RMP Lecture Notes

Algebraic Solution of Equilibrium Stage Problems: The Kremser Equation

Solving problems involving equilibrium stage separations requires simultaneous solution of the equilibrium and operating (component balance) expressions. Choice of a solution technique -- algebraic, graphical, or numerical -- depends on the form of the expressions.

The Kremser Equation, an "absorption factor method", provides an algebraic solution for analyzing equilibrium cascades. It cannot be used for every problem, but is convenient for several cases, notably:

dilute gas absorption (when set up on "solvent free" basis)
distillation (use for the extreme ends of a high purity separation where the curvature of the equilibrium curve is not significant)
leaching

Modeling

The equations will be developed for a countercurrent cascade of N stages. Begin by writing the steady state component balance over n-1 stages:

The equilbrium expression will be written in terms of a "K-value" (MSH develop these equations starting with a linear equilibrium expression with slope m):

The absorption factor will then be defined. It is the ratio of the local slope of the operating curve to that of the equilibrium curve. Similar expressions can be defined to serve as "stripping factors", or "extraction factors", or "wash factors", etc.

The absorption factor thus varies from stage to stage.

These three expressions (component balance, equilibrium, absorption factor) are then combined and rearranged

If the same steps were taken for a balance over n-2 and n-3 stages, the results would be:

These expressions are then "nested" into the first to obtain

This process is repeated, until the balance over 1 stage is incorporated

The balance will be written one more time, over n stages

Then the last two equations are set equal and rearranged:

If the absorbent fed is pure, x₀=0, and the second term vanishes. It is then convenient to define the "fraction NOT absorbed", the ratio of solute leaving to solute fed

$fraction not absorbed$ which can sometimes be used to compact the notation. This equation allows calculation of the recovery; but it is unlikely that anyone would have all the required absorption factors.

Group Method Approximation

The absorption factor A varies from stage to stage as the liquid and vapor flows and equilibrium shift. The "group method" approximation says that we can assume an average, "effective" value of the absorption factor that is defined to be the same for all stages. Note, though, that if both the equilibrium curve and operating curve are straight lines, no approximation is involved.

This allows algebraic simplication of the recovery fraction

$fraction not absorbed$ provided one remembers the rules for geometric series from calculus

A similar simplification can be done on the L₀x₀ term, noting that the order is one less. The full equation is thus

(Save this -- we'll come back and use it again later)

The coefficient on L₀x₀ represents the consequences of both impure absorbent and the fact that vapor flow may do some stripping of the enriched absorbent. It thus makes sense to express this quantity in terms of the stripping factor:

Beginning by setting up a common denominator, the L₀x₀ coefficient can be rewritten to obtain

so that the overall equation is

This equation is useful in solving some problems.

Operating Equation Forms

Now go back to the form I said to save, and rearrange it in the operating equation form. From here on we will assume that the flow rates L and V and the equilibrium K-value are constants. This means that both the equilibrium and operating curves will be straight lines and that the absorption and stripping factors are constants.

Define the hypothetical equilibrium vapor composition, substitute, and rearrange.

Next we need a rearranged version of the balance over n stages

(Note that y_n^* = y_n). This can be used to calculate A from known endpoint compositions

which is the same as MSH equation 20.23

We might also combine the last two equations to get

(same as MSH eq. 20.24) which can be used to determine the number of stages needed to make a separation.

References:

McCabe, W.L., J.C. Smith, and P. Harriott, Unit Operations of Chemical Engineering (6th Edition), McGraw-Hill, 2001, pp. 632-38.
Seader, J.D. and E.J. Henley, Separation Process Principles, John Wiley, 1998, pp. 242-46.

R.M. Price
Original: 3/26/97; 8/8/2002
Modified: 4/3/97,3/15/99; 8/15/2002; 2/6/2003