3i8 CHAPTER XVIII 



Now a 2 = i tfj and t 2 = i t lf 

 whence h^ = k a^ t = k 2 (i aj (i zj, 

 or &j a^ ti = k 2 k 2 t k 2 a + k*. #1 t L . 

 k 2 (i - aj 



Wherefore *, _ *, ^ - 



Differentiating and equating to zero 

 dh 1 =d{[k 1 k 2 a (i- a 



* 



^ & 2 ) a +2 k 2 aj k 2 =o 



Solving, 0j = = 



ki ~\ \/ J? J? 



j /^l ~ v /EI /gfc 



whence^- 1 = ~^V^J 2 = Ik, 



a ' ^ 



and generally if 1} 2 , a 3 . . . . are the heating surfaces, and k v k 2 , k 3 . . . . the 

 coefficients of transmission, then for maximum efficiency or for the passage 

 of the greatest quantity of heat 



== j=r> . .... etc. 



a 2 V&! a 3 Vk 2 



Similar reasoning gives -*= p=?, r = ^, so that in all case^ for 



t z V*, *s V & 2 



maximum efficiency the division of heating surface and of temperature 

 difference is the same. 



As a numerical example, let the coefficients of transmission in the first, 

 second and third cells of a triple effect be 9, 4 and i. Then for maximum 



' . #j \/4 2 j a z A/I i 



efficiency = 7^ = - and ====-. 



2 A/9 3 s V4 2 



and if ^ + , + a, = i, then a, = ^, 0,= ^, ,= ^. 



2 q f) 



Also ^ = , ^., = A - / o = where ^, + ^ 9 + /?. = i 

 ii ii ii 



A ^. _ 3 6 



II II "~ 121 



which is the maximum value under the stated conditions. 



Within the limits that occur in practice, however, no great advantage 

 is to be found in dividing the heating surface as indicated above. Economy 

 in construction costs is obtained by building all vessels of equal size, and 

 there are reasons to believe that in the last cell where a very viscous material 

 is boiled, the coefficient transmission increases more rapidly than does the 

 temperature difference. It is well then to aim at having a large temperature 



