Measurement of Transient Heat Flux and Surface Temperature Using Embedded Temperature Sensors

Abstract

A method for resolving inverse heat conduction problems using approximating polynomials is described. The coefficients are obtained by constraining the even-numbered spatial derivatives of the polynomial at two points within the domain using the heat equation. The result is a linear time-invariant system that can be implemented as a digital filter and is suitable for application in a real-time sensor. The method places no restrictions on boundary or initial conditions, and temperature dependence of transport properties is accounted for in an approximate way. The performance of the model has been validated using analytical and numerical solutions, and results are presented in the frequency and time domains for several orders of polynomials. A propagation of error analysis is presented and is used to establish the optimum location of the sensors. Nomenclature a ij = matrix of location-dependent coefficients, a 13 , a 14 :m 2 ; a 21 , a 22 :m 1 ; a 23 , a 24 :m; a 43 , a 44 :m 1 b j t = vector of temperature measurements, b 1 , b 2 :K; b 3 , b 4 :K=m 2 c i t = vector of temperature profile polynomial coefficients, c 1 :K; c 2 : K=m; c 3 :K=m 2 ; c 4 :K=m 3 c p = heat capacity, J=kg K Im = imaginary part i = imaginary number, 1 p k = thermal conductivity, W=m K P n = temperature profile polynomial of degree n, K q = heat flux, W=m 2 Re = real part T = temperature, K _ T = rate of change of temperature, K=s t = time, s x = distance from the surface, m = thermal diffusivity, m 2 =s j = low-pass filter coefficient for time derivative, s 1 = gain, nondimensional j = low-pass filter coefficient for time average t = time step, s = uncertainty = density, kg=m 3 = length scale parameter, m 1 = phase angle, rad ! = angular frequency, rad=s j j = amplitude Subscripts i, j = polynomial and matrix indices m = low-pass filter index n = order of polynomial approximation for temperature 1 = position nearest to the measurement surface 2 = position farthest from the measurement surface % = nondimensional number Superscripts = frequency domain quantity = low-pass filtered quantity = complex conjugate

Experimental and Numerical Investigation of CH$$_4$$/O$$_2$$ Rocket Combustors

“…Inverse heat transfer methods have been successfully applied for the heat flux estimation in capacitively cooled engines where the temperature field is not stationary during the test operation and hence a transient inverse heat conduction method is needed [4,19,27]. The main concept behind an inverse method for heat conduction problems lies in trying to estimate the boundary conditions (causes) which best fit the measured temperature values (effects) while keeping the physics of the problem intact.…”

Analyzing impact of sensor coupling on measurement representativeness of wall surface temperature

“…Also, he analysed the accuracy of the results by using two sensors, one embedded into the object and one at the surface. He strongly believed on this quote: "Do not relying on surface temperature measurements but embed the sensors within the wall where they are protected from erosion" (Coy, 2010). Also, he suggests using more well calibrated sensor as these devices can control each other's performance and generate more accurate result.…”

Studies on Heat Flux Imparted on Thermal Insulation Inside Rocket Motor Containing Double Base Propellant

“…Simulation of thermal loading on rocket chamber needs dedicated and exclusive instrumentation set-up for the measurement of high heat flux observed in actual rocket operations. Thermocouples based method for transient heat flux measurement is well established (Ellison et al 1969;Coy 2008;2010).…”