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""UNITED STATES ATOMIC ENERGY
ESTIIMATING INTERSTATE FLOW IN A SEPARATING
CASCADE WITH A BYlPASSING REFLUX STREAM
September 2, 1952
[Site Issuance Date]
Carbide and Carbon Chemicals Company (K(-25)
Tehi cal,... Information Service, Oak Rid, Tennessee
A test procedure is presented which evaluates average flows along with
average separation factor of a uniform stagewitse cascade section
fractionating a binary mixture such as isotopes. Th procedure calls
for isotopic gradient measurements at the terminals of the isola~ted
section on total reflux and also with at least one selected small
rate of bypassing reflux stream.
In the interest of economy, this report has been
reproduced direct from copy as submitted to the
Technical Information Service.
PRIN~TED IN USA
PRICE: 20 CENTS
Available from the
Office of TCechnical Services
Department of Commnerce
Washington 25, D). C.
Work performed under
Contract NJo. W7405-eng-26.
AEC, Oak Rdge, Tenn.-W26812
ESTIMA1TIG UTINTERSTAGE FLOWE OF A_ SQUAR
SECTION ON PARTIAL :INVETRGE RECYCLE
In a square-x section of a stagevise se-paration. systemn, such, as that
of a gaseous diffusion cascade or a plate column., the numer of stages,
N, is defined. Thus, the two variables, 4 (Cr-1, stage separation factor)
and V (molar stage upflow rate), are the only uanJt~i-ties required to
define the complete, separating capacity of the section. Th~at is,
for given net transports of total and light .:0tupo.~nerit and, one given
point of concentration of light component in the section, the steady-state
performance of the section is then completely established.
.bP the separation factor, is either transla-ted frlo n vilot plant
measurements on total reflux or is obtained in palace fromr e l-test",
a measurement of the total reflux gradient in the squlare section of the
cascade of interest.
V, the interstate flow (upflow), can be mceaure~d at, the sectional
terminals where an ideal, reversiblec process with accurately "constant
molal overflow" is involved. However, in an irreversible process
(one involving individual stage driving forces and po~tential partial
refluxing at every stage) the terminal flows need not be representative
of the exact average interstate flows of the section; particularly,
since the terminal connections are almost by definition different from
the normal interstate connections. In such a sy;tem flows can either
be measured in representative interstate systems or calculated by
fluid flow circuit adalyes from test data of the major stage component
equipmet.ut Secondary- effects of minor componen~t~s, such2 as connections
and configuration, can be neglected or roughly taken into account.
For this type of stagewise system, the problem can arise that the sectional
perforace along with the measured stage separation factor and known.
number of stages ap-pear to make thfe calculated average stage flows
questionable. This is particularly true when the flov circuit is complex.
Some~times, after the effect has been established, it is possible to
modify a minor portion of the circuit and impro.e the overall ?eformance.
Representative flows can be measured in place in single stages only with,
complex additions of transve~rsing probes and instrumentation. It is the
purpose of this study to outline tests which can be 'used in such cases
to compute average flow along with the average separ-atifon factor of a
building or section in place.
Total and net flows as defined in this study refer to tra~nspor:ts of the
two components to be separated. Thus, they must be corrected for any
carrier or diluent content unless this content represents a constant
fraction of net and total flows of components thiroughiout the system.
" The term "square" refers to a uniform section of stages With constant
stage separation factor, 9r, and stage upflow, L,
II. Total Reflux Performance
SIt is assumed throughout this study that stage flow and stage separation
factor are independent variables. Actually, in most processes and
systems, large changes in flows or net flows produce changes in effective
separation factors. This point will be discussed further in regard to
the effect of stage "cut" ("cross-flow" to distillation) on separation
For the special case of total reflux measurements of gradients in
isolated uniform sections, the value of existing interstate flow
is irrelevant+, since the performance can be expressed by the usual
xT = mole fraction of light component in the downflow to
the top stage (N) of the section (top recycle),
xB = mole fraction of light component in the downflow from
the bottom stage (1) of the section (bottom recycle).
Thus, in a r-test", the number of
concentrations establish the value
neted not be considered at all.
stages and the measured terminal
of J. The rate of stage upflow, V,
III. Performance with Net Transports
For the general. case of net transport (finite reflux) through the section,
the performance of the section is given by the usual combination of the
following two relationships which must be met:
(a) Material Balace between stages n and n+1:
SThis and subsequent approximate forms of equations would apply to
difficult separations (small separtion factors, such as in isotope
Vyn = (V-D) xn~ + D;y
(b) Separation equation of stage n:
D = net upflow of
those of V,
both components, in units consistent with
D;Y = net uprflow of light component, in the same units,
y9; = mole fraction of light component in upflow from stage n to
xn+ = mole fraction
to stage n,
x = mole fraction
to stage n-1.
of light component in downflow from stage n+1
of light component in downflow from stage n
Therse two equations can be combined to yield the following exp-~re-ssion
for the x-gradient across stage n:
1 D E
Xn+1 xn D n V D n~~
xn (1-x), D-
Equation (4) can be treated in its exact form according to the meth~bbA
of Teller and Tour+ by calculus of finite differences to yield thy,
performance of the whole section**n, or it can be rewritten in the
differential form, as dx/dn at stage n, and integrated***-~ to approximate
the performance of a section with a small stage separation factor.
From an inspection of equation (4), it is apparent that thle stage
performance as expressed by the enrichment depends upon V and JI and
upon the net flows, D, and DyD. Upon integration of the enrichment
equation for the whole section, the numer of stages, N, is of course
essential for the overall section performance. If the performance is
to be expressed in actual concentrations, then the actual value of
one concentration moust also be defined.
STeller, F. M. and Tour, R. S., Transactions of the American Institute
of Chemical Engineers, 4O, 317 (1944).
**~ Burton, D. W., "Solutions of Enrichment Equations by Method offini)&
Differences", A-4l51, Mar~ch 28, 1947, and
Shacter, J. and Garrett, G. A., WAaL~ogies between Graso~us IDiffuion
ap2d: FraciL~tionlal :~D isti~ilt~rlation ZSI ..@}0190, Mvag 7i 1948;.
***t Shac~ter and Garrett (1bid.) or:
Cohen, K., Journal of Chemical Physics, 8, 588 (194o).
Squires, A. M., "Note on Method of Calculating the Separation
Performance of the K 25 Plant", October 25, 1944.
Henkin, L., Squires, A. M., and Montroll, E. W., "Method of C~alculating
Separation Performance of> the K-25 Plant", Se~ptember 14, 1951,
Garrett, G. A., "A New Treatment of Steady-State Enrichment Equations",
2.28.1, September 24, 1946.
Burton, D. W., "Steady-State Equations of Diffusion Cascade Based
Upon Abundance Ratios", A-3664, December 26, 1946.
It is interesting to note that for the case of total reflux
(D = O, Dyd = O), equation (4) is reduced to the form,
xn+1-xn = $| Xn (1-yn)
'If e xn (1-xn) (5)
Comparison betwJeen equations (4) and (5), or their integrated forms,
as represented by equations (U3.) and (1), reveals that the definition
of finite reflux performance involves the! same variables as that of the
total reflux performance plus the additional terms, D, Dy and V.
IV. Measurements with Net Transport "On-Stream"
Thieor~etically, any one of the terms, NJ, JI, x,, x D, DyD, or V,
can be obtained from the integrated steady-state performance equation
if' the other terms are known. From a practical. point of view, the usual
existence of losses, inventory changes, scri;ng, contaminant bubbles,
deviations from steady-state gradients, etc., will severely limit the
on-st~reaml performance data of any section to the point where estimates
based on such data would be much less accurate than estimates based
on independent sources of V and W.
The very general theoretical method of evaluating a section would
consist of measuring two sets of terminal concentrations at twro
different reflux ratios (values of D and Dy: ) and solving the two
recultingr equations simultaneously for V and 4. In a practical case,
howJever, 9 and V are both obtained independently (V from equipment
perfeormancezc data and fluid flowJ analyses, 9 from pilot planet asnd
laboratory datca and separation performance relations, or frcia a miore
direc-t "#-tiest" swi.th the section on total reflux), and the perf;olr~cac
of the section is calculated with the use of anl integ~rated form of equat~ion ( ).
V. Measurements with a Bypassing Reflux Stream
Although there are usually severe limitations to the usefulness of
on-stream performance data in large systems or cascades, these
limitations could be virtually removed if the one section of interest
can be isolated and operated independently. This is done in the case
of -tests" on total reflux.
However, the same type of test of an isola ted section could be extended
to operation with~ reflux, or D f 0, DyD O 0 It would be a somewhat
more complex test since it would require accurate knowledge of the value
of; net flow (`reflux stream), D, in addition to the usual requirements
of a total reflux test. This net flow would be removed from one,
say the top, terminal of the section and recycled (outside) to the other
end, where it would be reintroduced into the section, as illustrated
in Figure 1.
A simple test procedure would consist first of a total reflux r-test"
which would establish \ from the terminal concentrations, by equation (1)
or (5), and then of a "Iflow-test"' which would duplicate the r-test"
procedure with a well-measured bypassing reflux stream, D, from the
top stage to the bottom stage of the section. The value of D could
be chosen, for instance, to result in about half the total reflux
gradient. An estimate of the magnitude of flow which accomplishes this,
Dh, can be obtained from equation (4). It can be shown by algebra that
Dh= 2 (YD ~)
x is a weighted average concentration of the section.
In most cases of interest, this recycle rate, Dh, would only have to be
a small fraction of the interstate flow, V. Permanent or temporary
connections must be provided for the accurate measurement of the recycle
The enrichment equation for this type of oflow-test" vould be,
An integrated form of this equation would be solved for V. The value for
Jrin the equation would be obtained from the preceding -test", the
other terms of the equation would be measured.
In selecting the \ value for equation (7), care would have to be
exercised to correct the measured total reflux value of for different
stage efficiencies, such as stage cut (a term analogous to cross flow
in plate columns), if D is not very small in respect to the estimated
value of V. Thus
"flow test" ftJ-test" (RC)
IC) is a calculated ratio of cut corrections for the two situations.
a.nd where the cut 8 can be expressed in terms of the measured D and
In most cases, approximate knowledge of V and D is sufficient for this
correction. For very exact evaluation, the estimated value of V for this
cut correction should be equal to the selected V value for the3 trial-error
method described in subsequent paragraphs.
(1-Yn) (x -x)3
" Personal Communcation.
The exact value of V could be obtained from. the identical-stage circuit
analyses with net flow D, or it might even be permiserble to assume
V to be independent of D.
For instance, using t~he exact method of finite differences as listed in
AECD-1940, we obtain the integrated expression,
Nr (B-xO) (%1-9) (0
where the roots, xO and x1 are given by
xq,~ ~ ~~ I 1- XT) + L(1 f-y f( T (1
These equations apply even with large separa-tion factors, providing It is
still defined by the exact form of equation (3). The terms (1 1 x)
and (1 .) can be emitted only when they amount to very small corrections
on JI. must be properly corrected for cut whSesn D is large.
The solution for V involves a trial-error procedure of selecting two
or three values of V and solving equations (11) and (10) for N~. The
resulting values of N can be plotted against corresponding assumed
values of V, and the correct value of V can be found on that curve when
read against the actual, known number of stages, N, on partial inverse
A. de la GartaWe of this Division has analyzed the propagation of inaccuracies
and imprecisions on the measurements of the various items which, determine V
in a gaseous diffusion section, in order to get an estimate of the
comparative usefulness of this method. He estimates that average stage
flows of a 60-stage section should be obtainable with. an accuracy
of about 3 44% by this method, if the anall net flow, D, is measured
within about 1 2L$. As stated earlier, the procedure? should prove of
value as a check on section performance when calculations based on
individual equipment tests and applied to the actual stage circuit,
are under suspicion.
I Reflux Stream
Section I n-1
All upper-case letters refer to molar flow
rates of both components; all lower case
letters refer to mole-fractions of light
component; and all subscripts refer to stream
SCHEMATIC DIAGR ILLUTRATIN TH
UNIVERSITY OF FLORIDA
3 1262 08905 4224
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