By A. V. Luikov
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Additional info for Analytical Heat Diffusion Theory
7) where a is the thermal diffusivity and is inherently positive and k is some constant, determined from boundary conditions. 8) V29(x9 y, z) + k2&(x, y, z) = 0. 9) is often referred to as the Pockel equation and is well known in mathematical physics. Thus, applying the Fourier method, the heat conduction equation is reduced to the equation of the Pockel type, the solution of which is governed by the geometric shape of the body and the initial temperature distribution as well as by conditions of heat transfer between a body and the surrounding medium or surrounding bodies.
6) The total heat amount AQ which was supplied for heating for the time Ax will be found after integration over the whole volume AQ = Q2-Qi =
But it has already been established that for similar distributions ε is constant. Therefore, if ε is determined for some similar distributions, it is also valid for all similar distributions. Simple considerations, not given here, show that similar distributions are those which correspond to identical values of (α/λ)1. But this means that the factor ε is a single-valued function of the value (α/λ)1 and consequently, Eq. 7) This result is important because the solution is presented as a function of one argument, although it is quite obvious that the temperature distribution is caused by the effect of three parameters, a, /, and λ.
Analytical Heat Diffusion Theory by A. V. Luikov