A new multidimensional integral relationship between heat flux and temperature for direct internal assessment of heat flux

This paper derives a new integral relationship between heat flux and temperature in a transient, two-dimensional heat conducting half space. A unified mathematical treatment is proposed that is extendable to higher-dimensional and finite-region geometries. The analytic expression provides the local heat flux perpendicular to the front surface solely based on an embedded line of temperature sensors parallel to the surface. The relationship does not require apriori knowledge of the surface boundary condition. A new sensor strategy is analytically conceived based on the integral relationship for estimating the local, in-depth heat flux without surface instrumentation. It should further be clarified that the integral relationship requires only knowledge of the local, in-depth temperature and heating/cooling rate (time rate of change of temperature). The resulting formulation is mildly ill-posed and either requires digital filtering of the temperature signal to remove high frequency components of noise or the development of direct heating/cooling rate sensors. This paper (a) develops the new mathematical relationship; (b) demonstrates that the proposed relationship reduces to well-known (i) one-dimensional results under the appropriate assumptions; and, (ii) two-dimensional surface results; and, (c) provides a simple numerical example validating the concept.     

Local sensitivity analysis for the heat flux-temperature integral relationship in the half-space

This paper presents insight into the heat flux-temperature (q″ ? T) integral relationship based on constant thermophysical properties. This relationship is often used in one-dimensional, transient heat transfer studies involving null-point calorimetry and heat flux investigations. This study focuses on a short transient studies where energy has not fully penetrated the body as the result of an imposed surface heating condition. A full nonlinear heat transfer model is developed involving a half-space planar region. Temperature results are then introduced into the constant property integral relationship and a newly derived Kirchoff integral relationship for retrieving the local heat flux. Good agreement is observed between the fully nonlinear results and locally linearized system. Additionally, a sensitivity study is presented which involves perturbing the average thermophysical properties of thermal conductivity and heat capacity.     

A Stefan problem for a non-classical heat equation with a convective condition

We prove the existence and uniqueness, local in time, of the solution of a one-phase Stefan problem for a non-classical heat equation for a semi-infinite material with a convective boundary condition at the fixed face x = 0. Here the heat source depends on the temperature at the fixed face x = 0 that provides a heating or cooling effect depending on the properties of the source term. We use the Friedman-Rubinstein integral representation method and the Banach contraction theorem in order to solve an equivalent system of two Volterra integral equations. We also obtain a comparison result of the solution (the temperature and the free boundary) with respect to the one corresponding with null source term.     

Measuring heat transfer during spray cooling using controlled induction-heating experiments and computational models

This paper presents a methodology combining experimental measurements with computational modeling to find the heat flux extracted during spray cooling of a metal surface. Controlled experiments are performed to impinge air-mist spray onto a metal probe surface while applying induction heating to follow a desired temperature history. A transient axisymmetric computational model of induction heating which couples electromagnetics and heat conduction has been developed and validated with a test problem. The model is calibrated to match transient dry measurements and then used to simulate a steady-state air-mist spray cooling experiment in order to quantify the heat extracted from the probe surface by the boiling water droplets. A detailed example is presented to illustrate this approach.     

Heat transfer to power-law fluids in thermal entrance region with viscous dissipation for constant heat flux conditions

An analytical solution of the heat transfer problem with viscous dissipation for non-Newtonian fluids with power-law model in the thermal entrance region of a circular pipe and two parallel plates under constant heat flux conditions is obtained using eigenvalue approach by suitably replacing one of the boundary conditions by total energy balance equation. Analytical expressions for the wall and the bulk temperatures and the local Nusselt number are presented. The results are in close agreement with those obtained by implicit finite-difference scheme. It is found that the role of viscous dissipation on heat transfer is completely different for heating and cooling conditions at the wall. The results for the case of cooling at the wall are of interest in the design of the oil pipe line.     

Natural convection boundary layer flow of fluid with temperature-dependent viscosity from a horizontal elliptical cylinder with constant surface heat flux

This study deals with the temperature-dependent viscosity effects on the natural convection boundary layer on a horizontal elliptical cylinder with constant surface heat flux. The mathematical problem is reduced to a pair of coupled partial differential equations for the temperature and the stream function, and the resulting nonlinear equations are solved numerically by cubic spline collocation method. Results for the heat transfer characteristics are presented as functions of eccentric angle for various values of viscosity variation parameters, Prandtl numbers and aspect ratios. Results show that an increase in the viscosity variation parameter tends to accelerate the fluid flow near the surface and increase the maximum velocity, thus decreasing the velocity boundary layer thickness. As the viscosity variation parameter is increased, the surface temperature tends to decrease, thus increasing the local Nusselt number. Moreover, the local Nusselt number of the elliptical cylinder increases as the Prandtl number of the fluid is increased.     

Natural convection flow from a horizontal circular cylinder with uniform heat flux in presence of heat generation

Natural convection boundary layer laminar flow from a horizontal circular cylinder with uniform heat flux in presence of heat generation has been investigated. The governing boundary layer equations are transformed into a non-dimensional form and the resulting non-linear systems of partial differential equations, which are solved numerically by two distinct methods namely: (i) implicit finite difference method together with the Keller-box scheme and (ii) perturbation solution technique. The results of the surface shear stress in terms of local skin-friction and the rate of heat transfer in terms of local Nusselt number, velocity distribution, velocity vectors, temperature distribution as well as streamlines, isotherms and isolines of pressure are shown by graphically for a selection of parameter set consisting of heat generation parameter.     

An inverse hyperbolic heat conduction problem in estimating surface heat flux of a living skin tissue

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to solve the inverse hyperbolic heat conduction problem in estimating the unknown time-dependent surface heat flux in a living skin tissue from the temperature measurements taken within the tissue. The inverse solutions will be justified based on the numerical experiments in which three different heat flux distributions are to be determined. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The influence of measurement errors upon the precision of the estimated results is also investigated. Results show that an excellent estimation on the time-dependent surface heat flux can be obtained for the test cases considered in this study.     

Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition

In this paper the boundary layer flow over a flat plat with slip flow and constant heat flux surface condition is studied. Because the plate surface temperature varies along the x direction, the momentum and energy equations are coupled due to the presence of the temperature gradient along the plate surface. This coupling, which is due to the presence of the thermal jump term in Maxwell slip condition, renders the momentum and energy equations non-similar. As a preliminary study, this paper ignores this coupling due to thermal jump condition so that the self-similar nature of the equations is preserved. Even this fundamental problem for the case of a constant heat flux boundary condition has remained unexplored in the literature. It was therefore chosen for study in this paper. For the hydrodynamic boundary layer, velocity and shear stress distributions are presented for a range of values of the parameter characterizing the slip flow. This slip parameter is a function of the local Reynolds number, the local Knudsen number, and the tangential momentum accommodation coefficient representing the fraction of the molecules reflected diffusively at the surface. As the slip parameter increases, the slip velocity increases and the wall shear stress decreases. These results confirm the conclusions reached in other recent studies. The energy equation is solved to determine the temperature distribution in the thermal boundary layer for a range of values for both the slip parameter as well as the fluid Prandtl number. The increase in Prandtl number and/or the slip parameter reduces the dimensionless surface temperature. The actual surface temperature at any location of x is a function of the local Knudsen number, the local Reynolds number, the momentum accommodation coefficient, Prandtl number, other flow properties, and the applied heat flux.     

A nondestructive experimental approach for heat flux determination on direct‐chill cast aluminium alloy billet surface

In this paper a new nondestructive experimental method for the determination of the billet surface heat flux in direct‐chill casting of aluminium alloys is presented. The new measurement method is based on measurements of the water film temperatures along the billet surface and determination of the local heat flux from the energy balance. Experiments using the new technique are carried out in a real plant environment. The results are used for the validation of a computational model of the process.     

Unsteady Marangoni convection heat transfer of fractional Maxwell fluid with Cattaneo heat flux

Fractional shear stress and Cattaneo heat flux models are introduced in characterizing unsteady Marangoni convection heat transfer of viscoelastic Maxwell fluid over a flat surface. Governing equations and boundary condition are formulated firstly via the balance between the surface tension and shear stress. Numerical solutions are obtained by new developed numerical technique and some novel phenomena are found. Results shown that the fractional derivative parameters, Marangoni number and power law exponent have significant influence on characteristics velocity and temperature fields. As fractional derivative parameters increase, the temperature profiles rise remarkably and the viscoelastic effects of the fluid enhance with delayed response to surface tension, however the temperature profiles decline significantly with a thinner thickness of thermal boundary layer with the increase of Marangoni number. The average skin friction coefficient increases with the augment of Marangoni number, while the average Nusselt number decreases for larger values of power law exponent.     

Natural convective boundary-layer flow in a heat generating porous medium with a prescribed wall heat flux

The free convection boundary-layer flow on a vertical surface in a porous medium with local heat generation proportional to (T – T_∞)^p, where T is the local temperature and T_∞ is the ambient temperature, is considered when the surface is thermally insulated. The way in which the flow develops from the leading edge is seen to depend critically on the exponent p. For p ≤ 2 there is a boundary-layer flow for all x > 0, where x measures distance from the leading edge, with the internal heating having a significant effect at large x. For p ≥ 5 there is also a boundary-layer flow to large x but now the internal heating has an increasingly weaker effect as x increases. For 2 < p <  5 the boundary-layer solution breaks down at a finite x, with a singularity developing leading to thermal runaway at a finite distance along the surface.     

Computational treatment of free convection effect on flow of elastico-viscous fluid past an accelerated plate with constant heat flux

Effect of free convection on the visco-elastic fluid (Walter - B’ type) flow past an infinite vertical plate accelerating in its own plane with constant heat flux is examined analytically. It is found that for given values of Grashof number, Prandtl number and Newtonian parameter; flow velocity at any point increases with the increase in time and non-Newtonian parameter, however, it decreases with both, the heating and cooling of the plate.     

Fourier regularization method of a sideways heat equation for determining surface heat flux

We consider a non-standard inverse heat conduction problem in a quarter plane which appears in some applied subjects. We want to know the surface heat flux in a body from a measured temperature history at a fixed location inside the body. This is an exponentially ill-posed problem in the sense that the solution (if it exists) does not depend continuously on the data. A Fourier regularization method together with order optimal logarithmic stability estimates is given. A numerical example shows that the theoretical results are valid.     

On the heat flux vector for flowing granular materials—part II: derivation and special cases

Heat transfer plays a major role in the processing of many particulate materials. The heat flux vector is commonly modelled by the Fourier's law of heat conduction and for complex materials such as non‐linear fluids, porous media, or granular materials, the coefficient of thermal conductivity is generalized by assuming that it would depend on a host of material and kinematical parameters such as temperature, shear rate, porosity or concentration, etc. In Part I, we will give a brief review of the basic equations of thermodynamics and heat transfer to indicate the importance of the modelling of the heat flux vector. We will also discuss the concept of effective thermal conductivity (ETC) in granular and porous media. In Part II, we propose and subsequently derive a properly frame‐invariant constitutive relationship for the heat flux vector for a (single phase) flowing granular medium. Standard methods in continuum mechanics such as representation theorems and homogenization techniques are used. It is shown that the heat flux vector in addition to being proportional to the temperature gradient (the Fourier's law), could also depend on the gradient of density (or volume fraction), and D (the symmetric part of the velocity gradient) in an appropriate manner. The emphasis in this paper is on the idea that for complex non‐linear materials it is the heat flux vector which should be studied; obtaining or proposing generalized form of the thermal conductivity is not always appropriate or sufficient. Copyright © 2006 John Wiley & Sons, Ltd.     

Heat transfer in a visco-elastic fluid past a stretching sheet with viscous dissipation and internal heat generation

This paper deals with the study of heat transfer characteristics in the laminar boundary layer flow of a visco-elastic fluid over a linearly stretching continuous surface with variable wall temperature subjected to suction or blowing. The study considers the effects of frictional heating (viscous dissipation) and internal heat generation or absorption. An analysis has been carried out for two different cases of heating processes namely: (i) Prescribed surface temperature (PST) and (ii) Prescribed wall heat flux (PHF) to get the effect of visco-elastic parameter for various situations. Further increase of visco-elastic parameter is to decrease the skin friction on the sheet. The solutions for the temperature and the heat transfer characteristics are obtained in terms of Kummers function. Received: June 16, 2004; revised: February 8, 2005     

On the heat flux vector for flowing granular materials—Part I: effective thermal conductivity and background

Heat transfer plays a major role in the processing of many particulate materials. The heat flux vector is commonly modelled by the Fourier's law of heat conduction and for complex materials such as non‐linear fluids, porous media, or granular materials, the coefficient of thermal conductivity is generalized by assuming that it would depend on a host of material and kinematical parameters such as temperature, shear rate, porosity or concentration, etc. In Part I, we will give a brief review of the basic equations of thermodynamics and heat transfer to indicate the importance of the modelling of the heat flux vector. We will also discuss the concept of effective thermal conductivity (ETC) in granular and porous media. In Part II, we propose and subsequently derive a properly frame‐invariant constitutive relationship for the heat flux vector for a (single phase) flowing granular medium. Standard methods in continuum mechanics such as representation theorems and homogenization techniques are used. It is shown that the heat flux vector in addition to being proportional to the temperature gradient (the Fourier's law), could also depend on the gradient of density (or volume fraction), and D (the symmetric part of the velocity gradient) in an appropriate manner. The emphasis in this paper is on the idea that for complex non‐linear materials it is the heat flux vector which should be studied; obtaining or proposing generalized form of the thermal conductivity is not always appropriate or sufficient. Copyright © 2006 John Wiley & Sons, Ltd.     

2D internal flux compatibility equation of the flux Green element method for transient nonlinear potential problems

This article presents the derivation and implementation of the normal directional flux compatibility equation (relationship) at internal nodes when the Green element formulation that consistently provides accurate estimates of the primary variable, and its normal directional derivative (normal flux) is applied in 2D heterogeneous media to steady and transient potential problems. Such a relationship is required to resolve the closure problem due to having fewer integral equations than the number of unknowns at internal nodes. The derivation of the relationship is based on Stokes' theorem, which transforms the contour integral of the normal directional fluxes into a surface integral that is identically zero. The numerical discretization of the compatibility equation is demonstrated with four numerical examples using the six‐node quadratic triangular and the four and eight‐node rectangular elements. The incorporation of triangular elements into the current formulation demonstrates that the internal compatibility equation can be successfully implemented on irregular grids. The direct calculation of the fluxes significantly enhances the accuracy of the formulation, so that high accuracy, exceeding that of the finite element method, is achieved with very coarse spatial discretization. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Blow-up estimates for system of heat equations coupled via nonlinear boundary flux

This paper deals with a system of heat equations coupled via nonlinear boundary flux. The precise blow-up rate estimates are established together with the blow-up set. It is observed that there is some quantitative relationship regarding the blow-up properties between the heat system with coupled nonlinear boundary flux terms and the corresponding reaction–diffusion system with the same nonlinear terms as the source.     

The Reynolds analogy for the mixed convection over a vertical surface with prescribed heat flux

The steady mixed convection boundary layer flow over a vertical surface with prescribed heat flux is revisited in this Note. The subset of solutions which can be obtained with the aid of the Reynolds analogy is discussed in a close relationship with the dual solutions reported by Merkin and Mahmood [1] for impermeable, and more recently by Ishak et al. [2], for permeable surfaces.