3i8 CHAPTER XVIIT 



Now fl^ = I — a^ and ^2 = ^ — " h> 

 whence h^ = k^ a-^ t^ = k^ {1 — «i) (i — ^1), 

 or ki a^ t^ ^ k'2 — ^2 h — ^2 ^1 ~t~ ^^ ^1 ^i- 



c- , ■ k-2 (l — '^l) 



A, ^2 ^1 (^ ^1) 



Wherefore A, = k, a, t, = ^^^^ _ k^ ^ j,^ 



Differentiating and equating to zero 



dki = d {[Ajj ^2 « (I — «i)] K (^1 — ^2) + kJ\~^\ 



— -1 = (^j — ^2) «i" + 2 ^2 «i — k-i = 

 a a^ 



Solving, «! = — 



and fl!2 = I — flj = 



«j «2 



^1 4: V^i k.^ 



R>-i — /2. 



whence ^ = "^^±^^1^2 _ . /i, 



and generally if (^j, ag, ag . . . . are the heating surfaces, and k^, k.^, k^ the 



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

 of the greatest quantity of heat 



V^2 ^2 V^3 



etc. 



'•2 



^2 Vk^ H V^2 



Similar reasoning gives -=-y^, j-^-j^, so that m all case-, tor 



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 ceUs of a triple effect be 9, 4 and i. Then for maximum 



„ . fl-i V4 2 , ^2 Vi I 



efficiency — = -^ = -- and — = —-^ = ~ . 



«2 Vg 3 «3 V4 2 



236 



and if a^ + a., + ag = i, then a^ = — , a^— ~, %= — . 



236 

 Also L = — t.,= — /.= — where L + to -\- (3 = i, 

 '■ II - II ^ II 11- 



22 3 3,, 6 6 36 



and^,fli?,=9X~X--^2«2^':=4X^X^^ = ^3%^3=ix — X-= — 



which is the maximum value under the stated conditions. 



Within the Hmits 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 



