Thermal Measurement of harsh environments using indirect acoustic pyrometry

The inversion of a composite governing equation for the estimation of a boundary heat flux from ultrasonic pulse data is presented. The time of flight of the ultrasonic pulse is temperature dependent and can be used to predict the boundary heat flux. Sensitivities of the approach are examined, results from fabricated data are presented, and example solutions are provided with actual ultrasonic temperature measurement data. The results indicate that compared to the canonical inverse heat conduction problem, the additional step of resolving the time-offlight data to temperature degrades the sensitivities. Nevertheless, sampling the entire temperature distribution enhances the results. This method of using ultrasonic pulses to remotely determine heat fluxes is comparable in terms of accuracy to more common heat flux estimation methods. INTRODUCTION Remote sensing of temperature—and perhaps more importantly heat flux—is critical to a number of applications such as wind tunnel measurements, combustion chambers and large gun barrels. Each of these applications involve extremely harsh environments where sensors are not likely to survive or where measurement devices would interfere with the operation of the system. Furthermore, these applications involve high heat fluxes and fast transients. Consequently, high resolution transient characterization of these inaccessible thermal environments is difficult. Using ultrasonic pulses through a conducting wall and inverse methods, the present effort will demonstrate the feasibility Address all correspondence to this author. of measuring internal heat fluxes in a harsh environment from a remotely mounted sensor. Ultrasonic pulse measurements have been used in nondestructive evaluation (NDE) for decades with a great deal of success [1]. Furthermore, ultrasonic pyrometry has been used in many process control systems [2] because the sound speed is a strong function of temperature in most materials. This technique has proven effective for gases [3], fluids [4] and extrusions [5] as long as direct access to the material where the temperature is being measured is available. These applications are concerned with average temperature measurements and have not, in general, been used to extract transient heat fluxes. When a sound wave propagates through a material, its propagation speed will be a function of the local temperature. Therefore the time-of-flight for an ultrasonic pulse will be a function of the temperature distribution along the pulse path [6]. For uniform temperature distributions, the average temperature of the medium can be deduced easily from the time-of-flight calibration data. For non-uniform temperature distributions, the path integral over the unknown temperature must be performed. Although the solution is ill-posed, a priori knowledge of the functional form of the temperature distribution can provide reliable estimates of interior temperatures [7]. Inverse methods have been used to reconstruct steady temperature distributions [8], but transient effects have largely been ignored. The present analysis will show how ultrasonic temperature measurements can be made on solid structures to extract transient thermal conditions. Transient features are critical to many applications inherent to the aerospace industry. In combustion chambers, for example, internal instabilities need to be characterized 1 Copyright c © 2007 by ASME for effective design of new technologies. However, access to the interior is limited due to the harsh environment and because of possible disruption to the operation of the device. Yet, remote measurement without sacrificing the integrity of the combustion wall is simply not possible with modern measurement systems. Ultrasonic measurement systems allow the sampling of thermal loads in a wall from the presumably benign environment on the outside of the combustion chamber. In order for an ultrasonic measurement system to be viable for accurate characterization of heat fluxes, a technique to recreate the thermal history of time of flight data must be developed. The present work demonstrates this methodology and characterizes the uncertainty in the solution. A solution approach can be devised from standard inverse techniques. Solutions of the inverse heat conduction problem are predicated on the fact that a discrete number of interior temperatures are known and an unknown boundary condition is wanted. Unlike the traditional inverse heat conduction problem (IHCP), though, the time-of-flight data does not provide local temperatures. Instead, the data represent an integral over the temperature distribution along the pulse path. Consequently, the applicability of traditional methods is uncertain because most solutions are ill-posed in a particular way, and methods are designed to address the instability of particular problems [9]. The present effort will demonstrate the effectiveness of a function specification approach in solving this new class of problems. While actual ultrasonic measurements, where the interior boundary is unknown, are considered, three test cases with manufactured data are examined first. The manufactured data contain similar sample rates, geometry and magnitude of heating as the actual demonstration data. From the test cases, we can evaluate the error introduced by measurement noise and bias inherent to the function specification approach. With the real measured data, external influences will be identified and explained from features in the boundary estimates. FORMULATION The inverse solution requires a forward conduction solution to convert an approximate boundary condition to a temperature distribution in the material. Then the temperature distribution is used to predict the time required for an ultrasonic pulse to traverse the medium. The difference between the calculated time of flight and the measured time of flight is minimized by adjusting the approximate boundary condition. Heat transfer relationships For the forward conduction solution, consider a onedimensional solid wall bounded on one surface by a thermally and possibly chemically harsh environment and on the other by ambient conditions. In the present example, the wall is exposed unknown reacting environment heat flux benign conditions ultrasonic tranducer conducting wall harsh, Figure 1. Schematic of the one-dimensional conduction domain where the unknown boundary flux is on the side of the harsh environment and the measurement system (transducer) detects the interior boundary remotely through the wall. to combustion products whose thermal load is to be measured. The configuration is shown in Figure 1. The governing equation for constant properties is ∂2θ ∂x2 = 1 α ∂θ ∂t , 0 ≤ x ≤ L, t > 0, (1) where x is the position in the wall and θ is the temperature rise above ambient conditions. The internal boundary condition is a time-dependent function for the heat flux −k ∂θ ∂x = q(t), x = 0, t > 0, (2) with a specified temperature on the external surface θ = 0, x = L, t > 0. (3) Here, k is the thermal conductivity of the wall, and L is the thickness of the wall. The initial condition is homogeneous θ = 0, t = 0, 0 ≥ x ≥ L. (4) For a constant heat flux at the boundary (independent of time), the temperature solution can be written for constant properties as [10] θ(x, t) = 2q kL ∞ ∑ m=1 cos(βmx) βm exp(−αβmt), (5) 2 Copyright c © 2007 by ASME where q is constant, α is the thermal diffusivity, and βm is an eigen-value of the kernel function cos(βmx), given by βm = (2m−1)π 2L . (6) Because the interior temperature will span a wide range of values, the constant property assumption may introduce some error. However, this approximation is tolerable because 1) the properties don’t change dramatically over our temperature range (approximately 10%), 2) extreme temperatures are only seen in a very small location and for short times, so the impact is reduced, and 3) other approximations in the comparison to real gun data limit our accuracy anyway. In the present problem, the boundary function is arbitrary and unknown. Duhamel’s theorem can be used with a piecewise constant approximation to the heat flux to generate a general solution. The temperature can be written as a superposition of solutions for heat flux at each time step as θi(x) = i ∑ j=0 2 kL (q j −q j−i) ∞ ∑ m=1 cos(βmx) βm exp(−αβm(i− j)∆t), (7) where the heat flux at t ≤ 0 is zero and ∆t is the time step between measurement samples. Therefore the time when the temperature is calculated corresponds to the time when the measurements are taken. In general, the time step does not have to be constant, but the foregoing analysis does not require this added complexity. Realize that this time is somewhat ambiguous because to obtain a single measurement requires a pulse to be induced, traverse the medium and be detected by the sensor. However, during this traversal, we assume that the thermal transients are negligible. As such, the thermal transients must be smaller than the time of flight for the ultrasonic pulse. Based on an acoustic velocity of 5096m/s and wall thickness of 0.064m, the pulse transit time is of the order of 30μs (see Table 1). The measured temperature rise occurs over 3ms, which is two orders of magnitude greater, so our assumption is justified. Acoustical propagation The system considered here uses ultrasonic pulses to deduce the transit time of acoustic energy across a solid. The round-trip time for an acoustical pulse to traverse a wall is given by