Laminar forced convection in elliptic ducts

The problem of laminar forced convection heat transfer in short elliptical ducts with (i) uniform wall temperature and (ii) prescribed wall heat flux is examined in detail with the well known Lévêque theory of linear velocity profile near the wall. Moreover, consideration is given to the variation of the slope of the linear velocity profile with the position on the duct wall. A correction factor for the temperature dependent viscosity is included. Expressions for the local and average Nusselt numbers and wall temperatures are obtained. For the case of constant heat flux the Nusselt numbers are higher than for constant wall temperature.The results corresponding to the classical Graetz and Purday problems are deduced as special cases.Nomenclature a, b semiaxes of ellipse, b Graetz number (average), Re Pr D _e/Z - h _i ^o local heat transfer coefficient - J _n(x) Bessel function of order n - K thermal conductivity of the fluid - [X] Laplace transform of X - N _u ^o local Nusselt number, h _i ^o D _e/K - perimeter average Nusselt number - overall average Nusselt number - Nu _w wall Nusselt number - Nu Nusselt number at large distance from the inlet - p Laplace transform parameter - Pr Prandtl number, C _a/K - Re Reynolds number, D _e / _a - T temperature of the fluid - T ₁, T _W inlet and wall temperatures, respectively - u _z local isothermal velocity along the axis of the duct - average fluid velocity - x, y, z Cartesian coordinates, z-axis parallel to the axis of the duct (z=0 at duct inlet) - Z length of the duct - thermal diffusivity, K/C - * correction factor for the temperature dependent viscosity - (x) gamma function - coordinate measured normal to the wall of the duct - _a, _w viscosity of fluid at average and wall temperatures - , , z elliptic cylindrical coordinates - density of fluid - (z) heat flux     

Laminar forced convection in two-dimensional impacting tee junctions

This study presents numerical predictions of the laminar fluid-flow and heat-transfer characteristics in planar (two-dimensional) impacting tee junctions. The applicable Navier-Stokes equations and the energy equation were solved for airflow at two inlet Reynolds numbers (Re₁) and a wide range of the mass split ratio (β). The results include wall shear stress distributions, streamlines showing the number, location, and size of the re-circulation zones, the pressure loss coefficient, wall heat flux distributions, isotherms, and the overall rate of heat transfer. These results indicate that two re-circulation zones always form on the inside-bend wall of the tee at all values of β and Re₁. Two more re-circulation zones may form on the impacting wall of the tee depending on the values of β and Re₁. It was also found that the pressure loss coefficient reaches a minimum and the overall rate of heat transfer reaches a maximum at even mass split (β = 0.5).     

Laminar forced convection of power-law non-Newtonian fluids inside ducts

Thermal entrance region heat transfer for laminar forced convection of power-law fluids inside a circular tube and parallel plate channel for uniform wall temperature is solved exactly, and as many eigenvalues and eigenfunctions as needed for the solution are determined automatically and with high accuracy by using the recently advanced Sign-Count method. Results are presented for the local and average Nusselt number over a wide range of the Graetz number in both graphical and tabular forms. The present benchmark results are utilized to critically examine the accuracy of the approximate Leveque solution.

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Laminare Zwangskonvektion von nicht-Newtonschen Fluiden, die in Kanälen strömen und dem Potenzansatz folgen

Zusammenfassung Es werden exakte Lösungen für den Wärmetransport in der thermischen Einlaufzone in runden Rohren und zwischen parallelen Platten für Fluide nach dem Exponentialansatz bei laminarer Zwangskonvektion und mit gleichmäßiger Wandtemperatur angegeben. Unter Benutzung der jüngst verbesserten Sign-Count-Methode werden so viele Eigenwerte und Eigenfunktionen, wie für die Lösung benötigt, automatisch und mit großer Genauigkeit bestimmt. Ergebnisse werden in graphischer wie tabellarischer Form über einen weiten Bereich der Graetz-Zahl für die örtliche und mittlere Nusselt-Zahl vorgestellt. Die vorliegenden richtungsweisenden Ergebnisse werden dazu benützt, um die Genauigkeit der Levequeschen Näherungslösung kritisch zu prüfen.

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Nomenclature b radius of circular duct or half the spacing between parallel plates - C - D _h hydraulic diameter=4b for parallel plate, 2b for circular tube - f(r), F(r) temperature distribution at the inlet, dimensional and dimensionless, respectively - g (r), G (R) energy generation, dimensional and dimensionless, respectively - h (z) heat transfer coefficient - H _i(Z) defined by Eq. (11b) - k thermal conductivity - l₀,l _N reference lengths to nondimensionalizer andz coordinates respectively (l ₀=b andl _N= D_h) - n power-law index - Nu _av average Nusselt number=h _av D_h/k - Nu (Z) local Nusselt number=h (z)D _h/k - p 0 for parallel-plate duct, 1 for circular duct - r radial or normal coordinate - R = dimensionless radial coordinate - T(r, z) fluid temperature - T _av (z) average fluid temperature - T ^* reference temperature - T reference temperature difference - U(R) = normalized velocity profile - U ^*(R) - w (r) fully developed velocity profile - w _av average velocity - z axial coordinate - Z = dimensionless axial coordinate Greek letters thermal diffusivity - (R,Z) = dimensionless temperature - _w = dimensionless wall temperature - _i eigenvalues of the eigenvalue problem (10) - (_i,R) eigenfunctions of the eigenvalue problem (10)     

Two-dimensional adiabatic forced convection at low péclet number

This paper deals with heat transfer to two-dimensional laminar Poiseuille flow with arbitrary heat fluxes specified on the walls. The solution is first obtained for fluxes which are symmetric and antisymmetric delta functions and from these the general solution is obtained by integration. The solutions include the effects of axial conduction and also of preheating of the incoming fluid. Detailed solutions are given for the cases Pé=1 and Pé=.5.     

Laminar thermal convection in rotating cavity between two disks

In solving several technical problems it is necessary to know what takes place in a closed rotating axisymmetric cavity filled with a nonuniformly heated viscous fluid. Such cavities are encountered, for example, in the rotors of steam and gas turbines. The thermal convection in these cavities is studied for a definite temperature condition of the rotors: in [1, 2] some qualitative considerations are presented, and quantitative estimates are given for thermal convection in cavities of turbine rotors; in [3,4] there is presented a very approximate calculation using the method of integral relations of the heat transfer coefficients in the case of a narrow cavity between two rotating disks which have different temperatures. We note that the thermal convection effect in a rotating cavity may be utilized in various technical devices, for example, in equipment for separating isotopes, etc. [5], A solution is presented for the problem of laminar thermal convection in a narrow cavity between two disks which are rotating with the same velocity and which have different temperatures which are constant along the radius. In the case of the narrow cavity we can neglect the influence of the cylindrical cavity rim on the flow in primary portion of the cavity (see [6]); therefore it is sufficient to solve the self-similar problem for two infinite disks.In conclusion I would like to thank A. Z. Serazetdinov and V. L. Karaseva for carrying out the computer calculations.     

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     

Radiation-conduction interaction in mixed convection along rotating bodies

This paper investigates the interaction of the steady mixed convection boundary layer flow past a rotating impermeable body placed in a uniform stream moving opposite to the gravitational force and parallel to the axes of the body of revolution with uniform surface temperature and thermal radiation. The fluid considered here is a gray, absorbing-emitting but non-scattering medium, and the Rosseland approximation is used to describe the radiative heat flux in the analysis. The difficulty of having a unified mathematical treatment of this problem is due to the nonsimilarity nature of the governing equations arising from the buoyant force-field and the transverse curvature of the bodies. The important parameters of this problem are the radiation-conduction parameter R _d and the wall to free stream temperature ratio θ _w , for the case of a heated surface. Numerical simulations of the boundary layer equations are performed using the local nonsimilarity method as well as an implicit finite-difference method. The theory is applied to a rotating sphere for the gases with Prandtl number of 0.7. The results are shown graphically in terms of the local skin-friction coefficients and the local rate of heat transfer. Effects of the pertinent parameters R _d and θ _w are also shown on the components of the velocity distribution as well as on the temperature distribution in the boundary layer. Received on 14 January 1997     

Laminar forced convection in a channel with arrays of thermal sources

The problem of laminar forced convection in a two-dimensional channel with arrays of thermal sources is studied numerically. The surfaces of the thermal sources are assumed to be isothermal. In order to promote the accuracy of the result, numerical computation is performed with alternating direction implicit (A.D.I.) method and SIMPLER algorithm. The primary purpose of this paper is to study the fundamental heat transfer phenomena in L.S.I. packages, which are widely used in microelectric equipments recently. The contribution of this study is not only on academic purpose but also on industrial application. Influence of Reynolds number on Nusselt number of thermal sources is discussed. Calculations are made under the conditions of different arrangements of the thermal sources. Flow patterns, pressure and temperature distributions are demonstrated.

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Laminare Zwangskonvektion in einem Kanal mit einem Wärmequellen-Feld

Zusammenfassung Hier wurde das Problem der laminaren Zwangskonvektion in einem zweidimensionalen Kanal mit einem Wärmequellen-Feld numerisch untersucht. Die Oberflächen der Wärmequellen wurden als isotherm vorausgesetzt. Um die Genauigkeit des Ergebnisses zu verbessern, wurden die numerischen Berechnungen mit dem A.D.I.-Verfahren und dem SIMPLER-Algorithmus durchgeführt. Das grundlegende Ziel dieser Arbeit sind die wesentlichen Wärmeübertragungsphänomene in L.S.I.-Paketen zu untersuchen, welche vorwiegend in Geräten der Mikroelektronik verwendet werden. Diese Untersuchung liefert nicht nur einen Beitrag für akademische Zwecke, sondern ist auch für die industrielle Anwendung gedacht. Der Einfluß der Reynolds-Zahl auf die Nusselt-Zahl der Wärmequellen wurde besprochen. Die Berechnungen wurden für verschiedene Zusammenstellungen der Wärmequellen gemacht. Strömungsbilder, Druck- und Temperaturverteilungen sind dargestellt worden.

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Nomenclature B ₁ channel width - B ₂ horizontal distance between heating elements - g magnitude of gravitational acceleration - i numbers of the horizontal grid points - j numbers of the vertical grid points - k thermal conductivity of fluid - l ₁ distance between two thermal sources - L channel length - L ₁ distance between the entrance and the first column of heating element - N number of thermal sources in a horizontal column - Nu Nusselt number - Nu average Nusselt number - p, P dimension and dimensionless pressure - Pr Prandtl number - Re Reynolds number - T local temperature - T _H temperature of the top surface of the thermal source - T _L inlet-flow temperature - U _O average inlet-flow velocity - u, U dimensional and dimensionless horizontal velocities - v, V dimensional and dimensionless vertical velocities - x, X dimensional and dimensionless horizontal coordinates - y, Y dimensional and dimensionless vertical coordinates - W width of the thermal source - fluid thermal diffusivity - coefficient of thermal expansion - lateral pitch of thermal sources - viscosity - kinematic viscosity=/ - density - dimensionless temperature=(T–T _O)/(T _H–T_O)     

Laminar forced convection with sinusoidal wall heat flux distribution: axially periodic regime

Stationary and laminar forced convection in a circular tube with a sinusoidal axial distribution of wall heat flux is studied under the hypothesis that both axial heat conduction and viscous dissipation in the fluid are negligible. Two cases are considered: a sinusoidal wall heat flux distribution with a vanishing mean value; a sinusoidal wall heat flux distribution which does not change its sign. In both cases, the temperature field and the local Nusselt number are evaluated analytically in the fully developed region, i.e. where the local Nusselt number depends periodically on the axial coordinate. It is shown that, in the first case, the fully developed region presents an infinite sequence of axial positions where the local Nusselt number is singular. In these positions, the wall heat flux has a non-vanishing value even if the wall temperature equals the bulk temperature.     

Laminar forced convection in vertical pipes exposed to external natural convection and external radiation: uncoupled/lumped solution

This paper is concerned with the laminar forced convection flow in a vertical pipe exposed to either natural convection or simultaneous natural convection and thermal radiation external to the pipe. An uncoupled/lumped formulation enabled the determination of the mean bulk temperature distribution of the internal flow and the total rate of heat transfer. Average values for both internal and external Nusselt numbers have been taken from standard correlations reported in the literature, leading to the calculation of an effective average Nusselt number, which controls the thermal interaction process. Typical results for a selected combination of internal and external fluids are discussed at length and they compared favorably with others based on a conjugate/numerical formulation. This formulation necessitated a finite difference methodology where information was transferred between the two flows (internal and external), via their respective heat transfer coefficients. On the other hand, attention was focused on the uncoupled/lumped formulation in order to examine in detail the effects of the thermal boundary conditions, and consequently the important role of radiation as an enhancing heat transfer mechanism.     

Laminar fluid convection between concentric and eccentric heated horizontal rotating cylinders for low-Prandtl-number fluids

Numerical experiments are performed to study rotational effects on the mixed convection of low-Prandtlnumber fluids enclosed between the annuli of concentric and eccentric horizontal cylinders. The inner cylinder is assumed to be heated and rotating. The rotational Reynolds number considered is in the range where the effect of Taylor vortices is negligible. The Prandtl number of the fluid considered is in the range 0·01–1·0. The Rayleigh number considered is up to 10⁶. A non-uniform mesh transformation technique coupled with the introduction of ‘false transient’ parameters to the vorticity and streamfunction-vorticity expressions was used to solve the governing set of equations. Results show that when the inner cylinder is made to rotate, the multicellular flow patterns observed in stationary cylindrical annuli subside in a manner depending on the Prandtl number of the fluids. Eventually the flow tends toward a uniform flow similar to that of a solid body rotation. For a fixed Rayleigh number and with a Prandtl number of the order of 1·0, when the inner cylinder is made to rotate, the mean Nusselt number is observed to decrease throughout the flow. For lower Prandtl number of the order 0·1–0·01 the mean Nusselt number remained fairly constant when the inner cylinder was made to rotate. The mean Nusselt numbers obtained were also compared with available data from other investigators.     

Numerical and experimental study of the forced convection inside a rotating disk-cylinder configuration

In this paper, axisymmetric bulk flow patterns generated by moderate disk rotation and counter-rotation inside a coaxial disk-cylinder configuration with a fixed aspect ratio are obtained both experimentally and numerically. Experimental results are based on chronophotographic visualization and image processing techniques, while numerical results are computed using the full stationary Navier-Stokes equations assuming two different dynamic boundary conditions (no-slip and meridional free-slip) for all rigid walls. A comparative analysis between both numerical distributing and the patterns obtained experimentally is carried out in terms of streamfunction and vorticity meridional distributions.     

Laminar assisting mixed convection heat transfer from a vertical isothermal cylinder moving in water at low temperatures

In all studies concerning laminar mixed convection along a vertical isothermal moving cylinder a linear relationship between fluid density and temperature has been used and viscosity and thermal conductivity have been considered constant. However, it is known that the density-temperature relationship for water is non-linear at low temperatures and viscosity and thermal conductivity are functions of temperature. In this study the problem of water laminar mixed convection along a vertical isothermal moving cylinder has been investigated in the temperarure range between 20 °C and 0 °C taking into account the temperature dependence of μ, k and ρ. The results are obtained with the numerical solution of the boundary layer equations. The variation of μ, k and ρ with temperature has a strong influence on mixed convection characteristics.     

Numerical simulation of forced convection over a periodic series of rectangular cavities at low Prandtl number

Convective heat transfer in laminar conditions is studied numerically for a Prandtl number Pr = 0.025, representative of liquid lead–bismuth eutectic (LBE). The geometry investigated is a channel with a periodic series of shallow cavities. Finite-volume simulations are carried out on structured orthogonal curvilinear grids, for ten values of the Reynolds number based on the hydraulic diameter between Re_m = 24.9 and Re_m = 2260. Flow separation and reattachment are observed also at very low Reynolds numbers and wall friction is found to be remarkably unequal at the two walls. In almost all cases investigated, heat transfer rates are smaller than the corresponding flat channel values. Low-Prandtl number heat transfer rates, investigated by comparison with Pr = 0.71 results, are large only for uniform wall temperature and very low Re. Influence of flow separation on local heat transfer rates is discussed, together with the effect of different thermal boundary conditions. Dependency of heat transfer performance on the cavity geometry is also considered.     

Laminar free convection heat transfer from isothermal convex bodies of arbitrary shape: a new dynamic model

Calculation of free convection from bodies of arbitrary shape has been investigated previously. The Body Gravity Function (BGF) which accounts for the geometry of each body shape was considered to be a constant value. In the present study, it is shown that BGF is not a constant value in a wide range of Rayleigh number. Instead, its value changes as Rayleigh number increases. Therefore, by analytical modeling of Dynamic BGF and derivation of a new parameter called Body Fluid Function, a novel method is proposed to calculate laminar free convection heat transfer from isothermal convex bodies of arbitrary shape. Results for 24 different body shapes are compared with the available experimental and numerical data. Excellent agreement shows that the present simple method accurately predicts laminar free convection heat transfer from isothermal convex bodies of arbitrary shape in the whole range of laminar flow and for fluids of any Prandtl number.     

Laminar forced convection heat transfer from isothermal cylinders with active ends and different aspect ratios in axial air flows

In this article a semi-analytical approach is employed to obtain dimensionless heat transfer correlations for forced convection from isothermal circular cylinders with active ends and different aspect ratios (l/d ￡ 8) (l/d \le 8) in laminar axial air flows. Then, using the present results and previous works, the modeling is extended to higher aspect ratios (l/d 3 8) (l/d \ge 8) ) as long as the entire flow field remains completely laminar. Validations of the present work are done not only with the available data on drag coefficients but with previous works for long cylinders with inactive ends and long spheroids. Two general correlations are also developed for a rough estimate of forced convection heat transfer from isothermal cylinders with active ends and arbitrary aspect ratios in the range of \frac12 ￡ \fracld ￡ 8 \frac{1}{2} \le \frac{l}{d} \le 8 and l/d 3 8 l/d \ge 8 .