AUGUST, 1940
INDUSTRIAL AND ENGINEERING CHEMISTRY
Literature Cited (1) Farmer, R . C . , J. SOC.C h e m . I n d . . 50,751‘ (1931). (2) Gilman, H . , “Organic Chemistry. an Advanced Treatise”, Val. 11, p. 111, New York, John Wiley & Sons, 1938. (3) Grignard, V.,“Trait6 de chimie organique”, Vol. IV, p. 232, Paris, hlasson et Cie., 1936. (4) Groggins. P. H . , “Unit Processes in Organic Synthesis”, p . 23, New E-ork, McGraw-Hill Book Co., 1935. (5) Hantssch. A , , Ber., 50, 1422 (1917), 58, 612, 941 (1925). 60, 1933 (1927): Z.Elektrochem., 24,201 (1918). 29,221 (1923),30, 194 (1924), 31,167 (1925); 2 . p h y s i k . C h e m . , 134,406 (1928). (6) Hetherington, J. A , , and Masson. I., J . Chem. Soc.. 1933, 105. ( 7 ) Holleman, -1.F . , “Die direkte Einfuhrung Ton Puhstituenten in den Bensolkern”, pp. 71, 97, Leipeig, Verlag von Teit und Camp., 1910. Sal-age. W., and Van Malle. D. J . , C h e m . 12 .\let. ( 8 , Hough, -1., Eng., 23,666 (1920). (9) International Critical Tables, 1-01. 111, p. 307, S e i v York, McGraw-Hill Book Co., 1928. (10) Kullgren. C., Z . ges. Schiess- Sprengstoflw., 3 , 146 (1908). (11) Lauer, K., and Oda, R.,J . prakt. Chem., 144, 176; 146,61 (1936).
1101
(12) Martinsen, H . , Z . p h y s i k . Chem., 50, 385 (1904): 59,605 (1907). (13) Othmer. D. F . , ISD. EEG.CHEM.,20,743 (1928). (14) Parks, G . S., and Huffman, H. M.,“Free Energies of Some Organic Compounds”, A . C. S. Monograph 60, p. 221, Xew York, Chemical Catalog Co., 1932. (15) Pounder, F . E., and Masson, 1..J. C‘hem. SOC.,1934,1352. (16) Saposchnikow. A . W., 2. p h y s i k . C h e m . , 49, 697 (1904), 51, 609 (19051, 53, 225 (1905); Z. ges. Schiess- Spreng&o$w., 4, 441, 462 (1909). (17) Schaeffer, K . , Z. anorg. allgem. Chem., 97, 285; 98, 70, 77 (1916). (15) Bpindler, P . . B e r . , 16, 1252 (1883); Ann., 224, 283 (1884). (19) Suen. T . J . , M a s s . Inst. Tech., 9c.D. thesis, 1937. (20) Taylor, H . S.,“Treatise on Physical Chemistry”, 2nd ed., Vol. 11, p. 1026, Kew York. D. Van Nostrand Co., 1930. (21) Tronov, B. V. et al., J . Russ. Phys. Chem. SOC.,62, 2267 (1930); Ckrabz. Khem. Zhirr.. 7 , No. 1, Sci. pt. 55 (1932). (22) Walker, W.H . , Lewis, W. K., and McAdams. W. H., “Principles of Chemical Engineering”, 2nd ed., p. 128, New Tork, MrGraw-Hill Book Co.. 1927. (23) Wibaut, J . P . . Rec. trav. chim., 34,241 (1915); 54,409 (1935). (24) Wyler. O., Hela. C‘him. .4cta. 15, 23, 591, 956 (1932).
MULTICOMPONENT RECTIFICATION Minimum Reflux Ratio’ E. R. GILLILAND Massachusetts Institute of Technology, Cambridge, Mass.
.4pprosimate equations are derived for the estimation of the minimum reflux ratio for multicomponent mixtures, and these equations can be used to calculate upper and lower limits for the true minimum reflux
F
OR a given separation by rectification, the minimum number of theoretical plates a t total reflux and the minimum reflux ratio corresponding to an infinite number of plates render valuable aid to the designer, by defining the limits within which his calculations must be confined to give an operabIe design. Satisfactory methods for the calculation of both of these factors have been developed for binary mixtures. However, for the case of multicomponent mixtures, the problem is more complev than for binary mixtures, and only for the case of the minimum number of theoretical plates a t total reflux have suitable methods been developed ( I ) . Several methods for predicting the minimum reflux ratio for multicomponent mixtures have been published; of these, that of Underwood (6) has been the most widely used by other investigators. Underwood’s equation is:
This equation was derived on the basis of assumptions which were equivalent to assuming that for the condition of minimum reflux the concentrations of the key components mere the same on the feed plate as in the feed. Such an assumption i, true for binary mixtures of normal volatility but is seldom triie for multicomponent mixture xhk
511;
1 -
n
- s xi
5ik
1
=
The solution of this equation is the same as Equation 1.5, but instead of I and J , I ' and J', respect'ively, are used. The method of calculating the minimum reflux ratio by t,liese equations will be illustrated by the following examples.
ESARIPLE 1, A-B-C mixture; feed enters such that I:n
v,>l:
--
Feed
0.10 0.10 0.80
.I B
c
D @
l l o l e Fraction-?Distillate Residue 0.95 n ,00556 n.n5 0,10556 .. n ,5589
( 0 9 (2
- 0.05) - 1)
( O / D ) , , = 1.7
0.26
(0.95) ( l l 5 . 5 )
+ 1.26(0.05/0.25)(1/5.5)
= 0.57
The value of x h h in the last term of the denominator was asumed to be 0.25, but this term makes little difference in 3: . Then
These corrections give a R value of 0.594 and
n
10, Tt' = 90 per 100 moles of feed. = 1. By Equation 9,
+1-
=
-
Yulatilities Relative t o Component B 2.0 i n
=
();
Using this value of (O/D),*,the neglected terms can be calculat,ed. By Equation 2, xfi;can be approximated as
xh
= m
(O/D)," = 5.5
(2
By Equation 6,
+ 1) = 2.7
(O/D)., = 5.57
This value is essentially the same as before the recalculation was macle, and in general the correction can be neglected unless the difference between alk and a h i r is quite small. Equation 9 gives a value of ( O / D ) mequal to 5.4, and Equation 11 gives ( O / D ) , equal to 5.55. Using the assumptions of case 11, by Equation 5:
AUGUST, 1940 Zy = h
INDUSTRIAL A S D ESGINEERIKG CHEhlISTRY
-[
66.8 (0.663)(0.449) J77n (0.597) ~
+
( 0 . 3 9 4 )(0.224) _ 0.866 _
+
1103
limits for the true value but the limits are much wider than those obtained by the use of Equation 9 or 10 together with Equation 13 or 14.
~
1.173
EXAMPLE 3, gasoline qtabilization; example from Robinson and Gilliland (5, page 176) :
Assuming (O/D),, = 5 , then T-,,, = 199.2 and
zy
= 0.201; Z y = 0
1
h
=
35
+
C1H6 C3Hb C3Hs Iso-CdHm
1
( r o m j- 1) (66.8)(0.0524) = 1.95
1 l 5 + (1 - 0.204
+1 =
0 0 0 0 0
C b
CT C8 Residue
[(0.95/1.95) - 0.05][(1.26)(1.95) 0.26
+ 11
=
j.8
(O/D)r,z= 4.8
Recalculating 4' for ( O / D ) , , = 4.8 gives a value of 1.94 but does not change the value of 4.8 significantly. Equation 14 gives a value of (O,lD), equal to 4.7. Equation 15 gives a value of (OID),, equal to 6.0 for case I and 4.7 for case 11. By Equation 1, 1'26
0 035 0 1.50
n-CI C6
Using Equation 13,
();
Feed 0 02 0 10 0 06 0 125
c Ha
From Equation 12,
+'
Alole FractioiiDi.tillate 0 0683 0 1(16 0 190 0.390
7 -
(2)
0.35(0/0), 0.15(O/D), (OID), = 8.83 =
+ 0.95 + 0.05
Detailed stepwise plate calculations indicate that the true minimum reflux ratio is approximately 5.2. Equations 9, 10, and 11 give values about 6 per cent too high, and Equations 13 and 14 give values about 8 per cent too low. These percentage deviations from the true value are a function of the concentrations of the components other than the key components for the middle portion of the tower; the deviations are higher when these concentrations are high. The values given by Equation 15 show higher percentage deviations fro? the true minimum reflux ratio than the other equations. The value given by Equation 1 is again much higher than the true value. I n the derivation of Equations 13 and 14 it was pointed out that they could be used with+ calculated from Equation 6 instead of for Equation 12, but that the values ryere liable to be erratic. Using Equation 13 with equal to 2.33 gives a value for ( O I D ) , of 4.4, whereas Equation 14 gives ( O I D ) , = 6.3; and in general results of this type will be obtained. The equation based on conditions above the feed will give values on one side of the true value while the equation based on conditions below the feed will give values on the opposite side of the true value. Thus Equations 13 and 14 can be used with instead of +' to give upper and lower limits for the minimum reflux ratio, but in general the region defined is broader than the region defined by Equations 9 or 10 with and Equations 13 or 14 withqb'. If the feed to the column had been a vapor such that O,, = Q,,,,the calculations would require trial and error to obtain a n d 4'. Assuming ( O I D ) , for this case to be 7.0, then Vrt= 266 and = 166. By Equation 6 , + = 1.61. The value of ( O / D ) , by Equation 10 is calculated to be 7.03, and the assumed value of ( O I D ) , is satisfactory; but if the difference between the assumed and the calculated values had been larger, a recalculation mould have been necessary. Equation 9 gives a value for (O/D),, of 6.8 and Equation 13 gives a value of (OID),, equal to 6.3, which indicates that the true minimum reflux ratio is between 6.3 and 7.0. Equations 13 and 14, together with r$ instead of give values for ( O / D ) , of 5.6 and 7.0: and as indicated before, this method gives
+'
+
+
0 .'do.? 0.041 0.211 0 222 0 1H3 0 132 0 124 0 10"
0 022 0 019
...
152 113 090 08.3 070
\'olatilities Relative to n-C1Hu 26 G 2.6 2 .3 1.2:1 1.0
Redidue
...
... ... ,..
0.4,3 0.18
0.08 0.03 0.00.j
D = 31.6, TV = 68.4 per 100 mole; of feed; feed enters such that I??,,- i7,& = 0.42 F . This case differs from the previous examples in t h a t both light and heavy components are present, and in the fact that the component isobutane is between the two key components propane and n-butane. The distribution of the isobutane given is that used a t a value of ( O / D ) = 1.5 from Robinson and Gilliland, and this distribution will change as the reflux ratio is varied; but since the amount of this component is small, it mill not greatly influence the results and will be allowed to remain a t the values given. In using the equations there is the problem of whether isobutane should be a light or heavy component; but since its relative volatility is nearer to that of the heavy key component than that of the light key component, it will be considered as a heavy component. Assuming a value of (OID),, = 0.7, J',, = 53.7, V,,, = 9 5 . 7 , O,, = 22.1, and O,,, = 164.1. By Equation 4, (2.3)(0.21 1) (68.4/164.1) = 1.3
z h
A =
(26)(0.0633) (6)(0.316) 1.04 25 4.5 + 5
B =
(1.25)(0.041) 1.07
+
+
+
+
2.6(0.19) , 0.45
1.6
(0.43) (0.222) 1.87
+
o. 156 ~
o.6,
(0.18)(0.165) +
2.12
+
(0.08)(0.132) (0.03)(0.124) (0.005)(0.102) = 2.22 + 2.27 + 2.295
o.l
19
By Equation 6,
'
12.5 = 15.0
+ [(53.7/95.7) - 1](68.4)(0.0025) = 1.43j + [(53.7/95.7) - 1](68.4)(0.211)
Since R is less than -4,Equation 9 will be used in preference t l ) Equation 10. By Equation 9,
+
+
+',
(O/D),, = 0.68
Calculating the lower limit by Equation 13, and assuming (O/D),, = 0.4 V,L = 44.2; 0,, = 12.6 V,, = 86.2; O,, = 154.6
By Equation 5 ,
1106
INDUSTRIAL AND ENGINEERING CHEMISTRY
6(0.316)(1/1.4) zy = 26(0.0633)(1/1.4) 25 0.105 + 5 0.105 + 1 2.6(0.19)(1/1.4) = o.52
+
xwh
+
1.6
VOL. 32, NO. 8
+ 0.105
zy = 0.119
ai
h
-
ahk
+
al
-
By Equation 12,
=
(s9) (w9)
+ [( g ) 15.0 + [(46) 12.5
Q/
-
11(68.4)(0.0025)
=
2.75
- 1 1 (68.4)(0.211)
From Equation 13,
( O / D ) , = 0.39
By a similar calculation, Equation 14 gave a value of 0.42 for (OID),. These calculations would indicate that the true minimum reflux ratio would be between 0.39 and 0.68. Equation 15 gives upper and lower limits for the minimum reflux ratio of 0.79 and 0.39, respectively. Stepwise plate calculations indicated that the actual value was approximately 0.53. The use of Equation 1 gives a ralue of ( O / D ) , equal to 1.5, which is much higher than the correct value.
Conclusions Equation 9 or 10 when used with Equation 6 gives values for the minimum reflux ratio equal t o or greater than the true minimum reflux ratio; this is a conservative method of calculating the ratio. The use of Equation 13 or 14 with Equation 12 gives calculated values that are equal to or less than the true minimum reflux ratio and serves as a lower limit for this quantity. Equation 15 can be used to calculate the upper and lower limits for the minimum reflux ratio, but in general, Equations 9 to 14 are easier to use and give more sharply defined limits, and for these reasons the latter equations are preferred to Equation 15. Equation 1 gives values that are much higher than the correct values in most cases.
Nomenclature
D
=
moles of distillate per unit time
N
=
[elkc
11”
=
[elkc’
+
+
ahkl ahk]
0 or 0, = moles of overflow from a plate above feed plate, per
unit time
0, = moles of overflow from a plate below feed plate, per unit
time
(xwhk
T
=
b l k
-
7) Uhk)
V or V , = moles of vapor t o a plate above feed plate, per unit time V, = moles of vapor to a plate below feed plate, per unit time
W = moles of residue, or bottoms, per unit time = (OTh- O,)/F = mole fraction in liquid = mole fraction in vapor = average mole fraction ( O / D ) , = minimum reflux ratio a = relative volatility Q = term defined by Equation 6 6’ = term defined by Equation 12 Z = summation z = sum for all components less volatile than heavy key comh ponent = sum for all components more volatile than light key com1 ponent p z y z
Subscripts D = distillate F = feed i = intersection of operating line equations 1 = components more volatile than light key component lk = light key component h = components less volatile than heavy key component hk = heavy key component
Literature Cited F = moles of feed per unit time
(1) Fenske, IND.ENQ.CHEM.,24,482 (1932). (2) Gilliland, IND.ENQ.CHEM., 32, 918 (1940). (3) Lewis and Matheson, Ibid., 24,494(1932). (4) McCabe and Thiele, Ibid., 17,605 (1925). (5) Robinson and Gilliland, “Elements of Fractional Distillation”, New York, McGraw-Hill Book CO.,1939. (6) Underwood, Trans. I n a t . Chem. Enurs. (London), 10, 112 (1932).
