THE EXTRACTION OF THE JUICE BY MILLS 239 



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



simple imbibition by the expression C j I (- j K w, where C 



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

 the corresponding expression for compound imbibition is 



C < 



w 



W 



\ n 



) . 



~Kw 



W V I + Wf 



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 : 



maximum, 



C 1 -( n \ n -Kw = 

 \n + w) 



iv- 



or I n (n -f- w)~ n L w maximum, where L = r = constant. 



o 



Differentiating and equating to zero 



n 2 (n + w)~ ("+ 1 ) L =zero 



Solving, w = n 



For example, the maximum value of the expression 



J i 





o-i w 



will obtain when w = 2 I J ~ (0-025)*} = 



o 025* 



The general formula for compound imbibition may be written : 

 w 



_ T + w __ - L w, where L =~ as before. 



w f _ w \n C 



I + W \ I + 



This expression reduces to 



w (i + w)*~ l {w (i + w)"- 1 + i}- 1 - L w. 

 Differentiating and equating to zero 



{w (i + w) 1 + i}- l {(* + w) n ~ l + w (n-i) (i + w) 

 w(i+w) n ~ l {w (i + w) n ~ l + i}- 2 {(i+ w) H - l + w (n-i) 

 _ L = zero 



(i +^) n ~ 1 -{-w (n - i) (i -f w) n ~ 2 (i +i;)"- 2 (i +nw) 



{w (i + Z0)*- 1 + i} 2 : {w (i +w) n ~ l + i} 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 



by trial and error. 



Having now obtained the expressions indicating the economic limit, it 

 remains to find some values relating to actual practice. 



