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Hildebrand, J. H.; Prausnitz, J. M.; Scott, R. L. Regular and Related Solutions; Van Nostrand Reinhold: New York, 1970. Mukhopadyay, M.; Dongaonkar, K. R. Prediction of Liquid-Liquid Equilibria in Multicomponent Aromatics Extraction Systems by Use of the UNIFAC Group Contribution Model. Ind. Eng. Chem. Process Des. Dev. 1983, 22, 521-532. Pierotti, G. J.; Deal, C. H.; Derr, E. L. Activity Coefficients and Molecular Structure. Ind. Eng. Chem. 1959,51, 95. Rawat, B. S.; Gulati, I. B. Liquid-Liquid Equilibrium Studies for Separation of Aromatics. J . Appl. Chem. Biotechnol. 1976, 26, 425-435. Renon, H.; Prausnitz, J. M. Local Compositions in Thermodynamic Excess Functions for Liquid Mixtures. AIChE J . 1968, 14, 135-144. Scott, R. L. Corresponding States Treatment of Nonelectrolyte Solutions. J . Chem. Phys. 1956,25, 193-205.
Sorensen, J. M.; Arlt, W. Liquid-liquid Equilibrium Data Collection; DECHEMA Chemistry Data Series; DECHEMA: Frankfurt, 1980; Vol. V, Parts 1-3. Tripathi, R. P.; Raja Ram, A.; Bhimeshwara, Rao. P. Liquid-Liquid Equilibria in Ternary System Toluene-n-Heptane-Sulfolane. J. Chem. Eng. Data 1975,20, 261-264. Voetter, H.; Kosters, W. C. G. New Applications of the Sulfolane Extraction Process and Industrial Experiences with This Process. Erdoel Kohle 1966, 19, 267-271. Wilson, G. M. Vapor-Liquid Equilibrium. XI-A New Expression for the Excess Free Energy of Mixing. J . Am. Chem. SOC.1964,86, 127. Received for review November 30, 1988 Revised manuscript received April 4, 1989 Accepted June 19, 1989
Temperature-Heat Diagrams for Complex Columns. 1. Intercooled/Interheated Distillation Columns Brenda E. Terranova and Arthur W. Westerberg* Department of Chemical Engineering and Engineering Design Research Center,’ Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213
We show how to establish heating and cooling curves for distillation columns featuring interheating and intercooling. The approach is to construct a diagram that plots column pinch temperatures versus reboiler and condenser duties. No assumptions are needed on ideal behavior.
Introduction A distillation column is conveniently thought of as a device to degrade heat to produce separation work. Heat enters into the reboiler of a typical column, the hottest point in the column. It is removed from the condenser, the coldest point in the column. Often in industry, one will see columns in which heat has been removed from or added to a column on an intermediate tray. Figure l a illustrates this. Such heat addition and heat removal are termed interheating and intercooling, respectively. The advantage obtained for using interheating or intercooling is that the heat is added or removed at a temperature that is between the temperatures of the reboiler and condenser for the column. Thus, heat can be added a t a lower temperature than the reboiler temperature or removed at a temperature higher than the condenser temperature, each of which is advantageous from a second-law point of view. Andrecovich and Westerberg (1985) presented a diagram to illustrate the flow of heat in a column on a temperature versus heat (T-Q)diagram. The axes for this diagram are the same as used in heat-exchanger network synthesis calculations (Hohmann, 1971; Linnhoff et al., 19821, allowing this very useful “cascade“ representation to be used on the same diagram as one would use to design heat-exchanger networks. Figure l b illustrates such a diagram for an interheated/intercooled column. The top of the area representing the column corresponds to heat entering the column at two temperatures: the hotter is the reboiler and the other is the interheater. Similarly, heat is shown being removed at two temperatures: from the colder condenser temperature and from the intercooler. The main question to be answered in this paper is how does one construct this cascade diagram for a column in
* Author
to whom correspondence should be addressed.
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which there is interheating and intercooling taking place; i.e., how does one construct Figure I b given the column in Figure la? Temperature Profiles for Heating and Cooling. The temperatures for the heat entering ( Q R and Qk) and leaving (Qc and Q’c) are shown here to be constant. Since the material being vaporized or condensed will generally be a mixture, the temperature can in fact vary from the dew point to bubble point for condensing and the reverse for vaporizing. Often condensing or reboiling is carried out by having the liquid mixture in a pool on the shell side covering the tubes of the heat exchanger. In this case, the temperature is constant and is the bubble point for the liquid mixture in the pool. For the top and bottom of a column having a total condenser (not uncommon) and a partial reboiler (a little thought will show this to be the only reasonable type of reboiler), the temperature is the bubble point of the respective products. In both cases, the temperature is the bubble point, and our sketch is appropriate. For interheating or intercooling, one can imagine coils being placed in the trays of the column to effect the heat addition or removal; the heating and cooling would be at the bubble point of the liquid on the tray, again a constant temperature. For external heat removal from a tray, the temperature profile would be from the dew point to the bubble point. The bubble point is a safe “lowest” temperature at which to assume the heat is removed. For external heat addition to a tray, the liquid would likely be circulated and be on the shell side in a pool covering the tubes. Only a small fraction of the liquid would vaporize with each pass of the liquid and would again vaporize at essentially the bubble point temperature of the liquid on the tray. Finally, the diagram can easily be modified if desired to show a temperature range for any heat transfer. Approximately Equal Heats for Base Case. Andrecovich and Westerberg (1985) argue that the amount (C 1989 American Chemical Society
Ind. Eng. Chem. Res., Vol. 28, No. 9, 1989 1375
Figure 2. Material and heat balance envelops for deriving the effect of intercooling.
T
Y
e Figure 1. (a) Column with interheater and intercooler. (b) T versus Q diagram for interheated/intercooled column. I
of heat added to the reboiler of a conventional column is about equal to the heat removed from the condenser IF the column feed and top and bottom products are liquids at their respective bubble points. A heat balance around such a base case column gives hFF - h s - hBB + QR - Qc i= 0 (1) The sensible heat terms are small compared to the reboiler and condenser heat terms. Thus, they can be ignored to give QR-QcxO
(2)
For the same base case of feed and products as bubble point liquids, the total of heat added to a column through the reboiler and interheating will approximately equal the total of heat removed from the condenser and intercooling. Thus, the notched box in Figure l b has the same width top and bottom and can be shown conveniently with lines connecting heat in and heat out to enclose an area. If the column operates away from this base case, one can usually argue directly how to correct the construction of the 2’-Q diagram. For example, suppose the feed is added as a vapor (i.e., with a feed quality of q = 0). For this case, one will discover that the condenser duty will exceed the reboiler duty by an amount of heat approximately equal to that required to preheat a bubble point liquid feed to form a dew point vapor feed. By mentally “bringing” the feed back to the base condition of a bubble point liquid, one readily sees this approximate relationship.
Binary Columns Ho and Keller (1987) present an analysis that allows one to construct this diagram for a binary column. We sum-
XB
XD
XF X
Figure 3. McCabe-Thiele diagram for interheated/intercooled binary column.
marize their ideas here, as these ideas motivate part of our development for columns that are separating more than two-component feeds. We assume constant molar overflow for this analysis. When we look a t the multicomponent case, we shall relax such assumptions. Figure 2 is the top section of a distillation column where part of the heat is removed using a condenser and part using an intercooler. Two envelopes, I and 11, are illustrated, which will be used to develop the operating lines for the section of the column above the intercooler (I) and the section just below (11). Figure 3 shows the structure of a McCabe-Thiele diagram for this column, as we shall now show. The operating lines, which come from a component material balance around the column, are as follows: (3) (4)
By substituting y = x into both of these equations, one discovers that both pass through y = x = xD, as one should expect. By noting that 1 - LI/VI = D/VI (5) a similar relationship for operating line 11, and that LII is larger than LI, one can argue that the slope for operating
1376 Ind. Eng. Chem. Res., Vol. 28, No. 9, 1989
line 11,Lu/ VII, must be larger than the slope for operating line I. Thus, the general structure of the diagram in Figure 3 is consistent with these observations. What we find is that intercooling has changed only the slope of the operating line. To step off the number of trays for such a column, one steps down operating line I until the intercooler is encountered (which has to occur before this operating line hits the equilibrium line) and then switches to the other operating line as illustrated. How does this situation differ from that for a conventional column in which all the heat is removed in the condenser? If operating line I1 represents the operating line for a conventional column, then adding the intercooler will require us to use a column with more stages to effect the same separation. These extra stages occur because operating line I is closer to the equilibrium line than operating line 11. The total heat removed from the condenser and the intercooler will be exactly equal to the heat removed from the condenser of the conventional column. Thus, we require the same total cooling, require more plates, BUT gain by being able to remove some of the heat at a higher temperature. Another way to state this is we come out even in terms of the first law (same amount of heat) and lose on capital investment (more trays) but gain on the second law (can use cheaper cooling). To motivate the algorithm to generate the diagram we are after, we note that the equilibrium line has a temperature implicitly associated with each point along it. Specifically the temperature is the dew point temperature for the vapor composition, y, for the point that equals the bubble point temperature for the liquid composition, x , for it. The temperature increases as we move down the equilibrium line. The algorithm to construct our desired diagram is the following. 1. Choose R = 0. 2. Compute L / V = R / ( R + 1). 3. Sketch the operating line with this slope that passes through xD on the 45-deg line on the McCabe-Thiele diagram. Find the point where it intersects with the equilibrium line. Let T be the temperature associated with this equilibrium point (i.e., T = dew point (y) = bubble point (XI).
4. Estimate the condenser duty associated with this operating line, for example, as equal to approximately the following:
Qc == XD(R + 1)
(6)
where X is the heat of vaporization per mole for the top product. 5 . Plot T versus Qc on a T-Q diagram. 6. Increase R by some small amount and repeat from step 2. Keep repeating from step 2 until R is as large as desired for operating the column (say 1.2 times the minimum reflux ratio predicted for the column). The bottom curve in Figure 4a is such a plot for a suitable range of reflux ratios. The top half of Figure 4a is for doing a similar set of steps for putting heat into the reboiler, only here one would step through increasing reboil ratios S = P / B , from zero to the maximum desired for operating the column. It helps to remember that = P (1- q)F (7)
v
+
which is readily converted into D(R + 1) = S B + (1- q)F
(8)
to relate R to S . Given the plot in Figure 4a, we are now able to see how to construct the required notched T-Q cascade diagram
QR I
I
I
I
I
I
Figure 4. (a) Interheating/cooling diagram for binary column. (b) 2’ versus Q diagram for interheated/cooled column constructed from information in Figure 4a.
for a binary column. Select a temperature at which it is desired to remove heat from the column using an intercooler, say T *. Assume that the column operates with a total amount of heat removed, QC,min,corrresponding to a conventional column with R = kR/RmhRmin, with kRIRmi in the range 1.1-2.0. A t least the amount of heat labeled min Q, on Figure 4a must be removed in the condenser to allow the top operating line (operating line I in Figure 2) to get to the temperature T * before it intersects with the equilibrium line. To move to operating line I1 in Figure 2, which has a total heat removal above it of QC,act,says that no more than max Qc’ can be removed from the intercooler. Thus, min Qc is the minimum amount of heat to be removed from the condenser and max Qc’ the maximum removed from an intercooler operating at temperature T * . Figure 4b shows a typical notched diagram then for a column for which intercooling is done at temperature T* and interheating at temperature T * .
Multicomponent Columns To extend the Ho and Keller analysis to multicomponent columns, we proceed as follows. By analogy with the binary case, we appreciate that we want to find points where the operating line intersects with the equilibrium surface in multidimensional space for increasing values of the reflux ratio, R. The analysis here is not unlike that used by Underwood (1946, 1948) to establish conditions for the minimum reflux conditions for a column. However, here we will not limit ourselves to the ideal assumptions that he used to derive his results.
Ind. Eng. Chem. Res., Vol. 28, No. 9, 1989 1377 an enormous difference in the structure of the results, giving the multiple roots that one finds for the Underwood equations which do not occur for the flash calculation:
V
I
L r
Yi
or
Summing over all components gives L
aiXDi
-
?ai(R
xi
The algorithm to compute T versus Q is then as follows. 1. Given D, give x D ~for all components i. 2. Compute by whatever means are available Rminfor the column (for example, use existing simulation packages for this task). 3. Choose R = 0. Note, this is the value of R a t the pinch point rather than at the condenser; i.e., it gives the value of the liquid flow down the column to the distillate product rate at the pinch point. 4. Estimate ai for all components at the pinch point. 5 . Use eq 10 to compute ii! a t the pinch point. 6. Use eq 13 to compute x i for all components i at the pinch point. 7. Compute the temperature corresponding to the bubble point for composition x i for all i. This computation can be a nonideal bubble point calculation, which will result in a temperature and new relative volatility values,
V
D
+-----I xDI
I I
I
XI
aL*
(b)
Figure 5. (a) Material flow diagram for flash unit. (b) Material flow diagram for pinched top part of distillation column.
Equilibrium between the vapor and liquid leaving a tray can be written as follows: ai a
yi = - x i
(9)
where
n = caiq
+ 1) - nR = 1
(10)
i
is a mole fraction averaged relative volatility. In contrast to the usual treatment, we do NOT have to assume that relative volatiles are constant. Indeed, they are in general a function of temperature, pressure, and composition. They tend to be much less strong functions of temperature than K values. The component material balance for vapor and liquid streams passing between two trays is given by yiV = x ~ L + xD~D (11) Note, we have not written the tray subscripts here to keep the notation as simple as possible. The operating and equilibrium conditions intersect at a pinch point, as they did on the McCabe-Thiele plot above for a binary column, when y i and x i for one tray equals y i and x i for the next for all i. An infinite number of trays results. Combining these relationships gives us our desired equations for analysis. Figure 5 gives us an interesting insight into this calculation. It shows a conventional flash computation, Figure 5a, and this pinch calculation, Figure 5b. Note that they look very similar except for the direction of the feed flow and the vapor flow. These changes in flow direction introduce a minus sign in the resulting equations and cause
8. Interate from step 5 until the relative volatiles, ai, do not change from one iteration to the next. 9. From a heat balance around the column (envelop I in Figure 2), determine the condenser duty. Since the value of R and thus the liquid ( L = RD) and vapor flow ( V = ( R + 1)D)are at the pinch point, this computation is not approximate. 10. Plot T versus this heat duty, Qc. 11. Increment R and repeat from step 4. Keep repeating until R reaches the maximum value desired for operating the column (e.g., 1.2 times the minimum reflux ratio computed in step 2 above). Some discussion is appropriate for the first step of this algorithm. One might argue that the product compositions cannot be known for our column. However, a t minimum reflux, one really does know them if no components distribute. Typically the distribution of the key components is specified: e.g., 99% of the light key is to appear in the top product and 99.5% of the heavy in the bottom product. All of the components heavier than the heavy component cannot make it to the top of the column past the upper pinch point that will occur in the column; similarly, all the components lighter than the light key will not make it to the bottom of the column past the lower pinch point. Thus, the compositions are known. If a component distributes, then the compositions can be those obtained from the minimum reflux computation asked for in step 2 of the algorithm. Figure 6 is a typical plot that one might generate from this algorithm, showing a notch that corresponds to introducing an interheater into the column. It is for a column in which feed and products are bubble point liquids. It differs in form from the binary column in that the two curves (bottom for condenser and top for reboiler) do not generally intersect at the point where minimum reflux conditions occur. They will only intersect if all components
1378 Ind. Eng. Chem. Res., Vol. 28, No. 9, 1989
=o
q.1
Figure 6. Interheating/cooling diagram for multicomponent distillation column. Lower cooling line extends past q = 1 point, which is partially vaporized. Tie line locates points with the same feed quality on heating and cooling curves, as illustrated for q = 0 points.
appear in both products; then the two pinch points that will normally occur for the column will become one pinch point at the feed tray. This diagram has been extended for feed conditions that are other than bubble point liquids. It can be altered easily to accommodate products that are not bubble point liquids. To extend it for a feed with an arbitrary quality (q = 1 for bubble point liquid and q = 0 for dew point vapor), one must redo step 2 in the algorithm to find the minimum reboiler and condenser duties for whatever cases are of interest (e.g., for q = 0, 0.25, 0.50,0.75, and 1.00). These points can be noted along the curves. Only for bubble point liquid feed will the condenser and reboiler duties be equal. For all others, there will be a difference in these duties that equals the amount of heat that is required to change the feed from bubble point liquid to the feed quality being considered. Figure 6 illustrates both of these points. Note for q = 0 (dew point vapor) that the reboiler duty is reduce while the condenser duty is increased relative to q = 1. The same curves result irrespective of the thermal condition of the feed, so one has in fact proved that partially vaporizing the feed w ill always increase the condenser duty and decrease the reboiler duty relative to bubble point liquid feed. We show a “tie-line’’ which can be plotted on this figure to allow in-between values of q to be accommodated. To change the thermal condition of the products requires that one alter the plot. The current plot is for a total condenser. It shows that the minimum condenser duty, Qc in Figure 6, is the amount to condense the top product for a zero reflux ratio. If the top product is withdrawn as a dew point vapor, Qc will become zero and the curve for the condenser duty will shift to the left everywhere by Qc. Similarly, if one wishes to withdraw the bottom product as a dew point vapor, then the reboiler curve would have to shift to the right everywhere by the amount of heat needed to vaporize the bottom product. For a condenser from which both a vapor and liquid product is withdrawn, the value of Qc can be appropriately adjusted. The curved segments of the plot are only shifted; their shapes are not altered. Example In Figure 7, we show the plot constructed for a column fed with 2.5 kg/s of equal weight fractions of five components isobutane, n-butane, 2-methylbutane, n-pentane, and n-hexane. The feed is bubble point liquid. The light
260
1 0
I
I
I
I
I
100
200
300
400
500
Qkcdsec)
Figure 7. Interheating/cooling diagram for example problem. Heating curve for preheating feed from bubble point liquid to dew point vapor shown. 3
:clK’+ heat in
290
280
t
\
1
k interheater
heat out 260
I
I
I
I
I
I n d . Eng. Chem. Res. 1989, 28, 1379-1386
be put into the process at a temperature that exceeds the reboiler temperature of the column, and second, more heat passes through the process. For a multicomponent column where the heavy key is not the heaviest component, the dew point of the feed to a column can often exceed the bubble point of the bottoms product.
1379
Ho, F. G.; Keller, G. E. Process Integration. Recent Deuelopments in Chemical Process and Plant Design; Liu, Y.A., McGee, H. A., Epperly, W. G., Eds.; John Wiley: New York, 1987; pp 101-126. Linnhoff, B.; et al. A User Guide on Process Integration for the Efficient Use of Energy. Technical Report, The Institution of Chemical Engineers, 1982. Underwood, A. J. V. Fractional Distillation of Multicomponent Mixtures-Calculation of Minimum Reflux Ratio. J. Znst Petrol. 1946, 32, 614.
Literature Cited Andrecovich, M. J.; Westerberg, A. W. A Simple Synthesis Method Based on Utility Bounding for Heat-Integrated Distillation Sequences. AIChE J. 1985, 32(3), 363. Hohmann, E. C. Optimum Networks for Heat Exchange. Ph.D. Thesis, University of Southern California, 1971.
Underwood, A. J. V. Fractional Distillation of Multicomponent Mixtures. Chem. Eng. Prog. 1948,44, 603. Received for review November 28, 1988 Revised manuscript received May 19, 1989 Accepted June 15, 1989
Temperature-Heat Diagrams for Complex Columns. 2. Underwood’s Method for Side Strippers and Enrichers Neil A. Carlberg and Arthur W. Westerberg* Department of Chemical Engineering and Engineering Design Research Center,+ Carnegie-Mellon Uniuersity, Pittsburgh, Pennsyluania 15213
Insight into complex column heat flow facilitates extension of Underwood‘s method to columns with side-stream strippers and side-stream enrichers. This presentation is much simpler than the previous ones. The same insights show that, even though these configurations are more energy efficient, they require a larger temperature range for operation than analogous simple column sequences. When designing a distillation column, it is imperative that the minimum reflux ratio is known. This parameter is critical; it determines a lower limit to column operation. The reflux ratio sets the internal flow rates of the column, which, in turn, determine the utility consumption and column diameter. A t some point, as the reflux ratio decreases, one or more of the operating lines of the column will intersect the equilibrium surface. These intersection points are known as pinch points. An infinite number of trays are required to pass through a pinch point. Thus, the pinch points determine a minimum value for the reflux ratio. Normally, a column is operated just slightly above this minimum value. To find the minimum reflux ratio rigorously, a set of simultaneous nonlinear mass and energy balances and equilibrium relationships must be solved. This approach is often difficult. In order to find a solution quickly, several shortcut methods have been developed. The most notable of these shortcut calculations is the classic method of Underwood (1946,1948). For separations with constant relative volatility and constant molar overflow, an algebraic construction is used to obtain a simple solution procedure. Although Underwood only considers simple columns, the analysis can be extended to complex column configurations. Recent works address the issue of shortcut methods for complex columns. Glinos and Malone present the correct algorithm for analyzing a column with a side-stream stripper (Glinos, 1985; Glinos and Malone, 1985) and a column with a side-stream enricher (Glinos, 1985). The side-stripper analysis is developed for a ternary mixture; the generalization to the n component case is assumed to hold but is not proved. The authors base their results on estimating the location of the pinch point in the side column. A step in their development assumes that the composition of the liquid return stream from the side column is at the pinch composition. While true for the ternary case, this as-
* Author to whom correspondence
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sumption is not always true and in fact, as it will be shown, is not necessary to make. For complex columns, the overall minimum reflux is achieved only when each column is at its respective minimum. The authors observe and use this fact but do not prove it. In a later paper, Glinos and Malone (1988) formulate several design rules with regard to when to favor various complex column configurations. The criteria for these design rules is the overall reboil rate. It will be shown, however, that the temperature range over which a complex configuration operates is also an important design consideration. Fidkowski and Krolikowski (1987) present a method to find the minimum energy requirements for a side stripper and side enricher. They restrict their analysis to ternary mixtures, and the development is very complex algebraically. Underwood’s method is used as a basis for an optimization procedure. The appropriate vapor flow rate is minimized subject to internal mass balances and pinch point constraints. By choice of the proper objective function, the overall reflux is minimized. An analytical solution is obtained by observing the effect of the decision variables on the objective function. The authors then compare complex columns to the equivalent simple column sequences on an energy usage basis. Because the analysis is limited to ternary mixtures, their results are not generalized to an n-component mixture. With the complexity of the algebra involved in their derivation, such an extension would not be easy. There is a much cleaner and more general approach to obtain these results for multicomponent columns. This is a principle contribution of this work. This paper presents a straightforward generalized multicomponent Underwood analysis for several complex column configurations. First, the side stripper and side enricher are analyzed with Underwood’s basic principles. Insight into complex column heat flow allows for the formulation of a simple solution strategy. This strategy is readily extended to multiple side strippers and side enrichers. Finally, complex columns are compared to analogous simple column sequences. By use of the same heat flow insights, 0 1989 American Chemical Society
