## Distillation

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## Distillation VI: Enthalpy-Concentration Methods

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Distillation calculations can be performed graphically on an enthalpy-concentration (Hx) diagram. This approach is sometimes called the Ponchon-Savarit Method. Working on the Hx diagram is more general than a McCabe-Thiele construction, because it takes direct account of the thermal effects and does not require an assumption of equimolal overflow.

The very shape of the Hx diagram provides a clue as to the importance of the energy balances. If the dew point and bubble point lines are more or less straight and roughly parallel, it indicates that the latent heat of vaporization is basically constant with respect to composition. This is the prerequisite for assuming equimolar overflow, and so the energy balances may be neglected. If the saturation curves show significant changes in curvature or separation, it suggests that to assume equimolar overflow will introduce error.

Points for the feed and product can be located on the Hx diagram; for our purposes we'll call them the F, D, and B points. The coordinates are their composition and enthalpy. If the products are saturated liquid, as is the case for total condensers without subcooling and for partial reboilers, these points will lie on the bubble point curve on the Hx diagram.

### Overall Enthalpy Balance

The steady-state enthalpy balance for a distillation column is:

One of the tools of graphical solution is the notion of colinearity. This has been used before if you have used lever arm principles. For an adiabatic process, the feed and products will be colinear on an Hx diagram. Thus, it is useful to redefine our distillation system to be adiabatic, by bringing the condenser and reboiler inside the system boundary. Rearranging the enthalpy balance gives:

If we then define:

The enthalpy balance becomes:
Substituting from the overall material balance:

After this modification, the system is adiabatic, so a line can be drawn through the feed point, F, and the points (xD, hDp) and (xB, hBp). This line represents the system enthalpy balance, and so is called the overall enthalpy line. Remember: for a given separation only one of the reboiler and condenser duties is independent. So, you will probably want to pick one duty and then construct the line through the feed point to determine the other duty.

### Reflux Ratio

As before, the reflux ratio can be determined from the L/V ratio, and for this formulation is given by:

where hD is the enthalpy of the overhead product, Hy1 the enthalpy of the vapor entering the condenser, and hDp the adjusted enthalpy of the overhead. Notice that this represents the ratio of distances on the Hx diagram: the numerator is the vertical distance between the hDp point and the dew point saturation curve, while the denominator is the distance between the saturation curves.

A calculation often begins by using the overhead product composition and temperature to obtain hDand Hy1. These in turn are used with the reflux ratio to get hDp. Then the overall enthalpy line is drawn from hDp through the feed point, and the intersection with xB gives hBp.

### Stepping Off Stages

In the McCabe-Thiele procedure, operating curves were constructed to represent the component balances for the column and to relate the liquid composition on a stage to the composition of the entering vapor. These were then paired with the equilibrium curve, which relates the composition of liquid on a stage to the vapor leaving the stage.

A similar procedure can be followed on an Hx diagram, except instead of using component balances, enthalpy balances are used. The equilibrium data is represented by the equilibrium tie lines on the Hx diagram.

The operating lines are developed from enthalpy balances on the rectifying and stripping sections (just as in the McCabe-Thiele approach operating equations were developed from the equivalent component balances). The operating lines will connect the point representing the liquid on a stage (with coordinates xn, hn) to the point representing the adjusted enthalpy at the appropriate end of the column. It will cross the saturated vapor line on the Hx diagram at the point corresponding to the vapor leaving the stage (coordinates yn+1, Hn+1).

#### Summary of Procedure

1. Obtain enthalpy-composition diagram
2. Fix the feed point F, and product points D and B using stream compositions and enthalpies
3. Use the overhead product enthalpy and the reflux ratio to find the adjusted enthalpy of the overhead. Plot it as point D', on a vertical line with point D.
4. Construct the overall enthalpy line from point D' through the feed point. It intersects a vertical line drawn through point B at point B'.
5. Plot point V1. For a total condenser, the composition entering the condenser is the same as the overhead product, so this point will be vertically above point D on the saturated vapor curve.
6. Follow the tie line from point V1 to the saturated liquid curve. This intersection will be point L1.
7. Construct an operating line connecting points D' and L1. The intersection of the operating line with the saturated vapor curve will be point V2.
8. Repeat the two preceding steps until one of the V or L points is to the left of the overall enthalpy line. Once it is crossed, construct operating lines using points Li and B'.
9. When xi is less than xB, construction is finished.
The number of stages can be read as the number of complete triangles.

### Limiting Conditions

At total reflux, operating lines are vertical (infinite slope). This can be used to determine the minimum number of stages. Not that operating curves are not required to do this -- only the endpoint compositions.

While doing constructions on the yx diagram, a "pinch" was defined as the intersection point between the equilibrium curve and the operating curve. On an Hx diagram, there isn't an equilibrium curve -- it has expanded to a region, and each point from the xy equilibrium curve is represented by a tie line. The "pinch point" also expands, resulting in a single line where the operating and tie lines overlap.

Minimum reflux still corresponds to a pinch at the feed conditions, so to determine the minimum reflux a line must be constructed so that the overall enthalpy line coincides with the tie line that runs through the feed point.

### Example

A feed containing 40 mole percent n-hexane and 60 percent n-octane is fed to a distillation column. A reflux ratio of 1.2 is maintained. The overhead product is 95 percent hexane and the bottoms 10 percent hexane. Find the number of theoretical stages and the optimum feed stage. Assume that a total condenser is used. The column is to operate at 1 atm.

Step 1: Equilibrium data is collected.

VLE Data, Mole Fraction Hexane, 1 atm
x (liquid) 0.0 0.1 0.3 0.5 0.55 0.7 1.0
y (vapor) 0.0 0.36 0.70 0.85 0.90 0.95 1.0

Enthalpy-Concentration Data
Mole Fraction Hexane Enthalpy cal/gmol
Sat. Liquid Sat. Vapor
0.0 7000 15,700
0.1 6300 15,400
0.3 5000 14,700
0.5 4100 13,900
0.7 3400 12,900
0.9 3100 11,600
1.0 3000 10,000

This information can be used to create enthalpy-concentration and equilibrium diagrams.

Step 2: Plot the feed and product points. All three will lie on the saturated liquid line, B at xB=0.1, F at xF=0.4, and D at xD=0.95.

Step 3: From the data tables (or from the Hx diagram) find the enthalpy of the distillate, hD=3050 cal/gmol. Because a total condenser is used, the vapor leaving the top stage will have concentration y1=0.95. Consequently, it will have enthalpy HD=10,800. These values and the reflux ratio can be used to find the enthalpy coordinate for the D' point, HDp.

and so HDp=20,100. The D' point on the Hx diagram can then be placed at (0.95,20100).

Step 4: The overall enthalpy line is then drawn from D', through F. Its intersection with the line x=0.1 is the point B'.

Step 5: Because of the total condenser, the V1 point will lie on the saturated vapor curve at x=xD.

Step 6: Follow the tie line that passes through the V1 point back to the saturated liquid curve. The intersection is the point L1 (liquid on tray 1). If tie lines are not available, they may be constructed using the xy diagram.

Step 7: Construct an operating line through both the L1 point and the D' point. Its intersection with the saturated vapor curve will be the point V2.

Step 8: Continue the construction, alternating between tie lines and and operating lines until you have moved to the left of the overall enthalpy line at point L3.

Once the overall enthalpy line is crossed, construction continues, but operating lines are now drawn between the Li point and the B' point (instead of the D' point). Construction continues until L5 which is almost directly on top of xB.

Consequently, there are 5 ideal stages required for this separation -- with a partial reboiler, that means 4 ideal trays. The optimum feed tray is number 3.