234 CHAPTER XI 



the quantities of juice obtained are very different. Similarly the values 



of — ■—., or of the juice expressed per unit originally present for the 



example, are 0-8889 ^-ud 0*8095.* Now to — bagasse let w water be added 



and let the water mix completely with the — ^^ residual juice. t 



-^ m 



The weight of diluted juice now becomes 



f h. — m) , f 4- wm — fm 



-^ — i '- 4- w =— ■ — 



m m 



Let the wet bagasse be again crushed till it contains m fibre, when w 

 dilute juice is obtained and per unit of original juice present in the dry 



/ + wm — f m wm 



crushed bagasse there is obtained w -•- 



m / + wm — fm 



Now let the value of -. --. — be denoted by r for simplicity and let 



/ + wm— f m 



the juice in the dry crushed bagasse be unity. Then with the addition of 

 w water r juice is obtained and 1 — r passes away in the wet crushed bagasse. 



Again adding w water and again crushing to a fibre content m, the second 

 wet crushing affords r {1 —r) juice, the recovery at this point being 

 r -{• r {1 — r), and generally in n operations the recovery is given by the 

 expression 



r + r (i — r) -f r (i — r)2 + + r (i — r)""^ = i — (i — r)" 



This formula is the general expression for determining the recovery in simple 

 imbibition schemes, and its application is of most interest when w the added 

 water is constant ; that is to say, in a single process w water is used once, 



w 

 in a double process — water is used at both first and second wet crushing 



w 

 mills, and generally — water is used at each mill of a train of n mills. 



When single imbibition is used, the value of r in the series above is 



wm wm 



f + wm — fm wm + / (i — m) 

 Substituting — for w the value of r becomes 



n wm + nf (i — m) 



As n is the only variable in this expression, it may for convenience be 



written — ; where a and b are constants and the recovery with «-fold 



a -\- n b 



imbibition will be i — 1 i ; ^ ) , and when w is a positive integer this 



\ a -\-n b/ 



expression increases as n increases. 



• The exactness of this expression depends on the uniformity of the juice throughout the cane. This condition 

 does not obtain and the earlier expressed iuice is denser and sweeter than that obtained later. Accordingly 

 the actual recovery of sugar is greater than the formula indicates. To avoid quite unnecessary complications in 

 the establishment of certain principles, a uniform composition is accepted. 



t Complete admixture never takes place, but its assumption is required for the convenient development of the 

 theory. In applying the formulae obtained to actual practice it is necessary to introduce a coefficient of admixture 

 of value such as experience indicates. 



