Jay I. Frankel

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

A transformative methodology is presented for predicting surface heat fluxes based on interior temperature measurements. A physics-based calibration method is mathematically developed presently in the context of the linear heat equation and experimentally verified (in a later paper), indicating the merit and accuracy of the approach. Sensor characterization, sensor positioning, and thermophysical properties are inherently contained without being explicitly expressed in the final mathematical expression, relating the surface heat flux to interior temperature measurements. A unified theoretical basis is presently under development that encompasses one-, two-, and three-dimensional multiregion geometries possessing orthotropic thermophysical properties. Additionally, the mathematical formalism will recover either the local surface heat flux or total surface heat transfer. This paper represents the first presentation of the concept, illustrates its genesis, and presents insight toward developing a comprehensi...

Cumulative variable formulation for transient conductive and radiative transport in participating media

A new mathematical formulation is proposed for transient conductive and radiative transport in a participating gray, isotropically scattering plane-parallel medium. The methodology can be easily extended to include numerous additional effects. A systematic and unified treatment is presented using cumulative variables that allows for high-order integration using standard initial-value methods in the temporal variable while allowing for an effective orthogonal collocation method to be implemented in the spatial variable. A spectral approach is incorporated in the present context where Chebyshev polynomials of the first kind are used as the basis functions. This article illustrates the methodology and presents some comparisons with previously reported works.

Stabilization of Ill-Posed Problems Through Thermal Rate Sensors

Reliance on conventional temperature and heat flux sensors in transient situations can inhibit predictiveness and lead to unsatisfactory results that require extensive post-processing procedures for reconstituting usable results. A mathematical formalism is presented to motivate the development of thermal rate sensors. Rate-based temperature and heat flux sensors can be designed in a manner without requiring any form of data differentiation. The proposed temperature and heat flux rate sensors can enhance both real-time and postprocessing investigations. A new sensor hierarchy is proposed that reduces and, in some cases, removes the often encountered ill-posed nature observed in numerous heat transfer studies. Four diverse examples are presented illustrating the power of rate-based data for enhancing stability and accuracy. Numerical regularization that is normally required for assuring stability can effectively be eliminated by data from thermal rate sensors. Additionally, in some investigations, these data forms can assist in identifying optimal regularization parameters. Data from rate-based sensors can make an immediate impact on a variety of aerospace, defense, and material science studies. Nomenclature A = constant, ◦ C/s 3 C = heat capacity, kJ/(kg ◦ C) D = differential operator, ∂/∂t f (t) = specified temperature function, Eq. (15b) G = Green’s function, Eq. (2b) g(t) = specified heat flux function, Eq. (15c) H = Heaviside step function, Eq. (8d) K = integral operator, Eq. (3b) k = thermal conductivity, W/(m ◦ C) L = fixed position, m M = number of data points N = number of space terms N j = difference norm, Eq. (20) P = number of time terms popt = optimal number of temporal terms Q = dimensionless heat flux QNP = approximate heat flux q �� = dimensional heat flux, W/m 2 q �� = discrete heat flux, W/m 2

Inferring convective and radiative heating loads from transient surface temperature measurements in the half-space

The classical, one-dimensional, transient half-space heat conduction problem in an opaque material is revisited in order to highlight several important observations that can impact a variety of ill-posed problems. First, the differential statement is recast into an equivalent Abel integral equation that describes the penetrating (conductive) heat flux in terms of the surface temperature. In many aerospace studies, the surface temperature is measured and the surface heat flux is calculated. Though intuitively simple, using unfiltered, noisy temperature data leads to unstable heat flux predictions as the sample density increases. This article presents a clear mathematical proof using the Discrete Fourier Transform Method verifying that the root-mean square error of the surface heat flux grows as the square root of the sample size (i.e., it is ill-posed when based on surface temperature measurements containing white noise). Second, a digital filtering method is proposed to reduce the instability problem while permitting an accurate depiction of the surface heat flux. Third, the study indicates that it is possible to predict the radiative and convective heat loads based on surface temperature measurements without a priori specification of the heat transfer coefficient or emissivity. That is, the Abel formulation effectively uncouples the interior (conductive region) from the surface (convective and radiative regions). Finally, it is demonstrated that the average convective heat transfer coefficient and average emissivity can be determined through the decoupled formulation using a simple least-squares approach. Further, the effect of data filtering is illustrated on the predictions of both the convective and radiative heat loads. The proposed Gauss filter is well suited to this problem owing to (i) its inherent behavior as a low-pass filter in the frequency domain and (ii) maintaining smooth, analytic support in the time domain.

Generalizing the Method of Kulish to One-Dimensional Unsteady Heat Conducting Slabs

aj = values of (0 or 1), j 1, 2 b = slab width, m bj = values of (0 or 1), j 1, 2 C = heat capacity, kJ= kg C F = arbitrary heat source, C=m fmax = forcing function, C=m G = Green’s function g x = initial condition, C k = thermal conductivity, W= m C q00 = dimensional heat flux,W=m s = dummy time variable, s T = temperature, C To = initial temperature, C t = time, s to = dummy time variable, s u = dummy time variable, s wx = weight function x = spatial variable, m xo = dummy spatial variable, m = thermal diffusivity [k= C ], m=s = small increment, s = fixed position, m = density, kg=m

Rate-Based Sensing Concepts for Heat Flux and Property Estimation; and, Transition Detection

This paper begins by reviewing recent advances invo lving rate-based sensors for investigating a variety of aerospace heat transfer applications. These investigations include: (a) estimating transient, surface heat fluxes from in-depth sensor s; (b) estimating the location for the onset of transition in high Mach number flows (hypersonic fl ow studies); and, (c) estimating thermophysical properties by in-situ means. A pract ical view to inverse heat conduction is then described pertinent to ground-based studies. Numeri cal stabilization required for the inverse heat conduction study is obtained by (i) interrogat ing the frequency domain of the acquired temperature data; (ii) designing a digital filter t hat preserves accuracy in the time derivative of the temperature; and, (iii) imposing physical const raints generated by the experiment itself. Finally, a brief discussion on the effect of transd ucer lag times, inherent to very short-time studies, is presented. Nomenclature

Nonlinear Inverse Calibration Heat Conduction Through Property Physics

This paper proposes a nonlinear calibration approach for inverse heat conduction. A set of observations is put forth based on reformulating the heat equation in terms of thermophysical properties. An emerging pattern suggests a fundamental format for the development of a calibration integral equation that accounts for temperature-dependent thermophysical properties. An arbitrary kernel expansion is proposed in terms of a Taylor series expansion in temperature possessing undetermined coefficients. These undetermined coefficients are estimated using an additional calibration run. This concept preserves the features of the linear calibration methodology previously reported by the authors, while extending the methodology to include temperature-dependent thermophysical properties. This paper articulates an encouraging preliminary study that highlights the initial steps toward developing a nonlinear calibration inverse method. Preliminary numerical results based on simulated data using the proposed generalized ...

Design and control of interfacial temperature gradients in solidification

Abstract In a unidirectional solidification design problem, the solidification velocity and the liquid-side interfacial temperature gradient are of principle interest due to their effect on the morphology of the cast structure. The design challenge is prediction of the temporal conditions at the boundaries, such that the solidification velocity and the liquid-side temperature gradient at the solid–liquid interface follow a predetermined design scenario. The stated problem requires the resolution of two inverse problems: one, in an expanding solid domain and the second, in a shrinking liquid domain. An innovative solution technique is proposed and demonstrated for design of the liquid-side temperature gradient during unidirectional solidification. During the early transient, the control of the interfacial temperature gradient presents a challenge due to the diffusion time between the boundary and the interface. This challenge is met using a combination of initial condition design and time structuring, which allows independent control of the interfacial temperature gradient for the extent of the solidification process. The solution is developed in the context of a classic weighted-residual method, where the temporal variable is treated in an elliptic fashion.

Numerically stabilizing ill-posed moving surface problems through heat-rate sensors

Nomenclature B2 M = deterministic bias, Eq. (10c) b = constant in Gaussian function, s, Eq. (7) cp = heat capacity, kJ/(kg◦C) I j (z) = j th modified Bessel function K = convolution kernel, Eq. (1c) k = thermal conductivity, W/(m◦C) M = number of data points Ma = convolution kernel, Eq. (5a) Mb = convolution kernel, Eq. (5b) N = convolution kernel, Eq. (6b) q ′′ = dimensional heat flux, W/m2 q ′′ s = surface heat flux, W/m 2 q ′′ 0 = maximum Gaussian heat-flux value, W/m 2 T = temperature, ◦C Ti = discrete temperature, ◦C Ts = surface temperature, ◦C T0 = initial temperature, ◦C t = time, s tmax = maximum time, s t0 = dummy variable, s u = velocity of moving surface, m/s v = dummy variable, Eq. (3) x = spatial variable, m z = dummy argument α = thermal diffusivity, m2/s β = constant, Eq. (1c) i = noise factor, Eqs. (8a) and (8b) θ = temperature difference, Ts − T0, ◦C λ = constant, Eq. (1c) ρ = density, kg/m3 σ = constant, Eq. (7) σ 2 M = variance, Eq. (10d)

Global time method for inverse heat conduction problem

Traditional space-marching techniques for solving the inverse heat conduction problem are highly susceptible to both measurement and round-off error. This problem is exacerbated if the problem requires small time steps to resolve rapid changes in the surface condition, since this can cause instability. The inverse technique presented in this article utilizes a global time approach which eliminates the instability usually observed when using small time steps. It is demonstrated that a higher sampling rate (smaller time steps) in fact improves the inverse prediction. This is accomplished using a functional representation of the time derivative in the heat equation, and a physically based regularization scheme. A Gaussian low-pass filter is used with an analytically determined optimum cut-off frequency. The filter delivers an analytical function which has smooth, bounded derivatives. The inverse technique is demonstrated to accurately resolve the transient surface thermal condition in the presence of noise.