ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
Nornographs for Lewis-Matheson Distillation Calculations F. RODRIGUEZ1 Ferro Chemical Corp., Bedford, Ohio
T. J. WALSH Case lnsfifuteof Technology, Cleveland, Ohio
M
ETHODS of calculation for multicomponent distillation
are based on material balances, energy balances, and equilibrium calculations as originally proposed by Sore1 (6). These were modified for binary system calculations and adapted to a graphical solution by McCabe and Thiele ( S ) , using assumptions which lead to the concept of a constant molal overflow. Lewis and Matheson ( 2 ) extended this approach to multicomponent mixture problems, and a graphical technique using multiple plots v a s described by Lewis and Cope ( 1 ) . This latter was made the basis of a nomographic solution by Schotte and Selke (5).
Also
+ + zc 1 + + YC = K A X A+ K B Z B+ K C X C= 1 ( K A - Kc)xa + ( K B - K C ) Z E+ K c = I ZA
YA
Then
=
X B
YB
and
+ ~ B ( K-BK c ) / ( l - K c ) - 1 = 0
X A ( K A- K c ) / ( l - ZCc) This is in the form [f(T)lZA
+ [Y(2’)lxB - 1
Now let
Equilibrium Temperature Nomograph Eliminates Trial-and-Error Solution for Temperature
q-xB=o
p-xA=O
The major difficulty with the algebraic approach is that it requires a trial and error solution a t some point for each plate in the desired still. These solutions generally involve the assumption of a plate temperature for each plate and the calculations become time-consuming as the temperature must he checked. The time required for :t solution can he greatly reduced if a nomograph is prepared from which the temperature may be evaluated directly.
0
[S(2’)lP
+ idnln - 1 =0
If the last three equations are assumed to contain two variables, p and q, a simultaneous solution is possible only when the determinant of the coefficients of p and q and the constant1term is equal to zero. 1 ‘0
If(T
0 1 dT)
:;A, -1
1
=
0
If column two is added to column one, the columns are [rearranged, and the lowest row is divided by f ( T ) g( T )
+
Therefore, the three points
(54,0),
(xa, l), and
1
(f(T)
+ dT)’
) lie on a straight line. This condition is illustrated + !AT) in Figure 1. A similar situation exists with the vapor compo‘(*)
,ET)
sition equation
Figure 1.
Theory of Nomographs
and
That this may be done is shown as follows: Let
YA
=
KAXA
2 / ~= K B X B
YC =
KCXC
where x and y are the mole fractions of the conlponents A , B , and C in the liquid and vapor phases, respectively, at equilibrium, and K is the equilibrium constant. In general, K is a function of temperature and pressure. For the present, assume the pressure is constant so that K is a function of temperature only. 1
Present address, U. 8. 31d Armored Division, Fort Knox, Ky.
December 1954
This, again, is of the form YAf(T) f yBg(T)
-1=0
The position of the temperature scale can he determined by evaluating the functions of temperature given as coordinates
INDUSTRIAL AND ENGINEERING CHEMISTRY
2509
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT above. However, it is easier t o assign a value for temperature and either zero or one for X B and ? / B arid solve for ZA or X / A when
XU =
0
(I - K c ) / ( K A 0
X A =
US = IcSxB = UA = K A X A
tillen
Ig =
1
.%A
Kc)
- K B ) / ( K A - Kc?
= (1
yg = K B X B = ?;A
-
Iig
KAXA
=
1 .(I
a 0 c
0.9
!2J 0.e
W
0.7
5 2 c8
..
90
0.6
g
5 0.5
100 _,
t
t
110..
12c
.,
I
\\
__
L
a )I
0.2
\
\
f
0.1
0-0
I
I
Figure 2.
F 0.3
\
140
170
1-0
Equilibrium Temperatures versus Composition
Propane-isobutane-n-butane o f 125 Ib./sq. inch abs.
As an example, consider the system propane-isobutane-n-butane a t a pressure of 125 pounds per square inch absolute. I< data are available ( 4 ) . For the sake of convenience, A is tlie most volatile component and C, the least volatile component,. The limits of temperat,ure for the system are set by K A = 1 and l i c = 1. These occur a t 70" and 163' F., respectively. The data &ridralculated results are shown in Table I and plotted in Figure 2. I n using this nomograph, a knowli-dge of either liquid or vapor composition gives the equilibrium temperature for that composition.
Table !.
70 90 110 130 1.50
0.42 0.Rf 1.55 0.72 1.85 0.90
2510
Operating Line Nomograph
With the nomograph thus constructed, a calculation may be made for a three-component system a t constant pressure. Pressure variations within a column are normally neglected but may be included by adding extra temperature scales to the nomograph, each having a fixed pressure as a parameter. The use of the nomograph remains the same with the pressure being increased through the calculation. Four-component systems may also be handled with a parameter on the temperature line for a fixed mole fraction of the fourth component. However, if major components are considercd as A and B components, the effect of the other components is of the second order. ( I n some cases the network obtained using a fourth variable is inconvenient in that the network coordinates are a t a n acut'e angle. This can usually be remedied by moving the ZA, Y A axis to the left progressively with increasing networks are obt,ained, each values of t,hc parameter. TTVO cont,aining the extra parameter as a component.) Material Balance Nomograph Aids in Solution of Distillation Problems
The algebraic work in solving problems can be further qirnplified hy additional nomographs. These nornographs are not rcstrictcd as t o system. Consider the operating line equation obtained by a materid halttnw around a plate, n, above the feed and the top of the column
Calculations for Equilibrium Temperature Nomograph of Figure 2
1.00 0,700 0 450
1.00 1.25
0.31 0.RR
0,267
2 20
1.12 0.87 1 . 3 3 1.08
-- 0 . 0 3 0
170 2 . 5 3
Figure 3.
0.4
130 ,.
140
S e c t i o n below f e e d p l a t e
I
\
---4'---
5 j;'
\ 130
xn
Y
0.42
0.55
0 OR7
0.810 0 5'30 0 280 0,086 -0.090 -0,220
1.00 0.87s 0.700 0.495 0.211
0.077
0 0 0 0 0 0
0.840 0.608
0.435
0.159
-0.198 -0.562
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
0.42 0.J6 0.72 0.90
1.12 1.33
Vol. 46, No. 12
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
+
y n + i = xnLn/Vn+l
XD
D/V,+i
where L , refers to the liquid stream leaving plate n, V,+1 refers to the vapor stream leaving plate n 1 (plates are numbered from the top down), and D refers to the distillate stream leaving the column top.
+
Also
+D + D ) + Z D D / ( L+ I))
yn+l
= xnLn/(Ln
=
.EnR/(R
+ 1) +
ZDl/(R
A B
c
Yn+l
1/2
i:C
This leads to the determinant I
0.950 0.049 0.001
Tis,
’ F.
Kib 1.05 0.46 0.34
7s
YI/KI 0.905 0,107
XI =
o.003 1.015
f 1)
of
Plate 1
Let R = L,/D, the reflux ratio expressed in terms of liquid leaving the nth plate. z/n+1
Illustrative Problem of Determination T e m p e r a t u r e a n d Liquid Composition U1
Vn+i = L n
or
T a b l e II.
0.905
0.090
l7z,
F.
Plate 2 Kz 1.11 0.50 0.37
81
0.0025
52
0,815
0.180
$1
Corrected to z = 1 0,890 0.105 -0
0.998
x2 Corrected
. .
50. 1.0018
0 -1 R
Plate 3
1 l i = 0 1
us A
0.847 0.146
C
0.0061
B
I n this form the equation may be set up as a network nomograph * (Figure 3).
:c.0
5.0
Tz.
F.
86
K3 1.17 0.53 0.3s
53
0.712 0.272
0.0181 0.9971
a
From Figure 2 .
b From Scheibel and Jenny D
From Figure 3.
zs Corrected 0.714 0.273 0.0131 1,0001
(4).
..o.a
1.0
,,O.l
.
,o 6 1.0
Figure 4.
N o m o g r a p h f o r y = Kx
For this system x cannot be greater than 1 ( 7 )
Below the feed, a similar development leads to a similar chart. For convenience, this has also been included in Figure 3, w h c ~ e R’ = V,/W, the reboiler ratio. V mmust be determined to evaluate the calculation in any case. If the usual simplifying assumptions of constant molal overflow are applicable, the values of R and R’ are fixed throughout the calculation. Otherwise, a curve corresponding to variations in R with plate number may be plotted. If desired, a nomograph can be constructed for y = Kx. This is shown in Figure 4. A slide rule will generally be found as quick for this calculation. The use of these nomographs in the solution of a problem is as follows: Problem: From a feed consisting of propane, isobutane, and nbutane, an overhead product is being made which contains 95.0 mole % propane, 4.9 mole % isobutane, and 0.1 mole % n-butane
a t a pressure of 125 pounds per square inch absolute. Calculate the liquid composition and temperature on the third plate from the condenser. Reflux ratio is fixed a t 3.0. Solution: The vapor composition from the first plate is the same as the distillate composition-namely, y a = 0.950, y~ = 0 049, and yc = 0.001. The operations are summarized in Table 11. The temperature a t the first plate is selected from Figure 2. The R data are read from the nomograph of Scheibel and Jenny (4). The liquid composition is calculated and adjusted. The concentration of each component in the vapor from plate two is ascertained by using Figure 3. In using this graph, R is fixed at 3.0 and ZD is fixed for each component. The main advantage of this method is that it eliminates the trial-and-error solution without introducing the assumption of constant relative volatility. For repeated calculations on a given system in which reflux ratio and ratios of stream components are varied, the nornographs ma> w l l he of service. literature Cited Lewis, W. K., and Cope, J . Q., IND.EXC.CHGM.,24, 498 (1932). Lewis, W. K., and Rlathesoii. G. L., I h i d . , 24, 492 (1932). Ibid., 17, GO5 (1925). McCabe, W. L., and Thiele, E. W., (4) Scheibel, E. G., and Jenny, F. J., Ihid.. 3 7 , 81 (1945). (5) Schotte, W., and Selke, W.A., Zhid., 45, 472 (1953). (6) Sorel, “La Rectification de l’alcool.” Paris, 1893. ACCEPTED July 17, 1954. RECEIVED for review -4priI 12, 1954. Presented before the Division of Industrial and Engineering Chemistry a t the 125th Meeting of the . h E R l C A S CHEMICAL SOCIETY, Kansas City, Ma,
END OF ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT SECTION
December 1954
INDUSTRIAL AND ENGINEERING CHEMISTRY
2511
