Explicit approximations to estimate the perturbative diffusivity in the presence of convectivity and damping. I. Semi-infinite slab approximations

Abstract

In this paper, a number of new approximations are introduced to estimate the perturbative diusivity (χ), convectivity (V), and damping (τ) in cylindrical geometry. For this purpose the harmonic components of heat waves induced by localized deposition of modulated power are used. The approximations are based on semi-innite slab approximations of the heat equation. The main result is the approximation of χ under the inuence of V and τ based on the phase of two harmonics making the estimate less sensitive to calibration errors. To understand why the slab approximations can estimate χ well in cylindrical geometry, the relationships between heat transport models in slab and cylindrical geometry are studied. In addition, the relationship between amplitude and phase with respect to their derivatives, used to estimate χ, is discussed. The results are presented in terms of the relative error for the dierent derived approximations for dierent values of frequency, transport coecients, and dimensionless radius. The approximations shows a signicant region in which χ, V , and τ can be estimated well, but also regions in which the error is large. Also, it is shown that some compensation is necessary to estimate V and τ in a cylindrical geometry. On the other hand, errors resulting from the simplied assumptions are also discussed showing that estimating realistic values for V and τ based on innite domains will be dicult in practice. This paper is the rst part (Part 1) of a series of three papers. In Part 2 and Part 3 cylindrical approximations based directly on semi-innite cylindrical domain (outward propagating heat pulses) and inward propagating heat pulses in a cylindrical domain, respectively, will be treated.

Explicit approximations to estimate the perturbative diffusivity in the presence of convectivity and damping. II. Semi-infinite cylindrical approximations

“…However, a number of assumptions are necessary to derive direct expressions for v, which are standard in the literature. 2,4 These assumptions simplify (1) such that analytic solutions can be derived for (1). Only measurements are considered for which the transients due to the initial condition can be neglected.…”

“…Here, it is additionally assumed that the density n is constant with respect to q. Under these assumptions, (1) can be transformed into the Laplace domain yielding 3 2 s þ s inv ð ÞH q;s ð Þ ¼ 1 q d dq qv @H q;s ð Þ @q þ qVH q;s ð Þ…”

“…On the other hand, it is possible to avoid the calculation of the spatial derivatives by directly expressing the transport coefficients and q in terms of the measured amplitudes using the transfer function description. 1,[11][12][13] This description needs a second boundary condition to define D 1 (s), which basically defines the relationship between A 0 /A and A, and / 0 and /.…”

New evidence and impact of electron transport non-linearities based on new perturbative inter-modulation analysis

“…The long term goal is to match the estimated values of G (1) and G (2) , which we estimate here, to those derived from physics. To make it more concrete G (1) is simply the transfer function, which is specifically defined for various transport models as is explained [43]. However, as at this stage it is unclear what the underlying physics are we will not try to match G (1) and G (2) against simulations here, but will focus on their estimation from measurement data and derive conclusions from calculations using the general Volterra series.…”

“…The Volterra kernel G (1) equals the linear transport in terms of a transfer function as defined in [43]. Hence, this kernel G (1) is frequency dependent and represents the best true linearized dynamics whereas G (2) acts as a non-linear error on this measurement.…”