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I C~YC3 ~"` ' L~e JI /. Oc ""UNITED STATES ATOMIC ENERGY SK943 MAMRO}I ESTIIMATING INTERSTATE FLOW IN A SEPARATING CASCADE WITH A BYlPASSING REFLUX STREAM By J. ~Shacter September 2, 1952 [Site Issuance Date] Carbide and Carbon Chemicals Company (K(25) Tehi cal,... Information Service, Oak Rid, Tennessee ABSTRACT 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. CHEMISTRY 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. W7405eng26. AEC, Oak Rdge, Tenn.W26812 ESTIMA1TIG UTINTERSTAGE FLOWE OF A_ SQUAR SECTION ON PARTIAL :INVETRGE RECYCLE I. Introduction In a squarex section of a stagevise separation. 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 (Cr1, stage separation factor) and V (molar stage upflow rate), are the only uanJt~ities 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 steadystate performance of the section is then completely established. .bP the separation factor, is either translated frlo n vilot plant measurements on total reflux or is obtained in palace fromr e ltest", 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 appear 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 separatifon 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" ("crossflow" to distillation) on separation factor. 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 FenskeUnderwood formula: 1xT 1xB where 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 rtest", 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 separations). Vyn = (VD) xn~ + D;y (b) Separation equation of stage n: Yn x In ~~n so n x (x) n` n where 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 stage n+1, xn+ = mole fraction to stage n, x = mole fraction to stage n1. of light component in downflow from stage n+1 of light component in downflow from stage n and Therse two equations can be combined to yield the following exp~ression for the xgradient across stage n: 1 D E Xn+1 xn D n V D n~~ xn (1x), 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", A4l51, 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 K25 Plant", Se~ptember 14, 1951, Garrett, G. A., "A New Treatment of SteadyState Enrichment Equations", 2.28.1, September 24, 1946. Burton, D. W., "SteadyState Equations of Diffusion Cascade Based Upon Abundance Ratios", A3664, 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+1xn = $ Xn (1yn) 'If e xn (1xn) (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 "OnStream" Thieor~etically, any one of the terms, NJ, JI, x,, x D, DyD, or V, can be obtained from the integrated steadystate 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 steadystate gradients, etc., will severely limit the onst~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 direct "#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 onstream 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 rtest" which would establish \ from the terminal concentrations, by equation (1) or (5), and then of a "Iflowtest"' which would duplicate the rtest" procedure with a wellmeasured 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 ~) where 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 flow. The enrichment equation for this type of oflowtest" vould be, xn+1 xn 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" ftJtest" (RC) where 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 estimjlated V: 2VD 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 trialerror method described in subsequent paragraphs. 1 x D n (1Yn) (x x)3 " Personal Communcation. The exact value of V could be obtained from. the identicalstage 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 AECD1940, we obtain the integrated expression, In(XlX) (XTx0) Nr (BxO) (%19) (0 In 1+(X where the roots, xO and x1 are given by xq,~ ~ ~~ I 1 XT) + L(1 fy f( T (1 These equations apply even with large separation 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 trialerror 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 recycle. 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 60stage 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. Downi Op flowflow Byrpassing I Reflux Stream Stage L, n+1 stage n laolated Lx Cascade EXI Section I n1 Bot. Stage All uppercase letters refer to molar flow rates of both components; all lower case letters refer to molefractions of light component; and all subscripts refer to stream locations. Figure 1 SCHEMATIC DIAGR ILLUTRATIN TH "FLOW TEST" 4 UNIVERSITY OF FLORIDA .1111111111111111111111111111111111111111111111111111111111111 3 1262 08905 4224 I 
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