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Distillation - On the McCabe-Thiele method

 The method of McCabe-Thiele is one of the most used techniques in distillation. First, because it was published in 1925 and second due to its presentation in textbooks. In fact Warren L. McCabe has published a well known chemical engineering textbook (still in use in 2025!). You may read the paper of W. L. McCabe and E. W. Thiele here: [Ind. Eng. Chem. 1925 17 (6)].

However, there are more advanced techniques which are property of companies and it will take time for textbooks to present it. Therefore, this post is devoted to the graphical method of McCabe-Thiele.

 

CONTENTS

 Some basic equations on the distillation columns

 Lines on the x-y diagram

 Stages of the distillation column

Some basic equations on distillation columns

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On the diagram of mole fractionof the most volatile component in the liquid x versus the mole fraction of the same component in the vapor phase y, the rectifying and the stripping sections should be distinguished. First, from the rectifying section isolated in the figure below a mass balance produce the following equations.

Fig. 01 Variables in a rectifying section on a distillation column.


$G_{n+1}=D+L_n$        Eq. (01)

$G_{n+1}\,y_{n+1}=D\,z_D+L_n\,x_n$        Eq. (02)

Eq (01) is the global mass balance while Eq. (02) a mass balance on the most volatile component. Combination of Eqs. (01-02) and solving for $y_{n+1}$ gives,

$y_{n+1}= \dfrac{D}{G_{n+1}}z_D + \dfrac{L_n}{G_{n+1}}x_n$        Eq. (03)

At the heart of the McCabe-Thiele an assumption is made: there is constant overflow at each stage. This means that the the flow rate of liquid leaving (down) a stage and the flow rate of vapor leaving (up) a stage are the same for all stages. In other words,

$L_0=L_1=L_2=\dots=L_n=L$

$V_1=V_2=V_3=\dots=V_{n+1}=V$        Eq. (04)

Equations (04) reduce Eq. (03) to,

$y_{n+1}= \dfrac{D}{G}z_D + \dfrac{L}{G}x_n$        Eq. (05)

Following this idea, the global mass balance can be rewtitten as,

$G=L+D$        Eq. (06)

If the same exercise is performed for the stripping section the an analogous equation to Eq. (05) come up.

Fig. 02 Variables in a stripping section on a distillation column.


$y_{m+1}=\dfrac{\bar{L}}{\bar{G}}x_m - \dfrac{W}{\bar{G}}x_W$        Eq. (07)

Also, the global mass balance for the stripping section could be expressed as,

$\bar{G}=\bar{L}-W$        Eq. (08)

Now, a global mass balance on the whole distillation column is expressed as

$F=D+W$        Eq. (09)

Fig. 03 A sketch for a global balance on a distillation column.


Again, we can look for an expression related to the stages inside the column. This can be done combining the new found Eq. (09) with Eqs. (06,08) dealing with mass balances on each individual sections.

$F=G-L+\bar{L}-\bar{G}$

$\Rightarrow G-\bar{G}=L-\bar{L}+F$        Eq. (09)

where $W$ and $D$ were eliminated. Next, if the mass balance on the component of interest is made over the whole column, the equation is,

$Wx_W+Dz_D=Fz_F$        Eq. (10)

Finally, we may write the reflux equations for the rectifying and stripping sections. These are,

$R=\dfrac{L_0}{D}$        Eq. (11)

and

$S=\dfrac{\bar{G}}{W}$        Eq. (12)

respectively.

Lines on the x-y diagram

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Let us now translate the equations obtained from the different mass balances obtained in the previous section into the x-y diagram (see Fig. 04). The equilibrium curve y versus x is made of all points $(x_n,\,y_n)$.

Fig. 04 x-y diagram.


Also, the diagonal line on the x-y diagram in Fig. 04 represents the set of all points in which $y_{n+1}=x_n$. At this moment, a question arises: how can we represent the composition at the rectifying and stripping sections? This question is answer using the mass balances for each section.

In the case of constant molar overflow (see Eqs. (04), for example) the compositions for the rectifying and stripping sections can be presented as straight lines called (in the literature) as: operating lines (a more convenient name could be just: mass balance lines).

The mass balance (on the most volatile component) lines at the rectifying and stripping sections ar given precisely by Eqs. (05,07), respectively. You should recall that in Eqs. (05,07), the mole fractions in the feed $F$ and distillate $D$ are constant. In this way two diagonals appear in the x-y diagram (see Fig. 05 below).

Fig. 05 Component balance lines in the x-y diagram.


The two lines defined by Eqs. (05,07) cross at a point called $q$ which is easily found by the simultaneous solution of the equations. Also, the rectifying section component mass balance line cuts the 45$^\circ$ diagonal at $x=z_D$ while the stripping section component mass balance line cuts it at $x=x_W$, as shown in the sketch above.

The intersection of the two lines is at the point $(x_i,y_i)$. Then from Eqs. (05,07) it follows,

$x_i=-\dfrac{\dfrac{W}{\bar{G}}x_W+\dfrac{D}{G}z_D}{\dfrac{L}{G}-\dfrac{\bar{W}}{\bar{G}}}$        Eq. (13)

and

$y_i=\dfrac{D}{G}z_D+\dfrac{L}{G}x_i$        Eq. (14)

Stages of the distillation column

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The procedure to determine the number of plates in a distillation column by the McCabe-Thiele method is quite simple.

  • Start from the top of the column and proceed downwards.
  • Taking the condenser as the end of the distillation, the concentration at the top would be $(x_D,y_D)$. This point is located right on the 45$^\circ$ diagonal line. Of course, this composition correponds to the distillate.
  • Since in the plate, both liquid and vapor, are in equilibrium the composition we are looking for can be found by keeping $y_D$ constant and moving horizontally in the direction of the equilibrium curve. This means that in the plate #1 the composition of the vapor leaving is $y_D$ while that of the liquid is the reacently found data.
  • Moving vertically downwards to the diagonal line gives you the plate #2 but only the composition of the vapor. 
  • The composition of the liquid in plate #2 is found by moving horizontally to the equilibrium curve.
  • Repeat the process until the compositions at the plate are below that of the bottom stream $W$.
  • The number of times the equilibrium curve is touched says the number of stages or theoretical plates.

Finally, the number of stages above the intersection Eqs. (13-14) is the number of plates in the rectifying section. The same can be said for the stripping section.

This is the end of the post. I hope you find it useful.

Any question? Write in the comments and I shall try to help.

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Ildebrando.

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