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 mixed convection in the entrance region of horizontal rectangular ducts

A generally applicable finite element procedure for the prediction of laminar mixed convection in horizontal straight ducts of arbitrary cross-section is presented. The procedure, based on the parabolized simplification of the complete Navier-Stokes equations and on the Boussinesq approximation of the buoyancy terms, is validated through comparisons of computed results with the available literature data for mixed convection in the entrance region of a rectangular duct of aspect ratio a=2. Uniform heating at different sides is considered as the thermal boundary condition, although the proposed formulation allows specification of most thermal boundary conditions of practical interest.     

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.     

Axial heat conduction effects in the entrance region of circular ducts

This paper describes the transport of thermal energy within a small distance after an abrupt wall temperature change in a circular duct. In general, the axial conduction becomes significant when the Peclet number is small. The results indicate that the inclusion of axial conduction in the fluid substantially increases the wall heat flux at near the thermal inlet location. The exact series solution leads to a modified Graetz type problem. This exact solution is augmented by an asymptotic solution describing the wall heat flux near the thermal entrance location.     

Entropy generation in non-Newtonian fluids due to heat and mass transfer in the entrance region of ducts

A new solution for the Graetz problem (hydrodynamically developed forced convection in isothermal ducts) extended to power-law fluids and mass transfer with phase change at the walls is presented. The temperature and concentration spatial distributions in the corresponding entrance regions are obtained for two geometries (parallel-plates duct and circular pipe) in terms of appropriate dimensionless parameters. They are used to illustrate the effects of the fluid nature on the velocity, temperature and concentration distributions, on the axial evolution of the sensible and latent Nusselt numbers as well as on the local entropy generation rate due to velocity, temperature and concentration gradients.     

Convective heat transfer in the thermal entrance region of finned double-pipe

Numerical simulation of the steady and laminar convection in the thermal entry region of the finned annulus is carried out for the case of hydrodynamically fully developed flow when subjected to uniform heat flux thermal boundary condition. Finite difference based marching procedure is used to compute the numerical solution of the energy equation. The results to be presented include Nusselt number, as a function of dimensionless axial length and thermal entrance length for various configurations of the finned double-pipe. The numerical results show that Nusselt number has complex dependence on the geometric variables like ratio of radii, fin height, and number of fins. A comparison of the computed results for certain limiting cases with the results available in the literature validates the numerical procedure used in this work.     

Laminar flow in the entrance region of a pipe

Summary The laminar flow of an incompressible fluid in the inlet of a pipe is analyzed numerically. The numerical technique allows a closer approximation to the basic equations of fluid motion than has been possible in previous investigations. Significant differences are shown between the results of the numerical solution and previous work for both velocity profiles and development lengths.     

Laminar flow in the entrance region of a porous-wall channel

Summary The effect of fluid injection at the walls of a two-dimensional channel on the development of flow in the entrance region of the channel has been investigated. The integral forms of the boundary layer equations for flow in the channel were set up for an injection velocity uniformly distributed along the channel walls.With an assumed polynomial of the n-th degree for the one-parameter velocity profile a solution of the above boundary layer equations was obtained by an iteration method. A closed form solution was also obtained for the case when a similar velocity profile was assumed. The agreement between the entrance region velocity profiles of the present analysis for an impermeable-walled channel and of Schlichting¹) and Bodoia and Osterle²) is found to be very good.The results of the analysis show that fluid injection at the channel walls increases the rate of the growth of the boundary layer thickness, and hence reduces considerably the entrance length required for a fully developed flow.Nomenclature h half channel thickness - L entrance length with wall-injection - L ₀ entrance length without wall-injection - p static pressure - p=p/U ₀ ² dimensionless pressure - Re=U ₀ h/ Reynolds number at inlet cross-section - u velocity in the x direction at any point in the channel - =u/U ₀ dimensionless velocity in the x direction at any point in the channel - U _av average velocity at a channel cross-section - U _c center line velocity - U ₀ inlet cross-section velocity - _c =U _c /U ₀ dimensionless center line velocity - v velocity in the y direction at any point in the channel - v ₀ constant injection velocity of fluid at the wall - v=v/v ₀ dimensionless velocity in the y direction at any point in the channel - x distance along the channel wall measured from the inlet cross-section - x=x/hRe dimensionless distance in the x direction - y distance perpendicular to the channel wall - y=y/h dimensionless distance in the y direction - thickness of the boundary layer - =/h dimensionless boundary layer thickness - =/ dimensionless distance within the boundary layer region - =v ₀ h/ injection parameter or injection Reynolds number - kinematic viscosity - 1+ie - mass density of the fluid - parameter defined in (14)     

Convective heat transfer in the thermal entrance region of parallel-flow noncircular duct heat exchanger arrays

Convective heat transfer properties of a hydrodynamically fully developed flow, thermally developing flow in a parallel-flow, and noncircular duct heat exchanger passage subject to an insulated boundary condition are analyzed. In fact, due to the complexity of the geometry, this paper investigates in detail heat transfer in a parallel-flow heat exchanger of equilateral-triangular and semicircular ducts. The developing temperature field in each passage in these geometries is obtained seminumerically from solving the energy equation employing the method of lines (MOL). According to this method, the energy equation is reformulated by a system of a first-order differential equation controlling the temperature along each line.Temperature distribution in the thermal entrance region is obtained utilizing sixteen lines or less, in the cross-stream direction of the duct. The grid pattern chosen provides drastic savings in computing time. The representative curves illustrating the isotherms, the variation of the bulk temperature for each passage, and the total Nusselt number with pertinent parameters in the entire thermal entry region are plotted. It is found that the log mean temperature difference (T _LM), the heat exchanger effectiveness, and the number of transfer units (NTU) are 0.247, 0.490, and 1.985 for semicircular ducts, and 0.346, 0.466, and 1.345 for equilateral-triangular ducts.

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Konvektiver Wärmeübergang im thermischen Einlaufgebiet von Gleichstromwärmetauschern mit nichtkreisförmigen Strömungskanälen

Zusammenfassung Die Untersuchung bezieht sich auf das konvektive Wärmeübertragungsverhalten eines Gleichstromwärmetauschers mit nichtkreisförmigen Strömungskanälen bei hydraulisch ausgebildetet, thermisch einlaufender Strömung unter Aufprägung einer adiabaten Randbedingung. Zwei Fälle komplizierter Geometrie, nämlich Kanäle mit gleichseitig dreieckigen und halbkreisförmigen Querschnitten, werden bezüglich des Wärmeübergangsverhaltens bei Gleichstromführung eingehend analysiert. Das sich entwickelnde Temperaturfeld in jedem Kanal von der eben spezifizierten Querschnittsform wird halbnumerisch durch Lösung der Energiegleichung unter Einsatz der Linienmethode (MOL) erhalten. Dieser Methode entsprechend erfolgt eine Umformung der Energiegleichung in ein System von Differentialgleichungen erster Ordnung, welches die Temperaturverteilung auf jeder Linie bestimmt.Die Temperaturverteilung im Einlaufgebiet wird unter Vorgabe von 16 oder weniger Linien über dem Kanalquerschnitt erhalten, wobei die gewählte Gitteranordnung drastische Einsparung an Rechenzeit ergibt. Repräsentative Kurven für das Isothermalfeld, den Verlauf der Mischtemperatur für jeden Kanal und die Gesamt-Nusseltzahl als Funktion relevanter Parameter im gesamten Einlaufgebiet sind in Diagrammform dargestellt. Es zeigt sich, daß die mittlere logarithmische Temperaturdifferenz (T _LM), der Wärmetauscherwirkungsgrad und die Anzahl der Übertragungseinheiten (NTU) folgende Werte annehmen: 0,247, 0,490 und 1,985 für halbkreisförmige Kanäle sowie 0,346, 0,466 und 1,345 für gleichseitig dreieckige Kanäle.

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Nomenclature A cross sectional area [m²] - a characteristic length [m] - C _c specific heat of cold fluid [J kg^–1 K^–1] - C _h specific heat of hot fluid [J kg^–1 K^–1] - C _p specific heat [J kg^–1 K^–1] - C _r specific heat ratio,C _r=C _c/C_h - D _h hydraulic diameter of duct [m] - f friction factor - k thermal conductivity of fluid [Wm^–1 K^–1] - L length of duct [m] - m mass flow rate of fluid [kg s^–1] - N factor defined by Eq. (20) - NTU number of transfer units - Nu _{x, T} local Nusselt number, Eq. (19) - P perimeter [m] - p pressure [KN m^–2] - Pe Peclet number,RePr - Pr Prandtl number,/ - Q _T total heat transfer [W], Eq. (13) - Q ideal heat transfer [W], Eq. (14) - Re Reynolds number,D _h/ - T temperature [K] - T _b bulk temperature [K] - T _e entrance temperature [K] - T _w circumferential duct wall temperature [K] - u, U dimensional and dimensionless velocity of fluid,U=u/u - , dimensional and dimensionless mean velocity of fluid - w generalized dependent variable - X dimensionless axial coordinates,X=D _h ² /a ² x* - x, x* dimensional and dimensionless axial coordinate,x*=x/D _hPe - y, Y dimensional and dimensionless transversal coordinates,Y=y/a - z, Z dimensional and dimensionless transversal coordinates,Z=z/a Greek symbols thermal diffusivity of fluid [m² s^–1] - * right triangular angle, Fig. 2 - independent variable - T _LM log mean temperature difference of heat exchanger - effectiveness of heat exchanger - generalized independent variable - dimensionless temperature - _b dimensionless bulk temperature - dynamic viscosity of fluid [kg m^–1 s^–1] - kinematic viscosity of fluid [m² s^–1] - density of fluid [kg m^–3] - heat transfer efficiency, Eq. (14) - generalized dependent variable     

Laminar flow heat transfer for simultaneously developing velocity and temperature profiles in ducts of rectangular cross section

Summary A numerical method is used to solve the heat transfer equations for laminar flow in ducts of rectangular cross section with simultaneously developing temperature and velocity profiles, both for constant wall temperature and for constant heat input per unit length of the duct. Like the solutions for a fully developed velocity profile, the Nusselt number for each aspect ratio is found to increase from a limiting value at large distances from the entry plane to a maximum at the entry plane. The results also show a strong effect of the Prandtl number on the heat transfer coefficients with uniform and fully developed velocity profiles representing the upper and lower limits respectively. Comparisons are made with analytical solutions for circular ducts and parallel plates and with experimental data.     

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.     

Laminar heat transfer in parallel plate flow

An exact solution for the fluid temperature due to laminar heat transfer in parallel plate flow is found. The formulas obtained are valid for an arbitrary velocity profile. The basic problem encountered involves finding certain expansion coefficients in a series of nonorthogonal eigenfunctions. This problem is solved by passing to a vector system of equations having orthogonal eigenvectors. The method is applicable to more general problems.     

Analysis of laminar flow forced convection heat transfer in the entrance region of a circular pipe

A numerical solution, for incompressible, steady-state, laminar flow heat transfer in the combined entrance region of a circular tube is presented for the case of constant wall heat flux and constant wall temperature. The development of velocity profile is obtained from Sparrow's entrance region solution. This velocity distribution is used in solving the energy equation numerically to obtain temperature profiles. Variation of the heat transfer coefficient for these two different boundary conditions for the early stages of boundary layer formation on the pipe wall is obtained. Local Nusselt numbers are calculated and the results are compared with those given byUlrichson andSchmitz. The effect of the thermal boundary conditions is studied by comparing the uniform wall heat flux results with uniform wall temperature.     

A study on flow resistance in the entrance region of isosceles triangular ducts

In the present paper the variational solution of velocity profile for anincompressible laminar and fully developed flow in isosceles triangular ducts is derivedby applying the Kantovorich method.The theoretical and experimental results ofpressure loss are also given.The velocity distribution model,additional pressure losscoefficient and calculating method of inlet length in the entrance region of isoscelestriangular ducts are also derived,which are suitable for various kinds of vertex angles.The calculations and experiments are also performed for two models:the isoscelestriangular channels with vertex angles2α=45.1°and2α=60°.Comparisons aremade between the theoretical analysis in this paper and those of the other authors.Itcan be seen that the present analytical result is of high.accuracy and widepracticability,and agrees well with the authors’experiment.     

Finite difference analysis of forced-convection heat transfer in entrance region of a flat rectangular duct

Summary The laminar forced convection heat transfer in the entrance region of a flat rectangular duct is studied. In this region temperature and velocity profiles are simultaneously developed. The basic governing equations of momentum, continuity, and energy are expressed in finite difference form and solved numerically by use of a high speed computer for a mesh network superimposed on the flow field. All fluid properties are assumed to be constant. The cases of uniform constant wall temperature and of uniform constant heat flux from wall to fluid are considered. Nusselt numbers are reported for Prandtl numbers in the range of 0.01 to 50. The exact solution of the energy equation obtained by means of the numerical method is compared with the results of approximate solutions.Nomenclature A surface area of channel walls through which heat is being transferred - a duct half-height - C _p specific heat at constant pressure - D _e equivalent diameter for a duct, 4a - G _Z Graetz number, Re _d Pr/(x/D _e ) - h heat-transfer coefficient, Q/{A(t)} - k thermal conductivity of the fluid - Nu _m average Nusselt number, h _m D _e /k - Nu _x local Nusselt number, h _x D _e /k - Pr Prandtl number, C _p /k - p fluid pressure - p ₀ pressure at channel mouth - P dimensionless pressure, (p–p ₀)/u ₀ ² - Q heat-transfer rate - Re _a Reynolds number, u ₀ a/ - Re _d diameter Reynolds number, u ₀ D _e /=u ₀4a/ - t temperature - t ₀ temperature of fluid at entrance section of channel - t ₁ constant wall temperature - t _w wall temperature - u fluid velocity in x-direction - u ₀ fluid velocity at inlet - U dimensionless u velocity, u/u ₀ - v fluid velocity in y-direction - V dimensionless velocity, av/ - x coordinate along channel - X dimensionless x-coordinate, x/(a ² u ₀)=(x/a)/Re _a - X dimensionless x-coordinate defined as x/(D _e ² u ₀)=(x/D _e )/Re _d =X/16 - y coordinate across channel - Y dimensionless y-coordinate, y/a - thermal diffusivity of fluid, k/C _p - kinematic viscosity of fluid - fluid density - dynamic viscosity of fluid - dimensionless temperature, defined by (8), (t–t ₀)/(t ₁–t ₀) for constant wall temperature, k(t–t ₀)/(ag) for constant heat flux case - _b,x dimensionless bulk temperature at any location x, defined by (15) - _w dimensionless wall temperature defined by (8)     

Forced convection heat transfer in doubly connected ducts

Forced convection heat transfer in doubly connected ducts bounded externaly by a circle and internally by a regular polygon of various shapes is analysed using a finite element method. Hydrodynamically and thermally developed, steady, laminar flow of a constant property fluid is investigated. An insulated outer tube and constant heat flux at the inner core are considered. Temperature profiles as well as Nusselt numbers are presented. Salient characteristics of the temperature field in such passages are identified. Correlations for the Nusself number with aspect ratio are suggested.