Unknown Thermal Coefficients Research Articles

Relationship between two solidification problems in order to determine unknown thermal coefficients when the heat transfer coefficient is very large

Relationship between two solidification problems in order to determine unknown thermal coefficients when the heat transfer coefficient is very large

Determination of one unknown coefficient in a two-phase free boundary problem in an angular domain with variable thermal conductivity and specific heat

Determination of one unknown coefficient in a two-phase free boundary problem in an angular domain with variable thermal conductivity and specific heat

Determination of unknown thermal coefficients in a non-classical Stefan problem

We study a non-classical one phase Stefan problem with a particular control function which depends on the evolution of the temperature at the fixed face x = 0 and, we assume a Neumann boundary condition and an over specified Robin condition at the fixed face. Under certain restrictions on the data an explicit similarity type solution is given. Moreover, we determine the free boundary and one unknown thermal coefficient, or two unknown thermal coefficients depending on whether the direct or inverse Stefan problem is considered.

An Inverse Problem Solution for Thermal Conductivity Reconstruction

This work deals with the inverse problem of reconstructing the thermal conductivity coefficient of the (2+1)D heat equation from over–posed data at the boundaries. The proposed solution uses a variational approach for identifying the coefficient. The inverse problem is reformulated as a higher–order elliptic boundary–value problem for minimization of a quadratic functional of the original equation. The resulting system consists of a well–posed fourth–order boundary–value problem for the temperature and an explicit equation for the unknown thermal conductivity coefficient. The existence and uniqueness of the resulting higher–order boundary–value problem are investigated. The unique solvability of the inverse coefficient problem is proven. The numerical algorithm is validated and applied to problems of reconstructing continuous nonlinear coefficient and discontinuous coefficients. Accurate and stable numerical solutions are obtained.

Determination of unknown thermal coefficients in a Stefan problem for Storm’s type materials

Fil: Briozzo, Adriana Clotilde. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matematicas; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas; Argentina

Application of KF-RLSE algorithm for on-line estimating the time-dependent melting thickness and input heat flux in participating media

This study proposes an inverse heat transfer process to estimate the time-dependent melting thickness and input heat flux in participating media. The inversion algorithm comprises Kalman filter technique and recursive least squares estimator (KF-RLSE) for on-line and real time predict the input heat flux and position of the solid–liquid phase front from the temperature history by using a sensor mounted on the surface. The KF technique is employed to generate a regression equation between the biased residual innovation and the unknown thermal coefficient. The recursive least squares estimator is utilized in the regression equation to predict time-varying phase front position and input heat flux. Based on simulation results, the proposed algorithm exhibits good reconstruction capability. The influences of forgetting factor, process noise, measurement noise, and absorption coefficient on the stability and precision of the retrieval results are also analyzed. The accuracy of the estimation increases with decrease of the distance between the acting surface of input heat flux and the sensor location. Furthermore, the present algorithm cannot accurately determine the abrupt heat flux with large time step.

Determination of Two Unknown Thermal Coefficients Through an Inverse One-Phase Fractional Stefan Problem

Abstract We consider a semi-infinite one-dimensional phase-change material with two unknown constant thermal coefficients among the latent heat per unit mass, the specific heat, the mass density and the thermal conductivity. Aiming at the determination of them, we consider an inverse one-phase Stefan problem with an over-specified condition at the fixed boundary and a known evolution for the moving boundary. We assume that it is given by a sharp front and we consider a time fractional derivative of order α (0 < α < 1) in the Caputo sense to represent the temporal evolution of the temperature as well as the moving boundary. This might be interpreted as the consideration of latent-heat memory effects in the development of the phase-change process. According to the choice of the unknown thermal coefficients, six inverse fractional Stefan problems arise. For each of them, we determine necessary and sufficient conditions on data to obtain the existence and uniqueness of a solution of similarity type. Moreover, we present explicit expressions for the temperature and the unknown thermal coefficients. Finally, we show that the results for the classical statement of this problem, associated with α = 1, are obtained through the fractional model when α → 1—.

English

Fil: Ceretani, Andrea Noemi. Consejo Nacional de Investigaciones Cientificas y Tecnicas; Argentina. Universidad Austral. Facultad de Ciencias Empresariales. Departamento de Matematicas; Argentina. Universidad Nacional de Rosario; Argentina

Determination of One Unknown Thermal Coefficient through a Mushy Zone Model with a Convective Overspecified Boundary Condition

A semi-infinite material under a solidification process with the Solomon-Wilson-Alexiades mushy zone model with a heat flux condition at the fixed boundary is considered. The associated free boundary problem is overspecified through a convective boundary condition with the aim of the simultaneous determination of the temperature, the two free boundaries of the mushy zone and one thermal coefficient among the latent heat by unit mass, the thermal conductivity, the mass density, the specific heat, and the two coefficients that characterize the mushy zone, when the unknown thermal coefficient is supposed to be constant. Bulk temperature and coefficients which characterize the heat flux and the heat transfer at the boundary are assumed to be determined experimentally. Explicit formulae for the unknowns are given for the resulting six phase-change problems, besides necessary and sufficient conditions on data in order to obtain them. In addition, relationship between the phase-change process solved in this paper and an analogous process overspecified by a temperature boundary condition is presented, and this second problem is solved by considering a large heat transfer coefficient at the boundary in the problem with the convective boundary condition. Formulae for the unknown thermal coefficients corresponding to both problems are summarized in two tables.

Determination of One Unknown Thermal Coefficient through the One-Phase Fractional Lamé-Clapeyron-Stefan Problem

We obtain explicit expressions for one unknown thermal coefficient (among the conductivity, mass density, specific heat and latent heat of fusion) of a semi-infinite material through the one-phase fractional Lamé-Clapeyron-Stefan problem with an over-specified boundary condition on the fixed face . The partial differential equation and one of the conditions on the free boundary include a time Caputo’s fractional derivative of order . Moreover, we obtain the necessary and sufficient conditions on data in order to have a unique solution by using recent results obtained for the fractional diffusion equation exploiting the properties of the Wright and Mainardi functions, given in: 1) Roscani-Santillan Marcus, Fract. Calc. Appl. Anal., 16 (2013), 802 - 815; 2) Roscani-Tarzia, Adv. Math. Sci. Appl., 24 (2014), 237 - 249 and 3) Voller, Int. J. Heat Mass Transfer, 74 (2014), 269 - 277. This work generalizes the method developed for the determination of unknown thermal coefficients for the classical Lamé-Clapeyron-Stefan problem given in Tarzia, Adv. Appl. Math., 3 (1982), 74 - 82, which is recovered by taking the limit when the order .

A Comparative Study of Gradient-Type Methods for Recovering Time-Dependent Heat Sources

Gradient-based methods belong to offline techniques which use all the available data reading to set the unknown thermal coefficients. General application and rigorous mathematical investigation of these methods make them robust tools for analyzing inverse heat problems. In the present study, four search techniques in gradient-type methods through the use of adjoint problem are employed for the evaluation of heat sources in arbitrary geometries. The methods include Fletcher–Reeves version of the conjugate gradient method (CGM), Hestenes–Stiefel version of the CGM, Broydon–Fletcher–Goldfarb–Shanno version of quasi-Newton method and a novel form of the gradient-type method. The search direction in the novel method appears to have originated as a hybridization of a conjugate gradient and a quasi-Newton method. A conservative cell-based version of finite volume scheme is applied for the solution of heat-conduction problem in irregular domains. To highlight the flexibility and advantages of these algorithms, their performance and accuracy are presented and compared for two-dimensional heat-conduction problems.

Determination of a time-dependent diffusivity from nonlocal conditions

The inverse problem of determining the temperature and the time-dependent thermal diffusivity from various additional nonlocal information is investigated. These nonlocal conditions can come in the form of an internal or boundary energy, or, in the one-dimensional case, as a difference boundary temperature or heat flux so as to ensure the uniqueness of solution for the heat conduction equation with unknown thermal diffusivity coefficient. The Ritz-Galerkin method with satisfier function is employed to solve the inverse problems numerically. Numerical results are presented and discussed.

A sensitivity analysis for the determination of unknown thermal coefficients through a phase-change process with temperature-dependent thermal conductivity

In Tarzia, Int. Comm. in Heat and Mass Transfer, 25 (1998), 139–147, explicit formulas for the simultaneous determination of unknown thermal coefficients of a semi-infinite material through a phase-change process with temperature-dependent thermal conductivity were obtained. Moreover, ten different cases were studied: four cases of free boundary problems (i.e. Stefan-like problems) and six cases of moving boundary problems (i.e. inverse Stefan-like problems). The goal of this paper is to obtain a numerical sensitivity analysis of the mentioned ten cases for the simultaneous determination of unknown thermal coefficients and to determine the coefficients which are more sensitive with respect to the given parameters. We show numerical result for the aluminum.

A Novel Methodology for Combined Parameter and Function Estimation Problems

This article presents a novel methodology, which is highly efficient and simple to implement, for simultaneous retrieval of a complete set of thermal coefficients in combined parameter and function estimation problems. Moreover, the effect of correlated unknown variables on convergence performance is examined. The present methodology is a combination of two different classical methods: The conjugate gradient method with adjoint problem (CGMAP) and Box–Kanemasu method (BKM). The methodology uses the benefit of CGMAP in handling function estimation problems and BKM for parameter estimation problems. One of the unique features about the present method is that the correlation among the separate unknowns does not disrupt the convergence of the problem. Numerical experiments using measurement errors are performed to verify the efficiency of the proposed method in solving the combined parameter and function estimation problems. The results obtained by the present approach show that the combined procedure can efficiently and reliably estimate the values of the unknown thermal coefficients.

Determination of unknown thermal coefficients for Storm’s-type materials through a phase-change process

Abstract Unknown thermal coefficients of a semi-infinite material of Storm’s type through a phase-change process with an overspecified condition on the fixed face are determined. We follow the ideas developed in C. Rogers (Int. J. Non-Linear Mech. 21 (1986) 249–256) and in Tarzia (Adv. Appl. Math. 3 (1982) 74–82; Int. J. Heat Mass Transfer 26 (1983) 1151–1157). We also find formulae for the unknown coefficients and, the necessary and sufficient conditions for the existence of a similarity solution.

The determination of unknown thermal coefficients through phase change process with temperature-dependent thermal conductivity

Abstract We study the determination of unknown thermal coefficient of a semi-infinite material through a phase-change process with an overspecified condition on the fixed face with temperature-dependent thermal conductivity. We determine necessary and sufficient conditions on data in order to obtain the existence of the solution. We also give formulae for the unknown coefficients.

DETERMINATION OF UNKNOWN THERMAL COEFFICIENTS THROUGH A FREE BOUNDARY PROBLEM FOR A NON LINEAR HEAT CONDUCTION EQUATION WITH A CONVECTIVE TERM

We determinate unknown thermal coefficients of a semi-infinite material with an overspecified condition on the fixed face following the ideas developed in C. Rogers (ZAMP, 39, 122–128 (1988)) and in D.A. Tarzia (Adv. Appl. Math, 3, 74–82 (1982)). We also obtain formulae for the unknown coefficients and, the necessary and sufficient condition for the existence of solution. © 1997 Elsevier Science Ltd

Determination of one or two unknown thermal coefficients of a semi-infinite material through a two-phase stefan problem

This paper consists of two parts: 1.(1) We consider the two-phase (fusion) Stefan problem for a semi-infinite material with one unknown thermal coefficient. We have an overspecified condition of type k2∂T2∂x(0, t) = −ho/√t with ho > 0 on the fixed face x = 0 of the phase-change material. We obtain: (i) if ρ is unknown, then the corresponding free boundary problem always has a unique solution of the Neumann type; (ii) if one of the remaining five coefficients is unknown, then the corresponding free boundary problem has a unique solution of the Neumann type iff a complementary condition is verified.2.(2) We consider the inverse two-phase (fusion) Stefan problem for a semi-infinite material with an overspecified condition k2∂T2∂x(0, t) = −ho/√t with ho > 0 on the fixed face and with two unknown thermal coefficients.We obtain: (i) if (p, k2) are unknown, the corresponding moving boundary problem always has a unique solution of the Neumann type; (ii) if (l, k1), (l, c1) or (, k1, c1) are unknown, the corresponding moving boundary problem has infinite solutions whenever the complementary conditions are verified; (iii) in the remaining eleven cases, the corresponding moving boundary problem has a unique solution of the Neumann type iff complementary conditions are verified. (1) We consider the two-phase (fusion) Stefan problem for a semi-infinite material with one unknown thermal coefficient. We have an overspecified condition of type k2∂T2∂x(0, t) = −ho/√t with ho > 0 on the fixed face x = 0 of the phase-change material. We obtain: (i) if ρ is unknown, then the corresponding free boundary problem always has a unique solution of the Neumann type; (ii) if one of the remaining five coefficients is unknown, then the corresponding free boundary problem has a unique solution of the Neumann type iff a complementary condition is verified. (2) We consider the inverse two-phase (fusion) Stefan problem for a semi-infinite material with an overspecified condition k2∂T2∂x(0, t) = −ho/√t with ho > 0 on the fixed face and with two unknown thermal coefficients. We obtain: (i) if (p, k2) are unknown, the corresponding moving boundary problem always has a unique solution of the Neumann type; (ii) if (l, k1), (l, c1) or (, k1, c1) are unknown, the corresponding moving boundary problem has infinite solutions whenever the complementary conditions are verified; (iii) in the remaining eleven cases, the corresponding moving boundary problem has a unique solution of the Neumann type iff complementary conditions are verified. Moreover, in both parts, we obtain formulas for the unknown thermal coefficients.

Determination of unknown thermal coefficients of a semi-infinite material for the one-phase Lame-Clapeyron (Stefan) problem through the Solomon-Wilson-Alexiades' mushy zone model

Abstract We use the Solomon-Wilson-Alexiades' mushy zone model (Letters Heat Mass Transfer, 9 (1982), 319–324) for the simultaneous determination of unknown coefficients of a semi-infinite material through a phase-change problem with an overspecified condition on the fixed face. We also find formulas for the unknown coefficients and the necessary and sufficient conditions for the existence of a solution.

Simultaneous determination of two unknown thermal coefficients through an inverse one-phase Lamé-Clapeyron (Stefan) problem with an overspecified condition on the fixed face

Abstract Formulas are obtained for the simultaneous determination of two of the four coefficients, k (thermal conductivity), l (latent heat of fusion), c (specific heat), ρ (mass density), of a material occupying a semi-infinite medium. This determination is obtained through an inverse one-phase Lame-Clapcyron (Stefan) problem with an overspecified condition on the fixed face of the phase change material. To solve this problem, we assume that the coefficients h 0 , σ, 0 0 > 0 are known from experiments (where h 0 characterizes the heat flux through the fixed face, σ characterizes the moving boundary and 0 0 is the temperature on the fixed face). Denoting the temperature by 0 , the results we obtain concerning the associated moving boundary problem are the following: (i) When one of the triples { 0,k,l }, { 0,k,p } is to be found, the corresponding moving boundary problem always has a solution of the Lame-Clapeyron-Neumann type. (ii) If one of the triples { 0, k, c }, { 0,l, c }, { 0, l, p }, and { 0, c, ρ } has to be determined, the above property is satisfied if and only if a complementary condition for the data is verified. Formulas are also obtained for the simultaneous determination of other physical coefficients and the inequality σ 2 Ste /2( Ste :Stefannumber)for the coefiicient ζ of the free boundary s(t) = 2 a ζ t ½ of the Lame-Clapeyron solution of the one-phase Stefan problem without unknown coefficients.