By A. V. Luikov
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Extra resources for Analytical Heat Diffusion Theory
1) should converge. This imposes certain limitations on the function f(r) (for details see Chapter 14). [F(s)] = — - f ra+ioo F(s)e" ds. 2) Integration is performed in the complex plane, s = ξ + ίη along the straight line a = const, parallel to the imaginary axis. 2) lie in the left halfplane of the complex plane Res > sx > σ0. The technique of such an integration is described in detail in manuals on the function theory of a complex variable.
2) where at is the temperature drop over the plate thickness /. 3) 38 2. THEORY OF GENERALIZED VARIABLES the Eq. 4) So from properties which are determined from the basic Eq. 1), the ratio between the temperature drop and the temperature difference is defined immediately by the expression (α/λ)1. In general, with a variable temperature gradient (dt/dx = var), the temperature distribution is obviously different from linear and this simple result loses its validity because now it is impossible to represent the derivative (dt/dx)s as — ôt/l.
1= Bi. 8) Hence, Eq. 7) may be rewritten as dt/At = F(Bi). 9) A transition from initial variables a, /, λ to a new variable Bi, results not only in a decrease in the number of arguments; in parallel to this, the analysis itself changes. Actually, when a, l, λ are taken as basic quantities, some particular case for each set of the given values of these parameters is obtainable. In contrast to this, any particular case does not uniquely satisfy the given value of the Biot number, because this value may be realized by the infinite number of various combinations of quantities a, /, λ.