Majid Keyhani

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...

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.

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.

Heat Transfer Analysis via Rate Based Sensors

This paper presents an integrated rate-based sensor/numerical method for resolving the surface heat flux of a fundamental inverse heat conduction problem in a semi-infinite geometry that additionally accounts for thermocouple delay due to its intrinsic time constant. A series of well-designed voltage-rate interfaces, that directly generate the required signal from the in-depth thermocouple, are used in conjunction with the thermocouple calibration curve to provide higher-time derivatives of the thermocouples temperature. Using a lumped energy balance about the thermocouples bead, a first-order model is produced relating the thermocouple temperature to the positional temperature. The required thermocouples time constant is estimated with the aid of the Loop-Current Step Response method. Higher-time derivatives of the in-depth heat flux are produced using time integral relationships between the positional temperature and heat flux. Finally, the surface heat flux and temperature are estimated using the finite difference based Global Time Method. To verify this concept and acquire real data, an experiment was performed using a well-designed heater sandwiched between two identical plates for producing a symmetric temperature distribution in each plate with an accurately known heat input. Encouraging results are presented from this preliminary study indicating the merit of the methodology. Nomenclature AH = Heated Area of the Block, m 2

A new nonlinear surface heat flux calibration method based on Kirchhoff transformation and rescaling principles

A novel surface heat flux calibration method is presented applicable to nonlinear inverse heat conduction problems. Quasi-linearization of the nonlinear heat equation is achieved by combining the Kirchhoff transform with time-domain rescaling based on the local temperature measurement. At each time step, the thermophysical properties are held constant throughout the spatial domain though allowed to vary with advancing time. The rescaled forms are then resolved through a calibration framework. The proposed calibration formulation is expressed in terms of Volterra integral equation of the first kind. This functional equation relates the rescaled net unknown surface heat flux to the rescaled net calibration surface heat flux and their corresponding rescaled Kirchhoff transformed variables for the in-depth temperature measurements during the calibration test and unknown runs. Tikhonov regularization is introduced for generating a family of predictions based on the Tikhonov parameter. The L-curve strategy is used for selecting the proper regularization parameter. In this paper, favourable numerical results are demonstrated verifying both accuracy and robustness of the rescaling calibration approach in the presence of significant experiment noise. The methodology works well for a variety of practical isotropic materials. This approach does not require knowledge of the probe position but presently requires knowledge of the host material’s thermal diffusivity.

New In Situ Method for Estimating Thermal Diffusivity Using Rate-Based Temperature Sensors

This paper proposes an experimental methodology for estimating the thermal diffusivity, a in a one-dimensional, half-space geometry based on two in-situ positioned probes that can acquire temperature; and, the first- and second-time deriva tives of temperature. The thermal diffusivity is estimated at each sampled time by solving an n th -degree polynomial for the thermal diffusivity. The degree of the a-polynomial and required order of the time derivati ve sensors depend on the chosen spatial truncation of the Tayl or series. This approach does not require the specification of the imposed surface boundary condi tion. Additionally, a novel inter-sensitivity analysis is proposed for guiding sensor placement t hat ensures a maximum, absolute sensitivity between the two probes; and, develops a single, tim e-point estimation of thermal diffusivity at the maximum inter-sensitivity. As a preliminary ind icator of the newly proposed methodology, numerical simulation provides sufficient merit for further concept development and experimental verification.

Inverse Ablation Analysis and the Calibration Integral Equation Method

The calibration integral equation method has been demonstrated for resolving inverse heat conduction problems based on an invariant sample geometry. This paper proposes to extend its applicability to the investigation of ablation through an abstraction based on forming a fictive surface temperature poised at the nonrecessive origin. Resolving this time-dependent temperature history provides a boundary condition that can be used in a restricted space defined between the nonrecessive origin and an in-depth thermocouple. This fictive temperature (or heat flux) provides an equivalence-based formulation defined in the original spatial domain because the calibration integral equation method is based on conservation principles. The extraction of the fictive boundary condition does not require knowledge of the thermophysical properties or probe position(s) because it is based on calibration. The complexity of ablation requires a compromise between a fully calibrative technique and forward solving approaches. The ...

Nonlinear, Rescaling-Based Inverse Heat Conduction Calibration Method and Optimal Regularization Parameter Strategy

A novel surface heat flux calibration method applicable to nonlinear inverse heat conduction problems and a new strategy for obtaining an optimal regularization parameter independent of the regularization technique are presented herein. For the proposed nonlinear calibration integral equation, quasi-linearization of the nonlinear heat equation is achieved through both time domain and heat flux magnitude rescaling based on the local temperature measurement. The inverse heat conduction problem is then resolved in terms of rescaled variables through a calibration framework. The proposed calibration equation has a form of the Volterra equation of the first kind that relates the rescaled net unknown heat flux to the rescaled net calibration heat flux and their corresponding rescaled in-depth temperature measurements during both calibration and reconstruction tests. Because the functional form of a Volterra integral equation of the first kind is ill-posed, three different regularization methods are considered a...