Ptirnum Feed Tray in Multicomponent Distillation Calculations EDWARD G. SCHEIBEL' AND CHA4RLESF. 310NTROSS2 Polytechnic Institrite of B r o o k l y n , Brooklyn,S. 1..
I[II this
e
work the pre\ious criterion for determining the feed tray in multicomponent distillation calculations has been found to be inaccurate; empirical equations are presented by which the ratio of the key components in the optimum feed-tray liquid can be calculated directly. This ratio has been found to vary with the reflux ratio, and it has been expressed as a function of the limiting valuesthat is, the ratio of the key components in the feed-tra? liquid at total reflux and also at minimum reflux.
The ratio of the keys i n the feed-tray liquid at total reflux is calculated by a simple relation; the ratio of the keys in the feed-tray liquid at minimum reflux has been correlated to an empirical equation based on a study of the same cases used to develop a previous empirical equation €or minimum reflux i n multicomponent distillation. The location of the optimum feed tray by this method has been compared with that determined by actual tray calculations 011 about eighty systems. The agreement of the calculated optimum feed tray with that determined by the traj calculations was within the accuracy of determining the latter value. The equations have been found applicable to the range of conditions usually encountered in process design wrork.
Figure 2 shows the tray ca1culat)ions for a system in which a large fraction of the feed consists of components heavier than the heavy key as in the case of a dcpropanizer column. In surh ;I case the separation will be more difficult and may be impossible if the feed is introduced a t the intersection of the operating lines on this diagram. The best feed tray will be several trays aboyt. the intersection. This can he understood readily by considering the location of the equilibrium curve for the key components, as shown. If the components heavier than the keys were assunicd t'o be substantially nonvolatile, the multicomponent equilibrium rurve would coincide v-ith the equilibrium curve for the key c o ~ n ponents above the feed tray. For such a case, if t,he feed ic introduced at the intersection of the operating linea. t,he desircd separation is impossible. By introducing the feed hiphcr in idle column, it will be' possible to effect the desired separation at the given reflux ratio in a finit,e number of trays. This indicat,es t,hnt t,he optimum feed tray is not wholly dependent on the ronceritt,ation of the light comp0nent.s in the feed. The purpose of the present 1voi~kwas to oi)taiii a r o i d a t i o i l for the location of the optimum feed tray. This correlation was based on the ratio of the key components in the optimum fcetl tray and empirical correct>ion factors were introduced for the components in the feed which are lighter and heavier t'han the keys. In order t o develop this correlation, it was neccsswy first to devise a method for locat'ing esactly the optimum f t . c A t l tray for a multiconiponent distillation. DETERMINATION O F OIyrIR!KJ.M FEEI) THAY
I
S T H E design of distilling columns, the location of the feed tray is equally as important as the total trays required in the column. An improper point of introduction of the feed t o the column will make the separation appear more difficult than it would be at the proper feed location. This can be seen most readily by considering a binary system in which both components have the same molal latent heat. I n this case, the operating lines on the conventional McCabeThiele diagram (1) are straight, as shown in Figure 1. I n this figure, feed is introduced five trays above the intersection of the operating lines and it is apparent that if the feed had been introduced at the tray just below the intersection, only two additional fractionating trays would be required as conipared to the five additional stripping trays required, as shown. Thus, the optimum feed tray can be located easily for binary distillation because i t ie that tray which straddles the intersection of the operating lines. However, in the case of a multicomponent system, it is more difficult t o locate this optimum feed tray. Jenny and Cicalese ( 4 ) have proposed a method for representing the tray calculations on multicomponent systems. This method consists of combining all components lighter than the keys with the light kry and all components heavier than the keys with the heavy key, and constructing operating lines and equilibrium curves on a twocomponent diagram. The equilibrium curve must be based on a set of actual tray to tray calculations. 1 2
Present address, Hoffmann-LaRoche, I n c , Niitley. S J Present address, The Lummus Company, S e n TorA A Y
The optimum fced traj- for any distillation is that traj- upon which the feed can be introduced to give less total trays for t,lie desired separation than will be rcquired by introducing the feed on any other tray in the column. Figure 3 shows the results of H series of tray calculations using different feed-tray locations. This curve shows that the optimum feed tray cannot be located accurately from this type of plot and in order t o study this problem and to develop a reliable correlation, accuracy is the firet requisite. The exact optimum feed tray will be t h a t at which the slopc of the curve in Figure 3 is zero. Thus, the total t'rays may bc cspressed as the sum of the fractionat'ing and st,ripping tra? 7' = F S. Differentiating with respect, t o t,he numbel stlipping trays, S:
-+
dT F+I dS = -d&
=o
Hcnce, the optimum feed tray corresponds to the tray when:
AF By differentiating the data and plotting - as shown in Figuitl AS dF 4, the curve of - can be drawn and the point where this function dS is equal t o - 1 can be located easily. By superimposing a plot ot the fractionating trays and the ratio of the keys in the feed-tra\T liquid on this figure, the values of these functions a t the optimuni feed trav were also determined. This rigorous method is extiemely tedious, requiring many hours of calculation, and is not usually used in design work. Instead, other approximate mcthotlme uscd ( 2 , d ) . The results of thew methods have hcen fouutl T ~
1398
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INDUSTRIAL AND ENGINEERING CHEMISTRY
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1399
TABLEI. CALCULATIONS SHOWING VARIATIONOF OPTIIIIX FEED-PLATE COMPOSITION WITH REFLUX RATIO 1. R - R .VI Case Nan
RM 2 88
R
R
0 424 15.0 0.809 7’ 1.127 2.0 0,436 3.0 0.625 10.0 0.887 14 4.7’6 7.38 0.355 23.4 0 797 Caves are the same a$ 1 , 7, and 14 in Table 1
TRAY LOCATION REQUIRES THREE TRAYSMOREW OTHER LOCATION
5 0
rM 0 727
0.971
r
TM
0 652 0.570
0.898 0 785 0.819 0.700 0.569
0,795 0.68 0.553 0.385 0 476
0.310
1 241.
1 5.32
11.
Table I shows the summary of the calculations on three of the cases and the results are plotted in Figure 5.
R
- Rx
Extrapolation of these lines to total reflux, i.e., ___ = 1,
R
gave an intercept of
L
MOLE FRACTION OF LIGHT COMPONENT IN LIQUID
rT TM
and it was found that
TT = ratio of key components in optimum feed tray at total reflux; TF = ratio of key components in feed; and OIB = relative volatility of key components.
Figure 1. Effect of lrnproper Feed-Tray Location on -NcCabe Thiele Diagram for Two-Component System
diffei from the results of the rigorous treatment; a t reflux ratios close t o the minimum the previous criteria would make a separation appear impossible, whereas a more thorough investigation will show that the separation can be achieved. The purpose of this work was t o develop a dircwt method which gave the same results as the rigorous treatment without the large numbe? of celculations.
Figure 3. Effect of Feed-Tray Location on Total Trays Required i n Column
EFFECT O F REFLUX RATIO ON OPTIMUM FEED-TRAY LIQUID COMPOSITION
A study of about thirty different three-component mixtures at varying reflux ratios indicated that a straight line was obtained when the function of
rM
was plotted against
R
- RN
___.
R
LOCATION OF FEED TRAY TRAYS ABOVE BOTTOM
where
= ratio of light key to heavy key in optimum feed-tray liquid at reflux ratio, R; rail = ratio of light key t o heavy key in feed-tray liquid at minimum reflux; and R M = minimum reflux ratio = ratio of liquid reflux t o overhead product. T
The agreement of the observed values of -i- with the lines in r,lI
Figure 5 is considered within the accuracy of determining the optimum feed tray. This was confirmed further by applying this method of optimum feed-tray location t o several two-component systems, as mentioned in a previous article (6). At total reflux there is no actual feed t o the column. However, the value of r , is~ the limiting value that is approached as the reflux ratio is incrcascd t o infinity. Also, it might be considered as the optimum point for the introduction of a differential amount of feed to the column operating at total reflux. The minimum reflux ratios and thp ratios of the key coniponents in the feed-tray liquid a t minimum reflux were determine51 from the calculations discussed in a previous article ( 7 ) . These calculations are based on the concept of a theoretical tray-that is, a tray with a n efficiency of loo’%. This value is approached and can be exceeded in practice because of the liquid concentration gradient across the tray. A t low efficiencies, the value of TT for the optimum feed tray at total reflux would be nearly equal t o r F and the value of rz’ may be more accuratelv expressed :
*B
s
MOLE FRACTION OF LlOHT COMPONENT IN LIQUID
Figure 2.
= l /where z rp
rT
Effect of Improper Feed-Tray Location in iMulticomponent Distillation
’
where E , = Murphree efficiency ( 5 )expressed as a fraction. To use this concept the tray-to-tray calculations would have t o be carried out for every actual tray and the concentrations corrected according t o the Murphree efficiency of the tray. This method is tedious and never used in practical design calculations. Multicomponent tray calculations are based on the concept of a theoretical tray; this concept was used to develop the present correlations.
INDUSTRIAL AND ENGINEERING CHEMISTRY
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Vol. 40, No. 8
I
UA OB all
X.4 S B
XC r$ 1;
I I
I
e 8 IO Na OF STRIPPINO TFAYS
.64
Figure 4. Determination of Optiiriiiiii Feed Tra?
Figure j. Variation of Katio of Key Components in Optimum Feed-Tray Liquid with Kefliix Ratio
Thus, the ratio of the keys in the optimum feed tray a t any reflus is given. 7' =
r.,J
-
LIL2 12
()..,I
-
).T?)
= ,1' t
K_1i.1 R
(/
',J
- ?T)
(2)
FEED-TRAY LIQUID CO%IPOSITION A T :MINIMUM REFLUX
I n order to apply the previous relation to the determination of the optimum feed-tray composition, i t is desirable to determine ~ , > f directly. Thus, an empirical correlation was developed by a method somewhat similar to that which had been applied successfully to the correlation of R.u previously reported ('7). The same calculations for extreme cases x-ere used in developing tho correlation. It !?as found necessary to include an additional corrertion factor for the fmction of the feed entering the column a.s vapor. The filial correlation is a3 follows:
relative volatility of components lighter than l x > s with respect to heavy key = relative volatility of light key to heavy key = rclat,ivc volatility of components heavier tlian keys with rcspect t,oheavy key = concentration of each component lighter than keys in feed = concentration of light key in feed = concentration of heavy key in feed = concentration of each component hcavicr than keys in feed = ratio of keys in feed-tray liquid a t the pseudo niiiiimum reflux ratio obtained from the binary diagram (7) = i'raction of feed vaporized = fraction of feed as liquid =
This relation is empirical but JT-as developed for the same csti'cme range of conditioiis as in the previous article arid therefore is applicable generally t,o the cases encountered in practice. The equation was applied to 106 different cases varying the composition, relative volatility of components, sharpness of split, and condition of the feed. The cases studied represented a comprehensive survey of those encountered in process design calculations and t,he ayerage deviat,ioii of this equation varied from about 3% to 1oyoy1 the extreme cases. The ratio of t,he key components in the feed-tray liquid Tvas found to vary considerably with the minimum reflux ratio. Conse uently, this deviation was considered \Tithin the reliability of ?he determination of minimum reflux ratio. This latter error was about 1% in the usual cases encountered in practice and 7.755 in the extreme cases. Table I1 shows the agreement of the equation with the actual values for tn.erity of the caws studied. hPPLICAT1O.U OF EQUATIONS TO YETER.\.IINATION O F OPTI.MUBI FEED TRAY
The empirical method previously described for optimuni feedtray location was applied to about eighty difforent eases of varying feed compositions, relative volatilities, reflux rat,ios, arid sharpness of separation; average deviations were of the order of 0.1 tray. This was considered to be L ~ S accurate as the detcrmination of the optimum feed tray by graphical differentiation because. in many cases, only three calculations were made at different feed-tray locations and the deriva. .,' v2>r = (3) dF dS 1 + I.'> O D ( 1 - a u ) S o + I.'? - Fs tive curve of - or - was assumed straight in the d9 dF I', (1 - a B u ) c g X B (3a) region studied. This rvas considered sufficient!y accurate for (a3 - 1 ) ( S B + S C ) ( l - ZA-A) practical purposes. However, in practically all cases, the (ea - 1)*1 (3b) calculations a t additional feed trays gave derivative curves (sa+ S c ) ( 1 - Z S D ~ which \\ere in better agreement with the feed trays , a t the calculated values of r and the deviations were attributed TABLE11. C O h l P A R I S O S O F h ' l Y O O F K E Y C O h I P O S E N T S I N FEEII-TRAY LIQVIl> A T to insufficient calculation^ in deterMI XI^,^ REFLUX WITH T-ALUX CALCXLATED B Y EWATIOX 3 mining the theoretical optimum feed R a t i o of Key tray. &laterial Balance Comuonents i n Relative Rejection Retention of B , Feed-Tray a t filin. Reflux Liwid Table 111 shows the comparison of Case Feed Composition Volatilities 6 of c, the feed-tray locations for the same X 0 . O XA XB XC S O aA OB OD 75 % hctual Calcd. , . 0.30 0.40 0.30 . . 2.00 0.80 0.75 2.33 0.727 0.720 cases reported in a previous article (6). 1 2 , . 0.30 0.40 0.30 3.00 0.60 0.75 2.33 0.737 0,716 I n all t,hese cases where there is any 3 0.12 0.16 0.72 .. .. 2.00 0.50 0.63 7.50 0.650 0.650 4 . 0.15 0.55 0.30 .. 2.00 0.50 0.18 6.00 0.270 0.264 difference in the t,otal trays, the number 5 , . 0.05 0.05 0.90 . , 2.00 0.50 2.00 20.00 0.817 0.819 a t the calculated value of 1' in the feed 0.24 0.04 0.72 2.00 0.50 2.25 3.33 4.59 4.68 B 7 0130 0 30 0.40 . . 4:00 2.00 ., .. 1,50 1.33 0.971 0.958 trag is less than a t the previously 8 0 30 0.30 0.40 , , 4.00 1.30 1,50 1.33 0.868 0.840 9 0.72 0.12 0.16 .. 8.00 2.00 . . 1.00 10.00 2.01 2.09 determined optimum. This shows the 10 0.90 0 05 0.05 . , 4.00 2.00 .. 2.00 2.00 2.59 2.67 limit,ations of the method of determina11 0.90 0.05 0.0.5 2.50 2.00 2.00 2.00 1.495 1.41 12 0.30 0.20 0.20 0:dO 4.00 2.00 0:;0 2.50 2.50 1.322 1.30 t,ion of the exact optimum feed tray 13 0.40 0.10 0.10 0.40 4.00 2.00 0.50 LO0 8.00 1.9 1.75 14 , , 0.30 0.40 0.30 .. 2.00 0.50 0.75 2.33 0.310 0,311 and would indicate the necessitl of 15 0.24 0.04 0.72 2.00 0.50 Segligible Xegligible 0.633 0.589 more calculations to estahlish this value 16 0 : 3 0 0.30 0.40 .. 4:00 2.00 .. 1.50 1.33 0.399 0.381 17 0.72 0.24 0.04 4.00 2.00 Segligible Negligible 3.64 3.77 accurately. The calculated optimum 18 0.40 0.10 0.10 0:40 4.00 :.00 0 . h 0.345 0.325 0.143 0.1354 feed t,ray \vas located from the calcu0.30 0.30 0.40 0.00 0.50 19 0.471 0.472 20 0:72 0.12 0.16 _ . 4:OO 2.00 .. lated value of 1' on the curve of ratio of the keys in feed-tray liquid against a Cases 1 t o 13 are f o r l ~ y u i dfeed a t t h e feed-plate temperature; cases 14 t o 20 are for vapor feed at feed-plate temperature. stripping or fractionating travs reb In all cases ac = 1.00. quired.
c
, , ,
r.,
2%)
:yp
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INDUSTRIAL AND ENGINEERING CHEMISTRY
August 1948
,
.
assumed feed-tray temperatures indicated this t o be correct. However, the curve of total trays against feedtray temperature is very flat over the range of 195 ’ to 205 F. At this reflux ratio, the optimum feed-tray tempere ture does not have to be determined accurately. At reflux ratios close to the minimum the determination of optimum feed-tray temperature is very critical. It was found also that a t fzed-tray temperatures more than 10 F. outside of this optimum range, several extra trays will be required.
TABLE 111. FEED-TRAY LOCATION
___-_ Number Fractionating
of Theoretical Trays---Stripping Total Calcd. Actual Calcd. Actual Calcd. Botual R RAf rM rT 8.92 16.15 16.17 8.93 7.22 7.25 9 3.00 1.127 0.958 0.530 0.692 16.75 16.77 6.95 6.70 1.638 0.530 1.279 10.07 9.80 B 0.74 0.500 9.32 19.37 19.38 9.14 0.705 0,530 0.643 10.23 10.06 C 4.00 2.51 16.65 16.66 6.43 6.27 0.705 0.530 0.654 10.38 10.23 II 3.50 2.48 16.70 16.70 ‘6.93 7.09 9.61 9.77 0.530 0.613 0.650 E 2.85 4.05 7.21 7.20 16.45 16.43 9.25 9.24 0.311 0.530 0,382 F 7.38 4.60 14.26 14.30 7.10 7.20 7.06 7 . 2 0 0,707 1.010 G 1 78 0.913 1.30 Balance of cas@ based on liquid feed a t a Case F based on vapor feed a t feed-plate temperature. feed-plate temperature.
Case No.n
1’
CONCLUSION
The columns of r , and ~ TT in this table also indicate the wide variation of the ratio of the key components in the feed-tray liquid between minimum reflux and total reflux. According to the previous method of Gilliland (2),the optimum feed tray in cases A t o E would be that on which the ratio of the keys in the liquid was less than 0.75 and would require that this ratio be greater than 0.75 on the tray above the feed a t all reflux ratios. Thus, considerable error would result from the application 01 this criterion, particularly in the region close to minimum reflux. The previous method (8) applies exactly at total reflux and reasonably well at reflux ratios close to total reflux, The method of Underwood (8) gives essentially the same results as that of Gilliland and would include the same objections. APPLICATION OF EQUATIONS
As a sample calculation, the present empirical method can be applied to the multicomponent tray calculations of Jenny (23). The calculation of minimum reflux ratio for this case was demonstrated in a previous article ( 7 ) . At the feed tray, the following relative volatilities of the components were obtained (7) : XF 0.26 0.09 0.25 0.17 0.11 0.12
ff
20.6
5.0 2.0 1 .o 0.44 0.21
It was previously determined that the pseudo feed line intersects the binary equilibrium curve at 0.470, thus
The empirical equations previously presented give the ratio of the light to heavy key in the optimum feed-tray liquid. The equations have been developed for the range of conditions usually encountered in practice. Without doubt they have limitations in extreme cases for feeds consisting of less than 10% key components and particularly ,when the relative volatilities of all the components in this feed are very close (within 10%). The accuracy of the equations was found t o be within the reliability with which the optimum feed tray could be located by a series of tray calculations a t different feed-tray locations. The method is particularly applicable to tray calculations by the Jenny method (9) of assuming a feed-tray temperature and calculating the feed-tray liquid composition. By the use of these equations, the ratio of the keys in the optimum feed tray can be calculated directly. A comparison with the ratio a t the asspmed feed-tray temperature will indicate immediately whether the optimum feed tray occurs close to the assumed temperature or at a higher or lorver temperature. I t is thus possible to assume a second feed-tray temperature which will be closer to the optimum value. A final set of complete tray calculations can then be made Ft the optimum feed-tray temperature to determine accurately the trays required above and below the feed. The present method eliminates the tedious trial and error procedure of carrying out tray-to-tray calculations using different feed-tray locations to determine which location requires the smallest number of total trays for the desired separation. LITERATURE CITED
(1) B a d g e r , W. H., and McCahe, W. H., “Elements of Chemical Engineering,’’ New York, McGraw-Hill Book Go., 1931.
v
= 0.66
1 = 0.34 (1
aD(l
-
aD)XD = (2.0
- 2.0 X
0 6612.0 X 0.25
- 1)(0.25 + 0.17)(1 - 0.26 -
- 0 4410.11 f 0.21(1 - 0.21)0.12]
[0.44(1
0.09)
-0,0275
XA2= rrA
-1
-
(2 0 1)20 34 ( 0 . 2 5 + 0 17)(1 0 11 - 0 12)1-c
-
0 262 +
-1
RECEIVED April 25, 1947. 09*
=
0 0058
(2.0 - 1)0.66
XAZ
F a x
=
+
2,.0(0.25 0.17)(1
20.6 L O . 6 IM =
7
=
-2 0 - 1.0O
- 0.11 - 0.12) -
5.0 2.0 0.092] 26z + 5.0 - 1.0
0.887
1
- 0 0275 + 0.0058 - 0.0717
1,040
=
(2) Gilliland, E. R., IND. ENG.CHEM., 32,918 (1940). (3) Jenny, F. J., T r a n s . Am.Inst. Chem. Engrs., 35,635 (1939). (4) Jenny, F. J., and Cicalese, M. J., IND. ENG.CHEM., 37, 956 (1945). (5) Murphree, E. V., Ibid., 17,747,960 (1925). (6) Scheihel, E. G., Ibid., 38, 397 (1946). (7) Soheihel,E. G., a n d M o n t r o s s , C. F.,Ibid., 38, 268 (1946). (8) Underwood, A. J. V., T r a n s . Inst. Chem. Eng. ( L o n d o n ) , 10,112 (1932).
Heat Transfer-Correction =
0.0717
0 978
+ O 1.5 x 5 (0 978 - 1.040) = 1.000
Thus, the calculated optimum ratio of the key components in the feed-tray liquid is 1.000 and the ratio of the keys in the feed-tray liquid a t 205’ F. has been calculated by Jenny as 0.872. This indicates that the assumed feed-tray temperature ia too high and another trial a t 200” F. will give the desired o timum ratio. Accurate tray calculations on this separation a t Jfferent
In the article “Heat Transfer with Extended Surface. Determination of the Local Heat Transfer Coefficient from the Average Coefficient,” by C. F. Bonilla [IND. ENG.CHEM.,40, 1098-1101 (1948)1, the expression tan h should be replaced by tanh, or hgperbolic tangent, twice in Equation 4,and in Equations 5 and 6, and undyr example a. In the following article, “Heat Transfer with Extended Surface. No Mixing Parallel to the Extended Surface,” by W. E. Dunn, Jr., and C. F. Bonilla, the same misprint is present in Equation 3, three lines under Equation 3, and in example 6. The symbol ta 2 in examples a and b, and in the definition of T in the nomenclature, should have a bar over it to indicate that it is the average temperature of the fluid leaving.
*
