Stabilization of Ill-Posed Problems Through Thermal Rate Sensors

Abstract

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