Multicomponent Systems
J O H N D . L E S L I E
U niversity of B ritish Colum bia. V ancouver, C anada
I
X A R E C E N T article Sun an d S ilverm an1 developed two simple graphical m ethods for converting weight, volume, and mole fractions into one another, and also p a rts b y weight into weight, volume, and mole fractions, in binary systems.B y a simple extension, th e present paper m akes th e first of these m ethods applicable to m ulticom ponent systems. Al
though bin ary system s are perhaps m ore common, m ulti- component system s have increased greatly in im portance in recent years. T his should justify th e slight added compli
cations in th e following m ethod.
These graphical m ethods are very general in n a tu re b y virtue of th e fact th a t all th e above m entioned conversions can be expressed in tw o simple eq u a tio n s:
(1)
Pa =
Qa Sa QAj tQs + Sa Sb
Qa Sa’
+ 9 » S.v
Qa , Qb , Sa Sb' ^
(2)
Qa + Qb + Qc + . . . . + Q.v = 1 usually, b u t this is not a necessary condition. Our problem is to find Paor Pa know
ing the other quantities in th e equation. E qu atio n 2 is usually given in th e fo rm :
Pa' = QaSa
Q a S a + Q b S b + ■ - - - + QtrSy The alteration above is m ade so th a t b o th equations m ay be covered b y one graphical method.
G R A P H I C A L M E T H O D
We consider a system of four compo
nents; th e extension to a system of 2V components will be obvious. L ay out in one com er of th e graph paper, following Sun and Silverman, th e rectangle A BC D as in Figure 1. L et CD represent unity. Using suitable scales, plot points E(Qa, Sa) and F(Qb, Sb) as shown, w ith A and B as re
spective origins. C onnect A E and B F ; and through th e p o in t of intersection, G, draw line GH parallel to th e base A B . ith B as new origin, p lo t point J{Q c, Sc) to the right of BD . L et th e line joining B and J cut GH a t P . A t this stage the method m ust be altered slightly since we have no origin for th e next point, K(Qd, Sd).
1 In d. En g. Ch e m., 34, 682 (1942).
T h e m e t h o d o f S u n a n d S i l v e r m a n f o r t h e i n t e r c o n v e r s i o n s o f w e i g h t , v o l u m e , a n d m o l a r c o m p o s i t i o n s i n b i n a r y s y s t e m s h a s b e e n e x t e n d e d t o m u l t i p l e s y s t e m s . T h i s s h o u l d i n c r e a s e c o n s i d e r a b l y t h e u s e f u l n e s s o f t h e m e t h o d .
L a y off B M = Qd a n d B X = Sd as shown and join M X . T hen draw through P a line parallel to M Y , and let it intersect A B produced a t R. R is th e required origin, an d p o int K(Qd, Sd) is located as shown. Now join RD and extend to cut A C produced a t S. Join S to G ' an d let S G ’ cut CD a t T. T hen C T = P A P A can be obtained similarly using th e coordinates (Qa, Sa'), {Qb, Sb'), . . . . in place of (Qa, Sa), (Qa, S a ) , . . . I t will be noted th a t one new origin m ust be found b y th e procedure above for every pair of com ponents beyond two. A nother point w orth noting is th a t, in order to have point S a t a convenient height above A B , line CD can be raised or lowered after point R has been found. I n fact, CD need not be placed a t all u n til after th e rest of th e graph has been draw n. M erely choose S a t a suitable p o int on A C produced and get D by letting R S cut th e perpendicular on B .
Picking out pairs of sim ilar triangles we have:
(3)
(4) AG ’ A E ’ = 9a
GG’ E E ’ Sa
BG ’ B F ' Qb GG’ F F ’ Sb B P ’
p p . B J ’ j r
B P ’ GG’
Qc
Sc (5)
494 I N D U S T R I A L AND E N G I N E E R I N G C HE MI S T R Y Vol. 35, No. 4 S ubstituting E quation 6 in E quation 5,
C
2
+62
Cl
1 — ai
= m ( —\ c 2 + j t tbij) + 'ii.:
R earrangem ent of E quation 7 gives:
( ! ) ( h r r ) = m ( ¿ t t t 2) + k a very small value—i. e., high osmotic pressure ratio between the solvent and diluent. This would indicate th a t, as c2 ap
proaches zero, (1 — ai)/(cj + bi) approaches zero more rapidly th a n c2/ci approaches infinity so th a t in the lim it k would be zero. E quation 5 then reduces to
m = (bi + ci)/ c, Thus, since E quation 2 can represent a straight line only when fl is constant, we m ay make the generalization th a t the m ethod holds where the diluent increases proportionally to the solute in the solvent layer.
I n system s where only one pair of com ponents is com
pletely miscible, it is possible for the left-hand side of E qua
tion 8 to be greater th a n zero in the limit, as in the case of m ethylcyclohexane-aniline-n-heptane. /3, th e osmotic pres
sure coefficient, will then have positive values which can be
W ork along this line is being conducted in this laboratory.
A nother im p o rtan t use of this m ethod of tie line correla
tion is in solving the M aloney-Schubert diagram for theoreti
cal stages in an extraction system . Using three or four points on th e te rn a ry diagram of the system , a plot of th e solvent interpolation m ethods previously cited. T he M aloney- S chubert m ethod, in conjunction w ith th e tie line correlation presented in this paper, gives a m uch more rapid and accurate determ ination of theoretical stages required for a given separa
tion in a system where the binodal curve is pinched close to the solvent end of th e triangular diagram .
B I N A R Y V A P O R - L I Q U I D S Y S T E M S
In an analogous m anner vapor-liquid equilibrium in binary mixtures m ay be interpolated if equilibrium composi
tions are known for two points and if R a o u lt’s law applies; i. e., the relative volatility is approxim ately constant over the entire range employed.
If y = mole fraction of m ore volatile com ponent in vapor x = mole fraction of m ore volatile com ponent in liquid a = relative volatility
then the x-y relation in term s of relative volatility is:
y =
1 + (a - \ ) xa X (12) A rearrangem ent of E quation 12 results in:y/x = a - (a - 1 )y (13) Plots of y/x against y for systems such as nitrogen-oxygen (8), and benzene-toluene (9), as well as others following R ao u lt’s law, give straight lines on rectangular coordinates. These can be readily used in conjunction w ith th e Ponchon ch art to calculate theoretical plates in rectification.
A C K N O W L E D G M E N T
The author wishes to acknowledge w ith th a n k s th e sug
gestions of Joseph C. Elgin of Princeton U niversity and D onald F. O thm er of Brooklyn Polytechnic I n s titu te which were valuable in th e preparation of this paper for publication.
L I T E R A T U R E C I T E D
Graphical Interconversions for
Multicomponent Systems
J O H N D . L E S L I E
U niversity of B ritish Colum bia, V ancouver, C anada
I
N A R E C E N T article Sun and Silverm an1 developed two simple graphical m ethods for converting weight, volume, and mole fractions into one another, and also parts by weight into weight, volume, and mole fractions, in binary systems.By a simple extension, th e present paper makes the first of these m ethods applicable to m ulticom ponent systems. Al
though binary system s are perhaps more common, multi- com ponent system s have increased greatly in im portance in recent years. This should ju stify th e slight added compli
cations in th e following m ethod.
These graphical m ethods are very general in n atu re by virtue of th e fact th a t all the above m entioned conversions can be expressed in tw o simple equations:
Pa =
Qa Sa Q a , Q b ,
S A S b ^ ' + Qn
Sn
Pa' = S A'
Qa i Q a ,
Sa' Sb' ^ + Qn
Sn'
(1)
(2)
Q a + Q b + Qc + . . . . + Qn = 1 usually, b u t this is n o t a necessary condition. Our problem is to find P a or P a ' know
ing the other q uantities in the equation. E quation 2 is usually given in th e form:
Pa' =
QaSa
QaSa + QbSb + • • • ■ + QnSn
The alteration above is m ade so th a t both equations m ay be covered by one graphical method.
G R A P H I C A L M E T H O D
We consider a system of four compo
nents; th e extension to a system of N com ponents will be obvious. L ay out in one corner of th e graph paper, following Sun and Silverman, th e rectangle ABC D as in Figure 1. L et CD represent unity. Using suitable scales, p lo t points E(Qa, Sa) and F(Qb, Sb) as shown, w ith A and B as re
spective origins. Connect A E and B F ; and through th e p o int of intersection, G, draw line GH parallel to the base A B . W ith B as new origin, plot point J(Q c, Sc) to the right of BD . L et the line joining B and J cut GH a t P . A t this stage the m ethod m ust be altered slightly since we have no origin for th e next point, K(Qd, Sd).
1 In d. En g. Ch e m., 34, 682 (1942).
T h e m e t h o d o f S u n a n d S i l v e r m a n f o r t h e i n t e r c o n v e r s i o n s o f w e i g h t , v o l u m e , a n d m o l a r c o m p o s i t i o n s i n b i n a r y s y s t e m s h a s b e e n e x t e n d e d t o m u l t i p l e s y s t e m s . T h i s s h o u l d i n c r e a s e c o n s i d e r a b l y t h e u s e f u l n e s s o f t h e m e t h o d .
Lay off B M = Qn and B N = S d as shown and join M N. Then draw through P a line parallel to M N, and let it intersect A B produced a t R. R is the required origin, and point K ( Q d , S d ) is located as shown. Now join RD and extend to cut A C produced a t S. Join S to G ' and let S G ' cut CD a t T. Then CT = P A P A can be obtained similarly using the coordinates ( Q a , S a ' ) , ( Q b , S b ' ) , . . . . in place of (Qa, Sa), (Qb, Sb), . . . I t will be noted th a t one new origin m ust be found by th e procedure above for every pair of components beyond two. A nother point w orth noting is th a t, in order to have point S a t a convenient height above A B , line CD can be raised or lowered after point R has been found. In fact, CD need not be placed a t all until after the rest of the graph has been drawn. M erely choose S a t a suitable point on A C produced and get D by letting R S cut the perpendicular on B.
Picking ou t pairs of similar triangles we have:
(3)
(4)
(5)
AG' A E ' Qa
GG' E E ' Sa
BG’ B F ' — Ql GG' ~ F F ' Sb
B P ' P P '
B J ’ J J ’
B P ' GG'
Qc Sc
495
496 I N D U S T R I A L A ND E N G I N E E R I N G C H E MI S T R Y
R P ' R K / R P ' _ Qd
P P ' K K ' GG' Sd
S ubstituting E quations 3 to 6 in E q u atio n 1,
Vol. 35, No. 4 (6)
Pa =
A G ' GG'
A G ' BG ' B P ' R P ' GG' + GG' + GG' + GG' , . A G ' S A A R Again, C T s c CD
A G '
A R (7)
(8)
H ence A G ' / A R = C T/ C D, and from E q u atio n 7, P a = C T / C D = C T (if CD = 1)
Joining S to B and P', and letting S Band S P ' cu t CDa t Uand V, respectively, we can show sim ilarly th a t P b = TU, P c = UV, and then obviously P d = VD.
E X A M P L E
As an example we shall consider a sam ple of b last furnace gas having the composition
c o 2
C O j
In this example Sf Pi = X<:
Mco, = 44.01 M co = 28.01 M b, = 2.016 Mb, = 28.02
1 2 . 5 % 2 6 . 8
H , N ,
3 . 6 % 5 7 . 1
= M it Qi = Wi, and we wish to find
W co, = 0.125 TFco = 0.268 W B, = 0.036 TFNj = 0.571
Aco, = ? A co = ? A h, = ? A n, = 7 Figure 2 gives th e graphical solution of th e problem and follows the scheme of Figure 1. E, F, J, and K are the points with coordinates (TFCo„ Mco?), (Wco, M Co), (JFH„ Mb,) and (TFn*, Mb,), respectively, and th ey are p lotted from the origins as shown. N M is th e construction line parallel to K R . I t is purely coincidence th a t R K and R D S are nearly colinear in this diagram . T he results as read from CDare:
Mole fraction of C 02 = A co, = C T — 0.056 Mole fraction of CO = Aco = T U = 0.189 Mole fraction of H2 — X e , = U V = 0.353 Mole fraction of N2 = X b , = V D = 0.402 Although th e m ethod is more cum bersom e here th a n when dealing w ith binary system s, it should be capable of rapid m anipulation if carried o u t frequently, and should give sufficiently accurate results if done on large size ( 2 X 3 m eter) graph paper. I t should be noted th a t for tern ary system s the extra origin Ris unnecessary, and in this case the m ethod will be alm ost as rapid as th a t for binary systems.
I t will be necessary only to plot th e p o int J{Q c, Sc), drop the perpendicular PP' , and produce P ’Dto get S. T he rest will follow as above. F or all system s having odd num bers of components, th e last p a rt of th e construction will proceed likewise.
N O M E N C L A T U R E
M = molecular weight density
w eight fraction volume fraction mole fraction p a rt by weight
a q u a n tity which m ay be W , V, A , or U
a q u a n tity which m ay be M , p, (M /p ), (p / M ), or u n ity a q u an tity which is sim ilar to S, b u t is the product formed
from the rem aining (N — 1) of th e N com ponents— e. g., S c ' = MaMbMd Mn
a fraction which m ay be W , V, or A
Subscripts A , B, C, . . . . N = com ponents of m ulticom ponent system
pW V X u Q S S '
p =