Heat transfer to non-Newtonian fluids in laminar flow through rectangular ducts

Heat transfer to non-newtonian fluids flowing laminarly through rectangular ducts is examined. The conservation equations of mass, momentum, and energy are solved numerically with the aid of a finite volume technique. The viscoelastic behavior of the fluid is represented by the Criminale-Ericksen-Filbey (CEF) constitutive equation. Secondary flows occur due to the elastic behavior of the fluid, and, consequently, heat transfer is strongly enhanced. It is observed that shear thinning yields negligible heat transfer enhancement effect, when compared with the secondary flow effect. Maximum heat transfer is shown to occur for some combinations of parameters. Thus, there are optimal combinations of aspect ratio and Reynolds numbers, which depend on the fluid's mechanical behavior. This result can be usefully explored in thermal designs of certain industrial processes.     

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.     

On the steady laminar flow in a curved pipe of varying elliptic cross-section

The fully developed steady flow of a fluid through a curved tube with elliptic cross-section is studied, the cross-sectional area varying slowly with longitudinal distance. Using a perturbation scheme in terms of two small parameters (geometric parameter and curvature parameter), complete analytical solutions are obtained to the first order. The effects of constriction combined with constant curvature of the centre line are discussed. The phenomenon of secondary flow, shear stress and the increased impedance (due to the constriction) are calculated. The possibility of the application of the results to various engineering problems and physiological flows is indicated.     

Heat transfer in magnetohydrodynamic laminar radial channel flow

Heat transfer in the steady axisymetrical laminar source flow of an incompressible electrically conducting fluid between two parallel disks in the presence of a transverse applied magnetic field is analyzed. The energy equation is solved numerically for the temperature distribution, where both Joulean and viscous heating are included. Both local and average Nusselt numbers for the case of constant wall temperature are evaluated. For fluids of moderate and high Prandtl numbers, Nusselt number is seen to be a strong function of both Hartmann number and a heat generation parameter together with a modified Peclet number. However, for fluids of small Prandtl number, Joulean heating and viscous dissipation can be neglected without appreciable error.     

Heat transfer in unsteady laminar flow through a channel

The temperature distribution in unsteady laminar flow of a viscous incompressible fluid in a flat channel is investigated, when the pressure gradient is an arbitrary function of time. Two techniques are presented (i) explicit finite difference scheme, and (ii) Chebyshev polynomial solution. Using both the techniques, several cases of pressure gradient are considered, with special attention paid to the linearly varying case.     

Heat transfer by laminar flow in a vertical pipe with twisted-tape inserts

Heat transfer for laminar flow of water in an air-cooled vertical copper pipe with four twisted-tape inserts was determined experimentally. The tests were executed for laminar flow within 110 ≤ Re ≤ 1500, 8.1 ≤ Gz ≤  82.0 and 1.62 ≤ y ≤ 5.29. The correlation equation for heat transfer was defined for the tested range. The obtained results were compared to the results of other authors. Received on 28 April 1998     

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.     

Heat transfer solutions in laminar co-current flow of immiscible liquids

Thermal entry region solutions are analytically determined for horizontal, co-current laminar flow of immiscible liquids in direct contact, inside circular tubes and parallel plate channels. The related eigenvalue problem for such a composite media is readily solved by extending the ideas in the recently advanced sign-count method. It is assumed a core-annular flow configuration for circular tubes and sheat-core flow for the parallel plates channel, without consideration of interface instabilities and stratified flow. First, the velocity problem is solved for fully developed flow and pumping power expressions established for different operating conditions. Then, the temperature problem is analytically handled to yield expressions for quantities of practical interest such as total heat exchange rates, along the duct length and, again, for different flow rates and pressure drop requirements. The analysis is illustrated through consideration of an application dealing with pumping of a very viscous oil with the addition of an external thin layer of a less viscous fluid (water). Pumping power and total heat exchange are then evaluated for both geometries and critically compared to the single fluid flow problem.Hier sind Lösungen für die thermische Eintritts-strecke von horizontalen, laminaren Gleichströmungen in unvermischbaren Flüssigkeiten, die in direktem Kontakt untereinander sind, analytisch bestimmt worden. Die Lösungen gelten für Gleichströmungen in Röhren und in parallelen flachen Kanälen. Das betreffende Eigenwertproblem für solch ein zusammengesetztes Medium ist vollkommen mit dem Gedanken des kürzlich weiterentwickelten Zeichenzählverfahrens gelöst worden. Für die Rohre sind kreisring- und kernförmige Strömungen und für die parallelen Plattenkanäle Schichtkernströmungen angenommen worden. Hierbei ist die Grenzflächeninstabilität nicht in Betracht gezogen worden. Als erstes ist das Geschwindigkeitsproblem für eine vollkommen entwickelte Strömung gelöst und es sind Ausdrücke für die Pumpleistung für verschiedene Betriebsbedingungen ermittelt worden. Dann ist das Temperaturproblem analytisch behandelt worden, um Ausdrücke für die Größen von praktischem Interesse zu erzielen, wie die gesamte Wärmeaustauschrate über die Kanallänge für verschiedene Strömungsgeschwindigkeiten und Druckverlustanforderungen. Die Berechnung ist durch die Betrachtung eines Anwendungsbeispiels veranschaulicht worden, bei dem sehr zähflüssiges Öl mit einer zusätzlichen äußeren, dünnen Schicht, die weniger zähflüssig ist, gepumpt wurde. Die Pumpleistung und der gesamte Wärmeaustausch sind für beide Geometrien ausgewertet und kritisch mit dem einfachen Strömungsproblem von Fluiden verglichen worden.     

Heat transfer for non-Newtonian laminar flow in internally finned pipes with uniform temperature

An analysis is presented for fully developed laminar convective heat transfer of non-Newtonian power-law fluids in pipes with internal longitudinal fins and uniform outside wall temperature. The governing momentum and energy equations have been solved numerically, with the influence of fin conductance. The distributions of fin temperature, fluid temperature and local heat flux (both at finned and unfinned surfaces) are presented. These are shown to be strongly dependent on finned pipe geometry, fluid flow behavior index and the fin conductance. Values of overall Nusselt number indicated significant heat transfer enhancement over finless pipes. The flow behavior index affects the no. of fins which maximizes the overall Nusselt number.     

Heat and mass transfer in laminar flow between parallel porous plates

Summary The problem of heat transfer in a two-dimensional porous channel has been discussed by Terrill [6] for small suction at the walls. In [6] the heat transfer problem of a discontinuous change in wall temperature was solved. In the present paper the solution of Terrill for small suction at the walls is revised and the whole problem is extended to the cases of large suction and large injection at the walls. It is found that, for all values of the Reynolds number R, the limiting Nusselt number Nu increases with increasing R.Nomenclature stream function - 2h channel width - x, y distances measured parallel and perpendicular to the channel walls respectively - U velocity of fluid at x=0 - V constant velocity of fluid at the wall - =y/h nondimensional distance perpendicular to the channel walls - f() function defined in equation (1) - coefficient of kinematic viscosity - R=Vh/ suction Reynolds number - density of fluid - C _p specific heat at constant pressure - K thermal conductivity - T temperature - x=x ₀ position where temperature of walls changes - T ₀, T ₁ temperature of walls for x<x ₀, x>x ₀ respectively - = (T – T ₁)/T ₀ – T ₁) nondimensional temperature - =x/h nondimensional distance along channel - R ^* = Uh/v channel Reynolds number - Pr = C _p/K Prandtl number - _n eigenvalues - B _n() eigenfunctions - B _n ⁽⁰⁾ , () eigenfunctions for R=0 - B ₀ ⁽ⁱ⁾ , B ₀ ⁽ⁱⁱ⁾ ... change in eigenfunctions when R0 and small - K _n constants given by equation (13) - h heat transfer coefficient - Nu Nusselt number - _m mean temperature - C _n constants given by equation (18) - perturbation parameter - B _0i () perturbation approximations to B ₀() - Q = B ₀/ ₀ derivative of eigenfunction with respect to eigenvalue - z nondimensional distance perpendicular to the channel walls - F(z) function defined by (54)     

Heat transfer in vertical annular laminar flow of two immiscible liquids

The paper investigates heat transfer in annular laminar undisturbed flow of two immiscible liquids, with constant heat-flux generated at the wall of the tube. It presents an analytical solution for the fully developed temperature field. This is used to obtain a more general solution from a model, describing the temperature field as a superposition of the fully developed and the developing fields. This superposition model is solved by an orthogonal collocation method. An asymptotic model for short entry lengths is also described. Calculations for a kerosene-water system, show that the superposition solution converges to the entrance solution below 100 diameters and converges asymptotically to the solution of the fully developed temperature field beyond 5000 diameters. The effect of the wavy interface is assessed experimentally for annular kerosene-water flow, by comparing predicted and measured temperature profiles. It is found that experimental profiles are considerably flatter and measured Nusselt numbers for the kerosene phase are accordingly higher by 40–320% as compared to the undisturbed flow analyses.     

Heat transfer in channels in the laminar flow of a non-newtonian liquid with slip

We consider the case of quasi-isothermal heat transfer in the laminar flow of a non-Newtonian medium with slip in circular and plane channels. The calculation is based on the simultaneous solution of the equations of energy, motion, and a rheological relation with the specific boundary conditions which takes into account discontinuity in the velocity and temperature on the heat-transfer surface.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 146–151, September–October, 1971.     

Heat transfer and flow behavior around four staggered elliptic cylinders in cross flow

Experimental and numerical studies were carried out to investigate forced convection heat transfer and flow features around the downstream elliptic cylinder in four staggered cylinders in cross flow. The elliptic cylinders examined had an axis ratio (b/c) of 1:2, and they were arranged with zero angle of attack to the upstream flow. The present heat transfer measurements were obtained by heating only the downstream elliptic cylinder (test cylinder) under the condition of constant heat flux. The testing fluid was air and the Reynolds number based on the major axis length (c) was ranged from 4,000 to 45,570. The tested longitudinal spacing ratio (S_x/c) and the transversal spacing ratio (S_y/b) were in the ranges of 1.5 ≤ S_x/c ≤ 4.0 and 1.5 ≤ S_y/b ≤ 4.0, respectively. The air flow pattern and temperature fields around the four staggered elliptic cylinders were predicted by using CFD software package. Also, a flow visualization study was made to show the flow features around the elliptic cylinders. It was observed that Nu_m of the downstream elliptic cylinder in four staggered cylinders was higher than that of three in-line cylinders for all tested spacing ratios and Reynolds numbers except for Re = 4,000. It was clear that, at lower Reynolds number values (Re < 14,100), the average Nusselt number of the downstream elliptic cylinder in three staggered arrangement was higher than that of the downstream cylinder in four staggered arrangement for all tested spacing ratios. On the other hand, at Re > 14,100, the tested elliptic cylinder in four staggered arrangement had the higher values of the average Nusselt number. Moreover, in four staggered arrangement, the maximum average Nusselt number enhancement ratio (average Nusselt number of the tested downstream cylinder/average Nusselt number of a single elliptic cylinder) was found to be about 2.0, and was obtained for spacing ratios of S_x/c = 2.5, S_y/b = 2.5 and at Re = 32,000. Finally, the average Nusselt number of the tested cylinder in four staggered arrangement was correlated in terms of Reynolds number and cylinder spacing ratios.     

Heat transfer by laminar flow in a vertical and horizontal pipe with twisted-tape inserts

 This article presents the results of laboratory research on heat exchange while heating water in horizontal and vertical tubes with twisted-tape inserts. The scope of the research: 70 ≤ Re ≤ 4000 3.6 ≤ Pr ≤ 5.9 8.6 ≤ Gz ≤ 540 The research was held for three cases: – horizontal experimental tube – vertical experimental tube, the direction of flow according to the free convection vector – vertical experimental tube, the direction of flow not in accordance with the free convection vector For such cases the correlation equation was defined Nu_T=f(Gz; y), Nu = f(Gz) and the proportion Nu_T/Nu was analysed. Received on 30 March 2000