Effortless Application of the Method of Lines for the Inverse Estimation of Temperatures in a Large Slab With Two Different Surface Heating Waveforms

2008 ◽  
Vol 131 (2) ◽  
Author(s):  
Antonio Campo ◽  
John Ho
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Heat Conduction Problem ◽  
Method Of Lines ◽  
Time History ◽  
Numerical Differentiation ◽  
Heat Fluxes ◽  
Exposed Surface ◽  
The Method Of Lines ◽  
Temperature Time

The boundary inverse heat conduction problem (BIHCP) deals with the determination of the surface heat flux or the surface temperature from measured transient temperatures inside a conducting body where the initial temperature is known. This work addresses a BIHCP related to the spatiotemporal heat conduction in a large slab when a time-variable heat flux is prescribed at an exposed surface and the other surface is thermally insulated. Two different heating waveforms are studied: a constant heat flux and a time-dependent triangular heat flux. The numerical temperature-time history at the insulated surface of the large slab provides the “temperature-time measurement” with one temperature sensor. Framed in the theory of the method of lines (MOL) first and employing rudimentary concepts of numerical differentiation later, the main objective of this paper is to develop a simple computational methodology to estimate the temporal evolution of temperature at the exposed surface of the large slab receiving the two distinct heat fluxes. In the end, it is confirmed that excellent predictions of the surface temperatures versus time are achievable for the two cases tested while employing the smallest possible system of two heat conduction differential equations of first-order.

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.

scholarly journals Solving Nonlinear Thermal Problems of Friction by Using Method of Lines

2015 ◽  
Vol 9 (1) ◽  
pp. 33-37
Author(s):  
Ewa Och
Keyword(s):  
Heat Conduction ◽  
Differential Equations ◽  
Heat Conduction Problem ◽  
Method Of Lines ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Partial Linearization ◽  
The Method Of Lines ◽  
Third Stage ◽  
Independent Variable

Abstract One-dimensional heat conduction problem of friction for two bodies (half spaces) made of thermosensitive materials was considered. Solution to the nonlinear boundary-value heat conduction problem was obtained in three stages. At the first stage a partial linearization of the problem was performed by using Kirchhoff transform. Next, the obtained boundary-values problem by using the method of lines was brought to a system of nonlinear ordinary differential equations, relatively to Kirchhoff’s function values in the nodes of the grid on the spatial variable, where time is an independent variable. At the third stage, by using the Adams's method from DIFSUB package, a numerical solution was found to the above-mentioned differential equations. A comparative analysis was conducted (Och, 2014) using the results obtained with the proposed method and the method of successive approximations.

Transient Conjugated Conduction: External Convection With Front Face Imposed Wall Heat Flux

2007 ◽  
Author(s):  
Carolina P. Naveira ◽  
Renato M. Cotta ◽  
Mohammed Lachi ◽  
Jacques Padet
Keyword(s):  
Heat Flux ◽  
Integral Transform ◽  
Heat Conduction Problem ◽  
Solid Wall ◽  
Method Of Lines ◽  
Wall Heat Flux ◽  
Front Face ◽  
Flat Plates ◽  
Partial Differential ◽  
Differential Formulation

This work presents hybrid numerical-analytical solutions for transient laminar forced convection over flat plates of non-negligible thickness, subjected to arbitrary time variations of applied wall heat flux at the interface fluid-solid wall. This conjugated conduction-convection problem is first simplified through the employment of the Coupled Integral Equations Approach (CIEA) to reformulate the heat conduction problem on the plate by averaging the related energy equation in the transversal direction. As a result, a partial differential formulation for the average wall temperature is obtained, while a third kind boundary condition is achieved for the fluid in the heat balance at the solid-fluid interface. From the available velocity distributions, the solution method is then proposed for the coupled partial differential equations, based on the Generalized Integral Transform Technique (GITT) under its partial transformation mode, combined with the method of lines implemented in the Mathematica 5.2 routine NDSolve.

Nonlinear Inverse Heat Conduction With a Moving Boundary: Heat Flux and Surface Recession Estimation

1999 ◽  
Vol 121 (3) ◽  
pp. 708-711 ◽  
Author(s):  
V. Petrushevsky ◽  
S. Cohen
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Fourier Series ◽  
Heat Conduction ◽  
Moving Boundary ◽  
Heat Conduction Problem ◽  
Inverse Heat Conduction Problem ◽  
Variable Order ◽  
Inverse Heat Conduction ◽  
One Dimensional

A one-dimensional, nonlinear inverse heat conduction problem with surface ablation is considered. In-depth temperature measurements are used to restore the heat flux and the surface recession history. The presented method elaborates a whole domain, parameter estimation approach with the heat flux approximated by Fourier series. Two versions of the method are proposed: with a constant order and with a variable order of the Fourier series. The surface recession is found by a direct heat transfer solution under the estimated heat flux.

Applying Neural Networks to the Solution of the Inverse Heat Conduction Problem in a Gun Barrel

2008 ◽  
Vol 130 (3) ◽  
Author(s):  
Y. Hwang ◽  
S. Deng
Keyword(s):  
Neural Network ◽  
Heat Flux ◽  
Heat Conduction ◽  
Heat Conduction Problem ◽  
Nonlinear Function ◽  
Inverse Heat Conduction ◽  
Unknown Parameters ◽  
Acceptable Error ◽  
Gun Barrel ◽  
Inverse Heat Conduction Problems

The primary cause of gun barrel erosion is the heat generated by the shell as its travels along the barrel. Therefore, calculating the heat flux input to the gun bore is very important when investigating wear problems in the gun barrel and examining its thermomechanical properties. This paper employs the continuous-time analog Hopfield neural network (CHNN) to compute the temperature distribution in various forward heat conduction problems. An efficient technique is then proposed for the solution of inverse heat conduction problems using a three-layered backpropagation neural network (BPN). The weak generalization capacity of BPN networks when applied to the solution of nonlinear function approximations is improved by employing the Bayesian regularization algorithm. The CHNN scheme is used to calculate the temperature in a 155mm gun barrel and the trained BPN is then used to estimate the heat flux of the inner surface of the barrel. The results show that the proposed neural network analysis method successfully solves forward heat conduction problems and is capable of predicting the unknown parameters in inverse problems with an acceptable error.

scholarly journals Numerical Study Of Thermal Behavior Of Anisotropic Medium In Bidirectional Cylindrical Geometry

2019 ◽  
Vol 286 ◽  
pp. 08009
Author(s):  
Rabiâ Idmoussa ◽  
Nisrine Hanchi ◽  
Hamza Hamza ◽  
Jawad Lahjomri ◽  
Abdelaziz Oubarra
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Thermal Behavior ◽  
Anisotropic Medium ◽  
Numerical Study ◽  
Heat Conduction Problem ◽  
Inverse Transformation ◽  
Complicated Geometry ◽  
Alternating Directions Method ◽  
Transient Thermal

In this work, we investigate the transient thermal analysis of two-dimensional cylindrical anisotropic medium subjected to a prescribed temperature at the two end sections and to a heat flux over the whole lateral surface. Due to the complexity of analytically solving the anisotropic heat conduction equation, a numerical solution has been developed. It is based on a coordinate transformation that reduces the anisotropic cylinder heat conduction problem to an equivalent isotropic one, without complicating the boundary conditions but with a more complicated geometry. The equation of heat conduction for this virtual medium is solved by the alternating directions method. The inverse transformation makes it possible to determine the thermal behavior of the anisotropic medium as a function of study parameters: diagonal and cross thermal conductivities, heat flux.

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