Analytical solution of conjugate heat transfer and optimum configurations of flat-plate heat exchangers with circular flow channels

An analytical solution is developed for conjugate heat transfer in a flat-plate heat exchanger with circular embedded channels. The analysis was carried out for fully-developed conditions in the circular tube and uniform heat flux at the plate boundary. The results are applicable to cooling channels that are 50 μm or more in diameter with a large length–diameter ratio. The thermal characteristics of the heat exchanger have been examined for a wide range of the relevant independent parameters and optimum designs for three different sets of constraints have been presented. It was found that the overall thermal resistance increases with the depth of the tube from the heated surface, as well as the spacing between the tubes. For a given combination of tubes’ depth and spacing, there is a certain tube diameter at which the thermal resistance attains a minimum value.     

Simplified determinations of heat transfer data for solids, applications to quality testing of electrical insulations

Summary The first part of the paper describes a simple steady state apparatus for measuring the thermal conductivity of plates of electrical insulations, plastics, etc. The measurement is quick, requiring approximately 5 min. The apparatus also enables one to determine the thermal capacity of the sample. The second part of the paper describes the k-meter, a new device for measuring the heat transfer coefficient through the insulation on metal bars such as generator coils. The method does not require any access to the inner conductor. The suitability of the k-meter for nondestructive thermal ageing investigations of e.g. generator insulations and for quality control, is demonstrated. In this respect k-value measurements might be a valuable complement to dielectric loss measurements.An Appendix gives details of the theory and error estimates.     

Periodic heat conduction with relaxation time in cylindrical geometry

Steady-periodic heat conduction with relaxation time in an infinitely long hollow cylinder is considered. Four boundary value problems, with boundary conditions of the first and of the second kind, are solved analytically. The solution for a solid cylinder with a sinusoidally varying surface temperature is obtained as a special case of a solution found for the hollow cylinder. The effects of the relaxation time on the steady-periodic temperature field are analysed, in details, for a solid cylinder with a sinusoidally varying surface temperature and for a hollow cylinder with a sinusoidally varying heat flux at the inner surface and with a constant temperature at the outer surface. The results show that thermal resonances may occur and suggest that accurate measurements of the relaxation time could be obtained by means of experiments on steady-periodic heat conduction in cylindrical geometry. Received on 15 April 1997     

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.     

Modeling and simulation of first and second sound in solids

We investigate the propagation of mechanical and thermal waves in solids at cryogenic temperatures. The latter are known as the second sound phenomenon. It occurs, e.g., in dielectric solids and differs greatly from the classical case in which the heat transport proceeds by diffusion. Since Fourier’s law of heat conduction fails for modeling second sound, we apply a non-classical one. During the last two decades, the non-classical thermoelastodynamic theory of Green and Naghdi enjoys steadily growing research activities.     

Non-fourier heat conduction in a plane slab with prescribed boundary heat flux

The non-stationary heat conduction in an infinitely wide plane slab with a prescribed boundary heat flux is studied. An arbitrary time dependent boundary heat flux is considered and a non-vanishing thermal relaxation time is assumed. The temperature and the heat flux density distributions are determined analytically by employing Cattaneo-Vernotte's constitutive equation for the heat flux density. It is proved that the temperature and the heat flux density distributions can be incompatible with the hypothesis of local thermodynamic equilibrium. A comparison with the solution which would be obtained by means of Fourier's law is performed by considering the limit of a vanishing thermal relaxation time.     

Non-Fourier heat conduction in a finite hollow cylinder with periodic surface heat flux

Analytical solution of the non-Fourier axisymmetric temperature field within a finite hollow cylinder exposed to a periodic boundary heat flux is investigated. The problem studied considering the Cattaneo–Vernotte (CV) constitutive heat flux relation. The material is assumed to be homogeneous and isotropic with temperature-independent thermal properties. The standard method of separation of variables is used for solving the problem with time-independent boundary conditions, and the Duhamel integral is used for applying the time dependency. The solution is applied for the special cases of harmonic uniform heat flux and an exponentially pulsed heat flux with Gaussian distribution in outer surface for modeling a laser pulse, and their respective non-Fourier thermal behavior is studied.     

A new heat transfer analysis in machining based on two steps of 3D finite element modelling and experimental validation

Modelling machining operations allows estimating cutting parameters which are difficult to obtain experimentally and in particular, include quantities characterizing the tool-workpiece interface. Temperature is one of these quantities which has an impact on the tool wear, thus its estimation is important. This study deals with a new modelling strategy, based on two steps of calculation, for analysis of the heat transfer into the cutting tool. Unlike the classical methods, considering only the cutting tool with application of an approximate heat flux at the cutting face, estimated from experimental data (e.g. measured cutting force, cutting power), the proposed approach consists of two successive 3D Finite Element calculations and fully independent on the experimental measurements; only the definition of the behaviour of the tool-workpiece couple is necessary. The first one is a 3D thermomechanical modelling of the chip formation process, which allows estimating cutting forces, chip morphology and its flow direction. The second calculation is a 3D thermal modelling of the heat diffusion into the cutting tool, by using an adequate thermal loading (applied uniform or non-uniform heat flux). This loading is estimated using some quantities obtained from the first step calculation, such as contact pressure, sliding velocity distributions and contact area. Comparisons in one hand between experimental data and the first calculation and at the other hand between measured temperatures with embedded thermocouples and the second calculation show a good agreement in terms of cutting forces, chip morphology and cutting temperature.     

Performance analysis and optimization of convective annular fins with internal heat generation

An analysis of the thermal performance for convective annular fins, having a general trapezoidal profile and internal heat generation, is presented. The solution of the optimal problem is also given when either the heat dissipation rate or the volume of the fin is specified. The results are expressed in suitable nondimensional variables that are specified by the problem, and presented graphically. The effect of the fin's profile and thermal conductivity upon the optimum dimensions is discussed. It is shown that the presence of heat generation reduces the ability of the fin to convect heat. Furthermore, certain limiting values of the heat generation that may be imposed on the fin for a feasible optimization are also derived.     

Transient multiboiling in a pin fin with temperature dependent thermal conductivity

Transient heat conduction in fins undergoing different kinds of convective processes (film, transition, nucleated boiling and natural convection) as in multiboiling processes take place, is a strongly non-linear problem because of the abrupt changes in the heat transfer coefficient that occur at certain temperatures. Transient equations for the thermal fields and fluxes are solved simultaneously, giving the time constant of the process; the stationary solution is compared with the numerical or experimental values of other authors. Temperature dependencies of the heat transfer coefficient and the thermal conductivity is assumed due to the large interval of temperatures occurring. Network Simulation Method is used for the numerical solution, which gives simultaneously thermal field of temperatures and heat fluxes.     

Exact solution of conductive heat transfer in cylindrical composite laminate

This paper presents an exact solution for steady-state conduction heat transfer in cylindrical composite laminates. This laminate is cylindrical shape and in each lamina, fibers have been wound around the cylinder. In this article heat transfer in composite laminates is being investigated, by using separation of variables method and an analytical relation for temperature distribution in these laminates has been obtained under specific boundary conditions. Also Fourier coefficients in each layer obtain by solving set of equations that related to thermal boundary layer conditions at inside and outside of the cylinder also thermal continuity and heat flux continuity between each layer is considered. In this research LU factorization method has been used to solve the set of equations.     

On the axi-symmetric equations of a rigid heat conductor: discontinuity waves in a non-constant state

By requiring the invariance with respect to a stretching group of transformations the non-linear system of equations governing the heat propagation in a rigid conductor is mapped in a form which has a constant state solution in terms of suitable similarity variables. It has been shown that constant state solutions become special non-constant state solutions in the original variables. In particular, we prove that the temperature dependence of the propagation of discontinuity waves along a solid in an equilibrium state well reproduces the temperature behaviour of the second sound speed.     

Evaporator critical heat flux in the double-wall artery heat pipe

The fact that heat is transferred into a heat pipe through the liquid-saturated evaporator wick gives rise to the so-called boiling limit on the heat pipe capacity. The composite nature of the double-wall artery heat pipe (DWAHP) wick structure makes the prediction of the evaporator superheat (Δ T_crit) and the critical radial heat flux (q_r) very difficult. The effective thermal conductivity of the wick, the effective radius of critical nucleation cavity, and the nucleation superheat, which are important parameters for double-wall wick evaporator heat transfer, have been evaluated based on the available theoretical models. Empirical correlations are used to corroborate the experimental results of the 2 m DWAHP. A heat choke mounted on the evaporator made it possible to measure the evaporator external temperatures, which were not measured in the previous tests. The high values of the measured evaporator wall temperatures are explainable with the assumption of a thin layer of vapor blanket at the inner heating surface. It has been observed that partial saturation of the wick (lean evaporator) causes the capillary limit to drop even though it may be good for efficient convective heat transfer through the wick. The 2 m long copper-water heat pipe had a peak performance of 1850 W at 23 W/cm² with a horizontal orientation.     

Turbulent heat transfer in a flow of liquid metal near the wall

The article discusses turbulent heat transfer in media with small Prandtl numbers (Pr?1 for liquid metals). In this case, the thermal sublayer is Pr^-1 times thicker than the viscous sublayer. It is established that the coefficient of turbulent heat transfer varies in the thermal sublayer proportionally to the second power of the distance to the wall; the ratio of the coefficients of the turbulent transfer of heat and momentum in this region decreases in accordance with a linear law with approach to the wall. The conclusions of the theory are compared with the experimental data of other authors.     

Numerical nonlinear inverse problem of determining wall heat flux

The inverse problem of determining time-variable surface heat flux in a plane wall, with constant or temperature dependent thermal properties, is numerically studied. Different kinds of incident heat flux, including rectangular waveform, are assumed. The solution is numerically solved as a function estimation problem, so that no a priori information for the functional waveforms of the unknown heat flux is needed. In all cases, a solution in the form of a piece-wise function is used to approach the incident flux. Transient temperature measurements at the boundary, from the solution of the direct problem, served as the simulated experimental data needed as input for the inverse analysis. Both direct and inverse heat conduction problems are solved using the network simulation method. The solution is obtained step-by-step by minimising the classical functional that compares the above input data with those obtained from the solution of the inverse problem. A straight line of variable slope and length is used for each one of the stretches of the desired solution. The influence of random error, number of functional terms and the effect of sensor location are studied. In all cases, the results closely agree with the solution.     

Analytical study of heat transfer from circular cylinder in liquid metals

In this study the influence of a thin hydrodynamic boundary layer on the heat transfer from a single circular cylinder in liquid metals having low Prandtl number (0.004–0.03) is investigated under isothermal and isoflux boundary conditions. Two separate analytical heat transfer models, viscous and inviscid, are developed to clarify the discrepancy between previous results. For both models, integral approach of the boundary layer analysis is employed to derive closed form expressions for the calculation of the average heat transfer coefficients. For an inviscid model, the energy equation is solved using potential flow velocity only whereas for a viscous model, a fourth-order velocity profile is used in the hydrodynamic boundary layer and potential flow velocity is used outside the boundary layer. The third-order temperature profile is used inside the thermal boundary layer for both models. It is shown that the inviscid model gives higher heat transfer coefficients whereas viscous flow model gives heat transfer results in a fairly good agreement with the previous experimental/numerical results.     

Propagation of blast waves with exponential heat release and internal heat conduction and thermal radiation

The problem of reactive blast waves in a combustible gas mixture, where the heat release at the detonation front decays exponentially with the distance from the center, is analyzed. The central theme of the paper is on the propagation of reactive blast into a uniform, quiescent, counterpressure atmosphere of a perfect gas with constant specific heats. The limiting cases of Chapman-Jouguet detonation waves are considered in the phenomenon of point explosion. In order to deal with this problem, the governing equations including thermal radiation and heat conduction were solved by the method of characteristics using a problem-specific grid and a series expansion as start solution. Numerical results for the distribution of the gas-dynamic parameters inside the flow field are shown and discussed.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

The distortion of a cylinder with non-uniform axial heat conduction

Closed form expressions are developed for the thermoelastic curvature of the initially plane end faces of a traction free cylinder subjected to arbitrary axisymmetric heat flux, the curved surfaces being assumed insulated. The solution is developed from a potential function representation of displacement and temperature for an elastic layer. The reciprocal theorem is invoked to show that the tractions at the curved surface of the cylinder vary linearly along the axis and they are removed by superposition of biaxial bending. It is found that the curvature of the plane ends depends on the local heat flux and the mean heat flux, whilst the cylindrical face distorts into a cone.     

Thermoconductivity and thermoelasticity of thin isotropic shells heated by a moving heat pulse

The paper presents a solution to the problem of thermal conduction and thermoelasticity for a thin shallow spherical shell heated by a concentrated or local impulsive heat source moving over the shell surface. It is assumed that temperature is linearly distributed across the shell thickness and that the shell, on its sides, exchanges heat with the environment in accordance with Newton’s law of cooling. The Fourier and Laplace transforms are used to find an analytic solution. The dependence of the temperature field and stress/strain components on the type of heating and the form of heat source is studied __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 11, pp. 85–92, November 2006.