure forced convection, are also developed, and their implications are discussed.     

Chebyshev finite difference approach to modeling the thermoviscosity effect in a power-law liquid film on an unsteady stretching surface

The effect of thermoviscosity (temperature-dependent viscosity) on the heat transfer in a power-law liquid film over an unsteady stretching sheet is investigated. Similarity analysis is used to transform the governing equations for mass, momentum and energy into a system of ordinary differential equations, which contain a thermoviscosity parameter θ_r, unsteadiness parameter S, generalized Prandtl number Pr and power-law index n. The film thickness, the temperature distributions, the local heat transfer rate, and the local skin-friction coefficient were obtained using the Chebyshev finite difference method (ChFD). The results show that thermoviscosity significantly increases the film thickness and the local heat transfer rate while decreasing the local skin-friction coefficient as θ_r → 1. It is found that liquids with a higher power-law index will have a larger film thickness and a higher free-surface temperature, which indicate a lower local heat transfer rate, ?θ′(0).     

Mixed convection cooling of a heated, continuously stretching surface

 An numerical study of the flow and heat transfer characteristics associated with a heated, continuously stretching surface being cooled by a mixed convection flow has been carried out. The relevant heat transfer mechanisms are of interest in a wide variety of practical applications, such as hot rolling, continuous casting, extrusion, and drawing. The surface velocity of the continuously stretching sheet was assumed to vary according to a power-law form, that is, u _w (x)=Cx ^p . Two conditions of surface heating were considered, a variable wall temperature (VWT) in the form T _w (x)−T _∞=Ax ⁿ and a variable surface heat flux (VHF) in the form q _w (x)=Bx ^m . The governing differential equations are transformed by introducing proper nonsimilarity variables and solved numerically using a procedure based on finite difference approximations. Results for the local Nusselt number and the local friction coefficient are obtained for a wide range of governing parameters, such as the surface velocity parameter p, the wall temperature exponent n, the surface heat flux exponent m, the buoyancy force parameters (ξ for the VWT case and χ for the VHF case), and Prandtl number of the fluid. It is found that the local Nusselt number is increased with increasing the velocity exponent parameter p for the VWT case, while the opposite trend is observed for the VHF case. The local friction coefficient is increased for a decelerated stretching surface, while it is decreased for an accelerated stretching surface. Also, appreciable effects of the buoyancy force on the local Nusselt number and the local friction coefficient are observed for both VWT and VHF cases, as expected. Received on 11 January 1999     

Numerical investigation of mixed convection heat transfer past an in-line bundle of cylinders

Two dimensional unsteady Navier-Stokes and the energy equations are solved using finite element method for the case of flow past five row deep in-line bundle of circular cylinders with pitch to diameter ratios (PDR) of 1.5 and 2.0. Numerical solutions of governing equations have been obtained using Euler's explicit algorithm. Analysis have been made for Reynolds number of 100 and Prandtl number of 0.71. The effect of Richardson number (Ri=Gr/Re ²) on the flow and heat transfer have been investigated forRi=?1.0, ?0.5, 0.0, +0.5 and +1.0. Streamlines, isovorticity lines, pressure and temperature contours, local and average Nusselt numbers, pressure and shear stress distribution around the cylinders are presented. Results obtained for forced convection (Ri=0.0) agree well with the available experimental and numerical results. There is considerable effect of buoyancy over tube bundles both in buoyancy aiding and opposing flows.     

Statistical analysis of instantaneous turbulent heat transfer in circular pipe flows

Turbulent heat transfer in circular pipe flow with constant heat flux on the wall is investigated numerically via Large Eddy Simulations for frictional Reynolds number Re _τ  = 180 and for Prandtl numbers in the range 0.1 ≤ Pr ≤ 1.0. In our simulations we employ a second-order finite difference scheme, combined with a projection method for the pressure, on a collocated grid in cylindrical coordinates. The predicted statistical properties of the velocity and temperature fields show good agreement with available data from direct numerical simulations. Further, we study the local thermal flow structures for different Prandtl numbers. As expected, our simulations predict that by reducing the Prandtl number, the range of variations in the local heat transfer and the Nusselt number decrease. Moreover, the thermal flow structures smear in the flow and become larger in size with less sharpness, especially in the vicinity of the wall. In order to characterize the local instantaneous heat transfer, probability density functions (PDFs) for the instantaneous Nusselt number are derived for different Prandtl number. Also, it is shown that these PDFs are actually scaled by the square root of the Prandtl number, so that a single PDF can be employed for all Prandtl numbers. The curve fits of the PDFs are presented in two forms of log-normal and skewed Gaussian distributions.     

The effect of a nonlinear temperature dependent Prandtl-number on heat transfer of fully developed flow of liquids in a straight tube

If Nu_o is the Nusselt Number for a temperature-independent Prandtl number Pr, and Nu the Nusselt number for a temperature dependent Prandtl number, it is usual to define the correction factor Nu/Nu_o=C. A correction factor which has been defined in this form has, up to now, only been expressed as a function of two characteristic Pr numbers. Thus it was simply assumed that the Pr number was a linear function of the temperature. Fluids with very large Pr numbers show a (T;Pr) relationship which deviates considerably from a linear one. This leads to a very large difference (up to 70%) between the calculated and the measured values of the Nusselt number. In the following study we shall determine a so-called curvature parameter of the (T;Pr) curve and obtain a semi-empirical formula for C. This formula has a deviation from the actual relationship many times smaller than that of the formulae for a linear (T;Pr) relationship.

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Zusammenfassung Ist Nu_o die Nusseltzahl bei temperaturunabhangiger und Nu die Nusseltzahl bei temperaturabhangiger Prandtlzahl Pr, so ist es üblich, mit Nu/Nu_o=C den Korrekturfaktor zu bezeichnen. Ein in dieser Form definierter Faktor C ist bisher als Funktion nur zweier charakteristischer Pr-Zahlen ausgedrückt worden. Es wurde somit nur eine lineare Abhängigkeit von der Pr-Zahl von der Temperatur T vorausgesetzt. Flüssigkeiten mit großen Pr-Zahlen weisen eine (T;Pr)-Charakteristik auf, die sehr stark von der linearen abweicht. Sehr große Abweichungen (bis — 70%) der gerechneten von den gemessenen Nu-Zahlen sind eine Folge davon. In vorliegender Arbeit bilden wir mit einer dritten charakteristischen Pr-Zahl einen sogenannten Krümmungsparameter der Kurve (T;Pr) und leiten eine semiempirische Formel für C ab, die um ein großes Vielfaches kleinere Fehler aufweist, als die Formeln für den linearen (T;Pr)-Verlauf.

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Nomenclature

Material constants c_p specific heat at constant pressure [J/kgK] - k heat conductivity [W/mK] - density [kg/m³] - a temperature diffusivity, a=k/c_p [m²/s] - dynamic viscosity [Ns/m²] - kinematic viscosity [m²/s] Fluid dynamics D inner diameter of the tube [m] - L length of tube [m] - w mean speed of fluid [m/s] Heat transfer h coefficient of heat transfer [W/m²K] - T absolute temperature [K] - T_b bulk temperature (corresponding to the adiabatic mixing temperature) [K] - T_w tube wall temperature [K] - T_f=(T_b+T_w)/2 film temperature [K] - T=T_b-T_w temperature forcing difference of heat transfer [K] Characteristic quantities without dimensions Re=wD/ Reynolds number - Pr=/a Prandtl number - Nu=hD/k Nusselt number - related temperature - related Prandtl number - curvature parameter of the Prandtl number Various - C=Nu_b/Nu_o correction factor according to Eq.(5) - p exponent in Eq.(6), (a), (8) and (16) Indices o corresponding to the quasi-isothermal heat transfer - b,w,f with reference to quantities, including physical properties which correspond to the temperatures T_b, T_wor T_f - Pr,k,, for quantities calculated corresponding to the Prandtl number, the thermal conductivity coefficient, the density or the dynamic viscosity - H,C for heating or cooling exp for quantities calculated from experimental data     

The effect of suction or injection on the boundary layer flows induced by continuous surfaces stretched with prescribed skin friction

The boundary layer flow and heat transfer on a stretched surface moving with prescribed skin friction is studied for permeable surface. Three major cases are studied for isothermal surface (n=0) stretched corresponding to different dimensional skin friction boundary conditions namely; skin friction at the surface scales as (x ^?1/2) at m=0, constant skin friction at m=1/3 and skin friction scales as (x) at m=1. The constants m and n are the indices of the power law velocity and temperature exponent respectively. Similarity solutions are obtained for the boundary layer equations subject to power law temperature and velocity variation. The effect of various governing parameters, such as Prandtl number Pr, suction/injection parameter f _w , m and n are studied. The results show that for isothermal surface increasing m enhances the dimensionless heat transfer coefficient for fixed f _w at the suction case and the reverse is true at the injection case. Furthermore, for fixed m, as f _w increases the dimensionless heat transfer coefficient increases. Large enhancements are observed in the heat transfer coefficient as the temperature boundary condition along the surface changes from uniform to linear where the dimensional skin friction is of order (x) at m=1. This enhancement decreases as the suction increases.     

Mixed convection boundary layer flow over a horizontal circular cylinder with Newtonian heating

The steady mixed convection boundary layer flow over a horizontal circular cylinder, generated by Newtonian heating in which the heat transfer from the surface is proportional to the local surface temperature, is considered in this study. The governing boundary layer equations are first transformed into a system of non-dimensional equations via the non-dimensional variables, and then into non-similar equations before they are solved numerically using a numerical scheme known as the Keller-box method. Numerical solutions are obtained for the skin friction coefficient Re ^1/2 C _f and the local wall temperature θ _w (x) as well as the velocity and temperature profiles with two parameters, namely the mixed convection parameter λ and the Prandtl number Pr.     

The buoyancy effects on the boundary layers induced by continuous surfaces stretched with rapidly decreasing velocities

Heat and mass transfer characteristics of the self-similar boundary layer flows induced by continuous surfaces stretched with rapidly decreasing power law velocities U_w x^m, m < –1 are considered for mixed convection flow. The effect of various governing parameters, such as Prandtl number Pr, temperature exponent n, dimensionless injection/suction velocity f_w, and the mixed convection parameter = s Gr/Re² are studied. These parameters have great effects on velocity and temperature profiles, heat transfer coefficient, and skin friction coefficient at the moving surface. Results show that similarity solutions exist only when the condition n = 2m – 1 is satisfied. Critical values of , Nu/Re^0.5 and C_f Re^0.5 are obtained for predominate natural convection for different Prandtl numbers at m = –2, –6 and n = –5, and –13 respectively. Results also show that the effect of buoyancy is more significant for weak than for strong suction. Furthermore, critical Prandtl numbers where f_w profiles have minimums are obtained for m = –2 and –6. Finally, critical values of , C_f Re^0.5 are also obtained for predominate natural convection for both m = –2 and –6.     

Analytical and numerical study of double diffusive convection in a vertical enclosure

This paper presents an analytical and numerical study of natural convection of a double-diffusive fluid contained in a rectangular slot subject to uniform heat and mass fluxes along the vertical sides. Governing parameters of the problem under study are the thermal Rayleigh number, Ra _T ; buoyancy ratio, N; Lewis number, Le; Prandtl number, Pr and aspect ratio of the cavity, A. In the first part of the analytical study a scale analysis is applied to the two extreme cases of heat-transfer and mass-transfer-driven flows. In the second part, an analytical solution, based on the parallel flow approximation, is reported for tall enclosures (A?1). Solutions for the flow fields, temperature and concentration distributions and Nusselt and Sherwood numbers are obtained in terms of the governing parameters of the problem. In the limits of heat-driven and solute-driven flows a good agreement is obtained between the prediction of the scale analysis and those of the analytical solution. The numerical solutions are based on the complete governing equations for two-dimensional flows, and cover the range 1≤Ra _T ≤10⁷, 0≤N≤10⁵, 10^-3≤Le≤10³, 1≤A≤20 and Pr=7. A good agreement is found between the analytical predictions and the numerical simulation.     

Natural convection-radiation interaction on boundary layer flow along a thin vertical cylinder

A free convertion flow of an optically dense viscous incompressible fluid along a vertical thin circular cylinder has been studied with effect of radiation when the surface temperature is uniform. With appropriate transformations, the boundary layer equations governing the flow are reduced to local nonsimilarity equations. Solutions of the governing equations are obtained employing the implicit finite difference methods together with Keller box scheme as well the local nonsimilarity method with second order truncation for all ξ (nondimensional transverse curvature parameter) in the interval [0,10] and are expressed in terms of local Nusselt number for a range of values of the pertinent parameters. Effects of pertinent parameters, such as, the radiation parameter, R _d , the surface temperature parameter, θ _w , taking Prandtl number, Pr, equals 0.7 on the velocity and temperature field are also presented graphically. From the solution it is seen that increase of R _d , or θ _w leads to increase in the local rate of heat transfer coefficients. Results obtained by both the methods are obtained in excellent agreement between each other upto ξ = 10.     

Mixed convection heat transfer from three heated square cylinders in cross-flow at low Reynolds numbers

This paper presents the effects of cross buoyancy and Prandtl number on the flow and heat transfer characteristics around three equal isothermal square cylinders arranged in a staggered configuration within an unconfined medium. Transient two-dimensional numerical simulations are performed with a finite volume code based on the SIMPLEC algorithm in a collocated grid system. The pertinent dimensionless parameters, such as Reynolds, Prandtl and Richardson numbers are considered in the range of 1 ≤ Re ≤ 30, 0.7 ≤ Pr ≤ 100 and 0 ≤ Ri ≤ 1. The representative streamlines, vortex structures and isotherm patterns are presented and discussed. In addition, the overall drag and lift coefficients and average Nusselt numbers are determined to elucidate the effects of Reynolds, Prandtl and Richardson numbers on flow and heat transfer. The flow is observed to be steady for all the ranges of parameters considered. The drag coefficient is found to decrease with Re (for Ri = 0) and Ri at low Pr, whereas it increases with Pr at higher Ri. The lift coefficient decreases with Ri at low Pr and increases with Pr at higher Ri. The time and surface average cylinder Nusselt number is found to increase monotonically with Re as well as Pr while it remains almost insensitive to Ri at low Pr.     

Numerical study of steady magneto-convection around an adiabatic body inside a square enclosure in low Prandtl numbers

In this paper, the effects of Prandtl number on the steady magneto-convection around a centrally located adiabatic body inside a square enclosure are numerically investigated. Two-dimensional nonlinear governing equations are discretized using the control volume method and hybrid scheme. The equations are solved using SIMPLER algorithm. The results are displayed in the form of streamlines and isotherms when the Rayleigh number varies between 10³ and 10⁶, the Hartmann number changes between 0 and 100 and the Prandtl number ranges between 0.005 and 0.1. The ratio of the buoyancy force to the Lorentz force (Ra/Ha ²) is introduced as an index to compare the contribution of natural convection and magnetic field strength on heat transfer. The results obtained from numerical modeling show that the Prandtl number has not considerable effect on heat transfer at low Rayleigh numbers. The effect of magnetic field strength on convection is increased by increasing Prandtl number. The effect of Prandtl number on the average Nusselt number in the presence of a magnetic field is less than the case without a magnetic field.     

A note on magnetoconvection in a vertical enclosure

The combined effect of viscous and ohmic dissipations on magnetoconvection in a vertical enclosure heated at the vertical side walls in the presence of applied electric field parallel to gravity and magnetic field normal to gravity is investigated. The coupled non-linear equations governing the motion are solved both analytically valid for small buoyancy parameter N and numerically valid for large N. Solutions for large N reveal a marked change in velocity profile, mass flow rate, skin friction and rate of heat transfer. These results are presented for various Hartmann number M, electric field loading parameter E and buoyancy parameter N. It is shown in the case of open circuit (i.e. E ≠ 0) that the effect of magnetic field is to increase both the velocity and temperature in contrast with the short circuit case (i.e. E = 0). The results for the case when the walls are maintained at the same temperatures (i.e. T₁ = T₂) are obtained as a particular case.     

Boundary layer development on a continuous moving surface with a parallel free stream due to impulsive motion

The development of the momentum and thermal boundary layers over a semi-infinite flat plate has been studied when the external stream as well as the plate are impulsively moved with constant velocities. At the same time the temperature of the wall is suddenly raised from T_∞, the temperature of the surrounding fluid, toT _w and maintained at this temperature. The problem has been formulated in a new system of scaled coordinates such that fort?=0 it reduces to Rayleigh type of equation and fort? → ∞ it reduces to Blasius or Sakiadis type of equation. A new scale of time has been used which reduces the region of integration from an infinite region to a finite region which reduces the computational time considerably. The governing singular parabolic partial differential equations have been solved numerically using an implicit finite difference scheme. For some particular cases, analytical solutions have been obtained. The results show that there is a smooth transition from Rayleigh solution to Blasius or Sakiadis solution as the dimensionless timeξ increases from zero to one. The shear stress at the wall is negative for the friction parameterλ<0.5, positive forλ>0.5 and zero forλ=0.5. The zero shear stress at the wall does not imply separation but corresponds to the parallel flow. The surface heat transfer is strongly dependent on the Prandtl numberPr and increases with it. Also forPr<Pr ₀, the surface heat transfer is enhanced as the friction parameterλ increases, but forPr>Pr ₀ it get reduced.     

Computational study of combined effects of conduction-radiation and hydromagnetics on natural convection flow past magnetized permeable plate

The computational study of the combined effects of radiation and hydromagnetics on the natural convection flow of a viscous,incompressible,and electrically conducting fluid past a magnetized permeable vertical plate is presented.The governing non-similar equations are numerically solved by using a finite difference method for all values of the suction parameter ξ and the asymptotic solution for small and large values of ξ.The effects of varying the Prandtl number P r,the magnetic Prandtl number P r m,the magnetic force parameter S,the radiation parameter R d,and the surface temperature θ w on the coefficients of the skin friction,the rate of heat transfer,and the current density are shown graphically and in tables.An attempt is made to examine the effects of the above mentioned physical parameters on the velocity profile,the temperature distribution,and the transverse component of the magnetic field.