Differential distillation

 In this post, I will discuss distillation from a different mathematical perspective, namely, a differential equation formulation. This is called differential distillation and concerns a batch operation, as in the laboratory scale.

Fig. 01 Equipment for batch evaporation

A critical feature of differential distillation is the time. It should be performed for a very long time and is idealistic, as is the case with several other distillation modes. The only reason for using long times is the mathematical formulation. Differential distillation can also be applied to binary or multicomponent mixtures.

The mechanism of distillation

As you may remember, in batch mode operation, all raw materials are loaded into the reactor at the beginning, and from these products are obtained in a given stream. 

In differential distillation, the same occurs. The mixture is loaded into the still so that the most volatile component leaves the vessel from above, while the residue remains at the bottom. The product, obtained as condensate, is not necessarily pure, nor is the residue. Further operations may still be required. However, most of the product is composed of the most volatile component, and most of the residue is composed of the less volatile component. This would fit very well for a binary mixture.

A ternary mixture

Let us consider the distillation of a multiple-component solution, say a ternary one, in an evaporator like that shown in Fig. 01. Suppose the mixture is composed of A (the most volatile), B (having intermediate volatility), and C (the least volatile). The most volatile component, A, will evaporate first, so that, if you take samples of the condensate product and analyze them, you will find that concentrations are as A > B > C. After a certain time, once A is almost depleted in the bottom mixture, the concentrations as measured in the leaving vapor would be B > A > B. In the end, the component with the highest boiling point would remain in the pot bottom, where the concentration profile would be C > B > A.

Mathematical formulation (for a binary mixture)

Let us consider the formulation for a binary mixture being distilled in batch mode. A simple global mass balance for the distillation pot shown in Fig. 01 would be just,

$Feed\quad +\quad Product \quad =\quad Residue$        Eq. (01)

As you may already ascertain, there is no feed. The initial mixture is loaded once; no further feed is introduced. In other words, the feed is zero. The story of the product and residue terms is different.

The stream of product $D$ leaving as vapor has concentration $y$. For this stream, one statement is made: the rate $D$ may change, but its composition does not. This means that $y$ is constant during the distillation process. This assumption is the most difficult to apply in practice.

On the other hand, the residue, say $L$, will change during the distillation process since a stream $D$ is slowly taking the most volatile component from it. Also, the composition of $L$, say $x$, will change. These thoughts are at the heart of the theoretical analysis.

The global mass balance, in Eq. (01), can then be rewritten as,

$0\quad - \quad dD\quad = \quad dL$        Eq. (02)

in terms of the small differential changes in each stream. A global mass balance, for the pot, for the most volatile component would be something like,

$-d\left( yD \right)\quad =\quad d\left( xL \right)$

$-ydD\quad-\quad Ddy\quad =\quad xdL \quad+\quad Ldx$

which, since $y$ is constant, reduces to,

$-ydD\quad =\quad xdL \quad+\quad Ldx$        Eq. (03)

Further substitution of Eq. (02) into Eq. (03) gives,

$ydL\quad =\quad xdL \quad+\quad Ldx$        Eq. (04)

Although it is not obvious, Eq. (04) is a differential equation modelling the distillation process in the equipment shown in Fig. 01. This equation sets a relationship between the residue stream and the compositions of the vapor and liquid in equilibrium. One way to solve Eq. (04) is by direct integration, as follows,

$\dfrac{dL}{L}=\dfrac{dx}{y-x}$

$\int_{W}^{F}\dfrac{dL}{L}=\ln \dfrac{F}{W}=\int_{x_W}^{x_F}\dfrac{dx}{y-x}$        Eq. (05)

where $F$ is the stream of moles loaded at the beginning of the process. The composition of $F$ is given by $x_F$. Also, $W$ is the stream of residual moles in the pot with final composition $x_W$. The problem of Eq. (05) is that it must be solved numerically from experimental data.

According to Robert Treybal, any variable among $F$, $W$, $x_F$, and $x_W$ can be determined if you already know the other three. Also, once all these variables are known, a global mass balance on the most volatile component can be performed to estimate $y_{av}$ as follows,

$Fx_F=Dy_{av}+Wx_W$        Eq. (06)

Constant relative volatility $\alpha$

It is not obvious, but one assumption underlying Eq. (05) and the physical process is that the total pressure is constant. Also, one practical parameter of distillation estimations is the relative volatility $alpha$, which is given in terms of Eq. (01) (see post Relative volatility $\alpha$ for more details)

$\alpha=\dfrac{y_A/x_A}{\left(1-y_A\right)/\left(1-x_A\right)}$        Eq. (07)

and can be combined with Eq. (05) if $y_A$ is isolated from Eq. (07). This is,

$y=\dfrac{\alpha x}{1+x\left(\alpha-1\right)}$        Eq. (08)

The new form of Eq. (05) would then be,

$\ln \dfrac{F}{W}=\dfrac{1}{\alpha - 1}\ln \dfrac{x_F\left(1-x_W\right)}{x_W\left(1-x_F\right)}+\ln \dfrac{1-x_W}{1-x_F}$        Eq. (09)

or

$\log \dfrac{Fx_F}{Wx_W}=\alpha \log \dfrac{F\left(1-x_F\right)}{W\left(1-x_W\right)}$        Eq. (10)

The big issue with Eqs. (09-10) is that $\alpha$ is not contant! So, what do we do? Easy. Take an average value of $\alpha$. One important thing you should not forget while using Eqs. (09-10) is that ranges on which $\alpha$ is almost constant are preferred.

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

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

Other stuff of interest

• LE01 - AC and DC voltage measurement and continuity test
• LE 02 - Start and stop push button installation 24V DC
• LE 03 - Turn on/off an 24V DC pilot light with a push button
• LE 04 - Latch contact with encapsulated relay for turning on/off an AC bulb light
• LE 05 - Emergency stop button installation
• About PID controllers
• Ways to control a process
• About pilot lights
• Solving the Colebrook equation
• Example #01: single stage chemical evaporator
• Example #02: single stage process plant evaporator
• Example #03: single stage chemical evaporator
• Example #04: triple effect chemical evaporator

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Ildebrando.