Post‐Procesado

Find the optimum feed plate location and the total number of equilibrium stages.

B. Explore. Equilibrium data at 1 atm are given in Figure 2-2. An enthalpy-composition diagram at 1 atm will be helpful to estimate q. These are available in other sources (e.g., Brown et al., 1950, or Foust et al., 1980, p. 36), or a good estimate of q can be made from Figure 2-4 despite the pressure difference. In Example 4-1 we showed that CMO is valid. Thus we can apply the McCabe-Thiele method.

C. Plan. Determine q from Eq. (4-17) and the enthalpy-composition diagram at 1 atm. Plot the feed line. Calculate L/V. Plot the top operating line; then plot the bottom operating line and step off

stages.

D. Do It. Feed Line: To find q, first convert feed concentration, 20 mol%, to wt % ethanol = 39 wt

%. Two calculations in different units with different data are shown.

Thus small differences caused by pressure differences in the diagrams do not change the value of q.

Note that molecular weight terms divide out as in Example 4-2d. Then

Feed line intersects y=x line at feed concentration z=0.2. Feed line is plotted in Figure 4-13.

Figure 4-13. Solution for Example 4-3

Top Operating Line:

Alternative solution: Intersection of top operating line and y = x (solve top operating line and y = x simultaneously) is at y = x = x_D. The top operating line is plotted in Figure 4-13.

Bottom Operating Line:

We know that the bottom operating line intersects the top operating line at the feed line; this is one point. We could calculate / from mass balances or from Eq. (4-25), but it easier to find another point. The intersection of the bottom operating line and the y = x line is at y = x = x_B (see Problem 4.C9). This gives a second point.

The feed line, top operating line, and bottom operating line are shown in Figure 4-13. We stepped off stages from the bottom up (this is an arbitrary choice). The optimum feed stage is the second above the partial reboiler. 12 equilibrium stages plus a partial reboiler are required.

E. Check. We have a built-in check on the top operating line, since a slope and two points are calculated. The bottom operating line can be checked by calculating / from mass balances and comparing it to the slope. The numbers are reasonable, since L/V < 1, / > 1, and q > 1 as expected. The most likely cause of error in Figure 4-13 (and the hardest to check) is the equilibrium data.

F. Generalization. If constructed carefully, the McCabe-Thiele diagram is quite accurate. Note that there is no need to plot parts of the equilibrium diagram that are greater than x_D or less than x_B. Specified parts of the diagram can be expanded to increase the accuracy.

We did not have to use external balances in this example, while in Example 4-1 we did. This is because we used the feed line as an aid in finding the bottom operating line. The y = x intersection points are useful, but when the column configuration is changed their location may change.

4.6 Profiles for Binary Distillation

Figure 4-13 essentially shows the complete solution of Example 4-3; however, it is useful to plot

compositions, temperatures, and flow rates leaving each stage (these are known as profiles). From Figure 4-13 we can easily find the ethanol mole fractions in the liquid and vapor leaving each stage. Then x_W = 1

− x_E and y_W = 1 − y_E. The temperature of each stage can be found from equilibrium data (Figure 2-3) because the stages are equilibrium stages. Since we assumed CMO, the flow rates of liquid and vapor will be constant in the enriching and stripping sections, and we can determine the changes in the flow rates at the feed stage from the calculated value of q.

The profiles are shown in Figure 4-14. As expected, the water concentration in both liquid and vapor streams decreases monotonically as we go up the column, while the ethanol concentration increases.

Since the stages are discrete, the profiles are not smooth curves. Compare Figures 4-13 and 4-14. Note where the operating line and equilibrium curve are close together. When these two lines almost touch, we have a pinch point. Then the composition and temperature profiles will become almost horizontal and there will be very little change in composition from stage to stage. The location of a pinch point within the column depends on the system and the operating conditions.

Figure 4-14. Profiles for Example 4-3

In this ethanol-water column the temperature decreases rapidly for the first few contacts above the reboiler but is almost constant for the last eight stages. This occurs mainly because of the shape of the temperature-composition diagram for ethanol-water (see Figure 2-3).

Since we assumed CMO, the flow profiles are flat in each section of the column. As expected, > and V > L (a convenient check to use). Since stage 2 is the feed stage, L₂ is in the stripping section while V₂ is in the enriching section (draw a sketch of the feed stage if this isn’t clear). Different quality feeds will have different changes at the feed stage. Liquid and vapor flow rates can increase, decrease, or remain unchanged in passing from the stripping to the enriching section.

Figure 4-13 illustrates the main advantage of McCabe-Thiele diagrams. They allow us to visualize the separation. Before the common use of digital computers, large (sometimes covering a wall) McCabe-Thiele diagrams were used to design distillation columns. McCabe-McCabe-Thiele diagrams cannot compete with the speed and accuracy of process simulators (see this chapter’s appendix) or for binary separations with spreadsheets; however, McCabe-Thiele diagrams still provide superior visualization of the separation (Kister, 1995). Ideally, McCabe-Thiele diagrams will be used in conjunction with process simulator results for both analysis and troubleshooting.

4.7 Open Steam Heating

We now have all the tools required to solve any binary distillation problem with the graphical McCabe-Thiele procedure. As a specific example, consider the separation of methanol from water in a staged

distillation column.

Example 4-4. McCabe-Thiele analysis of open steam heating

The feed is 60 mol% methanol and 40 mol% water and is input as a two-phase mixture that flashes so that V_F/F = 0.3. Feed flow rate is 350 kmol/h. The column is well insulated and has a total condenser.

The reflux is returned to the column as a saturated liquid. An external reflux ratio of L₀/D = 3.0 is used. We desire a distillate concentration of 95 mol% methanol and a bottoms concentration of 8 mol% methanol. Instead of using a reboiler, saturated steam at 1 atm is sparged directly into the bottom of the column to provide boilup. (This is called direct or open steam.) Column pressure is 1 atm. Calculate the number of equilibrium stages and the optimum feed plate location.

Solution

A. Define. It helps to draw a schematic diagram of the apparatus, particularly since a new type of distillation is involved. This is shown in Figure 4-15. We wish to find the optimum feed plate location, N_F, and the total number of equilibrium stages, N, required for this separation. We could also calculate Q_c, D, B, and the steam rate S, but these were not asked for. We assume that the column is adiabatic since it is well insulated.

Figure 4-15. Distillation with direct steam heating, Example 4-4

B. Explore. The first thing we need is equilibrium data. Fortunately, these are readily available (see Table 2-7 in Problem 2.D1).

Second, we would like to assume CMO so that we can use the McCabe-Thiele analysis procedure.

An easy way to check this assumption is to compare the latent heats of vaporization per mole (Himmelblau, 1974).

ΔH_vap methanol (at bp) = 8.43 kcal/mol ΔH_vap water (at bp) = 9.72 kcal/mol

These values are not equal, and in fact, water’s latent heat is 15.3% higher than methanol’s. Thus, CMO is not strictly valid; however, we will solve this problem assuming CMO and will check our results with a process simulator.

A look at Figure 4-15 shows that the configuration at the bottom of the column is different than when a reboiler is present. Thus we should expect that the bottom operating equations will be different from those derived previously.

C. Plan. We will use a McCabe-Thiele analysis. Plot the equilibrium data on a y-x graph.

Top Operating Line: Mass balances in the rectifying section (see Fig. 4-15) are V_j+1 = L_j + D

y_j+1V_j+1 = L_j x_j +Dx_D Assume CMO and solve for y_j+1.

Since the reflux is returned as a saturated liquid,

Enough information is available to plot the top operating line.

Feed Line:

Intersection: y = x = z

Once we substitute in values, we can plot the feed line.

Bottom Operating Line: The mass balances are

Solve for y:

Simplifications: Since the steam is pure water vapor, y_s = 0.0 (contains no methanol). Since steam is saturated, S = V and B = L (constant molal overflow).

Then

(4-42)