ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
arch to St Column ROBERT F. JACKSON
AND
ROBERT L. PIGFORD
University of Delaware, Newark, Del.
T
RAKSIENT changes in composition within a countercurrent distillation column are of practical interest in three instances:
1. During start-up of a continuous column, especially if the separation is a very difficult one, as for mixtures of isotopes when the initial concentration is very small 2. During a batch distillation cycle 3. I n connection with the selection of a control system for which the rate of response of the column to small upsets in external conditions needs to be knonn
Case 1 is the simplest of these because it corresponds to a column that experiences a finite step-change in composition of a feed stream, after which the compositions of the distributed holdup in t h e column gradually approach their stezLdy-state distribution. A study of this case may shed some light on the other two, although it should not be expected that exactly the same frequencyresponse characterktics will be found in all three, owing to the essentially nonlinear dynamics of columns in which large differences in composition occur. I n starting the operation of a column the entire apparatus is filled with the feed liquid or with a vapor generated by boiling the feed liquid, and the column must be operated for some time before the countercurrent contact of the reflux and the vapor stream will be able t o produce the enrichment for which the apparatus is designed. Frequently the column is operated a t total reflux during this warm-up period, for even if a distillate were removed it would not be fully enriched; in fact, its removal would reduce the rate of approach to the desired steady state. For many ordinary distillations the time needed for this preliminary operation may be so small as to be of only minor influence in planning the start-up procedure for a plant as a whole, but in special cases such as the distillation of ordinary water to produce a stream enriched in deuterium oxide the time needed may be so great, because of the large number of stages employed and because of the very low concentration of heavy water in the natural water fed to the column, that production schedules may be significantly affected. Calculations of changing liquid and vapor compositions during this start-up operation are not difficult in principle. They involve the same material balances for liquid and vapor flows to an individual stage that are used.in designing continuous columns that operate at steady state, except t h a t transient conditions involve a redistribution of material in the liquid and vapor holdup on each tray. The rates of accumulation of a particular component are expressed in the material balances by means of derivatives of composition with respect to time. Thus, instead of having n simultaneous algebraic equations to express the concentration distribution in a column of n trays, one has n simultaneous ordinary differential equations, each of which may be nonlinear. These equations must be solved simultaneously, subject to boundaTy conditions a t zero time and a t the top and bottom of the column, Such a task is formidable from the formal mathematical viewpoint, although several approximate schemes for carrying it out in simplified cases have been suggested. The purpose of this article is t o report the results of numerical solutions of the exact problem for a range of conditions, the mathematical work having
1020
been carried out vith the help of a large scale digital computer t o avoid physically restrictive approximations. The results to be presented below are believed to be as sound as are the generally accepted results of tray-to-tray material balance calculations for steady-state operation, since the representation of transient conditions on the trays does not introduce any important additional physical assumptions.
Mathematical Background Consider a column composed of N actual trays, each having a biurphree efficiency ( E )and containing a two-component system that has a constant relative volatility (a). Neglect the holdup in the vapor phase and assume that the same number of moles of liquid mixture ( H ) are present on each tray. Assume also that the liquid is well mixed, so that the average composition of the holdup on each tray can be taken equal to that of the liquid leaving the tray. Assume further that the molar flow rates of vapor and liquid remain constant on all the trays of the column, that there is no feed stream, and that operation is a t total reflux. A material balance around one tray shows that T’(Y,-I
- yn)
+ L ( ~ n + l- x,) = H ( d z n / d t )
(1)
where z n and yn are the mole fractions of the low boiling constituent in the liquid and vapor leaving the nth tray, respectively. The trays are numbered from 1 a t the bottom of the column to N at the top. Sating that a t total reflux L = V and introducing T = L t / H , the time required for the liquid stream to fill a single tray, Equation 1 becomes ~(n-1
- yn
+
~ n + i- ~n
=
dxn/dT
(2)
The blurphree efficiency of each tray, assumed the same for all trays, leads to (1 - E )
y n = ya-1
+ y,*E
(3)
where y: is the composition of a vapor a t phase equilibrium with Xn.
z 4 = 1 + ( a x-n
1) xn
(4)
Equation 4 is the cause of nonlinearity in the system of equations and has heretofore been approximated by various simpler functions to permit formal solutions to be obtained. For example, for small xn the linear approximation to Equation 4 is
u?
=
ff
(5)
xn
This has been employed by Bardeen ( I ) , Cohen ( 3 , 4), Davidson ( 6 ) , and Marshall and Pigford (8). biontroll and Neve11 (9) were able to obtain a solution of a problem closely related to the one indicated by the above equations, although with a second term in the series approximation t o Equation 4 y.* = xn
+
- 1)2,(1
(ff
INDUSTRIAL AND ENGINEERING CHEMISTRY
- 2,)
(6)
Vol. 48, No. 6
PROCESS CONTROL An approximation very similar t o Equation 6 was employed by Pigford, Tepe, and Garrahan (10) in their application of material balance equations like 2 t o the batch-distillation problem. The boundary conditions that complete the mathematical definition of the problem stated above are
( a ) xn =
constant a t T = 0 b ) Z N + ~= y r i . e . , no holdup in the condenser C) dx,/dT = k(a1 - y?) ~i=
(sa)
where k = liquid holdup per tray/holdup in reboiler and x, = mole fraction of liquid in reboiler. Solutions were carried out for two general types of operation: Type I, in which the reboiler has an infinite capacity (IC = 0) and in which the initial liquid on all the trays IS obtained by condensing the vapor generated in the reboiler (zi = y:) Type 11, in which the reboiler has very small capacity (k = 1 or 2) and in which the column is filled initially with equal amounts of light and heavy components (2, = 0.5). Equation 2 can be regarded as a difference-differential equation with n and T as independent variables. Steady-state concentration distributions of concentration can be found by obtaining the solution of the difference equation derived by setting the right side of Equation 2 equal to zero. When E = 1 the well known solution of Equations 2, 3, and 4 is CY"+
yn =
Xn
1
xs
+ 1 = 1 + (an+1 - 1)x.
(7)
downward flux of heat from the top of the column will lead eventually to an exponential temperature distribution analogous to
x
=
xaa = axr e ( a
-1
(12)
) ~
The transient solution of Equation 11 is given most completely by Cohen (d), although Marshall and Pigford (8) have also discussed the problem. According to Cohen, the fractional departure of the composition of the instantaneous distillate, XD(= x,+ 1), from the steady-state value, Z D . ~ is ~ , given approximately by -
. ._
(13)
-
Where A [( 01 - 1 ) N ] lies between 0.81 and 0.90 for (a l)N from zero t o 1.2 and where B is related to the smallest root (rl) of the transcendental equation tan ( r ) = 2r/(a
-B = r :
+
-
l)+V
(14)
( a - 1)N2
2
As shown by Marshall and Pigford ( 8 ) , B is approximately ( ~ / 2 ) 2 if (a - l ) N is very small. (The error is 473 if ( a l)W = 0.1.) A more useful approximation is given by Cohen ( 4 ) for large, positive values of ( a - 1)N:
-
- B = [ ( a - 1 ) N J 2 e - ( a -1 ) N { l - e - ( a -
..]
-
[In this case Equation 2 becomes linear in the mole ratio, x/( 1 x).] A similar solution does not appear to have been obtained 1)x is small compared to unity on when E # 1, although if ( C Y all trays-Le., if the equilibrium line is straight, an approximate solution is
(16)
Cohen's work ( 4 ) also includes values of A and B for columna having finite holdup in condenser and reboiler. Even though the values of B according to Equations 14 and 15 can be computed for large values of the enrichment factor, (01 1)N, there is a serious practical limit t o the usefulness of such results because of the very strong curvature of the equilibyn = x n + i = ax. [l ( a - 1)E]* (8 ) rium curve (Equation 4 ) when the factor a is not small. Under these conditions the linearized form of the differential equation Transient-Diffusion Method of Analysis gives the wrong steady-state solution, as is seen readily by comparing Equations 7 and 8. (Equation 8 indicates that the mole For the study of the mathematical properties of these equations fraction in the distillate increases without limit as ( a l)N it is convenient to replace the finite-difference terms on the left approaches infinity, while X D according t o Equation 7 does of Equation 2 by their approximate differential-quotient equivanot exceed unity.) There is every reason to believe t h a t the lents. For simplicity we assume for this purpose t h a t E = 1, transient solution according to Equations 13 and 14 will also be and on introducing Equation 4,Equation 2 becomes incorrect. Although - the enrichment may be satisfactorily small in certain xn(1 - xn)x,,-,(l - x n - l ) ] (9) (a - l)xn 1 ( a - 1)Zrn-l isotope separation problems in which 1 Cohen was principally interested, the correct representation of the transient behavior of an ordinary or, approximately, distillation column, except possibly for small perturbations in bX the concentrations, requires that the nonlinear character of the (10) problem be taken into account. The first attempt to solve the nonlinear problem was made by If, now, the linearizing approximation ( a - 1)s. Figures I and 2 show typical results obtained from calculations for Type I and Type I1 operation, respectively. Figure 1 especially shows that the upper trays of the column approach their steady compositions more rapidly than do those lower in the column, probably because of the influence of the upper "pinch" composition that limits the mole fractions to unity as a maximum value. Figure 2 shows that when both top and bottom of a column are free to change in composition toward separate steadystate levels, the rate a t which the whole column reaches its equilibrium state is greater than when the composition at one end of the column is fixed, as in Type I operation. If the time required for a given degree of approach to steady state is proportional t o the square of the number of trays and if, as in the case shown on Figure 2, the compositions on the center tray of the column are essentially constant, Type I1 operation should require just one fourth as long as Type I operation of the same column. This is based on the idea t h a t the top (or the bottom) section of the column behaves under these conditions just as June 1956
\\
NO 3
REROILER
though its opposite section were replaced by an infinitely large reboiler capable of supplying vapor (or liquid) of the proper constant composition to the bottom (or top) tray of the section. Figure 3 shows the fractional departure of the distillate composition from steady state in a typical case plotted semilogarithmically versus T, as suggested by Equations 13 and 18. Two curves are shown, representing values of emand EX. Both curves are substantially straight after a slightly curved portion near zero time, although the E* line may be somewhat the straighter of the two. It is perhaps surprising in view of the essentially nonlinear characteristics of the operation that the rate of response of the column as a whole should be characterized by a single time constant, as suggested by the linearized equations. Whatever
Table 1. Summary of Results for Machine Computation o f Time Required for Equilibration o f Distillation Columns A.
RunNo. 1 2 6 7 10a
11 101
103 104 105 109 112
Type I1 Operation (Very Small Reboiler)
N 15
15
15 15 15 30 30 30 30 15
30 30
CY
1.3593 1.3593 2.5119 2.5119 1.3593 1.1659 1,1659 1.1659 1.1659 1.3593 1.1659 1.5846
E
k
0.5 1 0.5
2 2 2 2 1 1 2 1
1
0.5 0.5 0.5 0.5 1 1 1 0.5
1
2 2 2
xi
= 0.5
T I N at E , = 0.1 1.27 1 .Q5 0.57 0.30 1.34 2.77 2.67 2.73 4.00 1.93 3.94 1.21
X D . ~ ~
0.7767 0.9265 0.9648 0.9995 0.7835 0.7719 0.7685 0.7720 0.9211 0.9265 0.9183 0.9679
B. Type I Operation (Very Large Reboiler) k
= 0
T I N at
RunNo. 3 4 5 8 9 10 110 111 115 116
N 01 E 15 1.3593 0 . 5 15 1.3593 1 15 1.3593 1 15 2.5119 1 15 2.5119 0 . 5 15 2.5119 1 30 1.1659 0 . 5 30 1.1659 1 60 1.0798 1 10 2 .oooo 0 . 5
Xs
XD,W
0.1375 0.01 0.5709 0.001 0.1200 0.01 0.9999 0.001 0.758 0.001 0.9996 0.1125 0.01 0.01 0.5442 0.01 0.520 0.03 0.860 0.01
INDUSTRIAL A N D ENGINEERING CHEMISTRY
e, =
0.1
21.6 64.5 131.5 8.7 115 88.0 39.1 142 295 6.9
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ENGINEERING, DESIGN, AND
PROCESS DEVELOPMENT
1.0
Run No. 4 8
08
5
10
o.4
0.2 0
40
20
60
80
/ZO
/00
/40
T / N , CHANGES OF TOTAL COLUMN HOLD-UP
Typical semilogarithmic plot of d e p a r t u r e for s t e a d y - s t a t e composition of distillate
the explanation may be, however, the simple semilogarithmic relationship has the practical result t h a t incomplete calculations can be extrapolated and that the time needed for any value of e can be computed quickly from T for any other value. (For example, the time needed for E = 0.01 will be about twice the time for E = 0.1.) Straight lines were obtained when e, was plotted versus T from all the cases calculated. Table I lists values of T I N interpolated (extrapolated, in a few cases) from the computer results, corresponding t o = 0.1. (Copies of the original machine records showing instantaneous compositions on several trays can be obtained from the authors.)
Discussion ob Computer Results The calculated effect of changing the number of trays in the column may be seen by comparing selected cases from Table I. Consider, for example, the following three cases in which the composition in the reboiler, the column fractionation factor, and the tray efficiency are constant and in which only iV varies:
Run No. 4 111 115
a:
1.3593 1.1659 1.0798
Note that 58 = 0.01; (a).v
XD.ss
0.5709 0.5442 0.520 =
T / N 2a t
N 15 30 60
100; k = 0; E
E,
=
= 0.1 4.30 4.74 4.92
1
It is evident from the last column that, a t least under these conditions, the equilibrium time is proportional nearly to the equare of the number of trays. At the same time the value of T / W is about five times as great as expected from t h e oversimplified form of the linear theory. [Cf. Equation 13 using B = (a/2)2.] 9 similar conclusion is reached from runs 3 and 110, which differed from those listed only in that the Murphree tray efficiency mas 50 rather than 100%; t h e value of T / N 2 vas, however, only about one third as great as the values shown. The influence of a change in the relative volatility, all other conditions including the number of trays being held constant, is shown in the following comparison. 1024
loglo(a:).v xs 2 0.01 6 0.01 2 0.001 6 0.001
TIN2 a t zn.da E, = 0.1 0.5709 4.30 0.9999 0.58 0.1200 8.76 0.9996 5.79
T t is evident t h a t the equilibration time is affected by the column separation factor, especially when, as in run 8, the compositions on the upper trays approach unity. Comparison of runs 4 and 5 suggests t h a t T is also reduced by the larger net rate of flow of low-boiler into the column a t the larger reboiler composition. Neither of these influences is expressed by the linearized theory according to Equation 13. The influence of the holdup in the reboiler is indicated by a comparison of machine runs in which all other parameters were the same:
EX
Figure 3.
1.3593 2.5119 1.3593 2.5119
N 15 15 15 15
Note that E = 1, N = 15, k = 0.
0.6
OR
01
Run iVo. 4 2 8 7
xi 0.0135 0.5000 0.0251 0.5000
k 0 2 0 2
a:
1.3593 1.3593 2.5119 2.5119
T / W a t E, = 0.1 4.30 0.16 0.58 0.036
Note that E = 1, N = 15 throughout.
It should be evident from this comparison that the equilibration time decreases by more than a factor of 4 when the column is “cut in half’’ by eliminating the holdup in the reboiler, thus permitting the compositions in the lower half of the column to fall while those above the middle are rising. I n this case, although the steady-state enrichment produced by the column may be constant, there is no pinch in composition limiting the rate of flow of low-boiler from the lower into the upper half of t h e column; such a pinch does exist a t the bottom of the column in runs 4 and 8. Only a single comparison of the machine results with the linearized theory of Cohen ( 4 ) was made. Run 111 was selected for this purpose because in i t the range of liquid compositions a t steady state v a s only from 0.01 in the reboiler t o 0.5442 a t the top of the column, and for this reason the linearizing assumption, although not good, was probably better than for the other cases calculated. Using Cohen’s ( 4 ) tabulated values of B and A , it was estimated that E, = 0.1 should require T / X a = 12.5, compared with 4.73 computed accurately by the machine. Of greater interest is the comparison of the machine results with Berg and James’ ( 2 ) adaptation of Cohen’s linear theory. The following table compares accurate and estimated times for six selected runs in which the reboiler holdup was infinite.
Run h-0. 4 5 8 10 111 115
cy
1.3593 1.3593 2.5119 2.5119 1.1659 1.0798
N
58
0.01 0.001 0.01 0.001 0.01 60 0 . 0 1
1.5 15 15 15 30
TIN2 Corresponding to E , Berg and Huffman and James (a), T e y Eauation Lauation 19 26 3.06 2.35 3.87 3.56 0.46 1.29 4.54 9.90 4.77 2.66 5.67 2.84
-
(r),
A
0.1 From digital computer 4.30 8.77 0.58 5.86 4.73 4.92 =
hTotet h a t in all these runs E = 1 and (a)” = 10*or 108; ez = 0 for Huffman and Urey calculations. Although the approximate results are more nearly of the correct magnitude than was found for run 111 using the linear theory in mole fractions, there are rather large discrepancies,
INDUSTRIAL AND ENGINEERING CHEMISTRY
VoI. 48, No. 6
PROCESS CONTROL as in run 5, where Berg and James’ equation is more than 100% low. I n the values of T/N2 calculated from the material balance approach of Huffman and Urey (7),the T-values again are of the right order of magnitude, but they are generally too low, except for runs 8 and 10. These are the two cases among those selected for which t h e steady-state enrichment was very large [(a)N= 1061 and where, for this reason, many of the upper trays of the column reached into the upper pinch region. I n most cases, therefore, the degree of approach t o the steady-state composition of the distillate will be less than 90% complete when, according t o the Huffman and Urey approximations, it should have been 100% complete. I n view of these as well as more extensive comparisons of the machine results with approximate predictions, i t is clear that neither of the approximate methods can be relied on and t h a t either a n improved nonlinear theory or a n empirical expression of the computer results should be sought. I n view of the difficulty in the approach through formal mathematics, shown by the work of Montroll and Newel1 (9), a graphical representation of the results obtained from the computer was attempted; the hope was held t h a t it would turn out t o be general. Figure 4 is a plot of T/TH, the ratio of the time required for the instantaneous distillate composition t o achieve e, = 0.1 t o t h e time to fill the column liquid holdup with its steady-state amount of low-boiler according to Huffman and Urey’s assumptions us. the common logarithm of the reciprocal of mole fraction of high-boiler in the distillate at steady state. T h e numerical value of the abscissa represents the influence of the pinch in compositions on the top trays of the column; for a constant reboiler composition it depends directly on the column separation factor d . All the computer results from Type I runs, in which the vapor composition at the bottom of the column was fixed, fall on the single line for all values of N , a, XS, and E. Presumably the curve can be regarded as an empirical expression of the complete mathematical solution of the nonlinear problem, whatever form such a solution may have, a t least for the transient changes of the distillate composition. For the sake of completeness it must he noted that in locating points on Figure 4 in cases for which E = 1/52, the mean composition of the column’s liquid holdup at steady state was csmputed using Equation 25 with a number of theoretical trays equal t o EON. Although an exact formula does not exist for relating E and E , when the equilibrium line (Equation 4) is curved, a sufficiently accurate estimate of E, can be obtained from
log [l + E
E, =
(y
- l)]
log ( G V / L )
This is strictly valid only for the linear conditions t h a t lead to Equation 8 as a solution of the steady-state enrichment of composition on the trays. In Equation 27 % may be thought of as the mean slope of the equilibrium line over the range of compositions present in the column at steady state; for t h e purpose of constructing Figure 4 this was taken as equal to a,(01 1)/2, 1, (1 a - * ) / 2 , or CY-1, depending on the range of compositions involved. The figure also contains a few points resulting from machine calculations of Type I1 operation. I n these cases, as indicated on Figure 2, the composition on the center tray of the column remained constant and equal t o the composition of the initial charge. I n computing TH, therefore, t h e number of trays used in Equation 25 for computing 5,, was half the total number in the column, and the equivalent reboiler composition (58) was taken equal to t h a t of a liquid phase a t phase equilibrium with a vapor containing 0.5 mole fraction low-boiler. Although the open points on Figure 4 computed in this way do not fall precisely on
+
June 1956
+
2
T T”
I
0.6
0.6
0.4
1
I\
I
I
I
I
0.1’
0
I
I 2
I
I
4
I
I
6
Figure 4. Generalized plot of time needed to approach steady state--T based on = 0.1
the line drawn through the circular points from Type I operation, they are close enough in view of the approximations just described to consider t h a t they support the general significance of Figure 4. Figure 4 may be thought of as an empirical generalization of the conclusion reached by Davidson (6) for the linear problem, although the effect of N found by Davidson is slightly different. It is interesting to observe t h a t the trend toward larger time constants (BTHIN), found by Davidson for columns having larger separation factors ( a N )is qualitatively in accord with Figure 4, although the trend shown on the figure is greater than expected from Davidson’s work. A general conclusion from the success of the correlation in Figure 4 is t h a t the equilibration time ( T ) depends on the closeness of approach between vapor and liquid compositions both at the bottom of t h e column and at the top. If the two bottom compositions are close together T R is large because of a limitation in the rate a t which low-boiler can enter the column from the reboiler and be transported from tray t o tray by a diffusionlike mechanism; when the top compositions reach into the upper pinch the equilibration time is small because the large differences in vapor and liquid compositions on the lower trays makes the diffusive movement of low-boiler toward the upper trays very easy while the upper bound on holdup composition limits the amount of low-boiler t h a t must be transported upward. Thus, there are competing influences of pinch effects in the upper and lower extremities of the column, as shown in Figure 4.
Example Calculation Compute t h e times required for the distillate composition t o reach 90 and 99% of the way toward its steady-state value during the start-up of an enriching column designed t o fractionate a binary mixture having CY = 1.05 into a distillate containing 0.95
INDUSTRIAL AND ENGINEERING CHEMISTRY
1025
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT mole fraction lowboiler. The reboiler composition will remain constant a t 0.05 mole fraction. The Murphree efficiency of each tray will be 0.70, the vapor velocity will correspond to an F-factor (u 4%) of 0.7 p i t h a vapor density, p v , of 0.0367 lb./cu.ft, and each tray will contain liquid equivalent to a depth of 1 inch and a specific gravity of unity. The trays are filled initially with liquid pumped from the reboiler. The number of theoretical trays required is calculated from the Fenske formula to be [log(O.95 X 0.95/0.05 X O.O5)]/log (1.05) = 121. Using E = 1 because of the symmetry of compositions a t steady state and B = L owing t o total reflux operation, in Equation 27 E, = E = 0.70, so that 121/0.70 or 172 actual trays will be required. Soting t h a t aNf1 = 379, Equation 25 gives the following value for the increase in the mean composition of the liquid holdup at steady state:
x,, - xi
=
log
O”
j [l + (379 - 1)0.051 I1
1 [ l + (1.05
=
- 1)0.05] [l log (1.05)
+ (398 - l)0.05] + (1.0Zz - 1)0.05],t - 0.05
0.460 mole fraction low-boiler
Using Equation 26, the number of tray changes according to the rough material balance method is
T H = 172
(+) 0 00238
=
33,300
Note that in this equation yo - T I is taken as 21; ( a - 1)rs (1
1
+
(cy
-
- 58
=
- 5.9) = 1)ZS
(0.05) (0.05) (0.95)/[1
+ (0.05) (0.0511 = 0.00238
From Figure 4, using an abscissa of -log(l - 0.95) = 1.30, T / T H = 0.62, from which it follows t h a t T = (0.62) (33,300) = 20,650 tray changes required to achieve E, = 0.1. The superficial vapor velocity is 0.7/(0.0367) = 3.66 ft./sec.. which is equivalent t o a mass velocity of (3.66) (0.0367) or 0.134 lb./(sec.) (sq. ft.). The mass of liquid on each tray is ( l / l 2 ) (62.3) = 5.20 lb./sq. ft., so that the time of passage of the liquid across each tray is 5.20/(3600)(0.134) or 0.0107 hour. The operating time needed for e2 = 0.1 is therefore 0.0107 X 20,650 or 221 hours; a t this time the instantaneous X D will be 0.0738 O.Q(O.95 - 0.0738) = 0.8624 mole fraction. For E, = 0.01 the time will be about twice as great and the instantaneous x~ = 0.9413.
+
Acknowledgment The authors wish to express their appreciation to the Engineering Department, E. I. du Pont de h-emours & Co., who provided financial support for one phase of the work. Assistance
1026
in the preliminary programming of the problem for the computer was provided by the staff of the Applied Science Division, International Business Machines Co.
Nomenclature A = dimensionless coefficient (Equation 13), a function of ( a - 1)N B = time constant, dimensionless, a function of (a - 1)iV (Equations 14 and 15) E = Murphree tray efficiency, fractional E , = over-all column effieiency, fractional, defined as theoretical trays divided by actual trays for same steady-state separation F = superficial vapor velocity (ft./sec.) multiplied by squaie root of vapor density (lb./cu. ft.) H = holdup of liquid per tray, lb. moles IC = ratio: moles of liquid on one tray to moles of liquid in reboiler L = molar flow rate of liquid stream, Ib. moIe/hour % = mean slope of equilibrium line over range of compositions present in column a t steady state iV = KO.of top tray in column n = ordinal number of tray T = time, expressed as number of single-tray displacements by liquid stream t = time, hours u = superficial vapor velocity, ft./sec. V = molar flow rate of vapor stream, lb. mole/hour xu = mole fraction low-boiler in instantaneous distillate; subscript ss means steady-state value x 2 = mole fraction lowboiler in initial column contents z , ~ = instantaneous mole fraction low-boiler in liquid leaving n t h tray xs = mole fraction lowboiler in liquid contained in reboiler %s = steady-state value of mean composition of liquid holdup in column, mole fraction low-boiler y. = instantaneous mole fraction low-boiler in vapor leaving n t h tray y: = mole fraction low-boiler in vapor a t phase equilibrium with liquid leaving n t h tray oi = relative volatility of binary mixture E X = fractional departure of distillate composition from steadystate value, expressed in mole ratios (Equation 17) E , = fractional departure of distillate composition from steadystate value, expressed in mole fractions (Equation 13) p u = density of vapor phase, Ib./cu.ft.
Literature Cited Bardeen, J., Phys. Rev. 57,35 (1940). Berg, C., James, I. J., Chem. Eng. Proor. 44, 307:(1948). Cohen, K., J . Chem. Phus. 8,588 (1940). Cohen, K., “Theory of Isotope Separation as Applied t o Large Scale Production of U236,” McGraw-Hill, New York, 1951. Colburn, A. P., private communication, 1943. Davidson, J. F., Inst. Chem. Engrs., London, December 1955. Huffman, J. R., Urey, H. C., IND. EKG.CHEX29, 531 (1937). i’vlarshall, W. R., Pigford, R. L., “Application of Differential Equations t o Chemical Engineering Problems,” University of Delaware, Kewark, Del., 1947. RIontroll, E. W., Nemell, G. F., J. A p p l . Phys. 23, 184 (1952). Pigford, R. L., Tepe, J. E., Garrahan, C. J., IXD.EKG.CHEM. 43,2592 (1961). RECEIVED for review January 1, 1956.
ACCEPTEDApril 12, 1956.
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 48, No. 6
