Nomographic Calculation for Three-Cornponent
e a
e WILLIAM SCHOTTE AND W. A. SELKE Deporfment o f Chemicol Engineering, Columbia University, New York, N. Y.
T
HE previous graphical methods (5-6)which could be used for the calculation of the number of plates in ternary distillation have been based upon the McCabe-Thiele method (8). Although such methods have the advantage of facilitating visualization of the operation, the calculations are rather laborious. I n the Lewis and Cope method ( 6 ) , the most Tvidely known, a trial and error procedure is required to relate separate x - y diagrams for each component. By means of a differrnt type of diagram, as is proposed here, a direct solution can be obtained.
Nomographic Calculation
Rearranging the terms leads to:
/ = O
The condition, expressed now by the determinant, is that the three points,
(0, ‘ ! / A ) ,
Consider a mixture of three components A , B , and C. A is -the most volatile component, B is intermediate, and C is the least volatile. The equilibrium relationship, presented by Lewis .md ?\latheson( 7 ) ,can be written as:
lie on a straight line. shown on Figure 1.
ZC
= 1 -
-
X A
(1, U S )
On an X - y diagram the point,s would be
.%E
and
yc = 1 - yx -
(33
YX
where x and y are mole fractions in the liquid and vapor streams, respectively, leaving the same plate. T h e relative volatilities, defined as 01 = (y/yc)(zc/z), are based on CYC = 1. Substitution of Equation 3 in Equation 1 gives:
A nomograph of this equation will now be constructed. Let
p - g A = o
(4)
and
q - y g = O
(5) Figure I, First Step of Graphical Procedure Connect known vapor concentrations y A and y B on y-axes in order to determine third point on reference line Z,.
or
[XA
(2;
- 1) -
&] p + zx (A - 1) 4 +
ZA =
(6)
0
The third point lies on a line through the origin, Z A in the diagram, with a constant slope of -* U B
These three equations, 4,5 , and 6, may be considered t o contain two unknowns, p and q. According t o the theory of determinants a solution is possible only when the determinant of the coefficients of p and p and the constant term is equal to zero. 1 0 IXA
(& - 1)
- CyA 1
X A
0
-YAI
1
-YB,
(A -
1)
Ug
Connecting the third point with 1ine : XA
=
(1
-
-
(ZA,
1
0) will give the following
A)
ZA
z A
?4
0
xA1
472
INDUSTRIPL AND ENGINEERING CHEMISTRY
February 1953 Rearranged the equation is:
I n order that there may be a point, PA, on this line which is independent of ZA, the coefficient of X A must be zero. Then
1
v=
1
1
ffA
OIB
413
and ZB. The results up to this point are shown in Figure 2, Different origins have been chosen for components A and BThis simplifies the equations for the reference line, ZB,and reference point, P B ,for component B, making them similar to those for component A , Using Equations 7 , 8, 9, and 10,a diagram is easily constructed, and knowing Y A and Y B , X A a n d X B a r e readily found. OlB
SlopeofZA: ___ ffB- 1
ffA
Thus, the equilibrium relationship is found between Y A and I n a similar manner we can find the relationship between XA.
YP,
= YPA
(7)
ffB
2PB =
1-
(lo>
YPA
Equations 7 and 8 show that the slopes of Z A and Z B are positive when CYA and OIB are larger than unity, Only then can 2.4, and Z B be drawn within the boundaries of the diagram It i s therefore necessary always to base the diagram on the fmolighten components.
Figure
4. Second Step
Connect intersection of s B - l l n e and Z A with reference point PA This line intersects the x-axis et xA, which i s in equilibrium with yA: For xB repeat urine ZB and PB.
O R ~ N FOR A
Figure 4.
-xA
Xg--
ORICIfi FOR 6
Operating Lines for Components A and Enriching and Stripping Sections
B in
If necessary, xc and yc can bo determined as is shown in Figure 3. So far only the equilibrium relationship has been discusadI n order to relate the compositions of passing liquid and vapor streams, operating lines can be constructed. Using the first assumption, that of constant liquid and vapor rates, straight operating lines can be drawn similar to those used for a binary system. The equations for the operating lines are:
0
,
XA 0.5
0
Figure 3.
IM
-*-
,
xe
0
Mole Fractions of Component C in Liquid and Vapor Streams
-
Distance xAxB = 1
ON
'/2
-
- xA - x B
-
xc
' / % ( Y A f YB) = Y c / P
Enriching section:
2/ = L -- x
Stripping section:
Lz y ==
DXD
v +v
(11)
wxw v - -=-V
(12)
where x and y are the mole fractions in the liquid and vapor streams, respectively, passing each other between plates. T h e material balance or operating lines are shown in Figure 4. This figure may be superimposed upon Figure 2, which showed the equilibrium relationship, to give a complete diagram necessary to solve the problem. Procedures for different cases are indicated on Figures 5 and 6. Figure 5 illustrates the procedure for the case where the third component, A, is lighter than the key components, B and C ,
INDUSTRIAL AND ENGINEERING CHEMISTRY
414
Vol. 45, No. 2
First the Z-lines are constructed, the points P A and PB, and the operating lines for which Equations 7, 8,9, IO, 11, and 12 have been presented. Then 2nd (on the yA-axis) and znB (on the ys-axis) are connected, as is shown by line 1 on the diagram. P A is connected with the intersection of Z A and line 1. The intersection of this line 2 and the z-axis is xIA. Since yZA and x ? ~ determine a point on the enriching operating line of A , ySA 1s found by drawing lines 3 and 4. ytB is similarly found by construction of lines 2’, 3‘, and 4’. The operations are repeated for the next plate until the feed plate is reached. The location of the feed plate is found by Gilliland’s relationship ( 2 ) for the determination of the optimum feed plate. For the stripping section a similar procedure is followed; however, the stripping instead of the enriching operating lines are now used.
we ye
T , ‘DB
Figure
6. Procedure When Starting at Reboiler
Lines 5 and 6 are drawn from values of bottoms composition. Llne 7 gives values of vapor concentrations in equilibrium with bottoms. Lines 8 and 9 are drawn to sive values of composition on the plate above. Procedure i s repeated for subsequent plates.
.. Figure 5.
-XA
I
Xe-
Procedure When Starting at Top of Column
Line l’connects values of distillate composition, Lines 2 and 2’ are drawn to s i v e h ’ u e s X I A and X ~ B . Drawing liner 3 and 3 ’ to operating lines and 4 and 4 ‘ to y-axes gives values of y n A and y l B . Procedure i s then’ repeated lor subsequent plates.
When the third component, G, is heavier than the key components, A and B, the procedure illustrated in Figure 6 is followed. T h e 2-lines, P Aand P B ,and the operating lines are constructed Because the amount of C in the distillate will usually be negligible, calculaticns should be started a t the bottom of the column. xwA is connected with P A , which gives line 5 on the diagram, and zwB with Pg, mhirh gives the line marked 6. A line 7 is drawn through the intersections of 5 and 6 with Z A and Z g , respectively. T h e intersections of 7 with t h e y-axis are ywA and ywB. Using the stripping operation lines z I A and zIBare found (lines 8 and 9). The operations are continued until the feed plate is reached, after which the enriching rather than the stripping operating lincs are used.
a s described before.
*4third possibility is that the third component, B , is intermediate in volatility between the key components, A and C. Calculations can now be started at either end of the column. Thus both procedures described before are valid here. A trial and error procedure is necessary to find the distribution of B betn een the distillate and residue. An approximate relationship t o find this distribution is given by Hengstebeck ( 3 ) . Calculation of M e a n Relative Volatility For the sake of simplicity the relative volatilities were assumed t o be constant in the previous discussion. The calculation of a mean relative volatility is described below. A geometric mean, as was proposed by Fenske ( I ) for use in calculating the minimum number of theoretical plates, will in many cases lead t o erroneous results when applied to cases of finite reflux ratios. For example, when the number of plates
is infinite the mean relative volatility, C Y M , w 4 I approach the value of a a t the intersection of the equilibrium curve and the operating line, which here v, ill be called a m . The smaller the number of plates the closer is an1 to a geometric or arithmetic mean in the column section which is being considered. The variation of a p i t h the number of plates has been studied in order to find an empirical relationship between OL and n, the number of theoretical plates. -4theoretical equation would be too complex, as it vould have t o be based upon equations similar to those used for calculating the number of plates. The requirements for the relationship between a and n are twofold. First, it should be simple and easy to integrate. Secondly, it should contain few constants, since only the boundary values of a can readily be calculated. From these considerations the following equation was chosen:
where d, e, and j” are constants. When n = a, a = a,. Therefored = a,. For n = 1, CY = the enriching section or CY = CYWfor the stripping section where O ~ Dand a m are the values of a corresponding to the temperatures of the overhead vapor and the residue, respectively. Thus for the eririching section: e = a~ - a, and CYDfor
The value of the constant, ,f, is of no import,ance here. The above relationship is not exact, but i t is sufficiently accurate for evaluating a mean a. CYM
=
n-lfa.
dn
INDUSTRIAL A N D ENGINEERING CHEMISTRY
February 1953
The remaining difficulty is the determination of an, which is the value of a at the plate above the feed plate, the last plate in the enriching section. The value of 011, corresponding to the condition at the intersection of the operating lines, will usually be close to an. Since an cannot be determined, ai can be used instead. This will lead t o a small error, b u t a small variation in O ~ Mwill not change the number of plates noticeably. Moreover, the values of O(D and CY,, which are usually calculated from equilibrium constant charts, are not very accurate. Therefore using CY( the equation for CYMbecomes:
or
415
c;i"_'z_ V
=
B
V
K values can be found which satisfy Equation 14 and thus the value of C Y - at the pinch. The trial and error calculation can be simplified b y remembering t h a t for the heavy key ZD is usually small and thus
A similar relationship for the stripping section is:
x$"=-
The same equation holds for the stripping section, but CYDh a to be replaced by ayw. hText comes the question of determining a,. The equation for the operating line in the enriching section is:
V
-
where When n = m we can also write y = librium constant. Thus
KX
Kx,where K is
L KLK M =
the equi-
vL x f D
~ X Z J
W
=- K V
V
Since
CY
is given by the equation
n Kc
a = -
or
CY, can easily be determined from the equilibrium constants which satisfy Equation 14 or 15.
Since
zx
= 1 we can write
Example A mixture of benzene, toluene, and xylene is t o be separated according to the following specifications: Benzene (light key) Toluene (heavy key) Xylene (heavieet comp.)
XD
'feed
IW
0.99 0.01 0
0.70
0.020
0.20 0.10
0.645
0.835
A
E
c
The feed is saturated liquid and a reflux ratio of L/D = 1.75 will be used. Solution. From a material balance the rate of distillate formation is found t o be: D = 70.1 moles per 100 moles of feed
-
L = 1.75 X 70.1
Then
_-
+ 100 = 222.5
V = V = 2.75 X 70.1
and
=
192.6
Thus
5v = 1.155
and
- = 0.636
V
With these calculated slopes the operating lines can be drawn. The values of 01 corresponding to the top plate, reboiler, and feed are calculated for the given compositions and are found to be A B
C
aD
ai
QW
6.46 2.49 1.0
6.14 2.42 1 .o
5.10 2.24 1.0
Figure 7. Stripping Section of Example
Using Equation 14a for the enriching section
Unprimed and primed numbers refer to x and y values of component A and B, rerp.c€lvdy, at corresponding plater
K H K= KB = 0.636
INDUSTRIAL AND ENGINEERING CHEMISTRY
416
Since X D ~= 0.01, the above value is sufficiently accurate for the determination of a,. T h e corresponding values of K d and Kc are: K A = 1.56and Kc = 0.268. Thus 0~~ = 5.83. Substitution in Equation 13 gives: aAX=
5.83 f
6.46 - 6.14 = 6.26 6.46 - 5.83 In 6.14 - 5.83
The corresponding value of aB&f = 2.45. T h e reference lines and points for the enriching section can now be calculated: Slope of Z A : ___ o(B = 2.45 CYB - 1 1.45
1.69
~
Vol. 45, No. 2
Total number of theoretical plates: 12 Feed plate: 7th plate from reboiler. A plate-to-plate calculation, as outlined by Lewis and Matheson ( r ) ,gives the same answers. This is to be expected, because the graphical procedure is based on the rigorous Lewis and Slatheson method. Errors are introduced only by making the graph too small and through the assumption of constant relative volatilities. The inaccuracy of the mean relative volatility probably is not greater than that of the individual values of the relative volatilities used for the multicomponent system. For the sake of simplicity, the graphical procedure h a s been illustrated for only the stripping section. I n practice the lines do not have to be drawn in full, but only the intersections with the axes and reference lines need be marked This has been shown here for the operating lines. Discussion
1 1 6.28
PA,B: Y = 2:
= 1
- 0.698
1 2.45
= 0.698
= 0.302
For the stripping section the value of follows:
a m is
deterniined as
L
KLK = K A = = = 1.155 Tr
Correspondingly, Kc = 0.185, thus a d m = 6.25 Using Equation 13:
The method proposed here, like the Lewis-Cope method, is a graphical interpretation of the plate-to-plate calculations as derived by Lewis and Matheson. This gives the advantage t h a t z and y values of all components are known throughout the column. Whereas Lewis arid Cope use a separate z-y diagram for each component, here all components are related in one diagram. I n this manner the trial and error procedure, necessary to make all diagrams consistent, is avoided. Thus a more rapid calculation is possible. The proposed method may be evtended to any number of components, but i t was found that in cases of four or more components, the diagram would be evcessively complex. Notation A
= most volatile component
B = component intermediate in volatility
and C Y B , ~= 2.37 For the stripping section the following va.lues will have to be used : - = 1.73 Slope of Z A : 2’37 1.33
x
v =
= 1 - 0.712 = 0.288
Because the third component is heavier than the key components, calculations will be started a t the reboiler. Figure 7 shows the graphical steps to be used in the determinat,ion of the number of plates in the stripping section. Gilliland’s rule (5’) for the det,ermination of the feed plate states t h a t the ratio of the mole fraction of the light key to t h a t of the heavy key a t the intersection of the operating lines should be more than the ratio of light t o heavy key in the liquid coming from the feed plate, b u t less than t h a t in the liquid coming from the plate above the feed plate. I n our case
ZLK XHK
= 3.5, a t plate
c = least volatile component D = molar rate of distillate formation, moles per unit time K = equilibrium constant I T , = molar liquid rate in enriching section, moles per unit time L = molar liquid rate in stripping section, moles per unit time 71 = number of theoretical plates P = reference point for plate determination molar vapor rate in enriching section, moles per unit time vf i r = = molar vapor rate in stripping section, moles per unit time molar rate of withdrawal of residue, moles per unit time N = mole fraction in liquid y = mole fraction in vapor z = reference line for plate determination a = relative volatility
7:
-
xLK = ZHK
035
= 3.18 and a t
Subscripts A , B, C = various components D = distillate H K = heavy key component i = intersection of operating lines LK = light key component J f = appropriate mean W = residue m = intersection of operating line and equilibrium curve literature Cited (1) Fenske, M. R., IND. ENG.CHEM.,24, 482 (2) Gilliland, E. R., Ibid., 32,918 (1940).
(1932).
(3) Hengstebeck, R. J., T ~ n n s .Am. I n s t . Chem. Engrs., 42, 309
(1946). (4)Jenny, F. J., Ibid., 35, 635 (1939). (5) Jenny, F. J., and Chicalese, IT. J., IND.ENG.CHEW, 37, 956
(1945).
Thus plate 7 is the feed plate. For the enriching section the procedure is the same, except that now the first set of reference points (PA,P B ) and reference lines ( Z A , 2,) are to be used. Calculation shows t h a t another five plates are needed:
(6) Le&, W. K., and Cope, J. Q., Ibid., 24,498 (1932). ( 7 ) Lewis, W. K., and Matheson, G. L., Ibid., 24,494 (1932). ( 8 ) McCabe, W. L., and Thiele, E. W., Ibid., 17, 605 (1925). RECEIVED for review March 15, 1952. ACCEPTED October 21, 1952. Presented a t the F o u r t h Annual Meeting-in-Miniature of t h e North Jersey Section of the AMERICANCHEMICALSOCIETY, J a n u a r y 28, 19-52. Contribution 28 from t h e Chemical Engineering Laboratories, Engineering Center, Colunibia University, New York, S . Y .
