Heat experiment design to estimate temperature dependent thermal properties

Abstract

Abstract. Experimental conditions are studied to optimize transient experiments for estimating temperature dependent thermal conductivity and volumetric heat capacity. A mathematical model of a specimen is the one-dimensional heat equation with boundary conditions of the second kind. Thermal properties are assumed to vary nonlinearly with temperature. Experimental conditions refer to the thermal loading scheme, sampling times and sensor location. A numerical model of experimental configurations is studied to elicit the optimal conditions. The numerical solution of the design problem is formulated on a regularization scheme with a stabilizer minimization without a regularization parameter. An explicit design criterion is used to reveal the optimal sensor location, heating duration and flux magnitude. Results obtained indicate that even the strongly nonlinear experimental design problem admits the aggregation of its solution and has a strictly defined optimal measurement scheme. Additional region of temperature measurements with allowable identification error is revealed. IntroductionThe estimation of thermal properties as a solution of an inverse problem is constantly of great interest in the theory of ill-posed problems. A very large number of investigations, combing experimental and mathematical activities, have been devoted to the reconstruction of the specific heat and thermal conductivity [2-5, 9-12, 16, 24]. The final trend in the development of heat inverse theory is a design of experiments [1,6,7,13,14,[18][19][20][21]23]. The efforts in this direction are focused on the advance of numerical methods to determine the optimum conditions to conduct the experiment. The rules on the choice of various optimal experimental variables, including sensor locations, sampling times and heat loading are studied.To design experiment the investigation of the state function sensitivity over the sought quantities is mainly executed. The sensitivity approach gives very important information on mathematical model features. However, in the framework of the sensitivity approach a number of theoretically and practically important questions remain unanswered.The major open question is the existence of the structure of the design problem solution. The requirement of the structure seeking means that the typical classes of the problem solutions should be revealed. The basic feature of the structure is a similar behaviour of a certain set of solutions. By virtue of it, the respective change of the initial data will not vary the nature of the optimal design solution. Therefore, the absence of the structure knowledge limits the received solution only to a narrow particular field. On the other hand, the determination of the design solution structure indicates the typical cases of the identification error behaviour and specifies the ways of the design generalization. In total, the general solution of the design problem is divided into certain classes of the optimal experimental conditions, in the framework of which the best measurement ...