INDUSTRIAL AND ENGINEERING CHEMISTRY
2502
TABLE 111.
WIT11
SAMPLE
Silica-Reinforceda Acrylate Unreinforced Acr late Acrylonitrile Tripolymerb Acrylonitrile Tripogmerb Elongation, % Ed Elongation, % Rd 0 30.8 0 26.2 50 30.5 100 27.4 225 34.4 200 29.4 250 .51.1 250 31.1 450 55.2 350 32.6 550 59.4 300 33.9 600 63.1 650 33 2 625 63.9 ” 57.5 ethyl acrylate. 30% rutyl acrylate. 12.5% acrylonitrile. b 20 parts silica per 100 parts tripolymer.
near the ultimate elongation the samples were very white and opaque. The unloaded film, on the other hand, showed this effect only slightly. Some measure of the increase in whiteness may be observed from Table 111,in which the results of the determination of luminous reflectance, &, measured with a Hunter colorimeter (Gardner Laboratories, Inc., Bethesda, Md.), are listed at various elongations. A white opaque porcelain tile plate used as a standard has an Rd of 79.5 when measured on this instrument. It may be observed from Table III that with the loaded stock, there is over a twofold increase in luminous reflectance from 0 to 625% elongation, whereas in the case of the unloaded stock the increase is approximately 1.2-fold. It is also significant for the reinforced film that up to22570 elongation the increase in white-
Vol. 43, No, 11
ness is relatively small, but beyond 225% elongation there is a large increase in whitenesR as evidenced by the rise in Rd. Concerning the whitening of strained silica-reinforced films, two possibilities exist. In the first instance, the effect may be due to the stress-induced orientation of the polymer chains contributing to a change in refractive properties of the copolymer films. If such is the case, the refractive indexes of the silica and polymer film, initially equivalent, may be sufficiently different after elongation to permit observation of the silica particles in the film matrix. That the refractive properties of a copolymer film, per se, change as the film is elongated can be substantiated by viewing the unstrained and strained film between crossed polaroids. Unstrained the film is isotropic, whereas the elongated film is anisotropic. The second instance concerns the possibility that when the elastomeric copolymer film i s elongated sufficiently, the elastomer may pull away from the particles of silica in the axis of stress, causing separation of the silica from the copolymer film, and whitening (6). LITERATURE CITED (1) Allen, E. M., Gage, F. W., and Wolfe, R. F., Rubber Age, 65, 297 (1949).
(2) Blevins, T. B., Wright, W. S., and Leonard, F., Anal. Chem., 22. 1205 (1950).
(3) Chamot, E. M., and iMason, C. mi., “Handbook of Chemical Microscopy,” Vol. l, p. 362, New York, John Wiley & Sons, 1944. (4) Moore, R. L., IndiaRubber World, 118,232-4 (1948). (6) Schippel, H. F., IND. ENG.CHEM.,12,33-7 (1920). (6) Schmidt, E., Ibid., 4 3 , 6 7 9 (1951). (7) Wiley, R. H., J . Polymer Sci., 2, 10 (1947). RECEIVED M a y 25, 1951.
Digital Computers for Trial and Error Calculations of Distillation Design ARTHUR ROSE, THEODORE J. WILLIAMS, AND HARRY A. KAHN The Pennsylvania State College. State College, Pa. Trial and error calculations are frequently essential in the solution of a variety of engineering and scientific problems, and they often involve extensive computation. This paper describes the use of IBM computers for performing automatically the trial and error calculatione involved in the choice of the proper reflux ratio for a continuous distillation of a binary mixture. The same general method m a y be used for more complex distillation problems and for trial and error calculations of other unit operations.
T
HE advantages of commercially available punched card digital computers for various types of straightforward plateto-plate distillation calculations have been described (5). Methods for the stepwise, plate-to-plate calculation of batch distillation curves with appreciable holdup and for plate-to-plate calculations of steady-state or continuous distillation are applicable to multicomponent as well as binary mixtures, and to cases of variable relative volatility and nonadiabatic operation; plate efficiencies other than 100% can also be taken into account. These procedures offer great savings in time and labor as well as
the elimination of human errors, advantages of major impartance when many plates or many steps are involved in the calculations. The present paper describes the use of the recently developed IBM card-programed electronic calculator for some of the more It ngthy and difficult distillation problems which require a trial and error procedure for successful solution. As an example, there is chosen the problem of estimating the reflux ratio required for separation of a specified binary feed into specified products in a column with a certain definite number of plates. The primary example assumes constant relative volatility, adiabatic column operation, and a 100% plate efficiency. A Recond example, dealing with the corresponding case when relative volatility is not constant, illustrates the approach when one or more of the usual simplifying assumptions are not justified. Simple extension to multicomponent problems of the trial and error type is not possible because the limits of the storage capacity of the machines are reached, and because of the additional degrees of freedom inherent in the general multicomponent problem. The methods illustrated herein are entirely adequatc, however, for many multicomponent problems of practical interest. Thus, Lewis and Matheson-type calculations ( 1 , .$) can be Rolved with ease, as can dny multicomponent problem where all
November 1951
INDUSTRIAL AND ENGINEERING CHEMISTRY
but two of the components are absent from one of the products or are present in specified concentrations. The possibilities of the machine when used with the Underwood and related equations (4, 6) have not yet been explored. The card-programed electronic calculator can carry out these calculations automatically without aid from the operator other than the prepmching and occasional handling of the instruction cards. Testa have shown that solutions are obtained approximately ten times as fast as by a person using the latest type of desk calculator. The ability of the card-programed calculator to perform these lengthy complex trial and error-type calculations aa well as other types of long complicated problems a r b s from a new system of calculation control and direction aa compared to previously avdlable calculators. The previous system was to wire removable control panels in such a manner as to cause the machine to carry out the various steps of the particular calculation procedure involved, using numerical values read from punched cards and punching the answem in the same or different cards. The utility of such machines for lengthy calculations is limited by the relatively small number of different operations that can be performed because of a limited number of connections on the control board and by a limited capacity for storing in the machine those numbers needed later in the calculation. I n the card-programed calculator, on the other hand, the control panels are wired to carry out each of the basic operations of arithmetic-addition, subtraction, etc.-aa well aa the necessary operations for transfer of numbers from one storage unit to another within the machine. These same control panels are then used with minor variations for a11 different calculation procedures. This is made possible by using punched cards not only as a medium for introducing the required numerical values of the problem but also to carry other special punchinga which oauw the machine to carry out the various arithmetic steps of the problem in their proper order. A change in the calculation procedure can then be brought about merely by substituting new properly punched cards for those previously used. As there is no limit to the number of cards that may be fed into the machine, there is no limit to the length and complexity of the problem that can be calculated by the machine, provided sufficient storage space is available for all the intermediate factors involved. The storage facilities of the card-programed calculator are five times larger than those of any previous commercially available machine. The usefulness of the card-programed calculator, &a well as previously developed calculators, for the calculations described herein arises from inclusion in the mechanism of two features designated as selector switches and a negative balance detector. The former may be used to direct the machine to perform one or another of two predetermined sequences of operations. The negative balance detector may be used (through its ability to compare two numbers automatically) to determine the point at which a selector switch is activated. One further requirement for the automatic trial and error calculations is that there be some relation that permits the machine to calculate a new and more nearly correct trial value of the primary variable, based on the error resulting from use of the previous trial value of this primary variable. When this requirement can be met, the procedures illustrated herein may be used for trial and error calculations for unit operations other than distillation, and for solution of a great variety of other trial and error problems. BASIC PROCEDURE FOR TRIAL AND ERROR DETERMINATION OF REFLUX RATIO
The essential steps in the trial and error calculation procedure of the reflux ratio required for a particular separation are: The numbers corres onding to the basic conditions of the distillation are manuafy punched on standard IBM cards. This includes values for the feed conditions, the dirrtillate rate,
2sO3
and the specified roduct compositions. A punchin for tho initial tria value o f t h e enriching liquid rate is also in&ded The machine automaticall calculates by material balance relations the remaining p r d c t compositions and flow rates, using the trial liquid rate. Thus if a distillate composition is specified, there is obtained a triaf bottom composition corresponding to the initial trial value of the liquid rate. This trial bottoms composition is based on material balance relations, and the next ste is to calculate the corresponding trial bottoms corn osition gy plate-to-plate calculations. Comparison of thew two\ottoms com ositions is the basis for estimating a new trial value of the liquig rate. The machine automatically carries out the plate-to-plate calculations by alternate use of the equations for the enriching line and t,he vapor-liquid equilibrium relations. This is continued until a plate composition is reached that is less than that corresponding to the intersection of the enriahing and stripping lines. At this point, one of the selector switches is activated so that in subsequent plate-to-plate calculations the stripping line e uation is used. This is continued until calculations are comppete for the specified number of plates. The trial bottoms composition from the plate-to-plate calculations is compare$ with that from material balance, and the difference is used by the machine to calculate automat8icttlly, with the equation
K’ = 4 a new trial value for the liquid rate. This is then used in repetition of the procedure. Because the machine prints its answers aa it proceeds, the ma nitude of the error after each cycle of the calculations is easfiy observed, and when the error reaches negligible proportions, the machine is Rtopped. It is paesible to have the machine stop automatically when the error reaches a s ecified small value. There is usually little advantage to this re&ement, because 100 pards pass through the machine per minute, and thus the CYC~PS and the entire procedure are very short. The on1 manual operatiom involved are the initial punching of the car&, the transfer of the pack of cards to the feed hopper of the machine after each cycle, the observation of the printed valuesl and the final stopping of the machine. By automatic punching of replicate seta of cards for the several cyclea, and hy &roviding for automatic stopping of the machine, operation can e made completely automatio, but this ia seldom worth whilc because of the high epeed of operation. SOLuTlON O F SPECIFIC EXAMPLE
As a specific example suitable for illuatration of the details of the procedure, the following problem is used: Calcukate the reflux ratio required to achieve the product compositions indicated when other operating conditions are as specified. The correct answer happens to be R = 4.
TABLEI I. OPEBATINGCONDITIONS Condition of feed Dbtillate cornposition Operating oondjtiona Relative volatility Theoretical plates
P
--
1O.oooO
. I
q, =. 0.7895 D 1.oooO P 2.2300 U 6 (tot&
s,
E
0.2OOO
Q a
1,0000
0
The 108 cards required for this computation are punched aa indicated in the sucoeeding paragraphs, which also indicate the corresponding computations performed by the machine. CARDS1AND 2. The 6rst card cleara the machine of numbers present from previous calculations, The second card is unched with numerical values of F, 21, g, m, D,and U , and a &st trial value of L. For the example under discussion, this value of L was L1 = 9. Aa this second card passes through the machine, the various numerical values are transferred to seven primary storage units directly associated with the calculating and printing mechanism. CARDS 8 TO 9, INCLUSIVXL Thew cards merely serve the purpoee of traneferring the numerical d u e s for F, 21, etc., from
2504
Vol. 43, No. 11
INDUSTRIAL A N D ENGINEERING CHEMISTRY
5
01000000001000000000000000000~000000000000000000000000000000000000000000000000000 I 2 I I 5 6 1 8 9 IO ll 11 0 14 I5 I 6 I1 18 18 ?On 2 2 P l 7 ~ 2 S Z 6 2 1 7 6 2 P J o11 11131b1516311839b0414243~~451841(8(95(5157$154 U 5 6 S 1 1 6 5 8 6 0 6 1 6 2 6 1 6 ~ S 5 6 6 6 r 6 8 8 9 1 0II 17 1 3 1 4 15 16 11 18 19W
~IrIllllrlllllllllllrrlllllllllllllllllll1l11l111l111l1l111111l11l11l111111ll1lll
m22221112222222222222222222222222222122222~22222222221~22222222222222~2222~~~2222 3333333333333333333333~33333~3333333333333~3333333333~333333~3~3333333333~~33333
4444644444444414444444444444444444444444444444b~4444444444444444444444444444444 55555551555555555555555555555555555555555555555555555555555555555555555555555555 66666666666666666666666666666666666666666666666666666666666~6666~666~66666666666
l1l11l1l1l1l1l1l1ll1lllllllllllllllllllll~lllllll~llllllllllllllllllllllllllllll
aaa8a6a8a1aa8aaaaa8aaa8a6aaa~a8aaaaa8a6~~~aaauaaa6a68auaa8aa~a~aaaa~aaa~a~aaaaa8
!!1 f!!
~
~
~
~
-D
Figure 1. Card Punched for Calculation of W = F
the primary storage units to similar storage units in an auxiliary
device h a w g a capacity for storing 16 different numbers of up to 10 digits each. CARD^ 10 TO 32) INCLUSIVE. These car& are punched so as
CARDS46 TO 58, INCLUSIVE. These function like the previous group of 13, and give values of l/n =
to c&use the machine to carry out in proper sequence the calculations corresponding to the following equations.
w=F -D
ZW
-1=9 F z i - DXD = (10)(0.2) - (1)(0.7595) = = 10
W
v
=
L
+D
= 9
sn
0.1378
~
(3)
+ 1 = 10
(4)
54
0.4410
= 0.2613
u = 4-0.0612
(2)
9
=
~ / 4+i
CARDS59 TO 71, INCLUSIVD. These function like the previous group and give = 0.3111 23
+ 10 = 19 V' = v + (q - 1)F = 10 + 0 = 10 L' = L + q F = 9
= 0.1684
(5)
u = -0.0317 (negative)
(6)
As u is negative, the selector switch changes osition and sub-
sequent crtlculations for y are made with t i e stripping line equation. CARDS72 TO 84, INCLUSIVE. These are punched so that the machine calculates " = a
- (aX D- l ) S D - (2.21) -0.7595 (1.23)(0.7595) u
p.
X#
- (2.1
+ 0.0001)
0
= 0.5861 (8)
0.3860
urn
P
Y2
=
L'
19 9 lo (0.1684) - -- (0.1378) 10
p 2 2 - wsw
0.1959 (11)
(13)
These include the series of material balancecalculationa referred
sm= 21
= a
- (Yal- 1)yI - (2.23) -(0.1959) (1.23) (0.1959)
E
t,o in the previom discussion, as well as other valuea required
in mbsequent calculation. The reaulting numerical values are transferred to storage units where they are held until needed for eubsequent calculations. CARDS33 TO 45, INCLUSIPE. These are the first of N identical groups of 13 cards each. Each group of cards supplies the necessary information for the machine to make the Iateto-plate calculations for one plate. The cards are p u n c h 2 with values for both the enriching and stripping section. As long as the value of u (Equation 13) remains positive, the selector switch causes use of enriching section equations and values. As soon as u becomes negative, the selector shifts and Wuses stripping section calculations to be made. The results for cards 33 to 45 for the example under discussion are: gt-,-,l
v + -v zo
L
- Z(
D
. I
VI-
Z'-' = a
f
a
9 - (0.5861)
10
- ( a - 1)yt-1
-
+ a(0.7595) = 0.6034 1
st-1
No further tests are made for the value of u, as the selector is wired to remain in the stripping position until the plate calculstions are complete. These function like the previous CARDS85 TO 97, INCLUSIVE. groups and give MI 21
0
f0.2055
=
x;V
0.0293
(9)
TABLE IT. TYPICAL CALCULATION
- (1.23)(0.6034)
- ( x < + 0.0001)
= 0.0631
This completes the late-to-plate calculations. CARD98. This car! merely releases the selector switch
0.6034
(2.23)
0.4056
u =
0.0985 (12)
(10) (13)
As u is still positive, the selector switch remains in the position
directing use of the enriching operating line for subsequent calculations.
Trial
Trial L 9.00 6.28 4.79 4.22 4.03 4.00
zv Material Balance 0.1378 0.1378 0,1378 0.1378 0.1378 0.1378
z& Plateto-Plate 0.0293 0.0603 0.1037 0.1261 0.1350 0.1376
0ZW
0.1085 0.0775 0.0341 0.0117 0.0021 0.0002
November 1951
will need t o be punched in coldrnns 4 and 5 so as again to withdraw factor F from storage t h o u h channel A, and that columne 7 and 8 will need td be punches to withdraw z/ from siora through channel B, that column 6 will have a line 3 punching cause the neceasary multiplication, while columns 9 and 10 must be punched t o cause transfer of the product Fzj t o a storage r e p t e r where it will remain until after DXDis calculated. All su sequent card punchings are developed in a similar manner.
C m s 99 m 105. These are unched to cauw the calculator to obtain a new trial value of L,By means of the equationa e,
=
xw
- x;P = 0.1378 - 0.0293
-
0.1085
(14)
h + K'(e.,) -- 1 + 4(0.1085) = 6.28
(1)
8
and
Ln
9
1
2905
INDUSTRIAL A N D ENGINEERING CHEMISTRY
DERIVATION OF EQUATION FOR SUCCESSIVE TRIAL VALUES O F L
CARDS106 M 108. These are unpunched and serve merely to cause the completion of all calculating and printing operations initiated by the immediately preceding cards. At this point the first 9 cards are removed from the deck and the remainder are reintroduced into the feed hopper of the machine, which then repeats the cycle beginning with the 23 cards (10 to 32)required for the series of material balance calculations.
The equation used for this purpose is based on the empirical relation
The numerical results of a typical salculation are given in Table 11.
e,.
As R2must be smaller than RI if e,. is positive, r is positive and r < 1 when e,. is negative. Now if the equation for r takes the form
MECHANISM O F OPERATION
r = 1
To give some further information on a typical step in the com-
Thus, card 10 was punched as in Figure 1 in order to cause the simple calculation W = F - D . The combination of punchings in line 1 of column 4 and line 2 of column 5 causes the numerical value of F to be transferred through electrical channel A to the counter of the calculator from storage register 2 of auxiliary storage bank 1, where thia value of F had been introduced by previous cards. The combination of punchings in line 2 of column 7 and line 5 of column 8 causes a similar transfer to the counter through electrical channel B of the value for D, which had previously been placed in storage register 5 of auxiliary storage unit 2. The punching in line 2 of column 6 actuates the calculator to subtract the number D coming in on channel B from the number F coming in on channel A, and the remlt is that the numerical value of W appeara on the counter. The combination of punchinga on line 1 of column 9 and line 8 of column 10 causes the transfer of W from the counter of the calculator to storage register 8 of auxiliary storage unit 1, where it remains until required for the subsequent step. The punchings in columns 1, 2, and 3 designate the card number and the zero punching in column 11 prevents shifting of the decimal point as the numbers are transferred. Even a cursory study of Table I11 will indicate that the 6rat card required for the computation xt
1 when
+ K'(e,,)
and D is canceled out, a new trial value of L can be calculated aa fOllOW6:
putation, the first few operations in the material balance calculations are suitable. Tab,le I11 indicates the general plan for punching cards to achieve a desired calculation.
E
>
It has been empirically determined that a K' of 4 is the best value to use for the case considered here. Smaller values of H' require an excessive number of trials, whereas larger values caum overcorrection of the value of L2 and consequent high and low oscillations of the value of LI. CALCULATIONS FOR CASES INVOLVING VARIABLE RELATIVE VOLATILITY
While relative volatility is substantially constant in some c a m , in a great many instances this simplification is not possible. The automatic calculation procedure can be easily modified for cases in which relative volatility is known as a function of composition, or cases in which vapor and liquid compositions may be correctly related by a function of the form
+ by* + cy8
x = ay
The use of fictitious molecular weights to adjust for unequal heats of vaporization usually leads to relations of this form. As a specific instance, calculations were made for an example in
FZJ - DXD W
-
TABLE111. GENERAL PLANFOR PUNCHINQ CARDSFOR CARDPRQQRAMEID CALCULATOR 4
Punch 12 11 0
-----Channel
...... ......
Card, read in
6
A-
...... ......
Card, read in
Aux. stor. bank 1 Aux. stor. bank 2
...... ......
......
...... 7 8
9 a
Counter read dut Counter, read out and reset
......
Optional aeaimment.
6 Operation List' ' ' ' ' ' '
........
Add" A + B - C Bubtraot" A - B - C Counter group or storage, register within bank
A + B = C Sq~are'root
4x
=
c
Available" lor other operation Transfer " C-A Transfer"
C-B
Summary punch
C?nl,,m,.
7 -Channel
...... ......
Card, read in
8
...... ......
Card, read In
Am. stor. bank 1 Aux. stor. bank 2
...... ...... ...... ......
Counter read dut Counter read oh and reset
......
. Counter group or
atorage regia& within bank
10
9
-hannel
......
Eleotronio oounter read od Aux. stor. bank 1 Aux. ator. bank 2
......
......
......
Counter. add Counter aubtrs'ot
......
C
...... I . .
Eleotronio oounter read oui
11 Shift
... . .
...
Shift 1 Shift 2
Counter group or storage redate; within bank
Shift 3 Shift 4 Shift 5
.. ...
...
I N D U S T R I A L A N D ENGINEERING CHEMISTRY
2506
which all numerical values were the same as for the preceding detailed caw, except that the vapor-liquid equilibrium was given by the preceding equatioii with a = 0.1429
b
=
0.8571
c = O
TIME REQUIREMENTS
The punrhed cards pass through the computer at the rate of 100 per minute, SO that each trial cycle for the examples cited herein requires but a minute, and 5 or 6 minutes suffice for the entire solution. More time would be required if there were more plates in the column, a t the rate of about 10 wconds per plate per trial cycle. An experienced operator can complete the manual prepunching of cards in 15 minutes or less. DISCUSSION AND CONCLUSION
TABLE IV. CALCULATION Trial
zp iMaterid
zw Plate-
Balance 0.1295 0.1295 0.1295 0.1295 0.1295 0.1295 0.lag:,
to-Plate 0.0176 0.0396 0.0895 0.1352 0.1268 0.1301 0.1296
Trial L 9.0000 6,2200 4,5700 3.9400 4.0300 3.9900 4.0000
Vol. 43, No. 11
% ji.
0.1120 0.0899 0.0400 - 0.0057 +0.0027 - 0.0006 -0.0001
The procedure wm the same as before, except that the value of was omitted from card 2 and three new cards, 9A, 9B, and 9C, were introduced with punchings for introduction of the values of a and b, and transfer of these to storage registers. -411 the other cards that served for calculations by the relation
The examples and procedures cited in this paper have been kept as simple and brief as possible. They are the first attempt, to the best of the authors’ knowledge, to use commercially available digital computing equipment for saving time in complex trial and error ralculation~in chemical engineering. It is believed that the methods can be greatly extended ta include many multicomponent distillation calculatiom, as well a8 for complex, lengthy, and trial and error calculations for other unit opriations, and for other engineering calculations
o
a-(a-l%
D E
were replaced by car& calling for calculations by t.he relation
+ by2
I = nr/
ets
F
The calculation t,hen proceeded in the maiiner described. Table IV lists the results. NONADIABATIC OPERATION AND PLATE EFFICIENCY
Nonadiabatic operation results in variation of L, V , L’, and V‘ from plate to plate in the column. For the caw of hwt low from the column
H, i K’
I,
L’ m
N n
Q
and
q /In
=
Q + H,
IJ2,+1
where Q is quantity of heat lost per unit of time arid H , is the heat of vaporization of 1 mole of the miuture concerned. 111 order to perform such calculations, each group of 13 card8 tor the plate calculations must be preceded by a group of cards punched to carry out the calculation of the preceding equation, a q well 2 s cards for calculation of V,, = L
+I
I,,, = I,“
+ (p
+ ((I - 1)F’ (0, + + 1 N q - l _l r _n = I,,, + qi) V,’
X‘
+D
=
,T’
I))Z/
The proper values of L , and V,, are thus made availttble for use in calculation of ~ , ~ - - 1with the enriching line equation, if the plate is above the feed, or L,,, and V, are available for similar use for plates below the feed. The modified value of x2 is used for obtaining the value of u for the plate in question The remainder of the calculation proceeds as described. Plate efficiency may be taken into account in a fiimilar manner by introducing cards for the calculation of Zn
=
5n+1
- E(&+I
- si)
after the usual calculation of the equilibrium liquid composition x* by use of the vapor-liquid equilibrium relation.
constant coefficients used in equation for vaporliquid equilibrium relation RR subscript refers to distillate x n + 1 - xn q = plate efficiency as defined by E = x = difference in value of TW as calculated by two different methods = feed rate, moles per unit time = heat of vaporization of liquid mixture at point in question = as subscript refers to intereection of operating lines -Le., feed plate empirical constant used in Equation 1 = enriching section liquid return rate, moles per unit time = stripping section liquid rate, moles per unit time = as subscript refers to general value of plate number in stripping section = total number of plates in column as subscript refer8 to general value of plate number = in enriching section of column = amount of heat loss from column per unit time = condition of feed as a ratio of heat required to vaporize 1 mole of feed t o that required t o vaporize 1 mole of saturated liquid of same composition = reflux ratio as LID = divisor used in derivation of equation for estimation of second and subsequent trial values = as subscript refers to top plate = value of difference between liquid composition on plate in question and liquid compogition on feed plate enriching section vapor rate, moles per unit time = = stripping section vapor rate, moles per unit time bottoms take-off rate moles per unit time; aR subscript refers to bottoms = liquid composition as mole fraction more volatile component; subscript refers to location in column = value of liquid composition as predicted from vapor composition vapor-liquid equilibrium relation = vapor composition as mole fraction of more volatile component. subscript refers to location in column = feed composition a s mole fraction of more volatile componmt, , = relative volatilitv of mixture under study = symbol signifving rate of change of quantitv in question
a, 4 c
?!
? ’ E
’
NOMENCLATURE
R r
t u
vV‘
=
= distillate take-off rate, moles per unit time;
w =
x xi: ?/ Zf
a
A
LITERATURE CITED
( I ) Lewis and Matheson, I N D .E m . CHEM.,24, 494 (1932). ( 2 ) Robinson and Gillilnnd. “Elements of Fractional Distillation,” Chap. XIV, New York, MoGraw-Hill Book Go., 1939. (3) Rose, Arthur, and Williams, T. J., I N D . ENQ. CREM.,43, 2494 (1951). (4) Shiras, Hanson, and Gibson, Ibid., 42, 871 (1950). (5) Underwood, A. J. V., Chem. Eng. Pmgresn, 45, 609 (1949). R m x i v n n Aoril 12, 19.51
