The Inverse Problem in Transient Heat Conduction

1964 ◽  
Vol 31 (3) ◽  
pp. 369-375 ◽  
Author(s):  
E. M. Sparrow ◽  
A. Haji-Sheikh ◽  
T. S. Lundgren
Keyword(s):  
Heat Flux ◽  
Inverse Problem ◽  
Heat Conduction ◽  
Numerical Calculation ◽  
Initial Temperature ◽  
Transient Heat Conduction ◽  
Analytical Treatment ◽  
Initial Temperature Distribution ◽  
Plane Slab ◽  
Long Cylinder

A general theory is devised for determining the temperature and heat flux at the surface of a solid when the temperature at an interior location is a prescribed function of time. The theory is able to accommodate an initial temperature distribution which varies arbitrarily with position throughout the solid. Detailed analytical treatment is extended to the sphere, the plane slab, and the long cylinder; and it is additionally shown that the semi-infinite solid is a particular case of the general formulation. The accuracy of the method is demonstrated by a numerical example. In addition, a numerical calculation procedure is devised which appears to provide smooth, nonoscillatory results.

Transient Heat Conduction in Elliptical Plates and Cylinders

1959 ◽  
Vol 81 (1) ◽  
pp. 54-58 ◽  
Author(s):  
E. T. Kirkpatrick ◽  
W. F. Stokey
Keyword(s):  
Heat Conduction ◽  
Temperature Change ◽  
Outer Surface ◽  
Digital Computer ◽  
Numerical Evaluation ◽  
Initial Temperature ◽  
Transient Heat Conduction ◽  
Elliptical Cylinder ◽  
Mathieu Functions ◽  
Sudden Temperature Change

In 1945, N. W. McLachlan published the equations governing the problem of heat conduction in a long elliptical cylinder, including the solution for the case of a cylinder with a uniform initial temperature, subject to a sudden temperature change at the outer surface of the cylinder. This paper describes the numerical evaluation of McLachlan’s solution by the use of a digital computer and includes a table of the necessary zeros of the modified Mathieu functions. Tables of the temperatures in cylinders with eccentricities of 0.6, 0.7, 0.8, and 0.9 are given.

Time Estimate for the Incipient Surface Melting of Regular-Shaped Metals Receiving Uniform Heat Flux Inside a Metal Melting Furnace

2015 ◽  
Vol 8 (2) ◽  
Author(s):  
Marcelo D. Marucho ◽  
Antonio Campo ◽  
N. Ben Cheikh
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Melting Furnace ◽  
Continuous Heating ◽  
Uniform Heat Flux ◽  
Large Plate ◽  
Time Solution ◽  
Uniform Heat ◽  
Metal Melting ◽  
Long Cylinder

This article addresses the continuous heating of regular-shaped metals (large plate, long cylinder, and sphere) at ambient temperature placed in a metal melting furnace. Under the assumption of temperature-independent thermophysical properties of the metal, the heat conduction problem entails to unsteady one-dimensional (1D) heat conduction with a boundary condition of uniform heat flux. Based on the exact, analytic spatiotemporal temperature distributions for the regular-shaped metals, the objective of this study is to construct simple predictive formulas so that engineers can estimate the incipient melting of these metals when heated continually. The time at which melting at the metal surface is initiated, tmelt, corresponds to setting the surface temperature, Tsur, equal to the melting temperature, Tmelt. The analysis will be done under the premises of two asymptotic solutions: one a “large-time” solution and the other a “short-time” solution. A collection of six formulas of simple form for predicting the melting time, tmelt, will be developed for those regular-shaped metals (large plate, long cylinder, and sphere).

An Inverse Determination of Unsteady Heat Fluxes Using a Network Simulation Method

2003 ◽  
Vol 125 (6) ◽  
pp. 1178-1183 ◽  
Author(s):  
F. Alhama ◽  
J. Zueco and ◽  
C. F. Gonza´lez Ferna´ndez
Keyword(s):  
Heat Flux ◽  
Inverse Problem ◽  
Heat Conduction ◽  
Input Data ◽  
Heat Conduction Problem ◽  
Network Simulation ◽  
Simulation Method ◽  
Heat Fluxes ◽  
Incident Heat Flux ◽  
Network Simulation Method

This work addresses unsteady heat conduction in a plane wall subjected to a time-variable incident heat flux. Three different types of flux are studied (sinusoidal, triangular and step waveforms) and constant thermal properties are assumed for simplicity. First, the direct heat conduction problem is solved using the Network Simulation Method (NSM) and the collection of temperatures obtained at given instants is modified by introducing a random error. The resulting temperatures act as the input data for the inverse problem, which is also solved by a sequential approach using the NSM in a simple way. The solution is a continuous piece-wise function obtained step by step by minimizing the classical functional that compares the above input data with those obtained from the solution of the inverse problem. No prior information is used for the functional forms of the unknown heat flux. A piece-wise linear stretches of variable slope and length is used for each of the stretches of the solution. The sensitivity of the functional versus the slope of the line, at each step, is acceptable and the complete piece-wise solution is very close to the exact incident heat flux in all of the mentioned waveforms.

Effect of Brownian and Thermophoretic Diffusions of Nanoparticles on Nonequilibrium Heat Conduction in Nanofluids

2008 ◽  
Author(s):  
Yuwen Zhang ◽  
Ling Li ◽  
H. B. Ma
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Initial Temperature ◽  
Volume Fraction ◽  
Density Ratio ◽  
Lewis Number ◽  
Soret Coefficient ◽  
Transfer Enhancement ◽  
Initial Volume Fraction ◽  
Periodic Heat

Effects of Brownian and thermophoretic diffusions on nonequilibrium heat conduction in a nanofluid layer with periodic heat flux on one side and specified temperature on the other side are investigated numerically. The problem are described by eight dimensionless parameters: density ratio, heat capacity ratio, Lewis number, Soret coefficient, initial volume fraction of nanoparticles, initial temperature, Sparrow number, and period of the surface heat flux. Effects of Brownian and thermophoretic diffusions of nanoparticles on nonequilibrium heat conduction in nanofluid obtained by dispersing copper nanoparticles into ethylene glycol are investigated. The results showed that the Brownian and thermophoretic diffusions only affect the nanoparticle temperature but their effect on the heat transfer enhancement is negligible.

Investigation of the Initial Inverse Problem in the Heat Equation

2004 ◽  
Vol 126 (2) ◽  
pp. 294-296 ◽  
Author(s):  
Khalid Masood ◽  
F. D. Zaman
Keyword(s):  
Inverse Problem ◽  
Heat Conduction ◽  
Heat Equation ◽  
Initial Temperature ◽  
Final Temperature ◽  
Conduction Model ◽  
Heat Conduction Model ◽  
Ill Posed

We investigate the inverse problem in the heat equation involving the recovery of the initial temperature from measurements of the final temperature. This problem is extremely ill-posed and it is believed that only information in the first few modes can be recovered by classical methods. We will consider this problem with a regularizing parameter which approximates and regularizes the heat conduction model.

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