Minimum Reflux Ratio for Multicomponent Distillation J. A. MAY The Dow Chemical Company, Freeport, Tex. *
A method is presented for the precise calculationof minimum reflux ratio for multicomponent distillation based

upon the resolution of the multicomponent mixture into an equivalent series of binary mixtures. Within the limitations of constant molal overflow and constant relative volatility, this method holds for all feed conditions and can be applied to all types of separations except those involving “split keys.” Although it becomes somewhat timeconsuming for the general case, it is particularly adaptable to situations wherein the feed is a saturated liquid or a saturated vapor (y = 1 or y = 0) and there are no components present lighter than the light key or else heavier than the heavy key.
0
NE of the f i s t factors sought in the design of continuous die
tillation columns is the minimum reflux ratio. This value represents the lowest reflux ratio under which a given separation could be accomplished even in a column having an infinite number of plates. I n addition to establishing the lowest operable reflux ratio and thereby aiding in the selection of a value for an operating reflux ratio, the minimum reflux ratio often can also be used with the calculation for the minimum number of plates to estimate by means of certain empirical comlations (3, 6) the number of theoretical plates required a t the operating reflux. The minimum reflux ratio for binary systems is readily determined by graphical procedure or, for ideal mixtures, algebraically by means of the Underwood (18) or Fenske (6) equation. For multicomponent systems, however, the problem is much more complex, as the “equilibrium curve” is a function of the reflux ratio. I n the past the only completely reliable means for determining the minimum reflux ratio for multicomponent systems has been the tedious and timeconsuming trialstepwise method, variations of which have been described by Jenny (8),Brown and Holcomb (d), Lewis and Matheson (9),and others. Several attempts have been made to develop rapid empirical methods to calculate the minimum reflux ratio for multicomponent systems, but unforunately most of these were based upon an erroneous assumption as to the ratio of the keys in the “pinches.” The first really satisfactory and practical solution to the problem was presented by Colburn (4, whose empirical correlation is remarkably accurate and relatively easy t o apply. Another good empirical method has recently been developed by Scheibel and Montross ( I d ) modified by Bailey and Coates (1). A still different approach to the problem has been presented by Mayfield and May (l,O), whose method gives a rapid and mathematically exact solution for those cases in which the separation is essentially complete and there are no components present lighter than the light key or heavier than the heavy key. The most recent development is due to Underwood (1316)who has presented a rigorous mathematical analysis of the fractional distillation of multicomponent mixtures for those cases for which constant molal reflux and constant relative volatility may be assumed. By means of the relationships developed, minimum
reflux can be computed with greater accuracy than by any of the previous stepwise platetoplate methods and with no more effort than is required by any of the approximate methods. Harbert (7) and Murdoch (11) have presented similar developments for multicomponent distillation; though not pointed out by the authors, the equations presented presumably could be used to arrive at the minimum reflux ratio. METHOD OF THE PRESENT PAPER
Because of the simplicity of the binary system it has long been the goal of process design engineers to reduce the multicomponent system to an equivalent binary system or systems. In a previous paper (10) it was postulated by analogy and inductive reasoning that, for an ideal ternary system or other multicomponent system in which there are no components lighter than the light key or heavier than the heavy key, the minimum reflux ratio required for complete separation is the same as the common minimum reflux ratio for the binary mixtures into which the original mixture can be resolved. No proof for this hypothesis was given, but this method was shown to give results in very close agreement with platetoplate calculations. Underwood has since proved the validity of this hypothesis (16). I n the present paper it is shown, again without proof but by comparison with results computed by platetoplate procedure or by the Underwood method (IS) (see Table I), that the principles developed in the iirst paper may be extended to the general case for complete separation. These findings also have been subsequently shown by Underwood to be valid (17). It has been found that, for cases having no component lighter than the light key or else heavier than the heavy key, the pinch composition remains the same for both complete and incomplete separations. This fact enables one to calculate the pinch composition for an incomplete separation using the simple relationships already developed for complete separation, and then by substituting in the Fenske (6) or Underwood (18) equation to arrive a t the minimum reflux ratio for the incomplete separation. For such cases the method, besides giving an exact solution, is rapid. In a somewhat similar manner, the minimum reflux ratio can be computed for incomplete separation for the general case by first determining the proper resolution for the sharp separation of the original multicomponent mixture. .MINIMUM REFLUX RATIO FOR COMPLETE SEPARATION TERNARY AND CLOSELY RELATED SYSTEMS
m
For the complete separation of multicomponent mixtures of the
(.
..a,b, c/d) and (c/d,e,f . . .) types, where c and d are the light I
.. . ..
and heavy keys, respectively, . a, b are components lighter than the light key, and e, f, , . are components heavier than the heavy key, the minimum reflux ratio may be determined by resolving the multicomponent mixture into binaries, all of which require the same (LID)minimum or a / W ) minimum. The material balance for all the binary mixtures combined must be, of course, the same as the material balance for the multicomponent separation. For example, the (c/d,e, f .) mixture of n
..
INDUSTRIAL A N D E N G I N E E R I N G C H E M I S T R Y
2176
Vol. 41, No. 12
TABLEI. COMPARISON OF MINIMUM REFLUXVALUES CALCULATED BY UNDERWOOD AND SUGGESTED METHODS (L/D)m
Feed
XF
XD
X W
OL
State
Underwood
May

1.5 1.0 0.2
vv
4.1094
4.1094
1.5 1.0 0.2
v
v
3.1140
3.1140
0.1 0.0
.., ... ... ... ... ...
0.3
0.8
...
1.5
v=v
2.1186
2.1186
d 0.3
0.2
2.0 1.0 0.9
v
=
7
7.2040
7.2040
2.0 1.0
v
=
‘t;
6,2513
6,2513

0.5
3,2182
3.2182
=
0.5
2,3002
2,3002
4.7332
4 , 7332
4.1279
4.1279
1.4902
1,4902
1,2063
1.2063
0.3 0.3 0.4
1.0 0.0
c_ 0 . 3
0.9
d e
0.3 0.4
c
c
2 e
0.0
2
0.4
0.0
d
0.1 0.5 0.4
1.0 0.0 0.0
0.1 0.5 0.4
0.9 0.1 0.0
0.3 0.3 0.4
1.0 0.0 0.0
0.3 0.3 0.4
0.85 0.15 0.00
a
0.2 0.2 0.6
1.0 0.0 0.0
0 d
0.2 0.2 0.6
0.9 0.1 0.0
0.2 0.3 0.5
0.4
e
c e
c
2 e
c
2 e c E
e
6 c
d
0.6
0.0
... ...
... ..,
...
...
... ... . .,
0.2
0.5 2.0
1.0 0.5
...
2.0 1.0
... I
.
.
,,.
.. .. .. ... ... .
.
I
5,O
5.0
0.0 0.0 1.0
8.0 2.0
0.8
1. o
b
0.3 0.4 0.3
0.4286 0.5714 0.0
0.0 0.0 1.0
8 2
0.3 0.4 0.3
0.425 0.525 0.050
0.0 0.10 0.90
8 2 1
c
2 b 0 d b c
2 b E
2 b
c
2 b E
h E
b c
e b
2 e
v
V = 8
0.96680
0.96680
v 3 F
0.6144
0.6144
v = v
0.40491
0.40491
I
v
=
v
0.39060
0.39060
0.3 0.3
0.0 0.0 1.0
V
=
v+ F
1.92877
1.92877
0.4
0.50 0.50 0.0
0.3 0.3 0.4
0.48 0.42 0.10
0.0 0.10 0.90
4
V =
VCF
1.36794
1.36794
0.30 0.20 0.20 0.30
0.60 0.40 0.00
0.00 0.00 0.40 0.60
4.0 2.0 1.0 0.5
I/v
0.30 0.20 0.20 0.30
0.60 0.39 0.01 0.00
0.00 0.01 0.39
V 
0.60
4.0 2.0 1.0 0.5
0.25
0.5 0.5 0.0 0.0
0.0 0.0 0.5 0.5
4.0 2.0 1.0 0.5
v=.v 1.12 1.1198 (plate t o plate calculation f r o m
0.0 0.0 0.0 0.0
11.01 4.20 1.80 1.34
0.00
0.026 0.1844 0,013 0,0922 0.044 0,3121 0.058 0,4113 0.558 0 . 0 0.300 0 . 0
0.6508
0.3492
f
0.0 0.0 0.01000
0.63822 0.35178

components would be resolved into n follows:
1 binary mixtures as
Part of c ( s a y c’) with all of d Part of c (say c”) with all of e, Part of c (say c”’) with all off and so on,such that (c‘ c” c”’ . . . = c)
+
+
+
For each of the above binaries there is a minimum reflux ratio required to separate the binary completely: (L’/D’).., for ( c ’ . d ) binarv”~ = (L’’/D;Yjm for ( c ” , e ) binary I
IC
3
)m
(L”’/D”’)m for (c’”, f) binary = (L’”/C”’)~ (For complete separation D‘ = c’, D” = c”, D”‘ = c”’, and so on)
and for the binaries as specified above (LID),
=
(L’/D’)m
=
(L”/D”)m = (L’”/D’f’)m
=
.
I
Figure 1 illustrates the basic concepts encountered in the prcsent method. Figures 2 and 3 illustrate the principles involved. Figure 2 is a typical y  x diagram for component c (the light key) for the complete separation of the following ( c / d e ) ternary mixture : XF

0.0 0.10 0.90
0.0
8

0.54 0.46 0.0
k
f
=
v
0.3 0.3 0.4
8
b s
v

0.0 0.0 1.0
0.25 0.25
a
=
0.0
0.25
E
v
v+ F
0.4
c
2
=
0.3 0.3 0.4
0.0
Figure 1. Resolution of a Ternary Mixture into Two Binary Mixtures
\~
V
1.0
8.0 2.0
b
V = V + F
1.0 0.5
0.0 0.2
2
q
1.0 0.5
0.5333 0.4667 0.0
c
4
0.5
0.2 0.3 ?l 0.5
6
c  c’t c “ (L/c&)= (L/c’),= (Lvc’3,
1.0
...
.,.
=
a
C 
0.3 
d e
0.3 0.4
Feed state such that
Q
1.5 1.0 0.2
~~
=
1 or V
=
7
10
I
2 1
08


v
0,95189
0.45189
0.8861
0.8861
06
2 
04
Colburn, 4)
v = v
14.4503
14.4503
02
1.00
0.67
11.01 4.20 1.80 1.34 1.00 0.67
v = 7
11.498
11.498
a
02
a4
06
08
IO
XC
Figure 2.
Typical Phase Diagram for c/de Separation
2777
INDUSTRIAL AND ENGINEERING CHEMISTRY
December 1949
For the condition of minimum reflux there is a pinch section of infinite plates in the upper part of the column where the concentrations of all components heavier than the heavy key become negliglible. Thus, with only the two key components (c and d) present the y  z curve for component c in the muIticomponent column operating at minimum reflux coincides with the y  5 equilibrium curve for the (c’, d ) mixture; these two curves coincide from the pinch point, the composition of which is c‘/(c‘ d) as specified above for the (c’, d ) binary, on up to X , = Y , = 1.0. Figure 3 is a typical y x diagram for component d (the heavy key) for the complete separation of the following (bcld) ternary mixture
+

P
b
d
XF
a
0.2
8 2 1

0.3 0.5
C
Feed state such that
q =
1 or V =
(.
7
. a, b, c / d ) MIXTURE. Feed such that V
d = d‘
1
In this case for the condition of minimum reflux there is a pinch section of infinite plates in the lower part of the column where the concentrations of all components lighter than the light key become negligible. With only the two key components (c and d ) present the y  z curve for component d in the multicomponent column operating a t minimum reflux coincides with the equilibrium curve for the (c, d ’ ) mixture; these two curve8 coincide from the pinch point, the composition of which, in terms of the heavy key, is d’/(c d’), on up to Xd = Y d = 1.0. It is especially important to note that in this case ( L / W ) , = (z’/E’/),,, (z”/Wt’), . . . and ( L / D ) , * (L’/D’),,, * ( L ’ f / D ’ ’ ) m . The principles presented above apply for any feed state,but only in those cases for which q = 1 (V = or p = 0 (V = V F ) can the computations be conveniently reduced to substitution into simple algebraic equations. The calculation of ( L I D ) minimum for the feed state q = 0.5 is shown in the example calculations. By equating the ( L I D )minimum or ( Z / W )minimum (depending upon type and resolution) for each resolute binary and noting that the summation of the separate parts of the resolved key must equal the whole, the following relationships for complete separation were derived (IO)for those cases for which p 1 or p = 0.
+
b
[k”(, 2:):
d’)]
ff
1
=
7+ F
(Jmnn=
O?/J+9n
=
 l ) ( c f d ” ] (a’ l)d

1
0)
+
a [‘a‘”
(q =
 XfF)

+....
(8
+1 =
+
+
v)
(c/d, e, f c
e
c’
. . .) MIXTURE. Feed state such that V =
+ 
e
[‘a’\
I)(C‘
a’
f
 l)c
(q = 1).
and cL’D)m
=
. + ++c )c)( a ’  1)
d(a’d’ d’(. . a b
(10)
To solve for ( L / D ) m in the above equations, c’ or d’ is first found from the appropriate equation (1, 3,5,or 8) by substituting various values by trial and error until the correct value of c’ or d’ satisfies the equation. Generally no more than three trials are required t o find the correct value. The value of c’ or d‘ thus obtained is then substituted in the appropriate equation to obtain (LID)m. For ternary mixtures, only the first two terms in Equations 1, 3, 5 , and 8 appear and these equations reduce to simple quadratic equations for which the roots are: ( c l d e ) MIXTURE. Feed state such that V = 7(q = 1 ) .
41  , + [
f ;t
(a”’  l ) ( c ’ (a‘ 1)c

“’1  , + . . .
(1)
and
ic/d, e, f
. . .) MIXTURE. Feed state such that V
=
7+ F (q = 0).
a”(a’
 1)c’
1
+
f
+.
*
.
(3)
” / ( a ’  1)c’
and
Xd
(.
. .a, b, c/d)
MIXTURE. Feed state such that V
= 7 (q = 1).
Figure 3.
Typical Phase Diagram for bcld Separation
INDUSTRIAL AND ENGINEERING CHEMISTRY
2778
+
 ’.(NI
 l)(c +e) (CY”  l ) ( d  c)] 4 (a’’ a’j(a”  1)cd 2(a”  a’)
V
(c/de) MIXTURB. Feed state such that
 l ) ( c ie )
[Ka“(Ol‘
+
(a’)(“’’ 
dp1’+ 4(a’)(cY’’ 
e’ =
+
d/q2f
c’ =
2(a” 
=
..
\la)
7+ F
( q = 0).
..
a’)
11. ( b , c / d ) Mixture. Feed such that V = 7. Compute (L/Djm for complete separation of the following mixture: EXAMPLE
l)(d  e ) ] +
 1)cd
CY’)“’’
(3a’i
b
( b c / d ) MIXTURE. Feed state such that
V
=
‘;i
2
i
(q = 0)
+ d )  (a”  l ) ( d  e ) ] + + 4(a”  a’)(”’  1)cd . . . 2(a” 
(8%)
ai)
( I t is obvious, too, that for the C H S ~(I = 1 or q = 0, the Underwood relationship, IS, also reduces to a quadratic equation for ternary mixtures.) EXAMPLE I. (cld, e ) Mixture. Feed such that V = 7. Compute ( L / D ) , for complete separation of the following mixture : C
a e
XB
a
0.3 0.3 0.4
1.5
*’ +
7
0.5 = 0.40260
+
7
(a’  l ) ( c
.
0.2 (1.75) ( 0 . 7 0 2 6 0 ) 0.40260 
I
(1.a  l)(c’j
0.2 (1.75)(0.70263) 0.40263
1
==
]+ 0.09737 = 0.5(1000
Alternate solution ford’:
d’
 [%”(a’  l ) ( b + d )  a’“’’ ____ l ) ( d  e ) ]+ drs1z + 4(01”(a”  a’)(”’  l ) ( c ) ( d )

2(&’  a‘)
J
+
 (K(S)(2 1)(0.2 0.5)  2 (8  1)(0.50.3)] 4 d[NI2 f 4(2)(8  2)(8  1)(0.3)(0,5) d‘ = 2(8  ‘2)
For first trial, assume c’ = 0.28440. +
=
+ 0.09737 = 0.49997 # 0.50000
0.40263
0.4
7
3
1
0.4
[(7.5  l ) ( ~+’0 . 3 ) T  
L
+
Try d‘ = 0.40263.
0’5A 0’40263 +
,.’ + ___
0.2 ‘(1.75)(0.3 d ’ )
For first trial assume d’ = 0.40260.
0.40260
a’ = 1 . 5 m” = 7 . 5
1.0 0.2
e
0”3 = 0‘28440
a
a
 l)(h
[”I*
4
$‘
+F
XF 0.2
0.3 0.5
C
7(q = I ) .
(bcld) MIXTURE. Feed state such that V = 7 [“(a‘
Vol. 41, No. 12
_ I _
0.4
+ 0.3)
(13)(0.28440 0.28440
[
0.28440
1
~
=
0.40283

+ 0.01556 = 0.29996 Z 0.30000
Try c’ = 0.28444 0.4 1.4902
0.28444
+ 0.01566 = 0.30000
EXAMPLE 111. ( c l d e ) Mixture. Feed such that q = 0.5. Compute ( L / D ) m for complete separation of the following mixture:
4lternate solution for c’:
 [“(a’  l ) ( c + e ) + (a” c’ =
dp12
+ 4(a” a””
 l)(d  e ) ] +  l)(c)(d)
2(a”  a’)
XF
a
d C
0.3 0.3
2.0
e
0.4
0.5
1.0
a s =. 2 a” = 4
Resolve mixture into two binary mixtures as follows:  1)(0.3 z/I“]“44(7.5
 [N(7.5 E”
=
+ 0.4)  (7.5  1)(0.3  0.3)]+ 1.5)(7.5  1)(0.3)(0.3) 2(7.5  1.5)
=
Part of c (say c’) with all of d , and, Part of c (say c ” ) with all of e such that c’ $ c“ = c and ( L / D ) m= ( L ’ / D ‘ ) m= (L’‘/D“)wc
INDUSTRIAL AND ENGINEERING CHEMISTRY
December 1949
For complete separation of binary mixtures, (L/D),,, is given by
where X’ is feed plate composition. The equation for the binary equilibrium curve a t the feed plate is
Y’
=
From Example I, C’ x’F= c’__ +d
r+ ( a  1)Xf
2 / , =    qxJ
*
For incomplete separation:
aXf
and for binary mixtures XI must also correspond to the intersection of the operating lines so that
(L/D),
=
+
(L’/D‘)m =
For q = 0.5 YJ =1 
0’28444 = 0,48669 0.28444 0.30
EXAMPLE V. (bclrl) Mixture. Feed such that V = Compute ( L I D ) , for the following separation:
where X P is the composition of feed. 0.5Xj
=
XF
q1
q1
2779
xf + 2 x F
XF 0  j=
b
C d
Equating to equilibrium curve relationship
7.
XF
xw
XD
a
0.2 0.3
0.0
0.53333
8
0.2 0 8
0.46666 0.0
2
0.5
1
For incomplete separation:
Trial I. I&t c’ = 0.2. From Example 11,
X f F =  =C’ =
OS2 0.2 + 0 . 3
c’ + d
0.4
0.30 c + d‘  6 3 0 + 0.40263 = 0.42697
XtF
c
X j = 0.31775 (L/D)’m = (a’ C” =
1
1 = 3.147 (2  1)(0.31775)

 1)Xj 
c”
+e
+ 2(0.2)
=
0.1 0.1 0.4 = Oa2
+
~
(L/D)m=
4X‘j
1
+ (4  1)X”f
=
1 (a”  l ) ( X ’ j )

(4

%
1 = 3.411 1)(0.09772)
MINIMUM REFLUX RATIO FOR INCOMPLETE SEPARATION IN TERNARY AND RELATED SYSTEMS
For cases having no component lighter than the light key or heavier than the heavy key, it has been observed that the composition at that pinch which is not adjacent to the feed plate is the same for both complete and incomplete separations. I t is thus possible to determine the pinch composition for complete separation and then by substitution in the TJnderwood (18) or Fenske ( 5 )equation to determine the minimum reflux ratio for the incomplete separation. The minimum reflux ratio for the incomplete separation of the multicomponent mixture is, obviously, the same as that required for the incomplete separation of the binary composed of the two keys and having a feed composition the same as the pinch composition for the multicomponent mixture.

EXAMPLE IV. ( c / d , e ) Mixture. Feed such that V = V . for the following separation: Compute (L/D)m C
7 e
XF
XD
(I
0.3 0.3
0.8
1.5
m’
0.2 0.0
1.0
a”
0.2
0.4
L  F
D
=
1.45235  1 = 1.2063 0.375
( c / d e ) Mixture. Feed such that q = 0.5. Compute ( L / D I m for the following separation:
Additional trial computations together with graphical interpolation give for c’ = 0.19378 and c” = 0.10622 (for which X j = 0.31074, X’j 0.10357), ( L / D ) , = (L’/D’)m = ( L ” / D ’ f ) m = 3.218. w
 0.625
EXAMPLE VI.
X ‘ j = 0.09772
(LID)”,
0.5 1.0
+ (0.2/0.8)
(0.5) = 0.625 0.375 = (W) (2.32376) = (0.625) (2.32376) = 1.45235
= =

C” = __ =
X”,
W D L
0.3  0.2 = 0.1

1.5 7.5
C
2 e
XF
XD
a
0.3 0.3 0.4
0.85 0.15
2.0 1.0 0.5
0.0
a‘ (2“
= 2 = 4
For incomplete separation where q # 1:
where X) is feed plate composition. From Example 111,X’,= 0.31074
GENERAL CASE FOR COMPLETE SEPARATION
For the special case ( c / d e f ) which was developed in the previous paper and has been reviewed above, it was pointed out that above the upper pinch one has to deal only with the binary c/d and that the situatiofi in this section is identical for both the binary and multicomponent systems. Similarly, for the general case (abc/ dej’), the concentration of all components heavier than the heavy key becomes zero at the upper pinch and from this point on up to the condenser, the equilibrium “curve” and operating line coincide for the original mixture and the less complex mixture abcld, which can be solved by the relationships developed previously. I n short, any problem is solved by resolving the multicomponent system into less complex systems until the original is in effect
2780
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 41, No. 12
for ( L / D ) mor (E/&’)% for each of the resolute mixtures follows, of course, in the manner already described. For the general case, although it is possible t o develop equations for a successive approximation solution, they become so involved that it is easier to use a graphical procedure. Different ratios for (abc)’/d, (abc)”/e, (abc)”’/f. . . . are assumed for which values of ( L I D ) , are coinput,ed and are plotted against per cent or fraction of (abc). B horizontal line drawn so that (abc)’ (abc)” (abc)”’ 4 . . = abc gives, of course, the proper solution. If there are only t,wo components in the bottoms product the plot may bz simplified t o give two curves which intersect a t the minimum ( L I D ) for the original mixture. Figures 4 and 5 illustrate the graphical procedure involved in determining ( L / D ) m for the complete separation of a complex mixture.
+
EXAMPLE VII. ( b c l d e ) Mixture. Feed such that V = 7. Compute ( L / D ) m for complete Reparation of the following mixture:
Fraction of ( a b c ) Figure 4.
+
.
Graphical Determination of ( L / D ) m for (abcldef) Sharp Separation
F, XF
a
0.30 0.20 0.20 0.30
4.0 2.0 1.0 0.5
b
c d e
Resolve mixture int,o two ternary mixtures as follows: Part of ( b , c), say (b, c)‘ a i t h all of d Part of ( b , c ) , say ( b , c)” with all of e such that ( b , c ) ’ 4 ( b (b,c) and ( L / D ) , = (L’/DOm = ( L t f / D f ’ ) , ,
c)”
=
Trial I. Let ( b , c)’ = (0.8) ( b , c). F’ b’
01
(0.8)(0.30) = 0.24 (O.S)(O.ZO) = 0.16 0.20
C‘
d’
4.0 2.0 1.0
F’ = 0.60 D’ = 0.40 W’ = 0.20
(L’/D’),n = 0.9506 and is det,ermined as shown in Example 11. ( b , c)” 0
0.2
10.4 0.6 (abc) / ( a b c )
0.8
1.0
broken down into an equivalent binary system. The criterion for proper resolution is simply that the minimum reflux ratio or minimum boilup ratio be equal for each of the resolute mixtures, depending, of course, on the mode of rcsolution as to which applies. If in the pinch considered the components heavier than the heavy key become zero, then the minimum reflux for the original and resolute mixtures must be equal each to each; if on the other hand the components lighter than the light key bccome zero, than the minimum boilup ratio, z / W , must be the same for the original and each resolute mixture. The equilibrium curve for the fractional distillation of multicomponent mixtures is discontinuous in nature and depends upon both the reflux ratio and the composilion at a given point. For this reason it is imperative that the overhead composition of the resolute mixtures b e identical to that of the original; otherwise the operating line and equilibrium curve for the primary resolute mixture (abc)’/dcontaining both keys mill not be the same as for the original mixture. For the case (abcldsf) the original mixture is resolved into (abc)‘/d, (abc)”/e, and (abc)”’/f such that ( L I D ) , = ( L f / D ‘ ) m  ( L l t / D f /m)  ( L I I I / D I f)fm and (abc)’ (abc)” (abc)”’ = abc. The resolution must also be such that a,b, and c are always present in the same ratio for each resolute mixture as for the original mixture. An alternate resolution would bs c/(def)’, b/(def)”, and a/(def)”‘ such that (z’/W’)n, = ( z ” / W f ’ ) m  ( z ‘ t f / W ’ f ’ * ) (LIW),, and (dsf)’ (def)” (def)”’ = ( d e j ) . The solution
+
+
+
+
(0.2) ( b , c) F‘
(0.2)(0.30) (0.2)(0.20)
b”
Figure 5. Simplified Graphical Determination of ( L / D ) m for (abclde) Sharp Separation
=
”C
e
a
= 0.06 = 0.04
0.30
F“ = 0 . 4 0 D” = 0.10 W’ = 0.30
8 4 1
(L”/D”)m = 0.9639 and is also determined as shown in Example 11. Additional trial computations together with graphical interpolat’ion give for ( b , c)‘ = 0.797 ( b , c) and ( b , e)” = 0.203 (b, e ) , ( L / D ) , = (L’/D’), = (L”/D”)m = 0.95189. EXAMPLE VIII. ( k a b c l d e ) Mixture. Feed such that V = E Compute ( L / D ) , for complete separation of the following mixture : k a b
0
d e
F, X F
a
0,026 0.013 0.044 0.058 0.559 0.300
11.01 4.20 1.80 1.34 1.00 0.67
Resolve mixture into two other mixtures as follows: Part of ( k , a,b, c ) , say (k,a, b , c)’ w i t h all of d Part of (k,a, b , c ) , say ( k , a, b , c)” w i t h all of e s u c h that (k,a,b, c)’ (k,a, b, c)” = (IC, a, b, c) and (L/D),,, = ( L f / D ’ ) , ,= , (L’,/D”)m
+
Trial I. Let ( k , a, b, c,)’ = (0.8) ( k , a , b, c) F’ k‘ a’ b’

C’
d
0.0208 0.0104 0.0362 0.0464 0.5590
a 11.01 4.20 1.80 1.34 1.00
Zi“ = 0.6718
D‘ = 0,1128 W’ = 0.5590
(L’/D’)n, = 15.357 and is determined as shown in LCxamplc 11. ( k , a, b, e)” = (0.2) ( k , a, 6, c) F” k” a” 6“
e P‘’
0.0052 0.0026 0,0088 0,0166 0.3000
a 11.01 4.20 1.80 1.34 0.67
F”
= 0.3282
W“
= 0.3000
D” =
0.0282
(L“/D“)m = 10.577 and is also determined as shown in Example 11. Additional trial computations toget8her with graphical interpolation give for ( k , a , b, c ) ’ = (0.85368) ( k , a b, c ) and ( k , a , b, c)” = (0.14632) ( k , a, b, c ) , ( L / D ) , = ( L J ’ / D ~=) ~(L”/D”),,, = 14.4503.
Retaining the same overhead composition for the primary resolute mixture as for t? original mixture, D‘ and follow by material balance. X w follows as a consequence of the bottoms quantities. The solution for (z’/W’),,, and (L’/D’)m for the incomplete seoaration of the above resolute mixture is carried out as shown inExample V.
w‘
MINIMUM REFLUX RATIO FOR INCOMPLETE SEPARATION FOR THE GENERAL CASE
*
*
For mixtures of the type (a,b,c/d,e,f) it is obvious that the pinch compositions for incomplete and complete separation cannot be the same; or else different values of (LID) minimum would be obtained for different pairs of component compositions used i n the Underwood (18) or Fenske (6) equation. However, for t h e ternary and related types constant pinch composition for complete and incomplete separations corresponds to the same resolution of the original mixture. The latter condition holds not only in this case but also for the general case. The procedure to follow for an incomplete separation for the (abc/def) type is to determine first the proper resolution for a complete separation. I n so doing the lower pinch composition is computed for the complete separation of the primary resolute mixture (abc)’/d containing both keys. By material balance the bottoms composition for the corresponding incomplete separation of (abc)’ / d is calculated which gives the same overhead composition for the incomplete separation of (abc)’/d as for the incomplete separation of the original multicomponent mixture. Inasmuch as the pinch composition of the primary resolute mixture does not change with product composition, ( z ‘ / W t ) m can he computed for such incomplete separation of (abc)’/d, from whence (L‘/D’)m follows and is the same as (,C/O), for the incomplete separation of the original mixture. EXAMPLE IX. (bclde) Mixture. Feed such that V = V. Compute ( L / D ) , for the following separation: b
C d
e
XF
XD
xw
a
0.30 0.20 0.20 0.30
0.60 0.39 0.01 0.00
0.00 0.01 0.39 0.50
4.0 2.0 1.0 0.5
F’ 0.2391 0.1594 0.2000
b’ 0’
rr
x‘D 0.60 0.39 0.01
D’ 0.2391 0.155415 0,003985
~


0.5985
1.00
0.398500
W‘ 0.0 0.003985 0.196015
X’W
a
4 2 1
0.0
0.019925 0,980075
____ 0.200000 1.000000
Retaining the same overhead composition for the primary resolute ternary as for the original mixture, D’and W’ follow by material balance. X ’ w follows as a conseauence of the bottoms quantities. The solution for ( p / W ’ ) m and (L‘/D)’,,, for the incomplete separation of the above ternary is carried out as shown in Example V. (Z‘/W’),,, = 4.7581 and (L‘/D’)m = (L/D),= 0.8861 EXAMPLE X. ( k , a, b, c/de) Mixture. Feed such that V = 7. Compute ( L / D ) , for the following separation: XF B
XD 0.17864 0.08832 0.29893 0.33611 0.10000
0.026 0.013 0.044 0.058 0.559 0.300
fC
ae
0.0
xw
U
...
11.01 4.20 1.80 1.34 1.00 0.57
...
0.0 0.01000 0.63822 0.35178
Inasmuch as the resolution for incomplete separation is the same as for complete separation, from Example VIII, ( k , a,b, c)’ = (0.85368) ( k , a, b, c ) . k’ a’ b‘ c’ d

F’
x ‘D
0.022196 0.011098 0.037562 0.049513 0.559000
0.17664 0.08832 0.29893 0.33611 0.10000

__
D‘ W’ X’W 0.022196 0 011098 ... 0:037562 0.0’“ 0.0 0.042234 0.007279 0.01315 0.012566 0.546434 0.98685
....
...
_____ __
0.679369 1,00000 0.125656 0.553713 1.00000
@‘/W’),,,
a
11.01 4.20 1.80 1.34 1.00
=
3.8361 and (L’/D’)m = ( L / D ) , = 11.498 CONCLUSION
Although no mathematical proof has been attempted, it has been found by comparing results with computations following the Underwood method (18) that at the condition of minimum reflux, multicomponent systems may be resolved into a series of binary mixtures which allow the precise determination of minimum reflux ratio. The validity of the initial hypothesis has, however, been proved by Underwood (16, 17) and the proof of the additional principles presented in this paper should follow in the same manner. The principles presented herein have been applied to all types of separations except those involving distributed components or “split keys” and it seems probable that even this type might be solved by a modification of this general procedure. Although the principles involved are very simple, only for certain cases can the computations be reduced to substitution into simple algebraic equations. As a result, the calculations for the general case become somewhat timeconsuming as compared to the Underwood method (18) and the latter is to be preferred in such instances. When the feed state is such that p = 1 or q = 0 and there are no components lighter than the light key or heavier that the heavy key, the method presented herein requires no more time than the Underwood method, and either method is as rapid as any of the empirical procedures. NOMENCLATURE
. . . a, b
= components lighter than light key component
C
= =
light key component portion of light key in the primary resolute mixture consisting of all of d and a part of ( . . . a, b, c) c” = portion of light key in the secondary resolute mixture consisting of all of e and a part of ( , , , a , b, c.) c/” = portion of light key in the tertiary resolute mixture consisting of all off and a part of ( . . a, b, c ) d = heavy key component d‘ = portion of heavy key in the primary resolute mixture consisting of all of c and a part of ( d , e, f . . ) d“ = portion of heavy key in the secondary resolute mixture consisting of all of b and a part of ( d , e, f , . , ) d”’ = portion of heavy key in the tertiary resolute mixture consisting of all of a and a part of ( d , e,f , . ) e, f . . = components heavier than the heavy key component D = moles of overhead distillate per unit of time for the multicomponent column D’ = moles of overhead distillate per unit of time for the ( a , b, c)‘/d or the c / ( d , e,! . .)’ piimary resolute mixture D” = moles of overhead distillate er unit of time for the ( . . , a, b, c)’ / e or the b/rd, e, f . .)“secondary resolute mixture D/tt = moles of overhead distillate per unit of time for the ( . . . a, b, c)”’/f or the a / ( d , e, f . )”‘ tertiary resolute mixture F = moles of feed per unit of time to the multicomponent column F’, F”, F”’ = moles of feed per unit of time to the primary, secondar and tertiary resolute columns, respectively L, = moles o?;eflux per unit of time above and below the feed, respectively, for the multicomponent column L‘, L”, Z“, L“‘,z“‘ = moles of reflux per unit of time above and below the feed, for the primary, secondary, and tertiary resolute columns, respectively = difference in moles of overflow above and below the P feed plate divided by the moles of feed per unit of time V ,7 = moles of vapor per unit of time above and below the feed, respeotively W = moles of bottom roduct per unit of time for the multicomponent coPumn C’
Inasmuch as the resolution for incomplete separation is the same as for complete separation, from Example VIT, (bc)’ = 0.797 (bc).
lT
2781
INDUSTRIAL AND ENGINEERING CHEMISTRY
December 1949
.
.
.
.
.
.
..
xf,
2782
INDUSTRIAL AND ENGINEERING CHEMISTRY
W’, W”, W”’ = moles of bottom product per unit of time for the primary, secondary, and tertiary resolute columns, respectively XD = mole fraction of a given component in the multicomponent column distillat,e’ X’D, X”D, X”’D = mole fractionof a given component in the distillate from the primary, secondary and ternary resolute columns, respectively1 XF = mole fraction of a given component in the feed to the multicomponent column’ X’F = mole fraction of a given component in the feed to the primary resolute column’ X, = mole fraction of a given component in the liquid phase on feed tray of a multicomponent column’ X ; , X I ’ / = mole fraction of a given component in the liquid phase on the feed tray of the primary and secondary resolute columns, respectively’ Xw = mole fraction of a given component in the bottoms of the multicomponent column1 X‘W,X ” W , X”‘w = mole fraction of a given component, in the bottoms from the primary, secondary, and ternary resolute columns, respectively’ = mole fraction of a given component in the vapor ZlJ phase on feed tray of a multicomponent column’ a = relative volatility a’ = relative volatility of component c with respect to component d a” = relative volatility of component b with respect to component d for (. . . a,b, c / d ) mixtures; or of component e with respect to component e for ( c / d , e, . f a . ) mixtures
.
1
Rzferj t o light component for binary mixtures.
Vol. 41, No. 12
relative volatility of compcnent a with ~espectto component’ d for (. , . a , b, c i d ) mixtures; or of component e with respect, to component f for ( c / d , e , f . . .) mixtures Subscript 7n refers t,o the condition of minimum reflux i,,
=
LITERATURE CITED
Bailey, R. V., and Coatos, J . , Petroleum Refiner, 27, 30 (1948). Brown, G . G., and Holconib, D . E., Petroleum. Engr., 11, No. 9, 55 (1940).
Brown, G. G., and Martin, H . Z . , T r a m . Am. Inst. Chem. E n g r s . , 35, 679 (1939). Colburn, 4. P., Ibid., 37, 805 (1 941). Fenske. M. R.,I N D . ENG.CHEM., 24, 482 (1932). Gilliland. E. R., Zbid., 32, 1220 (1940). Harbert, W. D., Ibid., 37, 1162 (1948). .Jenny, Frank J., Trans. Am. Inst. Chem. Engrs., 35, 635 (1939). Lewis, W . K., and Matheson. G. L.. IKD.ENG.CHEM.,24, 494 (1932).
Mayfield, 3’. D., and M a y , J. A , , Petroleum R e f i n e r , 25, 141 (1946).
Murdoch, P. G., Chem. Eng. Progress, 44, 855 (1948). Schcibel, E. G., and Montross, C. F., IND.ENG.CHEM., 38,
268
(1946).
Underwood, A . J. V., Chem. Eng. Pl.ogress, 44, 603 (1948), Underwood, A . J. Tr., J . Inst. Petroleiim. 31, 111 (1948). Ibid., 32, 898 (1946). I b i d . , 32, 614 (1946).
(17) Underwood, A. J. V.. private oonimunication to J. A. May, Feb. 18, 1949. (1st Underwood, A . ,I. T’., Trans. Inst. Chem. Engrs. (London), 10, 112 (1932). RECEIYED March 28, 1949.
Surface Tensions of Pure Liquids and Liquid Mixtures H. P. MEISSNER AND A. S . MICHAELS Massachusetts Institute of Technology, Cambridge 39, Muss.
A knowledge of the surface tension of liquids is becoming increasingly important to the engineer for the determination of the efficiency of gasliquid contact in processes such as distillation and absorption. A method for estimating the surface tension of mixtures of organic liquids from the refractive index is presented. Empirical relations are formulated which permit estimation of surface tensions of aqueous solutions with comparable accuracy.
I
KCREASING importance is being attached to the influence
of surface tension on the efficiency of fractionation, absorption, and extraction processes in the chemical industry. 9 need exists, therefore, for a simple method of prediction for multicomponent liquids. The object here is t o propose such a procedure based in part upon the parachor and molar refraction. PARACIJQR
Sugden’s parachor (11) is related to 7 , the surface tension, Jf, the molecular weight, and D and d, the liquid and vapor density, respectively, by the basic expression:
nl
y’
‘1
( P ) = ___ D  d
The parachor is practically independent of temperature, so that the surface tension can be calculated at any temperature if the other ternis in Equation 1 arc known. Khen applied considerably below the critical temperature of the liquid, the term for vapor density becomes negligible compared to the liquid density, whereupon Equation 1 can be simplified as follows:
(P)
w
1 ‘4
D
Sugden n a s able to show that the parachor is both an additive and constitutive property arid could he computed directly from the structural formula of the compound in question by use of the atomic and structural values presented in Table I. I n other words, the parachor can be computed without recourse t o Equations 1 or 2 , which require data on densities and surface tensions. An example of such a computation is shown in Table 11. It is worth noting that the original parachor values of Sugden (11), as presented in Table I, have been refined and improved, first by Mumford and Phillips (9,10) and then by Gibling (2). While these improved values check experimentally determined parachors more closely than Sugden’s original values, the latter are nevertheless retained here because they actually result in gieater success when used in Equation 5 proposed below for estimating surface tensions of liquids. MOLAR REFRACTIOh
The familiar LorentzLorenz expression for (R)h, the mol tr refraction with light of wave length A, is as follows: (3) Here, n h is the index of refraction determined with light of wave lengthk, M is the molecular weight, and D is the density of the phase to which the optical index refers (almost always thc liquid phase). All values of the optical index and molar re