Apparent temperature in temperature-sensitive paint measurement and its effect on surface heat flux determination for hypersonic flows

Abstract

Non-zero temperature gradients in temperature-sensitive paint (TSP) cause the apparent temperature (the temperature measured by the TSP) to correspond to the temperature somewhere inside the TSP, which does not equal the top surface temperature. Treating the apparent temperature as the average temperature across the TSP layer is not always accurate, especially when there is a large temperature gradient in the TSP. In this paper, the apparent temperature is theoretically derived by integrating the luminescent intensity across the TSP layer and numerically determined using a Monte Carlo ray-tracing method. The results of a simulation show that the difference between the apparent and average temperatures increases with the temperature gradient in the TSP layer, which leads to a non-negligible error when determining the surface heat flux based on the average temperature. A modified Levenberg–Marquardt algorithm is used to accurately recover the surface heat flux based on the apparent temperature for typical optical conditions in the measurement system. The results are compared with those obtained using the average temperature of the TSP. The effects of heat flux, TSP thickness and base material on the errors in the determination of the heat flux are investigated in detail.

Review of Temperature Measurement Techniques

“…A correction factor was developed within Liu et al study, which was validated against simulation data [69]. Other issues pertaining to the error arising from the TSP apparent temperature relative to the actual wall temperature, as was suggested in a study by Liu, et al [70]. Such discrepancy becomes more prominent as the thickness of the TSP layer increases, relative to adjacent layers [70].…”

Time-resolved heat flux determination from fast-responding temperature-sensitive paint measurement in a shock tunnel using transient in-situ calibration

“…This temperature is a weighted average temperature of the TSP layer because each depth inside the TSP contributes to the overall excitation intensity. The specific definition of the apparent temperature given by Liu et al [13] is complex. For practical convenience, the mean temperature has been considered a reasonable approximation of the apparent temperature [14,17]:…”

“…where ϵ is the error threshold and ϵ = 1 is used in this work. It is noteworthy that the magnitudes of the parameters in β differ greatly, and a parameter with large magnitude has relatively low sensitivity in equation (13). The parameters are scaled to similar magnitudes to avoid suppressing the iteration of a parameter with large magnitude.…”

“…The error arises from differences in the thermal properties of the TSP layer and the base (especially for a metal base), which leads to the failure of the single-layer assumption. In addition, the temperature measured using TSP is not the top temperature of the model but the apparent temperature of the TSP layer, and the incorrect assumption of the temperature-sensing location leads to notable heat flux error [13].…”

A Green’s function approach to heat-flux estimation from temperature-sensitive paint measurements

“…Capable of handling problems with both temperature-dependent thermal properties and complex internal interfaces, this technique is likely applicable to the reduction of full-field temperature measurements, but its use has only been demonstrated for dispersed, discrete sensors. Finally, Liu et al [22] identified a disparity between the apparent temperature of a TSP and its actual mean temperature, a combined effect of the nonlinear luminophore sensitivity and the integrated emission from the luminescent molecules, and extend the iterative numerical approach of Cai et al [19] to compensate for this potential discrepancy.…”