Heat transfer in the thermal entrance region for flow through rectangular porous passages

The study of heat transfer in rectangular passages with prescribed wall heat flux is of practical interest. These passages could be open or filled with saturated porous materials. A solution that uses the Green’s function can accommodate the inclusion of heat flux over the entire surface area or over isolated sections of the boundary. Also, this solution permits the inclusion of frictional heating. Two different boundary conditions are considered: constant wall temperature and constant wall heat flux. The computed heat transfer coefficients show that the thermally fully developed condition may not be attainable in practical applications for very narrow passages with prescribed wall heat flux.     

Laminar tube flow heat transfer in non-newtonian fluids under arbitrary wall heat flux

Analytical solutions are developed for the wall temperature profile of a power law fluid in laminar flow in a circular tube. This profile is first developed for the boundary condition involving uniformly constant heat flux at the wall. This is next extended for the boundary condition involving an arbitrarily varying heat flux at the wall. The computed results are finally compared with measured values obtained from a horizontal recirculating flow experimental unit.     

Temperature Distribution in Flux Channels with Discrete Contact Boundary Conditions

An analytical approach for the thermal behavior of two-dimensional rectangular flux channels with arbitrary boundary conditions on the source plane is presented. The boundary condition along the source plane can be a combination of the first kind boundary condition (Dirichlet or prescribed temperature) and the second kind boundary condition (Neumann or prescribed heat flux). To model the boundary conditions along the source plane, the method of least squares is used. The proposed solution is in the form of Fourier series expansion and can be applied to both symmetrical and non-symmetrical channels. This method is more general than other approaches and there is no need to use equivalent heat flux distributions to model isothermal heat sources. The general approach for obtaining the multidimensional temperature profile in flux channels and the advantages of the least-square method is discussed. The proposed solution can be used to calculate the temperature at any specified point in the flux channel. Two case studies are presented. The first case study is a flux channel with five discretely specified contact temperatures along the source plane. The second case study has both of the first kind and second kind boundary conditions on the source plane. The analytical results for both systems are compared with finite element method using a commercial software package. It is shown that the proposed approach can precisely model the temperature profile over the flux channel.     

Heat transfer in a viscoelastic boundary layer flow over a stretching sheet

Studies are made on the viscoelastic fluid flow and heat transfer characteristics over a stretching sheet with frictional heating and internal heat generation or absorption. The heat transfer analysis has been carried out for the cases of prescribed surface temperature (PST) and prescribed surface heat flux (PHF). The momentum equation is decoupled from the energy equation for the present incompressible boundary layer flow problem with constant physical parameters. Exact solution for the velocity field and the skin-friction are obtained. Also, the solutions for the temperature and heat transfer characteristics are obtained in terms of Kummer’s function. The work due to deformation in energy equation, which is essential while formulating the viscoelastic boundary layer flow problems, is considered. This paper examines the effect of viscoelastic parameter, Eckert number, Prandtl number and non-uniform heat source/sink parameter on temperature distribution, wall temperature gradient for PST-case and wall temperature for PHF-case.     

Buoyant instability in a laterally heated vertical cylinder

This paper presents a linear stability analysis for the buoyant convection in a vertical cylinder with isothermal top and bottom walls at the same temperature and with an axisymmetric heat transfer into the liquid from the vertical cylindrical wall. Results are presented for Prandtl numbers between 0.0 and 0.1 and for two different thermal boundary conditions at the vertical wall: a prescribed parabolic temperature variation or a prescribed parabolic radial heat flux variation. The results are radically different for the two thermal boundary conditions.     

Flow due to a porous stretching/shrinking sheet with thermal radiation and mass transpiration

This study investigates the behavior of carbon nanotubes (CNT) approaching an unsteady flow of a Newtonian fluid over a stagnation point on a stretching surface employing porous media. It flows when the liquid begins to move with the progression of time. Heat exchange with the environment has an impact on the flow. The implicitly limited component technique is used to solve the nondimensional partial differential equation with an associated boundary layer, which is an unstable system. Analytically, the solutions, as well as the required boundary conditions, are obtained. The effects of mass transpiration, volume fraction, and heat radiation on Newtonian fluid flow through porous media are explored. Single- and multi-walled CNTs are used as well as water, as base fluids in the experiment. The impact of thermal radiation and heat source/sink is shown in the energy equation, which is solved under four different cases: uniform heat flux case, constant wall temperature case, general power-law wall heat flux case, and general power-law wall temperature case. By supplying distinct physical characteristics, a theoretical analysis of the existence and nonexistence of unique and dual solutions may be explored. These physical parameters determine the velocity distribution and temperature distribution. Prescribed surface temperature (PST) and prescribed wall heat flux (PHF) heat transfer solutions can be written using confluent hypergeometric equations, and generic power-law PST and PHF situations can also be expressed using confluent hypergeometric equations. The graphical representations assist in the discussion of the current study's findings.     

The effect of upper surface conditions on convection in a shallow cavity with differentially heated end-walls

The effect of upper surface boundary conditions on the flow structure in shallow cavities with differentially heated end-walls is examined. Matched asymptotic solutions, valid for small cavity aspect ratios are presented for the following cases; uniform shear stress with zero heat flux, uniform heat flux with zero shear stress, and heat flux linearly dependent on surface temperature with zero shear stress. It is shown that these changes in surface boundary conditions have an important influence on the temperature and flow structure within the cavity.     

Heat transfer in a viscoelastic boundary layer flow over a stretching sheet with viscous dissipation and non-uniform heat source

In this paper, visco-elastic boundary layer flow and heat transfer over a stretching sheet in presence of viscous dissipation and non-uniform heat source have been discussed. Analytical solutions of highly non-linear momentum equation and confluent hypergeometric similarity solution of heat transfer equations are obtained. Here two types of different heating processes are considered namely (i) prescribed surface temperature (PST) and (ii) prescribed wall heat flux (PHF). The effect of various parameters like visco-elastic parameter, Eckert number, Prandtl number, and non-uniform heat source/sink parameter on temperature distribution are analyzed and effect of all these parameters on wall temperature gradient and wall temperature are tabulated and discussed.     

Analytical investigations for heat conduction problems in anisotropic thin-layer media with embedded heat sources

This article develops the analytical rigorous solution of a fundamental problem of heat conduction in anisotropic media. The steady-state temperature and heat flux fields in a thin-layer medium with anisotropic properties subjected to concentrated embedded heat sources or prescribed temperature on the surface are analyzed. A linear coordinate transformation is used to transform anisotropic thin-layer problems into equivalent isotropic problems without complicating the geometry and boundary conditions of the problem. By using the Fourier transform and the series expansion technique, exact closed-form solutions of the specific problems are presented in series forms. The complete solutions of heat conduction problems for the thin-layer medium consist only of the simplest solutions for an infinite homogeneous medium with concentrated heat sources. The numerical results of the temperature and heat flux distributions are provided in full-field configurations.     

理想流体对流传热问题的理论解

研究理想流体受迫对流传热和自然对流传热问题的理论解。采用流体无垂直于壁面法线方向运动(即无穿透)的条件取代黏性流体在壁面无滑移条件，解决了流体在边界上有滑移时计算对流传热系数的困难，给出了理想流体与平壁受迫对流传热、理想流体与竖直壁面自然对流传热和理想流体在管内受迫对流传热的理论解。结果表明：理想流体的对流传热与黏性流体同样存在着热边界层。在外部流动的情况下，无论受迫对流传热还是自然对流传热，对流传热系数都与流体的导热系数、密度和比热三乘积的二分之一次方成正比。在管内受迫对流的情况下，当无因次长度大于0．05时，局部Nu和界面无因次温度分布都不再变化，对于恒热流边界条件，Nu等于8，截面无因次平均温度等于2；对于恒壁温边界条件，Nu等于5．782，截面无因次平均温度等于2．316。     

METHODOLOGY TO GENERATE ACCURATE SOLUTIONS FOR VERIFICATION IN TRANSIENT THREE-DIMENSIONAL HEAT CONDUCTION

This article describes the development of accurate solutions for transient three-dimensional conductive heat transfer in Cartesian coordinates for a parallelepiped which is homogeneous and has constant thermal properties. The intended use of these solutions is for verification of numerical computer programs which are used for solving transient heat conduction problems. Verification is a process to ensure that a computer code is free of errors and accurately solves the mathematical equations. The exact solutions presented in this article can have any combination of boundary conditions of specified temperature, prescribed heat flux, or imposed convection coefficient and ambient temperature on the surfaces of the parallelepiped. Additionally, spatially uniform nonzero initial condition and internal energy generation are treated. The methodology to obtain the analytical solutions and sample calculations are presented.     

Possible transformations for natural convection on a vertical flat plate embedded in porous media with prescribed wall heat flux

This study derived the transformation of boundary layer equations for two-dimensional steady natural convection on a vertical wall embedded in porous media. Three different kinds of thermal boundary conditions are prescribed for wall heat flux: uniform distribution, power law variation, and exponential variation. The flow pattern contains three subregions based on the distance along the flat plate. When inertia resistance is ignored, similarity solution exists in case wall heat flux is in linear distribution. The known relationships of uniform wall temperature relative to wall heat flux variation and uniform wall heat flux relative to wall temperature variation in both cases of the pure fluid flow and the pure porous flow can all be obtained in the present transformation.     

Heat Transfer Characteristics of Laminar Flow in Internally Finned Tubes under Various Boundary Conditions

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Natural convection of a Darcian fluid about a cone

The problem of natural convection about a cone embedded in a porous medium at high Rayleigh numbers is analyzed based on the boundary layer approximation and the Darcy's law. Local similarity solutions are obtained for a truncated cone with the prescribed wall temperature or surface heat flux being a power function of distance from the leading edge. It is shown that at a location near the leading edge, the solution behaves like that of an inclined plate. At a location far downstream of the truncated cone, the solution approaches that of a full cone where similarity solutions exist.     

MHD free-convection flow in open-ended vertical concentric porous annuli

Analytical solutions for fully developed MHD natural-convection flow in open-ended vertical concentric porous annuli are presented. Four fundamental boundary conditions have been investigated and the corresponding fundamental solutions are obtained. These four fundamental boundary conditions are obtained by combining each of the two conditions of having one boundary maintained at uniform heat flux or at uniform wall temperature with each of the conditions that the opposite boundary is kept isothermal at the inlet fluid temperature or adiabatic. Expressions for the flow and heat-transfer parameters are given for each case. These fundamental solutions may be used to obtain solutions satisfying more general thermal boundary conditions.     

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.     

Heat transfer with upstream thermal penetration in flow through porous plate passages

The objective of this study is to develop a series solution for determination of the temperature field in parallel-plate ducts with prescribed wall heat flux. Consideration is also given to ducts filled with fluid saturated porous materials. A simple transformation is used to improve the convergence of this series solution. Temperature variations and heat transfer coefficients are determined for infinitely long parallel-plate ducts when the walls undergo a step change in the applied wall heat flux. Axial thermal penetration near the applied wall heat flux location is studied. The solutions with the contribution of axial conduction in these fluid passages are acquired using a modified Graetz-type solution. Finally, this technique is augmented by the contribution of frictional heating.     

Viscoelastic boundary layer flow and heat transfer over an exponential stretching sheet

Viscoelastic boundary layer flow and heat transfer over an exponential stretching continuous sheet have been examined in this paper. Approximate analytical similarity solution of the highly non-linear momentum equation and confluent hypergeometric similarity solution of the heat transfer equation are obtained. Accuracy of the analytical solution for stream function is verified by numerical solutions obtained by employing Runge-Kutta fourth order method with shooting. These solutions involve an exponential dependent of stretching velocity, prescribed boundary temperature and prescribed boundary heat flux on the flow directional coordinate. The effects of various physical parameters like viscoelastic parameter, Prandtl number, Reynolds number, Nusselt number and Eckert number on various momentum and heat transfer characteristics are discussed in detail in this work.     

An analytical solution for two-dimensional inverse heat conduction problems using Laplace transform

An analytical method has been developed for two-dimensional inverse heat conduction problems by using the Laplace transform technique. The inverse solutions are obtained under two simple boundary conditions in a finite rectangular body, with one and two unknowns, respectively. The method first approximates the temperature changes measured in the body with a half polynomial power series of time and Fourier series of eigenfunction. The expressions for the surface temperature and heat flux are explicitly obtained in a form of power series of time and Fourier series. The verifications for two representative testing cases have shown that the predicted surface temperature distribution is in good agreement with the prescribed surface condition, as well as the surface heat flux.     

Forced Convection Heat Transfer Simulation Using Dissipative Particle Dynamics

Dissipative particle dynamics (DPD) with energy conservation was applied to simulate forced convection in parallel-plate channels with boundary conditions of constant wall temperature (CWT) and constant wall heat flux (CHF). DPD is a coarse-grained version of molecular dynamics. An additional equation for energy conservation was solved along with conventional DPD equations, where inter-particle heat flux accounts for changes in mechanical and internal energies when particles interact with surrounding particles. The solution domain was considered to be two–dimensional with periodic boundary condition in the flow direction and additional layers of particles on the top and bottom of the channel to apply no-slip and wall temperature boundary conditions. The governing equation for energy conservation was modified based on periodic fully developed velocity and temperature conditions. The results were shown via velocity and temperature profiles across the channel cross-section. The Nusselt numbers for CWT and CHF were calculated from the temperature gradient at the wall using a second order accurate forward difference approximation. The results agreed well with the exact solutions to within 2.3%.