By A. V. Luikov
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Extra resources for Analytical Heat Diffusion Theory
24 1. PHYSICAL FUNDAMENTALS OF HEAT TRANSFER fore neglected. It should, however, be noted that in transpiration cooling when the injected gas is hydrogen or helium, neglect of this term leads to substantial error. If all the coefficients are assumed to be constant, then for a fixed gas mixture (w = 0, d\dx = d/dr), the following system of differential equations will be obtained from Eqs. 16) A similar system of differential heat and mass transfer equations describes heat and mass transfer in solutions.
Thus as a result of the solution, we find the function t(x9y9z9T)=J[x9y9z9r). 20) The function f(x, y, z, τ) should satisfy the differential equation (when it is introduced into the heat conduction differential equation instead of t9 it should become an identity) and the initial and boundary conditions. From the uniqueness theorem (see Appendix II), if some function t(x9 y9 z, τ) satisfies the differential heat conduction equation and initial and boundary conditions, it is a unique solution of the problem.
In this case the temperature / becomes a function of only one variable, the x coordinate. 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.