Theory of distillation and other countercurrent processes II

Physica X V I I , no 5

Mei 1951

THEORY OF DISTILLATION AND OTHER COUNTERCURRENT PROCESSES II b y W. R. VAN W I J K Laboratorium voor Natuur- en Weerkunde, Landbouwhogeschool, Wageningen, • Nederland

Summary The theory of countercurrent processes presented in a previous article is further developed. An analytical solution of the basic equation, from which the product of absorption factors is calculated, is given. A theoretically sound procedure for reducing the distillation of a m u l t i c o m p o n e n t mixture to t h a t of two binary mixtures is developed. Algebraic as well as graphical methods can be used to perform the actual calculation. The influence of unspecified components on the absorption factor product is discussed.

In a previous article a theory of countercurrent processes has been developed e). The product of absorption - - e.g. extraction factors of a reference component is the essential variable in this theory and it is calculated for the different trays or stages of a countercurrent apparatus. Distillation has in particular been stuaied as an example of a countercurrent process. The theory is further developed in the present article and the analogy between a multicomponent mixture and a binary system is discussed. The previous article will be referred to as I in the following text. • For the stripping section of a distillation column an equation has been derived b y which the absorption factor product Ap(m), on. an arbitrary tray, m, is expressed in the generalized liquid rate LtbI and in two functions, u(m) and z(m), which depend on the distribution of the components in the residue. The equation is (I eq. 8) : A

U)

=

(1)

For the meaning of these and other symbols reference is made to I, and to the index of nomenclature at the end of this article. Equation 1 has been solved by successive calculation of Ap(m) in I. We shall now solve it analytically and then show how the - -

Physica XVII

485

- -

31"

486

W.R. VAN WlJK

calculation of a multicomponent mixture can be reduced to that of a two component system.

Analytical solution. Although eq. I is quite general, it will be first applied to the components for which a finite residue rate is specified and then one has Z = 27~ V(~0)/(1- al t) and U = 27~ Blil/(1 - - air). The dash indicates that the summation is only extended over the components with. a specified residue rate. The unspecified components will be taken irito account afterwards (I, p. 947). Inserting these expressions for Z and U in eq. 1 gives:

A = [ ~ V~,oll(1--a,t)]l[Lib>--X" B<,II(I--a,t)]

(2)

A = Z~ c,/(1 - - ~ ' t )

(3)

or

Eq. 3 is ±he expansion of A into its elementary fractions. A becomes infinite for t = ~ - 1 . These values are, therefore, the roots of the denominator in eq. 2. Though the numerator in eq. 2 is infinite for t = a~-I, A remains finite and becomes equal t o - V(i o/B(o since the denominator is also infinite. This property of A provides us with exactly the necessary number of linear equations to calculate the c~'s in eq. 3 b y inserting successively t = a~-I, a~-2 etc. in that equation. To obtain the 9~'s of eq. 3 we have to solve :

L~b~- - z" B~d(1 -- a~ t) = 0

(4)

This equation has been obtained in quite different.ways b y U nderwood, Murdock and S m i t h u y s e n 6 ) 3)~). These authors have discussed eq. 4 for distillation and given a graphical representation of the left hand side. The roots are all different in that case and all B(0's are positive. With other countercurrent operations, however, negative B(o values and double roots may eventually occur.

Since Ap(m) is the coefficient of t" in the expansion of A into a power series in t we have from eq. 3:

Ap,(m) = X; ci ~o~". The subscript l to Ap(m) is added to indicate that no unspecified components have been tal~en it/to consideration (I p. 948).

D I S T I L L A T I O N AND O T H E R C O U N T E R C U R R E N T PROCESSES

487

A similar equation is found for the stripping factor product Sp(n) in the topsection (I p. 944). Sp(n) is calculated from the equation: s = z/(v(o-

u)

(s)

with Z = X~ L(,0)/(l - - a7-I t), U ----- 27' D(O/(1 - - a7-l t) and n=oo

S = T, sp(n) t". The dash now indicates a summation over the corn,,=0 ponents, which are specified in the distillate. For the expansion of S intoelementaryfractions, the equation V(0 - - U = 0 must be solved and this leads again to the calculation of the roots of an algebraic equation. The calculation of.a distillation operation is thus reduced to the solution of two algebraic equations and after that, a correction tor the presence of the unspecified components has to be made. A special feature of the present theory is that the individuality of the components is lost to a high degree since they appear only as constituents of the functions u and z (cf. list of notations). An approximation of these functions by a smaller number of virtual components reduces the calculation of the actual multicomponent system to that Of a system containing fewer components (I p. 941). This provides a basis for the approximation of a multicomponent mixture by, for instance, two binary mixtures. Graphical methods can then be applied for the calculation of actual problems. It should, however, be emphasized that the direct numerical calculation of Apl(ra ) by I eq. 3g is quicker than when the calculation is performed by solving even a quadratic algebraic equation, unless the number of trays is about 20 or more in each section. Reduction to a binary mixture. Several attempts have been made to reduce the calculation of a multicomponent system to that of a binary mixture. Some of these methods have been applied with considerable success, but they are admittedly based upon empirical correlations or assumptions and it seems difficult to prove that t h e y are valid for arbitrarily chosen mixtures 1). In the present method the functions u(m) = X~ B(o a~ for the stripping section and u(n) = ~ DI~I a-i-" for the toppsection, are each approximated by two components. In the stripping section one of these two components must be the light key component which is b y definition the component with the largest a~ value in u(m). Since u(m) becomes asymptotically equal to the value of B~0 a~ of the

488

W.R.

VAN

WIJK

light key .for large m, a good approximation of u(m) can then be achieved only b y the light key component itself. The other component is a virtual component of which the generalized residue rate B(b) and the relative volatility ab are so chosen that u*(m) = BObI a'~ + + Bc~k)a~, where lk refers to the light key component, approximates u(m) as good as is possible for an expression of this kind, from m = 1 upwards. Deviations between u(m) and u*(m) which are of different sign will partly cancel in the further calculation of Apl(m ) (I eq. 3g). It i s n o t necessary that u*(0) should approximate u(0). On the contrary,, the exact value of u(0) can be retained in the approximation as will now be shown. For a reflux of the same composition as the residue, z(m) = R8 u(m), and) thus, z(m) is approximated at the same time as is u(m). One then has according to I eq. 3g: k=m--I

Apt(m ) LObI =

,•

u ( m - - k) Apt(k ) --~ U(0) Apl(m ) + R B u(m). (6)

k=0

Since l u* (m) approximates u(m) from m = 1 on, we obtain the corresponding approximated absorption factor product Ap*(m), if u* is inserted instead of u, except in the second term at the right. The above equation holds for m > I and in addition we define

Ap'~(O) = APt(O) -~ 1. Addition of u*(O)Ap*(m) on both sides in eq. 6, gives after a slight reduction : k~m

Ap*(m) [Lob) - - u ( 0 1 + u*(0)] =

27 u * ( m -

k)Ap~(k) +

+ (RB -F 1) u*(m).

(7)

Since m = 1, 2, 3 etc. we now introduce the generating function A* - - 1 = ~Ap~(m) t cml and U * - - u * ( 0 ) = ~,u*(m) t" and abbre1

I

viating L~b) - - u(O) + u*(0) to L* I, one sees b y the reasoning given in I p. 940, that eq: 7 is equivalent to: (A* - - 1) = (R B + 1) (O* - - u*(0)/(t~, - - O*).

(8)

Furthermore, one has U* = ~" u*(m) t" = BOb~(1 - - a b t) + Bud)l(1 - - a,k t) 0

and u*--

u*(0) = B(;) ab t/(1 - - ab t) + B.k) a,kt/(1 --

t).

DISTILLATION AND OTHER COUNTERCURRENT

PROCESSES

489

an expression in which A* - - I becomes infinite for the roots ~0~-1 and 9~1 Of the equation L ~ I - U* = O. So w e write A* - - 1 = c b ~0b t/(1 - - ~0~ t ) --~ Clk ~Olk t/(] -- ~lk t) in accordance with eq. 3. In a similar w a y one finds for the top section: ( S * - - 1) = (n D 4- 1 ) ( O * - - u *

(O))/(V*-- O*)=.

= c, 9, t/(1 - -

t) + c,k

--

t).

(9)

Here, u*(n) = hlo a 7 + Olhk~ ah-~"approximates u,(n) = ~'~ D{o a-~" from n = 1 on upwards and hk is the index for the h e a v y key compoltent which has the lowest value of a i in u t (n), and V* = = V,--u,(0) +u*(0), whereas U* = D~o/(1 - - aT I t) + D(hk)](l --a~lt). F r o m eqs 8 a n d 9 one obtains : Ape'(m) = Cb.9'b" + Clk 9;~ m " 1, 2 . . . . and Ap~(o) = 1

Sp*(n) =

c, 9;' 4- Chk 9~k

;n, =

I, 2 . . . .

and Sp(*0) =

1

Graphical methods. Eq. 7 is an equation to calculate Ap*(m) for a binary mixture except t h a t RB + 1 appears instead of R* + 1, with R~ = (L~, - - u*(O))/u*(O). Therefore Ap*(m) is (RB + 1)/(R* + 1) time~ the absorption f a c t o r product of the binary mixture acc. to eq. 8. Since the absorption factor A,, = Apl(m)/Apt(m--1), the absorpion factors of the binary mixture and those calculated from eq. 7 are equal, except A,, owing to Ap*(O) = Apl(o ) = 1 in either case. These absorption factors can therefore be found from a n y graphical m e t h o d in which the binary system consisting of component b and the light key is treated. Similarly, the stripping factors in the top section are found from those of the binary system consisting of component t and the h e a v y key. The stripping factor product of this system is equal to (R* + 1)/(R D + l) times Spa(n) from n = 1 on upwards. Here R* =

( V * - - ,**(o))/,,,*(o).

The absorption factors or more precisely the function ~ = = VIn~Ap1(M + 1)/L~bl Ap1(M) can immediately be read in e.g. the generalized Mc. C a b e and T h i e 1 e diagram. In the latter the generalized vapor and liquid concentrations are used as variables instead of the ordinary concentrations in M c C a b e- and T h i el e's original diagram 7). Let fig. 4a represent a generalized diagram ; the light key is the volatile component of the binary mixture; so

490

w.R.

VAN ~vVlJK

xcmI and Y{m}in the diagram are the generalized concentrations of the light key component at the m ~htray. One has, introducing its generalized and ordinary liquid and vapor rates Litkm ~, Llk,, , etc. : Xlml/yc,,, I = = L(lkm} V(b}/V(lkm ) L(b) = A~k L~k,, V(b}/Alk Vlkm L(b I = Xra Lb V(b)/Ym gb LCbl. H o w e v e r , xraLb/ymV b is the absorption factor for the light key on the mth t r a y and it is equal to Amai; I. Hence x~,o/ytm I = a ~ t ~(m). In a similar way, one finds for the top section, in which component t is the light component, xt~l/y~ I = a-;-I ~(n). Thus the functions and ~ are immediately read from the diagrams. They are then plotted against m and n respectively, to find the feed tray, for which ~ ----y. (I p. 945). A correction for the unspecified components must, of course, still be made. E x a m p l e . A separation which has been treated by other authors m a y also serve here to illustrate the procedure x). The separation scheme is given in table I: TABLE I

comp.

a~

i=1

6.0 4.0 2.0 1.0 0.5 0.3

2 3 4 5 6

feed (moles/hr) 100 200 200 200 200 10o 1000

Di

I I

distillate rate (moles/hr) 100 200 195.6 11.7

507.3

i

Bi

residue rate (moles/hr) m

4.4 188.3 2O0 100 492.7

Table I. Separation scheme of multicomponent distillation.

Total liquid rate in stripping section L b ~ 1700 Total liquid rate in top section Lt = 700. Vapor rate in top and stripping section Vb ---- Vt = 1207. R o = 1.38 R e = 2 . 4 5 . The reflux is constant. This is not an essential limitation in the present theory, since a variable reflux can be treated by the same method as a constant reflux 7) ,). *) The generalized liquid- or vapor rat~ is obtained by multiplication of the molar liquid or vapor rate, respectively, with Ai = Hiv + hlv/ai + h2v - - Hi t - - hl t --h2l cti. The partial molar vapor heat content of a component i is, for the present purpose, approximated by the expression Hiv + hlv Kim/a i + lu20[Kim, in the range of temperatures in which this component is present in a finite quantity. Its partial molar liquid heat content is a p p r o x i m a t e d b y Hit + hit Ktm + h21 ai/Kim. I n those eases where the latter may be neglected with respect to the latent molar heat of evaporation Ai, is simply the partial molar latent heat of evaporation.

DISTILLATION AND OTHER COUNTERCURRENTPROCESSES

491

The refluxes have the same composition as residue and distillate, respectively, so one has at the bottom V~o = R s B ~ = 2.45 B~ and at the top Lio = RDD i = 1.38 Di. For a reflux in equilibrium with the end products one has Vio = aiS.oB i and Lio = a-i-l AoDi; S o is the stripping factor of the reference component in the reboiler, A o its absorption factor in the condensor. The function Z for stripping and top section would have been S O,~,~ a~B~/(1 - - a i t) and A o ~,~ a~-ID~/ /(1 - - a71 t) in that case, instead of R 8 27~ Bd(1 - - a i t) and R D Z; DJ(1 - - a'i-' t). However, the coefficient of t" in the series 27~ a~Bd(l - - a ~ t) is equ M to the coefficient of t "+' in the series 27~ B d ( l - - ai t). Therefore', A p ( m ) for equilibrium reflux is proportional to A p ( m + I) for a reflux of the same composition as the residue: The p[oportionality factor is So/R s. Thus, the assumption of a reflux of similar composition as the end products does not restrict the generality of the conclusions reached. On the other hand, it is essential that the separation band consist of only two key components, since the residue rate 10 3

.i-'t,, -

u (m) m,, i

~

g

m

\

x' x

\

",,

//

"lots = 2.oi " ~

oc es= 0,5

,. /

/

v . r 3 B

/A\

lo( 6 = o : 3 ~

101

,/

\/

\

.al

mimmll

i- m iW i

102 ~ ' x ~ , ~ \\

~,,

\~.

f

\

",

\\ \ \

1 0

"r 1

'

' 2

3

4

5

6

7

8 nt

Fig. la. Characteristic function for stripping section.

492

w . R . VAN WlJK

of components of a-volatility intermediate between the two keys would vary with reflux ratio and number of plates. The.function u(m) = ,,~ B~a~' is represented in fig. la and u(n) = = ,,~,~D#-[ ~ is shown in fig. tb both. as heavy lines. The most volatile component in the bottom is 3. So u*(m) = Bba ~' + B3a'~. The crosses in fig. l a are the values of u ( m ) - B3a'~ which are approximated by the b r o k e n line Bba'~. On the horizontal axis the deviations u(m)- u*(m) expressed in percentage of u(m) are printed at the corresponding value of m. 10 3

' ,

u(ru

t

.

•

i N. ~'%

10 2

\\

IoC 3-- 2.0 I " ~ \

~

]

~_1 o.I

~ L~2=4.01 1°(t-2.27] "

o~.\~ \o~ -¢~1

,

o~ to

~

,,'-.o

",°oN

_

0

1

2

3

4

5

6

7

8

n Fig. lb. Characteristic function for t o p section.

The functions u(n) and u*(n) are plotted in fig. lb. The values of B b, D t etc. taken from these figures are collected in table II. Suffixes t or b have been added to symbols as u(0), u*(0) in this table to indicate t h a t they pertain to the top section or bottom section, respectively. The roots of the equation L* - - U * " = 0 are ~0~-] = 0.49757 and 9~-1 = 0.871. One has A * - - 1 = - - (R B + 1) = - - 3.45 according

DISTILLATION

AND

OTHER

COUNTERCURRENT

TABLE Data pertaining

binary mixtures

Ut*

to

=

Bottom

~ i ' Di = 5 0 7 . 3

,,b(O) = ~ ' i ' B i = 4 9 2 . 7

342

Bb = 323 ab = 0.907

2.27 342+

11.7 = 3 5 4

1207--507

+ 354 =

Vt*--354

= 700

700/354 =

1.97

493

II

to substitute

Top ut(O) = Dt = at = ut*(O ) = Vt* = Lt* = RD* =

PROCESSES

Ub*(0)= 323 + 4.4 = 327 1054

Dr/(1-at-'t )+ Dd(l--a4--'t)

Vb* Lb* RB*

=

= Lb*--327

1700--493+327=

=

1207/327 = 3.73

Ub*

~ Bb/(l----abt)+Bd(l--a,t)

=

1207 1534

eq. 6 for t = a~-1 a n d t - = a~-1. T h i s g i v e s c 3 ~ - 0 . 0 2 6 3 4 2

c b = 0.7748. So Ap*(m) = 0 . 0 2 6 3 ( 2 . 0 0 9 6 2 ) " + 0 . 7 7 4 8 (1.150)" F o r t h e t o p s e c t i o n one o b t a i n s : $t5"(n ) = 0 . 0 9 3 (1.0262)" -~ 0 . 6 8 9 (0.642)"

m = n =

1, 2, 3

and

.

.

.

1, 2, 3 . . .

101

A;C

Sp*Cn:

\\

.\ \

\ \ \

5

15

10 mORn

Fig. 2. Ap*(m) and Sp*(n) for six component system.

494

W . R . VAN W l J K

These functions are plotted in fig. 2, whereas the values ot ~ =

= V~Ap*(M + 1)/L~Ap*(M) and ,7 = V,Sp*(N)/L,Sp*(N + 1) are shown as the crosses in fig. 3.

OR~ X

~I/~I~/9/R/R/

-

0

10

15 oo M+IOR N+I

Fig. 3. • ,7 a n d } n o t c o r r e c t e d for unspec, x ~ a n d ~ n o t c o r r e c t e d for unspec, (~ *7 a n d ~ n o t c o r r e c t e d for unspec, gram. I A ,~ a n d ~ c o r r e c t e d using Vi2v+ l 1i[7] *7 a n d ~ c o r r e c t e d using Vilv+ ! I I I • '1 a n d ~ c o r r e c t e d using Vi2v+t

c o m p . d i r e c t c a l c u l a t i o n f r o m I eq. 3g. c o m p . c a l c u l a t e d f r o m u*. c o m p . f r o m M c C a b e-T h i e 1 e dia-

--- D i, LiM+t F Bi = Di(1 + a ~ ' L t ¢/Vt), etc. Di/(1 - - a ~ -I L t ~] Vt), etc.

The letters M and N are used in the latter figure instead of m and n to indicate t h a t this figure refers to the feed plate (I p. 944). The graphical treatment ot this problem is given in figs 4a and b. The values of ~(m) can be read immediately in the graph 4a as the intersection of a line from the origin drawn accross the m-th step point on the equilibrium line and the upper horizontal axis. One finds for instance ~(9) = 0.716, ~(oo) = 0.805 for the binary mixture.

495

D I S T I L L A T I O N AND O T H E R C O U N T E R C U R R E N T PROCESSES 100

,,. 70 o u

•

/ / ~/" .J o

~

/o// - -

/ /

s

~~, O0

10

/

20

30

40

50

/~; //

21 /'/.

~i/)"

I,y l/

/

/ / ,,/

.,I

/ //

Ig

/

GO

,V 70

80

90

100

M 0 1 ~o LIOUID OF 00MP. 3

Fig. 4a. Separation of components b and 3 in stripping section. Reflux ratio R B = 3.73. Ratio of volatilities 2.20. The ~ values for the reference component in the approximated multicomponent mixture are ( V b a 3 / L b ) / ( V ~ / L ~ ) = 1.804 times as great. The latter have been plotted in graph 3. They coincide practically with those obtained b y the previous method. ~(1) has of course first to be multiplied with (R B + 1)/(R* + 1). The ~ values from fig. 4b are also plotted in fig. 3 after multiplication with atV,L*/ / V * L t = 2.60. In the same figure also ~ and ~/ values calculated directly from the unapproximated functions u ( m - - k) and u ( n - - k) are shown (I eq. 3g). There is some deviation for the first point, but an excellent agreement is obtained for following points. This is due to the fact that the inaccuracies introduced by the approximation of t h e u functions b y two components, for higher values of m or n, are of different sign and, therefore, cancel partially.

496

w . R, VAN W I J K le ^

..ca

n.

81 tJ. o ~e

g,,

,.e.

MOL ~o LIOUID OF COMP.t Fig. 4b. S e p a r a t i o n of c o m p o n e n t s t a n d 4 in t o p s e c t i o n . R e f l u x r a t i o R h = 1.97. R a t i o of volatilities 2.27.

The graphical method enables one to obtain quickly a general survey of the entire problem by drawing lines corresponding to other reflux ratio's. It is particularly interesting to observe how and ~ for the light components in the binary mixture increase and decrease, respectively, with increasing number of plates or with increasing reflux.

Unspeci]ied components. The final step consists in applying the correction for the unspecified components to the points in fig.3 in the way set forth in I p. 947 ff. One has for the corrected value of (I eq. 15a): corrected = VbApt(M + 1)/LbAP~(M) [1 - - X 7 Ci +

+ (Ap~(m + 1)~,TCia~-t)/Apl(M)].

D I S T I L L A T I O N A N D O T H E R C O U N T E R C U R R E N T PROCESSES

497

The subscript j refers to a component unspecified in the residue k~m

and C j = V i N + I V b / V t [ L b -

X al.k-''l u ( m - - k ) ] .

Since there are

k=0

only two unspecified components, l and 2 in this case, it is h a r d l y worth while to replace t h e m b y a single subsitute component. One has: J ---- 1, a I = 6.0, X k a~k--~ u(m - - k) = 493 + 55 + 8 + 1 + . . . . and j = 2, a 2 = 4.0, X k a~k---m)u ( m - - k ) = 4 9 3 4 8 3 4 1 8 4 4 4 1 + . . . .

558 600.

F r o m m = 4 on the sums are practically constant. In I p.947 the distillate rate D i was used as a first approximation for V i N+I, which is always smaller t h a n V i ~v+,. The correct expression lor V i ~v+, is (I eq. 56): V i ~+, = D i [X k a-;-oN-k) Sp(k) + R v a-f'N]/Sp(N). The first approximation V i N+I = Di can be used when a i is so great t h a t the terms between the brackets which contain a7-I as a factor are all small compared with I. All the terms are positive and the unspecified components are, therefore, undercorrected for, if terms are omitted. A second, third etc. order correction is obtained if some more terms of the sum are taken into account. The next approximation for ViN +t is ViN +t = D i [1 + a~-l S p ( N - - 1)/ /Sp(N)], which contains the absorption factor product of the N-th t r a y in the top section. This is inconvenient for a correction to be applied to ~, since ~ contains only quantities pertaining to the stripping section. It can however be proven t h a t S p ( n ) / S p ( n + 1) decreases with n and therefore S p ( N 1)/Sp(N) is smaller t h a n S p ( N ) / S p ( N + 1). The latter is equal to Lt ~I/V, or Lt ~/Vt owing to ~ / = ~ at the feed tray. So the unspecified components are still undercorrected for if

ViN+I = D i (1 4 a71Lt~/Vt) is used and the calculation of the column is still on the safe side. F u r t h e r corrections can be made b y including more terms. As a limiting expression

Vi N+l = Di/(l - - a~ 1 Lt ~/Vt) , m a y be used. In these corrections it is assumed t h a t the term RDaT N is negligible as compared with 1. In the present example this. term is about Physica X V I I

32

498

w . R . VAN WlJK

0,01 f r o m N = 4 on for b o t h unspecified components. Since it in~/olves only a correction in a correction t e r m a value of 0.05 m a y generally be considered as sufficiently small. F o r practical problems the Second o r limiting expression for V i N+1 are at once used in the calculation of Ci, b u t here, the first correction is also shown b y w a y of example in fig. 3. One obtains: C t = 0.0874, C 2 = 0.181

I. Vi~r+ 1 = D i

II. V i N÷I = D i (1 . + a~-~ L, ~/V~) I I I . V i N+t = Di/(1 - - ai -I L, ~/Vt)

C~ =

0.0965,

c2 =

0.209

C~ = 0.0975, C 2 = 0.214

T h e corresponding curves are m a r k e d 8i, ~H, 8 m in fig. 3. The last two curves fall together. F o r the correction of the unspecified c o m p o n e n t s in the top section we h a v e (I eq. 15b): corrected = (1 - - Z~" Ci) V, S p I ( N ) / L , s p t ( N + I) + X 7 C i a i V , / L H e r e C i = L i M+1LdLb [V, - - Z h a7- k u ( n - - k)] and the index j, refers to a c o m p o n e n t u n s p e c i f i e d in the distillate, which are c o m p o n e n t 5 and 6. Inserting LiM+t = B i gives C 5 = 0.142 and C 6 = = 0:065. I n the second and t h i r d a p p r o x i m a t i o n one has, starting from eq. 5a, in an analogous w a y as for the correction of ~: L i M + l = Hi(1 + (xi Ld~lVb) and L i M + t = H i / ( 1 - - a i L d ~ V b ) . F o r c o m p o n e n t 5 with a i = 0.5 the t e r m R s aM is abt. 0.10 at M = 4. T h e corresponding curves are m a r k e d ~i, ~H, and ~ m respectively. A n y combination of equal values of ~/ and ~ is a possible solution for the location of the feed t r a y . Ak~owledgme~t. Mr. C h r . F . numerical calculations.

Kleiss

jr. has p e r f o r m e d the

Notation. The standard system of nomenclature for chemical engineering

operations (Trans. Am. Inst. chem. Engrs 4o, 251, 1944)) has been followed as far as possible. Generalized quantities are indicated by a subscript between brackets e.g. L(im) ~ A t Lira, B(O = AIB ~ etc. They are not included in the list. A * indicates that the quantity refers to the binary mixture approximating a multicomponent mixture. ¢~ (alpha)

Relative volatility of component i towards a reference component.

A

~- ~. ~4p(k) t~ Generating function of Ap(m).

Aim

Absorption factor for component i on tray m.

k~0

DISTILLATION AND OTHER COUNTERCURRENT PROCESSES

499

N N+I RD, RB

Absorption factor for reference component on t r a y m. -~-AIA 2 .... A m. Absorption factor product of reference component on t r a y m. Residue rate, moles/sec. Residue rate of component i, moles/sec. Subscript referring to b o t t o m section. Constants in analytical expression of A. Constant used in correction for unspecified components. Distillate rate, moles/sec. Distillate rate of c o m p o n e n t i, moles/sec. R o o t of equation to be solved to obtain poles of A or S. Molal heat conten.t of component i as a v ap o r or liquid resp., extrapolated or interpolated to its boiling points, cal/mole. Molal heat content of c o m p o n e n t i as a v ap o r or liquid respectively at t r a y n u m b e r m, cal/mole. Constants occurring in the approximation of the molal vapor heat content of c o m p o n e n t i by a function of K i. As above b u t for the molal liquid heat content. I n d ex denoting a certain component. Eq u i l i b r i u m constant for component i at t r a y m. Liquid rate below feed, moles/sec. Liquid rate above feed, moles/sec. Liquid rate of c o m p o n e n t i from t r a y m, moles/sec. = g i v - ga, cal/mole. I n d e x for t r a y below feed, m = 0 means reboiler. I n d e x for feed tray. I n d ex ior t r a y above feed, n = 0 means condensor. If in the symbols m is replaced by n t h e y refer to the t o p section. N u m b e r of components. In d ex for feed tray. Reflux ratio in top and b o t t o m resp.

S

=

Am

Ap(m)

B Bi b C

C D Di 9 (phi) g~v, ga H imv, Himl

niv, hlv, h2t, H w hi:, h~t i

K¢ra Lm, Lb L,v L t Lim A t (lambda) m

M+I

F~ Sp(k) tl~ Generating function of Sp(n). k=0

Sn

sp(~)

u(,n--k)

Stripping factor of comp. i at t r a y n. Stripping factor for reference component on t r a y n.

= S I S 2 . . . . S,, Stripping factor product of reference component on t r a y n. Variable in generating functions or subscript referring to top section. i=N

~. B~ a~(m-k)

~=I

u*(m--k)

= B,k ~i~.--k~ + Bb ,~,.--kl.

u(n--k)

= F~ Dia~ ('-k)

u*(n--k)

= Dj~k a ~ ('~-~) + D t a ~-("--~).

i=N i=1

500

DISTILLATION AND OTHER COUNTERCURRENT PROCESSES

U

Vm, Vb V,.Vt Vim (xi) Yi~, Yw (eta)

Z

~. ~(k) t k. G e n e r a t i n g f u n c t i o n of u(~n - - k) or u ( ~ - - k) k-~0 respectively. V a p o r r a t e b e l o w feed, moles/sec. V a p o r r a t e a b o v e feed, moles/sec. V a p o r r a t e of c o m p o n e n t i f r o m t r a y ~ , cal/sec. M o l a l l i q u i d c o n c e n t r a t i o n of c o m p o n e n t i a t t r a y m or in residual, r e s p e c t i v e l y . ,4p(M ~, l) V(b)/Ap(M ) L(bP p a r a m e t e r for feed t r a y . M o l a l v a p o r . c o n c e n t r a t i o n of c o m p o n e n t i a t t r a y m or in distillate, respectively. Sp(N) V(t)/Sp(N -~ 1) L(t ), p a r a m e t e r for feed t r a y . --

Z z(k)t k. kffi0

z(,,~)

~.. V~o ~

in bottom section.

/ffin ---- ~ L/o a~-n i n t o p s e c t i o n . Received 24- ! 1-50.

REFERENCES l) 2) 3) 4)

Hengstebeek, R. J.,Trans. Am. Inst. chem. Engrs.,42 (1946) 309. M u r d o c h, P. G., Chem. Eng. Progress, 44 (1948) 855; 46 (1950) 37. S m i t h u y s e n, H. B., private communication. Underwood, A . J . V . , J o u r n a l I n s t . Petr.,31(1945) I l l ; i b i d e m 3 2 ( 1 9 4 6 ) $98, 614; Chem. Eng. Progress, 44 (1948) 603. 5) v a n Wij k, W. R., Physica, 's-Gray., 15 (1949) 933. 6) v a n Wij k, W. R., Physica, 's-Gray., 15 (1949) 634.