The figure below represents a block B of high thermal conductivity fastened to a specimen S of length 1 parallel to the axis of X, and of lower conductivity k. At t = 0, all is at constant temperature.
Then, suddenly, the right hand side of S is brought to a temperature r_ and kept there.
If 8 is the excess of the temperature at some point over r_, the general problem consists in finding 0 as a function of x and 1.
The simple theory assumes that 8, at 2 -0, decreases so slowly with time that the temperature gradient in S is, at each instant, independent of x and equal to t?/L. This assumption leads to 0 = Ooe-"af, where 80 is the initial value of 0, and X, = k/Msl, M and s being, respectively, the total mass per unit area of B projected on the plane of contact, and the specific heat of block B.
The more exact theory gives where the constants A, are to be determined to give B = f?o at t = 0. Xa is given by R&/X, = ti.9 where R is the ratio of the thermal capacity of the specimen to that of B, $, is a solution of the equation G,fi, tan +. = 1, and G, is a constant function of s whose form depends upon the nature of the boundary conditions at x = %-the place of junction of B and S.
For the case where there is perfect thermal contact at x = 0, G, = l/R and is independent of s. For the case where there is thermal resistance at x = 0, G. = [R + k#8'J/hl]-1 where h is defined by hA0 = heat transferred across the boundary per cm2 per sec. and A0 is the temperature drop. It turns out that, at the junction, 0 is of the form e = Aoa-b' + A re-hrt + Aze-X2f + . .
where ~1, ~2. X1, etc. are increasingly larger than XO. and bv very substantial amounts, *Since this paper was completed, there came to the author's attention an unpublished copy (a