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Plate by plate calculations of multicomponent distillation columns using difference equations—I
Pergamon Press Ltd., Oxford. Printed in Great Britain.
Chemical Engineering Science, 1963, Vol. 18, pp. 697-709.
Plate by plate calculations of multicomponent distillation columns using difference equations-I Ideal cascades and constant flow columns operating at total reflux Z. FRAENKEL and A. RAVIV Isotope Department, Weizmann Institute of Science, Rehovoth
and w.
KLEIN
Israel Atomic Energy Commission (Received 18June 1962; in revisedform 31 December 1962) Abstract-In this paper we describe plateby-plate calculations of separation columns using finite difference equations. The calculations were performed on the WEIZAC digital computer. The purpose of the calculations was essentially twofold: (a) comparison of various ideal cascades among themselves and with constant flow cascades. (b) Preparing a series of graphs from which the behaviour of a given column with given input parameters can be predicted a priori. In addition, the results of the methods are compared with the results of calculations based on the solution of differential equations for one system for which such a solution is available. The calculations are also compared with experimental results on fractional distillation and thermal diifusion columns.
I. INTRODUCTION CALCULATIONS of separation
columns have been performed in the past using either differential equations or making plate-by-plate calculations with the aid of difference equations. The use of differential equations has the advantage that knowing all variables at any given point of the column, these variables can be computed directly for any other point of the column. This procedure is however restricted to those systems for which the differential equations have an analytic solution. For the majority of actual conditions such solutions are unfortunately not available. The use of finite difference equations is‘ strictly justified only if the column is not of a continuous type but has a finite number of separation plates. However, one may use difference equations as an approximation for continuous columns. A plateby-plate calculation using difference equations has the advantage that it is applicable even to the most general system. In this paper the behaviour of ideal cascades and of constant flow columns operating at total reflux is investigated. The primary motivation for this work was the need for a priori determination of the B
working characteristics of practical distillation columns in the design of a system of columns for the separation of oxygen isotopes l-3. Hence the numerical results which are quoted here refer to this particular problem. The method is however completely general and may be applied to an n-component mixture and to other methods of separation.
II. THE DIPFERENCE EQUATIONS For all the calculations the following working conditions were assumed. 1. Constant relative volatility CIand constant heat of vapourization (the relative volatility and heat of vapourization are assumed to be independent of pressure, temperature and concentration). 2. An adiabatic column (there are assumed to be no heat losses to the surroundings). 3. 100 per cent plate efficiency, (the vapour rising from a plate is assumed to be in equilibrium with the liquid descending from it). Fig. 1 represents a schematic drawing of a distillation column of varying flow. From material balance at the bottom of the
697
Z. FRAENKEL, A. RAVIV and W. KLEIN
the product. The equilibrium condition for each component on every plate is given by 4 aiN; & = -C c %Nh
(3)
i=l
where ai is the relative volatility of component \
I
1’ : t
t
FN:-
.\ \ \ \
Denoting
\ \
l/a, = l$j
I
\
i with
respect to component t (a, = 1) c is the number of components
/
\ \
f==&
\
k
\
one obtains from (l), (2) and (3) N;_, \ \ \ \ \
/ I
Y
PNi
FIG. 1. Schematic drawing of an ideal cascade.
column and any plate K below the feed point (the “enricher” section) we obtain Lx-1 =P+
Ix
= PN; + I&
= C aiN;
‘k 1+6,
Ni
p
(4)
Equation (4) can be used to compute the concentrations of the isotopes in the liquid phase of the (k - 1)-th plate when the concentrations in the liquid phase of the k-th plate and the ratio PII, are known. Hence a plate-by-plate calculation starting with the lower end of the column and up to the feed point can be performed. For the section above the feed point (the “stripper” section) a similar equation can be derived.
(I)
where Lx-,, 1, are the liquid and the vapour flow, respectively, leaving the (k - I)-th and the k-th plate, expressed in l/day of liquid. P is the product withdrawn from the liquid phase of the last plate of the enricher. From isotope balance in the same section one obtains &-,N&
l+&
aiN: i=l
/
T
1
=-.-+-
(2)
where A$_,, rzh are the mole fractions of the i-th component in the liquid stream L.k_, and vapour stream Zx, respectively. Ni is the mole fraction of the i-th component in
Here 8h denotes the ratio of the waste W (withdrawn from the vapour phase of the upper plate) to the liquid stream Lh descending from the h-th plate 61, = W/L,,. n$ is the mole fraction of the i-th component of the waste. If, instead of starting the calculations from either end of the column, it is desired to start the calculations from the feed point, one obtains, by an identical derivation, the following equations
698
Plate by plate calculations of multicomponent distillation columns using difference equations-I
for the section below the feed point (the enricher) and
for the section above the feed point (the stripper). The average concentrations of an isotope in a column of constant flow are computed with help of the equation
(8) where m is the number of plates for which the average concentration is computed.
It is seen from equation (11) that for a column ideal with respect to a component which reaches its highest concentration at some intermediate point along the cascade (such as 0” for a mixture of all three oxygen isotopes Oi6, 01’ and O’a), 6 = 0 at its maximum concentration, i.e. the flow is infinite. This means of course that such a column cannot be constructed in practice up to that point. However, in order to enable the theoretical comparison of such cascades with cascades ideal with respect to other components, 0 l7 ideal cascades were computed to a predetermined minimum value of 6 = 6,i, = (J-%%li”* The average flow in the cascade is proportional to the average value of 6-l k=v
c
K’ (12)
III. CHARACTERISTICS OF IDEAL CASXDES In an ideal cascade no mixing occurs for the component for which the cascade is ideal, i.e. the concentrations of the “ideal” components in the vapour and liquid streams reaching any given plate are equal [5]. Mathematically this condition is expressed by N{_, = ni+1 Ni+l = ni-1
for the enricher section for the stripper section
(9)
where m is the number of plates in the cascade for which the average flow is computed. The average concentrations in the cascade are given by
=
,=;k-+ l(N:lsk) c
(11) For an ideal column, to a first approximation 6v = 6~ = 1 (see Appendix). With given concentrations at the end of the column, one can therefore compute the concentrations at the following stage, N,!-1, by equation (4), and a,,_, by equation (11) (using N:‘_2 = n{). Equations (3), (4), (9) and (11) are then used alternatively for a plate-by-plate computation of the flow and the concentration. 699
tl/dk)
k=v-m+l
(10)
where j denotes the component for which the cascade is ideal. Equation (9) can be satisfied by varying the flow from plate to plate. Combining equations (l), (2) and (9) one obtains equation (11) which shows the desired change in flow along the cascade, from the lower end up to the feed point.
(13)
&xv
where is the average concentration of some component i in the m plates of the column. From equations (3), (5) and (9) an equation similar to (11) can be derived for the section above the feed point. 6, =
n$_, - nh-* nj
w - ni-1
(14)
Comparison of ideal and constant jiow cascades
One of the most important characteristics of a fractional distillation column for the separation of oxygen isotopes is its enrichment factor (the ratio of maximum to average concentration N-/N,,, for the 0” isotope). While a high concentration of the 0”’ isotope can normally be achieved without difhculty, the enrichment of 01’ represents a much more formidable problem for two reasons: (1) Its very low natural abundance (O-037 per cent);
Z.
FRAENKEL,
A. RAVIV and W. KLEIN
(2) Its position between the two more abundant isotopes 016 and 0”. The comparison of ideal cascades with constant flow cascades was hence directed primarily towards determining the column design which would yield the highest enrichment of the 0” fraction. Starting with various NFs a series of computations were made to compare cascades ideal with respect to Or6, 0” and 01* and constant flow cascaded. As already mentioned, for cascades ideal with respect to or7 6 = 0 at the maximum concentration of Or’, sd that the calculations had to be stopped at some predetermined 6,i,. For cascades ideal with respect to 0i6 and 0” and for constant flow cascades the calculations were made up to the point of maximum concentration of O17. The following variables were computed : (a) The 0” enrichment factor. (b) The Or*-concentration at the point of maximum 0” concentration. (c) The number of plates. which is proportional to the (d) GW,:, average cross-section of the ideal column. (e) The average concentration of 0”. (f) The average concentration of O1’. Variables (c)-(f) were computed for the section between the lower end and the point of highest 01’ concentration in the enricher column.
The calculations were made for the following range of concentration : Npo18= 90-98 per cent q”
= O-l-9*9 per cent
Each ideal column was compared with a constant flow column of cross section equal to the average cross-section of the ideal cascade where CF and I pertain to “Constant Flow” and “Ideal”, respectively. Thus the ratio of volumes Y of the two cascades is equal to the ratio of the number of plates v, -=-
vt
VCF
vCF
In Tables 1 and 2 we compare columns ideal with respect to 0” and terminated at 6,i, = 10M3, 10S4 and lo-‘, with constant flow columns. In Tables 3 and 4 we compare columns ideal with respect to 01* and 016 with constant flow columns. Examining Table l-2, we find that the enrichment factor of columns ideal with respect to 01’ decreases with decreasing 6,i,. This of course is a consequence of the rapidly diverging plate size as Bminapproaches zero. As a result the average concentration of all three components tends more and
Table 1 Nmaxol’/Nmeol’ Np=‘17 (“A
smin Ideal
Comt.
NOI3
Ideal
flow 0.1
1
10-s
1.08
10-4
1.00
10-s 10-3 10-4 10-5 10-3
9.9
10-4 10-5
NkW201a
Y
Const. flow
<6i-‘hve -1 Ideal Ideal Const.
Pow
Const.
NS,,@ Ideal
Const. flbw
Pow
1.00
1.08 1.15 1.15
63.41 54.37 53.49
68.77 61.94 59.26
452 523 529
356 353 351
297 601 905
73.87 64.42 60.73
79.46 77.21 76.38
0.13 0.14 0.14
0.13 0.13 0.13
1.08 1.03 1.02
1.10 1.13 1.14
63.26 53.77 52.76
68.57 61.67 58.57
470 546 553
375 369 367
302 605 954
73.92 64.24 60.05
79.39 77.15 76.18
1.35 1.43 1.45
1.28 1.32 1.34
140 1.22 1.23
1.34 1.50 1.59
58.70 33.61 31.25
61.14 45.17 38.72
1198 1593 1621
1267 1194 1169
468 918 1344
72.74 54.31 46.96
74.79 69.07 66.82
25.10 34.51 36.97
23.06 27.67 29.51
The enrichment factor of 017 (Nmsx017/NB~e017) for NP01*= 90 per cent; also shown are No’s, the concentration of O1s at Nmsrol’; the number of plates V; < 81-1>,ye -1; the average concentration of O1s and 017 for cascades ideal with respect to 017 for Li, = 10-3, 10-4, 10-5 and for equivalent cascades of constant flow.
700
Plate by plate calculations of multicomponent distillation columns using difference equations-1
Table 2 NO18
Nmar017/N~ve017
0.1
1
1.9
NWP
”
<61-1>.V,-1 Ideal Const. Ideal Pow
Const. frow
Ideal
Const. Bow
86.31 72.65 66.43
90.19 87.54 86.41
0.19 0.23 0.24
0.17 0.19 0.20
389 712 1044
87.40 73.21 66.47
90-68 88.00 86.80
2.22 2.67 2.82
1.96 2.21 2.33
455 758 1050
89.18
91.03 88.17 86.77
6.43 8.84 9.63
5.56 6.85 7.50
Ideal
Const. Pow
Ideal
Const. Pow
10-s 10-4 10-S
1.26 1.13 1.08
1.30 1.37 140
73.01 55.42 55.53
77.88 67.95 63.56
791 942 955
668 658
367 692 1026
10-s 10-S
1.31 1.17 1.10
i.30 1.42 1.47
73.82 54.16 52.04
78.11 67-66 62.82
915 1093 1108
823 789 775
10-s 10-4 10-5
1.58 I.31 1.20
1.51 1.71 1.80
75.22 50.13 47.56
77.28 65.57 59.80
1353 1621 1641
1312 1239 1208
10-4
692
NW&Q='
7440 66.92
The enrlcbrnent factor of 0’7 (Nmax017/NaVeo17)for Npo ls = 98 per cent; also shown are No’s, the concentration of 01* at Nms,O1’; the number of plates V;.~~-1; the average concentration of O1s and 017 for cascades ideal with respect to Oi7 for &in = 10-3, 10-4, 10-5 and for equivalent cascades of constant flow.
Table 3 Q17/N,,.eQ17 NUWX
NO'8
NWeQ=8
v
Np=a
NpQl’
(%I
(%I
Ideal
Const. flow
Ideal
Cons. Pow
I&al
Co&. Pow
90
0.1 1.0 9.9
1.08 1.06 1.32
1.08 1.08 1.37
65.71 65.34 46.80
73.83 73.65 58.61
520 543 1525
362 379 1254
98
0.1 1.0 1.9
1.17 1.22 1.42
1.19 1.22 140
14.28 74.89 74.77
8470 85.05 83.04
960 1100 1542
713 852 1356
NWP7
m-1 Ideal
Const. flow
Ideal
Const. Pow
195 202 515
75*79 75.69 66.83
81.24 81.18 73.86
0.12 I.31 28.23
0.12 1.24 23.80
252 273 363
87.48 88.42 8964
92.16 92.59 92.48
0.18 2.00 5.83
0.16 1.77 4.93
The enrichment factor of 017 (Nmaxo17/N,Veo’7)for Npol8 = 90 and 98 per cent for cascades ideal with respect to 018 and for cascades with constant flow (i.e. 8C.F = <61-~>~,~-~). Al so shown are the concentration of 01* at Nmaxol’; the number of plates V; <61-1>~~~1; the average concentration of 018 and 017.
more towards their respective concentrations in the last plate. Thus in the limit ~5,~”+ 0, the average concentration of all three components equals the respective concentrations at the top plate and the enrichment factor of 01’ equals unity. Moreover the 0” enrichment factor of the ideal column is generally smaller than the enrichment factor of the equivalent constant flow column whereas the number of plates of the ideal column is larger than that of the equivalent constant flow column. The smaller enrichment factor of 01’ ideal cascades is again the result of rapidly increasing plate size in 01’ ideal cascades as the point of maximum 0” concen-
tration is approached. As mentioned above, this causes the average 01’ concentration to approach the maximum 01’ concentration and results in a low enrichment factor for 0” ideal cascades. Thus, considered from the point of view of the enrichment factor for 01’, the constant flow columns seem to be definitely superior to columns ideal with respect to 0”. In Tables 3 and 4 we find that the enrichment factor of the ideal and equivalent constant flow columns is similar whereas the number of plates of the ideal column is always larger so that constant flow columns seem to be superior to columns ideal
701
Z. FIUENKEL, A. RAIN and W. KLEIN
Table 4 N
Np"la
Np017
(%I
(%I
Ideal
Const. Pow
90
o-1 ;:;
1.08 1.07 l-23
o-1 1-o l-9
1.17 l-21 1.32
98
NO'8
017/Nave017 mar
Ni4V*01a
Y
sve-1 Ideal
Const. flow
Ideal Const. Pow
1.08
65.73
73.85
520
362
1.08 l-21
65.46 64.20
73.76 72.77
544 1692
379 1338
1.19 l-21 l-28
74.41 75.54 79.81
84.77 85-85 88-63
960 1126 1730
713 8.55 1410
N .veol’
Ideal
Const. Pow
Ideal
Const. Pow
195 200 303
76.17 76.11 7546
81% 81.22 79.33
0.12 1.30 21.95
o-12 1.24 19.41
251 262 286
87.70 88.81 91.38
92.18 92.82 93.97
O-18 l-98 4.98
0.16 l-75 4.28
The enrichment factor of 017 (Nmar O”‘lNSBW 01’) for Npola = 90 and 98 per cent for cascades ideal with respect to 018 and for cascades with constant flow (i.e. 8c.p =,, -1). Also shown are the concentration of Ola at Nrn~Pl’; the 1; the average concentration of 01s and 01’. number of plates v; <8+>,,-
with respect to 016 and 01’ as well as to columns ideal with respect to 0 I’. The fact that the enrichment factor of the 016 and 018 ideal cascades is similar to that of the constant flow cascade, while the number of plates is larger, is again a result of the increasing plate size of the ideal cascade as the maximum concentration of 0” is approached. While in O“j and 01* ideal cascades the plate size does not diverge as the maximum concentration is approached (as in the case of 0” ideal cascades), this
particular shape of the ideal cascades, in which most of the content of the cascade is close to the point of maximum concentration, results in relatively low enrichment factors. It should however be remembered that these conclusions apply to the rather artifkial case of no product extraction at the point of highest concentration of 0”. For a non-zero product flow at the point of highest 01’ concentration, ideal columns may well be superior to constant flow cascades.
6
I
I
I
I
I
IO
20
30
I
I
I
I
I
50
60
70
80
90
I
40
IO ,FI~. 2.
I
100
(% 1
The mean concentration of 01’ as a function of the mean concentration of 01s for P = 0. Parameter: Nmsxol’ = maximum concentration of 017 in the column. Given
Cr~aol~&l~ :
~~HaOle/He0”
l%iO
= 1.0032
702
Plate by plate calculations of multicomponent distillation columns using differenceequations-I
IV. CHARACT~US~CSOF CONSTANTFLOW GASCADES As stated in Section III, in an ideal cascade the flow varies according to equation (11). In the constant flow cascade the flow remains constant from stage to stage, but the vapour and liquid streams entering a plate are not of equal concentration. A given enricher column of constant flow can be calculated by starting either from the lower end (bottom to top calculation) using equation (4) or by starting from the feed-point (top to bottom calculation) using equation (6). In both cases the boiler of the column is assumed to have zero holdup of liquid. 1. Bottom-to-top calculations The following problem was treated: Given a column with a fixed number of plates and I
)O-
I
I
I
I
I
I
,
I
I
I
2
3
4
5
6
7
6
9
N;”
I
(%)
FIG. 4. The concentration of 01* (NDo’“) at bottom of column as a function of concentration of CV7 (Nr,017) at the bottom of column for P = 0. Parameter: Nmsxol’ = maximum concentration of 017 Given : EH~O~~/HZO~~ = 1*0060 ccHa01~/Hz017 = 1.0032
IO-
IO -
ro -
io -
IO !
I
,
I
I
I
,
I
I
IO
20
30
40
50
60
70
60
90
0”
z- w.)
FIG. 3. The concentration of 01s (Npor8) at the bottom
of the column as a function of the mean concentration of Ols (No’s) in the hold-up of the column for P = 0. Parameter: NmrXol’ = maximum concentration of 017 Given Y =looo aHsOl6/HsO’* = 1~0060 orHsOrs/HsOrr = 1.0032
water of given average concentration in Or* and 017, what will be the final isotope concentrations at the top and bottom of the column, and what will be the maximum concentration of 017, if the column is operated at total reflux and with zero feed and zero production? It is seen from equations (1) and (2) that for a column operated at zero product the concentration of the liquid entering the column is equal to that of the vapour leaving it, so that the computations made are correct also for total reflux operation. In order to solve this problem a series of computations were made [using equations (4) and (811 for a column with 1000 plates in the range of N;” = 50-99 per cent and Nz” = 05-10 per cent. The results are shown in Figs. 2-5. Using these four figures in succession one determines first the maximum concentration of 0” (Fig. 2), the bottom concentration of 0” (Fig. 3), 703
Z. FRAENKEL, A. RAVIVand W. KLEIN
the bottom concentration of 0” (Fig. 4) and the top concentration of Or’ and 0” (Fig. 5). In this way the characteristics of a separation column can be predicted if the enriched product is only taken from the column after it reaches steady state and not as a continuous product flow. 2. Top-to-bottom
calculations
In the top-to-bottom calculations for a column of given flow and product and a given number of plates, the product concentrations have to be initially assumed and successive approximations are made to find the correct values. The input parameters are the top concentrations, the assumed bottom concentrations, the number of plates of the column, the flow and the product. With these parameters one computation is performed after which the assumed
I’
’
’
’
’
’
’
’
Ni’s are replaced by the arithmetic mean between the Nk’s assumed and obtained, and the calculation is repeated. One can make the first computation with P, = 0 and increase the production step by step until the desired P is reached (Option I). The arithmetic mean of the assumed (input) value and the calculated (output) value of NL is taken as the input for the next iteration. However, the convergence is rather slow. Another way to approach the correct A$,% is to start calculations with the desired P and to assume that Ngs are zero (Option II). In this case after ten iterations an accuracy of 0.1 per cent is obtained (see Table 5). Table 5 L = 10 I/day
Iteration no. 10
Npi’sassumed
N#s obtained
NPois
0.9663626345
0.9663212538
NPol’
0.0164789602
0*0164940367
Npol’
0.0171584053
0.0171847095
l*oooooooooO
l*oooooooooO
P = 3.2 x 10-a l/day
’ 1’
Comparison of assumed (input) values and calculated (output) values for NPs in the last (10th) iteration of a top-tobottom calculation in which NpI = 0 was assumed for the hrst iteration (Option II). V.
COMPARISON OF THE CALCULATIONS WITH THE
RESULTS OBTAINED USING DIFFERENTIAL EQUATIONS
Ideal cascades
- -1 P%_
01
N_ J
I2
3
4
5 N;”
6
7
6
9
(%)
FIG. 5. The concentration of O’s and 017 at the top of the column as a function of the concentration of 0’7 at the bottom of the column for P = 0. =lOOO Given Y OL HaO“J/HzO1* = 1.0060 OT Hs01’3/HaOi’ = 1.0032
The results obtained by the method of difference equations were compared with those obtained by VON HALLE [6]* using differential equations for a cascade ideal with respect to 0” which consisted of both enricher and stripper columns with the feedpoint between these sections. The computation was made using equations (3), (4), (9) and (11) for the enricher section, the inverse of equation (3) and equations (5), (10) and (14) for the stripper
* This report has been made available to us through the courtesy of Dr. G. A. GAMETT.
704
Plate by plate calculations
Table 6.
of multicomponent
distillation columns using difference equations-I
Comparison of results obtained using difference and dl@erential equations
Results obtained by deference equations
Results obtained by differential equations 0’8 mole fraction
4oi
800 1200 1600 2ooo 2400 2800 3200 3600 4000
0.980000 0.964815 0.935754 0.877086 0.760483 0.565081 0.334874 O-159358 O-065795 0.025355 0.009495 OGO3516 0.001295
017
Flow
018
017
mole fraction
units/unit time
mole fraction
mole fraction
units/unit time
0.050 41.915 22.065 25.128 28.211 36.520 60.027 124402 299.358 774602 2065.962 5575.691 4007466
0.980000 0.964883 0.936006 0.877808 0.762212 0.568114 0.338288 0.161796 0.067081 0.025938 oM9741 0.003617 OGO1262
0.017478 0.027970 O-042438 0.058429 0068754 0.064393 0.045561 0.024977 0.011606 0.004956 0.002034 OGOO819 owo399
0.050 14.904 22.062 25.129 28.187 36.439 61.487 128.059 308.528 797.535 2123.107 5716.510 4041.984
0.017478 0.028018 0.042579 0.058676 0.069021 0.064500 0.04545 1 0.024809 0.011487 0034892 0.002004 OGOO805 OGOO396
VI. COMPARISON WITH EXPERIMENT
section. The concentrations at the enricher end of the column were: q’s = 0*98oooo $I’
= 0.017473
ql’
= OGO2527
and the concentrations column were :
at the stripper end of the
nz” = 0*001021 now”= ONK)352 n216= O-998627 6 at the top and bottom of the column were taken as 1.0. The flow L was calculated taking the product P = O-05 units/unit time, and the waste W = 47.986 units/unit time. The concentrations of 01’ and 0” were printed out once every twenty plates. Detailed plate-to-plate print-outs were obtained at the feed point and at the maximum concentration of O1’. As seen from Table 6 the results obtained by the two methods agree quite well. The differences in the concentration for 01’ are of the order of maximum 2 per cent, for 01* of the order of maximum 3 per cent and for the flow, of the order of maximum 3 per cent.
Flow
1. Thermal diffusion Column
The calculations were compared with experimental results for a thermal diffusion column filled with oxygen gas. Though the calculations were intended for fractional distillation columns, they can also be used for thermal diffusion, if conditions equivalent to those stated in II but pertaining to thermal diffusion columns are assumed to hold. In such a column both (a - 1) and the number of equivalent ideal plates will be dependent on the operating characteristics such as pressure, temperature, etc. These two variables were determined in the following manner: the cascade was tilled with oxygen and operated at zero production until a steady state was reached. Assuming a certain value for CIand starting with the experimental concentrations at the bottom of the column, equation (4) was used to compute the theoretical gradient of concentrations up to the experimental top concentrations of O1*. By this computation one obtained the number of plates of the cascade v corresponding to the assumed CL It should be noted that in a column operating at total reflux and zero production it is the product (CZ- 1). v which appears in the calculations rather than cxand
705
Z.FRAENKEL,
A.R~vrvandW.
v independently. For the present calculations the following assumptions were made: (i) The separation factor of Ols and 016 as calculated from the thermal diffusion factor [7] was taken as ,(Oi6/01*) = 1.016
No. 1 the number of plates obtained was v = 393 whereas experiment No. 2 gave v = 374. 2. In another experiment a second cascade similar to the fist one was filled with oxygen of given mean concentrations and operated at total reflux and zero production until steady state was reached. The theoretical concentrations predicted by Figs. 2-5 were compared with the experimental ones as shown in Table 7.
(ii) ~$O~~/oi*) - 1 = ~.Lx(O’~/O”) - 2 The theoretical and experimental concentrations were compared in Figs. 6 and 7. For experiment O’l”/ I
450
I
2 I
3 I
4 I
5 I
0
6 I
7 I
6 I
9 I
KLEIN
Table 7. Comparison of theoretically predicted and experimentally obtained results in a d@ksion cascade filled with oxygen of mean concentrations Nz’:” = 45.10 per cent, Nzz = 3.09per cent
Experimental concentration
Theoretical concentration
92.30 1.86 5.50 1.86 3.91
92.10 2.16 4.40 1.65 4.10
NO”,concentrations at top of column; NPS, concentrations at bottom of column.
Distillation (1) A column was operated with a continuous feed of 172,000 cm3/day of natural water which was fed at the top and a product of 100 cm3/day which was withdrawn at the bottom of the column. In this case the column had no stripper section and the descending liquid stream was of the same amount as the feed stream. The calculated bottom concentrations as obtained by equation (6) are NF” = 1.31 per cent, Nz” = O-11 per cent while the experimental concentrations measured after a steady state had been obtained were Npl’ = 1.73 per cent, NF” = 0.13 per cent. (2) A distillation column, which was operated at total reflux and zero production was estimated to have the following characteristics : (a) 1200 theoretical plates. (b) 2800 cm3 hold-up and 135 cm3 boiler content. The mean concentrations at the start were
100I-
5c,-
FIG. 6. Comparison of the theoretical and experimental concentration of Or* and 017 in the diffusion cascade. OT Or“/O’* = 1,016 ti 01’3/017 = lGO8 The curves show the calculated distributions and the dots and triangles show the experimental results for 01* and 0’7 respectively. The calculated curves are normalized to the experimental values at the bottom of the cascade (plate 0). Experiment No. 1.
O'S N BYC = 35.50per cent
706
Plate by plate calculations of multicomponent distillation columns using difference equations-I
such manner the concentrations in the above mentioned columns were increased to eVy = 42.82 per cent and NE: = 3.12 per cent. The expected peak of 0” and the expected concentrations of 01* and 01’ at the ends of the column were computed from Figs. 2-5. Comparison with the experimental results for both the initial and the tial average concentrations is given in Table 8.
0'77 0
Table 8. Comparison of the theoretically predicted and experimentally obtained results in a distillation column operated at zero reJEux
iTheoretical result
Ekperimental result
Theoretical result
Experimental result
-N,v,@* Nave@+
Nmsxo17 N,oi’ Npor7 N,o=a Noor’
42.82
35.5
2.90 91.3 164 1.05 056
1.89 2.92 90.7 1.68 1.10 0.54
4.63 94.1 2.05 190 1.15
3.12 4.70 94.30 1.66 1.93 128
* The same values were used in the calculation as input parameters.
--_I/ IO
20
30
I
I
I
40
50
60
I 70
I 80
Figs. 6 and 7 and Table 8 show that the calculations give a reasonably accurate description of the experimentally observed behaviour of both thermal diffusion and fractional distillation columns. The differences between the theoretically predicted and experimentally obtained results are probably due to: 1. Inaccuracy in the estimation of the number of plates. 2. Inaccuracy in the estimation of the hold-up and boiler content.
I 9~
Ole% FIG. 7. Comparison of the theoretical and experimental concentrations of Ols and 017 in the diffusion cascade. OT O’6/O’8 = 1.016 a o’s/017 = 1408 The curvea show the calculated distributions and the dots and triangles show the experimental results for Ole and 017 respectively. The calculated curves are
normalized to the experimental values at the bottom of the cascade (plate 0). Experiment No. 2.
In order to increase the peak of 01’ concentrations one has to increase the mean concentration of 01’ and to change the mean concentration of 018, according to Fig. 2. These operations are performed by changing a fraction of the column holdup, i.e. adding to the column water of certain concentration of 0” and 0” and withdrawing the same amount of liquid but with different isotopic concentration. While planning such an experiment the boiler content has to be taken into account. In
SUMMARY AND CONCLUSION From the above comparison of calculated values with experimental results we may draw the conclusion that the theory of multicomponent separation colnmns is in good agreement with experimental results. With the aid of plate-by-plate calcula-
tions based on difference equations it is possible to calculate a priori the important characteristics of a separation column. One possible technique is the
707
R. FRAENKEL,A. RAXJIVand W. KLEIN Superscripts i Referring 01s Referring 017 Referring 016 Referring I Referring
construction of a series of graphs like those shown in Figs. 2-5 from which the pertinent parameters may be obtained. Though more lengthy than the solution of the differential equation when the concentrations at only a few points along the column are required, this method may well be faster when the concentration at many points along the column is of interest. However the main advantage of this method as compared to the use of differential equations, is the fact that it can be applied to the most general system. The calculations which have been described in this paper were mostly restricted to columns operating at total reflux. Yet the procedure is not restricted to these columns and its application to other systems will be discussed in a future publication. While no experimental results are available for comparison with calculations on “ideal cascades”, the calculated results seem to indicate that these cascades do not have substantial advantages over constant flow columns if a high enrichment factor for 01’ is the prime objective.
APPENDIX:
to to to to to
the component i the component 0’s the component 0i7 the component 016 an ideal cascade
6, FOR AN IDEAL COLUMN
Assume that the concentration gradient per plate over the first three plates AN’JAK for the ideal component is constant, an&denote it by AN’ E=s
(A.11
The mass balance yields (Fig. 8) L,_,N;(i
- s) = PN:, + I,N;(l - 2.9 (A.2) L,_,=P+l,
(A.3)
From equations (A.2) and (A.3) one obtains E(L,-1 - 22”) = 0
Acknowledgements-The authors wish to thank Professor I. D~~TROVSKYfor helpful discussions and guidance in this work, and Professor C. L. PEKERISfor his cooperation in providing time on the WEIZAC. The assistance of Mrs. Mm BERKOin tabulating data is also gratefully acknowledged.
NOTATION
I Vapour flow leaving a plate, l/day L Liauid flow leaving a plate, I/day nr Mole fraction of ‘Ehei-th &mponent in the vapour stream I Ni Mole fraction of the i-th component in the liquid stream L
(A.4)
6,+1=2
(A.5)
6“=l
(A@
It is seen that 6, = 1 is not strictly’correct but is true only to a first approximation. Whether 6, is smaller or bigger than 1 depends on the sign of the quadratic term of the expansion of Nl_2 as a function of NL. In view of the small value of E the first approximation may be used in all practical applications.
708
FIG. 8. The last two plates of an ideal cascade with the corresponding liquid and vapour streams of the ideal component.
Plate by plate calculations
of multicomponent
distillation
columns using difference equations-I
REFERENCES [l]
DOSTROVSKY, I., LLEWELLYN, D. R, and VROMEN,B. H.,J. Chem. Sot. 1952 3509,3517. [2] DOSTROVSKY, I., GILLIS,J. and LLEWELLYN, D. R., L. Farkas Memorial Volume, Res. Council Israel, Spec. Publ. No. 1, Jerusalem 1952. [3] DOSTROVSKY, I. and RAMV, A., Proc. Znt. Symp. Isotope Separation, Amsterdam 19% [4] TREYBALR. E., Mass Transfer Operations p. 271. McGraw-Hill, New York, 1955. [5] COHEN, K., The Theory of Isotope Separation. McGraw-Hill, New York, 1951. [6] VON HALLE, E. and KNIGHT, G. B., Internal Report, Oak Ridge Gaseous Diffusion Plant, Union Carbide Nuclear Company, Oak Ridge, Term., 1 August 1956. (Unpublished.) [7] GREW, K. E., and IBBS, T. L., Thermal Diffusion in Gases. University Press, Cambridge 1952.
R&urn&-Les auteurs d&crivent une methode de calcul plateau par plateau des colormes de separations au moyen d’equations aux differences tin&. Les calculs ont ettt effectuees sur calculatrice numerique WEIZAC. Les deux objets des calculs sont les suivants: a) Comparaison de cascades id&des entre elles et avec des cascades a flux constant b) Preparation de graphiques permettant la prediction a priori des performances dune colonne donnQ a partir des partir des parametres dent&. De plus, les r&hats sont compares avec ceux obtenus par la resolution d’equations differentielles dam le cas ou cette methode est possible. Enfin, les calculs sont confront&s aux resultats experimentaux de colonnes de distillation et de diffusion thermique.
709
Chemical Engineering Science, 1963, Vol. 18, pp. 697-709.
Plate by plate calculations of multicomponent distillation columns using difference equations-I Ideal cascades and constant flow columns operating at total reflux Z. FRAENKEL and A. RAVIV Isotope Department, Weizmann Institute of Science, Rehovoth
and w.
KLEIN
Israel Atomic Energy Commission (Received 18June 1962; in revisedform 31 December 1962) Abstract-In this paper we describe plateby-plate calculations of separation columns using finite difference equations. The calculations were performed on the WEIZAC digital computer. The purpose of the calculations was essentially twofold: (a) comparison of various ideal cascades among themselves and with constant flow cascades. (b) Preparing a series of graphs from which the behaviour of a given column with given input parameters can be predicted a priori. In addition, the results of the methods are compared with the results of calculations based on the solution of differential equations for one system for which such a solution is available. The calculations are also compared with experimental results on fractional distillation and thermal diifusion columns.
I. INTRODUCTION CALCULATIONS of separation
columns have been performed in the past using either differential equations or making plate-by-plate calculations with the aid of difference equations. The use of differential equations has the advantage that knowing all variables at any given point of the column, these variables can be computed directly for any other point of the column. This procedure is however restricted to those systems for which the differential equations have an analytic solution. For the majority of actual conditions such solutions are unfortunately not available. The use of finite difference equations is‘ strictly justified only if the column is not of a continuous type but has a finite number of separation plates. However, one may use difference equations as an approximation for continuous columns. A plateby-plate calculation using difference equations has the advantage that it is applicable even to the most general system. In this paper the behaviour of ideal cascades and of constant flow columns operating at total reflux is investigated. The primary motivation for this work was the need for a priori determination of the B
working characteristics of practical distillation columns in the design of a system of columns for the separation of oxygen isotopes l-3. Hence the numerical results which are quoted here refer to this particular problem. The method is however completely general and may be applied to an n-component mixture and to other methods of separation.
II. THE DIPFERENCE EQUATIONS For all the calculations the following working conditions were assumed. 1. Constant relative volatility CIand constant heat of vapourization (the relative volatility and heat of vapourization are assumed to be independent of pressure, temperature and concentration). 2. An adiabatic column (there are assumed to be no heat losses to the surroundings). 3. 100 per cent plate efficiency, (the vapour rising from a plate is assumed to be in equilibrium with the liquid descending from it). Fig. 1 represents a schematic drawing of a distillation column of varying flow. From material balance at the bottom of the
697
Z. FRAENKEL, A. RAVIV and W. KLEIN
the product. The equilibrium condition for each component on every plate is given by 4 aiN; & = -C c %Nh
(3)
i=l
where ai is the relative volatility of component \
I
1’ : t
t
FN:-
.\ \ \ \
Denoting
\ \
l/a, = l$j
I
\
i with
respect to component t (a, = 1) c is the number of components
/
\ \
f==&
\
k
\
one obtains from (l), (2) and (3) N;_, \ \ \ \ \
/ I
Y
PNi
FIG. 1. Schematic drawing of an ideal cascade.
column and any plate K below the feed point (the “enricher” section) we obtain Lx-1 =P+
Ix
= PN; + I&
= C aiN;
‘k 1+6,
Ni
p
(4)
Equation (4) can be used to compute the concentrations of the isotopes in the liquid phase of the (k - 1)-th plate when the concentrations in the liquid phase of the k-th plate and the ratio PII, are known. Hence a plate-by-plate calculation starting with the lower end of the column and up to the feed point can be performed. For the section above the feed point (the “stripper” section) a similar equation can be derived.
(I)
where Lx-,, 1, are the liquid and the vapour flow, respectively, leaving the (k - I)-th and the k-th plate, expressed in l/day of liquid. P is the product withdrawn from the liquid phase of the last plate of the enricher. From isotope balance in the same section one obtains &-,N&
l+&
aiN: i=l
/
T
1
=-.-+-
(2)
where A$_,, rzh are the mole fractions of the i-th component in the liquid stream L.k_, and vapour stream Zx, respectively. Ni is the mole fraction of the i-th component in
Here 8h denotes the ratio of the waste W (withdrawn from the vapour phase of the upper plate) to the liquid stream Lh descending from the h-th plate 61, = W/L,,. n$ is the mole fraction of the i-th component of the waste. If, instead of starting the calculations from either end of the column, it is desired to start the calculations from the feed point, one obtains, by an identical derivation, the following equations
698
Plate by plate calculations of multicomponent distillation columns using difference equations-I
for the section below the feed point (the enricher) and
for the section above the feed point (the stripper). The average concentrations of an isotope in a column of constant flow are computed with help of the equation
(8) where m is the number of plates for which the average concentration is computed.
It is seen from equation (11) that for a column ideal with respect to a component which reaches its highest concentration at some intermediate point along the cascade (such as 0” for a mixture of all three oxygen isotopes Oi6, 01’ and O’a), 6 = 0 at its maximum concentration, i.e. the flow is infinite. This means of course that such a column cannot be constructed in practice up to that point. However, in order to enable the theoretical comparison of such cascades with cascades ideal with respect to other components, 0 l7 ideal cascades were computed to a predetermined minimum value of 6 = 6,i, = (J-%%li”* The average flow in the cascade is proportional to the average value of 6-l k=v
c
K’ (12)
III. CHARACTERISTICS OF IDEAL CASXDES In an ideal cascade no mixing occurs for the component for which the cascade is ideal, i.e. the concentrations of the “ideal” components in the vapour and liquid streams reaching any given plate are equal [5]. Mathematically this condition is expressed by N{_, = ni+1 Ni+l = ni-1
for the enricher section for the stripper section
(9)
where m is the number of plates in the cascade for which the average flow is computed. The average concentrations in the cascade are given by
,=;k-+ l(N:lsk) c
(11) For an ideal column, to a first approximation 6v = 6~ = 1 (see Appendix). With given concentrations at the end of the column, one can therefore compute the concentrations at the following stage, N,!-1, by equation (4), and a,,_, by equation (11) (using N:‘_2 = n{). Equations (3), (4), (9) and (11) are then used alternatively for a plate-by-plate computation of the flow and the concentration. 699
tl/dk)
k=v-m+l
(10)
where j denotes the component for which the cascade is ideal. Equation (9) can be satisfied by varying the flow from plate to plate. Combining equations (l), (2) and (9) one obtains equation (11) which shows the desired change in flow along the cascade, from the lower end up to the feed point.
(13)
&xv
where
n$_, - nh-* nj
w - ni-1
(14)
Comparison of ideal and constant jiow cascades
One of the most important characteristics of a fractional distillation column for the separation of oxygen isotopes is its enrichment factor (the ratio of maximum to average concentration N-/N,,, for the 0” isotope). While a high concentration of the 0”’ isotope can normally be achieved without difhculty, the enrichment of 01’ represents a much more formidable problem for two reasons: (1) Its very low natural abundance (O-037 per cent);
Z.
FRAENKEL,
A. RAVIV and W. KLEIN
(2) Its position between the two more abundant isotopes 016 and 0”. The comparison of ideal cascades with constant flow cascades was hence directed primarily towards determining the column design which would yield the highest enrichment of the 0” fraction. Starting with various NFs a series of computations were made to compare cascades ideal with respect to Or6, 0” and 01* and constant flow cascaded. As already mentioned, for cascades ideal with respect to or7 6 = 0 at the maximum concentration of Or’, sd that the calculations had to be stopped at some predetermined 6,i,. For cascades ideal with respect to 0i6 and 0” and for constant flow cascades the calculations were made up to the point of maximum concentration of O17. The following variables were computed : (a) The 0” enrichment factor. (b) The Or*-concentration at the point of maximum 0” concentration. (c) The number of plates. which is proportional to the (d) GW,:, average cross-section of the ideal column. (e) The average concentration of 0”. (f) The average concentration of O1’. Variables (c)-(f) were computed for the section between the lower end and the point of highest 01’ concentration in the enricher column.
The calculations were made for the following range of concentration : Npo18= 90-98 per cent q”
= O-l-9*9 per cent
Each ideal column was compared with a constant flow column of cross section equal to the average cross-section of the ideal cascade where CF and I pertain to “Constant Flow” and “Ideal”, respectively. Thus the ratio of volumes Y of the two cascades is equal to the ratio of the number of plates v, -=-
vt
VCF
vCF
In Tables 1 and 2 we compare columns ideal with respect to 0” and terminated at 6,i, = 10M3, 10S4 and lo-‘, with constant flow columns. In Tables 3 and 4 we compare columns ideal with respect to 01* and 016 with constant flow columns. Examining Table l-2, we find that the enrichment factor of columns ideal with respect to 01’ decreases with decreasing 6,i,. This of course is a consequence of the rapidly diverging plate size as Bminapproaches zero. As a result the average concentration of all three components tends more and
Table 1 Nmaxol’/Nmeol’ Np=‘17 (“A
smin Ideal
Comt.
NOI3
Ideal
flow 0.1
1
10-s
1.08
10-4
1.00
10-s 10-3 10-4 10-5 10-3
9.9
10-4 10-5
NkW201a
Y
Const. flow
<6i-‘hve -1 Ideal Ideal Const.
Pow
Const.
NS,,@ Ideal
Const. flbw
Pow
1.00
1.08 1.15 1.15
63.41 54.37 53.49
68.77 61.94 59.26
452 523 529
356 353 351
297 601 905
73.87 64.42 60.73
79.46 77.21 76.38
0.13 0.14 0.14
0.13 0.13 0.13
1.08 1.03 1.02
1.10 1.13 1.14
63.26 53.77 52.76
68.57 61.67 58.57
470 546 553
375 369 367
302 605 954
73.92 64.24 60.05
79.39 77.15 76.18
1.35 1.43 1.45
1.28 1.32 1.34
140 1.22 1.23
1.34 1.50 1.59
58.70 33.61 31.25
61.14 45.17 38.72
1198 1593 1621
1267 1194 1169
468 918 1344
72.74 54.31 46.96
74.79 69.07 66.82
25.10 34.51 36.97
23.06 27.67 29.51
The enrichment factor of 017 (Nmsx017/NB~e017) for NP01*= 90 per cent; also shown are No’s, the concentration of O1s at Nmsrol’; the number of plates V; < 81-1>,ye -1; the average concentration of O1s and 017 for cascades ideal with respect to 017 for Li, = 10-3, 10-4, 10-5 and for equivalent cascades of constant flow.
700
Plate by plate calculations of multicomponent distillation columns using difference equations-1
Table 2 NO18
Nmar017/N~ve017
0.1
1
1.9
NWP
”
<61-1>.V,-1 Ideal Const. Ideal Pow
Const. frow
Ideal
Const. Bow
86.31 72.65 66.43
90.19 87.54 86.41
0.19 0.23 0.24
0.17 0.19 0.20
389 712 1044
87.40 73.21 66.47
90-68 88.00 86.80
2.22 2.67 2.82
1.96 2.21 2.33
455 758 1050
89.18
91.03 88.17 86.77
6.43 8.84 9.63
5.56 6.85 7.50
Ideal
Const. Pow
Ideal
Const. Pow
10-s 10-4 10-S
1.26 1.13 1.08
1.30 1.37 140
73.01 55.42 55.53
77.88 67.95 63.56
791 942 955
668 658
367 692 1026
10-s 10-S
1.31 1.17 1.10
i.30 1.42 1.47
73.82 54.16 52.04
78.11 67-66 62.82
915 1093 1108
823 789 775
10-s 10-4 10-5
1.58 I.31 1.20
1.51 1.71 1.80
75.22 50.13 47.56
77.28 65.57 59.80
1353 1621 1641
1312 1239 1208
10-4
692
NW&Q='
7440 66.92
The enrlcbrnent factor of 0’7 (Nmax017/NaVeo17)for Npo ls = 98 per cent; also shown are No’s, the concentration of 01* at Nms,O1’; the number of plates V;
Table 3 Q17/N,,.eQ17 NUWX
NO'8
NWeQ=8
v
Np=a
NpQl’
(%I
(%I
Ideal
Const. flow
Ideal
Cons. Pow
I&al
Co&. Pow
90
0.1 1.0 9.9
1.08 1.06 1.32
1.08 1.08 1.37
65.71 65.34 46.80
73.83 73.65 58.61
520 543 1525
362 379 1254
98
0.1 1.0 1.9
1.17 1.22 1.42
1.19 1.22 140
14.28 74.89 74.77
8470 85.05 83.04
960 1100 1542
713 852 1356
NWP7
Const. flow
Ideal
Const. Pow
195 202 515
75*79 75.69 66.83
81.24 81.18 73.86
0.12 I.31 28.23
0.12 1.24 23.80
252 273 363
87.48 88.42 8964
92.16 92.59 92.48
0.18 2.00 5.83
0.16 1.77 4.93
The enrichment factor of 017 (Nmaxo17/N,Veo’7)for Npol8 = 90 and 98 per cent for cascades ideal with respect to 018 and for cascades with constant flow (i.e. 8C.F = <61-~>~,~-~). Al so shown are the concentration of 01* at Nmaxol’; the number of plates V; <61-1>~~~1; the average concentration of 018 and 017.
more towards their respective concentrations in the last plate. Thus in the limit ~5,~”+ 0, the average concentration of all three components equals the respective concentrations at the top plate and the enrichment factor of 01’ equals unity. Moreover the 0” enrichment factor of the ideal column is generally smaller than the enrichment factor of the equivalent constant flow column whereas the number of plates of the ideal column is larger than that of the equivalent constant flow column. The smaller enrichment factor of 01’ ideal cascades is again the result of rapidly increasing plate size in 01’ ideal cascades as the point of maximum 0” concen-
tration is approached. As mentioned above, this causes the average 01’ concentration to approach the maximum 01’ concentration and results in a low enrichment factor for 0” ideal cascades. Thus, considered from the point of view of the enrichment factor for 01’, the constant flow columns seem to be definitely superior to columns ideal with respect to 0”. In Tables 3 and 4 we find that the enrichment factor of the ideal and equivalent constant flow columns is similar whereas the number of plates of the ideal column is always larger so that constant flow columns seem to be superior to columns ideal
701
Z. FIUENKEL, A. RAIN and W. KLEIN
Table 4 N
Np"la
Np017
(%I
(%I
Ideal
Const. Pow
90
o-1 ;:;
1.08 1.07 l-23
o-1 1-o l-9
1.17 l-21 1.32
98
NO'8
017/Nave017 mar
Ni4V*01a
Y
Const. flow
Ideal Const. Pow
1.08
65.73
73.85
520
362
1.08 l-21
65.46 64.20
73.76 72.77
544 1692
379 1338
1.19 l-21 l-28
74.41 75.54 79.81
84.77 85-85 88-63
960 1126 1730
713 8.55 1410
N .veol’
Ideal
Const. Pow
Ideal
Const. Pow
195 200 303
76.17 76.11 7546
81% 81.22 79.33
0.12 1.30 21.95
o-12 1.24 19.41
251 262 286
87.70 88.81 91.38
92.18 92.82 93.97
O-18 l-98 4.98
0.16 l-75 4.28
The enrichment factor of 017 (Nmar O”‘lNSBW 01’) for Npola = 90 and 98 per cent for cascades ideal with respect to 018 and for cascades with constant flow (i.e. 8c.p =
with respect to 016 and 01’ as well as to columns ideal with respect to 0 I’. The fact that the enrichment factor of the 016 and 018 ideal cascades is similar to that of the constant flow cascade, while the number of plates is larger, is again a result of the increasing plate size of the ideal cascade as the maximum concentration of 0” is approached. While in O“j and 01* ideal cascades the plate size does not diverge as the maximum concentration is approached (as in the case of 0” ideal cascades), this
particular shape of the ideal cascades, in which most of the content of the cascade is close to the point of maximum concentration, results in relatively low enrichment factors. It should however be remembered that these conclusions apply to the rather artifkial case of no product extraction at the point of highest concentration of 0”. For a non-zero product flow at the point of highest 01’ concentration, ideal columns may well be superior to constant flow cascades.
6
I
I
I
I
I
IO
20
30
I
I
I
I
I
50
60
70
80
90
I
40
IO
I
100
(% 1
The mean concentration of 01’
Cr~aol~&l~ :
~~HaOle/He0”
l%iO
= 1.0032
702
Plate by plate calculations of multicomponent distillation columns using differenceequations-I
IV. CHARACT~US~CSOF CONSTANTFLOW GASCADES As stated in Section III, in an ideal cascade the flow varies according to equation (11). In the constant flow cascade the flow remains constant from stage to stage, but the vapour and liquid streams entering a plate are not of equal concentration. A given enricher column of constant flow can be calculated by starting either from the lower end (bottom to top calculation) using equation (4) or by starting from the feed-point (top to bottom calculation) using equation (6). In both cases the boiler of the column is assumed to have zero holdup of liquid. 1. Bottom-to-top calculations The following problem was treated: Given a column with a fixed number of plates and I
)O-
I
I
I
I
I
I
,
I
I
I
2
3
4
5
6
7
6
9
N;”
I
(%)
FIG. 4. The concentration of 01* (NDo’“) at bottom of column as a function of concentration of CV7 (Nr,017) at the bottom of column for P = 0. Parameter: Nmsxol’ = maximum concentration of 017 Given : EH~O~~/HZO~~ = 1*0060 ccHa01~/Hz017 = 1.0032
IO-
IO -
ro -
io -
IO !
I
,
I
I
I
,
I
I
IO
20
30
40
50
60
70
60
90
0”
z- w.)
FIG. 3. The concentration of 01s (Npor8) at the bottom
of the column as a function of the mean concentration of Ols (No’s) in the hold-up of the column for P = 0. Parameter: NmrXol’ = maximum concentration of 017 Given Y =looo aHsOl6/HsO’* = 1~0060 orHsOrs/HsOrr = 1.0032
water of given average concentration in Or* and 017, what will be the final isotope concentrations at the top and bottom of the column, and what will be the maximum concentration of 017, if the column is operated at total reflux and with zero feed and zero production? It is seen from equations (1) and (2) that for a column operated at zero product the concentration of the liquid entering the column is equal to that of the vapour leaving it, so that the computations made are correct also for total reflux operation. In order to solve this problem a series of computations were made [using equations (4) and (811 for a column with 1000 plates in the range of N;” = 50-99 per cent and Nz” = 05-10 per cent. The results are shown in Figs. 2-5. Using these four figures in succession one determines first the maximum concentration of 0” (Fig. 2), the bottom concentration of 0” (Fig. 3), 703
Z. FRAENKEL, A. RAVIVand W. KLEIN
the bottom concentration of 0” (Fig. 4) and the top concentration of Or’ and 0” (Fig. 5). In this way the characteristics of a separation column can be predicted if the enriched product is only taken from the column after it reaches steady state and not as a continuous product flow. 2. Top-to-bottom
calculations
In the top-to-bottom calculations for a column of given flow and product and a given number of plates, the product concentrations have to be initially assumed and successive approximations are made to find the correct values. The input parameters are the top concentrations, the assumed bottom concentrations, the number of plates of the column, the flow and the product. With these parameters one computation is performed after which the assumed
I’
’
’
’
’
’
’
’
Ni’s are replaced by the arithmetic mean between the Nk’s assumed and obtained, and the calculation is repeated. One can make the first computation with P, = 0 and increase the production step by step until the desired P is reached (Option I). The arithmetic mean of the assumed (input) value and the calculated (output) value of NL is taken as the input for the next iteration. However, the convergence is rather slow. Another way to approach the correct A$,% is to start calculations with the desired P and to assume that Ngs are zero (Option II). In this case after ten iterations an accuracy of 0.1 per cent is obtained (see Table 5). Table 5 L = 10 I/day
Iteration no. 10
Npi’sassumed
N#s obtained
NPois
0.9663626345
0.9663212538
NPol’
0.0164789602
0*0164940367
Npol’
0.0171584053
0.0171847095
l*oooooooooO
l*oooooooooO
P = 3.2 x 10-a l/day
’ 1’
Comparison of assumed (input) values and calculated (output) values for NPs in the last (10th) iteration of a top-tobottom calculation in which NpI = 0 was assumed for the hrst iteration (Option II). V.
COMPARISON OF THE CALCULATIONS WITH THE
RESULTS OBTAINED USING DIFFERENTIAL EQUATIONS
Ideal cascades
- -1 P%_
01
N_ J
I2
3
4
5 N;”
6
7
6
9
(%)
FIG. 5. The concentration of O’s and 017 at the top of the column as a function of the concentration of 0’7 at the bottom of the column for P = 0. =lOOO Given Y OL HaO“J/HzO1* = 1.0060 OT Hs01’3/HaOi’ = 1.0032
The results obtained by the method of difference equations were compared with those obtained by VON HALLE [6]* using differential equations for a cascade ideal with respect to 0” which consisted of both enricher and stripper columns with the feedpoint between these sections. The computation was made using equations (3), (4), (9) and (11) for the enricher section, the inverse of equation (3) and equations (5), (10) and (14) for the stripper
* This report has been made available to us through the courtesy of Dr. G. A. GAMETT.
704
Plate by plate calculations
Table 6.
of multicomponent
distillation columns using difference equations-I
Comparison of results obtained using difference and dl@erential equations
Results obtained by deference equations
Results obtained by differential equations 0’8 mole fraction
4oi
800 1200 1600 2ooo 2400 2800 3200 3600 4000
0.980000 0.964815 0.935754 0.877086 0.760483 0.565081 0.334874 O-159358 O-065795 0.025355 0.009495 OGO3516 0.001295
017
Flow
018
017
mole fraction
units/unit time
mole fraction
mole fraction
units/unit time
0.050 41.915 22.065 25.128 28.211 36.520 60.027 124402 299.358 774602 2065.962 5575.691 4007466
0.980000 0.964883 0.936006 0.877808 0.762212 0.568114 0.338288 0.161796 0.067081 0.025938 oM9741 0.003617 OGO1262
0.017478 0.027970 O-042438 0.058429 0068754 0.064393 0.045561 0.024977 0.011606 0.004956 0.002034 OGOO819 owo399
0.050 14.904 22.062 25.129 28.187 36.439 61.487 128.059 308.528 797.535 2123.107 5716.510 4041.984
0.017478 0.028018 0.042579 0.058676 0.069021 0.064500 0.04545 1 0.024809 0.011487 0034892 0.002004 OGOO805 OGOO396
VI. COMPARISON WITH EXPERIMENT
section. The concentrations at the enricher end of the column were: q’s = 0*98oooo $I’
= 0.017473
ql’
= OGO2527
and the concentrations column were :
at the stripper end of the
nz” = 0*001021 now”= ONK)352 n216= O-998627 6 at the top and bottom of the column were taken as 1.0. The flow L was calculated taking the product P = O-05 units/unit time, and the waste W = 47.986 units/unit time. The concentrations of 01’ and 0” were printed out once every twenty plates. Detailed plate-to-plate print-outs were obtained at the feed point and at the maximum concentration of O1’. As seen from Table 6 the results obtained by the two methods agree quite well. The differences in the concentration for 01’ are of the order of maximum 2 per cent, for 01* of the order of maximum 3 per cent and for the flow, of the order of maximum 3 per cent.
Flow
1. Thermal diffusion Column
The calculations were compared with experimental results for a thermal diffusion column filled with oxygen gas. Though the calculations were intended for fractional distillation columns, they can also be used for thermal diffusion, if conditions equivalent to those stated in II but pertaining to thermal diffusion columns are assumed to hold. In such a column both (a - 1) and the number of equivalent ideal plates will be dependent on the operating characteristics such as pressure, temperature, etc. These two variables were determined in the following manner: the cascade was tilled with oxygen and operated at zero production until a steady state was reached. Assuming a certain value for CIand starting with the experimental concentrations at the bottom of the column, equation (4) was used to compute the theoretical gradient of concentrations up to the experimental top concentrations of O1*. By this computation one obtained the number of plates of the cascade v corresponding to the assumed CL It should be noted that in a column operating at total reflux and zero production it is the product (CZ- 1). v which appears in the calculations rather than cxand
705
Z.FRAENKEL,
A.R~vrvandW.
v independently. For the present calculations the following assumptions were made: (i) The separation factor of Ols and 016 as calculated from the thermal diffusion factor [7] was taken as ,(Oi6/01*) = 1.016
No. 1 the number of plates obtained was v = 393 whereas experiment No. 2 gave v = 374. 2. In another experiment a second cascade similar to the fist one was filled with oxygen of given mean concentrations and operated at total reflux and zero production until steady state was reached. The theoretical concentrations predicted by Figs. 2-5 were compared with the experimental ones as shown in Table 7.
(ii) ~$O~~/oi*) - 1 = ~.Lx(O’~/O”) - 2 The theoretical and experimental concentrations were compared in Figs. 6 and 7. For experiment O’l”/ I
450
I
2 I
3 I
4 I
5 I
0
6 I
7 I
6 I
9 I
KLEIN
Table 7. Comparison of theoretically predicted and experimentally obtained results in a d@ksion cascade filled with oxygen of mean concentrations Nz’:” = 45.10 per cent, Nzz = 3.09per cent
Experimental concentration
Theoretical concentration
92.30 1.86 5.50 1.86 3.91
92.10 2.16 4.40 1.65 4.10
NO”,concentrations at top of column; NPS, concentrations at bottom of column.
Distillation (1) A column was operated with a continuous feed of 172,000 cm3/day of natural water which was fed at the top and a product of 100 cm3/day which was withdrawn at the bottom of the column. In this case the column had no stripper section and the descending liquid stream was of the same amount as the feed stream. The calculated bottom concentrations as obtained by equation (6) are NF” = 1.31 per cent, Nz” = O-11 per cent while the experimental concentrations measured after a steady state had been obtained were Npl’ = 1.73 per cent, NF” = 0.13 per cent. (2) A distillation column, which was operated at total reflux and zero production was estimated to have the following characteristics : (a) 1200 theoretical plates. (b) 2800 cm3 hold-up and 135 cm3 boiler content. The mean concentrations at the start were
100I-
5c,-
FIG. 6. Comparison of the theoretical and experimental concentration of Or* and 017 in the diffusion cascade. OT Or“/O’* = 1,016 ti 01’3/017 = lGO8 The curves show the calculated distributions and the dots and triangles show the experimental results for 01* and 0’7 respectively. The calculated curves are normalized to the experimental values at the bottom of the cascade (plate 0). Experiment No. 1.
O'S N BYC = 35.50per cent
706
Plate by plate calculations of multicomponent distillation columns using difference equations-I
such manner the concentrations in the above mentioned columns were increased to eVy = 42.82 per cent and NE: = 3.12 per cent. The expected peak of 0” and the expected concentrations of 01* and 01’ at the ends of the column were computed from Figs. 2-5. Comparison with the experimental results for both the initial and the tial average concentrations is given in Table 8.
0'77 0
Table 8. Comparison of the theoretically predicted and experimentally obtained results in a distillation column operated at zero reJEux
iTheoretical result
Ekperimental result
Theoretical result
Experimental result
-N,v,@* Nave@+
Nmsxo17 N,oi’ Npor7 N,o=a Noor’
42.82
35.5
2.90 91.3 164 1.05 056
1.89 2.92 90.7 1.68 1.10 0.54
4.63 94.1 2.05 190 1.15
3.12 4.70 94.30 1.66 1.93 128
* The same values were used in the calculation as input parameters.
--_I/ IO
20
30
I
I
I
40
50
60
I 70
I 80
Figs. 6 and 7 and Table 8 show that the calculations give a reasonably accurate description of the experimentally observed behaviour of both thermal diffusion and fractional distillation columns. The differences between the theoretically predicted and experimentally obtained results are probably due to: 1. Inaccuracy in the estimation of the number of plates. 2. Inaccuracy in the estimation of the hold-up and boiler content.
I 9~
Ole% FIG. 7. Comparison of the theoretical and experimental concentrations of Ols and 017 in the diffusion cascade. OT O’6/O’8 = 1.016 a o’s/017 = 1408 The curvea show the calculated distributions and the dots and triangles show the experimental results for Ole and 017 respectively. The calculated curves are
normalized to the experimental values at the bottom of the cascade (plate 0). Experiment No. 2.
In order to increase the peak of 01’ concentrations one has to increase the mean concentration of 01’ and to change the mean concentration of 018, according to Fig. 2. These operations are performed by changing a fraction of the column holdup, i.e. adding to the column water of certain concentration of 0” and 0” and withdrawing the same amount of liquid but with different isotopic concentration. While planning such an experiment the boiler content has to be taken into account. In
SUMMARY AND CONCLUSION From the above comparison of calculated values with experimental results we may draw the conclusion that the theory of multicomponent separation colnmns is in good agreement with experimental results. With the aid of plate-by-plate calcula-
tions based on difference equations it is possible to calculate a priori the important characteristics of a separation column. One possible technique is the
707
R. FRAENKEL,A. RAXJIVand W. KLEIN Superscripts i Referring 01s Referring 017 Referring 016 Referring I Referring
construction of a series of graphs like those shown in Figs. 2-5 from which the pertinent parameters may be obtained. Though more lengthy than the solution of the differential equation when the concentrations at only a few points along the column are required, this method may well be faster when the concentration at many points along the column is of interest. However the main advantage of this method as compared to the use of differential equations, is the fact that it can be applied to the most general system. The calculations which have been described in this paper were mostly restricted to columns operating at total reflux. Yet the procedure is not restricted to these columns and its application to other systems will be discussed in a future publication. While no experimental results are available for comparison with calculations on “ideal cascades”, the calculated results seem to indicate that these cascades do not have substantial advantages over constant flow columns if a high enrichment factor for 01’ is the prime objective.
APPENDIX:
to to to to to
the component i the component 0’s the component 0i7 the component 016 an ideal cascade
6, FOR AN IDEAL COLUMN
Assume that the concentration gradient per plate over the first three plates AN’JAK for the ideal component is constant, an&denote it by AN’ E=s
(A.11
The mass balance yields (Fig. 8) L,_,N;(i
- s) = PN:, + I,N;(l - 2.9 (A.2) L,_,=P+l,
(A.3)
From equations (A.2) and (A.3) one obtains E(L,-1 - 22”) = 0
Acknowledgements-The authors wish to thank Professor I. D~~TROVSKYfor helpful discussions and guidance in this work, and Professor C. L. PEKERISfor his cooperation in providing time on the WEIZAC. The assistance of Mrs. Mm BERKOin tabulating data is also gratefully acknowledged.
NOTATION
I Vapour flow leaving a plate, l/day L Liauid flow leaving a plate, I/day nr Mole fraction of ‘Ehei-th &mponent in the vapour stream I Ni Mole fraction of the i-th component in the liquid stream L
(A.4)
6,+1=2
(A.5)
6“=l
(A@
It is seen that 6, = 1 is not strictly’correct but is true only to a first approximation. Whether 6, is smaller or bigger than 1 depends on the sign of the quadratic term of the expansion of Nl_2 as a function of NL. In view of the small value of E the first approximation may be used in all practical applications.
708
FIG. 8. The last two plates of an ideal cascade with the corresponding liquid and vapour streams of the ideal component.
Plate by plate calculations
of multicomponent
distillation
columns using difference equations-I
REFERENCES [l]
DOSTROVSKY, I., LLEWELLYN, D. R, and VROMEN,B. H.,J. Chem. Sot. 1952 3509,3517. [2] DOSTROVSKY, I., GILLIS,J. and LLEWELLYN, D. R., L. Farkas Memorial Volume, Res. Council Israel, Spec. Publ. No. 1, Jerusalem 1952. [3] DOSTROVSKY, I. and RAMV, A., Proc. Znt. Symp. Isotope Separation, Amsterdam 19% [4] TREYBALR. E., Mass Transfer Operations p. 271. McGraw-Hill, New York, 1955. [5] COHEN, K., The Theory of Isotope Separation. McGraw-Hill, New York, 1951. [6] VON HALLE, E. and KNIGHT, G. B., Internal Report, Oak Ridge Gaseous Diffusion Plant, Union Carbide Nuclear Company, Oak Ridge, Term., 1 August 1956. (Unpublished.) [7] GREW, K. E., and IBBS, T. L., Thermal Diffusion in Gases. University Press, Cambridge 1952.
R&urn&-Les auteurs d&crivent une methode de calcul plateau par plateau des colormes de separations au moyen d’equations aux differences tin&. Les calculs ont ettt effectuees sur calculatrice numerique WEIZAC. Les deux objets des calculs sont les suivants: a) Comparaison de cascades id&des entre elles et avec des cascades a flux constant b) Preparation de graphiques permettant la prediction a priori des performances dune colonne donnQ a partir des partir des parametres dent&. De plus, les r&hats sont compares avec ceux obtenus par la resolution d’equations differentielles dam le cas ou cette methode est possible. Enfin, les calculs sont confront&s aux resultats experimentaux de colonnes de distillation et de diffusion thermique.
709