Separations analysis fundamentals - Principles of Chemical Separations with Environmental Applications - page 66

when our control volume contains N stages, and

y N −

V x N − 1

L
V x N +

L

y N + 1 = (3.39)

for the N th stage only.
Note that each balance describes the relationship between passing streams on each
side of a stage ( y N + 1 , x N ) or sequence of stages. This is in contrast to the equilibrium
relationship that describes two outlet streams from the same stage ( y N , x N ).
If the ratio L V remains constant, then the above balances represent linear equations.

Values for the concentrations on one end of the cascade would normally be known and
the number of stages required for a given separation would be the variable of interest. As
demonstrated above, a mass balance must be applied at each stage in combination with the
equilibrium relationship to obtain the outlet concentrations for that stage. One method to
perform this task is graphically. This method is referred to as the McCabe-Thiele method
and the approach is commonly called “stepping off” stages. In Figure 3.30 the curved line
is the equilibrium line and the straight line is our mass balance (operating line). Note that
one is not limited to assuming a linear equilibrium relationship. The composition of one
set of the passing streams for Stage 1 is ( x 0 , y 1 ). Starting at this point, one can move to the
equilibrium line to obtain point ( x 1 , y 1 ), the composition of the exit streams from Stage 1.
One then moves back to the operating line to point ( x 1 , y 2 ). This is the composition of
the second set of passing streams for Stage 1. This “step” on the graph indicates one
equilibrium stage in the cascade. One can continue in the same fashion and count the
number of steps to obtain a given separation for the cascade.
The operating line can be above or below the equilibrium line but the stepping-off stages
procedure remains the same. Start on the operating line with the passing streams on one
side of the stage, go to the equilibrium line for the composition of the exit streams from
that stage, then back to the operating line for the passing streams on the opposite side of
the stage.
An equation can also be obtained to calculate the number of stages required. To simplify
the analysis, some assumptions are made which reduce the complexity of the calculations
and clarify the separation process:
(a) the flowrates of each stream are constant;
(b) the equilibrium relationship is a simple linear one (i.e., Henry's Law).
This situation corresponds to the transfer of a dilute solute between phases. Many en-
vironmental applications of separations involving extraction, distillation, and adsorption
fall into this category so it is not a hypothetical example.
The normal method of cascading the equilibrium stages is using countercurrent flow.
With two streams, they would enter the cascade at opposite ends of the cascade.
The most general form of a linear equilibrium relationship is

/

y = mx + b (3.40)