Abstract

I NHIGH heat load environments such as combustion chambers of rocket motors or atmospheric entry, the determination of the net surface heat flux is an essential problem. Inmost cases, only in-depth temperature sensor data is available and derivation of heat flux requires the solution of what is known as the inverse heat conduction problem (IHCP). During the last 50 years, a wide array of analytical and numerical solutions for the IHCP have been developed [1,2]. However, most solutions are based on strict initial and boundary conditions. For example, some analytical solutions are only applicable to particular settings such as the semi-infinite half-space [3]. Furthermore, the knowledge of the thermophysical properties of the materials in use is a prerequisite. Finally, the temperature sensor position is very sensitive to the solution of such problems. Although promising solutions concerning the one-dimensional IHCP have been published at least for the semi-infinite half-space, there is still uncertainty in the material properties, contact resistance, and exact temperature sensor position [4]. Furthermore, given more complex geometric bodies, in which heat conduction in all three space dimensions cannot be neglected, these tools can not be applied. In this paper, an approach named the noninteger system identification (NISI) method is extended to a three-dimensional medium. The NISI method for one-dimensional problems has first been published byBattaglia et al. [5]. It has been shown that the basic idea of applying system identification as known from automatics and control as an alternative approach to solve IHCPs holds for several analytical problems, e.g., semi-infinte, sphere. It can even be applied to more complex systems since the system identification step accounts for effects as for example deviations from one-dimensional behavior or assembly issues of the thermocouple. During identification, the sensor system, i.e., the temperature sensor as mounted inside the probe body, is calibrated. That means, adhesive layers possibly in between the sensor and the probe body affecting the thermophysical properties, thermocouple position deviating from the specified position, or thermal contact resistancies of different materials are included in the calibration step and do not further influence the data analysis. More recently, the NISI method has successfully been applied to analyze the classical null-point calorimeter developed in the 1970s [6–8]. It has been shown that an asymmetry in the heat flux profile is an artificial effect arising due to radial heat fluxes to the temperature sensor. As a result of this investigation, modifications towards a miniaturized sensor aiming to measure radial heat flux profiles in other plasma flows has been developed and successfully applied [9,10]. Again this modification became possible, because the NISI method inherently takes effects as thermocouple mounting or thermal resistancies into account through the calibration step.Within the planned hypersonic flight experiment HIFiRE, a ceramic fin has been equipped with thermocouples and first investigations towards a NISI application have been conducted [11]. In the present investigation, it is tried to extend the method to a three-dimensional problem. In a simplified form a similar analysis has been published by Battaglia [12]. By simplifying the engineering problem and some further assumptions, the multidimensional approach had been considered as a series of one-dimensional problems. In the present study, we do not assume any symmetry or similarity from the experimental point of view. Using finite element analysis of a half-ellipsoid as a generic model featuring several virtually embedded temperature sensors, the method is applied in theory. In Fig. 1, the principal procedure is shown. A half-ellipsoid is the exemplary three-dimensional geometry. There are five surfaces and five temperature sensors defined. Using a commercially available finite element modelling software (Ansys), the system is first identified according to the NISI method applying a heat flux profile to each surface element and monitoring all five temperature sensors to calibrate the system. In a second step, an exemplary heatflux profile is applied to 1) all surface elements and 2) to only two surface elements. Finally, the temperature signal, which has been numerically calculated is used to determine inversely the preset heat flux profile. The following section describes the theory as an extension of the one-dimensional problem to more sensors andmore surface areas. In Sec. III the calibration based on numerical analysis is presented followed by the application to the numerical problem to solve the inverse problem.

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