Numerical determination of heat exchanger response

In this paper there is presented approximative calculation of the temperature of fluid in fixed point of heat exchanger and in fixed time. In presented method are used Newton-Cotes formulas. Moreover there is evaluated the error which is made during determination of heat exchanger response. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Improved spatial decay bounds in generalized heat conduction

This paper deals with heat conduction in a semi-infinite cylinder using the generalized Maxwell-Cattaneo equations. Spatial decay bounds for the temperature and heat flux under two different types of boundary conditions are derived. For fixed time it is shown that in each case the solutions decay in appropriate measure like the exponential of a quadratic function of the distance from the base of the cylinder, whereas in previous work they had been shown to decay only at least as fast as the exponential of a linear function.     

热传导问题的一种近似解法

本文在给定的精度要求下,利用多项式及Fourier级数,给出了求热传导方程Cauchy问题在某处的近似解的方法,并把此方法推广成为求解任一有界域上满足给定精度要求的近似解的方法。     

Improved spatial decay bounds in generalized heat conduction

This paper deals with heat conduction in a semi-infinite cylinder using the generalized Maxwell-Cattaneo equations. Spatial decay bounds for the temperature and heat flux under two different types of boundary conditions are derived. For fixed time it is shown that in each case the solutions decay in appropriate measure like the exponential of a quadratic function of the distance from the base of the cylinder, whereas in previous work they had been shown to decay only at least as fast as the exponential of a linear function.Received: January 13, 2004     

The method of central difference for the inverse time-dependent heat source problem

This paper investigates the inverse problem of determining a heat source in a parabolic equation where data are given at some fixed location. This problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the data. A central difference regularization method is given and an order optimal stability estimate is obtained. Numerical results for several benchmark test problems indicate that the central difference regularization method is an accurate and flexible method to determine the unknown time-dependent heat source.     

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.     

Estimates of derivatives of the heat kernel on a compact Riemannian manifold

We give global estimates on the covariant derivatives of the heat kernel on a compact Riemannian manifold on a fixed finite time interval. The proof is based on analyzing the behavior of the heat kernel along Riemannian Brownian bridge.

     

Solitary wave and other solutions for nonlinear heat equations

A number of explicit solutions for the heat equation with a polynomial non-linearity and for the Fisher equation is presented. An extended class of non-linear heat equations admitting solitary wave solutions is described. The generalization of the Fisher equation is proposed whose solutions propagate with arbitrary ad hoc fixed velocity.     

General constitutive equations of heat transport at small length scales and high frequencies with extension to mass and electrical charge transport

A generalized heat transport equation applicable at small length and short time scales is proposed. It is based on extended irreversible thermodynamics with an infinite number of high-order heat fluxes selected as state variables. Extensions of Fick’s and Ohm’s laws are also formulated. As a numerical illustration, heat conduction in a rigid body subject to fixed and oscillatory temperature boundary conditions is discussed.     

A simplified Tikhonov regularization method for determining the heat source

This paper is to discuss the inverse problem of determining a spacewise dependent heat source in one-dimensional heat equation in a bounded domain where data is given at some fixed time. This problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the data. The regularization solution is given by a simplified Tikhonov regularization. For this regularization solution, the Hölder type stability estimate between the regularization solution and the exact solution is obtained. Numerical examples show that the regularization method is effective and stable.     

Heating load density optimization of an irreversible simple Brayton cycle heat pump coupled to counter-flow heat exchangers

This paper presents a theoretical investigation on the finite time thermodynamic performance for an irreversible Brayton cycle heat pump (BCHP) coupled to counter-flow heat exchangers. The heating load density, i.e. the ratio of heating load to the maximum specific volume in the cycle, is taken as the optimization objective. Relations between heating load density and pressure ratio and between COP (coefficient of performance) and pressure ratio for BCHP in which the irreversibilities of heat resistance losses in the hot and cold-side heat exchangers and non-isentropic losses in the compression and expansion processes are derived. The analytical expression obtained for the cycle performance enabled its optimization through addressing the effects of mechanical and thermal inefficiencies of all components comprising the cycle. The influences of the temperature ratio of the reservoirs, the efficiencies of the compressor and expander and the effectiveness of the heat exchangers on the heating load density are provided. The cycle performance optimizations are performed by searching the optimum distribution of heat conductance of the hot- and cold-side heat exchangers for the fixed total heat exchanger inventory and the optimum heat capacity rate matching between the working fluid and the heat reservoirs. The BCHP design with heat loading density optimization leads to a smaller size of all equipments comprising the heat pump.     

Plasmon resonance and heat generation in nanostructures

In this paper, the photothermal effects of plasmon resonance are investigated. Metal nanoparticles efficiently generate heat in the presence of electromagnetic radiation. The process is strongly enhanced when a fixed frequency of the incident wave illuminates on nanoparticles such that plasmon resonance happens. We introduce the electromagnetic radiation model and show exactly how and when the plasmon resonance happens. We then construct the heat generation and transfer theory and derive the heat effect induced by plasmon resonance. Finally, the heat generation under plasmon resonance in a concentric nanoshell structure is considered specially, and excited result is obtained. Copyright © 2015 John Wiley & Sons, Ltd.     

Spatial behaviour of solutions of the three-phase-lag heat equation

In this paper we study the spatial behaviour of solutions for the three-phase-lag heat equation on a semi-infinite cylinder. The theory of three-phase-lag heat conduction leads to a hyperbolic partial differential equation with a fourth-order derivative with respect to time. First, we investigate the spatial evolution of solutions of an initial boundary-value problem with zero boundary conditions on the lateral surface of the cylinder. Under a boundedness restriction on the initial data, an energy estimate is obtained. An upper bound for the amplitude term in this estimate in terms of the initial and boundary data is also established. For the case of zero initial conditions, a more explicit estimate is obtained which shows that solutions decay exponentially along certain spatial-time lines. A class of non-standard problems is also considered for which the temperature and its first two time derivatives at a fixed time T₀ are assumed proportional to their initial values. These results are relevant in the context of the Saint-Venant Principle for heat conduction problems.     

Discretized Tikhonov regularization by reproducing kernel Hilbert space for backward heat conduction problem

In this paper we propose a numerical reconstruction method for solving a backward heat conduction problem. Based on the idea of reproducing kernel approximation, we reconstruct the unknown initial heat distribution from a finite set of scattered measurement of transient temperature at a fixed final time. Standard Tikhonov regularization technique using the norm of reproducing kernel is adopt to provide a stable solution when the measurement data contain noises. Numerical results indicate that the proposed method is stable, efficient, and accurate.     

Numerical null controllability of a semi-linear heat equation via a least squares method

This Note deals with the computation of distributed null controls for a semi-linear 1D heat equation, in the sublinear and slightly superlinear cases. Under sharp growth assumptions, the existence of controls has been obtained in [E. Fernández-Cara, E. Zuazua, Null and approximate controllability for weakly blowing up semi-linear heat equation, Ann. Inst. Henri Poincaré Analyse non linéaire 17 (5) (2000) 583] via a fixed point reformulation; see also [V. Barbu, Exact controllability of the superlinear heat equation, Appl. Math. Optim. Optimization, Theory and Applications 42 (1) (2000) 73]. More precisely, Carleman estimates and Kakutani?s theorem together ensure the existence of fixed points for a corresponding linearized control mapping. In practice, the difficulty is to extract from the Picard iterates a convergent (sub)sequence. We introduce and analyze a least squares reformulation of the problem; we show that this strategy leads to an effective and constructive way to compute fixed points.     

The revised generalized Tikhonov regularization for the inverse time-dependent heat source problem

This paper investigates the inverse problem of finding a time-dependent heat source in a parabolic equation where the data is given at a fixed location. A conditional stability result is given, and a revised generalized Tikhonov regularization method with error estimate is also provided. Numerical examples show that the regularization method is effective and stable.     

The influence of two moving heat sources on blow-up in a reactive-diffusive medium

Two concentrated heat sources move through a one-dimensional reactive-diffusive medium. The sources, moving at constant speed in the same direction, are separated by a fixed distance. The leading source serves to preheat the medium ahead of the trailing source. Sufficient conditions, involving the speed and the separation distance, are found for blow-up at the site of the trailing source. Also, the asymptotic behavior of the solution near blow-up is examined.     

End effects in three-phase-lag heat conduction

In this article we study the spatial behaviour of solutions for the three-phase-lag heat equation on a semi-infinite cylinder. First, we investigate the spatial evolution of solutions of an initial boundary-value problem with zero boundary conditions on the lateral surface of the cylinder. Under null initial data, a Phragmen–Lindelof alternative is obtained. An upper bound for the amplitude term in terms of the boundary data is also established. For the case of decay solutions, an improvement is obtained. We prove that the decay can be controlled by the exponential of a second-degree polynomial in the distance from the finite end of the cylinder. A class of non-standard problems is also considered for which the temperature and its first two time derivatives at a fixed time T ₀ are assumed proportional to their initial values. These results are relevant in the context of the Saint-Venant Principle for heat conduction problems.     

Oscillatory convection and the Cattaneo law of heat conduction

The problem of thermal convection is investigated for a layer of fluid when the heat flux law of Cattaneo is adopted. The boundary conditions are those appropriate to two fixed surfaces. It is shown that for small Cattaneo number the critical Rayleigh number initially increases from its classical value of 1707.765 until a critical value of the Cattaneo number is reached. For Cattaneo numbers greater than this critical value a notable Hopf bifurcation is observed with convection occurring at lower Rayleigh numbers and by oscillatory rather than stationary convection. The aspect ratio of the convection cells likewise changes.