Mass balance on rectifying and stripping plates

This post deals with the mass balance applied to a distillation column. This concerns something called fractional distillation. As you may know, in this kind of operation, the column top section is called the rectifying section, while the bottom section is called the stripping section, provided the feed is in the middle plate.

Mass balance on the rectifying section

Or better said, on the plates of the top section. This balance applies to a binary mixture only. In Fig. 01 a general representation of this is presented. 

• All plates are numbered from the top
• $n$ is the last plate on the rectifying section. Of course, $n$ is also a counter going from 1 to the last plate.
• The referred component on the mol fractions $(x,y,z)$ is the most volatile

Fig. 01 Mass balance on the rectifying section.

In this way, a global mass balance can be easily written as,

$G_{n+1}=L_n+D$        Eq. (01)

Notice that, for Eq. (01), it was assumed that a vapor stream $G_{n+1}$ enters the top section from the bottom, while two liquid streams, $L_n$ and $D$, exit the section at the bottom and the top, respectively. On the other hand, a mass balance for the component of interest (say component $A$)  on each plate in this section can be easily written as,

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

Equation (02) was derived based on the sketch in Fig. 02.

Fig. 02 Mass balance on a plate

From Eq. (02), the mole fraction in the vapor entering the nth plate can be isolated to be,

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

One important simplification on Eq. (03) can be made on the nature of the flow rates entering and leaving the plates: the flow rates of the liquid and vapor are constant always. This is a very strong assumption and is at the heart of the method of calculation called after Sorel and Lewis.

In this way, Eq. (03) becomes,

$y_{n+1}=\dfrac{L}{G}x_n+\dfrac{D}{G}z_D$        Eq. (04)

Mass balance on the stripping section

In this case, we just repeat the same process as before. Fig. 02 shows a schematic of this section.

Fig. 03 Mass balance on the stripping section.

In. Fig. 03 variables for vapor and liquid have been barred $(\bar{V},\bar{L})$ to avoid confusion with those of the rectifying section. Also, the counter $m$ has been introduced to number the plates of the stripping section. The global mass balance is,

$\bar{L}_m=\bar{G}_{m+1}+W$        Eq. (05)

In the same way as before, from the mass balance for the component $A$, the following equations arise.

$\bar{L}_m x_m=\bar{G}_{m+1}y_{m+1}+Wx_W$        Eq. (06)

and

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

Again, using the idea of constant liquid and vapor flow rate at each plate, Eq. (07) reduces to,

$y_{m+1}=\dfrac{\bar{L}}{\bar{G}} x_m-\dfrac{W}{\bar{G}}x_W$        Eq. (08)

Recall that Eq. (08) comes from a rough approximation.