Real-Time Estimation of Time-Dependent Heat Flux for 3D Domain and Its Application

Abstract

The heat input estimation based on the inverse heat conduction problem(IHCP) arises in the modeling and control of the process of heat propagation in widespread application fields from mechanical manufacturing systems and power plant to a thermal system of electrical devices and even for a geothermal system modeling of a reservoir. However, most of the inverse problem is experiencing the instability for the solution because the unknown thermal parameters or their solution have to be determined from the indirect observable variable in different physics, including the measurement error. Therefore, the common inverse heat conduction problem in engineering fields is limited in applications with the complex geometry domain and real-time estimation. This is the major obstacle to the general ill-posed problem or ill-conditioning problem. This research is motivated by the limitations of significant obstacles to improving the accessibility to the inverse heat conduction problem of general multi-dimensional application. In the estimation modeling part, a novel method of surface heat flux estimation for the 3D finite domain is introduced. The thermal mode parameter is adopted to produce a mathematical relation between the function of surface heat flux history and the transient temperature response vector. The direct solution and heat flux estimation model (Inverse solution) is obtained by utilizing the modal superposition method employing thermal modes with eigenvalue for a direct solution and least square deconvolution method for inverse solution. The recursive concept of the least square deconvolution method is proposed to handle the problems in discontinuous heat input function and large computation time for long-term estimation. The recursive concept is based on short term solution of IHCP at each divided recursive time duration in total estimation time range which makes possible that the model can produce accurate estimation result in discontinuous and reduce the computation time. In the validation part, the proposed method is evaluated by numerical and experimental tests. Firstly, the accuracy of the proposed estimation is investigated by using several hypothetical heat flux history functions and their analytic modal solutions for heat flux estimation of numerical validation specimens. As a result, hypothetical input cases show promising results, with less than 1% of the root mean square error (RMSE), which is compared with the function value of hypothetical input. The same aluminum specimen of numerical validation was employed to experiment with temperature response measurement under the heat liberation from the ceramic heater at a specific location and actual transient heat flux history measurement by using the thermocouple and commercial heat flux sensor. The estimation result of the experiment presents the RMSE within 2.5% for the maximum magnitude of measured heat flux history, achieved in only 49 seconds of computing time for a long-term estimation of 100 minutes. In the application part, the machining error caused by the thermal deformation of the milling tool during the carbon-fiber-reinforced polymer/plastic(CFRP) milling process is presented. Generally, a higher temperature increase in milling tool and thermal deformation is occurred for the same machining condition of metal milling due to the characteristics of CFRP. In the experiment, the temperature change of the milling tool during the machining is measured by infrared(IR) thermal camera and IR thermal spot sensor. Estimation is performed by using measured temperature and validated by comparing the measured transient temperature responses with simulated one from finite element analysis result employing heat input boundary of estimated heat flux history. The measured cutting depth change was compared with simulated results based on the calculated temperature response of the FE model using estimated heat flux. As a result, the temperature comparison result of the milling tool experiment presents the RMSE within 0.68K for tool temperature distribution comparison. The RMSE of thermal deformation comparison was 0.7 μm.