Finite Péclet number forced convection past a sphere in a porous medium using a thermal nonequilibrium model

We study the finite-Péclet number forced convective heat transfer from a uniform temperature sphere placed in otherwise uniform fluid stream within a porous medium. A numerical study is undertaken to determine how the lack of local thermal equilibrium between the phases affects temperature fields of the two phases and the respective rates of heat transfer from the sphere. On the upstream side of the sphere the temperature field extends further from the sphere in the solid phase than it does for the fluid phase, but the opposite is true on the downstream side.     

Heat transfer from a heated sphere in a rarefied gas at rest

We consider the problem of heat transfer from a slightly heated sphere in a resting rarefied gas. We assume that the Krook equation is valid in this case. Two forms of the basic equations are presented, and relations are given which are obtained as a result of calculations of the heat flux and the temperature jump at the sphere surface as a function of a parameter which is inversely proportional to the Knudsen number. The results obtained are compared with results given by the known approximate theories.In conclusion the author wishes to thank M. N. Kogan for proposing the problem and for numerous discussions.     

Transient conjugate mixed convection from a sphere in a porous medium saturated with cold pure or saline water

In this paper, a numerical investigation of the transient conjugate mixed convection flow about a sphere embedded in a porous medium saturated with pure or saline water is carried out. The effect of density extremum is considered by using the nonlinear dependence of density on the temperature. The salinity effects are considered by assuming uniform saline concentration over the domain considered. The direction of the natural convection is changed either to aiding or to opposing the upcoming flow direction simulating the sphere is either hot or cold relative to the surrounding temperature. Results show that the initial temperature differences as well as the saline concentration alter the transient heat transfer rate in conceivable degree. It was found that the heat capacity ratio between the sphere and the surrounding media has more significant effect on the calculated heat transfer rate than the thermal conductivity ratio. The study is performed by using six nondimensional parameters and results are discussed in detail. Received on 10 November 1997     

Temperature field of a three-layer sphere

This paper presents a solution of the problem of unsteady heat transfer in a three-layer hollow sphere in a central-symmetric formulation with various time-dependent boundary conditions on the inner and outer surfaces. In each layer of the sphere there is heat release of known intensity which depends on the radial coordinate and time. The solution is obtained by a finite integral transform on the radial coordinate. A numerical solution is presented for one version of the boundary conditions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 1, pp. 78–84, January–February, 2009.     

Heat transfer from a slowly rotating sphere

The problem of steady state forced convection heat transfer in a viscous incompressible fluid occupying the annular region between two concentric spheres is considered. The inner sphere is maintained at a constant temperatureT ₀ and rotates slowly around an axis through the centre. The outer sphere is at rest and the temperature of its surface is prescribed as a function of the spherical coordinates and. It is shown that, when viscous dissipation is small, the overall rate of heat transfer from the rotating sphere into the fluid is unaffected by convection from the sphere surface, in case of a slow rotation, where the Stokes solution holds.     

Thermal convection at a melting benzene surface

Summary Heat transfer by thermal free convection at the surface of a sphere has been studied experimentally by melting a sphere of solid benzene in a large quantity of liquid benzene of homogeneous temperature. The influence of cold liquid produced by the melting process is taken into account to yield results that are representative for the pure effect of heat transfer without melting. In the general formula for heat transfer by thermal convection, =C(GrPr)^1/4, we found C=0.525.     

Heat Transfer in Supersonic Low-Re Flow Past a Sphere and a Cylinder

Supersonic perfect-gas flow past a sphere and a cylinder at a constant surface temperature is investigated on the basis of a numerical solution of the Navier-Stokes equations. The Re-dependence of the Nusselt number and the recovery coefficient is calculated for the Reynolds numbers ranging from 1 to 1000 and Mach numbers 3 (cylinder) and 5 (sphere) and compared with the experimental data. The influence of slip and no-slip conditions imposed on the body surface on the heat transfer parameters and the base flow is analyzed.     

Mass and heat transfer during solid sphere dissolution in a non-uniform fluid flow

We studied a nonisothermal dissolution of a solvable solid spherical particle in an axisymmetric non-uniform fluid flow when the concentration level of the solute in the solvent is finite (finite dilution of solute approximation). It is shown that simultaneous heat and mass transfer during solid sphere dissolution in a uniform fluid flow, axisymmetric shear flow, shear-translational flow and flow with a parabolic velocity profile can be described by a system of generalized equations of convective diffusion and energy. Solutions of diffusion and energy equations are obtained in an exact analytical form. Using a general solution the asymptotic solutions for heat and mass transfer problem during spherical solid particle dissolution in a uniform fluid flow, axisymmetric shear flow, shear-translational flow and flow with parabolic velocity profile are derived. Theoretical results are in compliance with the available experimental data on falling urea particles dissolution in water and for solid sphere dissolution in a shear flow.     

Non-similar Solution for Natural Convective Boundary Layer Flow Over a Sphere Embedded in a Porous Medium Saturated with a Nanofluid

A boundary layer analysis is presented for the natural convection past an isothermal sphere in a Darcy porous medium saturated with a nanofluid. Numerical results for friction factor, surface heat transfer rate, and mass transfer rate have been presented for parametric variations of the buoyancy ratio parameter N _r, Brownian motion parameter N _b, thermophoresis parameter N _t, and Lewis number L _e. The dependency of the friction factor, surface heat transfer rate (Nusselt number), and mass transfer rate (Sherwood number) on these parameters has been discussed.     

Similarity between the heat transfer to a model in an underexpanded dissociated-air jet of a high-frequency plasmatron and to a sphere in a high-velocity flow in the terrestrial atmosphere

On the basis of the local heat transfer modeling concept the parameters of supersonic flow past a cylindrical flat-faced model, 0.01m in radius, in an underexpanded dissociate-air jet of the VGU-4 high-frequency plasmatron are recalculated to the conditions of sphere entry in the terrestrial atmosphere. The heat transfer parameters, similar in the experiment and the atmospheric entry, are determined.     

Experimental investigations of local heat fluxes on a sphere and on the spherical blunt end of an axisymmetric body

This article presents the results of experimental investigations of local heat transfer on a sphere and on the spherical blunt end of axisymmetric bodies in a flow of low-density gas. The data were obtained in the region of Mach numbers from 3.3 to 8, at Reynolds numbers from 5 to 2.6·10³.     

Heat transfer by laminar flow from a rotating sphere

Summary The heat transfer problem for the flow of an incompressible viscous, heat-conducting fluid, due to uniform rotation about a diameter of a sphere, which is kept at a constant temperature, has been solved with viscous dissipation included. Due to inflow at the poles the cooler liquid is drawn from infinity towards the rotating sphere and this causes a lowering of the temperature there. After flowing in the boundary layer of the sphere the liquid gets heated up and causes a rise in temperature near the equator. Numerical results are given in case of water (Prandtl number σ=5), and it is found that the isothermals are surfaces of revolution flattened at the poles and elongated near the equator. The thermal and the velocity boundary layers turn out to be of the same order of magnitude.     

Conjugate heat transfer in the unbounded flow of a viscoelastic fluid past a sphere

This work addresses the conjugate heat transfer of a simplified PTT fluid flowing past an unbounded sphere in the Stokes regime (Re = 0.01). The problem is numerically solved with the finite-volume method assuming axisymmetry, absence of natural convection and constant physical properties. The sphere generates heat at a constant and uniform rate, and the analysis is conducted for a range of Deborah (0 ≤ De ≤ 100), Prandtl (10⁰ ≤ Pr ≤ 10⁵) and Brinkman (0 ≤ Br ≤ 100) numbers, in the presence or absence of thermal contact resistance at the solid–fluid interface and for different conductivity ratios (0.1 ≤ κ ≤ 10). The drag coefficient shows a monotonic decrease with De, whereas the normalized stresses on the sphere surface and in the wake first increase and then decrease with De. A negative wake was observed for the two solvent viscosity ratios tested (β = 0.1 and 0.5), being more intense for the more elastic fluid. In the absence of viscous dissipation, the average Nusselt number starts to decrease with De after an initial increase. Heat transfer enhancement relative to an equivalent Newtonian fluid was observed for the whole range of conditions tested. The dimensionless temperature of the sphere decreases and becomes more homogeneous when its thermal conductivity increases in relation to the conductivity of the fluid, although small changes are observed in the Nusselt number. The thermal contact resistance at the interface increases the average temperature of the sphere, without affecting significantly the shape of the temperature profiles inside the sphere. When viscous dissipation is considered, significant changes are observed in the heat transfer process as Br increases. Overall, a simplified PTT fluid can moderately enhance heat transfer compared to a Newtonian fluid, but increasing De does not necessarily improve heat exchange.     

The influence of the Henry number on the conjugate mass transfer from a sphere

Unsteady conjugate mass transfer between a sphere and a surrounding fluid flow has been analysed. Two sphere models, the sphere with rigid interface and the sphere with mobile interface and internal circulation, have been studied. Creeping flow is assumed. The mass balance equations were solved numerically by the ADI finite difference method. The influence of the parameters that characterize the coupling features of the conjugate mass transfer, diffusivity ratio (Φ_D) and Henry number (H) (0.01?≤?H (Φ_D)?≤?100), is studied at intermediate Pe numbers. The occurrence and evolution of mass wake phenomenon is analysed. In contradiction with the assumptions practiced in classical theories of interface mass transfer, the results obtained in this work show that the Henry number and the diffusivity ratio have a significant and distinct influence on the values of both overall and fractional (internal and external) Sh numbers. Using the additivity relation as an interpolation formula between two asymptotic solutions does not lead to accurate approximations of the numerical results on the entire domain of variation of H and Φ_D. An analogy between conjugate heat and mass transfer was established.     

Steady and transient liquid sphere heating in a convective gas stream

 A finite-difference scheme has been developed to solve the equations governing the laminar forced convection heat transfer around and inside a spherical fluid droplet moving steadily in another immiscible fluid for both steady and transient thermal conditions. For large values of the external flow Reynolds number (Re), results not available in the literature have been obtained for circulating droplets at intermediate and high interior-to-exterior viscosity ratios (μ^*). Detailed results over a wide range of viscosity ratio (μ^*) and for 200≤Re≤1000 are presented for the temperature profiles outside and inside the sphere, Nusselt number, the time required to attain a uniform surface temperature and the time required to reach the steady-state temperature. Results show that convective heating is dependent on the external flow Reynolds number (Re) and the interior-to-exterior viscosity ratio (μ^*) where increasing Re or decreasing μ^* result in increasing heat transfer rate convected to the liquid sphere. Received on 1 March 1999     

Self-similar solution of the unsteady mixed convection flow in the stagnation point region of a rotating sphere

An analysis is performed to present a new self-similar solution of unsteady mixed convection boundary layer flow in the forward stagnation point region of a rotating sphere where the free stream velocity and the angular velocity of the rotating sphere vary continuously with time. It is shown that a self-similar solution is possible when the free stream velocity varies inversely with time. Both constant wall temperature and constant heat flux conditions have been considered in the present study. The system of ordinary differential equations governing the flow have been solved numerically using an implicit finite difference scheme in combination with a quasilinearization technique. It is observed that the surface shear stresses and the surface heat transfer parameters increase with the acceleration and rotation parameters. For a certain value of the acceleration parameter, the surface shear stress in x-direction vanishes and due to further reduction in the value of the acceleration parameter, reverse flow occurs in the x–component of the velocity profiles. The effect of buoyancy parameter is to increase the surface heat transfer rate for buoyancy assisting flow and to decrease it for buoyancy opposing flow. For a fixed buoyancy force, heating by constant heat flux yields a higher value of surface heat transfer rate than heating by constant wall temperature.     

Unsteady compressible boundary layer flow in the stagnation region of a sphere with a magnetic field

Summary An analysis is performed to study the unsteady compressible laminar boundary layer flow in the forward stagnation-point region of a sphere with a magnetic field applied normal to the surface. We have considered the case where there is an initial steady state that is perturbed by the step change in the total enthalpy at the wall. The nonlinear coupled parabolic partial differential equations governing the flow and heat transfer have been solved numerically using a finite-difference scheme. The numerical results are presented, which show the temporal development of the boundary layer. The magnetic field in the presence of variable electrical conductivity causes an overshoot in the velocity profile. Also, when the total enthalpy at the wall is suddenly increased, there is a change in the direction of transfer of heat in a small interval of time. Received 15 January 1996; accepted for publication 21 November 1996     

Numerical solutions of free convection boundary layer flow on a solid sphere with Newtonian heating in a micropolar fluid

In this paper, the problem of free convection boundary layer flow on a solid sphere in a micropolar fluid with Newtonian heating, in which the heat transfer from the surface is proportional to the local surface temperature, is considered. The transformed boundary layer equations in the form of partial differential equations are solved numerically using an implicit finite-difference scheme. Numerical solutions are obtained for the local wall temperature, the local skin friction coefficient, as well as the velocity, angular velocity and temperature profiles. The features of the flow and heat transfer characteristics for different values of the material or micropolar parameter K, the Prandtl number Pr and the conjugate parameter γ are analyzed and discussed.     

Turbulent free convection heat transfer to power-law fluids from arbitrary geometric configurations

The problem of turbulent free convection heat transfer from curved surfaces to non-Newtonian power-law fluids has been investigated using the Nakayama-Koyama solution methodology. The scheme is designed to deal with bodies of arbitrary geometric configurations and hence can be viewed as a generalized version of the Shenoy-Mashelkar approach for turbulent free convection heat transfer from a flat vertical plate to a power-law fluid. The surface wall temperature is allowed to vary in the streamwise direction in an arbitrary fashion, and calculations are carried out for the turbulent free convection about the horizontal circular cylinder and sphere for illustrative purposes. Available theoretical and experimental data have been compared with the predictions of the present analysis and the comparison of results has been found to be reasonably good.     

Self-similar solution of the unsteady flow in the stagnation point region of a rotating sphere with a magnetic field

The unsteady flow and heat transfer of a viscous incompressible electrically conducting fluid in the forward stagnation point region of a rotating sphere in the presence of a magnetic field are investigated in this study. The unsteadiness in the flow field is caused by the velocity at the edge of the boundary layer and the angular velocity of the rotating sphere, both varying continuously with time. The system of ordinary differential equations governing the flow is solved numerically. For some particular cases, an analytical solution is also obtained. It is found that the surface shear stresses in x- and y-directions and the surface heat transfer increase with the acceleration, the magnetic and the rotation parameters whether the magnetic field is fixed relative to the fluid or body, except that the surface shear stress in x-direction and the surface heat transfer decrease with increasing the magnetic parameter when the magnetic field is fixed relative to the body. For a certain value of the acceleration parameter, the surface shear stress in the x-direction vanishes while the surface shear stress in the y-direction and the surface heat transfer remain finite. Also, below a certain value of the acceleration parameter, reverse flow occurs in the x-component of the velocity profile. Received on 18 May 1998