Surface Heat Flux Prediction Through Physics-Based Calibration, Part 1: Theory

Frankel, J. I.; Keyhani, M.; Elkins, Bryan

doi:10.2514/1.t3917

Cited by 48 publications

(13 citation statements)

References 28 publications

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“…10,48,49 Frankel et al follow the calibration idea of the actual sensor, but based on an analytical approach. 50 So far, they did not use it in terms of a heat flux sensor.…”

Section: A Null-point Calorimetersmentioning

confidence: 99%

Review of Heat Flux Measurements for High Enthalpy Flows

Löhle

2016

32nd AIAA Aerodynamic Measurement Technology and Ground Testing Conference

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This article reviews heat flux measurement devices for high enthalpy flows as used in plasma wind tunnel facilities. Only heat flux gages measuring heat flux on cold copper surfaces are considered. The steady state water-cooled calorimeter, the Gardon gage sensor, slug calorimetry and swept nullpoint calorimetry are described. Based on published data and sensor heritage the various designs in Europe and the US are reviewed and analyzed. The paper ends with some remarks on comparability testing in different facilities.

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“…10,48,49 Frankel et al follow the calibration idea of the actual sensor, but based on an analytical approach. 50 So far, they did not use it in terms of a heat flux sensor.…”

Section: A Null-point Calorimetersmentioning

confidence: 99%

Review of Heat Flux Measurements for High Enthalpy Flows

Löhle

2016

32nd AIAA Aerodynamic Measurement Technology and Ground Testing Conference

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“…For many practical problems, inversion of the Laplace transform can be accomplished with the aid of the convolution theorem [4] without the need for contour integration. Here, we state, for reader convenience, the appropriate two-function convolution between f (t) and g(t) aŝ…”

Section: K α β) Does Not Change Among the Tests Then We Can Expressmentioning

confidence: 99%

“…In this section, the corresponding calibration integral equations are derived for the surface temperature and net surface heat flux based on the one-probe CIEM formalism [4,8]. The linear, constant property, one-dimensional, transient heat equation described in the reduced temperature variable, θ(x, t) is given in Eqs.…”

Section: Surface Heat Flux and Temperature By Ciemmentioning

confidence: 99%

“…This leads to an ill-posed problem that is highly sensitive to the noise in the data. Recovery of the activeside boundary condition requires regularization [1][2][3][4] for stablization of the numerical method. In many cases, in-depth thermocouples are used while this paper proposes an ultrasonic transducer arrangement adhered to the sample's passive side surface.…”

Section: Linear Heat Conductionmentioning

confidence: 99%

“…However, an "inverse" approach produces an ill-posed mathematical statement [3] that requires careful regularization [1][2][3] for extracting the "best" prediction over the spectrum of regularization parameters. Recently, the Calibration Integral Equation Method (CIEM) has been proposed for resolving inverse heat conduction problems [4][5][6][7][8]. The CIEM does not require explicit knowledge of the thermophysical properties, probe locations or sensor characterization.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Frankel

Bottländer

2014

19th AIAA International Space Planes and Hypersonic Systems and Technologies Conference

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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

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