Viscous dissipation effects on thermal entrance region heat transfer in pipes with uniform wall heat flux

The analytical solution to Graetz problem with uniform wall heat flux is extended by including the viscous dissipation effect in the analysis. The analytical solution obtained reduces to that of Siegel, Sparrow and Hallman neglecting viscous dissipation as a limiting case. The sample developing temperature profiles, wall and bulk temperature distributions and the local Nusselt number variations are presented to illustrate the viscous dissipation effects. It is found that the role of viscous dissipation on thermal entrance region heat transfer is completely different for heating and cooling at wall. In the case of cooling at wall, a critical value of Brinkman number, Br _c=−11/24, exists beyond which (−11/24<Br<0) the fluid bulk temperature will always be less than the uniform entrance temperature indicating the predominance of cooling effect over the viscous heating effect. On the other hand, with Br < Br _c the bulk temperature T _b will approach the wall temperature T _w at some downstream position and from there onward the bulk temperature T _b becomes less than the wall temperature T _w with T _w > B _b > T ₀ indicating overall heating effect for the fluid. The numerical results for the case of cooling at wall Br < 0 are believed to be of some interest in the design of the proposed artctic oil pipeline.     

Effect of viscous dissipation on mixed convection heat transfer in a vertical tube with uniform wall heat flux

 The laminar and parallel flow of a Newtonian fluid in a vertical cylindrical duct with circular cross section has been analysed. Both the viscous dissipation effect and the buoyancy effect have been taken into account. The momentum balance equation and the energy balance equation have been solved by means of a perturbation method, in the case of a uniform heat flux prescribed at the wall of the duct. The velocity distribution, the temperature distribution, the Nusselt number and the Fanning friction factor have been evaluated analytically. Moreover, the velocity and temperature of the fluid have been compared with those obtained in two special cases: forced convection with viscous dissipation (i.e. negligible buoyancy effect); mixed convection with negligible effects of viscous dissipation. Received on 26 June 2000     

Effect of ion slip on the time-varying Hartmann flow of a non-Newtonian viscoelastic fluid with heat transfer

Ion slip in a time-varying Hartmann flow of a conducting incompressible non-Newtonian viscoelastic fluid between two parallel horizontal insulating porous plates is studied with allowance for heat transfer. A uniform and constant pressure gradient is applied in the axial direction. An external uniform magnetic field and uniform suction and injection through the surface of the plates are applied in the normal direction. The two plates are maintained at different but constant temperatures; the Joule and viscous dissipations are taken into consideration. Numerical solutions for the governing momentum and energy equations are obtained with the use of finite differences, and the effect of various physical parameters on both the velocity and temperature fields is discussed.     

Hall and ion slip effects on peristaltic flow and heat transfer analysis with Ohmic heating

The peristaltic transport of a magnetohydrodynamic （MHD） fluid is exam- ined for both symmetric and asymmetric channels. Hall and ion slip effects are taken into account. The heat transfer is analyzed by considering the effects of viscous and Ohmic dissipations. The relevant flow problems are first modeled, and then the closed form solutions are constructed under the assumptions of long wavelength and low Reynolds number. The solutions are analyzed through graphical illustration. It is noted that the velocity increases but the temperature decreases with the increases in the Hall and ion slip parameters. The axial pressure gradient is less in magnitude in the presence of Hall and ion slip currents. The Hall and ion slip effects are to decrease the maximum pres- sure against which peristalsis works as a pump. The free pumping flux decreases with the increases in the Hall and ion slip parameters. The increases in the Hall and ion slip parameters result in an increase in the size of the trapped bolus.     

Laminar thermally developing flow in rectangular channels and parallel plates: uniform heat flux

Numerical simulations were conducted for thermally developing laminar flow in rectangular channels with aspect ratios ranging from 1 to 100, and for parallel plates. The simulations were for laminar, thermally developing flow with H1 boundary conditions: uniform heat flux along the length of the channel and constant temperature around the perimeter. In the limit as the non-dimensional length, x* = x/(D _h RePr), goes to zero, the Nusselt number is dependent on x* to the negative exponent m. As the non-dimensional length goes to infinity the Nusselt number approaches fully developed values that are independent of x*. General correlations for the local and mean heat transfer coefficients are presented that use an asymptotic blending function to transition between these limiting cases. The discrepancy between the correlation and the numerical results is less than 2.5 % for all aspect ratios. The correlations presented are applicable to all aspect ratios and all non-dimensional lengths, and decrease the discrepancy relative to existing correlations.     

MHD flow and heat transfer over a stretching surface with prescribed wall temperature or heat flux

The flow and heat transfer over a stretching sheet with a magnetic field in an electrically conducting ambient fluid have been studied. The effects of the induced magnetic field and sources or sinks have been included in the analysis. Both non-isothermal wall and constant heat flux conditions have been considered. The governing equations have been solved numerically using a shooting method. It is observed that for the prescribed wall temperature the skin friction, induced magnetic field at the wall and heat transfer are enhanced due to the magnetic field, but in general, they reduce as the reciprocal of the magnetic Prandtl number increases. For constant heat flux case, the temperature at the wall reduces as the magnetic field increases, but it increases with the reciprocal of the magnetic Prandtl number. The heat transfer is strongly affected by the Prandtl number, wall temperature and sink. Whenm<–2 andPr>2.5 the unrealistic temperature distributions are encountered. The present analysis is more general than any previous investigation.

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MHD Strömung und Wärmeübertragung über eine gedehnte Oberfläche mit vorgeschriebener Wandtemperatur oder Wärmestrom

Zusammenfassung In dieser Studie ist die Strömung und Wärmeübertragung über eine gedehnte Fläche mit magnetischem Feld in einem elektrisch leitenden Fluid untersucht worden. Der Einfluß des induzierten magnetischen Feldes und der Quellen oder Senken sind in die Untersuchung einbezogen. Die beiden Fälle, nicht-isotherme Wand und konstanter Wandwärmestrom, sind betrachtet worden. Mit dem Eliminationsverfahren sind bestehende Gleichungen numerisch gelöst worden. Es ist beobachtet worden, daß für eine vorgeschriebene Wandtemperatur die Oberflächenreibung, das induzierte magnetische Feld und die Wärmeübertragung aufgrund des magnetischen Feldes verbessert sind. Aber im allgemeinen reduzieren sie sich im umgekehrten Maß wie die magnetische Prandtlzahl ansteigt. Für den Fall des konstanten Wärmestromes sinkt die Wandtemperatur, wenn das magnetische Feld stärker wird. Die Temperatur steigt jedoch reziprok zur magnetischen Prandtlzahl an. Die Wärmeübertragung ist sehr stark von der Prandtlzahl, Wandtemperatur und der Senke beeinflußt. Bei Werten vonm<–2 undPr2.5 sind unrealistische Temperaturverteilungen eingetreten. Die gezeigte Analyse ist allgemeiner als jede vorhergehende Untersuchung.

     

Momentum and heat transfer in the entrance region of a parallel plate channel: developing laminar flow with constant wall temperature

A new integral method of solution is presented for developing laminar flow and heat transfer in the entrance region of a parallel plate channel with uniform surface temperature. Unlike earlier Karman-Pohlhausen analyses, the new analysis provides solutions which are free from jump discontinuities in the gradients of the velocity and temperature distributions throughout and at the end of the entrance region. The hydrodynamic and thermal results from the present analysis therefore join smoothly and asymptotically to their fully-developed values. The heat transfer results obtained are further found to agree well with previously published numerical solutions.Nomenclature a half width of the channel, m - D _h hydraulic diameter (=4a), m - h local heat transfer coefficient,W/(m²·K) - h _m mean heat transfer coefficient defined by Eq- (9),W/(m²·K) - k thermal conductivity, W/(m·K) - L _H axial length of the hydrodynamic entrance region, m - L _T axial length of the thermal entrance region, m - L _in,H axial length of the hydrodynamic inlet region, m - L _in,T axial length of the thermal inlet region, m - Nu _x local Nusselt number,hD _h /k, dimensionless - Nu _m mean Nusselt number defined by Eq. (9),h _mD_h/k, dimensionless - P pressure, N/m² - P _O pressure at the entrance, N/m² - Pr Prandtl number,c _p /k, dimensionless - Re Reynolds number, 4aU _o /v, dimensionless - T absolute temperature, K - T _b fluid bulk temperature, K - T _c centerline temperature, K - T _w wall temperature, K - U _c centerline velocity, m/s - U ₀ velocity of the fluid at entrance, m/s - U core velocity, m/s - u velocity component inx direction, m/s - v velocity component iny direction, m/s - x spatial coordinate, axial distance, m - y spacial coordinate measured from channel wall, m Greek Letters molecular thermal diffusivity, m²/s - hydrodynamic shape factor, dimensionless - _T thermal shape factor, dimensionless - hydrodynamic boundary layer thickness, m - * /a, dimensionless - _T thermal boundary layer thickness, m - * _{T T} /a, dimensionless - dimensionless distance,y/ ory/a - Pohlhausen's shape factor, dimensionless - dynamic viscosity coefficient, kg/(m·s) - v kinematic viscosity,/, m²/s - dimensionless axial distance,x/(a·Re) - _H dimensionless axial length of the hydrodynamic entrance region (=L _H /(a·Re)) - _T dimensionless axial length of the thermal entrance region (=L _T /(a·Re)) - _in,H dimensionless axial length of the hydrodynamic inlet region (=L _in,H/(a·Re)) - _in,T dimensionless axial length of the thermal inlet region (L _in,T /(a·Re)) - fluid density, kg/m³     

Numerical simulation for laminar flow and heat transfer of gas in rectanglar micropassages with constant wall heat flux

Theoretical investigations were performed on the developed laminar flow and convective heat transfer characteristics for incompressible gases flow through rectanglar micropassages with constant wall heat flux. Mathematical models were proposed for considering the change in viscosity and thermal conductivity of gas in the wall-adjacent region from the kinetic theory. The dimensionless velocity distribution and corresponding pressure drop, the dimensionless temperature distribution and corresponding heat transfer characteristics were both simulated numerically, and the results were compared to other report simulations [10–12] with brief discussions.     

Blasius flow and heat transfer of fourth-grade fluid with slip

This investigation deals with the effects of slip, magnetic field, and non- Newtonian flow parameters on the flow and heat transfer of an incompressible, electrically conducting fourth-grade fluid past an infinite porous plate. The heat transfer analysis is carried out for two heating processes. The system of highly non-linear differential equations is solved by the shooting method with the fourth-order Runge-Kutta method for moderate values of the parameters. The effective Broyden technique is adopted in order to improve the initial guesses and to satisfy the boundary conditions at infinity. An exceptional cross-over is obtained in the velocity profile in the presence of slip. The fourth-grade fluid parameter is found to increase the momentum boundary layer thickness, whereas the slip parameter substantially decreases it. Similarly, the non-Newtonian fluid parameters and the slip have opposite effects on the thermal boundary layer thickness.     

Turbulent heat transfer for pipe flow with uniform heat generation

An Analytic solution is presented of the problem of turbulent heat transfer in pipes with internal heat generation and insulated wall by applying a recently-developed eddy conductivity model. The results agree closely with available experimental data for a wide range of Prandtl number (0.02–10.5).     

Binary diffusion and heat transfer in laminar mixed convection channel flows with uniform wall heat flux: extremely thin film thickness

A detailed numerical study has been performed to investigate the combined heat and mass transfer in laminar mixed convection channel flows with uniform wall heat flux. In an initial effort the liquid film on the channel wall is assumed to be extremely thin in thickness. Major dimensionless groups governing the present problem areGr _T,Gr _Mx,Pr,Sc, φ andRe. Results are specifically presented for an air-water system under various conditions. The effects of wall heating flux, the Reynolds number and the relative humidity of the moist air in the ambient on the momentum, heat and mass transfer in the flow are investigated in great detail.     

Heat transfer to laminar flow between parallel plates with a prescribed wall heat flux

Summary The first three eigenvalues and constants, as well as asymptotic expressions for these quantities, are presented for heat transfer to laminar flow between parallel flat plates with a symmetrically prescribed wall heat flux.     

Effect of slip velocity on Casson thin film flow and heat transfer due to unsteady stretching sheet in presence of variable heat flux and viscous dissipation

The aim of the present paper is to study flow and heat transfer characteristics of a viscous Casson thin film flow over an unsteady stretching sheet subject to variable heat flux in the presence of slip velocity condition and viscous dissipation. The governing equations are partial differential equations. They are reduced to a set of highly nonlinear ordinary differential equations by suitable similarity transformations. The resulting similarity equations are solved numerically with a shooting method. Comparisons with previous works are made, and the results are found to be in excellent agreement. In the present work, the effects of the unsteadiness parameter, the Casson parameter, the Eckert number, the slip velocity parameter, and the Prandtl number on flow and heat transfer characteristics are discussed. Also, the local skin-friction coefficient and the local Nusselt number at the stretching sheet are computed and discussed.     

Effects of heat and mass transfer on peristaltic flow of Williamson fluid in a non‐uniform channel with slip conditions

The combined influence of heat and mass transfer has been explored in a study of peristaltic transport of magnetohydrodynamic Williamson fluid in a non‐uniform channel with flexible walls. The slip conditions are paid due attention and long wavelength and small Reynolds number assumptions are adopted in the problem formulation. The obtained results are valid for small Weissenberg number. A detailed study of involved key parameters in the obtained solutions is made by the sketched graphs. Copyright © 2010 John Wiley & Sons, Ltd.     

Boiling heat transfer region with independence of the wall temperature

 Extensive measurements of the intensive cooling of hot-rolled wires with temperatures between 1000 ^°C and 1100 ^°C are analysed. The analysis proves the existence of a convection-controlled boiling region, which has been previously observed by few authors in the case of high mass fluxes and high liquid subcooling. This region is characterised by an independence of the heat flux of the surface temperature. The heat flux depends essentially on the Reynolds number, the main influence parameter of the single phase convection, and on the liquid subcooling. Received on 13 September 1999