Technical Note Transient surface heating rates from a nickel film sensor using inverse analysis Niranjan Sahoo ⇑ , Ravi K. Peetala Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati 781 039, India article info Article history: Received 14 June 2010 Received in revised form 11 November 2010 Accepted 19 November 2010 Keywords: Surface heat flux Cubic spline method Control volume Inverse analysis abstract Convective surface heat transfer measurements play an important role in many industrial, environmental and aerodynamic problems. In most of the cases, the flow is unsteady which results in temperature var- iation in the body. The surface heating rates are then predicted from the measured temperature histories by suitable heat transfer modeling. In this paper, the temperature history obtained from a nickel film sen- sor during a flight test is considered to study the effect of sensor thickness on surface heat flux measure- ments during the flight measurement. Inverse methods using analytical solutions as well as control volume approximations are used to infer the surface heat flux. The experimental temperature data are discretized using cubic-spline method to obtain the closed form solution which is used for inverse analysis. The results are compared with that of standard bench mark results with thin film gauge analysis based on semi-infinite one dimensional medium. No significant change in surface heat flux is observed between inverse and thin film analysis. However, when the thickness of nickel film is increased by 100 times during numerical sim- ulation of inverse method, it is seen that peak surface heat flux increases by 20%. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction Transient heat transfer problems have received considerable interest in numerous important applications particularly in aero- dynamic heating where the convective surface heating rates plays a major role [1–4]. The classical approach (called as direct method) is to use heat-conduction analysis from the temperature distribu- tion measured on the surface of the given body [5,6]. However, there are certain practical situations in which it is difficult to mea- sure the temperature data exactly on the surface rather they are predicted at some interior points. Thus, it is necessary to calculate the surface temperature first followed by estimating the heating rate measurements. They are considered as inverse problems [7,8]. The most popular direct methods for the transient temperature measurements include thin-film metal resistance sensors and sur- face thermocouples. Usually, in metal resistance temperature mea- suring devices, a sensing metallic element is mounted on a substrate exposed to the flow and the inference of convective sur- face heating rates from the flowing fluid mainly depends on heat transfer modeling methods [8]. In the first approach, the instanta- neous temperature is recorded for which the instantaneous heat flux rates are deduced from the one-dimensional heat-conduction solution for a semi-infinite substrate. When there is a step-change in heat flux, the temperature history is considered at the surface of substrate and one-dimensional heat flow can still be achieved by assuming the substrate to be well-insulated from the surroundings and negligible temperature drop across the substrate. The extent to which the one-dimensional approach can be applied, mainly de- pends on the thermal penetration distance during experimental run-times which should be small compared to linear dimension of the gauge. Sometimes, the thickness of the sensor plays a lead- ing role while inferring the one dimensional heating rates for long- er duration of the experimental time scale. This is mainly because of the fact that the temperature history on the surface of the sensor is not identical to that of interface of the sensor and substrate. Such problems are addressed by inverse heat transfer problems mainly to determine the unknown surface temperature and heat flux from the measured transient temperature data. Various inverse tech- niques such as finite element method, genetic algorithms, and con- jugate gradient methods have been discussed in the open literature where the transient heat equation is discretized and solution is ob- tained by space and time marching [9–11]. 2. Problem definition In the present work, a numerical method has been discussed using inverse analysis of the transient one-dimensional heat con- duction equation from the experimentally measured temperature history during a supersonic flight test [12]. This temperature data was recorded from a nickel film sensor (0.001 mm thick) mounted on a quartz substrate (2 mm diameter and 4 mm thick) and is shown in Fig. 1. The physical one-dimensional model considered in the analysis is shown in Fig. 2. It consists of a gauge (region 1) with certain thickness (L) which is usually thermal sensor and a 0017-9310/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2010.11.029 ⇑ Corresponding author. Tel.: +91 361 258 2665; fax: +91 361 269 0762. E-mail address: shock@iitg.ernet.in (N. Sahoo). International Journal of Heat and Mass Transfer 54 (2011) 1297–1302 Contents lists available at ScienceDirect International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt