### Introduction

Hello and thank you for viewing this instructable on the basics of column distillation! In this introduction, I would like to go over two things, the goals of this instructable as well as cover background information.

__Goals__

The goals of this instructable are as follows:

- Introduce the concepts pertinent to why the analysis can be performed
- Show how and why the McCabe Thiele analysis is performed

Keeping these goals in mind, let's jump right in!

__Background Information__

For the sake of brevity in this portion of the instructable, the why's, how's, when's, and where's of a distillation column will not be covered. Only background information that pertains to the analysis that will be performed will be covered. If you need more information on how a column works please view this link (Insert Link here).

The McCabe-Thiele Analysis is a graphical analysis procedure used when a new distillation column is being installed (It was developed by two men named McCabe and Thiele). It is used to determine the number of distillation stages needed within the column to get the desired purity of product. In order to perform the analysis, overall material balances are performed on the streams of materials going into and out of the column. This is to understand how the material we are separating reacts based on the inlet conditions.

The aforementioned analysis procedure is only valid if the three following assumption are true: the column is at constant molal overflow (CMO), the separation is adiabatic (no heat is lost in separation in the stages), and each stage of the column is operating at Vapor-Liquid Equilibrium. Each one of these concepts will now be explained.

Constant molal overflow can be described as follows: the same material leaving a stage is equal to the amount entering a stage within the column. This doesn't mean the output streams from the column overall are the same but means we do not need to focus internally within the column when using material balances. Only the overall column needs to be looked at.

Adiabatic means no heat is lost during the separation process. If heat were lost during each stages, these calculations would get exceedingly complex and a stage by stage analysis would be the only way to analyze the column.

Vapor-liquid Equilibrium means that the amount of Vapor and liquid in each stage is not changing with time, if it were then the first assumption of CMO would not be valid.

That rounds out the assumptions and the information needed to begin taking a look at the analysis itself!

## Step 1: Overall Balance and Equilibrium Data

As described in the introduction, the first step of a McCabe Thiele analysis is determine two things. Get equilibrium data (always from a book, data, or experimental data) and do an overall material balance on the column. There are various ways to do this however the problem will specify the composition of the what comes off the top and the bottom. The following must be found using the equations in the picture:

- D
- B

A short example of an overall balance is performed in the next step!

## Step 2: An Example of the Overall Balances Around a Column

This example illustrates how a balance is done using information normally given in a problem statement.

## Step 3: Plotting Data and Feed Line

As mentioned in the introduction, to do a McCabe Thiele analysis a graph must first be generated. The Equilibrium Data, a top operating line, a bottom operating line, and a feed line must be plotted to generate the full graph. What is a top operating line, bottom operating line, and feed line? These lines represent how the material is flowing throughout the column between the distillation stages. The feed is always inserted in the middle of column then there is a flow of vapor up the column (top operating line) and a flow of liquid down the column (bottom operating line).

The equilibrium data must be found through experimental data usually found in a book. Then the feed line can be found by analyzing the feed itself. There are 6 conditions to be analyzed in terms of the feed:

- If the feed is a Saturated Liquid, the slope of the feed line is vertical
- If the feed is a Saturated Vapor, the slope of the feed line is horizontal
- If the feed is a Superheated Vapor, the slope of the feed line is less than 1
- If the feed is a Supercooled Liquid, the slope of the feed line is greater than 1
- If the feed is a liquid, you need to find the quality and find the slope of the feed line from the quality
- If the feed is a vapor, you need to do the same as the liquid case above

For the Superheated and Supercooled cases, the equations in the photo can be used to determine the quality and the slope (note, these can be used to address all cases).

Once all of this is done, plot the Equilibrium data and the feed line in order to start assembling the graph needed to perfom the McCabe Thiele analysis.

## Step 4: Top Operating Line

The next thing normally found in the problem is the top operating line and are many ways to do this, however, the most simple method involves using the reflux ratio of the column. The Reflux Ratio is simply the ratio of the material exiting the top of column and the material that is cooled in a condenser that re-enters the column. There is a minimum reflux ratio the occurs in every column and can be found using the following equation which involves the y intercept of the top operating line (See photo).

This y intercept for the minimum reflux ratio always intersects where the feed line and equilibrium data meet, therefore the y intercept is not hard to find and xD is provided in the problem statement.

After the minimum reflux ratio is found a modifying factor is given by the problem adjust it to the conditions of the column (See photo above again). Then the reflux ratio of column is found and the same equation used to find the minimum reflux ratio can be used to find the y-intercept of the top operating line. Then using the y-intercept and the y = x point of the top operating line (the distillate composition xD defines this point), the full top operating line can be plotted.

## Step 5: Bottom Operating Line

The bottom operating line is much simpler than the top operating line due to the amount of information we have already found. After plotting the top operating line, you see that it intersects with the feed line and that point is a point on the bottom operating line. So using the point where the top operating line and the feed line intersect along with the bottoms composition point (This is the same as the distillate composition point but using the bottoms composition given), you can construct the bottom operating line. Please see the photo above.

## Step 6: Final Step, the Graphical Analysis

Now that we have the bottom operating line, top operating line, and equilibrium data all on the same graph, a McCabe-Thiele graphical analysis can be performed. Why? Since the top and the bottom operating lines model the upper and lower portions of the column and the equilibrium data indicates the what will actually happen at our current environmental conditions, we have in essence modeled the whole column in an ideal situation.

The analysis itself is simple. You will need a straight edged ruler. Now follow these steps below:

- Draw a straight line from the distillate composition point to the equilibrium data
- Draw a line straight down from the where your first straight line intersected the equilibrium data to the top operating line
- Repeat steps 1 and 2 until you pass the point where the top and bottom operating lines meet
- After passing the point where the top and bottom operating lines meet, the lines being drawn down must go to the bottom operating line rather than the top
- Continue until you reach the bottoms composition point

At the final line, you may find the line goes to far, this indicate a partial stage. Draw a straight line down to the bottoms composition point. Now measure the total length of the horizontal line (equilibrium point to bottom operating line). Then measure the distance between the intersection of the line going down to the bottoms composition point and the bottom operating line (let's call it partial length). Finally, divide the partial length by the total you measured first. That decimal number indicates the how much of a stage you would need. *Note: In a real life situation, a partial stage could not be used, but this is an ideal scenario so in this case it is allowed.*

Finally, count the number of steps developed while drawing lines. This indicates the number of stages needed in your distillation column. Congratuations, you completed a McCabe-Thiele Graphical Analysis!

## Discussions