The total inventory IE and the inventory of desired component IExE may be evaluated if the inventory per stage is known. The stage inventory Ht may be related to the stage feed rate M(+Nt by

Ht = h(Mt + TV,) (12.198)

where h is the stage holdup time, the time it takes material to flow through one stage. We shall assume that h is constant throughout the cascade. This will be strictly true of an ideal cascade made up of identical separating units and is often approximately true of an ideal cascade made up of stages of decreasing size.

The total inventory of the enriching section IE then is just h times the total flow rate in the enriching section; for a close-separation, ideal cascade,

as may be seen from Eq. (12.137) and the fact that heads and tails flow rates are approximately equal in a close-separation, ideal cascade.

The inventory of desired component in the enriching section is

di,

di — —— dx =
dx

from (12.134). this inventory is given by

With Eqs. (12.199) and (12.203), approximate equation (12.197) for the start-up time of a close-separation, ideal cascade becomes