The McCabe-Thiele Method
This method uses the equilibrium curve diagram to determine the number of theoretical stages (trays) required to achieve a desired degree of separation. It is a simplified method of analysis making use of several assumptions, but nonetheless a very useful tool for the understanding of distillation operation. Click here for more information on the analysis of this method.
The VLE data must be available at the operating pressure of the column.
Information required are the feed condition (temperature, composition), distillate and bottom compositions; and the reflux ratio, which, as we seen earlier, is defined as the ratio of reflux liquid over the distillate product. This is shown in the Figure below.
For example, a column is to be designed for the separation of a binary mixture. The feed has a concentration of xF (mole fraction) of the more volatile component, and a distillate having a concentration of xD of the more volatile component and a bottoms having a concentration of xB is desired.
In its essence, the method involves the plotting on the equilibrium diagram 3 straight lines: the rectifying section operating line (ROL), the feed line (also known as the q-line) and the stripping section operating line (SOL).
Each of these lines passes through the points representing the mole fractions of the more volatile component in the distillate, bottoms and feed (xD, xB and xF) respectively. These lines represent the relationship between the concentrations in the vapour phase (y) and the liquid phase (x).
The number of theoretical stages required for a given separation is then the number of triangles that can be drawn between these operating lines and the equilibrium curve. The last triangle on the diagram represents the reboiler.
To obtain the number of theoretical trays using the McCabe-Thiele Method, we shall used the "Parts-Whole Relationship": analysis is first carried out by partitioning the column into 3 sections: rectifying, feed and stripping sections as shown in left Figure below. These sections are then represented on the equilibrium curve for the binary mixture in question and re-combined to make a complete design, as shown in the right Figure
In the simplest case, the McCabe-Thiele Method to determine the number of theoretical stages follows the steps below:
When R is unknown (but instead specified as a multiple of the minimum rate), we will first determine the q-line.
Usually the SOL is the last line to draw, after both ROL and q-line are drawn. Fixing the ROL and the q-line automatically fixes the SOL.
On the completed design (equilibrium diagram): The number of triangles drawn = Number of theoretical trays + 1 Reboiler (last triangle).
As an example, see the Figure below. The number of theoretical trays = 6 + 1 Reboiler
The feed plate location can also be determined. In the example above, it is Tray #3.
The following flowchart - FChart1 - can be used to find the number of theoretical trays required for a given separation.
It covers various scenarios other than the general 5-step approach as shown above, such as unknown R, unknown q, etc. For more information of the importance of reflux ratio, click here.
For analysis using partial condenser, click here.