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     

Laminar flow and heat transfer in a periodically converging-diverging channel

A finite difference solution for laminar viscous flow through a sinusoidally curved converging-diverging channel is presented. The physical wavy domain is transformed into a rectangular computational domain in order to simplify the application of boundary conditions on the channel walls. The discretized conservation equations for mass, momentum and energy are derived on a control volume basis. The pseudo-diffusive terms that arise from the co-ordinate transformation are treated as source terms, and the resulting system of equations is solved by a semi-implicit procedure based on line relaxation. Results are obtained for both the developing and the fully developed flow for a Prandtl number of 0.72, channel maximum width-to-pitch ratio of 1.0, Reynolds number ranging from 100 to 500 and wall amplitude-to-pitch ratio varying from 0.1 to 0.25. Results are presented here for constant fluid properties and for a prescribed wall enthalpy only.     

Laminar flow development and heat transfer in converging plane-walled channels

The characteristics of flow development and heat transfer in converging plane-walled channels are studied by the finite difference method. The velocity and temperature profiles in both angular and radial directions, the average Nusselt number and the pressure drop are calculated for three different taper angles. The results show that the transport process is governed by three parameters: the inlet Reynolds number, the Péclet number and the taper angle. The increase of the taper angle yields an increase of the Nusselt number and a decrease of the pressure drop.     

Laminar natural convection heat transfer from vertical plate to power-law fluid

Summary Approximate solutions for laminar natural convection heat transfer between a vertical plate and a power-law fluid with high Prandtl number were obtained using an integral method for cases with various types of boundary conditions. The results were found in good agreement with available experimental evidence.Nomenclature a exponent defined by equations (28) and (29) - A, B, C, D, E constants defined by equations (15) to (19) - C ₁, C ₂, M ₁, M ₂ coefficients for Nusselt number expression defined by (32b), (33b) - f temperature difference, equal to T _s–T - f ⁺ dimensionless temperature difference - g gravitational acceleration - Gr Grashof number defined by (25), (50) and (66), respectively - H heat flux at plate surface - h _x local heat transfer coefficient - K consistency index for Power-law fluid - k thermal conductivity of fluid - K ₁, K ₂ constants defined by (50) and (51) - L height of plate - n flow behavior index for Power-law fluid - P a quantity defined by (54a) - T temperature - T _s plate temperature - T temperature of the bulk of fluid - s constant given by (35) - u velocity component along x-direction - u _x maximum velocity induced by natural convection current, (10) - v velocity component along y-direction - x distance measured along direction parallel to that of gravitational force - x ⁺ dimensionless quantity, defined as x/L - y distance measured away from plate - Nu _x local Nusselt number - Nu _av average Nusselt number - Pr Prandtl number defined by (24) - T temperature difference according to boundary conditions - thermal diffusivity of fluid - coefficient of thermal expression of fluid - boundary layer thickness - ⁺ dimensionless boundary layer thickness - dimensionless velocity profile - dimensionless variable, defined as y/ - dimensionless temperature difference     

Laminar flow heat transfer in the entrance region of circular tubes

Summary The asymptotic solution of laminar convective heat transfer in the entrance region of a circular conduit where velocity and temperature profiles are developing simultaneously, is obtained for fluids with high Prandtl numbers. Numerical values of local and average Nusselt numbers as functions of Pr and dimensionless longitudinal distances have been evaluated and presented in graphical forms.Nomenclature A ₀, A ₁ ... A _k coefficients defined by (40) - B ₀, B ₁ ... B _k coefficients defined by (39) - C _p heat capacity of fluid - I _n (x) = i ^–n J _n (ix) where J _n is the n ^th order Bessel function - k thermal conductivity of fluid - Nu _z local Nusselt number defined by (41) - Nu _av average Nusselt number defined by (44) - P pressure - Pr Prandtl number of fluid defined as C _p /k - q heat flux - Re Reynold number, defined as PR/ - R radius of pipe - r radial distance - r ⁺ dimensionless radial distance defined by (8) - T temperature of fluid - T ₀ initial temperature of fluid - T _w wall temperature - T ⁺ dimensionless temperature defined by (11) - T ₀ ⁺ , T ₁ ⁺ , ... T _k ^/+ ... functions related to T ⁺ by (22). - u dimensionless variables defined by (20) - v _r radial component of velocity - v _z z-component of velocity - v ⁺ dimensionless velocity defined by (10) - y ⁺ dimensionless distance defined by (8) - X dimensionless parameter defined by (38) - z longitudinal distance - z ⁺ dimensionless longitudinal distance defined by (9) - thermal diffusivity - dimensionless parameter defined by (12) - a parameter appearing in (46) - (x) gamma function - density - dimensionless variable defined by (28) - parameter defined by (19) - dimensionless variable defined by (32) - viscosity of fluid - kinematic viscosity of fluid     

Laminar flow and heat transfer over a two-dimensional triangular step

The velocity correction algorithm is used in the finite element method to solve forced convection problems between parallel plates with a triangular step, for Reynolds numbers up to 1000. Equal-order interpolation functions for velocity, pressure and temperature are used. The solutions show a smooth variation of pressure. The streamfunction, isotherms, isobars and velocity profiles are presented for a typical Reynolds number of 500. The skin friction and heat transfer results are presented for Reynolds numbers up to 1000.     

Magnetohydrodynamic heat transfer in two-phase flow between parallel plates

This study dealt with two-phase magnetohydrodynamic (MHD) flow and heat transfer in a parallel-plate channel. Both phases were incompressible and the flow was assumed to be steady, one-dimensional and fully developed. The present study was expected to be useful in the understanding of the effect of the presence of slag layers on the heat transfer characteristics of a coal-fired MHD generator.The problem was investigated, in which one of the two fluids was assumed to be electrically non-conducting. The transport properties of the two fluids were taken to be constant, and the plates were assumed to be maintained at constant and equal temperatures. In this case, the governing differential equations were linear, and an exact solution was obtained. Results were presented for various height and viscosity ratios for the two fluids and for two values of the electric field loading parameter. The governing equations were also solved numerically in order to verify the exact solution.     

Laminar heat transfer between parallel plates with an unsymmetrically prescribed heat flux at the walls

Summary The first four odd eigenvalues and constants, as well as asymptotic expressions for these quantities, are presented for heat transfer to laminar flow between parallel flat plates with unsymmetrical heat rates per unit surface area prescribed at the walls.     

Investigation of flow and heat transfer in corrugated-undulated plate heat exchangers

 An experimental and numerical investigation of heat transfer and fluid flow was conducted for corrugated-undulated plate heat exchanger configurations under transitional and weakly turbulent conditions. For a given geometry of the corrugated plates the geometrical characteristics of the undulated plates, the angle formed by the latter with the main flow direction, and the Reynolds number were made to vary. Distributions of the local heat transfer coefficient were obtained by using liquid-crystal thermography, and surface-averaged values were computed; friction coefficients were measured by wall pressure tappings. Overall heat transfer and pressure drop correlations were derived. Three-dimensional numerical simulations were conducted by a finite-volume method using a low-Reynolds number k–ɛ model under the assumption of fully developed flow. Computed flow fields provided otherwise inaccessible information on the flow patterns and the mechanisms of heat transfer enhancement. Received on 5 February 1999     

Convective heat transfer in a parallel plate channel with porous lining

The paper proposes a theoretical model for the study of flow and heat transfer in a parallel plate channel, one of whose walls is lined with non-erodible porous material, both the walls being kept at constant temperatures. The analysis uses Brinkman model in the porous medium and employs the velocity and temperature slips at the interface (the so called nominal surface). The influence of the thickness as well as the permeability of the porous medium on the flow field and Nusselt numbers at the walls is investigated.

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Konvektive Wärmeübertragung in einem Parallelplattenkanal mit porösem Überzug

Zusammenfassung Die vorliegende Arbeit befaßt sich mit dem Vorschlag eines theoretischen Modells, um die Wärmeübertragung in einem Parallelplattenkanal mit unauswaschbarem porösem Überzug zu studieren. Die Strömung innerhalb des porösen Überzugs ist mit Hilfe der Brinkmannschen Gleichung analysiert. An der Grenze (der sogenannten Nominalfläche) zwischen dem Überzug und der freien Strömung sind die Geschwindigkeitsgleitung und die Temperaturgleitung benutzt. Die Beeinflussung des Geschwindigkeitsfelds und die Nusseltschen Zahlen an den Wänden in Abhängigkeit von der Dicke und der Durchlässigkeit des porösen Überzugs ist untersucht.

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Nomenclature u streamwise velocity in Zone 1 (Fig. 1) - û streamwise velocity in Zone 2 (Fig. 1) - p pressure - coefficient of viscosity of the fluid - k absolute permeability of the material used for lining - density of the fluid - R Reynolds number - the average velocity in Zone 1 (Fig. 1) - T temperature in Zone 1 (Fig. 1) - T temperature in Zone 2 (Fig. 1) - K thermal conductivity in Zones 1 and 2 (Fig. 1) - M ₁ non-dimensional mass flow rate in Zone 1 (Fig. 1) - M ₂ non-dimensional mass flow rate in Zone 2 (Fig. 1) - (Nu)_U Nusselt number at the upper plate (Fig. 1) - (Nu) _L Nusselt number at the lower plate (Fig. 1) - _E experimental value of the temperature in the channel (with porous lining) at a specified point - _E/* experimental value of the temperature in the channel (without porous lining) at a specified point     

Laminar fluid flow and heat transfer in a duct of cross-shaped cross-section

Fully-developed flow calculations were performed on ducts of cross-shaped cross-section, which may be regarded as possible candidates for compact heat exchanger configurations. A parametric study was made on the friction and heat transfer characteristics in terms of the parameter α associated with the decrease in the cross-sectional area (namely, α=0 for a square duct and α→1 for infinite parallel plates). As increasing α, both the Nusselt number and friction coefficient decrease toward their minimum levels, but then, increase gradually, and overshoot those of a square duct. Consequently, the heat transfer coefficient shows a significant increase for α>0.4, suggesting an excellent performance of heat transfer surfaces in the duct of cross-shaped cross-section.     

Laminar flow and heat transfer characteristics of nanoparticle colloidal dispersions in water

Numerical analysis is performed to examine axisymmetric laminar flow and heat transfer characteristics of colloidal dispersions of nanoparticles in water (nanofluids). Effect of volume fraction on flow and heat transfer characteristics is investigated. Eight different materials, alumina, copper, copper oxide, diamond, gold, graphite, silver, and zirconia are considered. Heat transfer and property measurements were conducted previously for Alumina nanofluid and the results have shown that nanofluids behave as homogeneous mixtures. It is found that oxide-based nanofluids offer the least heat transfer enhancement compared to elements-based nanofluids. For a given volume flow rate, all nanofluids exhibited linear increase in heat transfer enhancement with increasing colloids volume fraction, up to 0.05. Furthermore, it is found that in the thermal entrance region, a hydrodynamically developing flow exhibits significantly higher heat transfer enhancement than fully-developed conditions.     

Seed bubbles stabilize flow and heat transfer in parallel microchannels

Seed bubbles are generated on microheaters located at the microchannel upstream and driven by a pulse voltage signal, to improve flow and heat transfer performance in microchannels. The present study investigates how seed bubbles stabilize flow and heat transfer in micro-boiling systems. For the forced convection flow, when heat flux at the wall surface is continuously increased, flow instability is self-sustained in microchannels with large oscillation amplitudes and long periods. Introduction of seed bubbles in time sequence improves flow and heat transfer performance significantly. Low frequency (∼10 Hz) seed bubbles not only decrease oscillation amplitudes of pressure drops, fluid inlet and outlet temperatures and heating surface temperatures, but also shorten oscillation cycle periods. High frequency (∼100 Hz or high) seed bubbles completely suppress the flow instability and the heat transfer system displays stable parameters of pressure drops, fluid inlet and outlet temperatures and heating surface temperatures. Flow visualizations show that a quasi-stable boundary interface from spheric bubble to elongated bubble is maintained in a very narrow distance range at any time. The seed bubble technique almost does not increase the pressure drop across microsystems, which is thoroughly different from those reported in the literature. The higher the seed bubble frequency, the more decreased heating surface temperatures are. A saturation seed bubble frequency of 1000–2000 Hz can be reached, at which heat transfer enhancement attains the maximum degree, inferring a complete thermal equilibrium of vapor and liquid phases in microchannels. Benefits of the seed bubble technique are the stabilization of flow and heat transfer, decreasing heating surface temperatures and improving temperature uniformity of the heating surface.     

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.     

Laminar source flow between parallel porous disks

An analysis is presented for laminar source flow between infinite parallel porous disks. The solution is in the form of a perturbation from the creeping flow solution. Expressions for the velocity, pressure, and shear stress are obtained and compared with the results based on the assumption of creeping flow.Nomenclature a half distance between disks - radial coordinate - r dimensionless radial coordinate, /a - axial coordinate - z dimensionless axial coordinate, /a - radial coordinate of a point in the flow - R dimensionless radial coordinate of a point in the flow, /a - velocity component in radial direction - u =a/, dimensionless velocity component in radial direction - velocity component in axial direction - v = a/}, dimensionless velocity component in axial direction - static pressure - p = (a ²/ ²), dimensionless static pressure - =p(r, z)–p(R, z), dimensionless pressure drop - V magnitude of suction or injection velocity - Q volumetric flow rate of the source - Re source Reynolds number, Q/4a - reduced Reynolds number, Re/r ² - critical Reynolds number - R _w wall Reynolds number, Va/ - viscosity - density - =/, kinematic viscosity - shear stress at upper disk - ₀ = (a ²/ ²), dimensionless shear stress at upper disk - shear stress ratio, ₀/( ₀)_inertialess - u = , dimensionless average radial velocity - u/u, ratio of radial velocity to average radial velocity - dimensionless stream function     

Laminar flow and heat transfer in confined channel flow past square bars arranged side by side

A numerical investigation was conducted to analyze the unsteady laminar flow field and heat transfer characteristics in a plane channel with two square bars mounted side by side to the approaching flow. A finite volume technique is applied with a fine grid and time resolution. The transverse separation distance between the bars (G/d) is varied from 0 to 5, whereas the bar height to channel height is d/H=1/8, and the channel length is L=5H. Different flow regimes develop in the channel due the interaction between the two mounted square bars, steady flow, flow with vortex shedding synchronization either in phase or in anti-phase, or biased flow with low frequency modulation of vortex shedding are found. Results show that the pressure drop increase and heat transfer enhancement are strongly dependent of the transverse separation distance of the bars and the channel Reynolds number.     

MHD mixed convective heat transfer flow about an inclined plate

Mixed convection heat transfer about a semi-infinite inclined plate in the presence of magneto and thermal radiation effects is studied. The fluid is assumed to be incompressible and dense. The nonlinear coupled parabolic partial differential equations governing the flow are transformed into the non-similar boundary layer equations, which are then solved numerically using the Keller box method. The effects of the mixed convection parameter R _i, the angle of inclination α, the magnetic parameter M and the radiation–conduction parameter R _d on the velocity and temperature profiles as well as on the local skin friction and local heat transfer parameters. For some specific values of the governing parameters, the results are compared with those available in the literature and a fairly good agreement is obtained.