34 research outputs found

    An unsteady-state method for determining the thermal conductivity of materials

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    Originally this work was initiated to develope [sic] a method for measuring the thermal diffusivity and thermal conductivity of metals. Due to the characteristics of the method and the equipment used, it was concluded that the technique is not suitable for materials of high conductivity. However, results show that the apparatus is valuable for the determination of the thermal diffusivity of relatively poor heat conductors. The favorable characteristics of the method are its rapidity and basic simplicity --Abstract, page ii

    Identification of conductivity in inhomogeneous orthotropic media

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    Purpose - The purpose of this paper is to solve numerically the identification of the thermal conductivity of an inhomogeneous and possibly anisotropic medium from interior/internal temperature measurements. Design/methodology/approach - The formulated coefficient identification problem is inverse and ill-posed and therefore, in order to obtain a stable solution, a nonlinear regularized least-squares approach is employed. For the numerical discretisation of the orthotropic heat equation, the finite-difference method is applied, whilst the nonlinear minimization is performed using the MATLAB toolbox routine lsqnonlin. Findings - Numerical results show the accuracy and stability of solution even in the presence of noise (modelling inexact measurements) in the input temperature data. Research limitations/implications - The mathematical formulation uses temporal tem- perature measurements taken at many points inside the sample and this may be too much information that is provided to identify a spacewise dependent only conductivity tensor. Practical implications - Since noisy data are inverted, the study models real situations in which practical temperature measurements recorded using thermocouples are inherently contaminated with random noise. Social implications - The identification of the conductivity of inhomogeneous and orthotropic media will be of great interest to the inverse problems community with applications in geophysics, groundwater flow and heat transfer. Originality/value - The current investigation advances the field of coefficient identification problems by generalising the conductivity to be orthotropic in addition of being heterogeneous. The originality lies in performing, for the first time, numerical simulations of inver- sion to find the anisotropic and inhomogeneous thermal conductivity form noisy temperature measurements. Further value and physical significance is brought in by determining the degree of cure in a resin transfer molding process, in addition to obtaining the inhomogeneous thermal conductivity of the tested material

    Identification of the time-dependent conductivity of an inhomogeneous diffusive material

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    In this paper, we consider a couple of inverse problems of determining the time-dependent thermal/hydraulic conductivity from Cauchy data in the one-dimensional heat/diffusion equation with space-dependent heat capacity/ specific storage. The well-posedness of these inverse problems in suitable spaces of continuously differentiable functions are studied. For the numerical realisation, the problems are discretised using the finite-difference method and recast as nonlinear least-squares minimization problems with a simple positivity lower bound on the unknown thermal/ hydraulic conductivity. Numerically, this is effectively solved using the lsqnonlin routine from the MATLAB toolbox. Regularization is included wherever necessary. Numerical results are presented and discussed for several benchmark test examples showing that accurate and stable numerical solutions are achieved. The outcomes of this study will be relevant and of importance to the applied mathematics inverse problems community working on thermal/hydraulic property determination in heat transfer and porous media

    Retrieving the time-dependent thermal conductivity of an orthotropic rectangular conductor

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    The aim of this paper is to determine the thermal properties of an orthotropic planar structure characterised by the thermal conductivity tensor in the coordinate system of the main directions (Oxy) being diagonal. In particular, we consider retrieving the timedependent thermal conductivity components of the an orthotropic rectangular conductor from nonlocal overspecified heat flux conditions. Since only boundary measurements are considered, this inverse formulation belongs to the desirable approach of non-destructive testing of materials. The unique solvability of this inverse coefficient problem is proved based on the Schauder fixed point theorem and the theory of Volterra integral equations of the second kind. Furthermore, the numerical reconstruction based on a nonlinear least-squares minimization is performed using the MATLAB optimization toolbox routine lsqnonlin. Numerical results are presented and discussed in order to illustrate the performance of the inversion for orthotropic parameter identification

    Reconstruction of the heat transfer coefficient at the interface of a bi-material

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    The knowledge of heat transfer behaviour of composite thermal systems requires the characterization of the heat transfer coefficient at the contact interfaces between the constituent materials. The present work is devoted to investigating an inverse problem with generalized interface condition containing an unknown space- and time-varying interface coefficient from non-invasive temperature measurements on an accessible boundary. The uniqueness of the solution holds, but the problem does not depend continuously on the input measured temperature data. A new preconditioned conjugate gradient method (CGM) is utilized to address the ill-posedness of the inverse problem. In comparison with the standard CGM with no preconditioning, this method has the merit that the gradient of the objective functional does not vanish at the final time, which restores accuracy and stability when the input data is contaminated with noise and when the initial guess is not close to the true solution. Several numerical examples corresponding to linear thermal contact and nonlinear Stefan-Boltzmann radiation condition are tested for determining thermal contact conductance and Stefan-Boltzmann coefficient, respectively. The numerical results in both one- and two-dimensions illustrate that the reconstructions are robust and stable

    Ignition Transients of Large Segmented Solid Rocket Boosters

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    A model is described which provides a means for analyzing the complexities of ignition transients and pressure peaks of large, high performance, segmented solid rocket boosters. The method accounts for: (1) temporal and spatial development of the flow field set up by the head end igniter discharge, (2) ignition and flame spreading coupled to chamber flow, (3) the steep velocity, pressure, and temperature gradients that occur during the early phases of ignition, and (4) the interactions that produce ignition spikes (i.e., compression of chamber gases during pressurization, erosive burning, and mass added effect of igniter discharge). The technique differs from earlier models in that the flow interactions between the slots and main chamber are accounted for, and the original computer program for monolithic motors is improved. The procedures were used to predict the ignition transients of the current design for the space shuttle booster

    Internal volumetric heat generation and heat capacity prediction during a material electromagnetic treatment process using hybrid algorithms

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    This work considers the estimation of internal volumetric heat generation, as well as the heat capacity of a solid spherical sample, heated by a homogeneous, time-varying electromagnetic field. To that end, the numerical strategy solves the corresponding inverse problem. Three functional forms (linear, sinusoidal, and exponential) for the electromagnetic field were considered. White Gaussian noise was incorporated into the theoretical temperature profile (i.e. the solution of the direct problem) to simulate a more realistic situation. Temperature was pretended to be read through four sensors. The inverse problem was solved through three different kinds of approach: using a traditional optimizer, using modern techniques, and using a mixture of both. In the first case, we used a traditional, deterministic Levenberg-Marquardt (LM) algorithm. In the second one, we considered three stochastic algorithms: Spiral Optimization Algorithm (SOA), Vortex Search (VS), and Weighted Attraction Method (WAM). In the final case, we proposed a hybrid between LM and the metaheuristics algorithms. Results show that LM converges to the expected solutions only if the initial conditions (IC) are within a limited range. Oppositely, metaheuristics converge in a wide range of IC but exhibit low accuracy. The hybrid approaches converge and improve the accuracy obtained with the metaheuristics. The difference between expected and obtained values, as well as the RMS errors, are reported and compared for all three methods.Este trabajo considera la estimación de la generación interna volumétrica de calor y la capacidad calorífica de una muestra esférica sólida calentada por un campo electromagnético homogéneo variante en el tiempo. Para tal fin, la estrategia numérica soluciona el correspondiente problema inverso. Tres formas funcionales (lineal, senoidal y exponencial) para el campo electromagnético fueron considerados. Ruido blanco fue agregado al perfil de temperatura teórica (i.e. la solución del problema directo) para simular una situación más realística. La temperatura se pretendió que fuera leída por cuatro sensores. El problema inverso fue solucionado a través de tres diferentes enfoques: usando un optimizador tradicional, usando técnicas modernas y usando una mezcla de ambos. En el primer caso, usamos un algoritmo determinístico tradicional como lo es el de Levenberg-Marquardt (LM). En el segundo, consideramos tres metaheurísticos estocásticos: El Algoritmo de optimización de la espiral (SOA), la Búsqueda en vórtice (VS), y el método de atracción ponderada (WAM). Para el caso final, proponemos híbridos entre el LM y los algoritmos metahehurísticos. Los resultados muestran que LM converge a la solución esperada solo si las condiciones iniciales (IC) están dentro de un rango limitado. Por otra parte, los metaheurísticos convergen en un amplio rango de IC pero muestra baja precisión. Los enfoques híbridos convergen y mejoran la precisión obtenida con los metaheurísticos. La diferencia entre los valores esperados y obtenidos, así como, los errores RMS son reportados y comparados para los tres métodos

    The starting transient of solid propellant rocket motors with high internal gas velocities

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    A comprehensive analytical model which considers time and space development of the flow field in solid propellant rocket motors with high volumetric loading density is described. The gas dynamics in the motor chamber is governed by a set of hyperbolic partial differential equations, that are coupled with the ignition and flame spreading events, and with the axial variation of mass addition. The flame spreading rate is calculated by successive heating-to-ignition along the propellant surface. Experimental diagnostic studies have been performed with a rectangular window motor (50 cm grain length, 5 cm burning perimeter and 1 cm hydraulic port diameter), using a controllable head-end gaseous igniter. Tests were conducted with AP composite propellant at port-to-throat area ratios of 2.0, 1.5, 1.2, and 1.06, and head-end pressures from 35 to 70 atm. Calculated pressure transients and flame spreading rates are in very good agreement with those measured in the experimental system

    Simultaneous reconstruction of space-dependent heat transfer coefficients and initial temperature

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    Many complex physical phenomena and engineering systems, e.g., in heat exchanges, reflux condensers, combustion chambers, nuclear vessels, etc., due to the high temperatures/high pressures hostile environment involved, possess certain properties which are inaccessible to measure and therefore their influence/determination using inverse analysis is very important and desirable. In this spirit, the purpose of this paper is to mathematically formulate and analyse a new inverse problem in which given measurements of temperature at two different instants, it is required to obtain the space-dependent heat transfer coefficients (HTCs) and the initial temperature. This simultaneous identification is challenging since it is both nonlinear and ill-posed. The uniqueness of solution is established based on the max–min principle for parabolic equations and the contraction mapping principle for the existence and uniqueness of a fixed point. The novel inverse mathematical model that is proposed offers appropriate scientific guidance to the polymer/heat transfer processing industry as to which data to measure/provide in order to be able to reliably determine the desirable HTCs along with the initial temperature, which is in general unknown. Furthermore, for the reconstruction, the surface HTC is determined separately, whilst the variational formulation is introduced for the simultaneous determination of the domain HTC and the initial temperature. The Fréchet gradient of the minimizing objective functional is derived. The numerical reconstruction process is based on the conjugate gradient method (CGM) regularized by the discrepancy principle. Accurate and stable numerical solutions are obtained even in the presence of noise in the input temperature data. Since noisy data are invented, the study models realistic practical situations in which temperature measurements recorded using sensors or thermocouples are inherently contaminated with random noise

    Coefficient Identification Problems in Heat Transfer

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    The aim of this thesis is to find the numerical solution for various coefficient identification problems in heat transfer and extend the possibility of simultaneous determination of several physical properties. In particular, the problems of coefficient identification in a fixed or moving domain for one and multiple unknowns are investigated. These inverse problems are solved subject to various types of overdetermination conditions such as non-local, heat flux, Cauchy data, mass/energy specification, general integral type overdetermination, time-average condition, time-average of heat flux, Stefan condition and heat momentum of the first and second order. The difficulty associated with these problems is that they are ill-posed, as their solutions are unstable to inclusion of random noise in input data, therefore traditional techniques fail to provide accurate and stable solutions. Throughout this thesis, the Crank-Nicolson finite-difference method (FDM) is mainly used as a direct solver except in Chapter 7 where a three-level scheme is employed in order to deal with the nonlinear heat equation. An explicit FDM scheme is also employed in Chapter 10 for the two-dimensional case. The inverse problems investigated are discretised using the FDM and recast as nonlinear least-squares minimization problems with simple bounds on the unknown coefficients. The resulting problem is efficiently solved using the \emph{fmincon} or \emph{lsqnonlin} routines from MATLAB optimization toolbox. The Tikhonov regularization method is included where necessary. The choice of the regularization parameter(s) is thoroughly discussed. The stability of the numerical solution is investigated by introducing Gaussian random noise into the input data. The numerical solutions are compared with their known analytical solution, where available, and with the corresponding direct problem numerical solution where no analytical solution is available
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