Analytic transient solutions of a cylindrical heat equation with a heat source

The method of separation of variables is applied in order to investigate the analytical solutions of a certain two-dimensional cylindrical heat equation. In the analysis presented here, the partial differential equation is directly transformed into ordinary differential equations. The closed-form transient temperature distributions and heat transfer rates are generalized for a linear combination of the products of Fourier-Bessel series of the exponential type. Relevant connections with some other closely-related recent works are also indicated.     

On heat exchange and heat transport in a geothermal plant

In the long term, the only way to address the challenging task of power supply, is to make renewable energy sources economically attractive and to use them efficiently. In particular, geothermal energy is promising to take over the base load of the power supply. Nevertheless, a lot of investigations needs to be made to use the almost inexhaustible source of thermal energy in the interior of the earth effectively. Starting from the initially isothermal state, a cold fluid is injected through a borehole into a rock. By the rising pressure gradient, the fluid flows through the porous rock and escapes through another borehole. While the fluid passes the micro cracks in the hot rock, the water is heated by the rock due to the heat exchange between the constituents. This process is simulated based on the Theory of Porous Media (TPM). The presented modelling approach of the heat transport and the flow processes in a fully saturated subsurface includes two non-isothermal constituents: an elastically deformable, materially incompressible solid skeleton where thermal expansion is neglected, and a viscous, materially incompressible fluid constituent. To solve the initial-boundary-value problem, the governing primary variables of the coupled model are spatially approximated by mixed finite elements, and the time-discretisation is carried out by an implicit Euler time-integration scheme. The aim of the presented numerical simulations is to study the heat transport and to evaluate the efficiency by varying flow rates. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The determination of the thermal properties of a heat conductor in a nonlinear heat conduction problem

This paper considers the inverse determination of the positive unknown thermal properties K(T), C(T) and the unknown temperature T(_{x, t}) in the nonlinear transient heat conduction equation. In addition to prescribed initial and/or boundary values, specified continuously differentiable temperature data T(x₀, t) with non-zero derivative at a single sensor location x = x₀ is given. When K(T) and C(T) obey a certain relationship which enables one to linearise exactly the nonlinear heat equation then their dependence upon T is obtained explicitly, whilst the unknown temperature T(x, t) is obtained implicitly and is then calculated numerically. Results are presented and discussed for infinite, semi-infinite and finite slabs.     

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.     

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.     

Symmetry analysis of a heat conduction model for heat transfer in a longitudinal rectangular fin of a heterogeneous material

We consider a heat conduction model arising in transient heat transfer through longitudinal fins of a heterogeneous (functionally graded) material. In this case, the thermal conductivity depends on the spatial variable. The heat transfer coefficient depends on the temperature and is given by the power law function. The resulting nonlinear partial differential equation is analyzed using both classical and nonclassical symmetry techniques. Both the transient state and the steady state result in a number of exotic symmetries being admitted by the governing equation. Furthermore, nonclassical symmetries are also admitted. Both classical and nonclassical symmetry analysis results in some useful reductions and some remarkable exact solutions are constructed.     

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.     

Analytic solutions to heat transfer problems on a basis of determination of a front of heat disturbance

With the use of additional boundary conditions in integral method of heat balance, we obtain analytic solution to nonstationary problem of heat conductivity for infinite plate. Relying on determination of a front of heat disturbance, we perform a division of heat conductivity process into two stages in time. The first stage comes to the end after the front of disturbance arrives the center of the plate. At the second stage the heat exchange occurs at the whole thickness of the plate, and we introduce an additional sought-for function which characterizes the temperature change in its center. Practically the assigned exactness of solutions at both stages is provided by introduction on boundaries of a domain and on the front of heat perturbation the additional boundary conditions. Their fulfillment is equivalent to the sought-for solution in differential equation therein. We show that with the increasing of number of approximations the accuracy of fulfillment of the equation increases. Note that the usage of an integral of heat balance allows the application of the given method for solving differential equations that do not admit a separation of variables (nonlinear, with variable physical properties etc.).     

Unsteady thermal stresses near a curvilinear hole in a plate with heat transfer under its warming by a heat flow

We describe an algorithm of determining quasistatic thermal stresses in multiply connected plates with heat transfer, induced by the disturbance of heat flow near holes. Our approach is based on the Laplace transformation and a modified relation of its numerical conversion. The boundary-value problems for the Helmholtz equation, from which the Laplace transform is determined, are solved using the method of boundary integral equations. We solve the integral equations by the method of mechanical quadratures. The results of calculation of nonstationary temperature fields and stresses induced by them in a strip with small holes of different shape are also presented. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 105–111, January–March, 2008.     

Fourier regularization for a backward heat equation

In this paper a simple and convenient new regularization method for solving backward heat equation—Fourier regularization method is given. Meanwhile, some quite sharp error estimates between the approximate solution and exact solution are provided. A numerical example also shows that the method works effectively.     

The heat equation on a matrix space

On the space of real matrices, an analog of the heat equation is studied. On the basis of properties of a heat source, the solution of the Cauchy problem with an initial radiation defined at the vertex of the cone of positive definite symmetric matrices is constructed. A relation between the Riesz potentials associated with composite power functions on a matrix space and the solutions of the heat equation is established. Bibliography: 9 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 268–285.     

Additional solutions of a weighted heat equation

We obtain a simple algorithm for computing additional solutions of a weighted heat equation.     

Determination of a spacewise dependent heat source

This paper investigates the inverse problem of determining a spacewise dependent heat source in the parabolic heat equation using the usual conditions of the direct problem and information from a supplementary temperature measurement at a given single instant of time. The spacewise dependent temperature measurement ensures that the inverse problem has a unique solution, but this solution is unstable, hence the problem is ill-posed. For this inverse problem, we propose an iterative algorithm based on a sequence of well-posed direct problems which are solved at each iteration step using the boundary element method (BEM). The instability is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented for various typical benchmark test examples which have the input measured data perturbed by increasing amounts of random noise.     

Steady-state heat conduction in a circular cone

Zusammenfassung Diese Arbeit befasst sich mit der Bestimmung des stationären Temperaturfeldes, das in einem Kreiskegel durch eine unstetige Verteilung der vorgegebenen Oberflächentemperatur hervorgerufen wird. Die Temperatur der Mantelfläche wird oberhalb und unterhalb einer festen Entfernung von der Kegelspitze konstant angenommen. Dieses Problem wird mit Hilfe der Mellin-Transformation streng gelöst. Die so erhaltene Lösung wird in einer später erscheinenden Arbeit auf die Bestimmung der zugehörigen Temperaturspannungen angewendet.

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The results communicated in this paper were obtained in the course of an investigation conducted under Contracts Nonr 562(20) and Nonr 562(25) of Brown University with the Office of Naval Research in Washington, D.C.     

Temperature waves in a rigid heat conductor

The propagation of acceleration-temperature waves in a rigid heat conductor is investigated. The theory employed allows temperature to travel with a finite wavespeed, and the full nonlinear theory is analysed. It is shown that various types of behaviour are possible for the amplitude of the wave, including one for which the amplitude becomes infinite in a finite time. Higher order temperature waves are also briefly discussed.