Combined radiative and convective heat transfer in a three-dimensional rectangular channel at different wall temperatures

 This paper deals with a numerical study of combined convective and radiative heat transfer in a three-dimensional rectangular duct with hydrodynamically and thermally developing laminar flow. The gas is assumed to be an incompressible, absorbing, emitting, isotropically scattering, gray medium. Isothermal, gray, diffuse boundary walls at different temperatures are assumed. The finite-volume method (FVM) is adopted to describe both convective and radiative heat transfer. The coupled continuity and momentum equations are solved by means of SIMPLER algorithm. Numerical results for the radiative flux show very good agreement with the available data. The effects of aspect ratio, optical thickness, scattering albedo and wall emissivity on the mean bulk temperature are also investigated. By splitting the heat flux into convective and radiative contributions, the relative importance of these components is assessed for a typical range of values of the parameters. Received on 9 February 1999     

Turbulent boundary layer on a mildly curved convex surface

Extensive single point turbulence measurements made in the boundary layer on a mildly curved heated convex wall show that the turbulence heat fluxes and Stanton number are more sensitive to a change in wall curvature than the Reynolds stresses and skinfriction coefficient, and that downstream, as the flow adjusts to new curved conditions, the St/c _f ratio of Reynolds analogy is appreciably lower than in plane wall flow for the same conditions. Details of the turbulence structure in unheated flow have been documented in an earlier paper; temperature field measurements now described comprise mean temperature distributions, the streamwise variation of wall heat flux, profiles of the temperature variance, transverse and streamwise heat fluxes, and triple correlations. Turbulent diffusion of heat flux is drastically reduced even by mild curvature; changes in the heat fluxes are of the same order as changes in the shear stress, that is, an order of magnitude greater than the ratio of boundary layer thickness to wall radius of curvature. The data include plane flow measurements taken in a developed boundary layer upstream of a change in wall curvature.     

Nonmonotonic dependence of the convective heat flux on the rate of blowing of opaque admixtures into a radiating H-He shock layer

The results are given of calculations of radiative and convective heat transfer in a radiating H-He shock layer in the neighborhood of the stagnation point of a blunt body when graphite ablation products are blown from the surface. It is found that under the conditions in the shock layer characteristic for motion of the body in the atmosphere of Jupiter [3] the dependence of the convective flux on the blowing rate is essentially nonmonotonic. The maximal value is comparable with the radiative flux to the surface under these conditions. It is shown that a decisive part in the mechanism which increases the convective flux is played by the presence near the surface of particles which effectively absorb radiative energy in the spectral regions in which an appreciable radiation flux reaches the boundary layer; the difference between the transport properties of the blown and the oncoming gases is also important.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 106–113, May–June, 1981.     

Conjugate heat transfer with radiation from a vertical circular pin in a non-newtonian ambient medium

An analysis is presented for the heat transfer characteristics of a steady, laminar, mixed convective flow of power-law type non-Newtonian fluid over a circular pin by the conjugate convection-conduction theory including radiative effects under optically thick limit approximation. Numerical results are presented for the dimensionless heat transfer coefficients, local and overall heat fluxes and temperature distribution of the pin by a simultaneous solution of the convective boundary layer equations of the fluid and the energy equation of the pin.     

Thermal radiation effects on MHD flow past a semi-infinite inclined plate in the presence of mass diffusion

MHD mixed free-forced heat and mass convective steady incompressible laminar boundary layer flow of a gray optically thick electrically conducting viscous fluid past a semi-infinite inclined plate for high temperature and concentration differences is studied. A uniform magnetic field is applied perpendicular to the plate. The density of the fluid is assumed to reduce exponentially with temperature and concentration. The usual Boussinesq approximation is neglected due to the high temperature and concentration differences between the plate and the ambient fluid. The Rosseland approximation is used to describe the radiative heat flux in the energy equation. The boundary layer equations governing the flow are reduced to ordinary differential equations, which are numerically solved by applying an efficient technique. The effects of the density/temperature parameter n, the density/concentration parameter m, the local magnetic parameter M_x and the radiation parameter R are examined on the velocity, temperature and concentration distributions as well as the coefficients of skin-friction, heat flux and mass flux.     

Composite heat transfer in the case of a steady laminar flow of a gray fluid with small optical density past a horizontal plate embedded in a saturated porous medium

The problem of composite heat transfer in a boundary layer over a flat plate embedded in a saturated porous medium involving a gray fluid of small optical density is analysed. The analysis is based on treating the convective and radiative transport processes separately. Two flux model has been used for the radiative transport process. The results of the analysis show that the radiation flux at the wall increases with increase in porous parameter whereas, the opposite behaviour is observed at the edge of the boundary layer for a given value of the optical density and the temperature ratio between the wall and the free stream. However, the radiation flux always decreases with increase in optical density when the porous parameter and the temperature ratio are kept constant. The total heat flux which is the sum of the radiative and convective transport process increases as conduction to radiation parameter AT decreases.Das Problem von zusammengesetzter Wärmeübertragung in einer Grenzschicht über einer von einem grauen Fluid kleiner optischer Dichte umströmten flachen Platte, die in einem gesättigten porösen Medium eingebettet ist, wird untersucht. Die Berechnung basiert darauf, die Konvektion und den Transport durch Strahlung getrennt zu betrachten. Zwei Strom-Modelle wurden für den Prozeß des Transportes durch Strahlung benutzt. Die Ergebnisse der Untersuchung zeigen für eine gegebene optische Dichte und Temperatur zwischen Wand und freier Strömung, daß der Strahlungsstrom an der Wand mit dem Ansteigen des Porositäts-Parameters zunimmt, wohingegen das entgegengesetzte Verhalten an dem Rand der Grenzschicht beobachtet wird. Der Strahlungsstrom nimmt immer mit zunehmender optischer Dichte ab, wenn der Porositäts-Parameter und die Temperatur konstant gehalten werden. Der gesamte Wärmestrom, der die Summe aus Konvektion und Strahlung darstellt, wächst mit abnehmendem Verhältnis N von Leitung zu Strahlung.     

An experimental technique for measuringtransient natural/forced convective heatfluxes in a vertical channel

A series of systematic experiments for measuring transient natural andforced convected heat fluxes in a one-sided heated vertical channel have beenconducted. The total heat input in transient natural and forced experiments iscomposed of four kinds modes: radiative heat loss, conductiveheat lossm, thermal capacity of the plate, and convective heat transfer in thetest channel. In the transient periods, the generalized correlations for estimatingtransient convective heat flux are proposed in natural and forced convectionexperiments. By using the proposed q_c^″ distributions, satisfactory agreementsare achieved between the transient Nu_x predictions and experimentaldata.     

Experimental investigation of heat transfer in vertical upward and downward supercritical CO2 flow in a circular tube

An experimental investigation of turbulent heat transfer in vertical upward and downward supercritical CO₂ flow was conducted in a circular tube with an inner diameter of 4.5 mm. The experiments were performed for bulk fluid temperatures from 29 to 115 °C, pressures from 74.6 to 102.6 bar, local wall heat fluxes from 38 to 234 kW/m², and mass fluxes from 208 to 874 kg/m² s. At a moderate wall heat flux and low mass flux, the wall temperature had a noticeable peak value for vertical upward flow, but increased monotonically along the flow direction without a peak value for downward flow. The ratios of the experimental Nusselt number to the value obtained from a reference correlation were compared with Bo^* and q⁺ distributions to observe the buoyancy and flow-acceleration effects on heat transfer. In the experimental range of this study, the flow acceleration predominantly affected the heat-transfer phenomena. Based on an analysis of the shear-stress distribution in the turbulent boundary layer and the significant variation of the specific heat across the turbulent boundary layer, a new heat-transfer correlation for vertical upward and downward flow of supercritical pressurized fluid was developed; this correlation agreed with various experimental datasets within ±30%.     

Calculation of convective heat flux in the vicinity of the stagnation point for partially ionized gas flow past a body

From numerical solutions of the boundary layer equations for a four-component gas mixture (E, N⁺, N₂, and N) with gas injection, approximate formulas for the heat flux as a function of the variation of λρ/c_p and h^* across the boundary layer and the magnitude of the objection are obtained (λ is the thermal conductivity of the mixture,ρ is density, c_p is the specific heat, and h^* is the enthalpy of the ideal gas state of the mixture). An effective ambipolar diffusion coefficient D^(a)(i) is introduced, making possible finite formulas for the convective heat fluxes in the “frozen” boundary layer. We study the behavior of these coefficients within the boundary layer. A formula is obtained for convective heat flux to the wall from partially ionized air for a nine-component mixture (E, O⁺, N⁺, NO⁺, O, N, NO, O₂ N₂). Even for simpler four-component gas model three effective ambipolar diffusion coefficients are necessary: $$\begin{gathered} D^{(a)} (A) = D (A, M) D^{(a)} (I) = 2D (A, M), \hfill \\ D^{(a)} (M) = [ 1 + c_e (I)] D(A, M). \hfill \\ \end{gathered} $$ Here D(A, M) is the binary diffusion coefficient of the atoms into molecules, and c_e(I) is the ion concentration at the outer edge of the boundary layer. The assumption of an infinitely large charge-exchange cross section and the other simplifying assumptions used in [1] lead to overestimation of the magnitude of the dimensionless heat flux by 7–15% for the “frozen” boundary layer case.     

Numerical study of turbulent flow and heat transfer of Al2O3–water mixture in a square duct with uniform heat flux

Turbulent flow and convective heat transfer of a nanofluid made of Al₂O₃ (1–4 vol.%) and water through a square duct is numerically studied. Single-phase model, volumetric concentration, temperature-dependent physical properties, uniform wall heat flux boundary condition and Renormalization Group Theory k-ε turbulent model are used in the computational analysis. A comparison of the results with the previous experimental and numerical data revealed 8.3 and 10.2 % mean deviations, respectively. Numerical results illustrated that Nu number is directly proportional with Re number and volumetric concentration. For a given Re number, increasing the volumetric concentration of nanoparticles does not have significant effect on the dimensionless velocity contours. At a constant dimensionless temperature, increasing the particle volume concentration increases the size of the temperature profile. Maximum value of dimensionless temperature increases with increasing x/D_h value for a given Re number and volumetric concentration.     

Radiative heat transfer in absorbing, emitting, and anisotropically scattering boundary-layer flows with reflecting boundary

The heat transfer in absorbing, emitting, and anisotropically scattering boundary-layer flows with reflecting boundary over a flat plate, over a 90-deg wedge, and in stagnation flow is solved by application of the Galerkin method with the particular solution boundary condition I _p (τ₀,ξ,?μ) of the equation of radiative transfer for an inhomogeneous term and the Box method. The exact integral expressions for the radiation part of this problem are developed. The coupling between convective and radiative heat transfer in boundary-layer flows is described by a set of nonlinear simultaneous equations including differential equations and integrodifferential equations. The Galerkin method and the particular solution boundary condition I _p (τ₀,ξ,?μ) are used to analyze the radiation part of the problem. The nonsimilar boundary-layer equations are solved by the Box method. The present numerical procedure solutions are compared in tables with the other exact treating results, the P-3, and P-1 approximation methods for the case of isotropically scattering boundary-layer flows. The effects of linearly anistropically scattering and reflecting surface are taken into account. It is found that the present method is a reliable and efficient numerical procedure and scattering leads to a reduction in the total heat flux. The influence of the forward-backward scattering parameter on the total heat flux decreases with the increase of the surface reflectivity.     

Solution of the equations of motion for a selectively radiating gas in a shock layer

Numerical solutions are obtained for the system of integro-differential equations describing the flow of a viscous, heat-conducting, selectively radiating gas in the region between the shock wave and a blunt body. The calculations are made for bodies of radius from 0.1 to 3 m with stagnation temperature from 6000° to 15 000° K. As a result of the calculations the convective and radiative thermal fluxes in the vicinity of the stagnation point are obtained. The effect of injection on convective and radiative heat transfer is studied.The first calculations of radiative thermal fluxes in air were made about 10 years ago in [1,2]. However, the results did not take account of the effects of emission and reabsorption, nor the interaction of the convective and radiative heating processes. These effects have been studied primarily with the use of simplified models of a radiating gas. Most often the approximation used is that of a gray gas with absorption coefficient which is independent of wavelength ([3–6] and others).The appearance in the literature of quite detailed data on the selective spectral absorption coefficients of air over a wide temperature range [7,8] has made it possible to solve the direct problem of calculating the flow field of a selectively radiating gas behind a shock wave with account for all the effects mentioned above.     

Laminar forced convection conjugate heat transfer over a flat plate

Coupled heat transfer between laminar forced convection along and conduction inside a flat plate wall is theoretically studied. The laminar convective boundary layer is analyzed by employing the integral technique. The energy equations for the fluid and the plate wall are combined under the condition of the continuity in the temperature and heat flux at the fluid-solid interface. The analysis results in a simple formal solution. Expressions have been obtained for calculating local Nusselt number, wall heat flux and temperature along the plate, all are functions of the local Brun number, Br _x , which is a measure of the ratio of the thermal resistance of the plate to that of the convective boundary layer. The results indicate that for Br _x ≥0.15, neglecting the plate resistance will results in an error of more than 5% in Nusselt number. Comparison of the present solution with other previous studies has been made. The solution may be of a considerable theoretical and practical interest. Received on 19 August 1998     

Heat transfer in a hydromagnetic flow over a stretching sheet

Exact solutions are obtained for the heat transfer in an electrically conducting fluid past a stretching sheet subjected to the thermal boundary with either a prescribed temperature or a prescribed heat flux in the presence of a transverse magnetic field. The solutions for the heat transfer characteristics are evaluated numerically for different parameters, such as the magnetic parameterN, the Prandtl numberPr, the surface temperature indexs, and the surface heat flux indexd. It is observed that for the prescribed surface temperature case the fluid temperature increases due to the existance of the magnetic field, and decreases as the Prandtl number or the surface temperature index increases; for the prescribed surface heat flux case, the surface temperature decreases as the Prandtl number of the surface heat flux index increases, and the magnetic parameter decreases. In addition, varying the prescribed surface temperature indexs affects the mechanism of heat transfer.     

Planar solidification of a warm flowing-liquid under different boundary conditions

Planar solidification of a warm flowing liquid with the convective heat transfer to the growing solid layer, has been analysed for the boundary conditions of constant temperature, constant heat flux and convective heat flux at the surface respectively. The mathematical formulation of the problem resulted in a coupled set of two differential equations in temperature and solid thickness as function of position, time and the problem parameters. Analytical expressions for the temperature distribution within the growing solid layer, the rate of solidification and the solidification time are obtained. The perturbation techniques employed here is simple and straight forward in contrast with the earlier techniques. Good agreement between the experimental results and the present solutions is obtained for the convective heat flux boundary condition. The results of this analysis are useful in the design and analysis of experiments dealing with freezing/melting in one dimension. The role of the parameter Stefan number which is small for phase change materials, is discussed in context with the storage of thermal energy.     

Velocity and temperature distributions on a semi-infinite flat plate embedded in a saturated porous medium

In this paper the velocity and temperature distributions on a semi-infinite flat plate embedded in a saturated porous medium are obtained for the governing equations (Kaviany [7]) following the technique adopted by Chandrashekara [2] which are concerned with the interesting situations of the existence of transverse, velocity and thermal boundary layers. Here the pressure gradient is just balanced by the first and second order solid matrix resistances for small permeability and observed that by increasing of the flow resistance the asymptotic value for the heat transfer rate increases. Further we concluded that the transverse boundary layers are thicker than that of axial boundary layers. Hence we evaluated the expressions for the boundary layer thickness, the shear stress at the semi-infinite plate and _T (the ratio of the thicknesses of the thermal boundary layer and momentum boundary layer). The variations of these quantities for different values of the porous parameterB and the flow resistanceF have been discussed in detail with the help of tables. The curves for velocity and temperature distributions have been plotted for different values ofB andF.Lastly we have evaluated the heat fluxq(x) and found that it depends entirely upon the Reynolds numberRe, Prandtl numberPr,B andF.     

A simple equilibrium model for shallow-cumulus-topped mixed layers

A new equilibrium model for shallow-cumulus-topped mixed layers is presented. A variant on the w _* closure for the shallow-cumulus mass flux is applied that retains the convective area-fraction in its formulation. As opposed to being constant, the fraction is explicitly modeled using a statistical closure as a function of the saturation deficit and humidity variance at cloud base. As a consequence, important new interactions are introduced between the convective transport, humidity, and depth of the mixed layer. This mechanism, which we call the mass-flux humidity feedback, helps determine the character of the equilibrium state such that the mixed-layer top is maintained close to the cloud-base height. Due to the strong sensitivity of the mass flux to the area fraction, the latter thus acts as a regulator or valve mechanism on moist convective transport. As a consequence, the mixed-layer model is able to explain the robustness of many aspects of the shallow-cumulus boundary layer that is typically found in observations and large-eddy simulations (LESs). The model is evaluated for a single-LES case as well as for global climatology obtained from a 40-year reanalysis of meteorological data by the European Centre for Medium-range Weather Forecasts (ECMWF). LES characteristics of convective mass flux, cloud fraction, humidity variance, cloud-base height, and surface fluxes of heat and humidity are reproduced. The solution on reanalysis fields reproduces the spatial structure of mixed-layer temperature and humidity and their associated surface fluxes in the subtropical Atlantic and Pacific trade wind regions. Furthermore, the spatial structure of the convective area-fraction matches that of synoptic surface observations of frequency of occurrence of shallow cumulus. Particularly striking is the smooth onset of the convective area-fraction and mass flux along the trade-wind trajectory that is reproduced, from zero to typical trade-wind values. The cumulus onset represents the necessity for shallow-cumulus mass flux to occur in order to close the mixed-layer budgets of heat, moisture, and mass, as a response to the changing magnitude of large-scale subsidence and free tropospheric humidity along the trajectory. Finally, the mass flux model is implemented in an intermediate-complexity tropical climate model to study its behavior when fully interactive with the larger-scale flow. A climate run then shows that the model is stable, due to the mass-flux humidity feedback acting to keep the shallow-cumulus boundary layer close to its equilibrium state for long, climatological timescales.     

Investigation of heat flux at the forward critical point of bodies placed in a high-temperature subsonic stream

We present the results of measurements of the total heat flux at the forward critical point of axially symmetric models placed in a high-temperature subsonic stream of air and nitrogen in an apparatus which uses a high-frequency inductive discharge [1] to heat the gas. The heat fluxes were measured for cylindrical models whose forward part had one of three possible shapes: a hemisphere, a hemisphere with a blunt nose, or a flat circular end-face. A water-cooled calorimeter sensor was set up at the forward critical point of the model; the calorimeter sensor was made of different materials, so that it was possible to estimate the radiant and convective components of the total heat flux and determine the effect of the sensor material on the heat flux measured. The convective component of the heat flux was compared to a calculated value obtained by Fay and Riddell's formula. The heat-flux values found for two shapes of models were used in determining the effective radius of streamline flow for a model with a plane end-face.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 133–141, September–October, 1973.The authors are grateful to N. I. Nesterov for his help with the work.     

Effect of surroundings on the transient energy transfer in a layer of radiating gas

Summary An analysis is presented for the transient cooling or heating of a stagnant layer of hot radiating gas surrounded by a cold gas capable of absorbing and emitting radiation. Scattering of radiation is neglected, and energy transfer by conduction and convection is considered to be negligible compared with radiation. The gas is assumed to be perfect and in local thermodynamic equilibrium. The heating of a cold gas by a diffuse and a collimated radiation flux incident on the boundary of the gas from some external source is considered, and the dependence of physical and radiative properties on temperature is taken into account. The problem is formulated exactly using radiative transfer theory. A scheme is developed for the numerical solution of the nonlinear integrodifferential equations of energy conservation. Starting with arbitrary, but given, initial temperature profiles, temperature distributions and local radiative fluxes are predicted as a function of time for a wide range of physically interesting conditions.     

Problems of heat transfer and thermal protection of entry vehicles for an unmanned mission to mars

The problems of heat transfer and thermal protection for an orbitalMartian lander touching down on the planetary surface are investigated. A broad scenario of the mission is given and several possible aerodynamic shapes are considered. Several versions of the landing on the planetary surface are studied. The version with intermediate orbiting of an artificial Mars satellite using aerodynamic deceleration in the atmosphere is adopted as the main variant. The landing on the planetary surface is realized from satellite orbit. This landing pattern requires reusable thermal protection. The convective and radiative heat fluxes are calculated at characteristic points on the surface of a vehicle of the chosen shape. For this shape the necessary weight of thermal protective coating consisting of indestructible reusable TZMK material, used previously for shielding the Buran orbiter, is determined.