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. Nomenclaturespecified 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 p opt = optimal number of temporal terms Q = dimensionless heat flux Q NP = approximate heat flux q = dimensional heat flux, W/m 2 q i = discrete heat flux, W/m 2 q i = discrete heat flux rate, W/(m 2 s) q max = maximum heat flux, W/m 2 s(t) = location of moving front, m T = temperature, • C T i = discrete temperature, • Ċ T i = discrete heating/cooling rate, • C/s T i = discrete second time derivative of temperature, • C/s 2 T melt = melt temperature, • C T m (χ) = mth Chebyshev polynomial of the first kind T 0 = initial temperature, • C = discrete time, s t max = maximum time, s t pen = penetration time, s t 0 = dummy variable, s x = spatial variable, m x 0 = dummy variable, m α = thermal diffusivity, m 2 /s β = shape factor γ = constant in definition of penetration time i = imposed noise level, Eq. (6) η = fixed position θ = dimensionless temperature θ i = dimensionless discrete temperaturė θ i = dimensionless discrete heating/cooling rate λ = constant [1/ √ (kρCπ)] λ 0 = t max /2, s ξ = Chebyshev temporal coordinate ρ = density, kg/m 3 σ i = noise level, Eq. (17) χ = Chebyshev spatial coordinate mn = trial function defined, Eq. (18a) ω i = noise level, Eq. (13) ω m (χ) = spatial trial function, Eq. (18c)

A new multidimensional integral relationship between heat flux and temperature for direct internal assessment of heat flux

“…In one-dimensional analysis [13,14,17,18,22], the strategy involved Abel regularization owing to the structure of the integral relationship and the nature of the kernel (i.e., weak singularity, 1/ √ t − t 0 ). Regularization in such cases follow classical doctrine [9].…”

“…(17) indicates that it is mildly ill-posed due to the appearance of the local heating/cooling rate, dT /dt and the first kind nature of the integral equation. This is well known [13,14,17,18,22] and thus requires either digital filtering of the discrete temperature data or development of a sensor capable of the direct measurement of the heating/cooling rate.…”

“…The one-dimensional analog to the present study is well investigated [13,14,17,18]. Again, Kulish et al [20,21] developed the one-dimensional temperature-flux integral relationship, using Laplace transforms, as…”

“…Frankel et al [13][14][15][16][17][18] have inverted this equation using regularization principles associated with weaklysingular integral equations to arrive at…”

Regularization of inverse heat conduction by combination of rate sensor analysis and analytic continuation

“…The optimal value of this parameter is often difficult to acquire and affects the quality of the prediction. Frankel and colleagues [11][12][13][14][15][16][17] have taken a different point of view. Through analysis, they have identified the culprits in the projection process.…”