# Summary

A McCabe-Thiele plot is a simplified tool to assist in understanding distillation. It is a method for calculating the number of theoretical trays required for the distillation of a binary mixture. This article describes how to apply the McCabe-Thiele method.

# Definitions

: | molar flow rate of top product | |

: | molar flow rate of reflux | |

: | Slope of line | |

: | mol fraction liquid in feed | |

: | feed boiling temperature | |

: | feed dew point temperature | |

: | feed temperature |

# Introduction

McCabe-Thiele plots provide a simple, graphical tool for the analysis of binary distillations, specifically the determination of the number of trays required. Many real world applications are too complex for the McCabe-Thiele method however it provides a great tool for learning the basic thermodynamics of tray distillation, as well as understanding the impact of reflux rate, feed composition, product composition and vapor-liquid equilibrium on distillation column design.

To understand the terminology and discussing distillation in general please see our article on Distillation Basics.

# Assumptions

The McCabe-Thiele method is a very simplistic model of distillation and subject to the following assumptions:

- The mixture is binary i.e. there are only two components.
- The heat of vaporization of the two components are equal (When one mole of the heavier component is vaporised, one mole of the lighter component is condensed).
- Other thermodynamics effects are negligible, e.g. heat of dissolution is negligible.
- 100% tray efficiency. Trays calculated using the McCabe-Thiele method are called "theoretical trays". In practice fluids do not reach equilibrium on each tray, and so the "tray efficiency" is used to determine the number of actual trays required to make a given separation.

# The McCabe-Thiele Method

In this section we describe the McCabe-Thiele method step by step, however prior to starting several pieces of information are required:

- Vapour-liquid equilibrium data for the feed components.
- Feed composition.
- Dew point, boiling point and actual temperature of the feed (If the feed is sub-cooled or super-heated).
- Target product quality (composition) of the top and bottom products.

## Step 1: Establish the Plot

We begin with a basic x-y plot. On the x-axis is mole fraction of the lighter component in the liquid phase. On the y-axis is the mole fraction of the lighter component in the vapor phase.

## Step 2: Add 45 Degree line

Draw a 45° line tuning from the origin to the top right corner, (0,0) to (1,1).

## Step 3: Add Equilibrium Data

Plot the vapor-liquid equilibrium or VLE curve of the binary mixture.

## Step 4: Choose Product Compositions

Mark the composition of the feed, top and bottom products on the x-axis. Then plot a vertical line from each of these points to the 45° line as shown below.

## Step 5: Add the Feed Condition Line

The design of the distillation column is effected by the vapor fraction and temperature of the feed. This is shown on a McCabe-Thiele plot using the feed condition line. The feed condition line passes through the intersection point of the 45° line and the feed composition line, and its gradient can be calculated as a function of the properties of the feed.

Where the feed is 50 mol% liquid the line will be perpendicular to the 45° line. Where the feed is saturated liquid the line is vertical. Where the feed is a saturated vapor the line is horizontal.

For feeds lying between a saturated vapor and a saturated liquid we can use a simple formula to determine the slope of the feed condition line:

Where q is the mole fraction of liquid in the feed.

For feeds lying outside this range (sub-cooled or super-heated) we first determine the extent using the formula below and then feed the result (q) into the previous formula to determine the slope of the feed condition line.

For a sub-cooled liquid the slope of the line is between 45° and vertical. For a super-heated vapour the slope of the line is between 45° and horizontal.

Once slope of the line feed condition line is calculated it can be drawn on the McCabe-Thiele plot from the point where the feed line meets the 45° line until it intersects the VLE-curve.

## Step 6: The Rectifying Section

When vapor leaves the top of the column it is cooled and liquefied. Some of this stream is taken away as the top product while the rest is returned to the column as reflux. The reflux liquid travels down the column in the opposite direction to the rising vapor. The liquid "swaps" heavy components in the vapor for light components in the liquid, concentrating the light component in the vapor.

The rectifying section operating line describes the amount of liquid sent back down the rectifying section as reflux. Due to assumptions of the McCabe-Thiele method, the operating line estimates how much the composition can change at each tray, increasing reflux results in bigger steps and thus less trays.

**Step 6.1: The Slope of the Operating Line**

To determine the slope of the operating line we need to know the molar flow rate of both the top product (D) and the reflux returned to the column (L).

We use the reflux ratio as slope of our operating line:

**Step 6.2: Drawing the Rectifying Section Operating Line**

Using the reflux ratio as the gradient, the rectifying section operating line can be drawn as a straight line starting at the intersection of the vertical top product line with 45° line and ending at the feed condition line as shown below.

## Step 7: The Stripping Section

Similarly to the rectifying section, the operating line of the stripping section represents the gas travelling back up the column after leaving the reboiler. This hot gas vaporises light components in the liquid travelling down the column in exchange for condensing heavy components from the gas.

To draw the operating line for the stripping section we start at the point where the vertical bottom product line meets the 45° line and draw a line to the point where the rectifying section operating line mets the feed condition line as shown below.

## Step 8: Stepping Down the Plot

Once the rectifying and stripping section lines are completed the theoretical trays are drawn on the plot. Starting from the point where the top product line meets the 45° a horizontal line is drawn until it intersects the VLE curve. A vertical line is then drawn down from this point until it meets one of the two operating lines. This process is repeated until the last vertical line falls to the left of the bottom product line.

## Step 9: Counting the Trays

The number of theoretical trays can now be determined by counting the number of times the horizontal 'steps' touch the VLE curve (include the line that passes the bottoms product line). These trays are counted starting at the top right and moving to the bottom left.

# Effect of Varying the Parameters

## Reflux Rate

Reflux rate has a significant impact on the performance of a distillation column. With a higher reflux rate, the number of trays required decreases. Note that this increase in efficiency doesn't come for free, as in practice you will need a wider column to handle all of this liquid, as well as a larger reboiler to make enough vapour to send back through the liquid.

## Bottom and Top Composition

Pushing for a finer separation (i.e. increasing the light fraction in the top product or decreasing the light fraction in the bottom product) drives us further and further into the narrow parts of the gap between VLE curve and the 45° line. This results in rapid increase in the number of trays for ever smaller increases in purity.

## Relative Volatility

The difficulty of a separation is determined by the similarity of the two components. Separating two very different components, for example ethane (C_{2}H_{6}) and decane (C_{10}H_{22}) is easy. Separating two very similar components, for example ethane (C_{2}H_{6}) and ethylene (C_{2}H_{4}) is difficult. Difficult separations require taller columns with more trays.

## VLE Curves and Azeotropes

Some mixtures form an azeotrope or constant boiling mixture. Azeotropes have a composition at which both components have the same boiling point and therefore contacting the vapor and liquid phases will not cause phase changes of either component.

For example when setting up a VLE Curve for propanol and water we can see that something strange is happening at around x = 0.65. Here the VLE curve crosses the 45° line indicating the mixture is an azeotrope and cannot be further separated by simple distillation alone.