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

Abstract

This paper derives a new integral relationship between heat flux and temperature in a transient, two-dimensional heat conducting half space. A unified mathematical treatment is proposed that is extendable to higher-dimensional and finite-region geometries. The analytic expression provides the local heat flux perpendicular to the front surface solely based on an embedded line of temperature sensors parallel to the surface. The relationship does not require apriori knowledge of the surface boundary condition. A new sensor strategy is analytically conceived based on the integral relationship for estimating the local, in-depth heat flux without surface instrumentation. It should further be clarified that the integral relationship requires only knowledge of the local, in-depth temperature and heating/cooling rate (time rate of change of temperature). The resulting formulation is mildly ill-posed and either requires digital filtering of the temperature signal to remove high frequency components of noise or the development of direct heating/cooling rate sensors. This paper (a) develops the new mathematical relationship; (b) demonstrates that the proposed relationship reduces to well-known (i) one-dimensional results under the appropriate assumptions; and, (ii) two-dimensional surface results; and, (c) provides a simple numerical example validating the concept.

Locating Sudden Changes in Heat Flux Using Higher Temporal Derivatives of Temperature

“…For simplicity, let T i 0 C, then the resulting surface temperature distribution is analytically provided as [13] 0; y; t q 00…”

A new three-dimensional heat flux–temperature integral relationship for half-space transient diffusion

“…A numerical example is also provided in Ref. [2] involving noisy data (data with only random noise, i.e., no bias).…”

“…This is possible if the general law (first law of thermodynamics) and constitutive equation for heat flux (Fourier's law) are combined to obtain the classical heat equation in the temperature variable. Frankel and his colleagues [1][2][3][4][5][6][7] have developed a unified mathematical treatment for obtaining these new integral relationships that do not involve any knowledge of spatial gradients. Hence, only time histories of temperature are required.…”

“…The thermal diffusivity is given by a. The twodimensional, transient heat flux-temperature integral relationship in the half space subject to the trivial initial condition i.e., T 0 =0 (or henceforth T(x,y,z,t) may be interpreted as the temperature above its initial value) is [2] q x 00 ðx; y; tÞ ¼ k 2pa…”

“…This relationship must be discretized in some sense when sensors are used at the in-depth position. Frankel et al [2,7] have developed such a discretization in two-dimensions. The number of sensors required will physically be based on the imposed surface condition.…”