A lower bound for the equilibrium time of an ideal cascade may be found by determining the length of time it would take the plant to produce its own steady-state composition gradient if at all times during the start-up period it was possible to prevent loss of separative work through mixing of streams of different composition. Conceptually, this might be done if the plant consisted of a large number of small separating units whose connection in parallel and in series could be changed continuously during the start-up period. We assume that no product is withdrawn until the steady-state composition gradient has been established. Then product withdrawal may be started at the steady-state rate.

During the start-up period prior to product withdrawal it is necessary to bring in enough feed of composition zF and withdraw enough tails of composition Хці to provide the increase in inventory of desired component from its initial value of IzF to its steady-state value of lx. By material balance, the required amount of feed EF is

Z7 _ f(x ~ zf)

bp—————-

Zp-Xy,

During the start-up period, the change in the plant’s inventory of separative work is /(0 — фр), where

all

/0= ) 7(0 0i

If the duration of the start-up period is t, the amount of separative work S done by a plant whose separative capacity is D is

Dr = Ef(4>w — фр) + І(ф — фр) (12.208)

This may be solved for the equilibrium time r, with EF given by (12.205):

This expression gives a lower bound for the equilibrium time, which can be attained only if mixing of streams of different composition can be prevented during the entire start-up period. It provides a lower bound for the equilibrium time in somewhat the same way that consideration of a thermodynamically reversible process provides a lower bound for the amount of work needed to carry out a given change of state.