Acoustic Interferometry and the Calibration Integral Equation Method for Inverse Heat Conduction

Abstract

This paper expands the recently proposed Calibration Integral Equation Method (CIEM) for resolving inverse heat conduction problems to include an alternative instrumentation scheme based on ultrasonic interferometry. Conventional approaches to inverse heat conduction problems use local temperature data obtained from in-depth thermocouples. In contrast, the mathematical framework developed in this paper is based on an pulse-echo transducer arrangement for estimating the active-side surface heat flux and temperature. The resulting formulation leads to a first kind Volterra integral equation involving only calibration data and the unknown surface condition (either net surface heat flux or temperature). This approach eliminates the need to specify any thermophysical properties associated with the sample and hence reduces systematic errors. The linearized formulation can make use of the identical computational procedures as previously developed in the context of in-depth thermocouples. Nomenclaturespeed of sound, m/s c p = specific heat capacity, kJ/(kgK) f = arbitrary function f = Laplace transform of arbitrary function g= arbitrary function g = Laplace transform of arbitrary function g oi = G o − ∆T oi w/λ o , s G(t) = time of flight function, s G i (t) = time of flight function in discrete region, s G o = time of flight at reference temperature, s ∼ G (t) = approximate time of flight function, s h = effective heat transfer coefficient, W/(m 2 K) k = thermal conductivity, W/(mK) L = Laplace transform operator, s M = frequency transfer function, K/s M 1 = frequency transfer function, J/(m 2 ) N = number of discrete elements q = heat flux, W/m 2 1 Downloaded by NANYANG TECHNICAL UNIVERSITY on October 1, 2015 | http://arc.aiaa.org | AIAA Aviation q o = active side heat flux, W/m 2 q w = passive side heat flux, W/m 2 ∼ q = approximate heat flux, W/m 2 ∼ q = transformed approximate heat flux, J/m 2 s = frequency variable, Hz t = time, s T = temperature, K T i = discrete elemental temperature, K T ic = initial condition temperature, K T o = reference temperature, K T w = temperature at x = w, K w = width of plate, m x = spatial coordinate, m z = parameter in geometric seriesGreek α = thermal diffusivity, k/(ρc p ), m 2 /s β = linear temperature expansion coefficient,mK/s θ = reduced temperature, T − T o , K θ = transformed reduced temperature, Ks ρ = density, kg/m 3 ∼ Ψ = approximate reduced time of flight, ŝ ∼ Ψ = transformed approximate reduced time of flight, s 2