THE EXTRACTION OF THE JUICE BY MILLS 239 



The net profit to the producer will therefore be given in the case ot 



sunple unbibition by the expression C J i — ( — ^ — y\ — Kw where C 



\ \n -\- wJ \ 



and K are constants and n is the number of wet crushing mills. Similarly 

 the corresponding expression for compound imbibition is 



C 



1 -\- w 



~ Kw 



w_ / w \n 



\t -\-W \ I + w/ , 



The economic limit of extraction will be obtained when w is chosen, so that 

 these expressions are a maximum. Solutions of this problem are given 

 for completeness. 



The general formula when using simple imbibition may be written : 



{ \n + wJ J 



maximum. 



or I — n {n -\- w)~" — L w = maximum, where L = --= constant. 

 Differentiating and equating to zero 



n^ {n + z^;)— (»+i) — L = zero 



c 1 • w ( I - L'^^7 

 Solving, w = — ^!^ <~ 



For example, the maximum value of the expression 

 . (_ 2 



(2 + ^)^ 



0-1 w 



■^^ ^ . ■ , 2{l — (O ' 025)* [ 



Will obtain when w = — ' ^ — j = 4-84. 



0-025* ^ ^ 



The general formula for compound imbibition may be written ; — 



w 



' '^ — L w, where L =--^ as before. 



w , ( _ w \n ' C 



+ ( 



1 -{- W \ T -\- W^ 



This expression reduces to 



w (i + w)" ~^ {w (i + 1^)" ~^ -\- i}~^ — L w. 

 Differentiating and equating to zero 



(w (i + w)"-'^ + i}-^{(i + w)"-^ -\-w {n — i) (i + lay-^ } 



— w{i -j-w)"-'^{w (i + w)"-'^ + i}~''^{(i + tf')"-^ -j- w {n — i) (i + w)"~-} 



— L = zero 



{i ■i-w)"-''^ +ze; (« — i) (i + w)"-- _ (i + w)"-^ (i + nw) 

 Solving L = {w (I -{-wy-' + i}2 = {w (i+ze;)«-i+ 1)2 



If desired the roots of this equation may be found by Horner's method, 

 but generally the maximum value of w will be obtained with less labour 

 hy trial and error. 



Having now obtained the expressions indicating the economic limit, it 

 remains to find some values relating to actual practice. 



