Vol. 15, NO. 4
INDUSTRIAL AND ENGINEERING CHEMISTRY
402
Calculations on Fractionating Columns from Theoretical Data' By W. A. Peters, Jr. E. I. DU
PONT DE
NEMOURS & CO.,WILMINQTON,
DEL,
N an article on the gram molecule and A n equation is derived giving the relation between the compoHz, we get the ratio of the efficiency and capacsition of the vapor and of the liquid in equilibrium with it for ity Of fractionating binary mixtures of homologs, isomers, and other similar liquids. vapor tensions at the Co1UmnS,2it was suggested perature With the aid of this equafion the distillation equation giving the that the efficiency Of a composifions of a binary mixfure of liquids, before and after a be measured by certain portion has been distilled off, in terms of the weights of 'Omparing the performance material at the beginning and end of the distillation, can be fi of a definite unit or length infcgrafed. pz - e of the given column with A n equation is derioed which gioes the height of a fractionating the performance column required to make any given separation of a binary mixfor perfect ture of liquids under given conditions of heat expenditure, compoplate column under similar sition of distizlafc, eft. conditions. The theoretiFor mixtures of homologs, cally perfect plate column isomers, and many other i s defined as one in which the vapor rising from any similar liquids, H1 may be considered equal to H2, and H2 plate is in equilibrium with the liquid on the plate, all of equal t o HI, which is constant over a short temperature range. the liquid on the plate being of the same composition. H b-a The justification of this definition lies in the experimental (1) 1.99 ab = constant 91 data, which prove that, for the separation of any given liquid $2 = e mixture, the height of any column of uniform construction equivalent to a theoretical plate is constant and does not This cmstant will be called K (K < 1). It will usually be depend on the ratio of heat supplied a t the base of the column found more convenient to determine K directly from known to distillate withdrawn, to the composition of the liquid in data on the vapor tensions Of the liquids considered than to the section of column considered, or to the composition of use Equation 1, since the vapor tensions are more accurately the distillate or still charge. This ratio varies greatly for known than the heats Of vaporization. different liquids and must be determined separately for each From Equation $ = K t the vaPor-liquid composition new mixture. The constant will be called the height - of. equivr* dent theoretical plate, or H. E. T. P. Several values for it are relation is derived as follows: given in the article mentioned above and a method for calculating it from experimental data is also given. A similar Let X = concentration of low-boiling material in the liquid Y = concentration of low-boiling material in t h e vapor method of calculation was developed by W. K. Lewis13 M = total pressure under which the mixture is boiling but both methods are somewhat tedious. A shorter and A = vapor tension of the low-boiling material at the boiling more exact method can be used in certain cases where the temperature, all concentrations and ratios being expressed in terms of the latent heats of vaporization equation expressing the relation between the composition of of the materials the liquid and that of the vapor in equilibrium with it for mixtures of the two liquids under Consideration is known. I n For mixtures of homologs and isomers, we can write such cases, an equation expressing the rate of enrichment in M = AX AK (1 X) (2) the column can be developed and integrated, the resulting equation giving directly the height of column required to Now, AX is the partial pressure of the low-boiling material make the separation required with the. specified heat con- in the vapor. Therefore, sumption. y = -A X M VAPORCOMPOSITION EQUATION The equation for vapor composition in terms of the com- Eliminating A between Equations 2 and 3, we have position of the liquid in equilibrium with it is derived as X (4) follows : = K + X ( l - K) The Clapeyron equation, expressing the vapor pressure of a If K is chosen as the ratio between the vapor pressures of the substance in terms of its latent heat of vaporization and the two materials a t a temperature which is the average between absolute temperature, is the boiling points of the two a t the pressure considered, this equation will give results closer than the results taken from anx graph. The equation will first be applied to the case of i n which simple distillation. p = vapor pressure at temperature T P = vapor pressure at temperature a APPLICATION TO SIMPLE DISTILLATION H = latent heat of vaporization in calories per gram molecule 1.99 = the gas constant, R The equation expressing the weights of a given mixture For two materials whose boiling points are a and b a t the before and after a portion has been distilled off, in terms same pressure, and whose latent heats of vaporization per of the composition of the vapor and liquid, is
I
'
)
(
u
+
Received September 1 5 , 1922. Peters, THISJOURNAL, 14 (1922),476. a I b i d . , 14 (1922), 492. 1
p
In-
= W
O
-
(5)
IXDUSTRIAL A N D EhTGINEERING CHEMISTRY
April, 1923
where W, = weight of original mixture of concentration xo W = weight remaining a t a time when the concentration of the liquid is x4
Substituting in Equation 5 the value of Y given in Equation
4, and integrating, gives
whereA
=:
K 7 -K
This equation will give accurate results for all mixtures of isomers and homologs.
CALCULATION OF HEIGHT OF FRACTIONATING COLUMN I n any fractionating column, the relation between the vapor passing up through a horizontal plane a t the point n and liquid passing down is given by the equation Yn+I =
(n above n
+ 1)
PX,
+ (1 - P) Yc
(7)
where Y , + I = concentration of low-boiling material in the vapor X, = concentration of low-boiling material in the liquid Y, = concentration of low-boiling material in the distillate removed from the column p = ratio of vapor passing up, to liquid coming down
As before, all concentrations and weights are measured in terms of the latent heats of vaporization of the respective materials. From the definition of the theoretical plate column, it follows that a t a point higher than n by a distance equal to the H. E. T. P., the concentration of low-boiling material in the vapor will be Y,. The difference between the composition a t this higher point and a t the point n will be Y, - Y,+l, and this difference gives the rate of enrichment in the column. UTe can write Yn
- Y n + l = AY
(8)
and since it has been proved experimentally that the ratio between the enrichment and the height of the column is constant dk AY
a
dY =
Substituting in (10) for (Y, + 1 ) from (7) and (4),solving for dh, arid dropping subscripts gives dh=adY 1 Write A =
-
Y
1
- P ) Y o -!- I-K
-
C
ln(A+BY-Y2)+-
In -2Y
+ B - d-q
¶ : / .
As an example of the use of this equation, consider the case of the separation of benzene and toluene in a continuously operated plate column. It has been determined that two standard bubbler captype plates are approximately equivalent to one theoretical plate for this separation. If we make a: = 2, H will be given in actual plates. From data on the vapor tensions of benzene and toluene, average for K is found to be 0.4. It is desired to obtain a distillate running 99 per cent benzene with a steam consumption of about 1 lb. per lb. of distillate, making Y, = 0.99 p = 0.8
Let it be required t o find the number of plates in the column between the top and a point a t which the molecular concentration of benzene in the vapor is 0.4. The constants will be B
A = -0.3333
C
1.3313
= 2.0
-0.439
q
= 0.663
d-4
+ 1.3180 - 0.9801 21n -1.98 + 1.3313 - 0.663 -0,3333 + 0.5325 - 0.1600 0.663 -1.98 +1.3313+0.663 21n - 0 8 + 1.3313 - 0.663 -k 0.66.1 - 0.8 + 1.3313 + 0.663 ’Iates
= In -0.3383
Similar equations can be developed for the exhausting section of the column. Let R = ratio of liquid down t o vapor up (R>1) X, = composition of effluent Y , = composition of vapor from mth plate [mth plate above (m 1)th plate)
+
All compositions and ratios in terms of equivalent latent heats of vaporization. -Xs(R-l) l--K
Then AY =
+ 1-1-RK + (R-l)X8-1-
1-K
__ Y
(131
The equation for H for the exhausting section will be similar to (12). I n it B =-1 - R K 1-K
x,I(R-1) -K
A=--
+ (R-l)X,
(9)
where (Y = H. E. T.P. d h = an increment in the height of the column measured in theoretical plates Substituling in (9) the value of AY from (S), dk dYs= (Yn-Yn+l) (10) Q!
1 F K
403
1 - P 1-K
It is desirable to introduce the feed in any continuous column a t the plate where the rate of enrichment in the rectifying section is equal to that in the exhausting section. This point is easily found by equating (13) and the value of AY from (11). This gives X,(R - 1) - y J 1 - p ) - R) + (1 - K) [(R- 1)
Y = K(P
If,as is usually the case, X, Y =
- ~lY.1
(14)
= 0 or nearly so
- y , (1 - P ) K ( P- R)- (1 - K)(1 - P)Y,
(15)
The minimum height of column which will make a given separation with infinite expenditure of heat is given by making p
-
and R = 1 A then = 0 B = l
e = -1 + K 1-K
c=-
1-K
9 = 4A
- Ba
Substituting these values in between the limits-Yl andAYz 4
- B
q = -1
and Equation 12 becomes
tion 11 and integrating
C. S.Robinson, “Elements of Fractional Distillation,” p. 45.
The point in a column a t which no separation takes place is given by equating (10) to 0. This gives A BY Y2 = 0.
+
-
