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 applies to a binary mixture. 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)

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)

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. 02 Mass balance on the stripping section.

In. Fig. 02 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.

A simple mass and enthalpy balance on single stage distillation

 Here, the case of single-stage distillation is presented. This is a simple case involving only boiling the mixture and then separating it in the column. Of course, neither plates nor packing is needed. Despite its simplicity, it is worth discussing the physics of the process.

Instantaneous evaporation

In the sketch above, a mixture feed $F$ is passed to a heat exchanger that boils the liquid and then to a separator. In the separator, the vapor rises to the top, while the liquid settles to the bottom. A mass balance for this very schematic case is twofold: a global balance and a balance for the component of interest (which may be more than one). For the global mass balance, we have,

$F=D+W$        Eq. (01)

and for the component of interest, the balance is written as,

$Fz_F=Dy_D+Wx_W$        Eq. (02)

An energy balance based on the enthalpy of each stream can be written as,

$FH_D+Q=DH_D+WH_W$        Eq. (03)

where $Q$ is the heat supplied in the heat exchanger to boil the feed F. Equations (01-02) can be combined to give,

$\dfrac{W}{D}=\dfrac{y_D-z_F}{z_F-x_W}$        Eq. (04)

which can be a more useful relationship. Also, Eqs. (01,03) can also be combined. The result is,

$-\dfrac{W}{D}=\dfrac{H_D-\left( H_F+Q/F \right)}{H_W-\left( H_F+Q/F \right)}$        Eq. (05)

This is the end. Enjoy.