UNITED STATES ATOMIC ENERGY 
 
 K-943 
 
 ESTIMATING INTERSTAGE FLOW IN A SEPARATING 
 CASCADE WITH A BYPASSING REFLUX STREAM 
 
 By 
 
 J. Shacter 
 
 b 
 
 September 2, 1952 
 [Site Issuance Date] 
 
 Carbide and Carbon Chemicals Company (K-25) 
 
 Technical Information Service, Oak Ridge, Tennessee 
 
ABSTRACT 
 
 A test procedure is presented which, evaluates average flows along with 
 average separation factor of a uniform stagewise cascade section 
 fractionating a binary mixture such as isotopes. The procedure calls 
 for isotopic gradient measurements at the terminals of the isolated 
 section on total reflux and also with at least one selected small 
 rate of bypassing reflux stream. 
 
 CHEMISTRY 
 
 In the interest of economy, this report has been 
 
 reproduced direct from copy as submitted to the 
 
 Technical Information Service. 
 
 PRINTED IN USA 
 
 PRICE 20 CENTS 
 
 Available from the 
 
 Office of Technical Services 
 
 Department of Commerce 
 
 Washington 25, D. C. 
 
 Work performed under 
 Contract No. W7l|05-eng-26. 
 
 AEC, Oak Ridge, Tenn.-W26812 
 
ESTIMATING INTERSTAGE FLOW OF A SQUARE 
 SECTION ON PARTIAL INVERSE RECYCLE 
 
 I, Introduction 
 
 In a square* section of a stagewise separation system, such as that 
 
 of a gaseous diffusion cascade or a plate column, the number of stages, 
 
 N, is defined. Thus, the two variables, ty (a-1, stage separation factor) 
 
 and V (molar stage upflow rate), are the only quantities required to 
 
 define the complete, separating capacity of the section. That is, 
 
 for given net transports of total and light component and one given 
 
 point of concentration of light component in the section, the steady-state 
 
 performance of the section is then completely established. 
 
 \(f, the separation factor, is either translated from pilot plant 
 measurements on total reflux or is obtained in place from a "\ir-test" , 
 a measurement of the total reflux gradient in the square section of the 
 cascade of interest. 
 
 V, the interstage flow (upflov), can be measured at the sectional 
 terminals -where an ideal, reversible process with accurately "constant 
 molal overflow" is involved. However, in an irreversible process 
 (one involving individual stage driving forces and potential partial 
 refluxing at every stage) the terminal flows need not be representative 
 of the exact average interstage flows of the section; particularly, 
 since the terminal connections are almost by definition different from 
 the normal interstage connections. In such a system, flows can either 
 be measured in representative interstage systems or calculated by 
 fluid flow circuit analyses from test data of the major stage component 
 equipment. Secondary effects of miner components, such 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 
 performance along with the measured stage separation factor and known 
 number of stages appear to make the calculated average stage flows 
 questionable. This is particularly true when the flow circuit is complex. 
 Sometimes, after the effect has been established, it is possible to 
 modify a minor portion of the circuit and improve the overall performance. 
 
 Representative flows can be measured in place in single stages only with 
 complex additions of transversing 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 separation factor of a 
 building or section in place. 
 
 Total and net flows as defined in this study refer to transports 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 throughout the system. 
 
 * The term "square" refers to a uniform section of stages with constant 
 stage separation factor, ty, and stage upflow, L 
 
II. Total Reflux Performance 
 
 For the special case of total reflux measurements of gradients in 
 isolated uniform sections , the value of existing interstage flow 
 is irrelevant*, since the performance can be expressed by the usual 
 Fenske -Underwood formula : 
 
 <*♦>■ " T^l 
 
 (1) 
 
 •where 
 
 x,^ = mole fraction of light component in the downflow to 
 the top stage (N) of the section (top recycle), 
 
 x_ = mole fraction of light component in the downflow from 
 the bottom stage (l) of the section (bottom recycle). 
 
 Thus, in a "i|r-test", the number of stages and the measured terminal 
 concentrations establish the value of \|r. The rate of stage upflow, V, 
 need not be considered at all. 
 
 It 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" in distillation) on separation 
 factor. 
 
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 
 folio-wing two relationships which must be met: 
 
 (a) Material Balance between stages n and n+1: 
 
 Vy n = (V-D) x n+1 + Dy D (2) 
 
 (b) Separation equation of stage n: 
 
 * 
 
 y - x y - x 
 "'n n .%/ J n n 
 
 1 ■" J-l •%» Ai JUL f \ 
 
 * = ~ (l-y ) = x (l-x ) (5) 
 
 x 
 n * "n n N n 
 
 where 
 
 D = net upflow of both components, in units consistent with 
 those of V, 
 
 Dy = net upflow of light component, in the same units, 
 
 y = mole fraction of light component in upflow from stage n to 
 stage n+1, 
 
 x . = mole fraction of light component in downf low from stage n+1 
 to stage n, 
 
 and 
 
 x = mole fraction of light component in downflow from stage n 
 to stage n-1. 
 
 * This and subsequent approximate forms of equations would apply to 
 difficult separations (small separation factors, such as in isotope 
 separations ) . 
 
These two equations can be combined to yield the following expression 
 for the x-gradient across stage n: 
 
 X n + r X n = ^[* X n (l " 3r n ) " I V^ 
 
 Equation (k) can be treated in its exact form according to the method',' 
 of Teller and Tour* by calculus of finite differences to yield the_ 
 performance of the whole section*-*, 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 (k) , it is apparent that the stage 
 performance as expressed by the enrichment depends upon V and \J/ and 
 upon the net flows, D, and Dy-p. Upon integration of the enrichment 
 equation for the whole section, the number 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 must also be defined. 
 
 * Teller, F„ M„ and Tour, R, S„, Transactions of the American Institute 
 of Chemical Engineers, kO, 317 (l°M). 
 
 ** Burton, D. W«, "Solutions of Enrichment Equations by Method of (Finite- 
 Differences" , A-U151, March 28, 19^7, and • . / / 
 
 Shacter, J. and Garrett, G. A,, "Analogies between Qa«eOu« Diffusion 
 tod F»c.tionar.Di*tlllatioo' , J ( AECD-l^O, May 7, 19^8, ... 
 
 *** Shacter and Garrett (ibid.) or: 
 
 Cohen, K., Journal of Chemical Physics, 8, 588 (19UO). 
 
 Squires, A, M , "Note on Method of Calculating the Separation 
 Performance of the K-25 Plant", October 25, 19Mk 
 
 Henkin, L„, Squires, A, M., and Montroll, E„ W , "Method of Calculating 
 Separation Performance <of the K-25 Plant", September Ik, 1951. 
 
 Garrett, G. A,, "A New Treatment of Steady-State Enrichment Equations", 
 2.28.1, September 2k, 19k6. •'■.'"..' • 
 
 Burton, D. W„, "Steady-State Equations of Diffusion Cascade- Based 
 Upon Abundance Ratios", A-$66k, December 26, 19^6. 
 
 (k) 
 
It is interesting to note that for the case of total reflux 
 (D = 0, Dy = 0), equation {h) is reduced to the form, 
 
 x n -x = ty x (l-y ) 
 n+1 n Y n v J n 
 
 -tx (1-x ) (5) 
 
 T n v n w/ 
 
 Comparison between equations {k) and (5), or their integrated forms, 
 as represented by equations (ID) and (l) , 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" 
 
 Theoretically, any one of the terms, II, \]r, x , X- 9 D, Dy _, or V, 
 can be obtained from the integrated steady-sxate performance equation 
 if the other terras are known. From a practical point of view, the usual 
 existence of losses, inventory changes, surging-, contaminant bubbles, 
 deviations from steady-state gradients, etc., will severely limit the 
 on-stream 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 \j/. 
 
 The very general theoretical method of evaluating a section would 
 
 consist of measuring two sets of terminal concentrations at two 
 
 different reflux ratios (values of D and Dy ) and solving the two 
 
 resulting equations simultaneously for V ana \|r. In a practical case, 
 
 however, \[r and V are both obtained independently (V from equipment 
 
 performance data and fluid flow analyses, \jc from pilot plant and 
 
 laboratory data and separation performance relations, or from a more 
 
 direct "i|r-test" with the section on total rex lux), and the performance 
 
 of the section is calculated with the use of an integrated form of equation (k] 
 
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 "i]/-tests" on total reflux. 
 
 However, the same type of test of an isolated section could be extended 
 to operation with reflux, or D / 0, Dy f °- I_t vould 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 topj 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 "y-test" 
 which would establish \|r from the terminal concentrations, by equation (l) 
 or (5), and then of a "flow-test" which would duplicate the "i|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, 
 D,, can be obtained from equation (k) . It can be shown by algebra that 
 
 D h " 2 (y D -x) 
 * x (l-y) 
 
 - 1 
 
 V 
 2 (^-x) 
 
 i|r x (1-x) " 1 
 
 (6) 
 
 where 
 
 x is a weighted average concentration of the section. 
 
 In most cases of interest, this recycle rate, D,, would only have to be 
 a small fraction of the interstage flow, V. Permanent or temporary 
 connections must be provided for the accurate measurement of the recycle 
 flow. 
 
10 
 
 The enrichment equation for this type of "flow-test" would be, 
 
 - x = =■ \lr x (l-y ) - 77 (x -x ) I 
 
 n i 5. |_ n n ^ ^ n I 
 
 V 
 
 " 171 L * n (1_Xn) " ^ (XT " Xn) J 
 
 (7) 
 
 An integrated form of this equation would be solved for V. The value for 
 \|r in the equation would be obtained from the preceding "\|r-test", the 
 other terms of the equation would be measured. 
 
 In selecting the \|c value for equation (7), care would have to be 
 exercised to correct the measured total reflux value of ty 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" = Vtest" ^ (8) 
 
 where 
 
 (rc) is a calculated ratio of cut corrections for the two situations. 
 
 and where the cut can be expressed in terms of the measured D and 
 estimated V: 
 
 q. v 
 
 2V-D 
 
 2-8 
 
 V 
 
 (9) 
 
 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 the trial-error 
 method described in subsequent paragraphs. 
 
11 
 
 The exact value of V could be obtained from the identical-stage circuit 
 analyses with net flow D, or it might even be permissible to assume 
 V to be independent of D. 
 
 For instance, using the exact method of finite differences as listed in 
 AECD-19^0, we obtain the integrated expression, 
 
 N « 
 
 In 
 
 <*1 
 
 -x B ) 
 
 ( Xrp -Xq ) 
 
 
 (*B 
 
 - x o) 
 
 Ul-x T ) 
 
 In 
 
 1 + 
 
 4> xi 
 
 
 1 + Ip Xq 
 
 where the roots, Xq and Xj_ are given by 
 
 (10) 
 
 Xq, X X = 
 
 i ci - S V + 1 
 
 V(i -f-*r) + Dj . i^ (l -|)|x T 
 
 2 <!-$)* 
 
 (11) 
 
 These equations apply even with large separation factors, providing ip is 
 
 still defined by the exact form of equation (3). The terms (1 - - Xj.) 
 
 and (1 - — ) can be omitted only when they amount to very small corrections 
 
 on tp. ip must be properly corrected for cut when D is large. 
 
 The solution for V involves a trial-error procedure of selecting two 
 or three values of ? and solving equations (11) and |l0j)rfor 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, W, on partial inverse 
 recycle. 
 
 A. de la Garza* of this DiviBion 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 - ^ by this method, if the small net flow, D, is measured 
 within about 1 - 2$. 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. 
 
 * Personal Communication. 
 
12 
 
 Isolated 
 
 Cascade 
 
 Section 
 
 T^ 
 
 Top N 
 Stage 
 
 n 
 
 Down- 
 flow 
 
 ;u P - 
 
 .flow 
 
 Stage 
 n±l_ 
 
 viT_t y ° 
 
 Stage n 
 
 ■TIL 
 
 1 » 
 
 J_l 
 
 Bot. 
 
 Stage 
 
 *B 
 
 Flow Meter 
 
 Bypassing 
 Reflux Stream 
 
 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 
 locations. 
 
 Figure 1 
 
 SCHEMATIC DIAGRAM ILLUSTRATING THE 
 "FLOW TEST" 
 
UNIVERSITY OF FLORIDA 
 
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