- 9.0 Summary-Objectives
- 9.1 Introduction to Batch Distillation
- 9.2 Batch Distillation: Rayleigh Equation
- 9.3 Simple Binary Batch Distillation
- 9.4 Constant-Mole Batch Distillation
- 9.5 Batch Steam Distillation
- 9.6 Multistage Binary Batch Distillation
- 9.7 Multicomponent Simple Batch Distillation
- 9.8 Operating Time
- References
- Homework
- Chapter 9 Appendix A. Spreadsheet for Simple Multicomponent Batch Distillation, Constant Relative Volatility

## 9.6 Multistage Binary Batch Distillation

The separation achieved in a single equilibrium stage is often not large enough to obtain both the desired distillate concentration and a low enough bottoms concentration. In this case a distillation column is placed above the reboiler, as shown in Figure 9-2. The calculation procedure is detailed here for a staged column, but packed columns can easily be designed using the procedures explained in Chapters 10 and 16.

For multistage systems x_{D} and x_{W} are no longer in equilibrium. Thus, the Rayleigh equation, Eq. (9-7), cannot be integrated until a relationship between x_{D} and x_{W} is found. This relationship can be obtained from stage-by-stage calculations. We assume that there is negligible holdup on each plate, in the condenser, and in the accumulator. Then at any specific time we can write mass and energy balances around stage j and the top of the column, as shown in Figure 9-2A. These balances simplify to

Input = Output

since accumulation was assumed to be negligible everywhere except the reboiler. Thus, at any given time t

In these equations V, L, and D are molal flow rates. These balances are essentially the same equations we obtained for the rectifying section of a continuous column except that Eqs. (9-23) are time dependent. If we can assume constant molal overflow (CMO), the vapor and liquid flow rates will be the same on every stage, and the energy balance is not needed. Combining Eqs. (9-23a) and (9-23b) and solving for y_{j+1}, we obtain the operating equation for CMO:

At any specific time Eq. (9-24) represents a straight line on a y-x diagram. The slope will be L/V, and the intercept with the y = x line will be x_{D}. Since either x_{D} or L/V will have to vary during the batch distillation, the operating line will change continuously.

### 9.6.1 Constant Reflux Ratio

The most common operating method is to use a constant reflux ratio and allow x_{D} to vary. This procedure corresponds to a simple batch operation in which x_{D} also varies. The relationship between x_{D} and x_{W} can now be found from a stage-by-stage calculation using a McCabe-Thiele analysis. Operating Eq. (9-24) is plotted on a McCabe-Thiele diagram for a series of x_{D} values. Then we step off the specified number of equilibrium contacts on each operating line starting at x_{D} to find the x_{W} value corresponding to that x_{D}. This procedure is shown in Figure 9-6 and Example 9-2.

**FIGURE 9-6** *McCabe-Thiele diagram for multistage batch distillation with constant L/D, Example 9-2*

The McCabe-Thiele analysis gives x_{W} values for a series of x_{D} values. We can now calculate 1/(x_{D} – x_{W}). The integral in Eq. (9-7) can be determined by either numerical integration, such as Simpson’s rule given in Eqs. (9-12a) to (9-12c), or by graphical integration. Once x_{W} values have been found for several x_{D} values, the same procedure used for simple batch distillation can be used. Thus, W_{final} is found from Eq. (9-7), x_{Davg} from Eq. (9-8b), and D_{total} from Eq. (9-8a). If x_{Davg} is specified, a trial-and-error procedure will again be required.

*EXAMPLE 9-2. Multistage batch distillation*

*EXAMPLE 9-2. Multistage batch distillation*

We wish to batch distil 50.0 kmol of a 32.0 mol% ethanol and 68.0 mol% water feed. The system has a still pot plus two equilibrium stages and a total condenser. Reflux is returned as a saturated liquid, and we use L/D = 2/3. We desire a final still-pot composition of 4.5 mol% ethanol. Find the average distillate composition, the final charge in the still pot, and the amount of distillate collected. Pressure is 1 atm.

*Solution*

Define. The system is shown in the following figure.

Find W

_{final}, D_{total}, and x_{D,avg}.and C. Explore and Plan. Since we can assume CMO, a McCabe-Thiele diagram (Figure 2-2) can be used. This diagram relates x

_{D}to x_{W}at any time. Since x_{F}and x_{fin}are known, the Rayleigh Eq. (9-7) can be used to determine W_{final}. Then D_{total}and x_{D,avg}can be determined from Eqs. (9-8a) and (9-8b), respectively. A guess-and-check procedure is not needed for this problem.

Do it. The McCabe-Thiele diagram for several arbitrary values of x

_{D}is shown in Figure 9-6. The top operating line iswhere

The corresponding x

_{W}and x_{D}values are used to calculate x_{D}– x_{W}and then 1/(x_{D}– x_{W}) for each x_{W}value. These values are plotted in Figure 9-7 (some values not shown in Figure 9-6 are shown in Figure 9-7). The area under the curve (going down to an ordinate value of zero) from x_{F}= 0.32 to x_{fin}= 0.045 is 0.608 by graphical integration.**FIGURE 9-7***Graphical integration, Example 9-2*Then from Eq. (9-7),

W

_{final}= Fe^{–Area}= (50)exp(–0.608) = 27.21From Eq. (9-8a),

D

_{total}= F – W_{final}= 22.79and from Eq. (9-8b),

The area can also be determined by Simpson’s rule. However, because of the shape of the curve in Figure 9-7, it will be less accurate than in Example 9-1 unless the area is split into two or more parts. Simpson’s rule gives

This can be checked by breaking the area into two parts and using Simpson’s rule for each part. Do one part from x

_{fin}= 0.045 to x_{W}= 0.10 and the other part from 0.1 to x_{F}= 0.32. Each of the two parts should fit relatively well with a cubic. ThenTotal area = 0.6196

Note that Figure 9-7 is very useful for finding the values of 1/(x

_{D}– x_{W}) at the intermediate points x_{W}= 0.0725 (value = 2.03) and x_{W}= 0.21 (value = 2.23). The total area calculated is closer to the answer obtained graphically (1.9% difference compared to 4.7% difference for the first estimate).Then doing the same calculations as previously [Eqs. (9-7), (9-8a), and (9-8b)] with Area = 0.6196,

Check. The mass balances for an entire cycle, Eqs. (9-1) and (9-2), should be and are satisfied. Since the graphical integration and Simpson’s rule (done as two parts) give similar results, this is another reassurance.

Generalize. Note that we did not need to find the exact value of x

_{D}that corresponds to x_{F}or x_{fin}. We just made sure that our calculated values went beyond these values and then used Figure 9-7 to interpolate to find values for Simpson’s rule. This is true for both integration methods. Our axes in Figure 9-7 were selected to give maximum accuracy; thus, we did not graph parts of the diagram that we did not use. The same general idea applies if fitting data—fit the data only in the region needed. For more accuracy, Figure 9-6 should be expanded. If Simpson’s rule is to be used for very sharply changing curves or curves with maxima or minima, accuracy is better if the curve is split into two or more parts. Comparison of the results obtained with graphical integration to those obtained with the two-part integration with Simpson’s rule shows a difference in x_{D,avg}of 0.008. This is within the accuracy of the equilibrium data.

Once x_{fin}, D, and W_{final} are determined, we can calculate the values of Q_{c}, Q_{R}, and operating time (see Section 9.8).

### 9.6.2 Variable Reflux Ratio

Batch distillation columns can also be operated with a variable reflux ratio to keep x_{D} constant. Operating Eq. (9-24) is still valid, the intersection with the y = x line will be constant at x_{D}, but the slope varies. The McCabe-Thiele diagram for this case relating x_{W} to x_{D} is shown in Figure 9-8. Since x_{D} is kept constant, the calculation procedure is simplified.

**FIGURE 9-8** *McCabe-Thiele diagram for multistage batch distillation with constant x _{D} and variable reflux ratio*

With x_{D} and the number of stages specified, trial and error is required to find the initial value of L/V that gives the feed concentration x_{F} with the specified number of stages. The operating line slope L/V is then increased until the specified number of equilibrium contacts gives x_{W} = x_{fin}. This gives (L/V)_{final}. W_{final} is found from mass balance Eqs. (9-1) and (9-2).

The required maximum values of Q_{c} and Q_{R} and the operating time can be determined next (see Section 9.8). Note that the Rayleigh equation is not required when x_{D} is constant. However, if used, the Rayleigh equation gives exactly the same answer as Eq. (9-25) (see Problem 9.C1).