McCabe Thiele Method For Distillation
It is a graphical procedure of obtaining theoretical plates.
McCabe Thiele Method Assumptions
The assumptions made in this method are :
1. Constant molal heat of vaporisation.
2. No heat losses.
3. No heat of mixing.
The above cited assumptions leads to a concept of constant molal vapour flow and constant molal liquid flow in any section of the column (i.e., the molal flow rates of vapour and liquid are constant in any section of the column). In other words, for one mole of liquid vaporised, one mole of vapour is condensed in any section (rectifying or stripping).
Thus,
Vn = Vn+1, Ln = Ln+1 … etc.
The subscripts n, n + 1, m, m + 1 … etc. may be dropped and the operating lines then plotted on the equilibrium diagram are straight.
Therefore, operating line for the rectifying section becomes
y= L/(L+D)x + D.xd/(L+D) .....(1)
With R = L/D, Equation (1) becomes
y =R/(R + 1)x +xD/(R + 1) ..... (2)
On the xy diagram, it is a straight line with a slope equal to R/(R + 1) and an intercept equal to xD/(R + 1).
The point of intersection of the operating line for rectifying section and the diagonal y =x is obtained as
y=R/(R + 1)x +xD/(R + 1)
Substituting y = x
∴ x =R/(R + 1) x +xD/(R + 1)
(R + 1) x = R·x + xD
∴ x = xD
and y = xD
Thus, the operating line of the rectifying section is to be drawn on the equilibrium diagram through point (xD, xD) on the diagonal and with a slope equal to R/(R + 1) or an intercept on the y-axis equal to xD/(R + 1).
The operating line of the stripping section by dropping the subscripts m, m + 1 becomes
y =L'/(L' – W) x –W.xW/(L' – W) ..... (3)
On the xy diagram, i.e., on the equilibrium diagram, it is a straight line with a slope of
L'/(L' – W) and an intercept of W.xW/(L' – W).
The point of intersection of the operating line of the stripping section and the diagonal is obtained as follows:
y =L'/(L' – W) x –W.xW/(L – W)
Diagonal : y = x
∴ x = L'/(L' – W)x – W.xw/( L' – W)
∴ L'·x – W·x = L'·x – W.xW
∴ x = xW
and y = xW
Thus, the operating line of the stripping section is to be drawn through point (xW, xW) on the diagonal with a slope equal to L'/(L' – W).
Stepwise procedure for obtaining theoretical plates
1. By material balances, evaluate the terms D, W, L, etc.
2. Draw the equilibrium curve and diagonal with the help of x – y data given.
If the relative volatility is given, generate the x – y data needed using the relativity value provided.
3. Draw the operating line of rectifying section through a point (xD, xD) on the diagonal and with an intercept equal to xD/(R + 1) or DxD/(L + D) or a slope equal to R/(R + 1).
4. Draw the operating line of stripping section through a point (xW, xW) on the diagonal and with a slope of L'/(L' – W).
5. Starting from the point (xD, xD) on the diagonal, draw a horizontal line to meet the equilibrium curve at point 'a'. Drop a vertical from the point 'a' to meet the operating line at point 1.
6. Proceed in this way, that is constructing triangles (i.e., stages) between the equilibrium curve and operating line of the rectifying section, till we are above the point of intersection of the two operating lines. Once we cross this point of intersection, construct triangles (i.e., stages) between equilibrium curve and operating line of the stripping section, i.e., dropping verticals on the operating line of the stripping section.
7. Proceed in the same manner till we reach/across the point (xW, xW).
8. Count the number of triangles constructed between xD and xW. Each triangle on the x – y diagram represents a theoretical plate.
9. If the number of triangles are 'n', then 'n' represents the theoretical number of plates including the reboiler and n–1 represents the number of theoretical plates in a column. Since the vapour and liquid leaving the reboiler are in equilibrium, it is equivalent to one theoretical plate.
The number of theoretical/ideal plates in a column is equal to the number of theoretical stages minus one.
Limitations of McCabe Thiele Method for Distillation
1. It is not used when the relative volatility is less than 1.3 or greater than 5.
2. It is not used when more than 25 theoretical stages/plates are required, and
3. It is not used when the operating reflux ratio is less than 1.1 times the minimum reflux ratio.