The McCabe-Thiele Method Applied to Single Section Contactors

The McCabe-Thiele method (i.e., the stage-to-stage method) described previously for a two-section staged distillation column (i.e., where the feed is introduced in the middle of the staged cascade) can also be applied to a single section separator where the feed is introduced at one end of the staged cascade.  Examples of this type of process include gas absorption (where a solute in a gas feed is to be removed by contacting with an absorbing liquid separating agent), stripping (where a solute in a liquid feed is to be removed by contacting with a stripping gas separating agent), and liquid extraction (where a solute in a liquid mixture is to be removed by contacting with a liquid solvent separating agent).

For a dilute solute, the counter-current flows in the processes just mentioned will be constant, so that the operating line (i.e., a line consisting of points defined by the composition of streams passing each other) will again be a straight line on a y-x diagram.  The  McCabe-Thiele graphical construction is therefore performed exactly as described previously. 

Knowing the following five basic priciples will facilitate McCabe-Thiele calculations for absorbers, strippers, and extractors:

1.  The location of the equilibrium and operating lines on a y-x diagram are the same for staged or continous (i.e., packed) columns.

2.  The equilibrium line on a y-x diagram is always between the operating line and the axis for the phase the receives the solute.  Thus for a stripper the operating line is below the equilibrium line and for an absorber the opposite applies.

3.  The minimum flow of separating agent corresponds to an infinite number of plates (for a staged column) or an infinite height (for a packed column) and to a "pinch" at the feed end of the column.

4.  Countercurrent flow is more efficient than cocurrent flow since in the former an inlet stream can come to equilibrium with an outlet stream.  For the latter, it is only possible for outlet streams to come to equilibrium.

5.  Straight operating lines are convenient because they can be located by two points.  Often, even if the solute is not dilute, appropriate flow bases and composition variables can be chosed so that the operating line is still straight.

The Kremers-Sauders-Brown Equation

For straight operating and equilibrium lines on a y-x diagram, there is an analytical solution to the McCabe Thiele graphical method, termed the Kremser Sauders Brown (KSB) equation.  In particular, for an absorber the following form of the KSB equation is convenient to use:

            (y_out - y_out*)/(y_in - y_out*) = (1 - L/(K V))/(1 - (L/(K V))^(N+1))

while for a stripper the following form is convenient to use:

           (x_out - x_out*)/(x_in - x_out*) = (1 - KV/L)/(1 - (KV/L)^(N+1))

Selecting the Optimal Conditions

Generally, the design that minimizes the combination of operating costs (i.e., the consumption of separating agent) and the capital costs (i.e., the number of plates) occurs when KV/L is chosen to be approximately 1.25 for the case of a stripper, and when L/(K V) is chosen to be approximately 1.25 for an absorber. For the case of a stripper, an error tolerant design results if the number of plates is chosen based on KV/L = 1.25 while the diameter is chosen based the value of V (the flowrate of the stripping gas) according to KV/L = 2 so that the column diameter is oversized to some extent. Similarly, for an adsorber an error tolerant design results if the number of plates is determined using L/(KV) = 1.25 while the diameter is based on L/(KV) = 2.

KSB Equation for a Two-Section Absorber, Stripper, or Extractor

The McCabe Thiele analysis and the Kremser Sauders Brown equation can also be applied to a two section stripper-absorber or extractor which has a feed stream on an intermediate plate in the contactor.  Such a process has two separating agents (e.g., two different and immiscible solvents for the case of an extractor) fed at opposite ends of the contactor.  Processes of this type are useful for separating two components present in the feed stream so that each component exits the contactor in different product streams.  To properly design such a process to fractionate components A and B present in a feed stream, it is necessary to select L/V (i..e, the flow rates of countercurrent flowing streams) so that  K_B > L/V > K_A , where K_i denotes the equilibrium ratio of component i.  Although a graphical method can be used to solve problems of this type using a stage to stage calculation, it is generally easier to use the multisection version of the Kremser Sauders Brown equation.