Approximate solutions of heat and momentum transfer in laminar plane wall jet

In the present paper approximate solutions for the fluid and thermal boundary layers in an incompressible laminar plane wall jet with isothermal and adiabatic walls have been studied respectively, and comparisons with the known exact solutions have been made wherever possible. It is found that the present method is simple and straightforward, and gives results being in good agreement with the exact solutions. For moderate values of the Prandtl number the method may be used for calculating the heat transfer from an isothermal wall and temperature recovery factor for an adiabatic wall respectively.Nomenclature a* dimensionless temperature gradient at the wall - c _p specific heat at constant pressure - K momentum flux through a cross-section of the jet - Q volume flux through a cross-section of the jet - r* temperature recovery factor - T temperature of the fluid in the boundary layer - T _r adiabatic wall temperature - T temperature of the fluid at rest - u, v velocity components along and normal to the plane wall respectively - x, y rectangular coordinates along and normal to the plane wall respectively - z Greek symbols fluid boundary layer thickness - _t, _T thermal boundary layer thickness for an isothermal and an adiabatic wall respectively - dimensionless y-coordinate - dimensionless temperature difference (T–T )/T - coefficient of thermal conductivity - coefficient of viscosity - coefficient of kinematic viscosity - Prandtl number - _w shearing stress on the plane wall     

A note on heat transfer in laminar flow through a gap

Summary The problem of heat transfer in laminar flow through a gap between two semi-infinite parallel plates at constant temperature was recently studied by Agrawal¹⁾. He solved this problem with the use of infinite Fourier sine series and derived an expression for the local temperature profile and the local Nusselt number as a function of the distance along the gap. A detailed solution for Péclèt number Pe=1 is given. Far enough from the entrance of the gap the local temperature profile of the fluid is almost independent of it's initial temperature. In this paper this limit temperature profile is expressed with the confluent hypergeometric function and the corresponding Nusselt number as a function of Pe is calculated.     

Convective heat transfer characteristics of laminar pulsating pipe air flow

 Heat transfer characteristics to laminar pulsating pipe flow under different conditions of Reynolds number and pulsation frequency were experimentally investigated. The tube wall of uniform heat flux condition was considered. Reynolds number was varied from 780 to 1987 while the frequency of pulsation ranged from 1 to 29.5 Hz. The results showed that the relative mean Nusselt number is strongly affected by pulsation frequency while it is slightly affected by Reynolds number. The results showed enhancements in the relative mean Nusselt number. In the frequency range of 1–4 Hz, an enhancement up to 30% (at Reynolds number of 1366 and pulsation frequency of 1.4 Hz) was obtained. In the frequency range of 17–25 Hz, an enhancement up to 9% (at Reynolds number of 1366 and pulsation frequency of 17.5 Hz) was indicated. The rate of enhancement of the relative mean Nusselt number decreased as pulsation frequency increased or as Reynolds number increased. A reduction in relative mean Nusselt number occurred outside these ranges of pulsation frequencies. A reduction in relative mean Nusselt number up to 40% for pulsation frequency range of 4.1–17 Hz and a reduction up to 20% for pulsation frequency range of 25–29.5 Hz for Reynolds numbers range of 780–1987 were considered. This reduction is directly proportional to the pulsation frequency. Empirical dimensionless equations have been developed for the relative mean Nusselt number that related to Reynolds number (750 < Re < 2000) and the dimensionless frequency (3<Ω<18) with about 10% rms. Received on 16 May 2000 / Published online: 29 November 2001     

Convective heat transfer in a locally heated plane incompressible fluid layer

The problem of convection in a plane horizontal layer of incompressible fluid with rigid boundaries when the temperature is constant on the lower boundary and has a parabolic profile on the upper boundary can be reduced to solution of a system of time-dependent one-dimensional equations. An analytic solution of the problem is obtained directly at the extremum point. Together with the wellknown solutions which describe heat transfer for the linear temperature distribution on the boundaries, the results obtained make it possible to calculate the heat flux through a thin slit for an arbitrary given heating of a thin fluid layer between heat-conducting bodies.     

The effect of velocity distribution on forced convection laminar flow heat transfer in a pipe at constant wall temperature

In this paper, a theoretical study of heat transfer to a fluid of vanishing viscosity in laminar flow in a pipe is made. The constant wall temperature boundary condition is considered in order to facilitate comparison with other classical solutions. Using velocity profiles of simple geometrical shape, the dependence of the heat transfer on velocity distribution is illustrated. Because of the nature of the idealised flow and heat transfer models, the theoretical results are applicable to all axisymmetric flows. Accordingly, some account of the possible effects of swirl on heat transfer in real flows is given.

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Zusammenfassung Es handelt sich um eine theoretische Untersuchung des Wärmeübergangs in laminarer Rohrströmung bei verschwindender Viskosität. Zum Vergleich mit anderen klassischen Lösungen wurde konstante Wandtemperatur als Randbedingung vorgegeben. Unter Benutzung von Geschwindigkeitsprofilen einfacher Geometrie wurde deren Einfluß auf den Wärmeübergang ermittelt. Diese Ergebnisse sind wegen der gewählten Strömungs- und Wärmeübergangsmodelle auf alle achsensymmetrischen Strömungen anwendbar. Die mögliche Wirkung einer Wirbelströmung auf den Wärmeübergang wird diskutiert.

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Nomenclature =(k/c) Thermal diffusivity - C, C ₁, C₂, C₃, C_n Constants - c Specific heat at constant pressure - D=(2r_w) Diameter - k Thermal conductivity - M _n Root of Bessel Equation,J ₀(M_n)=0 - r Radius - T Temperature - u, Velocity, average velocity - x Axial distance - X, R Function ofx, (r) alone - _n (= 2M _n/r _w ² ) Eigen value - Dynamic viscosity - (=/) Kinematic viscosity - Density - (=(T-T _w)/(T₁-T_w)) Dimensionless temperature - (=(T–T _w)/(T ₁–T _w)) Nusselt number - Pe (=Re·Pr) Péclet number - Pr (= c/k) Prandtl number - Re(=2r_w·v) Reynolds number Suffixes b Bulk - 1 Inlet - w wall     

Similarity conditions for heat transfer in incompressible two-dimensional laminar boundary layer flow

Summary Similarity conditions are presented for the solution of some problems of heat transfer in incompressible two-dimensional boundary layer flow. The treatment holds for forced convection as well as for free convection. For free convection no a priori restriction is made with respect to geometry or temperature distribution of the solid surface. For forced convection the treatment is restricted to uniform bulk flow parallel to a flat surface of non-uniform temperature or heat flux. The results are summarized in some tables that facilitate comparison with older work.     

Effect of flow pulsation on laminar heat transfer between two parallel plates

The problem of heat transfer in viscous laminar pulsatile flow between two parallel plates is solved by means of a finite difference method. Boundary conditions of constant wall temperature and constant wall heat flux are considered separately. The numerical results show that flow pulsations change the instantaneous Nusselt number, but do not have any significant effect on the time-averaged values. A trend in reduction of timeaveraged Nusselt number is observed when the amplitude of flow pulsation increases and the frequency decreases. The validity of the result is limited to the case when no flow reversal exists.     

Anomalous heat transfer and drag in laminar flow of viscoelastic fluids

A boundary-layer approach is used to derive an expression for the heat transfer coefficient when a viscoelastic fluid flows past a cylinder. The heat-transfer coefficient becomes independent of velocity at large values of the latter, thus explaining the experimental results of James and Acosta. The agreement with the observations of these authors is, however, only qualitative, since their experiments were carried out at relatively low Reynolds numbers. By changing the exponent of the Reynolds number from the boundary-layer approximation value to one which is valid in the range used by James and Acosta, a correlation is suggested which is in satisfactory agreement with their data. Similar considerations are used to explain the additional experimental observation of James and Acosta, i.e. that the drag coefficient for the flow past a cylinder also becomes independent of velocity at large values of the latter quantity.     

Optimization of heat transfer through finned dissipators cooled by laminar flow

In the present work, the problem of optimizing the shape and the spacing of the fins of a thermal dissipator cooled by a fluid in laminar flow is studied. For a particular finned conduit, the velocity and temperature distributions on the transversal section are determined with the help of a finite element model and a global heat transfer coefficient is calculated. A polynomial lateral profile is proposed for the fins and the geometry is optimized in order to make the heat transfer coefficient as high as possible with the smallest dimensions or the lowest hydraulic resistance to the flow. The optimum fin profile and spacing, obtained by means of a genetic algorithm, are finally shown for different situations. Increases of 45% are obtained in the heat transfer coefficient referring to the maximum values which can be obtained with rectangular fin profiles.     

Local convective heat transfer for laminar and turbulent flow in a rotor-stator system

Abstract The aim of this work is to show a better comprehension of the flow structure and thermal transfer in a rotor-stator system with a central opening in the stator and without an airflow imposed. The experimental technique uses infrared thermography to measure the surface temperatures of the rotor and the numerical solution of the steady-state heat equation to determine the local heat transfer coefficients. Analysis of the flow structure between the rotor and the stator is conducted by PIV. Tests are carried out for rotational Reynolds numbers ranging from 5.87×10⁴ to 1.4×10⁶ and for gap ratios ranging from 0.01 to 0.17. Analysis of the experimental results has determined the influence of the rotational Reynolds number, the gap ratio and systems geometry on the flow structure, and the convective exchanges in the gap between the rotor and the stator. Some correlations expressing the local Nusselt number as a function of the rotational Reynolds number and the gap ratio are proposed.     

Melting heat transfer in steady laminar flow over a moving surface

The steady laminar boundary layer flow and heat transfer from a warm, laminar liquid flow to a melting surface moving parallel to a constant free stream is studied in this paper. The continuity, momentum and energy equations, which are coupled nonlinear partial differential equations are reduced to a set of two nonlinear ordinary differential equations, before being solved numerically using the Runge–Kutta–Fehlberg method. Results for the skin friction coefficient, local Nusselt number, velocity profiles as well as temperature profiles are presented for different values of the governing parameters. Effects of the melting parameter, moving parameter and Prandtl number on the flow and heat transfer characteristics are thoroughly examined. It is found that the problem admits dual solutions.     

Effect on the flow and heat transfer characteristics for sinusoidal pulsating laminar flow in a heated square cylinder

Two-dimensional numerical simulation is performed to understand the effect of flow pulsation on the flow and heat transfer from a heated square cylinder at Re = 100. Numerical calculations are carried out by using a finite volume method based on the pressure-implicit with splitting of operators algorithm in a collocated grid. The effects of flow pulsation amplitude (0.2 ≤ A ≤ 0.8) and frequency (0 ≤ f _p  ≤ 20 Hz) on the detailed kinematics of flow (streamlines, vorticity patterns), the macroscopic parameters (drag coefficient, vortex shedding frequency) and heat transfer enhancement are presented in detail. The Strouhal number of vortices shedding, drag coefficient for non-pulsating flow are compared with the previously published data, and good agreement is found. The lock-on phenomenon is observed for a square cylinder in the present flow pulsation. When the pulsating frequency is within the lock-on regime, time averaged drag coefficient and heat transfer from the square cylinder is substantially augmented, and when the pulsating frequency in about the natural vortex shedding frequency, the heat transfer is also substantially enhanced. In addition, the influence of the pulsating amplitude on the time averaged drag coefficient, heat transfer enhancement and lock-on occurrence is discussed in detail.     

The effect of blowing or suction on laminar free convective heat transfer on flat horizontal plates

In the present paper laminar free convective heat transfer on flat permeable horizontal plates is investigated. To assess the effect of surface suction or injection on heat transfer a correction factor, provided by the film model (or “film theory”), is applied. Comparing the film model predictions with numerical results of previous boundary layer analyses yields good agreement for a wide range of dimensionless transpiration levels.     

Conjugate heat transfer in thermally developing laminar flow with viscous dissipation effects

In this study, thermally developing laminar forced convection in a pipe including viscous dissipation and wall conductance is investigated numerically. The constant heat flux is assumed to be imposed at the outer surface of the pipe wall. The finite volume method is used. The distributions for the developing temperature and local Nusselt number in the entrance region are obtained. The dependence of the results on the Brinkman number and the dimensionless thermal conductivity are shown. The viscous heating effect on the wall is shown. Significant viscous dissipation effects have been observed for large Br.     

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.     

The conjugate heat transfer from an internal heated small strip in a forced laminar flow

 An asymptotic and numerical investigation was conducted for the cooling process, by a forced laminar flow, of a small strip with a non-uniform heat source. The nondimensional temperature distribution in the strip has been obtained as a function of the following parameters: (a) the intensity and distribution of the internal heat sources, (b) the aspect ratio of the strip, (c) the longitudinal heat conductance of the strip and (d) the Prandtl number of the fluid. Both the thermally thin as the thick wall approximations were considered in this paper. The total thermal energy or averaged temperature of the strip is found to decrease as the influence of the longitudinal heat conduction effects in the strip decreases in the thermally thin wall regime. After reaching a minimum, it increases again in the thermally thick wall regime. Received on 19 May 2000     

Numerical simulation of laminar flow and heat transfer over banks of staggered cylinders

A study is conducted to investigate forced convective flow and heat transfer over a bank of staggered cylinders. Using a novel numerical formulation based on a non‐orthogonal collocated grid in a physical plane, the effects of Reynolds number and cylinder spacing on the flow and heat transfer behaviour are systematically studied. It is observed that both the Reynolds number and cylinder spacing influence the recirculatory vortex formation and growth in the region between the cylinders; in turn, the rates of heat transfer between the fluid and the staggered cylinders are affected. As the cylinder spacing decreases, the size and length of eddies reduce. For sufficiently small spacings, eddy formation is completely suppressed even at high Reynolds number. Pressure drop and Nusselt number predictions based on numerical study are in excellent agreement with available correlations. The study provides useful insight on the detailed flow and heat transfer phenomena for the case of a bank of staggered cylinders. Copyright © 2002 John Wiley & Sons, Ltd.     

Calculation of heat transfer in laminar pipe flow of liquids with structural viscosity

Expressions have been obtained for the dimensionless heat-transfer criteria of liquids with structural viscosity for the conditions t_w=const and q_w=const in the case of laminar quasi-isothermal flow.     

Three-dimensional laminar flow and heat transfer in the entrance region of trapezoidal ducts

The laminar convective flow and heat transfer in a duct with a trapezoidal cross-sectional area are studied numerically. The governing equations are solved numerically by a finite volume formulation in complex three-dimensional geometries using co-located variables and Cartesian velocity components. Details of the numerical method are presented. The accuracy of the method was also established by comparing the calculated results with the analytical and numerical results available in the open literature. The Nusselt numbers are obtained for the boundary condition of a uniform wall temperature whereas the friction factors are calculated for no-slip conditions at the walls. The asymptotic values of the Nusselt numbers, friction factors. incremental pressure drops, axial velocity and momentum rate and kinetic energy correction factors approach the available fully developed values. Various geometrical dimensions of the cross-section are considered.