**Binary Distillation using McCabe-Thiele Method**

### Basics of Binary Distillation

Distillation is a separation method of two chemical components by utilizing the difference in boiling point. The goal of this calculation is to obtain theoretical stage by determining mole fraction of the top product and the bottom product. The assumptions modelled in this method are as follows:

*Molar geat of vaporization of the feed components are equeal*, which means that there are no reaction takes place,*Molar vapor-liquid equilibrium*, which means that for every mole of liquid vaporized, a mole of vapor is condensed, and*Ideal gas law is applied*, which means that heat effects such as heat of solution are negligible.

### McCabe-Thiele Diagram

There are several components in a McCabe-Thiele Diagram:

##### x-y diagram

The x-y diagram depicts vapor-liquid equilibrium data of the binary components, which any points on the graph shows the amount of liquid that is equilibrium with vapor at certain temperature. The diagram will help to determine the relative volatility (\(\alpha\)) to predict the amount of separation possible. When \(\alpha < 1.05\), the separation process will take more trays as the components are difficult to seperate using this method. Relative volatility is as follows: \[\alpha= {{y_i/x_i}\over{y_j/x_j}}\] Where \(y\) is the vapor fraction, \(x\) is the liquid fraction, and \(i, j\) is the chemical components. In the binary system, the above equation can be expressed as \[\alpha= {{y_i/x_i}\over{(1-y_i)/(1-x_i)}}\]##### Operating line in rectifying section

Operating line is a simple line equation to determine amount of liquid per amount of vapour (\( \mathcal{L} / \mathcal{V}\)) in the distillation process. If it is assumed that the heat of mixing is negligible and that the difference in the molar latent heats of vaporization of the binary system also differs by a negligible amount, then the observed relationship between any two passing streams is simplified to streams is simplified to: \[y = {{\mathcal{L}}\over{\mathcal{V}}} x + {{x_D D}\over{\mathcal{V}}} \] where \(y\) is the mole fraction of the more volatile component of the vapor, \(x\) is the mole fraction of more volatile component in the liquid, \(x_D\) is the mole fraction of the more volatile component in the distillate, \(D\) is the distilate flow rate (mol/time), \(\mathcal{L}\) is the total flow rate of the liquid stream in the rectifying section (mol/time), and \(\mathcal{V}\) is the total flow rate of the vapor stream in the rectifying section (mol/time).This is known as the operating line for the rectifying section, or the upper operating line, in short. Remembering that the x-y diagram is a plot of vapor (\(\mathcal{V}\)) vs. liquid (\(\mathcal{L}\)), it can be seen that the operating line is a simple \(y=mx+b\) equation. Therefore, with the point (\(x_D,y_1\)) and the slope (\(\mathcal{L}\)/\(\mathcal{V}\)), this operating line can be plotted.

##### Operating line in stripping section

With the similar assumptions with the operating line in rectifying section, the observed relationship between any two passing streams is simplified to: \[y = {{\overline{\mathcal{L}}}\over{\overline{\mathcal{V}}}} x + {{x_B B}\over{\overline{\mathcal{V}}}} \] where \(y\) is the mole fraction of the more volatile component of the vapor, \(x\) is the mole fraction of more volatile component in the liquid, \(x_B\) is the mole fraction of the more volatile component in the bottom, \(B\) is the bottom flow rate (mol/time), \(\overline{\mathcal{L}}\) is the total flow rate of the liquid stream in the stripping section (mol/time), and \(\overline{\mathcal{V}}\) is the total flow rate of the vapor stream in the stripping section (mol/time).This is known as the operating line for the stripping section, or the lower operating line, in short. Again, it can be seen that the operating line is a simple \(y=mx+b\) equation.

##### Feed Line

Feed line, also known as q-line represents the intersections of the upper and lower operating lines. The stage that crosses the q-line is the optimum feed plate. The q-line is that drawn between the intersection of the operating lines, and where the feed composition lies on the diagonal line.q = 0 (saturated vapour)

q = 1 (saturated liquid)

0 < q < 1 (mix of liquid and vapour)

q > 1 (subcooled liquid)

q < 0 (superheated vapour)

The q-lines for the various feed conditions are shown in the diagram on the top. If the slope of the q-line is not directly known,
look at the feed plate enthalpic conditions.

### References

- McCabe, W. L., Harriott, P., & Smith, J. C. (2004). Unit Operations of Chemical Engineering (7th ed.). McGraw-Hill.
- Geankoplis, C. J., Hersel, A. A., & Lepek, D. H. (2018). Transport Processes and Separation Process Principles (5th ed.). Prentice Hall.

**McCabe-Thiele Binary Distillation Calculator**

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