D • CONDUCTION OF HEAT 



damping due to the factor e~'^n^ as the mode number increases. Insofar as 

 the fundamental mode n = 1 is the dominant term in the Fourier series, 

 the physical role of the material and heating parameters {h, T^) in the 

 development of temperatures in the slab can be deduced from the ex- 

 pression for this mode. 



If only the fundamental mode is retained in the Fourier series of Eq. 

 4-7, the space average of the temperature ratio d{^, r) is 



Kr) = r ei^, T)dk = 1 + Ai^^5^ e-v + • ■ • (6-1) 



Jo Ml 



For small Biot numbers N <Kl the eigenvalue equation yields 

 N ^ HI tan Ml = Ml = y 



and therefore the exponential time term becomes 



2 _hd d _ ht 

 ^"^ ~ Td^~ '^ 



while the amphtude in the mean temperature formula (Eq. 6-1) is 



_ i^f^ ^ _i (6.2) 



Mi(2mi + sm 2iu0 



Hence for N <^1, the mean temperature is given by 



u_ 



e(j) = 1 - e p"^ (6-3) 



showing that, for low Biot numbers, the slab has a time constant 



U = ^ A^ « 1 (6-4) 



which, in contrast with time constants of vibratory mechanical or electri- 

 cal systems, depends not only upon the system structure (thermal proper- 

 ties and thickness) but also upon an external factor, namely the heat 

 transfer coefficient. 



For larger N the amplitude given by Eq. 6-2 is still approximately — 1, 

 and a more general expression, analogous to Eq. 6-4, is obtained for the 

 time constant: 



d^ 

 k = —, (6-5) 



where the heat transfer coefficient enters t^ via the Biot number implicit 

 in n\. The dependence on A^" becomes weak for A^ > 2 as ^i approaches its 

 asymptotic value of 7r/2 for large N. 



In appUcation to rocket wall design, Eq. 4-7 is useful in determining 



( 264 > 



