The extended Graetz problem with piecewise constant wall temperature for pipe and channel flows

Axial heat conduction effects within the fluid can be important for duct flows if the Prandtl number is relatively low (liquid metals). In addition, axial heat conduction effects within the flow might also be important, if the heating zone is relatively short in length. The present paper shows an entirely analytical solution to the extended Graetz problem with piecewise constant wall temperature boundary conditions. The solution is based on a selfadjoint formalism resulting from a decomposition of the convective diffusion equation into a pair of first order partial differential equations. The obtained analytical solution is as simple to compute as the one without axial heat conduction. The analytical results are compared to available numerical calculations and good agreement is found.     

The effect of wall conduction for the extended Graetz problem for laminar and turbulent channel flows

Axial heat conduction effects within the fluid can be important for duct flows if either the Prandtl number is relatively low (liquid metals) or if the dimensions of the duct are small (micro heat exchanger). In addition, axial heat conduction effects in the wall of the duct might be of importance. The present paper shows an entirely analytical solution to the extended Graetz problem including wall conduction (conjugate extended Graetz problem). The solution is based on a selfadjoint formalism resulting from a decomposition of the convective diffusion equation into a pair of first order partial differential equations. The obtained analytical solution is relatively simple to compute and valid for all Péclet numbers. The analytical results are compared to own numerical calculations with FLUENT and good agreement is found.     

Forced convection with viscous dissipation in the thermal entrance region of a circular duct with prescribed wall heat flux

The thermal entrance forced convection in a circular duct with a prescribed wall heat flux distribution is studied under the assumptions of a fully developed laminar flow and of a negligible axial heat conduction in the fluid, by taking into account the effect of viscous dissipation. The solution of the local energy balance equation is obtained analytically by employing the Laplace transform method. The effect of viscous dissipation is taken into account also in the region upstream of the entrance cross-section, by assuming an adiabatic preparation of the fluid. The latter hypothesis implies that the initial condition in the entrance cross-section is a non-uniform radial temperature distribution. Two special cases are investigated in detail: an axially uniform wall heat flux, a wall heat flux varying linearly in the axial direction.     

Non-axisymmetric forced and free flow in a vertical circular duct

The fully developed laminar mixed convection in a vertical circular duct is studied analytically, with reference to non-axisymmetric boundary conditions such that the fluid temperature does not change along the axial direction. The Boussinesq approximation is applied by taking the average temperature in a duct section as the reference fluid temperature. The dimensionless momentum and energy balance equations are solved by employing Fourier series expansions of the temperature and the velocity fields. The solution shows that the temperature field is not influenced by the velocity distribution and that the Fanning friction factor is not affected by buoyancy. On the other hand, the velocity field is strongly influenced by the buoyancy forces and may display flow reversal phenomena. Two special cases are studied in detail: a duct with a sinusoidal wall temperature distribution; a duct subjected to an external convection heat transfer with two environments having different reference temperatures.     

Combined forced and free flow in a vertical circular duct subjected to non-axisymmetric wall heating conditions

The fully developed mixed convection flow in a vertical circular duct is investigated analytically, under the assumption of laminar parallel flow. A wall heat flux uniform in the axial direction and dependent on the angular coordinate is considered. As a consequence, the fluid temperature is three dimensional, since it changes in the radial, axial and angular directions. An analytical method based on Fourier series expansions of temperature and velocity fields is adopted to determine the velocity and the temperature distributions as well as the friction factor and the average Nusselt number. The general solution, expressed in terms of Bessel functions, is applied to study a case that has a special importance in technical applications: a duct whose wall is half subject to a uniform heat flux and half adiabatic. The positive and negative threshold values of the ratio between the Grashof number Gr and the Reynolds number Re for the onset of the flow reversal phenomenon are determined. A comparison between the average Nusselt number for the considered non-axisymmetric case and that for the case of a duct subject to a uniform wall heat flux is performed.     

Experiments on laminar flow and heat transfer in an elliptical duct

Experiments were performed to study the laminar developing and fully developed flow and heat transfer inside an elliptical duct having an aspect ratio of 0.5. The working fluid was air and two thermal situations were investigated, the first with the duct at a uniform temperature and the second when the wall temperature distribution is linear in the axial direction and does not vary transversely. The hydrodynamic results are presented in the form of a sequence of velocity profiles on the major and minor axes, measured at axial locations extending from the duct entrance to the fully developed regime. The axial drop in the static pressure due to the combined effect of the flow development and wall friction is also reported. The extended length necessary for static pressure development, expressed as x/Re D_h, was found to be 0.0345. The thermal information depicts the temperature development in the duct entrance by a series of temperature profiles on the major and minor axes. The thermal results encompass as well the Nusselt number and the thermal entrance length in each of the above two thermal situations. To the author's knowledge, theoretical solutions for the hydrodynamic flow development in the entrance of elliptical ducts do not exist. The present experimental fully developed dimensionless velocity and friction factor were compared to the analytic value of L. N. Toa [On some laminar-forced-convection problems, ASME J. Heat Trans. 83, 466–472 (1961)]. The percentage difference in the friction factor is 0.78%. The thermal development of the flow in the elliptical duct was studied analytically by N. T. Dunwoody [Thermal results for forced heat convection through elliptical ducts, J. appl. Meal. 29, 165–170 (1962)] and V. Javeri [Analysis of laminar thermal entrance region of elliptical and rectangular channels with the Kantorowich method, Wärme Stoffubert. 9, 85–98 (1976)] for the uniform and linear wall temperature ducts respectively. Both analyses assume either uniform or fully developed velocity profiles in the developing regime.     

Laminar forced convection in circular and flat ducts with wall axial conduction and external convection

An analytical solution is obtained for laminar forced convection in circular and flat ducts with the presence of axial duct wall conduction and external convection at the outer surface of the duct wall. The eigenvalues for the problem are determined using the solution for the constant temperature boundary condition. The heat transfer results depend on four nondimensional numbers. The wall and fluid temperatures depend strongly on the wall conductance parameter while the heat flux enhancement due to wall conduction is large at short distances from the duct inlet.     

EFFECT OF BUOYANCY ON FORCED CONVECTION IN A CUSPED DUCT

ABSTRACT

In the event of a loss of coolant accident in a pressurized water reactor, swelling of the fuel rod cladding will lead to reduction of the subchannel flow area and worsening of the core heat transfer in the region of the blockage. The four-cusped duct is an ideal geometry for the simulation of such a channel blockage. Understanding the characteristics of flow and heat transfer in the cusped duct is essential for better design of the emergency core cooling system. Thus, in this paper, combined natural and forced convection in a vertical cusped duct has been investigated in the region of both hydrodynamically and thermally fully developed flow. The thermal boundary condition imposed on the cusped duct is the axial uniform heat flux with peripheral uniform temperature. The results indicate that the fluid flow and heal transfer in the comer region of the cusped duct are improved because of the influence of natural convection. As the Rayleigh number increases, the friction factor and Nusselt number increase accordingly. It was also found that the critical Rayleigh number is 1200, at which flow reversal occurs in the buoyancy-assisted flow ( heated upflow). The velocity, temperature, and local Nusselt number distribution are presented for a range of Rayleigh numbers.     

Physical Effects of Variable Thermophysical Fluid Properties on Flow and Thermal Development in Micro-Channel

Micro-scale cooling is an efficient and effective cooling technique to achieve the goal of higher heat removal capabilities. The present research focuses to find the physical effects of fluid property variations on flow and thermal development in micro-channel. The effects of temperature-dependent density, viscosity, and thermal conductivity variations on single-phase laminar forced convection are numerically investigated. The problem is especially simulated for hydrodynamically and thermally developing water flow in micro-channel with no-slip, no-temperature jump, and constant wall heat flux boundary conditions. It is observed that the density variation induces radially inward flow due to continuity, which sharpens the axial velocity profile and decreases Nusselt number compared to constant property solution. The axial velocity profile significantly alters due to viscosity variation. This alteration varies along the micro-flow and it induces radially flow due to flow continuity. The reducing rate of Nusselt number for viscosity variation is substantially lower than constant property solution due to a significant flattening effect of the axial velocity profile, which augments the Nusselt number. Thermal-conductivity variation across the flow induces radial conduction, which enhances convection compared to constant property solution. Additionally, the effects of thermophysical fluid property variations on static gauge pressure drop are also investigated.     

Axial conduction in laminar pipe flows with nonlinear wall heat fluxes

A numerical analysis to determine the heat-transfer parameters of a fluid flow rejecting heat to the surrounding medium by convection and radiation is developed. The influence of axial conduction is included and the velocity profile is taken as nonuniform in the transverse direction. Use of a transformation eliminates the required boundary conditions at infinity. Approximate numerical techniques are employed to solve the nonlinear conjugate problem. As Péclet number increases, the temperature fields simplify to those where axial conduction is excluded. The computed results indicate that the effects of axial conduction are strongly altered by the parameters responsible for the convection and radiation. Bulk fluid temperatures, wall heat fluxes and Nusselt numbers are plotted against Graetz numbers. Critical Péclet numbers for a variety of cooling conditions are presented using the bulk fluid temperature as a reference.     

SIMULATION OF SWIRLING TURBULENT FLOWS AND HEAT TRANSFER IN AN ANNULAR DUCT

The article presents a numerical simulation of swirling turbulent flows and heat transfer in an annular duct. The time-averaged governing equations are solved, which are closed by a new algebraic Reynolds stress model (ASM). The simulation is performed under different flow conditions. The calculated results of gas axial and tangential velocities, turbulent kinetic energy, temperature, and local heat transfer coefficients on the inner and outer walls of the annulus are provided. They illustrate the effect of swirl number, inlet axial velocity, and ratio of inner to outer radius on the mean flow and turbulence properties, as well as on enhancing heat transfer in the annular duct.     

Conjugated heat transfer in circular ducts with a power-law laminar convection fluid flow

This work deals with the study of the steady-state analysis of conjugated heat transfer process for the thermal entrance region of a developed laminar-forced convection flow of a power-law fluid in a circular tube. A known uniform heat flux is applied at the external surface of the tube. The energy equation in the fluid is solved analytically using the integral boundary layer approximation by neglecting the heat generation by viscous dissipation and the axial heat conduction in the fluid. This solution is coupled to the Laplace equation for the solid, where the axial heat conduction effects are taken into account. The governing equations are reduced to an integro-differential equation which is solved by analytical and numerical methods. The results are shown for different parameters such as conduction parameter, α, the aspect ratio of the tube, ε and the index of power-law fluid, n.     

Experimental investigation of coupled conduction and laminar convection in a circular tube

Wall heat conduction effects on laminar flow heat transfer are experimentally investigated. The steady flow of water through a uniformly heated copper pipe is considered in the experiment, which covers a range of Reynolds numbers from 500 to 1900. The thermal behaviour of the test section is simulated numerically and the influence of conduction along the pipe wall is therefore accounted for in the reduction of the data. Fully developed flow results satisfactorily compare with predictions by a theoretical method previously developed by the authors [Heat Technol. 2,72 (1984)]. Results are also reported for the case where the velocity profile is partially developed at the inlet of the heat transfer section. The combined effects on heat transfer of flow development and of wall axial heat conduction are discussed.     

Direct numerical simulation of turbulent flow and combined convective heat transfer in a square duct with axial rotation

Direct numerical simulations (DNS) of turbulent flow and convective heat transfer in a square duct with axial rotation were carried out. The pressure-driven flow is assumed to be hydrodynamically and thermally fully developed, for which the Reynolds number based on the friction velocity and hydraulic diameter is kept at constant (Re_τ = 400). In the finite length duct, two opposite walls are perfectly insulated and another two opposite walls are kept at constant but different temperatures. Four thermal boundary conditions were chosen in combination with axial rotation to study the effects of rotation and Grashof number on mean flow, turbulent quantities and momentum budget. The results show that thermal boundary conditions have significant effects on the topology of secondary flows, profiles of streamwise velocity, distribution of temperature and other turbulent statistic quantities but have marginal effects on the bulk-averaged quantities; Coriolis force affects the statistical results very slightly because it exerts on the plane normal to main flow direction and the rotation rate is low; Buoyancy effects on the turbulent flow and heat transfer increase with the increase of Grashof number (Gr), and become the major mechanism of the development of secondary flow, turbulence increase, and momentum and energy transport at high Grashof number.     

An approximate solution to the graetz problem with axial conduction and prescribed wall heat flux

A finite integral transform technique is used to obtain an approximate solution to the Graetz problem with axial conduction and prescribed wall heat flux for an arbitrary velocity profile. Unlike the exact results of earlier works, this method requires minimal computational time yet compares excellently with the exact solutions for laminar flow of a Newtonian fluid for the entire range of Peclet numbers.     

Viscous dissipation effects on the asymptotic behaviour of laminar forced convection for Bingham plastics in circular ducts

The present study concentrates on the effects of viscous dissipation and the yield shear stress on the asymptotic behaviour of the laminar forced convection in a circular duct for a Bingham fluid. It is supposed that the physical properties are constant and the axial conduction is negligible. The asymptotic temperature profile and the asymptotic Nusselt number are determined for various axial distributions of wall heat flux which yield a thermally developed region. It is shown that if the asymptotic value of wall heat flux distribution is vanishes, the asymptotic value of the Nusselt number is zero. The case of the asymptotic wall heat flux distribution non-vanishing giving a value of the Nusselt number dependent on the Brinkman number and on the dimensionless radius of the plug flow region was also analysed. For an infinite asymptotic value of wall heat flux distributions, the asymptotic value of the Nusselt number depends on the dimensionless radius of the plug flow region and on the dimensionless parameter which depends on the asymptotic behaviour of the wall heat flux. The condition of uniform wall temperature and convection with an external isothermal fluid were also considered. The comparison with other existing solutions in the literature in the Newtonian case is analysed.     

Numerical study of fluid flow and heat transfer in microchannel cooling passages

Numerical investigation was conducted for fluid flow and heat transfer in microchannel cooling passages. Effects of viscosity and thermal conductivity variations on characteristics of fluid flow and heat transfer were taken into account in theoretical modeling. Two-dimensional simulation was performed for low Reynolds number flow of liquid water in a 100 μm single channel subjected to localized heat flux boundary conditions. The velocity field was highly coupled with temperature distribution and distorted through the variations of viscosity and thermal conductivity. The induced cross-flow velocity had a marked contribution to the convection. The heat transfer enhancement due to viscosity-variation was pronounced, though the axial conduction introduced by thermal-conductivity-variation was insignificant unless for the cases with very low Reynolds numbers.     

An experimental and theoretical investigation into the hyperbolic nature of axial dispersion in packed beds

The concept of hyperbolic axial dispersion of heat in a flowing fluid which is known as `third sound wave' has been examined taking packed bed as an example. This technique analyse fluid flow and heat transfer by introducing an axial dispersion term in the one-dimensional energy equation to take care of flow maldistribution and backmixing. The present approach models this dispersion in terms of two parameters which are proposed to follow hyperbolic conduction law. A regenerator bed consisting of stainless steel wire mesh packing has been used to carry out experiments for the purpose of validating the concept. The analytical model presented uses a Laplace transform technique for the solution of simplified energy equation. The computed outlet fluid temperature is compared with experimental output and the two model parameters, dispersive Peclet number (Pe) and its propagation velocity (c<sup>*</sup>) are estimated. The present model, the parabolic (Fourier) dispersion model and the non-dispersive plug flow model are compared with the experimental result which brings out the closeness of the present proposition to reality compared to other models. A standard experimental technique is suggested which can be used for the measurement of parameters related to axial dispersion. The present study is the first experimental evidence of `third sound wave' in fluids and lays foundation of this proposition.     

Heat Transfer Analysis for a Concentric Tube Heat Exchanger Including the Wall Axial Conduction

The effect of wall axial conduction on the heat transfer in a concentric tube heat exchanger is examined for the inner flow laminar flow regime. The procedure used for the current analysis combines the analytical solution for the inner fluid with a numerical approximation for the wall conduction and has the capability of handling the temperature variation for the outer fluid. Both parallel and counterflow cases are evaluated for the analysis, and results are presented in terms of the axial variations of fluids and wall temperatures. Effects of the heat capacity rate ratio of the fluids on the temperature variations and on the mean heat flux are also pointed out. The effect of the exchanger length is included for the analysis. It is concluded that the total heat transfer between the fluids is greatly influenced by the wall axial conduction for the counterflow arrangement and is not ignorable when the heat capacity rate ratio of fluids are smaller than unity.     

Laminar Flow and Heat Transfer to a Non-Newtonian Fluid in an Entrance Region of a Square Duct with Prescribed Constant Axial Wall Heat Flux

Abstract

Forced-convection heat transfer information as a function of the pertinent nondimensional numbers is obtained numerically for laminar incompressible non-Newtonian fluid flow in the entrance region of a square duct with simultaneously developing temperature and velocity profiles for constant axial wall heat flux with uniform peripheral wall temperature. The power-law model characterizes the non-Newtonian behavior.

Finite-difference representations are developed for the equations of the mathematical model, and numerical solutions are obtained assuming uniform inlet velocity and temperature distributions. Results are presented for local and mean Nusselt numbers as functions of the Graetz number and the Prandtl number in the entrance region. Comparisons are made with previous analytical work for Newtonian fluids. The results show a strong effect of the Prandtl number on the Nusselt numbers with fully developed and uniform velocity profiles representing the lower and upper limits, respectively. The results provide a new insight into the true three-dimensional character of the pseudoplastlc fluid flow in the entrance region of a square duct and are accurate.