Minimum Number of Stages, Total Reflux: It is very useful to get an idea of minimum number of stages required to achieve a desired separation. This can be achieved by using the diagonal line as the operating line. The top- and bottom-operating lines merge when the column is operating under total reflux conditions. No product is withdrawn.

 

We can draw the number of steps as follows:

C                      Draw a vertical line from point (xW, xW) till it touches the equilibrium curve.

C                      Draw a horizontal line till it touches the diagonal line.

C                      Repeat the steps (1) and (2) till we reach point (xD, xD).

C                      The number of triangles represents the number of stages.

 

We may not get a whole number.

 

If relative volatility is fairly constant, then minimum number of stages can be calculated as

Where the subscript m, W, and D stand for minimum, waste, and distillate, respectively. And αWD is average relative volatility between the residue- and distillate-sections. The number of stages computed in this way includes 1/2 stage for condenser and 1/2 reboiler.

 

The minimum number of stages above the feed point can also be calculated by replacing xW with xF in the above equation as follows:


Where the subscript m, F, and D stand for minimum, feed, and distillate, respectively. And αFD is average relative volatility between the feed- and distillate-sections.

 

A ratio of NmWD/NmFD gives a measure of the percentage of rectification section above the feed location.

 

Theoretical Number of Stages:

 

If the reflux ratio at which the distillation column is to operate is known, then we can determine the number of stages required to achieve the desired separation using the following steps.

 

C                      Draw the equilibrium curve from the data.

C                      Determine the value of the intercept, I = xD/(R+1).

C                      Locate point (1) having coordinates (0, I).

C                      Locate point (2) having coordinates (xD, xD).

C                      Draw the top-operating line (TOL) by connecting points (1) and (2).

C                      Locate point (3) having coordinates (xF, xF).

C                      Locate point (4) having coordinates (xF/q, 0).

C                      Draw q-line by connecting points (4) and (3).

C                      Extend the q-line 4-3 to intersect the rectification line 1-2 at point (5).

C                      Locate point (6) having coordinates (xW, xW).

C                      Draw bottom-operating line (BOL) by connecting points (6) and (5).

 

Now we can draw the number of steps as follows:

C                      Draw a vertical line from point 6 till it touches the equilibrium curve.

C                      Draw a horizontal line till it touches the BOL.

C                      Repeat the steps (1) and (2) till we cross point (5).

C                      Draw a vertical line from the point obtained from previous step till touches the equilibrium line.

C                      Draw a horizontal line till it touches the TOL.

C                      Repeat the steps (4) and (5) till we cross point (2).

C                      The number of triangles represents the number of stages.

C                      Do not round off the number of triangles.

 

We can avoid drawing the number of steps, and find the number of stages directly by using a graphical presentation of Molokanov equation.


Z is plotted against ψ, where

R is the reflux ratio and subscript m represents the minimum.

 

Example 6.10: Calculate the number of theoretical stages when feed to a column contains a binary mixture with a composition of 36 % more-volatile component on a molar basis. Distillate withdrawn is 91.4 % rich on a molar basis. The residue is 0.565 % impure. The column is operated at a reflux ratio of 1.5 times the minimum reflux ratio. Use the following information:

Feed quality, q = 1.04

Relative volatility, α = 4.12

Minimum reflux ratio, Rm = 0.606

Minimum Number of Stages, Nm = 5.33

 

Solution: R = 1.5 0.606 = 0.909.

 

Using the technique of Molokanov, we can find ψ equal to 0.159.

This gives a value of Ζ to be 0.497.

Finally, the number of stages, N, are found to be 11.6.

 

Actual Number of Stages: If the efficiency of the stages is given, then the actual number of stages can be found by dividing the total number of stages by the efficiency of each stage. However, the condenser and reboiler are not equilibrium stages. We must reduce the number of stages by one and then correct for the efficiency of the stages.

 

Liquid and Vapor Quantities Inside the Tower: The liquid flow rates in the rectification- and stripping- section are given by the following relationships.


Where subscript n and m represent rectification- and stripping- section respectively. Vapor flow rates in the rectification- and stripping-sections are given by the following equations.

Condenser Load: If the condenser utilized in a distillation column to condense the vapors is a total condenser, then the composition of the condensate, xD, is the same as the composition of vapors, yD, entering the condenser. Heat removed in the condenser can be calculated by subtracting the enthalpy of the condensate from the enthalpy of the vapors. And the enthalpy of the condensate is comprised of enthalpy of the distillate and reflux. Enthalpy of the vapors can be calculated by adding sensible- and latent- heats. Sensible heat of the vapors, HGL, is given by the following equation:

 

 

Where Td is the dew point and To is the reference temperature. The latent heat of vaporization of the vapors, H, is given by the following equation

 

 

The enthalpy of the condensate, HDL, can be found by knowing the bubble point of mixture having composition of the distillate and is given by the following equation.

 

 

The condenser load, Qc, can be found to be