An inverse problem in simultaneously measuring temperature-dependent thermal conductivity and heat capacity

An inverse analysis utilizing the conjugate gradient method of minimization and the adjoint equation is used for simultaneously estimating the temperature-dependent thermal conductivity and heat capacity per unit volume of a material. No prior information is used for the functional forms of the unknown thermal conductivity and heat capacity in the present study, thus, it is classified as the function estimation by inverse calculation. The accuracy of the inverse analysis is examined by using simulated exact and inexact measurements obtained within the medium. Results show that the CPU time used on a VAX-9420 computer is within 1.4–4.46 s for all the test cases considered here. Moreover, excellent estimations on the thermal properties can be obtained when a good initial guess of either thermal conductivity or heat capacity is given before the inverse calculations.     

An inverse problem in simultaneous estimating the Biot numbers of heat and moisture transfer for a porous material

A conjugate gradient method based inverse algorithm is applied in the present study in simultaneous determining the unknown time-dependent Biot numbers of heat and moisture transfer for a porous material based on interior measurements of temperature and moisture.It is assumed that no prior information is available on the functional form of the unknown Biot numbers in the present study, thus, it is classified as the function estimation in inverse calculation.The accuracy of this inverse heat and moisture transfer problem is examined by using the simulated exact and inexact temperature and moisture measurements in the numerical experiments. Results show that the estimation on the time-dependent Biot numbers can be obtained with any arbitrary initial guesses on a Pentium IV 1.4 GHz personal computer.     

One-step GPS for the estimation of temperature-dependent thermal conductivity

In this paper we are concerned with the estimation of temperature-dependent thermal conductivity of a one-dimensional inverse heat conduction problem. First, we construct a one-step group-preserving scheme (GPS) for the semi-discretization of quasilinear heat conduction equation, and then derive a quasilinear algebraic equation to determine the unknown thermal conductivity under a given initial temperature and a measured temperature perturbed by noise at time T. The new method does not require any prior information on the functional form of thermal conductivity. Several examples are examined to show that the new approach has high accuracy and efficiency, and the number of iterations spent in solving the quasilinear algebraic equation is smaller than five even in a large temperature range.     

COMPUTATION OF TURBULENT NATURAL CONVECTION IN A RECTANGULAR CAVITY WITH THE k–ϵ– –f MODEL

A hybrid numerical method involving the Laplace transform technique and finite-difference method in conjunction with the least-squares method and actual experimental temperature data inside the test material is proposed to estimate the unknown surface conditions of inverse heat conduction problems with the temperature-dependent thermal conductivity and heat capacity. The nonlinear terms in the differential equations are linearized using the Taylor series approximation. In this study, the functional form of the surface conditions is unknown a priori and is assumed to be a function of time before performing the inverse calculation. In addition, the whole time domain is divided into several analysis subtime intervals and then the unknown estimates on each subtime interval can be predicted. In order to show the accuracy and validity of the present inverse scheme, a comparison among the present estimates, direct solution, and actual experimental temperature data is made. The effects of the measurement errors, initial guesses, and measurement location on the estimated results are also investigated. The results show that good estimation of the surface conditions can be obtained from the present inverse scheme in conjunction with knowledge of temperature recordings inside the test material.     

An inverse biotechnology problem in estimating the optical diffusion and absorption coefficients of tissue

An inverse algorithm for biotechnology problem utilizing the conjugate gradient method is applied in the present study in determining the unknown spatial-dependent optical diffusion and absorption coefficients of the biological tissue based on irradiance and temperature measurements. The accuracy of this inverse problem is examined by using the simulated exact and inexact irradiance and temperature measurements in the numerical experiments. Results show that the estimation on the spatial-dependent diffusion and absorption coefficients can be obtained with any arbitrary initial guesses on a Pentium IV 1.4 GHz personal computer for the test cases considered in the present study.     

Estimation of Heat Flux and Thermal Stresses in Functionally Graded Hollow Circular Cylinders

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown time-dependent heat flux at the inner surface of a functionally graded hollow circular cylinder from the knowledge of temperature measurements taken within the cylinder. Subsequently, the distributions of temperature and thermal stresses in the cylinder can be determined as well. It is assumed that no prior information is available on the functional form of the unknown heat flux; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements, and the effect of the errors in these measurements upon the precision of the estimated results is also considered. Results show that an excellent estimation on the time-dependent heat flux, temperature distributions, and thermal stresses can be obtained for the test case considered in this study.     

Simultaneously estimating the contact heat and mass transfer coefficients in a double-layer hollow cylinder with interface resistance

Based on the conjugate gradient method, this study presents a means of solving the inverse boundary value problem of coupled heat and moisture transport in a double-layer hollow cylinder. While knowing the temperature and moisture history at the measuring positions, the unknown time-dependent contact heat and mass transfer coefficients can be simultaneously determined. It is assumed that no prior information is available on the functional form of the unknown coefficients. The accuracy of this inverse heat and moisture transport problem is examined by using the simulated exact and inexact temperature and moisture measurements in the numerical experiments. Results show that excellent estimation on the time-dependent contact heat and mass transfer coefficients can be simultaneously obtained with any arbitrary initial guesses.     

Inverse Estimation of the Thermal Conductivity in a One-Dimensional Domain by Taylor Series Approach

The Taylor series approximation is developed for the inverse estimation of thermal conductivity in a one-dimensional domain. The differential governing equation of heat conduction is converted to a discrete system of linear equations in matrix form using the temperature measurement and heat generation at the grid points as well as surface heat flux. The unknown thermal conductivity is estimated by solving the linear algebraic equations directly without iterations. The features of the present method are that no prior information about the functional form of the thermal conductivity is required, nor are any initial guesses or iterations in the calculation process needed. The accuracy and robustness of the present method are verified by comparing the results with the analytical solutions for constant, spatial- and temperature-dependent thermal conductivities. The results show that the inverse solutions are in good agreement with the exact solutions.     

Inverse Thermo-Mechanical Analysis of a Functionally Graded Fin

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown time-dependent base heat flux of a functionally graded fin from the knowledge of temperature measurements taken within the fin. Subsequently, the distributions of temperature and thermal stresses in the fin can be determined as well. It is assumed that no prior information is available on the functional form of the unknown base heat flux; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The influence of measurement errors and measurement location upon the precision of the estimated results is also investigated. Results show that an excellent estimation on the time-dependent base heat flux, temperature distributions, and thermal stresses can be obtained for the test case considered in this study.     

Inverse problem of coupled thermoelasticity for prediction of heat flux and thermal stresses in an annular cylinder

This study presents a means of solving the inverse problem of coupled thermoelasticity in an annular cylinder. While knowing the temperature history at any point of the body, the boundary time-varying heat flux can be computed, and subsequently the distributions of temperature and thermal stresses in the cylinder can be determined as well. The accuracy of the inverse analysis is examined by using simulated exact and inexact measurements obtained on the surface and at an interior location of the cylinder. Numerical results demonstrate that excellent estimation for the heat flux and thermal stresses distributions can be obtained for all the test cases considered in this study.     

Estimation of heat flux on the surface of an initially hot cylinder cooled by a laminar confined impinging jet

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown space-and time-dependent heat flux at the surface of an initially hot cylinder cooled by a laminar confined slot impinging jet from the knowledge of temperature measurements taken on the cylinder’s surface. It is assumed that no prior information is available on the functional form of the unknown heat flux; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements, and the effect of the errors in these measurements upon the precision of the estimated results is also considered. The results show that an excellent estimation on the space-and time-dependent heat flux can be obtained even the distributions of thermal properties inside the cylinder is unknown.     

Non-iteration estimation of thermal conductivity using finite volume method

The finite volume approach is developed for the inverse estimation of thermal conductivity in one-dimensional domain. The differential governing equation of heat conduction is converted to a system of linear equations in matrix form using the temperature data and heat generation at the discrete grid points as well as surface heat flux. The unknown thermal conductivities are obtained by solving the system equations directly. The features of the present method are that no prior information about the functional form of the thermal conductivity is required and no iterations in the calculation process are needed. The accuracy and robust of the present method are verified by comparing examples of inverse estimation of spatially and temperature-dependent thermal conductivities with the exact solutions.     

GENERALIZED THERMOELASTICITY OF AN INFINITE BODY WITH A CYLINDRICAL CAVITY AND VARIABLE MATERIAL PROPERTIES

ABSTRACT

The equations of generalized thermoelasticity with one relaxation time in an isotropic elastic medium with temperature-dependent mechanical and thermal properties are established. The modulus of elasticity and the thermal conductivity are taken as linear function of temperature. A problem of an infinite body with a cylindrical cavity has been solved by using Laplace transform techniques. The interior surface of the cavity is subjected to thermal and mechanical shocks. The inverse of the Laplace transform is done numerically using a method based on Fourier expansion techniques. The temperature, the displacement, and the stress distributions are represented graphically. A comparison was made with the results obtained in the case of temperature-independent mechanical and thermal properties.     

An iterative regularization method in simultaneously estimating the inlet temperature and heat-transfer rate in a forced-convection pipe

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown space- and time-dependent inlet temperature and heat-transfer rate on the external wall of a pipe system using temperature measurements at two different locations. It is assumed that no prior information is available on the functional form of the unknown inlet temperature and heat-transfer rate; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The accuracy of the inverse analysis is examined by using simulated exact and inexact temperature measurements. Results show that an excellent estimation on the space- and time-dependent inlet temperature and heat-transfer rate can be obtained for the test case considered in this study.     

An iterative regularization method in estimating the base temperature for non-Fourier fins

An inverse non-Fourier fin problem is examined in the present study by an iterative regularization method, i.e., conjugate gradient method (CGM), in estimating the unknown base temperature of non-Fourier fin based on the boundary temperature measurements. Results obtained in this inverse problem will be justified based on the numerical experiments where three different temperature distributions are to be determined. Results show that the inverse solutions can always be obtained with any arbitrary initial guesses of the base temperature. Moreover, the drawbacks of previous study for this identical inverse problem, such as (1) the inverse solutions become poor when the frequency of base temperature is increased, (2) the estimations depend strongly on the size of grids, (3) the estimations are sensitive to the measurement errors and (4) the uncertainty of using the concept of future time step, can all be avoided by applying this algorithm. Finally, it is concluded that accurate base temperatures can be estimated in the present study.     

A simplex search method for a conductive–convective fin with variable conductivity

An inverse problem is solved for simultaneously estimating the convection–conduction parameter and the variable thermal conductivity parameter in a conductive–convective fin with temperature dependent thermal conductivity. Initially, the temperature field is obtained from a direct method using an analytical approach based on decomposition scheme and then using a simplex search minimization algorithm an inverse problem is solved for estimating the unknowns. The objective function to be minimized is represented by the sum of square of the error between the measured temperature field and an initially guessed value which is updated in an iterative manner. The estimation accuracy is studied for the effect of measurement errors, initial guess and number of measurement points. It is observed that although very good estimation accuracy is possible with more number of measurement points, reasonably well estimation is obtained even with fewer number of measurement points without measurement error. Subject to selection of a proper initial guess, it is seen that the number of iterations could be significantly reduced. The relative sensitiveness of the estimated parameters is studied and is observed from the present work that the estimated convection–conduction parameter contributes more to the temperature distribution than the variable conductivity parameter.     

Application of grey prediction to inverse nonlinear heat conduction problem

The purpose of this research is to estimate the thermal conductivity with the inverse method which is modified by grey prediction; herein the thermal conductivity is a nonlinear function. When the thermal conductivity is the function of position and temperature, if one would try to obtain the thermal conductivity with the inverse method, then the measuring points of the temperature shall be distributed in whole object, consequently there would be a large number of measuring points for the relevant temperatures. The method of grey prediction will be able to dramatically decrease the number of measuring points for the temperature accordingly. However, the method of grey prediction should be accompanied with the prediction errors, thus the estimation of inverse method will produce a major deviation. This paper adopts the methods of the “rolling grey prediction” and the “comparison of temperature measurement” to correct the errors of grey prediction, and then proceed the inverse method to estimate the thermal conductivity. The estimated value obtained by the proposed method and the actual value compares very well.     

Identifying heat conductivity and source functions for a nonlinear convective-diffusive equation by energetic boundary functional methods

Abstract

In the article, we solve the inverse problems to recover unknown space-time dependent functions of heat conductivity and heat source for a nonlinear convective-diffusive equation, without needing of initial temperature, final time temperature, and internal temperature data. After adopting a homogenization technique, a set of spatial boundary functions are derived, which satisfy the homogeneous boundary conditions. The homogeneous boundary functions and zero element constitute a linear space, and then a new energetic functional is derived in the linear space, which preserves the time-dependent energy. The linear systems and iterative algorithms to recover the unknown parameters with energetic boundary functions as the bases are developed, which are convergent fast at each time marching step. The data required for the recovery of unknown functions are parsimonious, including the boundary data of temperatures and heat fluxes and the boundary data of unknown functions to be recovered. The accuracy and robustness of present methods are confirmed by comparing the exact solutions with the identified results, which are obtained under large noisy disturbance.     

HYPERBOLIC STEFAN PROBLEM WITH APPLIED SURFACE HEAT FLUX AND TEMPERATURE-DEPENDENT THERMAL CONDUCTIVITY

Abstract

The hyperbolic Stefan problem with an applied surface heat flux and temperature-dependent thermal conductivity is solved numerically for a semi-infinite slab using Mac-Cormack's predictor-corrector method. Solutions are presented for cases where the melt temperature is both below and above the instantaneous jump in surface temperature at time t = O⁺. The interface condition, surface temperature, and internal temperatures are presented for different Stefan numbers and melt temperatures, as well as thermal conductivity both increasing and decreasing with temperature. The results obtained from the hyperbolic solution are compared with those obtained from the parabolic solution.     

An iterative regularization method in estimating the transient heat-transfer rate on the surface of the insulation layer of a double circular pipe

In this study, a conjugate gradient method based inverse algorithm is applied to estimate the unknown space- and time-dependent heat-transfer rate on the surface of the insulation layer of a double circular pipe heat exchanger using temperature measurements. It is assumed that no prior information is available on the functional form of the unknown heat-transfer rate; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements. The accuracy of the inverse analysis is examined by using simulated exact and inexact temperature measurements. Results show that an excellent estimation on the space- and time-dependent heat-transfer rate can be obtained for the test case considered in this study.