## Distillation

To reduce load times, this material is divided into seven files, corresponding to the numbered points below. The present file (distill4.html) contains point 4 only.

## Distillation IV: Calculations

Before beginning most distillation calculations, a decision must be reached: does equimolal overflow apply? If so, the operating equations are lines and you have one set of options -- notably the McCabe-Thiele method. If not, energy balances must be explicitly considered.

There are several ways of incorporating the energy effects. The Ponchon-Savarit method is a graphical approach that does not require an assumption of equimolal overflow. Graphical construction is done on the enthalpy composition diagram. Your text does not include this method, but it may be useful.

In all cases, one can use a "stepping" approach. Starting from one end of the column, the component, material, and energy balances can be solved simultaneously. After a stage is determined, you step up (or down) to the next and calculate that stage. Depending on what information is known, the form of the equilibrium relation, etc., this approach may require an iterative solution.

### McCabe-Thiele Method

The graphical McCabe-Thiele Method can be used to determine the number of ideal stages and feed tray location. To do this, you make a plot showing the equilibrium curve, feed line, and operating lines for the rectifying and stripping sections (all on the same axes), and then find answers by graphical construction.

A standard (beginner) distillation problem provides you with xF, q, xD, xB, and RD; although not necessarily directly. You may need to use the overall material balances to find some of the compositions; to calculate the q factor from temperature and composition data; and/or determine the reflux ratio based on the limiting minimum value. Once you have these values, the solution procedure is:

1. Plot the equilibrium curve.
2. Calculate the slope q/(q-1) of the feed line.
3. Plot the feed line using x=xF and the slope.
4. Calculate the y-intercept of the rectifying line xD/(RD+1).
5. Plot the rectifying line using (xd,xd) and the intercept.
6. Draw the stripping line connecting the intersection of the rectifying and feed lines and the point (xb,xb).
7. Correct for stage efficiency by drawing the "effective" equilibrium curve between the equilibrium curve and the operating lines.
8. "Step off" equilbrium stages.
The number of stages is found graphically by constructing triangles on the diagram. You can start from either the top or the bottom. From a product composition on the operating line (either (xd,xd) or (xb,xb)) move horizontally to the equilibrium curve, then vertically back to the operating line, horizontally to the equilibrium curve, etc., constructing triangles along the way. Always make the steps between the equilibrium curve and the lower of the two operating lines (the rectifying line in the top half of the column, the stripping line in the bottom half -- if you're looking for the bottom line, the switchover point will be clear). The number of triangles you draw is the number of stages in your column. If you are using the equilibrium curve to step from, you are determining ideal stages.

If you don't want to draw, you can do the same thing, iteratively solving for the various equation intersections. You'll need to be pretty careful with your "bookkeeping" if you try this.

To apply Murphree tray efficiencies, construct an effective equilibrium curve between the equilibrium and operating curves, and step using the effective curve to determine actual separation stages. Remember that a partial reboiler or partial condenser is by definition an ideal stage, so you use the ideal equilibrium curve (not the effective) for these stages.

The optimum feed tray is the triangle with one corner on the rectifying line and one on the stripping line. Putting the feed anywhere else increases the number of stages needed to make the separation. To visualize this, notice that the closer the operating line is to the equilibrium curve, the smaller the stepping triangles become. Introducing the feed at the intersection of the rectifying and stripping lines maximizes the size of the triangles and so leads to the fewest steps.

When analyzing existing distillation systems, the actual feed entry point may not be at the optimum (the tray where the operating lines intersect). In this case, the tray stepping should switch from the rectifying line to the operating line at the actual feed tray location.

An example of McCabe-Thiele analysis has been prepared. It is available for download as a Mathcad 5.0+ file.

### Limiting Cases

Frequently, when analyzing or designing a process, it is useful to look at limiting cases to assess the possible values of process parameters. In distillation analysis, separation of a pair of components can be improved by increasing the number of stages while holding reflux constant, or by increasing the reflux flow for a given number of stages. This tradeoff sets up two limiting cases:

1. Total Reflux (minimum ideal stages)
2. Minimum Reflux (infinite ideal stages)
The design tradeoff between reflux and stages is the standard economic optimization problem chemical engineers always face -- balancing capital costs (the number of trays to be built) vs. the operating cost (the amount of reflux to be recirculated). A good design will operate near a cost optimum reflux ratio.

#### Total Reflux

The total reflux condition represents operation with no product removal. All the overhead vapor is condensed and returned as reflux. Consequently, the reflux ratio (L/D) is infinite. This, in turn, makes the operating lines the 45 degree line (prove it to yourself by setting D=0, and noticing that consequently L=V). With the operating lines on the diagonal, they are as far as they can get from the equilibrium curve, so if the number of plates are stepped off using the diagonal and the equilibrium curve, the number of theoretical stages will be a minimum.

Often, columns are operated at total reflux during their initial startup, and product is not withdrawn until a separation close to that desired is achieved.

The Fenske Equation is another method for determining the minimum number of trays required for a given separation. It is an example of a "shortcut" distillation method. There are a number of these approximate methods available to get initial estimates of distillation requirements.

The Fenske equation applies to distillation systems with constant relative volatility. Note that the form of the Fenske equation shown calculates the minimum number of plates; it does not include the reboiler (hence the -1 on the right hand side). Other texts may use a form for the minimum number of stages and not subtract the reboiler.

If the relative volatility varies through a column because of temperature effects, it is possible to use a geometric mean value of the relative volatility (as is done for multicomponent distillation) and the Fenske equation to get an approximate value for the number of stages.

#### Pinch Points

The intersection of an operating line and the equilibrium curve is called a pinch point. A simple column will have two pinch points (because there are two operating lines). The points change when the operating lines do. An existing column can "pinch" if its operating line is too close to its equilibrium curve. This means that there are several stages doing very little separation and wasting resources.

To cure a pinch, the most direct solution is to move the feed entry point. This is often an expensive proposition. In such cases, the reflux and boilup ratios can be increased to change the operating lines. This will increase operating costs and energy consumption, but may be the only realistic option.

A pinch at the intersection of the feed line and the equilibrium curve indicates that the column is operating at minimum reflux.

#### Minimum Reflux

The minimum reflux condition represents the theoretical opposite of total reflux -- an infinite number of ideal separation stages. In this case, the intersection of the operating lines lies on the equilibrium curve itself. Thus, the distance between the equilibrium curve and the operating lines is at its minimum, the stepping triangles become very small, there is no gap between the equilibrium curve and the intersection point, so you cannot step past the feed point.

The minimum reflux rate can be determined mathematically from the endpoints of the rectifying line at minimum reflux -- the overhead product composition point (xD,xD) and the point of intersection of the feed line and equilibrium curve (x', y').

The derivation of this formula is given later. One important thing to realize: the formula only applies when the feed line is the breakpoint for the operating curve in the top portion of the column. If there are intermediate product draws between the reflux and the feed, the formula does not apply. In this case, you must calculate the liquid flow down the column at the pinch point, and then work it back up the column to find the reflux flow at minimum reflux conditions.

If the equilibrium curve has an inflection point, it may not be possible to construct a line between the overhead product point and the feed/equilibrium intersection without passing outside the equilibrium envelope. Operating curves must always intersect within the equilibrium envelope and cannot cross outside (in either half of the column). In this case, minimum reflux occurs at a tangent pinch and the operating line is tangent to the equilibrium curve. Calculations are based upon the intersection of the tangent operating line and the feed point.

When designing columns, it is common to define the design reflux ratio as some multiple of the theoretical minimum reflux. The cost optimum reflux ratio is typically in the 1.1 to 1.5 range depending on energy costs, condenser coolant, and materials of construction. The rule of thumb reported most often suggests that a reflux ratio of about 1.2 times the minimum is a good design value.

It may also be necessary to distinguish between returned reflux (the reflux stream flowing from the accumulator to the column) and the effective reflux flowing down the column. This is a concern if the reflux is subcooled. In this case, the effective reflux will consist of the returned reflux plus whatever additional liquid is condensed when the cold liquid contacts vapor on the reflux tray. It is probably best to use effective reflux in minumum reflux calculations.

##### Derivation of Minimum Reflux Formula

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Minimum reflux corresponds to a pinch at the intersection of the feed line and the equilibrium curve. From its formula, the rectifying line has slope and will connect the intersection point (xint, yint) and (xD, xD). Consequently, we can express the slope in terms of "rise over run", or

Algebraic rearrangement gives the desired formula:

Clearly, this formula doesn't apply if there are more than two operating regions. In these cases, it is probably smarter to calculate the reflux ratio from the ratio of the liquid and vapor flow rates.

The heating and cooling loads in the condenser and reboiler can be calculated from straighforward energy balances. The chief difficulty is in getting good values for the heats of vaporization -- an enthalpy-concentration diagram is very useful!

For the reboiler, energy must be added equal to the sum of the sensible heat needed to raise the liquid to its boiling point and the latent heat of vaporization. The steady-state energy balance on the process side of the reboiler is then:

This equation is expressed in terms of average heat capacities and an average latent heat and depends on the vapor boilup rate. Usually, the sensible heat transfer in a reboiler is relatively small, so that the heat load can be calculated from

The heating medium requirements can be calculated from an energy balance on the heating side of the reboiler. If saturated steam is the heating medium, then

so that (neglecting thermal capacitance in the reboiler and heat losses) the steam rate can be obtained from
Similarly, if a liquid heat transfer fluid (hot oil, etc.) is used, the equations
(htfs = heat transfer fluid supply, htfr = heat transfer fluid return) or
may be used.

A similar analysis provides the condenser load as

(neglecting any subcooling of reflux), so that when cooling water is used in the condenser
(cws = cooling water source, cwr = cooling water return) and since the specific heat of water is 1.0 for common units
When calculating cooling loads, you may need to adjust the vapor rate or condenser duty to account for vapor condensed by direct contact with cold reflux on the reflux return tray. Watch for this whenever the condenser temperature is significantly below the expected tray temperature.

It is possible to imagine a case where ; that is, when . The case when the vapor rates at the top and bottom of the column are most likely to be the same occurs when the feed is at its bubble point, and . In this situation, the condenser and reboiler loads will be approximately equal.

### Stage Efficiencies

Mass transfer limitations prevent the vapor leaving a tray from truly being in precise equilibrium with the liquid on the tray; consequently, the assumption of ideal stages is only an approximation.

An efficiency is used to represent the deviation from equilibrium. There are three types of efficiencies we will consider:

1. Overall efficiency
2. Local efficiency
3. Murphree efficiency

An overall efficiency is the simplest choice. It is the ratio of the number of ideal stages to the number of actual stages.

A single efficiency can thus be used for the entire column, but is only accurate enough for prelimary design. Some improvement can be achieved by using separate efficiencies for each section of the column. Accuracy is limited because effectiveness of mass transfer is constrained by geometry and design of the trays, flowrates and paths of all streams, compositions, etc. The problem is really too complex to lump into a single parameter, so when overall efficiencies are used they should be based on performance data from similar columns or laboratory tests.

The local efficiency is the most accurate option, but also the most difficult to use. It is defined at only a single point on a specific tray.

It is most necessary on large diameter columns where position dependence is significant.

A Murphree efficiency is probably the most common choice, since it represents a workable compromise between accuracy and ease of use. It has the same form as a local efficiency but is based on tray average compositions.

Values between 0.6 and 0.75 are common for sieve trays. We know that the liquid leaving a tray is not really the same as the tray average, so a Murphree efficiency effectively assumes perfect mixing on the tray. In practice, we normally measure the liquid composition and get the vapor composition from an equilibrium calculation or diagram. In the case of multicomponent systems, the efficiencies are different for each component.

The overall efficiency and the Murphree efficiency are not directly related. You cannot use an average Murphree efficiency in place of an overall value.

To use a local or Murphree efficiency with a graphical method, the true equilibrium curve is replaced with an effective equilibrium curve located between the true curve and the operating curves. The effective curve is used to count stages. Note, however, that the efficiency doesn't apply to the reboiler, so the true equilibrium curve should be used for the last stage of the stripping section.

To construct an effective equilibrium curve is not difficult. The effective curve is given by:

where y represents the operating curve. A plot of yeff will produce an interior line on the equilbrium diagram construction.