Theory Of Isotope Separation

NATIONAL NUCLEAR ENERGY SERIES Manhattan Project Technical Section Division III-Volume

1B

THE THEORY OF ISOTOPE SEPARATION AS APPLIED T O THE LARGE-SCALE PRODUCTION OF

u~~~

THE THEORY O F ISOTOPE SEPARATION AS A P P L I E D T O THE LARGE-SCALE PRODUCTION O F U235

KARL COHEN Director, Atomic Energy Division, The H. K. Ferguson Company; formerly Director, Theoretical Division, S A M Laboratories

Edited by

GEORGE M. MURPHY Washington Square College, New York University; formerly at SAM Laboratories, Columbia University

F i r s t Edition

New York

- Toronto . London

McGRAW-HILL BOOK COMPANY, INC. 1951

FOREWORD THE THEORY OF ISOTOPE SEPAMTION AS APPLIED TO THE LARGE-SCALE PRODUCTION OF

Copyright, 1951, by the McGraw-Hill Book Company, Inc. P r i n t e d i n t h e United S t a t e s of A m e r i c a

Copyright assigned, 1951, to the General Manager of the United States Atomic Energy Commission. All rights reserved. This book, o r parts thereof, may not be reproduced in any form without p e r mission of the Atomic Energy Commission.

uZ3'

The wartime project f o r development of atomic energy was a r e markable feat of cooperation and accomplishment by government, science, industry, labor, and the military s e r v i c e s aimed exclusively at the military application of atomic energy. Our present national atomic energy program, expanding upon the previous developments, i s directed not only toward the assurance of national security but also toward the realization of the immense potential benefits atomic energy holds f o r our civilization. The record of progress and t h e r e sults of extensive scientific investigation~and engineering development a r e contained in the National Nuclear Energy Series. This knowledge, which offers the basis of world -wide benefits from nuclear science, i s being published in the established scientific tradition, not solely to meet the precise needs of science but a l s o in support of the high goals of the American people s e t forth in the Atomic Energy Act. The work reported in this s e r i e s is a tribute to all the scientists engaged in both the Manhattan Project and the postwar Atomic Energy Commission program. Gordon Dean, Chairman U. S. Atomic Energy Commission

Lithoprinted by Edwards Brothers, Incorporated Ann Arbor, Michigan

PREFACE

ACKNOWLEDGMENT

The Manhattan Project Technical Section of the National Nuclear Energy Series embodies results of work done in the nation's wartime atomic energy program by numerous contractors, including Columbia University. The arrangements for publication of the s e r i e s volumes were effected by Columbia University, under a contract with the United States Atomic Energy Commission. The Commission, for itself and for the other contractors who contributed to this series, wishes to record here its appreciation of this service of Columbia University in support of the national nuclear energy program.

This volume is one of a series which has been preparedas a record of the research work done under the Manhattan Project andthe Atomic Energy Commission. The name Manhattan Project was assigned by the Corps of Engineers, War Department, to the far-flung scientific and engineering activities which had a s their objective the utilization of atomic energy for military purposes. In the attainment of this objective, there were many developments in scientific and technical fields which a r e of general interest. The National Nuclear Energy Series (Manhattan Project Technical Section) is a record of these scientific and technical contributions, a s well a s of the developments in these fields which a r e being sponsored by the Atomic Energy Commission. The declassified portion of the National Nuclear Energy Series, when completed, is expected to consist of some 60 volumes. These will be grouped into eight divisions, a s follows: Division I - Electromagnetic Separation Project Division I1 - Gaseous Diffusion Project Division I11 - Special Separations Project Division IV - Plutonium Project Division V - Los Alamos Project Division VI - University of Rochester Project Division VII - Materials Procurement Project Division VIII - Manhattan Project Soon after the close of the war the Manhattan Project was able to give its attention to the preparation of a complete record of the research work accomplished under Project contracts. Writing programs were authorized at all laboratories, with the object of obtaining complete coverage of Project results. Each major installation was requested to designate one o r more representatives to make up a committee, which was first called the Manhattan Project Editorial Advisory Board, and later, after the sponsorship of the Series was assumed by the Atomic Energy Commission, the Project Editorial Advisory Board. This group made plans to coordinate the writing programs at all the installations and acted a s an advisory group in all matters affecting the Project-wide writing program. Its last meeting was held on Feb. 9, 1948, when it recommended the publisher for the Series. vii

viii

ix

PREFACE

PREFACE

The names of the Board members and of the installations which they represented a r e a s follows:

Many difficulties were encountered in preparing a unified account of Atomic Energy Project work. F o r example, the Project Editorial Advisory Board was the f i r s t committee ever organized with r e p r e sentatives from every major installation of the Atomic Energy Project. Compartmentation for security was s o rigorous during the war that it had been considered necessary to allow a certain amount of duplication of effort rather than to permit unrestricted circulation of r e s e a r c h information between certain installations. As a result, the writing programs of different installations inevitably overlap markedly in many scientific fields. The Editorial Advisory Board has exerted itself to reduce duplication in s o f a r a s possible and to eliminate discrepancies in factual data included in the volumes of the NNES. In particular, unified Project-wide volumes have been prepared on Uranium Chemistry and on the Analysis of Project Materials. Nevertheless, the reader will find many instances of differences in results o r conclusions on similar subject matter prepared by different authors. This has not seemed wholly undesirable for several reasons. F i r s t of all, such divergencies a r e not unnatural and stimulate investigation. Second, promptness of publication has seemed more important than the removal of all discrepancies. Finally, many Project scientists completed their contributions s o m e time ago and have become engrossed in other activities s o that their time has not been available for a detailed review of their work in relation to similar work done a t other installations. The completion of the various individual volumes of the Series has also been beset with difficulties. Many of the key authors and editors have had important responsibilities in planning the future of atomic energy research. Under these circumstances, the completion of this technical s e r i e s has been delayed longer than its editors wished. The volumes a r e being released in their present form in the interest of presenting the material a s promptly a s possible to those who can make use of it.

Atomic Energy Commission Public and Technical Information Service Technical Information Division, Oak Ridge Extension Office of New York Operations

Alberto F. Thompson Brewer F. Boardman Charles Slesser, J. H. Hayner, W. M. Hearon

*

Brookhaven National Laboratory

Richard W. Dodson

Carbide & Carbon Chemicals Corporation (K-25)

R. B. Korsmeyer, W. L. Harwell, D. E. Hull, E z r a Staple

Carbide & Carbon Chemicals Corporation (Y-12) 'f

Russell Baldock

Clinton Laboratories 1

J. R. Coe

General Electric Company, Hanford

T. W. Hauff

General Electric Company, Knolls Atomic Power Laboratory

John P. Howe

Kellex Corporation

John F. Hogerton, Jerome Simson, M. Benedict

Los Alamos

R. R. Davis, Ralph Carlisle Smith

National Bureau of Standards

C. J. Rodden

Plutonium Project Argonne National Laboratory

R. S. Mulliken, H. D. Young

Iowa State College

F. H. Spedding

Medical Group

R. E. Zirkle

SAM Laboratories

Â

G. M. Murphy

Stone & Webster Engineering Corporation

B. W. Whitehurst

University of California

R. K. Wakerling, A. Guthrie

University of Rochester

D. R. Charles, M. J. Wantman

*Represented Madison Square A r e a of the Manhattan District. The Y-12 plant a t Oak Ridge w a s operated by Tennessee Eastman Corporation until May 4 , 1947, a t which time operations were taken over by Carbide & Carbon Chemicals Corporation. $Clinton Laboratories w a s the f o r m e r name of the Oak Ridge National Laboratory. §SA (Substitute Alloy Materials) was the code name for the laboratories operated by Columbia University in New York under the direction of D r . H. C. Urey, where much of the experimental work on isotope separation was done. On Feb. 1, 1945, the administration of these laboratories became the responsibility of Carbide & Carbon Chemicals Corporation. Research in progress t h e r e was t r a n s f e r r e d to the K-25 plant a t Oak Ridge in June, 1946, and the New York laboratories were then closed.

The Editorial Advisory Board

COLUMBIA UNIVERSITY PROJECT FOREWORD

The Manhattan Project Technical Section of the National Nuclear Energy Series is intended to be a comprehensive account of the scientific and technical achievements of the United States program for the development of atomic energy. It is not intended to be a detailed documentary record of the making of any inventions that happen to be mentioned in it. Therefore, the dates used in the Series should be regarded a s a general temporal frame of reference, rather than a s establishing dates of conception of inventions, of their reduction to practice, o r of occasions of first use. While a reasonable effort has been made to assign credit fairly in the NNES volumes, this may, in many cases, be given to a group identified by the name of its leader rather than to an individual who was an actual inventor.

Government-supported research on nuclear energy f i r s t occurred in early 1940 when certain funds made available by the Army, Navy, and National Bureau of Standards were used for experiments a t Columbia University. The subsequent history of this project a s it expanded in many other universities and industrial laboratories is told in detail in the Smyth Report. Two of the major programs began a t Columbia University: the uranium-graphite pile reactor and the gaseous diffusion method of uranium 235 separation. Lack of sufficient laboratory space a t Columbia for the development of both programs led to the transfer in the spring of 1942 of the uranium-graphite reactor program to the University of Chicago. The program of separation of uranium 235 by gaseous diffusion, and various ancillary projects, developed rapidly until not only all available space in the Columbia University buildings was occupied but, in addition, an even greater amount of rented space. The project for the separation of uranium isotopes by gaseous diffusion was first carried forward through various government cont r a c t s under the direction of Harold C. Urey, Professor of Chemistry, and John R. Dunning, Assistant Professor of Physics. In 1943 the work was unified under a single contract with the Manhattan Engineer District with Professor Urey as Director. An account of the research on the major assignment, the separation of the uranium isotopes, is to be found elsewhere in the National Nuclear Energy Series. The volumes that this preface introduces deal with the general mathematical theory of isotope separation, with some new experimental methods of separating isotopes, with spectroscopic properties of uranium compounds, and with the chemical and physical properties of heavy water and other deuterium compounds. Most of this work was done at Columbia, but parts of it were carried on a t the National Bureau of Standards and a t The Johns Hopkins University. George B. Pegram Chairman, Columbia University Division of War Research, 1941-1945

AUTHOR'S PREFACE

PUBLISHER'S NOTE

Although every effort has been made to ensure accuracy in references, at the time of publication of this book some of the other volumes of the Series had not been completed. It is therefore possible that some of the references to other volumes a r e in e r r o r . It is hoped that the extensive c r o s s checking which has been done in the preparation of this volume has resulted in keeping such e r r o r s to a minimum.

by the theoretical division of the SAM Laboratories and its antecedents, which the author was privileged to lead. The work was done over a period of years, from 1940 to 1945, under contract to various government agencies. However, we have in some instances drawn on outside sources for special topics, and some of the material in the volume is new. Since the reports have not been and probably will not be published, we have avoided distracting the reader with references in the body of the text which would be meaningless to him. At the end of each chapter a reference section of relevant reports is appended for purposes of record. Specific references to published articles a r e footnoted a s they occur in the text. Although the theory was developed piecemeal in response to various demands, it is exposed a s a logical whole according to our present understanding and experience. In Chaps. 1 to 5, on various aspects of cascade theory, certain generalities that a r e common to many different separation methods a r e collected. Chapter 6 treats the centrifugal method. Chapter 7 is composed of sections briefly outlining two-phase separations (including chemical exchange and distillation) and the thermal-diffusion method, and a section presenting some remarks on the separation of deuterium. The studies given a r e a selection from a much larger volume of material. More emphasis is put on general principles and concepts than on an exhaustive collection of particular results, no matter how elegant. It is thought unlikely that any detailed problem will recur in exactly the same form and that a thorough grasp of the fundamentals will be more useful to the reader. In any event, the most important applications of the theory cannot yet be discussed. While the theory itself is intact, the reader may find that the motivations and explanations a r e occasionally somewhat sketchy. This is a result of security deletions. We did not feel that they were important enough to warrant extensive rewriting. xiii

AUTHOR'S PREFACE

XIv

There is no reference to direct experimental proofs of the theory, although they a r e not lacking. This is only partly a question of space and security. In a large measure the theory is self-evident, in the fashion of demonstrations in thermodynamics. Karl Cohen 1

August 1951

ACKNOWLEDGMENTS Development of the theory in this volume was influenced by many discussions with people whose names appear on no theoretical report. It is a pleasure to acknowledge indebtedness to Drs. J. R. Dunning, E. T. Booth, and H. A. Boorse, all of Columbia University, and Dr. F. G. Slack, of Vanderbilt University, for ideas and criticisms. A British team, composed of Drs. F. Simon, R. Peierls, K. Fuchs, and N. Kurti, worked out independently the theory of separation a s it applied to a particular separation. They obtained substantially the same results, in a different chronological order, corresponding to the different order in which the problems were attacked. Adequate liaison was accomplished in early 1942, at which time a very stimulating comparison of results and viewpoints was made. This and subsequent contacts contributed toward clarity of thought and expression. Much of the credit for the success of these investigations i s due to Dr. H. C. Urey: on an administrative plane, for his unwavering support of the theoretical group; and on a scientific plane, for imparting to us his fine intuitive grasp of the essentials of separation processes. The concept of separative work, which i s basic for the comparison of different methods, was first used by him in 1939. The author is personally indebted to E. V. Murphree, of the Standa r d Oil Development Company, for allowing him the time to write this volume. The role of our colleagues in the theoretical division does not really need to be mentioned here; their contributions will be amply demonstrated by the reference sections. Above all, Dr. I. Kaplan, H. Mayer, and Dr. R. D. Present have contributed substantially to the success of this work. The elegant derivation of the value functions and the separative power in Chap. 1, Sec. 3, is due to Prof. P. A. M. Dirac. The value function itself, Eq. 1.43, was first determined by Prof. R. Peierls. The relaxation-time phenomena for square cascades were worked out with the constant aid of Dr. I. Kaplan. The method of determining the fundamental constants in the steady state i s due to Dr. W. I. Thompson, of the Standard Oil Development Company. The mathematical method employed in Chap. 4 on losses was shown to the author by Dr. R. P. Feynman, then at Princeton University. The solution of the

mi

ACKNOWLEDGMENTS

fluctuation problem, which avoids solving the differential equation f o r the second o r d e r , was devised by Dr. K. Fuchs. Without the aid of Mrs. L. W. Goodhart, of the Standard Oil Development Company, i n editing and in all phases of preparation of the manuscript, it could never have been finished; special thanks a r e due to h e r f o r this important s h a r e of the work. 1

Karl Cohen

CONTENTS Page

.................................... v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii . . . . . . . . . . . . . . . . xi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv .................................. 1

Foreword Preface Columbia University Project Foreword Author's Preface Acknowledgments. Introduction

CHAPTER 1 Ideal Cascades.

................................

5

CHAPTER 2 Square Cascades

...............................

30

CHAPTER 3 Equilibrium Time of a Square Cascade

.................

39

CHAPTER 4 Determination of Cascade Constants

..................

62

CHAPTER 5 The Control Problem

............................

79

CHAPTER 6 Centrifuges

...................................

103

Ss

CHAPTER 7 Other Separation Methods

......................... xvii

126

xviii

CONTENTS

Appendix A-Roots

of a Transcendental Equation

Appendix B-Equilibrium for N 1

-

.........

139

Time of Square Cascades

................................... Appendix C -The Holdup Function . . . . . . . . . . . . . . . . . . . Appendix D -Rayleigh Distillation . . . . . . . . . . . . . . . . . . . Appendix E-Properties of Concurrent Two-phase Elements. . Index.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

144 147

INTRODUCTION

i

150 154 163

The separation of isotopes may be accomplished in many ways. In this volume consideration will be given, in more, or l e s s detail, to centrifuges, electromagnetic methods, electrolysis, chemical exchange, thermal diffusion, and distillation. Within each of these categories there is considerable variety. For example, chemical exchange may take place between a liquid phase, between a liquid phase and a vapor phase by countercurrent scrubbing, or between two liquid phases with concurrent contacting. All these methods, except the mass-spectrograph method in isolated instances, have the common feature that one elementary process does not produce the desired large change in isotopic abundance. The problem of efficiently multiplying elementary processes to reach large end results is thus of utmost practical importance, and indeed of greater importance the smaller the change produced by the elementary process. Fortunately, although the methods differ both physically and mathematically in fundamental ways, they a r e sufficiently alike so that their multiplication problems a r e very closely related. It will therefore be useful to examine thoroughly the multiplication problem per se before taking up any of the several separation schemes in detail. The purpose of the first five chapters is to develop the theory of cascades a s generally a s possible in order to permit the widest possible application to future separation problems. Existing applications of the theory will b e found in the subsequent chapters on particular separation methods. The point of view taken is that of a development engineer, and the main concern is the significance of the results in terms of cascade design. For example, the effect of losses is investigated in terms of the problem "How much larger do we have to build our stages to get the same quality and quantity of yield for a given loss?" rather than "In a given cascade how will the yield vary with loss?" Consideration is seldom given to problems of the second kind-problems that occur after the cascade is built-although naturally the same equations govern both questions.

3

THE THEORY OF ISOTOPE SEPARATION

INTRODUCTION

In Secs. 1 to 4 of Chap. 1 the concept of an ideal cascade is examined from three radically different points of view; Section 5 introduces the idea of equilibrium time. Section 6 gives the general par-

They a r e Secs. 2 and 3 of Chap. 2; Secs. 1 and 2 of Chap. 3; Secs. 1,2, and 3 of Chap. 5; and Appendixes A and B. In the Appendixes a r e grouped a number of extensions and amplifications that have considerable practical interest. Appendixes D and E in particular contain special results upon which some work in the later chapters is based. The many graphs which a r e included were designed for use a s well a s illustration, and particular pains were taken to ensure their accuracy. The same applies to the tables, which were in some cases recalculated entirely for this book. Principles of Notation and Terminology. Because of the wide variety of topics that will be covered in this review, it will not be possible to set up a system of notation that will be general enough to cover every case and at the same time describe each special case naturally. However, certain principles of notation will be followed. A mole fraction of a particular isotope will always be denoted by the letter N. In our case of binary mixtures, the two mole fractions a r e of course N and (1-N). In two-phase systems, all lower-case letters will refer to one phase, upper-case letters to the other. Thus N and n will refer to mole fractions in different phases. With every binary mole fraction N we will associate the corresponding molecular abundance ratio R, where

2

tial differential equation showing the dependence of mole fraction in a cascade on time and stage number. The first portion of Chap. 2 is intended to show the steady-state behavior of single countercurrent columns, or square cascades. In Secs. 2 and 3 of the chapter, cascades of countercurrent columns a r e considered. Section 4 covers the squaring off of the top of a cascade. Time-dependent phenomena in square cascades a r e dealt with in Chap. 3, along with the methods for experimental determination of the fundamental constants of a separating element. Chapter 4 considers the increase in size of a cascade with losses, the efficient combination of plants using different separation processes, and the prime operating problem of the control of a cascade. Chapter 5 deals with the mathematical theory of the effect of disturbances in operation on the production of the cascade. Chapter 6 describes the different types of centrifuges and gives the partial differential equation of the centrifuge. Other separation methods a r e presented in Chap. 7. Only for certain kinds of processes-"infinitesimal" processes, countercurrent columns, and the mass spectrograph-is the theory developed explicitly, in such a way a s to permit immediate application. In the last chapter, which includes the deuterium problem, a r e introduced cascade problems that a r e not treated in Chaps. 1 to 5. But no one who has absorbed the first five chapters will be at a loss to solve the problems. Likewise, among the infinite variety of nonideal cascades, only square cascades have been selected a s being of universal interest. The reader will be able to work out for himself such variants a s cascades in the form of a lozenge (the "cascade-ofcascades" in British terminology). The generality of the treatment still leaves something to be desired, since the entire treatment is restricted to binary isotope mixtures. This is a consequence of the practical origin of the theory. In a general way an attempt has been made to avoid lengthy mathematical demonstrations, except when they a r e indispensable to the argument. For this reason there is only a very general treatment of the control problem, which is beset with many purely mathematical difficulties. A detailed study of the equilibrium-time problem of an ideal cascade has also been omitted; here again the problem has more mathematical than physical interest. Some of the sections that a r e mostly mathematical might perhaps be omitted on a first reading.

Thus corresponding to n there will be r = n/(l -n). AS material flows through a separating element, its mole fractions a r e changed. The mole fractions at the various orifices of a separating element will be distinguished by primes (for example, see Fig. 1.1). Quantities of material-generally expressed in moles per second-are designated by L's. The term "cascade') is used to mean any connected arrangement of elements. The "elementsv a r e the separative units-individual centrifuges o r fractionating columns. A "stagey7 of a cascade consists of all units operating in parallel on material of the same mole fraction. (In the special case of one element per stage, the terms "element" and "stage" will be used interchangeably.) The feed material of a cascade, generally of natural abundance ratio, is denoted by the subscript zero (No); the amount of feed is F moles/sec. The "stage number" (counting the bottom stage a s zero) is denoted by s. The mole fraction of material fed into the sth stage is Ng, and the amount is Lg.The subscript S indicates the top stage of a recti-

4

THE THEORY O F ISOTOPE SEPARATION

fying cascade. The mole fraction a t the product end of a cascade is Np ;the amount of product is P. The mole fraction of the waste i s Nw; the amount of waste is W. Other general terminology and notation will be covered in the subsequent introductory discussion of cascades. Equations, figures, and tables a r e numbered decimally (viz., Eq. 6.4); the f i r s t digit indicates the chapter, the number after the decimal the order in the chapter. Chapter 1 IDEAL CASCADES

Consider a separating element that divides an entering stream of L moles/sec of a bicomponent mixture, with the mole fraction of one component designated a s N, into two s t r e a m s of 0L moles and mole fraction N', and (1 - Q)L moles and mole fraction N", respectively (Fig. 1.1). Of course 0 < 0 < 1, where 0 is known as the "cut." Further suppose that the action of the unit is such that

The t e r m a is called the "simple-process factor" of the separation.' F r o m the conservation of matter

Now suppose that a cascade is built of such elements, the size of each stage being graduated s o that the flow entering the sth stage is L c . F o r purposes of this discussion it is irrelevant whether each stage is built of a single element, whose size differs from stage to stage, o r of identical small elements in parallel, whose number v a r ies. The elements a r e connected a s shown in Fig. 1.2, s o that the enriched fraction from each element is fed into the stage next above and the other fraction is fed into the stage immediately below it. At the top of the cascade, material is withdrawn continuously a t a rate P moles/sec. It should be observed that there is considerable latitude This t e r m i s usually reserved f o r t r u e equilibrium ratios of abundance ratios (cf. Eq. 6.7 and Eq. 8 of Appendix D) which, however, have no universal relationship to the a of Eq. 1.1. If necessary to distinguish, a of Eq. 1.1 i s referred to a s the "effective"

6

IDEAL CASCADES

THE THEORY OF ISOTOPE SEPARATION

about the choice of the flows, Ls. For the instant they a r e completely arbitrary. The problem now at hand is one of choosing flows Ls and cuts 13~to get the most efficient cascade. Clearly, the remixing of materials of different mole fractions must be avoided. That is, i t is imperative that R&-1= Rs = R;+l

7

Equation 1.11 gives the flow required in the sth stage of a cascade producing P moles/sec of material with mole fraction Np, and Eq. 1.6 gives the required cut to ensure that no mixing takes place in the

(1.3)

From Eqs. 1.1 and 1.3, Rs = RL-I = a R s - i , whence

ENRICHED OUTPUT e L,N1

follows immediately by induction. Likewise R;

=

1 Rs

STRIPPED OUTPUT (4-9)L.N''

a

From Eqs. 1.2 and 1.5,

Fig. 1.1-A

Considering that section of the cascade above an imaginary line drawn between the 8th and (s + 1)th stages (see Fig. 1.2), we must have balance between the amount of materials entering and leaving this whole section, i.e.,

Substituting Eq. 1.7 in Eq. 1.8,

Equation 1.9, which i s a consequence of the conservation of matter, holds generally for all cascades, even if mixing takes place. Introducing Eq. 1.3, the condition of no mixing, in Eq. 1.9,

Using the properties of the separating element itself and eliminating N; by Eq. 1.1 and eS by Eq. 1.6,

simple

separating element.

cascade. Such a cascade is called an "ideal" o r "no-mixing" cascade. In exactly what sense this is the most efficient cascade will be made clear later. Equation 1. l l can be written a s

by replacing the N's by R's and remembering that Rs = asRo and that R Ro. In this form the Lie may be summed to get the total flow p in an ideal cascade. Noting that

it readily follows that

IDEAL CASCADES

THE THEORY OF ISOTOPE SEPARATION

2

Since the size, o r number of elements, of a stage is proportional a the flow into the stage, the total flow measures the magnitude of the whole cascade. 1. THE INFINITESIMAL CASE

Because of the special importance of cascades when the simpleprocess factor is very nearly 1, the limiting f o r m s of the equations just found will be written for a - 1 = e

Nt-N

+

-_

1

Equation 1.1 can be expressed a s

I FROM s 2 STAGE

Ã

=-

R' 1 + R'

R R' - R = (a 1 + R - (1 + R') (1 + R)

s+4

which in the limit is the s a m e as

_

-

_ _ __I__-

(IMAGINARY DIVIDING LINE)

----------

--

In the no-mixing case Eq. 1.15 becomes Ns+l - Ns = eNs (1 - Ns )

(1.17)

and Eqs. 1.4, 1.6, and 1.11 a r e replaced by the following: Rs = e^R,

-- -- -- -

ENRICHED FRACTION STRIPPED FRACTION

1 6 - - + ^ - ( 2 ~ ~- 1) 2 4

(1.18) (1.19)

FEED MIXTURE

The total flow in the cascade may be determined from Eq. 1.20 a s follows: I

i

I

I Now Eq. 1.17 may be written

Fig. 1.2-A

cascade of simple

elements.

10

THE THEORY OF ISOTOPE SEPARATION

IDEAL CASCADES

11

ap

Inserting in Eq. 1.21 and dropping the now superfluous subscript s,

which could have also been obtained directly from Eq. 1.13. Equation 1.19 shows that in the infinitesimal case Os varies very slightly with Ns Even if -as they must be - e and 6 a r e related by the physical nature of the separating element, e will be constant throughout the cascade, to t e r m s of the o r d e r of e2, and the preceding analysis holds. But for the finite case, if an internal relation between a and 6 exists which differs from Eq. 1.6, a will vary from stage to stage, and the equations break down. Practically, this means that the cascade equations will not be universally applicable unless (a - 1) is sma1l.l

very close to $$. Passing over to the limit, which i s clearly legitimate since Ns varies slowly,

Equation 1.27 gives the r a t e of increase of N per stage a s a function of the flow per stage, here considered a s an arbitrary function. Now consider a function such a s the total flow in the cascade

.

Total flow =

L'L = l N p ~ ( ~ ) ds

ds

dN

0

o r the total number of elements, each of which i s capable of processing G moles/sec, Number of elements =

1

N~

L(N)

ds

dN

0

2. RELATION BETWEEN IDEAL AND NONIDEAL CASCADES

Perhaps the best way of understanding the r e a l significance of ideal cascades is to consider their relation t o nonideal cascades. To do this the infinitesimal case must again be treated, this time allowing mixing. F r o m Eqs. 1.2 and 1.16,

o r more generally, the integral of any property proportional to the flow per stage

where Na and Nb a r e arbitrary;
Combining Eqs. 1.24 and 1.25,

Now, from Eq. 1.7, if P is much smaller than Ls and if the values of Ls vary slightly from stage to stage, all the values of OS will be

' Cf. Appendix D:

Rayleigh Distillation.

the ideal flow of Eq. 1.20. The asterisk (*) i s used in this volume to denote the Laplace transform of any function. The design problem f o r cascades may then be expressed a s follows: Of all possible cascades that produce P moles of product with a mole fraction Np, starting from raw material of mole fraction No, which cascade makes I a minimum? Or, since each function L(N) completely determines a cascade, what function L(N) makes I stationary? Clearly, I will be a minimum when

12

IDEAL CASCADES

THE THEORY OF ISOTOPE SEPARATION

that is, when 3. VALUE FUNCTIONS AND SEPARATIVE POWER

L =L* Therefore the ideal cascade minimizes the number of separating elements, and related quantities, in any section of a cascade. When L = L*, from Eq. 1.27,

is half the maximum concentration gradient (which occurs when P/L = 0). Optimum rate of production of a stage i s that rate which reduces the concentration gradient to half its value at no production. It is well to bear in mind that the concept of an ideal cascade is a s sociated with the form of I. For example, to minimize expressions such a s Total number of stages

= Js

ds =

LoNp

L 2L - L*

dN mFm

(1.34)

requires L = m and not L = L*. In practice it will be necessary to strike a balance between reducing the number of separating elements and reducing the number of stages. Furthermore an ideal cascade requires each stage to be of a different size. F o r these reasons actual cascades will deviate somewhat from ideal cascades. Nevertheless, the properties of ideal cascades a r e of great practical importance because of the proposition that follows. Developing I about L = L *,

Equation 1..23, for the total flow in an ideal cascade, is the ratio of a term

which depends only on the amount and mole fraction of material produced, and a t e r m e2/2, which is characteristic of the separating element. This suggests that each t e r m separately has physical significance and also that Eq. 1.23 might be established in a more gene r a l way. An attempt should now be made to derive a function U that represents the value of a quantity of separated material, i.e., a function proportional to the number of separating elements required to produce it. Obviously U = P V(N), where P is the number of moles of material. Suppose G moles of material is passed through a separating element of the type shown in Fig. 1.1. There results a net change in value

Taking (a- 1) = e s e r i e s about N,

-K

1 and expanding V(N1) and V(Nt1) in Taylor's

I

...

[ ( N ~ ' - Nd2v ) ](N) } + ~ [ ~ G ( ~ ' ~ ~ ) ' +(N"( ~ - ~ )+ G The first term on the right is called I*, the value of I for the ideal case. If IL - L* 1 =s y L * for y c 1, Eq. 1.35 gives, to higher powers of Y, (1.36) I - I* 5 y21*

The coefficients of V(N) and d v ( ~ ) / d N vanish by the conservation of matter. From Eq. 1.2, 6(Nr -N) = - (1 - 6) (N" -N)

Expressed in words: If the flow in a cascade at no point differs from that in the corresponding ideal cascade by more than 20 per cent, integrals of the type in Eq. 1.30 will not differ from the ideal integrals by more than 4 per cent. It is this stationary property of integrals for the ideal cascade which makes them useful for all cascades.

(1.38)

and from Eq. 1.16, N1 - N = N(l - N). Thus Eq. 1.38 becomes

(1.39)

THE THEORY O F ISOTOPE SEPARATION

14

IDEAL CASCADES

Now, specifying that this change in value be independent of N, the following equation must hold:

Then Eq. 1.40 becomes

The t e r m 6 U i s called the "separative power" of the element.' The differential expression of Eq. 1.41 defines V(N) except f o r two integration constants. Following Dirac, these may be chosen by r e quiring that V(No) = ~ v ( N ~ ) / =~ 0, Nwhere No i s the natural abundance. The f i r s t condition is obvious; the second makes V(N) a minimum at N = No and s o e n s u r e s a positive value f o r V over the entire range of concentrations. Thus it is found that

15

GUp - 2P V(Np) 1 - 6 Total flow = 6U c2 6 should be pointed out that, in o r d e r f o r Eq. 1.44 to hold, there t be no process taking place in the cascade which changes U exthe separation process; that i s , t h e r e must be no mixing.l Equa1.46 therefore gives the total flow in an ideal cascade, a s found viously f o r the special case2 6 = $$. A different choice of the integration constants is of course permisble. Taking V(0.50) = dv(0.50)/dN = 0, which means taking the equiolar mixture instead of the natural abundance a s the fiducial point, ves the simple formula

the value, which has the obvious advantages of depending on only mole fraction and of being the same for all isotopes. However, equation like Eq. 1.46 takes the form

G

Total flow = - (Up + U& - U;) 6U

V will be called the "value function." Obviously it i s the value p e r mole. Since one element produces a change in value 6U, r e g a r d l e s s of the mole fraction of the material on which it operates, the number of elements required to produce a change in value AU is Total number of elements = A U / ~ U

,

(1.44)

and Total flow = G

X

(number of units) = G

AU

=

2AU 1 - 0 7 7

(1.45)

F o r the cascade previously considered

where the subscripts r e f e r to the product, waste, and feed points. With the convention chosen, Uo = Uw = 0 and Eq. 1.45 is merely 'Note that e must be a function of 6. a s otherwise bU

- - as 6

1.

and since U' i s always used in such differences, with two relations between the p a r a m e t e r s (Eqs. 1.61 and 1.62), i t s numerical value i s l e s s illuminating. F r o m the standpoint of calculations, if there is one particular isotope-separation problem, with No known, values of V a s defined by Eq. 1.43 a r e most useful. These a r e found in Fig. 1.3 f o r the m a aium problem, No = 1/140. F o r a survey of many different separation problems, values of V a r e required. They a r e found in Table 1.1 and Fig. 1.4. V will be called the "elementary value function," to distinguish it from V, which is the value function relative to natural abundance. U' will be called the "elementary value, " and U will be defined a s the "value of a quantity of material." The value functions and the separative power have been derived f o r infinitesimal p r o c e s s e s . They may also be extended to cover the case of large values of a. 'Mixing reverses the process in Eq. 1.37 and hence decreases U. 'Since 8 has not been restricted, in general a more intricate cascade than the one shown in Fig. 1.2 will be required. F o r example, if 6 = '/5, the enriched fraction from stage s will have to be fed into stage s + 3, while the depleted fraction is fed into stage s-2.

THE THEORY O F ISOTOPE SEPARATION

16

I

I

I

I

I

I

I

IDEAL

I

CASCADES

46 0

Table 1.1-The

Elementary Value Function

F o r N rel="nofollow"> 0.50 use V'(N) V = (2N

0

0.2

Fig. 1.3-Functions

0.4 0.6 M O L E FRACTION

0.8

1 .O

associated with infinitesimal cascade. No= 0007143.

Consider a symmetric process1 where (cf. Eqs. 1.5 and 1.6)

Then corresponding to Eq. 1.37

= constant

Now introduce s by the equation R = Roos and s e t (1 + R) V(R) = g(R) = g(RoCt3) = F(s) 'The problem i s also soluble for asymmetric processes, where quite complicated functions a r e found.

-

= V'(1

N 1) In 1 -N

IDEAL CASCADES

THE THEORY OF ISOTOPE SEPARATION

[

a + 1 2N-1 R V(R) = C - -in-+ a - 1 lna Ro

]

a - N o ( a + l ) (N-No) a - 1 No(l - NO)

(1.55)

Now take the constant C = 6u/G s o that

Thus R lna a-(a+l)No (2N - 1) In - + (N-No) a-1 N ~ ( ~ - N ~ ) RO

0

0.10

0.20

0.30

0.40

0.50

MOLE FRACTION Fig. 1.4-The elementary value function. For values of N > 0.50,use v'(N) = V f ( l- N); for N > 0.46, V = 2(2N - I)'.

With these definitions Eq. 1.50 becomes

The general solution of Eq. 1.52 is

where A and B a r e constants, which gives a + 1 2N-1 R AN V(R) = C + -+ a-1 l n a In RO Ro

B ( l -N)

(1.54)

Setting V(Ro) = 0, and establishing the convention that the waste m a t e r i a l from the feed stage, of abundance ratio Ro/a, has zero value also, the following relation holds:

I

(1.57)

Equations 1.56 and 1.57 a r e closely analogous to Eqs. 1.42 and 1.43. The variation of U f fwith a exhibited in the second t e r m is caused by the varying concentration of by-product waste material and has no theoretical significance. The elementary value for l a r g e a is of course still U ' . The separative power v a r i e s a s ( a - 1)' f o r small (a - I ) , but f o r large a increases only a s In. Since the number of elements in a c a s cade varies inversely a s the separative power of an element, it i s important to design and operate the elements s o that their separative power (Eq. 1.42 o r Eq. 1.56) is a s l a r g e a s possible. This problem will be considered in detail in subsequent discussions on particular separation schemes. It i s also possible to calculate the separative power for some more general types of separating elements. If it i s assumed that instead of two s t r e a m s leaving the apparatus of Fig. 1 . I , t h e r e a r e many, each , furthermore of magnitude L ~ and

20

THE THEORY O F ISOTOPE SEPARATION

The f i r s t two t e r m s onthe right vanish by the conservation of matter. Hence

ally both Â

of which Eq. 1.42 is a special case. For infinitesimal separating elements of the countercurrent type, 6U may be evaluated a s follows: Consider two large reservoirs (see Fig. 1.5) containing, respectively, M moles of material in which the

21

IDEAL CASCADES T

- NP

and N' - N a r e proportional to N(l - N); if

6u =

( T - NP)

(N' - N) ~ 2 (1~ ) 2

SO

(1.60)

quantity T , i.e., the amount of desired isotope moved forward, is ed the "desired material transport." The factor T - NP is called "net transport." Equation 1.60 is an exceedingly important fora for the theory of countercurrent processes. he separative powers for two other important special types of ents a r e fully discussed in Appendix D (Rayleigh Distillation) and ndix E (Properties of Concurrent Two-phase Elements). 4. CASCADES WITH STRIPPING SECTIONS

1 Fig. 1.5-Separative

SEPARATING ELEMENT

1

In addition to a rectifying section, it will usually be desirable to e a stripping section in the cascade.l The function of the stripping tion is to economize raw material by recovering the desired isoe from the reject stream of the rectifier. Figure 1.6 shows the ions between rectifier and stripper. r any cascade, ideal o r otherwise, provided there is no loss of terial anywhere in the system, the following relations hold:

work of a countercurrent element.

mole fraction of desired material is N, and M' moles of slightly different mole fraction N' > N. The separating element puts P moles of mixed isotopes containing T moles of (pure) desired isotope into the upper reservoir, withdrawing the same amounts from the lower r e s ervoir. The upper reservoir increases in mole fraction by an amount

r e F, P , and W refer to the amount of feed, product, and waste, pectively, and No, Np, and Nw a r e the corresponding mole fracs. aking the mole fractions a s given and eliminating f i r s t F and then etween Eqs. 1.61 and 1.62, it is found that

W = P N~ - No

No - Nw

(1.63)

and the lower decreases by the amount (T - PN)/(M - P). Then he terms "rectifier" and "stripper" are taken from fractionating-tower usage. tower for producing alcohol from wine, for example, the section above the feed t rectifies the alcohol, i.e., concentrates it, while the lower section strips the aste water, i.e., recovers the last traces of alcohol from the reject.

23

IDEAL CASCADES

THE THEORY OF ISOTOPE SEPARATION

0

which relate the size of the waste stream and the feed stream to the rate of production.

resemble closely those for an ideal rectifier. A moment's conderation, referring again to Fig. 1.2, shows that Eqs. 1.4, 1.9, and .ll also hold for the stripping stages of a cascade, with -W and Nw replacing P and Np, s running from -1 to -B instead of from 0 to S, and Rw = ~ ~ / a ~Equation + l . 1.11 f o r the flow in the sth stage of a rectifier becomes Ls =-

a + 1 W(N, -Nw) a - 1 Ns(l-Ns)

(1.65)

Equation 1.13 becomes Total flow (stripper)

= s=-1

a+1 =-W[ a-1

1 =

1 (aB+s*l - 1) + -(aB+l - a"' ) Ro

a -1

I

1-2Nw R In a In Rw - a

WASTE W, N,.

a + 1 ( a-1)lna

-N(,(a + 1) No-Nw (a-1) No(l-No)

I

(1.66)

u'

where U$ is the value defined by Eq. 1.57. This expression vanishes a accordance , with our convention of counting stage 0 for Rw = ~ ~ / in a s part of the rectifier. The total flow in a cascade composed of both rectifier and stripper, combining Eqs. 1.13, 1.64, and 1.66, takes the simple forms

FEED

Total flow (rectifier and stripper)

=

a+1 [P(2Np-1)ln- RP ( a - 1) In a Ro

+ W(2Nw - 1) lnFig. 1.6-Above, connections between rectifier and stripper. Below, detail

of con-

nections.

Since a stripping section is essentially no different from a rectifying section, except that i t produces material at a different rate and concentration, the equations for an ideal stripper should be expected

-

a+1 ( a - 1) In a (up + uw

--

a +1 (UP + U&- Uo) (1.67) (a - 1) In a

24

IDEAL CASCADES

THE THEORY O F ISOTOPE SEPARATION

The t e r m s Up + Uw and Up + U& - Ug a r e of course AU = AU" and AU', respectively. Note that in Eq. 1.67 the distinction between U" and U has disappeared, since the t e r m s that differ because of different conventions have canceled out. F o r the infinitesimal case, in place of Eq. 1.65,

In place of Eq. 1.66 Total flow (stripper) =

Nw2

f.

7

2 W(N-Nw) dN NO - N ) N(I-N)=F~W

and in place of Eq. 1.67 Total flow (rectifier and s t r i p p e r ) =

2 â

2

(up+ Uw)

= - (up + e2

uy, - UO)

F o r completeness the formulas f o r the number of stages may be added. Number of rectifying stages = S + 1 = In RP/RO In cr Number of stripping stages = B = In ~ o h _w 1 In a

will be shown l a t e r the operation of a cascade f r o m starting-up might be a s follows: At z e r o time the mole fraction a t the top the cascade i s No; hence operations begin with no withdrawal. The centrations increase gradually over the entire cascade, the top of cascade leading. The product end reaches design mole fraction st, while the r e s t of the cascade is still below design. Withdrawal a slow r a t e is then begun, maintaining the top mole fraction a t Np d gradually increasing to design production a s mole fractions in the t of the cascade build up to their steady-state values (Fig. 1.7). eneral definition of the equilibrium time that has meaning f o r this e of operation o r any other is the number of days production that lost between starting-up time and steady -state production. A useful approximate expression for the equilibrium time of a n eal rectifier by simple considerations will now be developed. The total amount of material in p r o c e s s in a section of a cascade called the "holdup" of the section.' In most cascades the amount holdup in a stage i s proportional to the flow in the stage. This i s rtainly t r u e if the cascade i s built up of identical small elements, that successive stages consist of different numbers of elements in rallel. It also holds when the stages a r e similar single units scaled size. The holdup p e r unit flow will be denoted by h, which i s a l s o e average p r o c e s s time p e r stage, expressed in seconds. The "net desired material holdup" (N.D.M.H.) is the amount of s i r e d isotope in a going plant over and above that which would be contained in a like quantity of material a t normal abundance. It is, the infinitesimal case,

N.D.M.H. = y " " ' h ~ ,(N, -No) d s

Number of stages (rectifier and stripper) = lnRp/Rw - 1 In a F o r the infinitesimal case, In a i s merely replaced by

f r o m that of the product to that of the feed. Before the plant can oduce concentrated material a t the top i t must build up these inntories of partly enriched material. The time spent in this procis known as the "equilibrium time" o r "relaxation time" of the

(1.73)

-[ ( N P - 2 N p N o + N o ) l n ~ - 2 ( ~-No)] p

(1.74)

RO

e.

5. EQUILIBRIUM TIME

Obviously some cascades f o r separating isotopes a r e very l a r g e and contain considerable material in p r o c e s s a t concentrations rang-

he equilibrium t i m e in any c a s e is equal to the N.D.M.H. divided by a suitable average transport of desired isotope over the initial period of operation. Without f o r the moment specifying too closely the operNomenclature again borrowed from fractionating-tower terminology.

THE THEORY O F ISOTOPE SEPARATION

IDEAL CASCADES

27

may be applied to other than ideal cascades. The usefulness of the expression will depend on the appropriateness of the choice of the average net transport, which will not in general be P(Np - No). The N.D.M.H. i s one of the functions that i s a minimum for ideal cascades, a s is the equilibrium time function E(Np) derived from it. E(Np) i s independent of P . 6. FUNDAMENTAL EQUATIONS O F ISOTOPE SEPARATION

The partial differential equations describing the time-dependent ef fects in any infinitesimal cascade a r e easily derived by a slight extension of the previous analysis.' Ls and 6s a r e taken a s known functions that may vary with time. The mole fraction Ns i s also a function of time: No = Ns(t). Referring again to Fig. 1.2, Eq. 1.7 takes the more general form

TIME

where Ts(t) i s the amount of (mixed) material transported into the cascade above the sth stage. Explicitly, if t h e r e a r e no leaks o r losses,

Fig. 1.7ÑProductio a s a function of time.

ating schedule, this average transport may be taken a s the net t r a n s port of the rectifier in the steady state, that i s , P(Np - No). This gives

à [(N,, Equilibrium time =

2

RP Ro

H i is the holdup in the jth stage and is a l s o a given function of time. Ts (t) i s defined in t e r m s of known functions (by Eq. 1.77 o r 1.78) and therefore is known. Likewise Eq. 1.8, the equation of conservation of desired material, holds in the form

- 2NpN0 + No) In -- 2 ( ~ p- No)]

(Np - No)

where rS(t) is the amount of desired material transported into the section of the cascade above the sth stage. Unlike T, T is unknown. Explicitly

It should be remarked that the formula for the equilibrium time in the f o r m Equilibrium time =

N.D.M.H. average net transport

' F o r finite a it i s difficult toformulate the problem without specifying the separation method.

IDEAL CASCADES

THE THEORY O F ISOTOPE SEPARATION

28

In using N,'(t) on the right the difference between the mean concentration of the stage and Ni has been tacitly neglected. Combining Eqs. 1.77 and 1.79

29

Because special cases of Eqs. 1.81 and 1.82 describe the properties of apparatus in widely differing separation methods and because all the theory s o f a r developed for infinitesimal cascades can be deduced from them by purely formal operations, Eqs. 1.81 and 1.82 a r e called the "fundamental equations of isotope separation. " If L is not a function of t , and H = hL, then T = P, and Eqs. 1.81 and 1.82 combine to give

Assuming Eq. 1.24 t o hold unchanged (what is involved is neglecting s

the holdup of one stage compared to

2

Hj) and introducing Eq. 1.24

in the last equation,

On passing over t o the limit of continuous functions, OsLs K ( t ) N(s,t), etc., this becomes

-

-

%L(s,t),

Equation 1.83, with the initial condition N(s,O) = No, describes the approach to equilibrium of a cascade. F o r an ideal cascade, L is the extremely complicated function defined by Eqs. 1.18 and 1.20. Equation 1.83 will not be solved f o r this case, since the utility of the solution i s slight. In Chap. 3 it will be solved for the practical (and considerably easier) case of a square cascade, taking L a s a constant. Should L (and also H and e ) be a functionof t, fluctuating about some mean value, then Eqs. 1.81 and 1.82 describe the effect of fluctuations on separation, which is covered in Chap. 5, on the control problem.

REFERENCES

which is the required generalization of Eq. 1.27. F r o m Eqs. 1.78 and 1.80 i t is found1 that

Differentiating Eq. 1.81 with respect t o s and using Eq. 1.82, 9 ~ / 9 s may be eliminated and a second-order partial differential equation f o r N will result. Thus Eqs. 1.81 and 1.82, with appropriate boundary conditions, completely determine N. ' T h e integral form of Eq. 1.78, for example, i s ~ ( s , t =) P

+f

a

H(j,t) dj

and differentiating with respect t o s gives Eq. 1.82.

Cohen, K., Report A-60, October 1941;Report A-96, Jan. 5, 1942. Cohen, K.,Columbia Ser. No. 4L-X28, Dec. 22, 1942 (letter to J. 8.Dunning). Cohen, K., and I. Kaplan, Report A-98, Jan. 14, 1942. Dirac, P. A. M., British MS, 1941. Fuchs, K., British MS 29, Dec. 18, 1941. Fuchs, K., and R. Peierls, British MS 12A, February 1942; British MS 47, June 26, 1942. Kurti, N., and F. Simon, British MS, 1940. Peierls, R., British MS 12, summer 1940; British MS 13, September 1940.

SQUARE CASCADES

31

lies to both square cascades and countercurrent units. Indeed, it o preserve this generality that the notation of Eq. 2.1 has been

1. STEADY STATE OF A SQUARE CASCADE (OR COUNTERCURRENT COLUMN)

In the steady s t a t e t h e f i r s t integral of Eq. 2.1 i s

Chapter 2 SQUARE CASCADES

PNp

Ideal cascades a r e characterized by a smoothly varying flow function which i s next to impossible to realize in practice. Fortunately, a s the discussion in Chap. 1 of the relation between ideal and nonideal cascades has shown,it is by no means necessary to adhere strictly to the shape of the ideal cascade. The outlines of this cascade roughly follow those of an ideal cascade, and the properties of the blocked-off cascade closely resemble those of the ideal cascade. A cascade thus obtained from an ideal cas cade is called a "squared-off cascade." Each section of a squaredoff cascade i s itself a square cascade. F o r a square cascade Eq. 1.83 takes the form

a~ a2N c 6 Ãa‘tà ‘ = c 5 à ‘

'

as2

a

as

=

ch separates to give a second integral,

ls iNs + ds =

S =

1 tanh-I 6

A($)

c1N2

(2.1)

where cl, P, c5, and c6 a r e constants. Specifically c, = hL = holdup p e r stage c5 = L/2 c1 = e L Proof will be given later that precisely the same equation governs the behavior of a countercurrent centrifuge, a fractionating column, a thermal-diffusion column, and most other countercurrent devices. In each of these cases c, and c5 have different but analogous meanings, and s is replaced by z , the coordinate along the length of the column. A countercurrent unit i s therefore equivalent to a square cascade of simple elements of the type of Fig. 1.1, and a cascade of countercurrent units i s like .a squared-off cascade. The ensuing discussion

A($) 6

c, dN (c1 + P)N - PNp

(Ng - No) A(+) [(Ns + No) (1 + $) - 2NsNo - 2Np+

^= [PN + c ~ N -( N)] ~

dN PN + c1N(l - N) - c ds

(2.5)

P/C,

[I + 2$(1 - 2 = c1/2c5 5

]

~+ ~

)

(2.6)

The use of <= in Eq. 2.5 agrees with Eq. 2.2; f o r countercurrent colmnns in which the constants c, and c5 a r e fundamental, l i s understood to be defined by 6 = c1/2c5. The dimensionless quantity is thought of as a normalized rate of production. Equation 2.5 gives the distribution in the cascade, for given Np and P,a s a function of s. At the top of the cascade, Ns = No and s = S = number of stages (or length of column, depending on the definition nd units of f ). Equation 2.5 becomes

+

THE THEORY OF ISOTOPE SEPARATION

g=-

1 e A(@)tanh-l

f

33

SQUARE CASCADES

-N~ - (1 + @)e2*^^

(NP - No) A(@) (Np - 2NpNo - No) - (Np - No)@

(2.14)

1 + i/fe2N1++)

N~

If values a r e assigned to Np and @, Eq. 2.7 i s an explicit equatio which determines the necessary number of stages (or length of countercurrent column). If S is given, Eq. 2.7 i s an implicit relatio between Np and @. When P = 0, then 0 = 0, A ( @ )= 1, and Eq. 2.5 becomes -

s

1 e

NS - NO 1 =In RsFt,, Ns -2NsNo + No 2e

= - tanhW1

-

or

ezfS =FRACTIONATION AT NO PRODUCTION

Rs = ~~e~~~

$ = YC,= NORMALIZED

-

Equation 2.9 is in no way peculiar to square cascades, since at P = 0 a l l cascades, regardless of their shape, have the s a m e distribution of concentrations, because Eq. 1.27 with P = 0 i s dN ds

2eN(l - N)

-=

-dR - 26R

ds

-

-

--------------

2

4 6 10'

2

4

6

10"

2

4 6

1

2

4 6

10

N O R M A L I Z E D PRODUCTION

whence Eq. 2.9 follows. However, although the distribution in an ideal cascade with production a t the designed rate i s again exponential (with half the slope of Eq. 2.9), Rs = RoeEs square cascades have the much more complicated distribution of Eq. 2.5. In the special c a s e when both No
s = 2c(l 1+ @ )

No l n N 0 -@(Np - No)

Fig. 2.1-The

effect of production on fractionation.

Referring to Eq. 2.3, since the t e r m s PN + c,N(l - N) increase with s o also must dN/ds. The smallest value of dN/ds is therefore a t foot of the cascade where N i s smallest, N = No. The maximum

@

=

No(l - No)

NP - No F o r this value of @ Eq. 2.7 becomes g=6

A(@)

tanh'll=-

34

Equation 2.7, for a given value of Np , may be satisfied by an infinite range of pairs of values of S and ip, ranging from S = m, ip = N ( 1 - No)/ (Np - No), to S = 1/26 In R ~ / % , $!J = 0. The optimum length of cascade and rate of production a r e that pair of values (S,$!J)which makes the average transport per stage P/S a maximum o r , what is the same thing, makes s/c1ip a minimum. From Eq. 2.7

whose integral is (cf. Eq. 2.5)

where

4s = A(&)

The right-hand side i s now a function of ip alone. Since differentiating Eq. 2.17 gives a transcendental equation, the minimum i s best found by direct computation.

35

SQUARE CASCADES

THE THEORY O F ISOTOPE SEPARATION

(2.20)

pAsc1

= [l +

Us (1 - 2Np) + @

The length of columns in the entire sth stage which changes mater i a l of mole fraction Ns to material of mole fraction Ns+l, while the cascade a s a whole produces material of mole fraction Np a t the r a t e P, is

2. SQUARED-OFF CASCADES (CASCADES OF COUNTERCURRENT ELEMENTS)

In the problem of squaring-off an ideal cascade which was previously considered, i t was possible to use square sections of any des i r e d length1 and breadth, i.e., Z and c5 arbitrary. The usual cascade problem is to make a cascade from elements that a r e identical countercurrent columns o r small-capacity square cascades. Each element has the same constants, cl and c5, and the same length, Z ? Subsequent examples will show that cascades that a r e very close to ideal may be expected, provided the square elements a r e sufficiently small2 Instead of solving the problem just s e t up, i.e., the best Ns with Zs given, i t is much more convenient to proceed indirectly. Suppose that the steps of the cascade, i.e., the Ns, a r e given, and solve for the best lengths Zs to fit this cascade. Then it will be seen how the solution of the second problem solves the desired one. The number of columns in parallel in the sth stage is denoted by vs. The mole fraction fed into the sth stage is Ns , as before. In place of Eq. 2.3, for each of v columns,

vs Zs is a function of <*sand its minimum (e.g., at A, = &) gives the minimum length of column for stage s. From
the number of columns in parallel, and also of each individual column,

Es, the appropriate

length

3. RAPID APPROXIMATE SOLUTION FOR SQUARED-OFF CASCADES

The solution, Eqs. 2.21 to 2.23, takes a simple form for moderate values of Rs+i/Rs. It may be presumed, subject to later check, that the argument of the arctanh is small. Since tanh-I x = x + Ysx3 + , neglecting t e r m s of third and higher order,

...

'Use is made of Z instead of S for the length of the square sub-cascades, retaining s a s the stage number of the grand cas.cade. "'Sufficiently small" is here used to mean the transport per element a cascade net transport, and length of stage a ideal cascade length [qNo(l- No)Ã P(N - No) and z a 1/e ~n RJRJ.

36

THE THEORY OF ISOTOPE SEPARATION

SQUARE CASCADES

which i s a minimum for

A column of length ZS operating with no production will, according to Eq. 2.11, produce an enrichment of e2&. Consequently the theor e m has been proved that a column o r square cascade in the body of a cascade should be operated a t the square rootof i t s maximum f r a c tionation. The foregoing result is really another manifestation of the s i m i l a r relation found for ideal cascades (Chap. 1, Sec. 2). Although the approximate expressions (Eqs. 2.25 and 2.26) a r e fairly inaccurate unless (Rs+l - Rs ) / R ~ is small, experience has shown that the final relation

With this value of As the argument of the arctanh becomes

-

-

F o r any stage of a cascade except the top, a s Rs+l Rs, As Ns (1 - Ns ) / 2 ( ~ p- Ns ) = finite, and the argument 0. But f o r the topmost stage the argument is not s m a l l even for Rp very nearly equal t o Re . F o r , when NS+, = Np,

-

37

and the square-root theorem, which follows f r o m it, both hold for

-

~ s + l / ~ s2.

-

-

-

and, a s Rp R e , A (&.) &., and the argument 1. Thus, except for the last stage of a cascade, the approximations in Eqs. 2.24 and 2.25 will be valid f o r sufficiently s m a l l values of R ~ + ~ / R ~ . Using the value Eq. 2.25 for As, from Eq. 2.24 i s obtained

F r o m Eqs. 2.22 and 2.25,

The recommended rapid procedure f o r finding optimum lengths and numbers of columns when the Ns a r e given i s therefore a s follows: The value of Zs is found from Eq. 2.30. Using this value in Eq. 2.23, (ps i s found. (This gives a better value than Eq. 2.25.) Then Vs may though Zs be found from Eq. 2.22. It should be remarked that even from Eq. 2.30 will be slightly i n e r r o r , and likewise V g , vsZs will be indistinguishable from the t r u e minimum because of the stationary properties of a minimum. The solution to the original problem of finding optimum Ns and us when the Z a r e given follows immediately. The Ns a r e found from Eq. 2.30, and the r e s t of the procedure i s the same a s in the preceding paragraph. The final proof that the procedure outlined gives a satisfactory answer is the comparison of the resulting cascade with an ideal (infinitesimal-step) cascade. As NS+l Ns, from Eq. 2.26,

-

and from Eqs. 2.26 and 2.27,

and Solving Eq. 2.28 f o r R ~ + ~ /leads R ~ to

where V i s the value function of Eq. 1.43. Equation 2.32 shows that, for a square cascade o r column defined by Eq. 2.31, the separative power p e r unit length o r per stage i s

THE THEORY OF ISOTOPE SEPARATION

In the case of the square cascade whose elements a r e of the type shown in Fig. 1.1, where c, = L/2 and c, = EL,Eq. 2.33 reduces to

Chapter 3 which coincides withthe previous expressionfor the separative power of such elements, Eq. 1.42. 4. SQUARING-OFF THE TOP O F A CASCADE

In the problem and example of squaring-off treated in the last two sections, a scheme was adopted which would make the squaring-off uniform over the entire cascade. If the cascade is built up of small standard elements, this represents no difficulty. F o r example, i f the whole cascade is assumed to consist of 10,000 centrifuges, then the top stage i s made of 9 centrifuges, the twelfth of 28, etc. Consider a cascade that is ideal up to a mole fraction NA and then square up to the top. The square section is taken to have the same flow a s the ideal section a t the point of intersection. It is necessary to compare the properties of an ideal cascadefrom NA to Np with that of a square cascade from NA to Np in which the flow is

EQUILIBRIUM TIME OF A SQUARE CASCADE

The equilibrium-time phenomenon s e e m s a t f i r s t glance to be merely an obstacle to be overcome in successful cascade operation. Indeed, it causes serious production losses in large plants by causing delays when operation i s changed, a s well a s in the initial incubation period. Even for small experimental cascades, it prevents rapid attainment of steady-state conditions and thus slows up the accumulation of data and necessitates long continuous operation for each experiment. However, it is possible to use the r a t e of approach to equilibrium a s a powerful tool f o r determining the constants c, and c5 of the fundamental equation (Eq. 2.1), and consequently to turn the disadvantage into an asset. The theory will be expounded below primarily with this end in view. Solutions will be found for those c a s e s which occur experimentally, and the results will be developed with the aid of tables and graphs for immediate application. The fundamental partial differential equation (Eq. 2.1)

The equilibrium time increases about twenty-five times a s rapidly a s the total flow and is usually the factor that decides the number of stages that can be squared-off. The squared-off cascade is shorter than the part of the ideal cascade which it replaces. is quasi-linear and parabolic and can be solved by an iteration method. When N <s: 1, Eq. 2.1 becomes linear, reducing to REFERENCES Cohen, K., Report A-530, Feb. 4, 1943. Cohen, K., Packed Fractionating Columns and the Concentration of Isotopes, J. Chem. Phys., 8: 588 (1940). Cohen, K., Columbia Ser. No. 4L-X28, Dec. 22, 1942 (letter to J. R. Dunning). Cohen, K., and I. Kaplan, Report A-101, Jan. 28, 1942.

Equation 2.1 also becomes linear even for large mole fractions provided the change in mole fractions is small. The substitution N = No + x gives the linear equation

EQUILIBRIUM TIME O F A SQUARE CASCADE

THE THEORY O F ISOTOPE SEPARATION

whose solution closely resembles that of Eq. 3.1. PRODUCTION

RESERVOIR

4

t HOLDUP

1 1

HOLDUP

p-

RESERVOIR TOP

41

column (by convention the "top"). The other end i s maintained a t a constant composition by supplying fresh feed continuously, which is equivalent to connecting the bottom of the columns to an infinite r e s ervoir. There i s usually a certain amount of external holdup a t the product end in connecting piping, surge tanks, etc. The holdup in this top reservoir i s denoted H. In Type I1 operation (Fig. 3.1) no material i s withdrawn. Both ends feed into and out of finite reservoirs. The holdup of the bottom r e s ervoir (at s = 0) i s called H'. Type I operation with P = 0 coincides with Type I1 operation with H' = w . A still more general type of operation (H and H ' finite; P # O), of which both Types I and I1 a r e special cases, is of no physical significance, since the system is gradually emptied of material and there i s no steady state. Before solving Eq. 3.1 it is convenient to divide through by c5, giving

where

+

CONSTANTCOMPOSITION FEED-

WASTE TYPE

Fig. 3.1-Diagrammatic columns.

I

I

I

HOLDUP

H' TYPE

-BOTTOM RESERVOIR

The definitions of 0 and c. a r e repeated f o r convenience. F o r square cascades of elements of the type illustrated in Fig. 1.1, A. = 2h by Eq. 2.2. If s i s dimensionless, A. i s expressed in units of time. In Type I1 operation, = 0 in Eq. 3.2. The boundary conditions for Type I operation are1 J! , I

It

representation of the two main types of experiment on

-

Consider f i r s t the solution of Eq. 3.1, with N ç 1. (For N 1, s e e Appendix B.) There a r e two main types of experiments: In Type I operation (Fig. 3.1) material i s withdrawn continuously from one end of the

'The difference in concentration between the reservoirs and the ends of the column may be neglected if K/X? << 1, which i s almost always the case.

THE THEORY OF ISOTOPE SEPARATION

EQUILIBRIUM TIME O F A SQUARE CASCADE

The boundary conditions under Type I1 operation a r e Ats=O,

aN --2eN=K

Ats=S,

aN -aasN- 2.5N = - K - a t

Â

as

aN at

-

F o r , integrating by parts,

N = No

At t = 0,

In Eqs. 3.4 and 3.5 the factors introduced were

The remaining boundary conditions likewise transform. F o r Type I operation Eq. 3.4 becomes

K =-H c5 H' c5

K ' =-

By f a r the most expeditious method of solving Eq. 3.2 under these boundary conditions i s the use of the Laplace transform. Briefly stated, the method i s a s follows: By the transformation

F o r Type I1 operation Eq. 3.5 becomes

the partial differential equation (Eq. 3.2) i s transformed into an ordinary differential equation for G with p a s a parameter. The solution of the ordinary differential equation gives G(s,p) explicitly. The inv e r s e operation to Eq. 3.7,

The general solution of Eq. 3.9 is

where Ci and C2 a r e constants and ml and m2 a r e the solutions of then gives the required solution. In Eq. 3.8, B r , i s the f i r s t Bromwich contour in the complex p plane, a line from -im to +i-, deformed if necessary to pass to the right of the poles of G(s,p). Equation 3.2 will now be solved according to the above outline. Multiplying Eq. 3.2 by e-Pt dt and integrating from 0 to m gives the equation

The constants Cland C2a r e determined by substituting in Eq. 3.10 f o r Type I operation o r Eq. 3.11 for Type I1 operation. Omitting these details, the results a r e a s follows. Type I Operation:

with Here again b,r = 0 in Type I1 operation. In establishing Eq. 3.9 the boundary condition a t t = 0 has been used to evaluate

EQUILIBRIUM TIME OF A SQUARE CASCADE

THE THEORY OF ISOTOPE SEPARATION

44

where the p j a r e the roots of

Type I1 Operation: No 2eN0ecs sin y s - f(S,p) g(s,p) G(s,p) = - + PD(S,P) P PY

1

+

g(s,p) = ecs (cos y s D(S,P) s

(K' - K) 6

Y

r

~ ( - A1 ) - KP

I

( e - pK) sin yS - cos yS

pK1 + e

-p

=

Type I1 Operation:

where the functions f , g, and D a r e defined by

P

tan yS

(3.16)

Y

sin y s)

~ - A~

(3.17) where the p j a r e the roots of D(S,p) = 0, o r

' sin y~ - (K + K') cos yS

tan yS

and y is the same a s in Eq. 3.15, but with $ = 0,

=

(K + K' )Y E ( K ' - K) - ~ K -KA

and

The inversion of either expression for G(s,p) can be carried out in two ways, giving two different infinite s e r i e s for the same function N(s,t). The difference between the two methods is that one s e r i e s converges rapidly for small values of t, the other for large values. Although either expansion can be used over the entire range of t by carrying enough t e r m s , it i s more convenient to use both expressions, each in i t s appropriate range. It i s possible in this way to find N(s,t) with considerable accuracy for all values of t, using only the first two t e r m s of the series.

1, ~ ( s , t )FOR VALUES O F t rel="nofollow"> h s 2 / l 0

The inverse of G(s,p) suitable for calculations for large t i s the usual Heaviside expansion. It i s obtained by evaluating Eq. 3.8 A s the sum of the residues a t the poles of G. Carrying out the indicated operations, the following i s obtained. Type I Operation:

+s i

2e sin yjs ec(l-irttS-~) p, sin y, S

AS A K ++-[Kpj~-e(l-$)~ 2 2y7

-

AS cos y j s

7

[(K' - K)E - P ~ K K '- A ]

Equations 3.19 and 3.21 describe N a s a function of stage number and time for square cascades (N Ç1). A discussion of the roots of Eqs. 3.20 and 3.22 i s given in Appendix A. It i s shown there that all the roots pj (there a r e an infinite number of them) a r e negative. Thus the t e r m s in the sums a r e transient, and the constant t e r m s represent N(s,m). The second root is shown to be approximately

Since the coefficient of epzt i s much smaller than that of eW, the second transient t e r m i s negligible when o,t < -2; that is, when epjt

+ 1] [Kp-e(1-I))]

(3.19)

46

THE THEORY O F ISOTOPE SEPARATION

EQUILIBRIUM TIME O F A SQUARE CASCADE

Consequently, f o r most purposes only the leading transient t e r m need be considered after t = ASVlO. Equations 3.19 and 3.21 a r e valid both for strippers and rectifiers. When the cascade o r column i s acting a s a stripper, c5, P, and S remain positive, but cl is negative. Consequently e and >/Ã a r e also negative. The same formal solutions hold. It i s quite incorrect to analyze a stripper a s if it were a rectifier of the s a m e length, since the equations a r e not symmetrical about eS = 0. Particular examples of the solutions given by Eqs. 3.19 and 3.21 may be found in the reports listed in the references for this chapter. The most important special case of the two types of experiment, namely K ' = -, P = 0, has been considered in detail. This is the case of a column operated under total reflux (no production), with the concentration at the bottom maintained a t No. The concentration a t the top of the column, o r in the reservoir a t the top, varies with time according to the relation

47

Accordingly Eq. 3.24 takes the form

Tables of the functions A and B for K/AS between 0.0 and 0.5 and for 0<2cS<1.20 a r e given in Table 3.1 and a r e plotted in Figs. 3.2 and 3.3. A wider range of these functions for the special case K = 0 is given in Table 3.2, where values of 2eS ranging from -5.0 to +4.0 a r e covered. 2. ~ ( s , tFOR ) VALUES O F t < X

S

~

Of interest here a r e the inverses of Eqs. 3.14 and 3.16 in the s,t plane near t = 0, that is, the expressions for G(s,p) in the s,p plane f o r p near + W. Dropping t e r m s of the order e-2@small compared to unity, Eqs. 3.14 and 3.16 may be written a s follows. Type I Operation:

neglecting the higher transients. Now Type 11 Operation:

and yl S is the smallest root of

Therefore ylS is a function of the form

6s and K/\S,

and pl may be exhibited in

and s o on. Equations 3.29 and 3.30 hold to within a few per cent provided 0S = 6 s > fi;that is, replacing p by 1/t,

Likewise 2<

AS

The t e r m 6 is defined by Eqs. 3.15 and 3.18. This f o r m is obtained by writing

A

1

(Kp,S - 6 S + 1) (Kp, - e )

= -A(2eS, K/kS)(ezes - 1)

(3.27)

48

EQUILIBRIUM TIME O F A SQUARE CASCADE

THE THEORY O F ISOTOPE SEPARATION

Thus they hold in precisely the range of t where the Heaviside expansion begins to become inconvenient. The inversion of Eqs. 3.29 and 3.30 a s they stand, in all their generality, is difficult and not illuminating. Instead, consideration will 0.98 Table 3.1 -Functions

for the Time Equation

0.96

N(S,t) = e 2 t s - ~ ( ~ 2 6 sÂ-¥ i ) e - ~ t / ~ s NO 2eS = In of steady -state enrichment

0.94

-K- - holdup in reservoir AS

holdup in column

-

'0.92

If)

.4 \

x

K/XS

Fig. 3.2-A

be given various special cases one by one, thereby reducing the complexity of the equations before inversion. For the moment attention should be confined to the mole fractions at the endsof the column (s = 0 and s = S). These a r e usually the only measurable mole fractions anyway, and the restriction greatly simplifies the inversi0ns.l 'The rest of the column will be considered in Sec. 3.

=

HOLDUP IN RESERVOIR HOLDUP IN COLUMN

function for the time equation A(ZeS,K/XS).

F i r s t consider special cases of Type I1 operation. Type 11, Case I: K = K' = 0 (no reservoirs at either end)

49

EQUILIBRIUM TIME OF A SQUARE CASCADE

THE THEORY O F ISOTOPE SEPARATION

Table 3.2-Functions

..,

74

,.- =

HOLDUP IN RESERVOIR HOLDUP IN COLUMN

f o r Approach to Equilibrium of a Square Cascade with No External Holdup

M(s^

3

51

= e 2 ~ S- Ate%s - l l e - ~ t / ~ ~ '

NO Rectifier

Stripper

= LN OF STEADY-STATE ENRICHMENT FACTOR

2eS

Fig. 3.3-A

function for the time equation B(2cS,K/AS).

whence

or

with M (-V2, %; - e2t/X) a confluent hypergeometric function. Values of the function a r e listed in Table 3.3.

tables of inverses, see G. A. Campbell and R. M. Foster, Fourier Integrals f o r Practical Application, Bell Telephone System Monograph B-584 (1931).

whence

Equations 3.32 and 3.33 a r e the desired expressions. Type 11, Case 11: K ' = -, K = 0 (infinite reservoir a t bottom, no reservoir on top)

52

EQUILIBRIUM TIME OF A SQUARE CASCADE

THE THEORY OF ISOTOPE SEPARATION Table 3.3-The

Confluent Hypergeometric Function

M(-Y2,%;

-x)

53

The condition in Eq. 3.36 is more stringent than in Eq. 3.31 if 2eS > 1; but if 2eS 5 1 it i s automatically satisfied when Eq. 3.31 holds. Type 11, Case 111: K ' = W, K = finite; 2e S < 1 (infinite reservoir at bottom, finite reservoir on top; short column)

whence

Type 11, Case IV: K , K ' both finite; 2eS < 1 (finite r e s e r v o i r s a t both ends; short column) whence whence

whence

whence Simple expressions f o r more complicated c a s e s can only be found by further specialization. Appreciable simplification of a l l formulas would be obtained if e 2 with respect to Ap could be neglected. The condition for validity of this procedure to a t least 2.5 per cent i s

54

EQUILIBRIUM TIME O F A SQUARE CASCADE

THE THEORY OF ISOTOPE SEPARATION

Some c a s e s of Type I operation will now be considered. In the most general case,

whence

If 2.5(1+ i^) S < l , s o that e 2 ( l + iA)' with respect to Xp may be neglected, the r e s u l t is precisely the s a m e expressions f o r G a s in Type 11, Case 111. To show the essential difference between operation with production and operation without, the following case is given: Type I, Case 1: k = 0 (no holdup in the column itself)

whence

N(0,t) is given by Eq. 3.42. 3 . DESCRIPTIVE MATERIAL ON RELAXATION PHENOMENA FOR COLUMNS

If the results of the preceding two sections a r e combined, the approach to equilibrium of square cascades over the entire period of operation may be followed. Figures 3.4 and 3.5 illustrate Type 11, Case 11, which i s a cascade of a r b i t r a r y length, with no r e s e r v o i r a t the top and an infinite r e s e r v o i r a t the bottom, operating under total Vi reflux. Plotted in these graphs i s N(s7t) against for No N(S 4. various values of 2eS = In 2 The initial p a r t s of the curves (for NO t/hs2 < 0.1) a r e plotted from Eq. 3.34, and the subsequent sections

(&)

THE THEORY OF ISOTOPE SEPARATION

EQUILIBRIUM TIME OF A SQUARE CASCADE

58

THE THEORY OF ISOTOPE SEPARATION

come from the two-term Heaviside expansion (Eq. 3.28). The two different p a r t s of the curve coalesce quite satisfactorily. The curves a r e straight lines n e a r the origin, the deviation appearing sooner on this s c d e for long columns (large 2eS) than for short columns. Another way of presenting the same r e s u l t s may be obtained by noting'that Eq. 3.34 gives N(S,t)/No a s a function of e2t/A. The expression breaks down a t ~ / A s ^ o . ~ , o r e2t/A= (2&)'/40. Figure 3.6,

EQUILIBRIUM TIME OF A SQUARE CASCADE

K

--

\S

holdup in top r e s e r v o i r holdup in column

At high ratios this gives practically a straight-line graph. (But compare with Figs. 3.4 and 3.5, where the abscissa i s the square root of t/\S2, and a straight line i s found when there i s no holdup on top.) Obviously N increases l e s s and l e s s rapidly a s K increases.

Fig. 3.8-Variation

of enrichment with production. Constants: 2eS = 0.8, S = 100 cm,

A = 2 sec/sq cm, K = 0.

Fig. 3.7-Graph of a short column o r cascade (finite reservoir at top; infinite r e s e r voir at bottom).

N(S,t) -- No

, shows how columns with differing NO 2eS follow the s a m e curve until they break away to approach their equilibrium values. Figure 3.7 gives a graph of Eq. 3.37 (infinite r e s e r v o i r a t the bottom, finite r e s e r v o i r a t the top) for short columns. This i s Type 11, Case 111, o r Type I for 2eS(l + V / ) < 1. Here N(S,t) - No has been (2eS)No plotted against t/AS2 for differing values of the ratio

which gives

In Fig. 3.8 the effect of changing r a t e of production on equilibrium time i s shown. Here 2eS = 0.8, \ = 2 sec/sq cm, S = 100 cm, and K = 0. F o r this graph the solution of Eq. 3.19 was used, keeping two transient t e r m s . Increasing the production r a t e d e c r e a s e s both the eventual enrichment and the relaxation time of the column. A cascade operating a t a lower production r a t e will reach any particular value of the enrichment sooner than one operating a t a higher production rate, except in the very earliest stages of the operation, in which the concentration increase i s independent of 1/1. It is of interest to follow theoretically the development of the concentrations along the entire column, especially since it i s very diffi-

THE THEORY OF ISOTOPE SEPARATION

I

I

I

I

I

EQUILIBRIUM TIME OF A SQUARE CASCADE

61

cult to do s o experimentally. Referring, therefore, to the master equations (Eqs. 3.29 and 3.30), an infinite r e s e r v o i r i s taken a t the bottom, no r e s e r v o i r on top, and 2.5s < 1 (Type 11, Case I1 f o r short columns), and it i s found that

The inverse of Eq. 3.44 i s

erfc

2eS(l - s/S) 4 lGq%-

d$

exp - [ 2 e ~ ( +l s/s)12 16e2t/A

+ 2eS(l + s/S)

erfc

2eS(l - s/S)

4Ve^tA

1

(3.45)

which holds a s long a s &/A < (2e~)'/40. F o r large values of e2t/A Eq. 3.28 may be used with K = 0. Figure 3.9 shows the behavior of such a column whose equilibrium fraction is twofold. Thus 2eS = In 2 = 0.693, and the critical value of e 2 t A i s 0.0125. Here N(s't) is plotted a s a function of s / S for No various values of 4e2t/A. The upper curves (e2t/A 5: 0.0125) were plotted from Eq. 3.28, the lower (e2t/A =s 0.0125) from Eq. 3.45. The two s e t s of curves fit admirably. It should be added that, during the early stages of the experiment, enriched material accumulates only a t the top of the column; when this enriched slug has extended itself down to the foot of the column, the nature of the p r o c e s s changes, and there is a change from one form of asymptotic equation to the other.

REFERENCES

FRACTIONAL LENGTH OF COLUMN

Fig. 3.9-Cascade of fixed length (2eS = In 2) (infinite reservoir at the bottom, no reservoir at the top).

Cohen, K., Report A-530, Feb. 4, 1943. Cohen, K., and I. Kaplan, Report A-195, June 17, 1942; Report A-395, Nov. 19, 1942; Report A-1276, May 1, 1944; Report A-1292, May 29, 1944. Kaplan, I., Columbia Ser. No. 4L-X136, Oct. 12, 1942 (letter t o C. A. Hutchison).

DETERMINATION O F CASCADE CONSTANTS

63

An experiment should be considered in which the withdrawal of product at the top i s balanced by the introduction of an equal amount of feed at the same place (Fig. 4.1). In this way the flows up and down a r e equa1,so that the operation of the unit resembles that of a column in the base stages of a cascade; and the equations a r e appreciably simp1ified.l

Chapter 4

PFg;CT

FEED

DETERMINATION OF CASCADE CONSTANTS

A complete analysis of certain transient phenomena of square cascades was given in the last chapter. This amount of material would not be justified were its purpose purely descriptive. The r e a l need for the detailed results just obtained i s given in this chapter, in which it i s shown how to turn the equilibrium effects into a tool for deter mining the cascade constants. It i s always desirable todetermine the fundamental constants of the steady state, cl and c5, by direct experiment. In some cases, such a s in thermal diffusion, the constants cannot be predicted because of ignorance of the theory. In other cases, such a s centrifuges, in which the constants a r e known functions, subsidiary parameters (viscosity, diffusion coefficients, etc.) a r e not known to sufficient accuracy. In all cases it i s useful to have experimental confirmation that cascade operation follows calculation. This restriction i s mainly responsible for the elegance of the r e sults-the generalization to N 1 will be much more cumbersome. Take the holdup per stage o r per unit length, c , the production, P , and the number of stages, S, a s known; c and c a r e to be determined. Consider f i r s t the evaluation of the constants along conventional lines, i.e., by experiments on the steady state. Equation 2.13, which may be written

-

Fig. 4.1 -An

experiment for the determination of column constants.

Following out the same kind of analysis a s in Chap. 1, Sees. 2 and 6, this case gives, analogous to Eq. 2.3, contains the two parameters e and c , which ostensibly may be determined by any two observations of P and Np. Such a naive procedure i s subject t o considerable experimental e r r o r , and another method, allowing the use of many data, will be presented. his device furnished

by W. I. Thompson.

64

THE THEORY OF ISOTOPE SEPARATION

65

DETERMINATION O F CASCADE CONSTANTS

The solution of Eq. 4.1 i s

which gives directly the separative work. At the same time there i s experimental determination of the optimum production r a t e and fractionation. Next, consideration should be given to the methods based on the equilibrium phenomenon. These a r e chiefly useful in c a s e s where the time to reach complete equilibrium i s of the order of days o r weeks, so that a s e r i e s of steady-state experiments would be excessively time -consuming. T o illustrate the use of relaxation phenomena for determining cl and c5, consider an experiment of Type I1,Case 111 (K' = m, K = finite, 2eS < I).' The pertinent equation for short t i m e s i s (Eq. 3.37)

(compare with Eq. 2.14). Now, s e t up f o r this system the function

which i s a generalization of the work function of Eq. 1.60, defined for infinitesimal concentration changes. The function i s plotted against P, from the experimental values of N p / ~ oand P. The maximum value of W i s a t

1N(S t) + NO

4

2cK e -" (exp

$ erfc ^-y- - 1)

( )

= l + % ( ~ Y + 2 e s -

6

xs2

XS

[ (yy AS^

(

exp - - e r f c AS t\^_ K AS

H, the holdup in the top r e s e r v o i r , i s known, a s well a s c6. Thus XS/K = c6S/H i s known, but e, \, and K individually a r e unknown. Equation 3.37 holds until t/& = 0.1. Unless XS/K Ã 1 (holdup in column much greater than holdup in r e s e r v o i r ) , the argument of the e r r o r function

and the maximum is

?h 3 (L )

It is worth noting that at P =

K

AS2

will be small a s long as the equation is valid. Expanding the e r r o r function in ascending powers of the argument, Eq. 3.37 behaveslike The values of

and

W

a r e chosen f r o m the curve. Then c1 = 2 V P W

and

F o r small enrichments, eZEs= 1 + 2 4 and the maximum of W is

(4.7)

-

0. The deviation from this formula becomes l e s s and l e s s a s t On the other hand, should the holdup in the r e s e r v o i r be negligible (e.g., XS/K > l o ) , then unless t/AS2 i s very small the argument of the e r f c will be large,

\ S l t \ " K ( W - / i S f t erfc - - - exp - - A S rt K xS2 AS2

he sample size i s supposed to be negligibly small.

p ~ :i-sr_*-

%, 9

%-

66

THE THEORY OF ISOTOPE SEPARATION

and Eq. 3.37 behaves like

DETERMINATION OF CASCADE CONSTANTS

67

As an example of how simple the whole procedure can be made, take the special case K = 0. From a graph of N(S,t)/No against t , the equilibrium fractionation may be found, along with tv,, the time when

over a considerable region. Depending: on which i s the appropriate case, N's9t' - 1 i s plotted No against t or t%. The slope of the straight-line portion of the curve is found, which i s either 2e/K = ci/H, o r 4e/* = 2 c 1 / i G . In either case a relation for the desired constants is obtained. The data for long times, which a r e more abundant, a r e worked up in quite a different manner. The pertinent equation i s (Eq. 3.28)

Here the procedure i s to find e2" by extrapolation and then to plot 1

[ e 2 â‚

No] against t. No

If the initial choice of e2" is good,this

graph will be a straight line1 whose intercept i s In [ ~ ( e -~I)]. ' ~ (A is found from Fig. 3.2.) If this i s not the case, a slightly different choice of ezCs must be made. From the straight line finally obtained, pl i s found a s the slope. A consistent set of values of 2eS and p, i s thus found. From the former, c1/c5 = 2e. From pi and the relation

The nomograph (Fig. 4.2) then gives c,.c5/c2 on the right-hand line when a straightedge i s laid against N(S,-)/Noon the left-hand line and ti/, on the center. 1. IDEAL CASCADE WITH LOSSES

So f a r , cascades, both ideal and otherwise, in which no losses of material occur have been considered. In actual cascades, sources of loss a r e always present. In most chemical processes the effect of small losses i s not serious. It is obvious that, in a given cascade, a s the losses increase the production diminishes, so that a cascade which has losses will have to be bigger to produce the same quantity and quality of material. Consider the problem of constructing a no-mixing cascade that will accommodate losses. To solve it a different procedure1 from that of Chap. 1 should be employed. It i s necessary to introduce the following notation (see Fig. 4.3): Ls = esLsN; = flow of desired isotope i n enriched fraction Me = (1 - es)LsN? = flow of desired isotope in stripped fraction

(4.12)

ZS = Os L s ( l - NL) = flow of other isotope in enriched fraction

ms where B is found from Fig. 3.3, cs is determined. The weak point of this procedure is the choice of e2cs,the equilibrium fractionation, which i s not always unambiguous. However, the value of the separative power of the cascade per stage, which i s the really important property of the cascade for design purposes,

i s unaffected by considerable e r r o r s in 26s when determined in this manner. 'Except, of course, for small values of t.

= (1 - & ) L s ( l

- Ni') = flow of other isotope in stripped fraction

From the conservation of matter, Ls + Me

=

LsNs = flow of desired isotope into the separator

ls + me = Ls(l - Ns) = flow of other isotope into the separator

(4.13)

THE THEORY OF ISOTOPE SEPARATION

DETERMINATION OF CASCADE CONSTANTS

ENRICHED FRACTION

t

LsMOLES I s MOLES

OF DESIRED ISOTOPE OF OTHER ISOTOPE

STRIPPED FRACTION

Me MOLES ms

MOLES OF DESIRED ISOTOPE

FEED

< ( L ~ +M ~ ) ( l s tmsl

OF DESIRED ISOTOPE MOLES OF OTHER ISOTOPE

MOLES OF DESIRED ISOTOPE MOLES OF OTHER ISOTOPE

Fig. 4.3-Diagramof a cascade with losses. Only the desired isotope flows areshown. Flows of the other isotope are obtained by using lower-case rather than capital italic letters.

The no-mixing condition leads to the analogues of Eqs. 1.3 to 1.5, the most convenient of which i s

From Eqs. 4.14 and 4.15, eliminating rns/L, Ls = a M s Now the cascade i s connected in such a way (Fig. 4.3) that Ls + Ms Fig. 4.2 -Nomograph

for determining column constants.

=

Ls.l + M g + 1 - losses of desired isotope

(4.17)

The losses per stage a r e often proportional to the flow in the stage. Thus the losses from the sth stage of a cascade a r e set a s yLs,where y i s a constant and Ls i s a s usual the flow entering the stage. Equation 4.17 becomes

DETERMINATION O F CASCADE CONSTANTS

THE THEORY O F ISOTOPE SEPARATION

71

o r , introducing Eq. 4.16, &+I

- (a + 1) (1

+ y) Ls + G - 1 = 0

(4.19)

Equation 4.19 i s a second-order difference equation for Lg.The general solution i s

That
where u1 and u2 a r e the roots of the characteristic equation

^ - ( a + 1) ( 1 + y ) u + a = o and can be expressed a s with A defined a s

The boundary conditions a r e

Using the form of Eq. 4.27 to simplify Ls and Is,

Solving for A and B,

- 2 P N p ( f f + 1 ) (1 + A)a(s-s-1)/2 sinh ( S [ a ( l + A)' - l ] N s

Now from Eqs. 4.14 and 4.15,

while Eq. 4.18 holds with Z and m substituted for L and M. Therefore Z S i s the same function of l / a that L s i s of a , and in place of Eq. 4.23 the result i s

+ 1 - s) In

(1 + A)

-/a (4.28)

This i s the required flow per stage with a fractional loss per stage y. The total flow in a cascade i s obtained by summing the Ls with r e spect to s from 0 to S. This gives

72

THE THEORY O F ISOTOPE SEPARATION

DETERMINATION O F CASCADE CONSTANTS

-

As before, Rs = o^Ro and R p = crS+lRo. When A 0, Eqs. 4.28 and 4.29 reduce to the familiar expressions of Eqs. 1.11 and 1.13. Equations 4.28 and 4.29 take more easily understandable forms if a = 1 + e. For then

73

The above results a r e quite sensitive to the mole fraction produced by the cascade. Figure 4.5 shows that value of y/e2 which demands a compensatory increase of 20 per cent in the total flow of the cascade a s a function of Np. A cascade producing a concentration of 10 per cent can afford more than five times the losses of one producing 90

It is found convenient to define an auxiliary symbol 6 by

From Eqs. 4.30 and 4.31, place of Eq. 4.28,

6 = A/â‚

4PNp Ls = Ns e ( l + 26) exp [$(s-s)]

Using these relations gives, in

sinh

[i( S

- s) (1

+ 2611

(4.32)

and for Eq. 4.29,

The relative increase in size of a stage caused by the loss of material i s given by the ratio

Fig. 4.4-Curve showing the effect of losses on total flow (the loss per stage i s proportional to the flow per stage). N p = 0.90, y = fraction of flow lost per stage, e = enrichment per stage.

which i s a function of 6. Since 6 i s in turn a function of y/e2 = yL/Le2, it i s seen that the increase in size of a stage i s a function of the ratio of l o s s to separative work. The bottom stages a r e enlarged propor tionally more than the upper stages. The increase in total flow for a (rectifying) cascade i s found by comparing Eq. 4.33 with Eq. 1.23, which i s its limit for y = 0. Figure 4.4 shows the percentage increase in total flow a s a function of y/e2 for the standard case. It will be seen that a 20 per cent increase in size of the cascade i s required when y = 0.025 e 2 .

per cent. Intuitively it should be understood that this effect is due to the increasing penalty for losing high-concentration material on which much work has been done. Obviously it i s worth while to take considerable pains to avoid losses in the small upper stages of the cascade. It may well turn out to be economical to use expensive methods and materials in the upper stages if by so doing losses can be cut and thus the amount of overbuilding required in the bulk of the cascade can be diminished.

THE THEORY O F ISOTOPE SEPARATION

DETERMINATION OF CASCADE CONSTANTS

The effect of losses on a stripping section i s somewhat different. The usual objective i s not that of producing a given quantity of waste, but rather that of economizing feed. A loss in stage s of the stripper means that a small amount of material i s stripped down to Ns and not

The amount of material transferred between two cascades depends on the mole fraction of the return stream from the upper cascade a s follows: Let PC and Nc be the rate of production and mole fraction of the product from the upper cascade, respectively. Let PB and NB be

74

75

MIXI N G

F I N A L M O L E FRACTION FOR WHICH P L A N T WAS DESIGNED

Fig. 4.5-Loss of process material resulting in 20 per cent increase in total flow (the loss per stage i s proportional to the flow per stage). The symbol e denotes the enrichment per stage, and y represents the fraction of flow lost per stage.

N w . The result i s to decrease the size of the stripper and increase the required feed. This problem will not be pursued any further here. The problem solved above (loss proportional to flow) is about the only one that admits a closed analytical solution. But, since in practice the only cases of interest a r e those in which the effect of losses is small, iteration methods a r e reasonably successful in dealing with more complicated conditions (see References).

Y

LOWER CASCADE

Fig. 4.6-Diagram

showing the combination of cascades.

2. COMBINATION O F PLANTS USING DIFFERENT SEPARATION PROCESSES

In this section consideration will be given the connections between two plants with different separation processes, giving expressions with whose aid the factors of plant size and amount of material transferred between the plants may be balanced.

the corresponding quantities for the lower cascade. Pn i s at the same time the feed to the upper cascade. Pp and NÃ a r e the waste stream and its concentration from the upper cascade; Pp i s the return stream to the lower cascade (refer to Fig. 4.6).

76

THE THEORY OF ISOTOPE SEPARATION

The conservation of matter gives the relations

77

DETERMINATION O F CASCADE CONSTANTS

Furthermore the return stream to cascade A i s made up of two parts so that

Substituting f o r lA- lBfrom Eq. 4.41, From Eqs. 4.35 and 4.36, and rearranging, LANA- lAnA= LBNA- lBnB- P-N

C A

In these two expressions, P C and Nc a r e given (they a r e the specifications on the final product), and No i s the mole fraction at which the change from the lower to the upper cascade is made. NA, the mole fraction of the return stream, i s the only variable. If the upper cascade is an ideal rectifier, Eq. 1.5 gives

For (a - 1) small, (Nn - NA) is very small, and PB and Pc a r e very large. Thus in general it will be advisable to add a stripper to the upper cascade, a s indicated in Fig. 4.6. The connections of the return stream, between p a r t s A and B of the lower cascade, a r e also shown. The upper cascade consists of an ordinary rectifier -stripper combination. Cascade B i s clearly a rectifier, receiving a feed of mater i a l at mole fraction NA and producing material of mole fraction NB. From the conservation of matter

= PBNB - P- N C

A

Equations 4.42 and 4.43 show that cascade A is just the part up to NA of a cascade that produces material (Pc,Nc). Expressions for the flows and holdups in these sections may easily be found on the assumption that they a r e ideal cascades. The total flow in cascade section A is

i f 1 + â ‚ is the simple-process factor in the lower cascade. The total flow in cascade section B is

The total flow in cascade section C is

The conservation of matter in the reservoir (actually i t s size i s immaterial) requires that

where 1 + ec is the simple-process factor in the upper cascade. The total flow in the stripper section A is

78

THE THEORY OF ISOTOPE SEPARATION

and in the stripper section C is

Analogous expressions hold for the finite case. Expressions for the desired material holdup and the contribution to the equilibrium time of the various sections above the feed point can also be obtained without any trouble (see Appendix C). Cascades A, A, and C a r e the same a s the corresponding sections of a single ideal cascade producing (Pc,Nc). But it will b e observed that cascade B is not the same as the section between NA and NB of a single cascade. B and C together a r e equivalent to (and slightly larger than) this section. An inefficiency in the combined cascades caused by the mixing of the constituents of the return stream to cascade A causes a l o s s in value of

e.

which is made up for by the increased size of B and It will be remarked that a combined cascade, for maximum efficiency, i s considerably different from two cascades designed for other duties and connected together. There is a considerable premium on foresight in this problem. The separation into two cascades which i s illustrated here can clearly be carried on further, giving any number of sub-cascades. The chief feature which makes continued subdivision attractive, even when the sub-cascades use identical equipment, i s that the flow between cascades (each of which i s a rectifier-stripper) i s much l e s s than the flow between stages of a single cascade (cf. Eqs. 4.37 and 4.38). This is an advantage in removing accumulated impurities (caused by inleaks, decomposition, etc.) o r in control.

REFERENCES Cohen, K., Report A-60, October 1941; Report A-530, Feb. 4, 1943; Report A-781, Aug. 13, 1943. Cohen, K., Columbia Ser. No. 4R-X143, May 13, 1942. Cohen, K., and W. I. Thompson, Columbia Ser. No. 4R-104, Sept. 4, 1943. Kaplan, I., Columbia Ser. No. 4M-133, Jan. 18, 1944 (memorandum to K. Cohen); Columbia Ser. No. 4M-160, Apr. 17, 1944 (memorandum to M. Kilpatrick). Lefkowitz, H., and B. Schwartz, Columbia Ser. No. 4R-18, Feb. 4, 1943. Lefkowitz, H . , B. Schwartz, and H. Mayer, Columbia Ser. No. 4R-42, Apr. 12, 1943. Mayer, H., I. Kaplan, and K. Cohen, Report A-396, Dec. 2, 1942. Thompson, W. I., Experimental Operation of a Single Centrifuge, S.O.D. Report, June 14, 1943.

Chapter 5 THE CONTROL PROBLEM

The problem of maintaining certain specified variables a t preassigned values is common to any planned operation. In a chemical reaction, for example, these quantities a r e usually the temperature, the pressure, and the amount of material in process. The control problem for isotope-separation plants differs f r o m that of other manufacturing operations in the degree of complexity. The cascades considered in this volume contain thousands of different interdependent stages or elements. Methods of analysis which a r e successful with systems containing, for example, a dozen related units will fail here. F o r this reason the control problem for isotope-separation cascades must be treated a s a distinct subject. Furthermore it is often possible to break up a cascade into smaller units. For example, a cascade of 10,000 centrifuges with nine centrifuges in the top stage (Chap. 2, Sec. 4) might be broken up into nine completely independent cascades in parallel, o r a cascade might be broken down into sub-cascades along the lines discussed in Chap. 4, Sec. 2. The desirability of such expedients will not be clear until the difficulties of controlling a single large cascade a r e known. In the discussion that follows, therefore, consideration will be given to a single large rectifier -stripper. It was shown in Chap. 1, Sec. 2, that for an ideal plant deviations up to 20 per cent from the calculated flows a r e without appreciable effect on the separation. This also holds true to a considerable extent for nonideal plants of intelligent design. Thus to have a cascade operate according to plan, it is only necessary to s e e that the flows throughout the plant a r e approximately correct. This question, of maintaining the temperatures, pressures, flows, etc., near their design values, of seeing that material does not accumulate in o r drain out of any part of the plant, and of seeing that small distrubances do not build up to large ones, is one part -the hydrodynamic part-of the control problem.

80

THE THEORY OF ISOTOPE SEPARATION

There is in addition a second part which deals not with the average conditions but with the effects of fluctuations from the average. A slight change in the operating conditions may produce a variety of effects: a change in the transport of material from stage to stage, a mixing of material of different concentrations from different stages, o r a change in the separating power of the element. In general, a fluctuation causes mixing, and mixing means a lowering of the concentration of the product. Both parts of the control problem a r e essential. Without the hydrodynamic part, the types, magnitudes, and frequencies of variations that a r e to be expected a r e unknown. Without a knowledge of the effects of fluctuations on concentration an evaluation cannot be made of the effectiveness of the hydrodynamic control measures s o far a s plant productivity is concerned. By i t s nature, the hydrodynamic problem is intimately associated with the particular separation method, equipment, and control instruments used. Since this problem cannot be generalized without destroying its illustrative value, an explanatory example will be presented. The problem of the effect of fluctuations on separation is more susceptible of general treatment. Even s o there a r e aspects which differ from one case to another. A cascade of centrifuges, for instance, has certain problems in common with all squared-off cascades; i t also has effects due to variations in the behavior of parallel units which a diffusion cascade lacks. But the theory developed in the succeeding sections is basic to all the problems. In Sec. 1 consideration is given to equations governing the change of mole fraction with fluctuations in the operating constants. Their solution in the general case is developed in such a way that the results of the hydrodynamic problem may be used in their most compact form (in the s,p plane of the Laplace transform). In Sec. 2 the average change in transport for periodic fluctuations is derived. Section 3 treats a particular example: the effect of variations in the rate of withdrawal of product on a square cascade. Sinusoidal variations a r e considered, and the effect of batch (instead of continuous) withdrawal of product is treated.

THE CONTROL PROBLEM

81

The effect of perturbations in the fundamental quantities cl, c5, c6, and P on the separation performed by the cascade is found by solving Eqs. 5.1 and 5.2 with these four quantities preassigned functions of time. Of primary interest a r e the small fluctuations, which if imposed on a system already in the steady state will give small changes in concentration. Gross upsets and phenomena associated with the approach to equilibrium a r e excluded from consideration. The desired solution may thus be obtained by ordinary perturbation theory, with the steady state a s the unperturbed system. The quantities c,, c5, and c6 a r e given a s functions of stage number. In cascades with more than one element per stage, they a r e to be understood a s averages over the elements. The mixing losses caused by deviations of parallel elements from their average a r e not included here. Suppose that there is a primary disturbance of some nature from an outside source. It will be written a s

The parameter u is not necessarily small-its function is to distinguish between the various o r d e r s of t e r m s resulting from the primary disturbance. As a consequence of of there will be induced disturbances in cl, c5, c6, and P. Thus c,, which before the disturbance is a function of s only, becomes a function of s , t, and u. Expanding in powers of a,

1. E F F E C T O F FLUCTUATIONS ON CONCENTRATION

The fundamental equations of isotope separation a s developed in Chap. 1, Sec. 6, using the generalized notations of Chap. 2, are1 The terms c,, cc, and c. were introduced in Chap. 2, Square Cascades, by the schedule in Eq. 2.2,but there i s no reason why they cannot be employed a s functions of s and t, a s indicated.

where cm(s) is the undisturbed value of c,. Generally the symbol c1(s,t;u) will be shortened to cl(s,t). Likewise

82

THE CONTROLPROBLEM

THE THEORY O F ISOTOPE SEPARATION

The notation is self-explanatory and defines the coefficients c5,, cS2, etc. As a result of Eq. 5.5 and the f i r s t line of Eq. 5.2,

.

The functions c,(s,t) . . T(s,t) a r e considered known, a s well a s the coefficients of their expansions. They a r e in fact the solutions of the hydrodynamic part of the control problem. Now, inserting Eqs. 5.4, 5.5, and 5.6 in Eqs. 5.1 and 5.2, it is desirable to find N(s,t) and r(s,t). These will also be functions of a, which will be written asi N(s ,t)

=

T(s,~)=

No(s) + aNl (S ,t) + a2N2(s ,t) + . . . To

(5.7)

+ C T T ~ ( S+ o, ~' T) ~ ( s , ~ )+ . . -

Of course

where N p is the undisturbed mole fraction of the product. The convergence of Eq. 5.7 is now of concern. The natural period of a cascade with respect to concentration disturbances is the equilibrium time, which we have seen in Chap. 3, Sec. 1, is of the order c.-n~50/cfo.Physically this means that Eq. 5.7 will converge \/e2 provided the average values of the disturbances on most of the stages, over periods of time smaller than \/e2, a r e small compared to the undisturbed quantities. Introducing the expansions of Eqs. 5.4 to 5.7 into Eqs. 5.1 and 5.2, , the following sysand separating the coefficients of uO, ul, 02, tems of equations a r e obtained:

and s o on for higher orders. The boundary conditions for each order equation can likewise be found by expansion. This will be done later in the section (Eqs. 5.23 e t seq.). When the boundary conditions a r e known the equations may be solved successively. In the paragraphs immediately following an investigation will be made of the properties of the differential equations, and method of solving them will be considered. Equation 5.9 gives the steady state of the unperturbed system. The expression for T , has the form

where D is the linear operator, independent of t,

-

and

...

,- cioNo(l- No)

PoNpo = PoNo(s)- Cgo aNo s

(5.9)

+

i s , after substituting No(s), which is a solution to Eq. 5.9, an explicit function of s and t. Combining Eq. 5.12 with the second equation of Eq. 5.10 gives

a

+

TPo

-

'51

9No

'60

+

cllNo(l

-

No) (5.10)

..

'The symbols N,,(s), N,(s,t) . introduced in Eq. 5.7 a r e not in contradiction with the previously used symbols of Nc, which have been supplanted by N(s,t) and have no

aQ

at a N l + -a D s Nl=-+Nas

o

" a1 s

(5.15)

The left-hand side of Eq. 5.15 is a linear homogeneous expression in N17with coefficients independent of t; the right-hand side is an explicit function of s and t. It is therefore easily solvable by standard methods. In particular the Laplace transform of Eq. 5.15,

84

THE THEORY O F ISOTOPE SEPARATION

is an ordinary linear second-order differential equation in s. It be remembered that the asterisk (*) was introduced to denote Laplace transform of any function. Thus if x(s,t) is a function and t,

The chief advantage1 of using the Laplace transform of Eq. 5.15besides the fact that i t solves the equation without trouble in spite complicated boundary conditions -is that the transforms TT , cz , a c * a r e used on the right-hand side. Since the hydrodynamic par the control problem will also be solved by using the Laplace t r a form, these quantities a r e immediately available. In fact, s i the inversion of T* to TI, etc., i s the last and hardest step of hydrodynamic solution, very considerable difficulties a r e avoi by using Eq. 5.16. The solution of Eq. 5.15 or 5.16 may be facilitated by transforming the dependent and independent variable^.^ The expression DNl becomes, using the integrating factor,

[ Ls

J ( s ) = exp -

'lo('

'50

+ 0'

ds]

The dependent variable is then chosen a s JN,. In many cascades the quantities clo, cco, etc., a r e most conveniently expressed in terms of the No. For example, an ideal cascade is defined in terms of the No so that c

-i-^- =

26 = constant

'50

This suggests changing the independent variable from s to No, which can be done with the aid of Eq. 5.9. The expression for T, has the form

85

THE CONTROL PROBLEM

where D i s the linear operator of Eq. 5.13 and

Combining Eq. 5.19 with the second part of Eq. 5.11,

The Laplace transform of Eq. 5.21 is1

The process can, a t least in principle, be continued indefinitely f o r NS, N4 . . . . The boundary conditions for the differential equations will now be obtained. First of all the initial condition N(s,O) = No(s) leads to

Assume that there a r e reservoirs a t each end of the plant, of holdup H (product end) and H' (waste end). The product and waste a r e withdrawn from the reservoirs. The mole fractions in the reservoirs a r e therefore called Np(t) and Nw(t), respectively. i t is not legitimate to neglect the difference between Np and N(S,t) . A rapid fluctuation in N(S,t), for example, leaves Np relatively unchanged because of the buffer effect of the holdup.] The waste stream is called W(t). The flow from the cascade into the top reservoir i s called c5(P,t);the flow from the cascade into the bottom reservoir is c5(W,t). These quantities also have expansions a s powers of o.

e ere

'The formula for the transform of the product of two functions i s 1 [x1(t)x;(t)]* = %/ x: (P - q)x;* (q) dq 'This feature was pointed out by H. Mayer. 2Following the practice of K. Fuchs.

Br,

The contour i s to the right of the poles of x;*(q) and to the left of the poles of x* (p - q).

THE THEORY O F ISOTOPE SEPARATION

THE CONTROL PROBLEM

.. . + . ..

+

H = Ho

+ 0% (1) + u ~ , ( t ) + u2H2(t)

H1=H;

+uH'(t)

+... + ...

W(t) = Wo

Np(t) = Npo

+ a w l (t)

+u2Hn(t)

+ uNpl (t) + o w 2(t)

Nw(t) = Nwo + uNwl(t)

87

The boundary conditions a t the top of the rectifier a r e obtained from the relations

(5.24)

+ cr2Nw2(t) + . . .

c5(P,t) = c ~ ~ ( +P uc5,(P,t) ) + u ~ c , ~ ( P +, ~.)

. c5(W,t) = cso(W)+ ucgl(W,t) + U ~ C ~ ~ + (W . . ,. ~ ) a

The mole fractions, N(s,t), and the transports, T(s,t) and ~ ( s , t ) , will r e f e r to the stripper if s has negative values. Equations 5.9 to 5.22 hold for the stripper if Po and N p a r e replaced throughout by -Wo and N W . The stage number of the lowest stage of the stripper will be called "B" (a negative number). The discontinuity in T and T a t s = 0 implied in the change in the equations is accounted for by the introduction of feed (F,Np).

On expanding in powers of o and collecting, it i s found that

These obvious relations have already been incorporated in the preceding equations (cf. Eqs. 5.6 and 5.8). Likewise, combining the f i r s t two equations of Eq. 5.29,

Note, however, that Np does not have to be the same a s N(O,t), o r even N F the s a m e a s No(0). F and Np may both vary with time (the cascade under consideration might be theupper cascade of a combination of plants). Expanding Eq. 5.25 in powers of u,

The boundary conditions a t the bottom of the stripper a r e very similar. Corresponding to Eq. 5.29,

88

THE CONTROL PROBLEM

THE THEORY OF ISOTOPE SEPARATION

In z e r o order

F r o m Eq. 5.31, The first-order t e r m s a r e

and the second-order t e r m s a r e

and from Eq. 5.35,

This elaborate s e t of conditions is very expeditiously condensed by using the Laplace transform. With the aid of Eq. 5.23, Substituting for 7, from Eq. 5.12 in the boundary conditions (Eqs. 5.27, 5.31, and 5.35), a complete s e t of boundary conditions may be found for the first-order differential equation (Eq. 5.25). F i r s t a n expression for

and therefore N: and N: may be eliminated altogether by the second p a r t s of Eqs. 5.39 and 5.40, which give

is needed. This can be obtained from Eq. 5.26 by substituting Eq. 5.9 and its analogue with -Wo for Po and N W for N p . It is found that1

with the definitions Then from Eq. 5.27, for the boundary condition at s = 0,

I N p - No(0) i s found by substituting the solution of Eq. 5.9, giving N p in terms of

N,(O), and N,,, in terms of No(0), into Eq. 5.26

a s in Chap. 3. With Eq. 5.41, the transforms of Eqs. 5.39 and 5.40 a r e

90

THE THEORY O F ISOTOPE SEPARATION

The transform of Eq. 5.38 i s

Similar expressions can be found for the boundary conditions of the higher-order differential equations. It must seem to the reader that the work required has been increased by carrying the intermediate quantity r all through the expansions and then eliminating it at the last step, instead of eliminating i t at once from Eqs. 5.1, 5.25, 5.29, and 5.33. The procedure was adopted to prepare for the work of the next section, where i t s usefulness will become apparent. With this cataloguing of the boundary conditions all the development that can be made without special assumptions has been completed. The treatment has been more detailed than the novelty of the content strictly merits; i t i s hoped that i t s reference value will make up for the difference.

THE CONTROL PROBLEM

91

answer. Usually1 the leading t e r m in the answer is the average second-order change in concentration o r in transport. Of course the average second-order quantities can be found by solving straightforwardly for N,(s,t), a s indicated in Sec. 1, and then averaging. But this procedure requires considerable work to get superfluous detail; it is possible to obtain2 T Z and Nz(s) directly without solving the second-order differential equation. The method for doing this, which was devised by K. Fuchs, follows. Suppose that the solution N,(s,t) o r N*(s,p) of the equation of the f i r s t order in u has been determined. Taking the average of Eq. 5.19 and using the relation Eq. 5.18 for D,

F r o m Eq. 5.2, a ~ / &= - a(c,.~)/at.The right-hand side must be zero; otherwise there i s no steady state. Therefore T i s constant, except for a discontinuity a t s = 0, and s o a r e all o r d e r s of i t s expansion, including G. The solution of the problem r e s t s on this property of r Z , which permits i t s evaluation f r o m the boundary condi4' .ions. Equation 5.46 integrates out to give, for the rectifier,

and, for the stripper (negative values of s ) ~

2. AVERAGE TRANSPORT LOSS FOR PERIODIC FLUCTUATIONS

There a r e two aspects of the effect of perturbations which a r e i m portant. The f i r s t is to predict the concentration changes in detail, s o that the success of corrective m e a s u r e s can be verified. The second i s to find the l o s s of production over long periods of time. If the primary disturbance uf(s,t) is nonperiodic, like a simple pulse o r a damped oscillation, both aspects a r e of interest. The leading t e r m in the answer is N,(s,t), but often N2(s,t) is also required to give a correct evaluation of the production loss. If the primary disturbance i s periodic, a new average steady state, with concentrations lower than before, will result. In this c a s e we a r e more interested in the new steady state-the l o s s of production and average change in concentrations-than in the full detail of the

'It may be supposed without loss of generality that af(s,t) has a vanishing time a v e r age. It will then usually be t r u e that the first-order t e r m s in the fundamental quantities c,, cs, co, P . have vanishing time averages:

..

-

Should this b e the case, then N,(s,t) = T , = 0, for, on averaging Eq. 5.10 and the bounda r y conditions (Eqs. 5.27, 5.31, and 5.35), i t will be found that % satisfies a n equation almost exactly like Eq. 5.9, but with the feed concentration z e r o instead of NFo- Note, however, that in the solution to b e given for N3 and T2 we shall not assume that c,,(s,t), etc., and hence q , equals zero. 2 T h e time average of x(s,t) will be written a s X(s) instead of the cumbersome x(s,t). The average of the product xl(s,t)x,(s,t) will, however, be written a s x,(s,t)x,(s,t).

92

THE THEORY O F ISOTOPE SEPARATION

THE CONTROLPROBLEM

93

- These linear relations will be shown, but the process of eliminating

It will be recalled that (Eq. 5.17)

N2(S) and N,(B) and solving for the % will not be undertaken here.

In the second equation of Eq. 5.32,

for the rectifier. J' has been used to denote the corresponding function for the stripper.

2) 1 [ lS ~ l O (l

J' (s) = exp -

- Wo dS

It now remains only to evaluate the a s yet unknown constants T2(S) and T(B) from the boundary conditions, and the problem is solved. From the boundary conditions of Eq. 5.28, at s = 0, on averaging,

o r else N p (t) increases o r decreases indefinitely. Thus on averaging i t is found that

The f i r s t equation of Eq. 5.32 gives

where A i s a known constant. Also Eliminating Np2 between Eqs. 5.53 and 5.54 gives N2(S) in t e r m s of

-

N2(0+)

Now J(0)

=

= N2(0-)

T , (S), a s required. Similarly, from the boundary condition (Eq. 5.36)

at s

=

B,

Jr(0) = 1. Thus with the aid of Eq. 5.51, (JN2Io = (J'N,).

Setting s

=

S in Eq. 5.47 and s

=

B in Eq. 5.48 and subtracting,

It should be understood that J in the last integral is given by Eq. 5.49 from B to 0 and by Eq. 5.17 from 0 to S. Now the boundary conditions (Eq. 5.32) a t the top of the rectifier give a linear relation between N2(S) and T2(S); those a t the bottom of the stripper (Eq. 5.36) give another between N,(B) and T,(B). Using ) Eq. 5.52, a linear relation these to eliminate &(s) and N 2 ( ~from between ~ ~ ( and 8 ) T ~ ( Bis ) obtained, which, together with Eq. 5.52, allows the determination of both quantities.

For many purposes Tz(S), which is the average (second-order) change in production of desired material, is a sufficient answer, and it is unnecessary to return to Eqs. 5.47 and 5.48 to find N2(s). The solution just given requires the averaging of some secondorder quantities, and of products of first-order quantities. If these functions a r e known explicitly in t e r m s of t, the averaging is done by elementary integration. However, they will usually be given a s Laplace transforms. To perform the averaging without taking the inverse the following theorem: which holds if a steady state exists, can be used: If lim px* = C(s) as p 0 by positive r e a l values, then X(s) = C(s); otherwise ~ ( s =) 0. ->

'

G. Doetsch, "Theorie und Anwendung der Laplace Transformation," p. 193, Dover Publications, New York, 1943.

THE THEORY O F ISOTOPE SEPARATION

3 . VARIATIONS IN PRODUCTION RATE FOR A SQUARE CASCADE

In this section the theory of the preceding sections will be illustrated and clarified by applying it to a special case. The abstractness of the demonstrations just given will be relieved by discussing further properties of the solutions with a particular example. The example chosen i s that of a square cascade-practical comparison may be made with a fractionating column-on which a r e imposed a r bitrary variations in P , the rate of production. Suppose that the hydrodynamics of the cascade a r e such that a t the same time everything else is invariant: the holdups, the flows and separation factors, and the feed rate and composition. To simplify the arithmetic a rectifier alone will be considered, with the feed concentration held fixed. It is further supposed that N i s always small. A reservoir i s included at the top. The t e r m s cm, ego, and ego do not vary with stage number. Let cc0(P) = cHO(flow into reservoir = flow between stages). Introducing the s a m e notation a s in Chap. 3,

THE CONTROL PROBLEM

95

The boundary conditional the top is still Eq. 5.29 and i t s descendants, notably Eq. 5.43. The solution of the zero-order differential equation (Eq. 5.9) contains nothing new. As in Chap. 2, Sec. 1, it is found that

The first-order differential equation (Eq. 5.16) requires the evaluation of QQ:/QS. From Eq. 5.14, in the present c a s e

and Eq. 5.16 becomes

where

The general solution of Eq. 5.62 i s N:

The new constant e' is used to condense the notation. The expansions of the coefficients reduce to

Everything else (except of course N and r ) has only zero-order terms. In place of the boundary conditions a t s = 0 and s = B previously used, there is the simple expression N(0,t) = Np an.d hence

=

=

eCts(Acosh Bs

CP* + B sinh /3s) - à ‘Ñ P

e2~Is

(5.64)

As in Chap. 3,

The constants A and B a r e evaluated from the boundary conditions (Eqs. 5.43 and 5.59). The latter in the present example reduces to

constant where

Performing the indicated operations and collecting,

THE THEORY O F ISOTOPE SEPARATION

96

N,'(S,~)=

"*

P

[

ec,S I M - c t ) sinh ;3(S- s) + 13 cosh ^(S

(M-

+ eels

el)

sinh 8s + 8 cosh

(3s

~ e sinh " 0s ~ (M - ct) sinh (3s + 13 cosh

THE CONTROL PROBLEM

(This is a particular example of the general result proved in Sec. 2.) To get the average loss of production, the second-order t e r m s a r e thus necessary. Equation 5.52 of Sec. 2 reduces now t o

- s)

(3s

-

97

1

a 2 c ~

and the boundary condition which determines Tn i s , from Eqs. 5.53 and 5.54, If the disturbance is taken to be sinusoidal, i.e., P, = cos u t

and

Now from Eq. 5.17,

and then Eq. 5.68 is inverted, contributions a r e obtained from the poles a t 2 iw, also from the roots of (M - e ' ) sinh )3S + )3 cosh BS = 0, which lie on the negative r e a l axis. There is no pole at p = 0. Thus

J

=

exp

(- f ' 2 ~ cis) = exp (-2ets)

(5.75)

Combining Eqs. 5.73, 5.74, and 5.75 and simplifying the result with the aid of Eq. 5.60, then

i.e., t e r m s of the same period a s the disturbance, plus transients. This is a general result. Note also that the form of Eq. 5.68, which is linear in P r , allows superposition of solutions, i.e., a Fourier analysis of the variations in P , c , c , etc., is permissible. In what follows a type disturbance more complicated than the sinusoidal will be discussed. Suppose that the product is withdrawn in batches a t regular intervals a , instead of continuously. Then the disturbance function is P I (t) = - Po + pulses including a quantity aPo a t t

=

0, a, 2a.

..

(5 .TI) This integral i s equal1 to the sum of the residues a t the poles of P^(q). These a r e a t q = 2k3ri/a (k = 51, ± .). Each residue has the value

The transform of Pl i s easier to write

PF(p) has no pole a t p 5.68 and 5.72:

=

Accordingly it i s necessary to evaluate the expressions on the right-hand side of Eq. 5.76. From Eq. 5.20, Q2 = NIPl,

..

0. The average of N,(s,t) i s found from Eqs. 'Strictly speaking

-

Nl (s) = lim pN;(s,p) P- 0

=

0

Q: =

lirn v-m

+

9 k=-v

(residues)

98

THE CONTROL PROBLEM

THE THEORY O F ISOTOPE SEPARATION

99

In the same way the effect of disturbances of the same frequency which a r e out of phase by an angle a can be realized. Corresponding to Eq. 5.70, Nl is

Then

and likewise, using the relation (Eq. 5.41) for NG,, -

The meaning of Eq. 5.78 will be clear by returning to the disturbance Pl = cos u t and following through the s a m e procedure. From Eq. 5.70 and .On = NIP,, this time averaging by direct integration with respect t o t,

c

F(s,iu)eiwt - -F(s, -iu)e-iwt] cos u t dt 2iu

Equation 5.80 is the contribution to D2(s) of a sinusoidal variation of frequency w/27r. Now, if the (improper) disturbance function (Eq. 5.71) is analyzed, representing batch withdrawal a s

and on calculating NIPl (or similar t e r m s ) there will result the contributions of the separate disturbances plus a cross-product t e r m with cos a a s a factor. Now if the two disturbances a r e unrelated cos a i s , on the average, zero, and the resultant is purely additive. Note, however, that fluctuations in P and, for example, c5 caused by a common primary disturbance a r e correlated through the hydrodynamic response of the plant and cannot be treated separately. The r e m a r k s of the preceding two paragraphs contain the results necessary f o r the composition of disturbances of different sources and types. Returning t o the derivation of i-according to Eq. 5.76 and picking up the thread of the argument a t the expressions Eqs. 5.78 and 5.80, consider ( l / i u ) \ ~ ( s , i u ) - F(s, -iu)] a s a function of u. T e r m F(s,p) is defined by Eq. 5.68. The t e r m e 2 " ~does not depend on u and hence gives nothing. Then F = (M- e') sinh (3(S - s) + (3 cosh (3(S - s) + (M - e') sinh /3S + (3 cosh (3s

(5.81)

where

M and if contributions of the individual frequencies given by Eq. 5.80 a r e summed, the result is precisely Eq. 5.78. This is a general result, which can be understood a s follows: In a disturbance, two different frequencies (in Ply for example) contribute independently to N , each giving a t e r m of its own frequency. In taking time averages, t e r m s like e^itlu:'t always vanish. The result is a sum of contributions from each frequency acting as if it were alone. This whole process i s taken c a r e of automatically by the use of the averaging theorems1

sinh (3s

=

2 e @ 5 iuky

l Âiuky

F o r C.A à e2(l + i/È)2 a condition fulfilled by all but the lowest f r e quencies because of the smallness of e , (3-

sinh (3(S- s )

Vd>x(i±i

- cosh'(3(~ s) - %e^

F(s, 5 iu)

-

-.

(s-s)(W

e-sV^(l?i)

Then 'In comparing the solution of thedifferential equation by the functionally closely r e lated methods of the Laplace transform o r the Fourier expansion, i t should be r e marked that the former method gives a t the same time the solution for the transients.

1 2 -[F(s,iu)-F(~,-iu)]=--e-~^ lu

Ill

sinsYt^X"

(5.82)

100

THE CONTROLPROBLEM

THE THEORY O F ISOTOPE SEPARATION

The contribution to T J ' P ~ N ~from the disturbance Pi = cos u t is, by Eq. 5.76,

101

In case B, T2 is already quite large; in fact it may be suspected that higher-order t e r m s a r e needed. In case A, Tn becomes large if a i s taken much larger, even though ae2/^ is still small. Thus in both cases T becomes appreciable a t a = -I/&, although pi does not appear in the formula (Eq. 5.85). This can be understood after an examination of the transient t e r m s in N, (s,t). The f i r s t transient has the factor

(5.84) The integral was evaluated by Eqs. 5.80 and 5.83 and by the definitiorof C , Eq. 5.63. The contribution from the term Pl (t)Np/Pg vanishes in the same manner a s e - s G / u and can b e neglected. The disturbance in Eq. 5.71 caused by batch withdrawal gives likewise, comparing Eq. 5.78 with Eq. 5.80,

This factor increases rapidly for fixed pl a s apl becomes large. For example, taking pl = -1: for a = 0, f = 0; for a = 0.1, f = 0.048; for a = 1, f = 0.42; for a = 10, f = 100. Thus the transients get large a s a -l/pl, implying that the new steady state differs significantly from the initial state. Holdup a t the top increases -l/pl and must therefore permit larger values of a , even though this is not exhibited by the rough formula, Eq. 5.85.

-

Table 5.1 -Production

provided a <s: 2rA/e2(l + fl. The holdup in the reservoir has no influence to this degree of approximation. For larger values of a, numerical summations a r e needed. The factors in the solution (Eq. 5.85) a r e a t e r m in the ratio of a to the natural period, X/e2; a factor
Equilibrium enrichment (no production) 2tS k/e2

Loss Due to Batch Withdrawal Case A

Case B

2-fold

30-fold

0.693 40 days

3.40 1 day

First transient period:'

1 day -3.5 %

1 day -29 %

'Taking the holdup in the top reservoir a s zero.

The evaluation of the loss in production by the single term T2 will b e correct only if the higher-order t e r m s a r e negligible, i.e., only if T i s quite small. This slightly discouraging conclusion does not deprive the calculation of i t s usefulness, which is mainly to assign permissible upper limits t o the disturbances. It i s difficult to generalize from the problems just solved to other problems. Even the most interesting result, that the effect of highfrequency disturbances vanishes a s w-%, is only true for a disturb-

102

THE THEORY OF ISOTOPE SEPARATION

ance in P alone. It does not subsist when fluctuations in the coefficients c6, c5, and c, a r e taken into consideration. The correlations to be expected between disturbances of the various coefficients a r e different from one separating apparatus to another. Hence this is about a s far a s the investigation can proceed without specifying the type of equipment in question. The reader may have noted that, leaving out the explanations and digressions, the solution to the problem of this section was obtained fairly easily. This is not generally true; problems on more intricate cascade shapes and for intermediate frequencies run into heavy computation. However, it should be remembered, in spite of the space required to expose the theory, that small correction terms a r e involved. It is therefore quite proper to reduce the numerical work by rather drastic approximations.

REFERENCES Fuchs, K., British MS 71, January -November, 1943. Fuchs, K., British Supply Council reports: MSN 4, Feb. 11, 1944; MSN 5, Mar. 4,1944; MSN 8, Mar. 7 and Mar. 17, 1944; MSN 10, Apr. 6, 1944; MSN 12, June 6, 1944; MSN 14, June 28, 1944; MSN 15, July 11, 1944; MSN 17, July 20, 1944.

Chapter 6 CENTRIFUGES

1. HISTORICAL INTRODUCTION

The effect of the earth's gravitational field on the composition of the gases in the atmosphere at various heights has long been recognized. For, if pi is the partial pressure of a gas of molecular weight Mi, according to the Boltzmann distribution law pi(h) = pi(0) exp where h is the height; g is the gravitational constant; R is the gas constant, 8.314 X lo7 ergs/OK/mole; and T is the absolute temperature. Then for two gases of different molecular weights M, and M,

The suggestion that gravitational o r centrifugal fields a r e suitable for isotope separation was made by Lindemann and Astonl in 1919. In fact, they proposed that samples of neon taken at ground level and at 100,000 ft would show a sensible shift in the isotopic abundance ratio. In a centrifuge, vastly more powerful fields a r e available. The energy per mole at radius r in a centrifuge rotating with angular velocity u is%Mi(ur),. This expression for the energy replaces -Migh in Eq. 6.1, giving pi(r) = pi@) exp

Mi (u r)2 2RT

104

THE THEORY O F ISOTOPE SEPARATION

Between the periphery r = r, and axis r = 0, for two different gas from Eq. 6.3, Pl(r2) _ PI(O) exp (M, - M2) ( u r 2 ) 2 ~ ~ ( 0 ) 2RT

~2b-2)

where w r , i s the velocity of the inside wall of the rotor, the peripheral velocity. In a binary system, if N is the mole fraction of the isotope ha m a s s Mi, Pi

+

P2 = P

N 1- N

= Pl/P =

p2/p

and p - p(r) i s the total pressure. Then Eq. 6.4 becomes

exp

= (&)o

(Ml - 1%) (urn)' 2RT

It is usual to take M, > Ml, and accordingly

a n = exp

(Ma - MI) ( u r J 2 2RT

so that Eq. 6.6 takes the form R(0) = a o R ( r 2 ) where a ni s the (equilibrium) simple-process factor of the separat The value of the simple-process factor for a peripheral velocity 300 meters/sec, T = 300°K and a m a s s difference of one unit i s , Eq. 6.6, a. = 1.0183. Shortly after the suggestion of Lindemann and Aston was made, tempts were made, in rather crude apparatus, by Joly and Poole1 by Mulliken2 to realize this separation factor, but these were uns cessful. The idea of the evaporative centrifuge was introduced at same time by Mulliken. In this method a small amount of liquid introduced into the centrifuge, forming a layer a t the periphery. DuPhil. Mag., 39:,, 372 (1920). 'J. Am. Chem. Soc., 44: 1033 (1922).

CENTRIFUGES

105

ing the spinning of the rotor, vapor is removed slowly through a shaft along the axis. In this way a Rayleigh distillation (Appendix D) of the liquid i s accomplished, with simple-process factor given by Eq. 6.31. An increased concentration change in the residual liquid i s thus obtained. The development by Beams1 of vacuum-chamber centrifuges, which were f r e e from vibration and thermally isolated to eliminate convection currents, finally permitted successful centrifugal isotope s e p a r a tion to be obtained in 1939. Beams and Skarstrom2 a t the University of Virginia, using the evaporative centrifuge method on CC14,reported a 13 p e r cent change in the C135/C137ratio. Shortly thereafter Hump h r e y ~ ; using the s a m e technique on ethyl bromide, altered the ~ r ~ ' / ~ abundance r " ratio by 11 p e r cent. The simple-process factor of centrifugal separation depends only on the m a s s difference. Unlike other methods (e.g., diffusion through b a r r i e r s , where a v a r i e s a s AM/M) it i s no more difficult for heavy elements than for light. In addition, it is possible to calculate a p r i o r i the separation to be expected from centrifugal apparatus without any undetermined constants o r theoretical uncertainties. The simple-process "flow-through centrifuge," a s the concurrent flow type was called, i s a n application to gaseous flow of the continuous c r e a m separator. As originally conceived, a single s t r e a m of g a s e n t e r s one end of a rotor through a hollow shaft, and two s t r e a m s a r e taken off the other end, one from the periphery and the other near the axis. This method produces a s m a l l change in concentration per machine. The countercurrent flow type was designed to attain considerable separation in a single centrifuge, thus reducing the number of stages required and the amount of material circulated between stages. As originally proposed by u r e y , circulation was to be established by continuous distillation of UFs from the bottom cap of the rotor. The vapor was to be condensed on the top cap. The heavy liquid would then be forced out to the periphery and would flow down the walls to the bottom cap, countercurrent to the vapor flow, and complete the cycle. Bramley and Brewer5 and Martin and Kuhn6 in Germany proposed to establish countercurrent flow by thermal convection. ' ~ e v s .Modern Phys., 10: 245 (1938); Rev. Sci. Instruments, 9: 413 (1938). 'Phys. Rev., 56: 266 (1939). 3 ~ h y s Rev., . 56: 684 (1939). Reports on P r o g r e s s in Physics, VI: 72 (1939). 'Science, 92: 427 (1940). '2. physik. Chem., A189: 219 (1940).

106

107

THE THEORY OF ISOTOPE SEPARATION

CENTRIFUGES

The aim of this chapter is to expose the theory of centrifugal separation for various types of centrifuges. The important question of mechanical design will be dealt with only in so far a s it is essential to the understanding of the process design. The partial differential equation of centrifugal separation is derived in Sec. 2. In Sec. 3 the theoretical maximum separative power of a centrifuge is found. The solution for the evaporative centrifuge i s given in Sec. 4, that for the concurrent centrifuge in Sec. 5, and that for the countercurrent centrifuge in Sec. 6. Section 6 investigates at length the properties of the countercurrent centrifuge. The writer was introduced to the a r t of centrifugation by C. Skarstrom, formerly of the University of Virginia and later with the Standa r d Oil Development Company. Harold C. Urey participated actively in the early work, until his other responsibilities became too heavy, contributing ideas and physical interpretations of mathematical results. Irving Kaplan shared in all the developments subsequent to December 1941, notably the work on cascades of centrifuges (incorporated largely in Chaps. 1 to 5) and the design problem.

The density distribution (Eq. 6.10), considered from a molecular point of view, is a dynamic equilibrium between the effect of random motion of the molecules (diffusion), which tends to erase density gradients, and the effect of the centrifugal field, which tries to pile all the molecules at the periphery. The diffusion counter to the density gradient gives a current density (in grams per second per square centimeter)

Here D is the coefficient of self-diffusion. The outward current per square centimeter caused by the centrifugal force, which balances this inward current, must therefore be

In mixtures, Eq. 6.11 holds for each component separately

2. PARTIAL DIFFERENTIAL EQUATION OF CENTRIFUGE

As a preliminary to the derivation of the differential equation, consider for a moment a centrifuge containing a single pure gas. The gas is supposed to rotate uniformly at the same speed a s the rotor, i.e., w radians/sec; to have no other motions; and to be isothermal. Choose a cylindrical coordinate system, rotating with the gas, with the origin at the axes of rotation. The effect of the rotation is to set up a centrifugal force w 2 r per unit mass. Under the influence of this force, a pressure gradient is established according to the hydrodynamic equation

Here p i s the density of the gas mixture in grams per cubic centimeter. The quantities p and p a r e related by the gas law

Combining Eqs. 6.8 and 6.9 gives Eq. 6.3 of its equivalent ~ ( r =) ~ ( 0exp )

Mw2r2

(6.10)

Di i s the diffusion coefficient for molecules of type i through the whole mixture. For isotopic mixtures the exact meaning of the diffusion coefficient is of no significance, and henceforth the subscript will be dropped. Equation6.12 i s the desired expression for the flux of molecules caused by a centrifugal field, and it i s now possible to derive the differential equation. Consider then a centrifuge containing a perfect binary gas mixture, isothermal and rotating uniformly. In order to accomplish separation, the gas i s made to flow radially and axially. In t e r m s of the cylindrical coordinate system chosen a s before, the motion of the gas a s a whole (drift velocity) is given by the components

(Radial)

u = u(r,z)

(Azimuthal)

v =0

(Axial)

w = w(r,z)

Because of this motion of the gas, the equilibrium distribution (Eq. 6.3 o r 6.10) i s continually upset. The system moves material in an effort to reestablish equilibrium; in this way a transport of the desired material i s set up.

CENTRIFUGES

Imagine a small annular region in the rotor, bounded by the cylind e r s r = roand r = r o+ 6r and by the planes z = z0 and z = z0 + 6z. In a short time interval 6t t h e r e flows a c r o s s the cylindrical face r = rn an amount of desired isotope (denoted by the subscript 1).

In this equation p,u is the flow caused by the drift velocity, p$cl i s the flow impressed by the centrifugal field, -D ap1/ar i s the contribution f r o m diffusion against the density gradient, and 2 r r O6z i s the a r e a of the cylindrical face. Likewise the flow a c r o s s the plane face z = z0 i s

which i s the desired partial differential equation of centrifugal separation. The three kinds of centrifuge described in Sec. 1 a r e governed by special c a s e s of Eq. 6.19. In the evaporative centrifuge w = 0 and r u = constant; in the concurrent and countercurrent centrifuges u = 0, and w describes the respective types of flow. The solution of Eq. 6.19 for these c a s e s i s the subject of the subsequent sections devoted to the particular centrifuge types. 3 . MAXIMUM SEPARATIVE POWER OF A CENTRIFUGE

The change in content of desired material in the volume element i s therefore

Introducing Eqs. 6.13 and 6.14 into 6.15 and making use of the relation of Eq. 6.12 for u C (dropping the subscript O),

Now

and

Substituting Eq. 6.17 in Eq. 6.16, using Eq. 6.18 and noticing that Dp constant, it i s found that

=

In this section the maximum separative work that can be performed by a centrifuge1 will be determined. It will be used a s a standard to evaluate the efficiency of the various centrifuge designs. Using the result of Sec. 6 (Eq. 1.60),

The net transport of desired material (in moles per second) a c r o s s a surface element do is, according to the analysis given in a preceding section,

where To i s a unit vector along a radius through do, TIcl i s the vector velocity created by the centrifugal field directed along F0 and with magnitude given by Eq. 6.12, and the symbol v is the vector gradient operator. Thus the separative power for a s m a l l slab of thickness ds, with f a c e s do normal to VN and sides parallel to VN, i s

Following Dirac.

110

Ill

THE THEORY O F ISOTOPE SEPARATION

CENTRIFUGES

The symbol indicates the scalar product. Taking VN a s a variable and do d s and everything else constant. Eq. 6.21 i s a maximum when

dependent of the radius. The radius affects the holdup, and in counter current centrifuges the flow into the rotors, but is otherwise irrelevant. It is usually chosen by mechanical considerations.

that is, along the direction of the centrifugal field, and one-half the equilibrium value. Then

SURFACE do-

\ \ \

Integrating over the whole centrifuge, of length Z and radius r2,

Note that the expression in brackets is the exponent of an (Eq. 6.7). The minimum number of centrifuges of length Z and peripheral velocity w r 2 required to perform a given separation is, by Eqs. 1.44 and 6.23, AU/(6U)rnax. The value of (i5U)max depends on the peripheral velocity of the fourth power, which puts a premium on high periphe r a l velocities. The separative power is also proportional to the length. But for constant peripheral velocity1 the separation is in-

\I

Fig. 6.1-Stresses diagram.

A

,/

in a thin rotating cylinder. (a) S t r e s s e s in a wall sector. (b)Vector

4. THE EVAPORATIVE CENTRIFUGE 'The peripheral velocity-not the angular velocity-is determined by the tensile strength of the rotor wall, s o that it i s the fundamental characteristic of the machine. The demonstration follows: Consider the equilibrium of forces acting on a sector of the rotor wall, subtending a small angle 4. Then the triangle of f o r c e s gives (Fig. 6.1;

In the evaporative centrifuge N is independent of z, and w = 0. Considering the steady state, a ~ at/ = 0 and Eq. 6.19 reduces to



r + d u u 2 r = 2 T d u s i n - = T dff 4 2 *

where where T is the s t r e s s in the wall and p i s its density. The rotor will break when T = p d r 2 i s greater than the tensile strength of a metal.

112

113

THE THEORY O F ISOTOPE SEPARATION

CENTRIFUGES

Suppose vapor i s removed from the axis a t a r a t e L moles/sec. Then

ducti0n.l If it i s assumed that the production i s only a small fraction of the charge, o r that successive batches of product a r e kept separate, the separative power of the centrifuge i s (cf. Appendix D)

-J L2mzu = L

(6.26j

RT

5U = -2L

Multiplying Eq. 6.24 by the constant 2 i i Z / R ~and integrating, 3N y r

+ 2flAr2 N ( l - N)

I

= L M(0) - L N(r)

(6.27~

where DP RT

(6.28)

The integration constant was evaluated by noting that the left side of Eq. 6.27 vanishes a t r = 0. Replacing N by R a s dependent variable,

+ 2AAr2 R

) [

= L R(0) - R(r)

] [ + K%] 1

È6s29

- R(0) s - AAr2 N(O), which can be neglected with respect to 1 + R(0) unity. After thus simplifying Eq. 6.29, the solution is readily found to be

Now

I)&(

a n y+L - 1 ) 2 = ~ L 2( ~~ )L+2 ( l n a o ) 2(6.32)

The value of 6U is a maximum when L = 2 ~ : (

y = 27rZ - (moles/sec)

( ($1 5

Y

~ = -(ln ) ~ a0)2 ~ =~Dp- (ln 4 RT

z

u 2

(6.33)

which i s the same a s the theoretical maximum separative power of a centrifuge (Eq. 6.23). In the development just given the gas was assumed to be isothermal. This picture i s f a r too simple. When vapor i s withdrawn a t the axis, the gas cools by expansion and a temperature gradient i s induced. The gradient i s determined by the balance between the r a t e of cooling of the g a s and thermal conductivity. As L increases, the gradient increases until the gas reaches the dew point o r the adiabatic g r a dient. The mean effective temperature Teff i s therefore reduced a s a function of L. An attempt may be made to account f o r this effect by writing in place of Eq. 6.31

fc^) T

2

R(0) = R ( r 2 ) a o

(6.34)

R = R(0) exp - A A r 2

and on introducing Eq. 6.7, which in t e r m s of AA i s a. = exp [ + A A ~ ; ]

(6.30)

this expression may be written

(&)

R(0) = R(r2)ffo

(6.31)

Equation 6.31 gives the relation for mole fraction of product N(0) in t e r m s of the composition of the charge N(r2) and the r a t e of pro-

Since T / T ~increases ~ ~ with L, the separation will not fall off quite so fast with L in practice a s Eq. 6.31 would indicate. A unit weight of gas rotating near the periphery with angular velocity w has the angular momentum r & ; at the axis it has none. When g a s i s moved from t h e periphery toward the axis, means must be provided to absorb this excess angular momentum, which otherwise tends to increase the angular velocity of the gas a t smaller radii. This i s accomplished by radial baffles2 (looking down along the axis, a s t a r ) which prevent the g a s from rotating a t angular velocities dif ferent from that of the rotor. 'This result was found by Humphreys, Phys. Rev., 56: Yearns and Skarstrom, Phys. Rev., 56: 266 (1939).

684 (1939).

114

CENTRIFUGES

THE THEORY O F ISOTOPE SEPARATION

Although the evaporative centrifuge, with L = 2y, delivers 100 per cent of the theoretical separative power, it is not easily adapted to continuous operation. 5. THE CONCURRENT CENTRIFUGE

Like the evaporative centrifuge, the concurrent centrifuge is a simple-process machine since the enrichment factor per machine is limited to a,,. In this device axial flow i s used to obtain an axial a s well a s a r a dial concentration gradient. The gas enters (Fig. 6.2) in two streams a t one end of the rotor and flows axially to the other end, where the s t r e a m s a r e removed separately. During the passage through the centrifuge the s t r e a m s tend to assume the (radial) equilibrium distribution (Eq. 6.6). The flow pattern adopted for the concurrent centrifuge consists of two thin cylindrical s t r e a m s located a t radii r, and r (periphery) and flowing parallel to the z axis. (The optimum position for the inner stream i s not a t the axis, a s will be seen subsequently.) There i s no radial m a s s flow. A rotor with a double entrance (so that entering streams of two different concentrations can be used) will be considered; previously the literature has covered only single-entrance machines. The doubleentrance centrifuge i s more flexible than the latter. Cascades of these elements, which a r e more general than the element .of Chaps. 1 to 5, a r e studied in detail in Appendix E. Among the more interesting features of this study i s the result that the no-mixing case (both entering s t r e a m s of the same composition) i s not an optimum. The solution given here follows roughly the demonstration by Murphree, which for the case considered i s superior to the more general method that was originally used. The value of u i s zero, and the flows a r e generally so large that back-diffusion is negligible. Then Eq. 6.19 reduces for the steady state to

In the region between r grates to

=

rl and r

=

WALL

r2,w = 0, s o that Eq. 6.35 inte-

Fig. 6.2-Simple

model of a gaseous concurrent centrifuge.

THE THEORY OF ISOTOPE SEPARATION

CENTRIFUGES

where f (z) is the transport of desired isotope a c r o s s a cylindrical element of unit length between the two streams. But also, if L(rJ i s the flow in the stream at r = r2,

The separative power when N(r2,0) and N(rl,O) a r e most favorably related1 is, according to Eq. 11 of Appendix E,

116

117

and the separative power per unit length is

and L(r2) N(r2,z) + L(rl) N(rl,z) = L(r2) N(r2,0) + L(rl) N(rl,O) =

B

(a constant)

(6.38)

Integrating Eq. 6.36 between rl and r2, considering AA N(l - N) constant to t e r m s of the order of (AA)',

which i s a maximum with respect to rl/r2 for any value of bz when rl/r2 = 0.534,

and substituting for f (2) from Eq. 6.37 and for N(r2,z) from Eq. 6.38,

and a relative maximum with respect to bz when bz = 0, namely,

(F)

max

Integrating this ordinary differential equation for N(rl,z) and r e arranging the result,

RT

If this result is compared with Eq. 6.23, it is seen that this i s 81.45 per cent of the theoretical efficiency. For a single entrance centrifuge, it is necessary to take N(r2,0) = N(rl,O). The separative power is then, according to Eq. 8 of Appendix E,

and per unit length where

= 0 . ~ 0 7 2 4 7DP r-(~nit~)~

THE THEORY OF ISOTOPE SEPARATION

CENTRIFUGES

The maximum with respect to rl/r2 is the same a s before. The expressionl (1 - e-^)'/bz has a true maximum a t bz = 1.25643 and thus

6. THE COUNTERCURRENT CENTRIFUGE

118

(y)

max

=

27rDp (0.40724)' -(In aJ2 KT

which has an efficiency of 66.34 per cent. The condition bz = 1.25643 gives a definite relation between the length of the rotor and the flow, namely,

If it i s remembered that the separative power i s the product of the concentration difference and the transport, the maximum of 5U with respect to rl/r2 is easily understandable. The concentration difference is the maximum when r,/r2 = 0, but the transport (through the cylinder of radius zero) is zero. The transport i s a maximum when r1/r = 1, but the concentration difference is zero. The maximum of the product thus lies somewhere in between. A fuller discussion of the significance of the dependence of the r e sults on 1 - e-^, the degree of equilibrium attained, is given in Appendix E. The phenomenon i s a quite general one. Notice that b contains the factor 0(1 - 0) (L, + LJ. Decreasing 0(1 - 0) and increasing L, + Ln leaves this factor unchanged, but small total flows a r e desirable and 0(1 - 0) should obviously be made a s large a s possible, which requires 0 = % (cf. Eq. 6.44).

Further properties2 of the concurrent centrifuges may be obtained from the equations given. 'Note that the correspondence e-b2

-
'For example, for any r l / r 2 the dependence on z/(L1 + L2) i s the same, and the maximum for the single entrance centrifuge is at t h e s a m e place, bz = 1.256 [with the appropriate value of In (r2/rl) in b]. But for z/(Ll + LJ preassigned (and not bz) the maximum with respect to r 1 / r varies.

Countercurrent separating devices in general have the property of multiplying the equilibrium simple-process factor many times in one unit. Because of the nature of the flow, a concentration gradient whose size i s limited only by back-diffusion i s established in the direction of the flow. Since a large separation i s obtained in one unit, much of the recycling between units which is otherwise necessary to multiply the simple-process factor i s avoided. Furthermore, the problem of cascade operation i s very considerably simplified since the number of units in s e r i e s required to effectuate a given fractionation decreases enormously. The cascade becomes broad instead of long and i s easily broken down into independent parallel sections. Countercurrent flow i s also efficient from a process standpoint because it i s possible to maintain maximum separative power through the entire unit.' F o r all these reasons, which have long been appreciated by chemical engineers, countercurrent processes a r e preferred if at all possible. The differential expression of Eq. 6.19 becomes for the countercurrent centrifuge

The solution of Eq. 6.45 may be found by following the standard methods for solution of partial differential equations. This method was utilized in June and July of 1940. However, Eq. 6.45 i s particularly simple since the gradients of the concentration in all directions a r e small. Thus the equation for a countercurrent centrifugal column can also be solved by the method used by Furry, Jones, and Onsager2 for the thermal-diffusion column. This possibility was first pointed out, and partially worked out, by B r a m l e ~ .A~ similar procedure was employed by Martin and ~ u h n . ~ The results of the two methods a r e of course identical. The second method will be demonstrated here because it i s easier and because it '1n contrast to concurrent devices, in which aN/ar changes with z, s o that if the separative power i s an optimum in one place it is necessarily l e s s in every other place. v h y s . Rev., 55: 1083 (1939). 'Science, 92: 427 (1940). Z . physik. Chem., A189: 219 (1940).

120

121

THE THEORY OF ISOTOPE SEPARATION

CENTRIFUGES

permits the c a s e in which the mole fraction of the product i s large to be included without added complications. The method depends essentially on the fact that aN/ar i s of the o r d e r of N(l - N)2AAr, which i s a s m a l l quantity. Thus the variation of N with respect t o r, a s compared to the variation of r, p, o r pw, may be neglected. F o r example

pw r d r , which i s just a s useful for present purposes. Substiar o tuting Eq. 6.46 in Eq. 6.49, it i s found that

CJ^L

pw r d r

- ?a-!z

r

pw r d r + t e r m s of the o r d e r of A A ~

rr

where

Consequently Eq. 6.45 may be integrated with respect to r. The f i r s t integration gives directly, dropping a2N/az2 and aN/at,which a r e of second o r d e r ,

The integration constant ( a function of z and t) w a s evaluated by r e membering that r ( a N / a r ) 0 a s r 0. The second integration with respect to r r e q u i r e s a little more artifice in o r d e r to get a form in which the integration constant can be evaluated. The boundary conditions a r e

-

-

c6 =

Equation 6.48 may be transformed by integrating by p a r t s the f i r s t t e r m on the left-hand side,

277

/

'Â¥

p r d r = holdup p e r unit length (moles/cm)

Equation 6.50 h a s the f o r m of Eq. 2.1, the typical equation f o r isotope separation in a square cascade o r column. The properties of such columns, individually and in cascades, f o r fixed values of c,, c , and c6 were studied in Chaps. 2, 3, and 4. The following discussion covers the investigation of how these coefficients vary a s a result of changes in operating conditions. The material in the e a r l i e r chapters will be r e f e r r e d to constantly. In order to examine properties of the countercurrent centrifuge, attention will be confined for the moment to a single centrifuge. F o r convenience it will a l s o be assumed that N<1, although t h i s i s by no means essential to the argument. Then the steady-state solution of Eq. 6.50 i s (Eq. 2.14)

Equation 6.49, although it does not give the integration constant that would permit the integration of a ~ / a ritself, tells how to integrate

122

THE THEORY O F ISOTOPE SEPARATION

CENTRIFUGES

123

and the maximum possible value of N(Z)/N(O) for a given length of centrifuge and for a given type of countercurrent flow i s

where

-N(z) NO)

- exp 2cOz

where 2e0 is the maximum of c2 + a3L2 with respect to L and i s

The quantities cl and c5 depend on the function pw, which measures the up-and-down flow, in two ways: They depend on the flow pattern, o r the locations and relative size of the s t r e a m s , and on the absolute magnitude of the flow. The pattern i s built into the rotor by the position of orifices in the end caps and position (or absence) of baffles. But the magnitude of the flow i s an operating variable and can be adjusted f r o m outside. This independence of flow rate and internal construction in a properly conceived centrifuge gives it more flexibility than the otherwise analogous thermal-diffusion column. Therefore a new quantity L is introduced.

which is the flow up the column plus the flow down the column. Then

does not depend on L, and Eq. 6.52 becomes

The corresponding value of L i s

The existence of a maximum in e i s physically due to the phenomenon of back-diffusion. The concentration gradient is proportional to e . When L, the total flow in the centrifuge, is large and the concentration gradient i s small, back-diffusion i s negligible (c, < a,L2) and e v a r i e s inversely a s L. But a s soon a s L is small and e and the concentration gradient a r e large, back-diffusion plays an important role, finally limiting the attainable fractionation. It will be found convenient to measure all flows in t e r m s of Lo. Therefore s e t

Substituting in e,

where a, = c,/L and a 3 = C J L ~ do not vary with L. N(z)/N(o) i s now a function of L and ip. F o r any value of L, N(z)/N(o) is greatest when the rate of, production i s zero. Then -N(Z) - exp (c2 ' t L 2

NO)

In t e r m s of the dimensionless parameters m and

z,

P = m^P0

ib.

P is given a s (6.61)

* 124

THE THEORY O F ISOTOPE SEPARATION

CENTRIFUGES o h K., memorandum to H. C. Urey, May 14, 1940. Cohen, K., and I. Kaplan, Report A-101, Jan. 28, 1942. Cohen, K., and C. Skarstrom, Columbia S e r . No. 4R-X138, Feb. 6, 1941. Dirac, P. A. M., British MS, 194'1. Kaplan, I., and K. Cohen, Report A-195, J u n e 17, 1942. Murphree, E. V., Separation of G a s e s by Diffusional Methods, February 1942.

where

Equations 6.57 to 6.62 constitute a complete description of the behavior of the countercurrent centrifuge. When the type of flow pw i s specified, co, Lo, and Po a r e determined. L and P a r e the conditions imposed and a r e measured on flowmeters by the experimenter. By Eqs. 6.59 and 6.61, m and $ a r e therefore known. And finally, substituting co, m, and ~L,J in Eq. 6.60, the fractionation of the column i s predictable. The maximum separationfactor occurs a t L = Lo, but this i s clearly not the most efficient circulation rate f r o m the standpoint of separative power, since the parasitic phenomenon of back-diffusion is very important at this flow. The relative weight of back-diffusion i s reduced by increasing L (thus decreasing the concentration gradient). These qualitative considerations a r e confirmed by examining SU. From Eq. 2.33,l

-

whose largest value i s reached asymptotically a s m m. It should be noticed that the limiting value of 6U i s approached very rapidly: When m = 3 the separative power i s 90 per cent, and when m = 5 it i s 96per cent of c;Z/4c3. At these valuesof m the concentration gradient i s 60 and 38 p e r cent, respectively, of the r n a x i m ~ m . ~ Thus it i s not necessary to discard the advantages of large separation in one unit, o r to increase the flows inordinately, to get adequate separation efficiency.

REFERENCES Bramley, A., The Centrifugal Separator, Oct. 7, 1940. Cohen, K., Report A-54, October 1940; Report A-52, Jan. 21, 1941; Report A-50, Jaz. 30, 1941.

C o n s i s t e n t with the e a r l i e r sectionsof t h i s chapter, the symbol 6U is being used k? the total separative power, and not the separative power p e r unit length a s in Chap. Z Sec. 3. 'The f a c t o r is 2m/(l + m2).

OTHER SEPARATION METHODS

127

isotopic solutions a r e perfect solutions and because the simpleprocess factor is nearly one; and (2) the inclusion of the equilibriumtime phenomenon, which is generally omitted although it is more often significant than is commonly susnpftedIt i s clear that the two phases need- - -not be liquid and gas; two liquids, o r a liquid and a solid, would be equally suitable. The phase Chapter 7 OTHER SEPARATION METHODS

1, TWO-PHASE SEPARATIONS

The classic case of a two-phase separation is fractional distillation, one of the most widely used unit operations of chemical industry. The earliest use of this process in isotope separation was the fractionation of Ne20 from liquid neon by Keesom. The method was later used to r a t e deuterium from liquid hydrogen andin the fractionation of H,0 to give 018(Urey e t al.). The scope of this process was much enlarged by Urey's invention of the chemical-exchange method. Here the driving force of the separation is the chemical equilibrium constant of a reaction such a s

Such a reaction can be carried out entirely analogously to ordinary fractional distillation (refer t o Fig. 7.1). In the example shown, HCN gas is scrubbed countercurrently with a water solution of NaCN in a packed tower, just a s vapor and liquid a r e contacted in fractional dis tillation. The condenser of the fractionating tower is replaced by NaOH, which converts the gas phase to the liquid phase. Although the theory of such processes was fairly well understood before the war, an account of it is included here by reason of its probable great future importance. Also, the generality of development of cascade theory now allows the theory to be given in a clearer and m e concise form. The theory has some points of resemblance with the ordinary theory of fractional distillation. The two chief differences a r e (1)the closed form of the results, which is possiblebecause 126

Fig. 7.1 -A

chemical-exchange tower.

in which enrichment of desired isotope takes place will be denoted by small letters and the other phase by capitals. Thus n, 1 , h, and d a r e , respectively, the mole fraction,flow in moles per second, holdup in moles per centimeter, and diffusion coefficient in one phase; and N, L, H, and D a r e the corresponding quantities for the other phaseAt equilibrium1 'A complete discussion of the theory of the equilibrium constant may be found in 3. Bigeleisen and M. G . Mayer, J. Chem. Phys., 15: 261 (1947).

THE THEORY O F ISOTOPE SEPARATION

OTHER SEPARATION METHODS

Eliminating ni and N, between Eqs. 7.5 and 7.6,

Countercurrent contacting (Fig. 7.1) will be considered first. The direction of flow of the enriched phase will be taken a s positive, and length along the column will be measured in this direction. Then if T is the rate of transfer of desired isotope to the enriched phase across the interface per unit length of column,

1 k

B DCu

-=-+-

b dcu

The rate of transfer T depends on two processes : the rate of at tainment of the chemical equilibrium and the rate of diffusion through fluid boundary layers. If the equilibrium process is rate determining, then T has the form of the rate constant for a bimolecular reversible reaction

where c and C a r e the concentrations in moles per liter of the proce s s material in two phases. Usually the slow process is the diffusion through the boundary layers. Referring to Fig. 7.2 and denoting the mole fractions a t the interface by ni and Ni, and the a r e a of the interface per unit length of column by a , then

The differential coefficients may be replaced k y (Ni - N)/B and (n - ni)/b, respectively, when B and b a r e the film thicknesses. Thus

Fig. 7.2-Relations at a gas-liquid interface. a relation that is quite familiar to chemical engineers. Equation 7.7 can be simplified by the assumption that the interfacial concentrations do not differ appreciably f r o m the concentrations in the body of the fluid, giving T = - k [(n - N) - (a- 1)N(1 - n)] = - k [n(l - N) - aN(1 - n)]

At the same time ni and N, a r e related by Eq. 7.1, s o that

(7.9)

which is the s a m e a s Eq. 7.3 but with a different meaning for k. It should be noted that in deriving Eq. 7.9 it has not been assumed that, '.a - 1) i s small. Inserting Eq. 7.9 in Eq. 7.2,

OTHER SEPARATION METHODS

THE THEORY OF ISOTOPE SEPARATION

h

an an + 1 -= - k az at

[(n - N) - (a - 1)N(1 - n)] (7.10)

13 1

The solutionof these equations i n t h e steady state (aN/at = &/at = 0) f o r ( a - 1) small has been treated at length in Appendix E, a s well a s the associated cascading problem. 2. THERMAL DIFFUSION

These two equations may now be combined, using the assumption that ( a - 1) i s very s m a l l (for the c a s e of large values of a , s e e Sec. 3). Noting that the operator a/at i s proportional t o (a - 1)' and that 9/9z i s of the o r d e r (a- 1).

The history and theory of the thermal-diffusion process has been the subject of a number of recent papers which have been widely read.' The treatment here, therefore, will be confined to a rapid review of the theory to show how i t fits in with other separation methods in the general scheme. The equation f o r thermal diffusion corresponding to Eq. 6.45 that was derived f o r a centrifuge is easily a r r i v e d a t by replacing

which is the form of the fundamental equation of isotope separation (Eq. 2.1) with the correspondences by the corresponding expression for thermal diffusion2

The theory of square cascades developed in Chap. 2 can thus be applied directly. The exact solution of Eq. 7.10 in the steady state for values of ( a - 1) that a r e not small i s also of some interest. Eliminating N by the relation In - LN = Pnp, defining $J = P / ( a - 111, (. = k(o1 - 1 ) / 2 ~ , the following is obtained:

where A($) = {[I + $ + ( a - l)$np)2 - 4a$Jnp}% and no, nz, and n~ are the mole fractions a t the bottom of the column, the top of the column, and the top of the cascade, respectively. Equation 7.13 is directly analogous to Eq. 2.5, to which it reduces f o r small values of ( a - 1). The equations for concurrent contacting a r e easily obtained f r o m Eq. 7.10 by changing L to -L. This gives 3N + L 3N = k [ ( n - N ) - ( a - l ) N ( l - n ) I Haz at h -a+n at

a n = - k [(n - N ) - ( a - 1)N(1 - n ) l 1az

(7 .14)

where a is known a s the thermal-diffusion coefficient. Then, i n place of Eq. 6.45,

where x is the distance from one wall in a thin annular space between two concentric cylinders. It must be remembered that because of the severe temperature drop a c r o s s the annulus pD varies with x. Proceeding now precisely a s i n Sec. 6 of Chap. 6, it is found that Eq. 7.15 resolves to

*L. Waldmann, 2. Physik, 114: 53 (1939); H. Furry, R . C. Jones, and L. Onsager, Phys. Rev., 55: 1083 (1939); P . Debye, Ann. Physik, 36: 284 (1939); K. Wirtz, Ann. Physik, 36: 295 (1939); A Bramley, Science, 92: 477 (1940); J. Bardeen, Phys. Rev., 57: 35 (1940), 58: 94 (1940). 'see S. Chapman and T. G.Cowling, "Mathematical Theory of Non-Uniform Gases," p. 244, Cambridge University Press, London, 1939.

132

THE THEORY O F ISOTOPE SEPARATION

OTHER SEPARATION METHODS

where now

133

which requires a catalyst. It was f i r s t proposed by Urey in late 1940 to use this p r o c e s s by countercurrent washing of water and hydrogen in a tower packed with catalyst. At the time of writing, water must be converted into hydrogen a t the product end of a cascade using the process of Eq. 7.17. The steady-state equations corresponding to Eq. 1 of Appendix E a r e , from Eq. 6.14,l

an

I - = - k [(n - N) - (a - 1)N(1 - n) ] az

In Eq. 7.17, p is the density, r i s the radius of the annulus, and a is the spacing a c r o s s the annular gap. The relations of Eqs. 7,16 and 7.17 hold f o r any fluid, liquid a s well a s gas.

Appreciable simplification occurs when N -s: 1; since this i s the major part of the Dn problem (initial concentration = 1/7,000) this case will be considered. Then

Values of cl and c,; depend on the spacing, the temperature drop a c r o s s the gap, and the intrinsic properties of the fluid. The further theory of the thermal-diffusion column closely r e s e m bles that of the centrifuge, except that in the present instance the m a s s flow is caused by convection and i s dependent on the temperat u r e drop and the spacing. Corresponding to ^>O, the flow that gives the maximum fractionating factor in a centrifuge, there i s a charact e r i s t i c width an for which the fractionation in a thermal-diffusion column i s a maximum. However the maximum efficiency from the standpoint of separative work per unit of heat loss i s at a spacing other than a. which corresponds to the maximum separative efficiency in a centrifuge f o r flows l a r g e r than &,. F r o m the standpoint of flow geometry, it i s easy to show, by the method of Chap. 6, Sec. 6, that the model described has an absolute efficiency of about 71 per cent and that the most efficient flow pattern corresponds to flow immediately adjacent to the walls of the annulus, but with no flow in the middle.

The solutions of Eq. 7.20 a r e

The cascade is shown in Fig. 7.3. Then f o r the (s + 1)th stage, the following identifications can be made:

Equation 7.2 1 becomes

3 , CONCENTRATION OF DEUTERIUM

A possible basis f o r the production of deuterium i s the exchange reaction HD + HgO=

+ HDO

(7.18)

'Since (a- 1 ) i s large, we may no longer s e t N(l

-

OTHER SEPARATION METHODS

THE THEORY O F ISOTOPE SEPARATION

135

where

ft = 1 - e x p

[-

k(L ~

~

1= fraction ) 2 of equilibrium attained

(7.23)

Now Eq. 7.22 can be combined with the conservation equation

to give three equations in the four quantities Ns+l, Ns, ns, and ns.i which can be solved in t e r m s of any one of them. For example

The difference equation (Eq. 7.25) is easily solved to give

where

and No is the concentration of HDO in the feed. The number of sections in the cascade, a s a function of the initial and final concentration, is accordingly

m KEY

(GAS) LIQUID

Since L - 1 = P, A and B (and hence S) a r e functions of P/L, the percentage withdrawal1 The volume of catalyst required can be obtained by writing

The

reciprocal, L/P, is known as the

"reflux ratio."

136

THE THEORY O F ISOTOPE SEPARATION

where V i s the volume of catalyst p e r stage and k' i s the transfer coefficient per unit of catalyst. Then Crom Eq. 7.23,

and the total volume i s VS, a function of (3 and P . The total volume of catalyst has i t s smallest value when 0,the fractional attainment of equilibrium, i s zero. Without proof, this limiting volume i s taken to be the same a s that f o r the operation of a countercurrent column and corresponds to an infinite number of stages. The minimum number of stages o c c u r s for (3 = 1 and of course requires an infinite amount of catalyst. It can also be shown that t h e r e i s a maximum in the function -P -

VS

production volume of catalyst

which gives the economic operating conditions when the catalyst volume i s limiting. However, it is usual.1~expedient to run a t higher values of P and get more complete recovery of heavy water from feed. REFERENCES Cohen, K., Report A-118, May 1941; R e p o r t A-530, Feb. 4, 1943; Report A-531, Feb. 5, 1943. Cohen, K., Columbia S e r . No. 4M-X148, June 25, 1941; Columbia S e r . No. 4R-X133, August 1941. Cohen, K., Columbia S e r . No. 4L-35, M a r . 17, 1943 (letter t o P. C . Keith). Cohen, K., Packed Fractionating Columns, J. Chem. Phys., 8: 588 (1940). Cohen, K., and I. Kaplan, Columbia S e r . No. 4R-6, Nov. 5, 1942. Hutchison, C. A,, J r . , "Chemical Separation of the Uranium Isotopes," Division 111, Volume 3, National Nuclear Energy S e r i e s . Kaplan, I., and K. Cohen, Report A-335, Oct. 19, 1942; Report A-1276, May 1 , 1944. Thompson, W. I., and K. Cohen, S e r . No. 4R-104, Sept. 3, 1943.

APPENDIXES Appendix A -Roots

of a Transcendental Equation

Appendix B-Equilibrium Appendix C -The

Time of Square Cascades for N

-1

Holdup Function

Appendix D-Rayleigh Appendix E -Properties

Distillation of Concurrent Two-phase Elements

Appendix A ROOTS OF A TRANSCENDENTAL EQUATION

The features of the solution of the characteristic relations of Eq. 3.20 o r Eq. 3.22 for the roots pi, will be made clear by completely discussing the simplest special case, namely K' = -, K = 0, and P = 0. In this case it is necessary to solve for all the roots pi, of the equation YS tan Y S = â

s

where

If both s i d e s of Eq. 1 a r e plotted a s functions of yS (Fig. I), the solutions of the equation a r e the intersections of the curves y = tan yS and y = y ~ / c ~If. cS is positive and l e s s than 1, there i s a root yS = (yS), in the f i r s t branch of the tangent function, a s well a s other roots " which approximate to

in all succeeding branches. Since p depends on (yS)', the symmetrical negative roots may be neglected. The smallest root is i n the f i r s t

APPENDIX A

THE THEORY O F ISOTOPE SEPARATION

140

branch and gives pi; the other roots in the l a t e r branches give pa,

141

Plotting both sides of Eq. 4 a s functions of flS (Fig. 2)- a root of (fiS), i s observed. Since tanh (fiS), i s always l e s s than 1, s o also i s (fi~},/es, and consequently p, i s negative. F o r negative eS, there i s no root of Eq. 4. All the roots yS a r e r e a l and satisfy Eq. 1. Referring to Fig. 1, there i s a root tyS), between TS/Z and n in the f i r s t branch of the tangent, and t h e r e a r e l a t e r roots which approximate to (2K - l)n/2 (K = 2.3.4.. ) i n all succeeding branches. Again, p i s negative. h

.

Q3 Fig. 2-Graph

of both sides of the equation tanh BS = SS/CS a s functions of BS.

Approximate expressions may be obtained f o r the roots (yS), when eS (of either sign) i s small o r large, o r when S i s positive and nearly 1. eS Large and Positive. The value of tanh (PS), is very nearly 1. Equation 4 i s expressed a s tanh pS = 1 - 2e-2'3s + 2e-4'3s + Fig. 1 :Graph

13S . . . -- eS

(6)

of both sides of the equation tan yS = @ / C S a s functions of y S .

Solving by successive approximations, If eS is positive and g r e a t e r than 1, the root in the f i r s t branch disappears (dash-dotted line, Fig. 1). In this c a s e t h e r e is a pure imaginary root, y , = ip,. Substituting /3 in Eq. 1, PS

tanh PS = cs and

Equation 7 to the number of t e r m s given i s good to 0.002 f o r eS > 2. eS Large and Negative. The value of (yS), i s nearly TS. We set (YS), = TS + x. Then Eq. 1 becomes

142

THE THEORY O F ISOTOPE SEPARATION

and solving by successive approximations,

APPENDIX A Table 1 -Roots

143

of the Equation yS =

(6s) tan y~

Equation 9 also gives ( Y S )for ~ eS very l a r g e and positive, and to the number of t e r m s given i t i s good to 0.06 for - eS > 2. eS Small, Positive o r Negative. The value of tan (yS), is nearly infinite, and ( Y S ) ~ i s nearly 7r/2. Set (yS), = (7r/2) - x. Then Eq. 1 becomes IT

expanding the cotangent. The s i z e of x i s very small and the root is accordingly

Equation 11 to the number of t e r m s given is good to 0.001 for l e ~ lS 0.20. The expression also gives the succeeding roots ($Ik by substituting (2K - l)7r/2 for 7r/2. Explicitly,

eS Nearly 1. Since (YS), is s m a l l , Eq. 1 may be written as

Equation 14, derived h e r e f o r eS < 1, can be shown to hold for ES > 1, giving the expected imaginary values f o r It holds to 0.001 for I s - 1 =Â 0.1. Roots of Eq. 1 for eS = [-2.0(0.1) - 1.0; -1.0(0.5) + 1.0; 1.0(0.1)2.0; 5D] a r e listed in Table 1.

(a.

whose solution is REFERENCES Cohen, K., Packed Fractionating Columns and the Concentration of Isotopes, J.Chem. Phys., 8: 588 (1940). Cohen, K., Report A-530, Feb. 4, 1943.

APPENDIX B

145

Now it is known f r o m the theory of partial differential equations1 that equations of the form

Appendix B EQUILIBRIUM TIME OF SQUARE CASCADES FOR N

-1

This section outlines the procedure that will solve Eq. 2.1 in the general case when N is not small. The differential equation i s , when P = 0,

with boundary conditions of the type in Eq. 3 a r e solvable by iteration. It remains t o choose the most efficient procedure. The expression in parentheses on the right-hand side of Eq. 4 i s , in t e r m s of N, aR 2eR - - =

2 ( -~ NT)

according to Eq. 1.77, where T - NT is the net transport of desired material. Since a ~ / a sis always positive and since the transport f o r the initial conditions chosen is also positive, the entire expression Consider the simple case (K'= 0 0 , K = 0, P = 0) -a cascade operated under total reflux, with no reservoir on top, and the concentration a t the bottom maintained constant. The boundary conditions a r e

The substitution of R instead of N as dependent variable makes the boundary conditions linear.

is essentially positive. Thus an iteration process in which we begin by overestimating this t e r m will converge to the solution from low estimates of the equilibrium time; whereas if we begin by underestimating the t e r m we will overestimate the equilibrium time. 2 aR The expression -- is the difficult term; the most efficient l+Rds iteration would seem to be performed onthis expression alone, getting successive equations of the form

to solve, where Rk is the kth approximation to R and

At the same time, Eq. 1 takes the form

^Maurice Gevrey, "Sur l e s equations aux deriv6es partielles du type parabolique," Gauthier-Villars. Paris. 1913.

146

THE THEORY O F ISOTOPE SEPARATION

We might take f o r fo either zero o r 4e%e2's/(l+ b e 2 " ) corresponding to the initial and final values of R and aR/as (underestimates and overestimates of f ) . However, unless fir is particularly simple, Eq. 5 also has to be solved by iteration. Thus practically the solution by way of Eq. 5 stoo fails after RO. Even the choice f,, = 4 e ~ o e ~ ' ~ /+( ~l ~ e ~ ' ~ ) i cumbersome; f i r s t - o r second-degree rational algebraic functions of s which lead to special c a s e s of the confluent hypergeometric equation a r e more convenient. An iteration method that can be continued indefinitely is obtained by solving equations of the f o r m

Appendix C THE HOLDUP FUNCTION

-

The flow in any section of an ideal cascade i s equal to ~AU/E'. Over a s m a l l section AU d ' v / d ~ ~ .Consider, instead of U = PV, the expression H = PhK, where h i s the holdup p e r unit flow and

where

Here the homogeneous p a r t of the equation has constant coefficients, which permits the particular solutions to be obtained operationally.

i t is clear that (2/e2) AH for the small section will be equal to the holdup of desired isotope. Now a large cascade section may be considered a s composed of small sections, the product of one section being the feed of the next. Likewise AH f o r the l a r g e section i s the sum of the AH'S of the s m a l l sections. Hence (~/E')AHover any s e c tion of an ideal cascade, l a r g e o r small, represents the desired mater i a l holdup in that section. The integral of Eq. 1 with the constants chosen s o that K(0.50) = d ~ ( 0 . 5 0 ) / d=~0 i s

This is called the "elementary holdup function." It i s the analogue of V (not V). To illustrate the use of H, consider the net desired material holdup i n the cascade section B of Fig. 4.6. Evaluating AU' around this section,

THE THEORY O F ISOTOPE SEPARATION

148

APPENDIX C

149

If we define a general function

F2(N1,N2,N3,N4) = N1 ln

L ~ v'(NA) ) + Z B ~ B - NA)

VN( A )

i U f = PBVfINB) - PB V f ( N ~-) PB (NB - NA) dN V' (NA)

+ (N1 - 1) (R2 - R3) +

Nl (R, - Rd RyRa

2h (N.D.M.H.)~ = 7PC F2(Nc,N~,No,No) â‚

(N.D.M.HJB

In exactly the s a m e manner,

hPB K(NB) - hPB K(N,i) - hpefNa - NA)

R Fl(Nl,N2,Ns) = (2N1 - 1 ) In Ñ R3

Then in a s i m i l a r fashion it i s found that

F r o m Eqs. 4.39 and 4.40, this reduces to

AH

Fl(Nl,N2,N3)

where the function Fl i s

gives the expression

i u l = pBV ! ( N ~ )+ ( I B -

-"2+ (N1 - 1) (R2 - R3) - N. Ra

2h

= 7 PB F2(NB9NB,NA,No) â‚

d

K(NA)

Substituting for V = (2N - 1 ) In R , K from Eq. 2, and

(N.D.M.H.)~ =

(N.D.M.H.)~ =

2h pCF2(Nc,Nc,NB,No) ec 2h

PA F~(N~,N*,N,,,N~)

â‚

(N.D.M.H.)?

gives

2h cc

= 7Pp F 2 ( N A 9 N A 9 N B , ~ J

REFERENCES Lefkowitz, H., B. Schwartz, and H. Mayer, Columbia Ser. No. 4R-42, Apr. 12, 1943. Mayer, H.,I. Kaplan, and K. Cohen, Report A-396, Dec. 2, 1942.

and

Then the equation for the net desired material holdup i s

2 2 N.D.M.H. = -2- AH - 2 (hNo) AUf

(5)

APPENDIX D

151

which integrates to give Q" ( a o - l ) lp-=Q

R" ln-+R

1-N ( a o - 1 ) in- 1 - N"

Here Q i s the initial quantity of the charge in the distilling flask, and N i s i t s mole fraction. Introducing the cut 6 = Q'/Q = (Q - Q")/Q, where Q' i s the volume of the distillate,

Appendix D RAYLEIGH DISTILLATION

cascades which was presented in Chaps. 1 to 5 was The theory built upon elements that separate according to the law R' = QR R~~=

$R

The mole fraction of the destillate, N', i s given by

a = constant) ( ft = constant)

Another common type of element which does not obey this simple law will now be considered. An example of such an element was f i r s t encountered by Lord Rayleigh while he was investigating batch distillation. Consider the course of a distillation. At any instant the vapor (mole fraction n) i s in equilibrium withthe composition of the liquid remaining in the flask (mole fraction N") according to the relation

Since the composition of the vapor leaving the liquid is different from that of the liquid, the composition of the residue changes a s the distillation proceeds. This is expressed by the conservation relation ~ ( N " Q)" = n d ~ ~ '

It is evident that Eq. 5 is fundamentally different from Eq. 1 and represents a new kind of separating element. However if a. - 1 i s small, then R"

R = 1 + (ao - 1 ) In (1 -6)

+ order

R' 1-6 -R= I + ( % - 1 ) - i n 9

1-6

1

of (ao -

(7)

+ o r d e r of ( a o -

Provided 6 i s constant o r nearly so,' Eq. 7 is approximately of the form of Eq. 1. Thus the infinitesimal c a s e of Rayleigh distillation can be treated according to the theory of Chaps. 1 to 5, with effective simple-process factor a = 1 + e given by

(3)

where Q" i s the quantity (in moles) of the liquid residue a t any instant. Combining Eqs. 2 and 3,

The change in concentration of the residue (N") and distillate (N') relative to the initial concentration has characteristic peculiarities. F r o m Eq. 7, '1t was seen in Eq. 1.19 that it is so in an ideal infinitesimal cascade.

THE THEORY OF ISOTOPE SEPARATION

APPENDIX D

- I)'] for 0 = 0, decreasing steadily because of mixing of the successive distillate fractions (which a r e of changing mole fraction). The value of 6V itself (column 4) increases initially a s 6 increases, in spite of the mixing effect, just because more is distilled: a s 0 approaches 1 the mixing is so serious that it causes a decrease in 6U in spite of the increase in amount distilled. (If everything is distilled, evidently no separation is accomplished.) The maximum separative power per unit distillate may be obtained for the entire charge by dividing the distillate into many small portions, instead of collecting it all together. Then the separative power developed reaches G ( f f o- 1)'/2. It is perfectly plain that any separation process that obeys Eqs. 2 and 3 is formally equivalent to Rayleigh distillation. Of interest is the evaporative centrifuge (Chap. 6, Sec. 4), where the process is indeed a distillation, but the concentration change (Eq. 2) is caused by the centrifugal field instead of by a difference in vapor pressures. is a maximum [=

from which Table 1 is constructed. Table 1 -Separation

and Separative Power in Rayleigh Distillation

REFERENCES Cohen, K., Report A-50, Jan. 30, 1941; Report A-60, October 1941. Cohen, K., and I. Kaplan, Report A-163, Apr. 25, 1942. Rayleigh, J. W. S., Phil. Mag., 42: 493 (1896).

6 = cut; fraction of charge in distillate N = mole fraction of charge N1=mole fraction of distillate N" = mole fraction of residue a,,= equilibrium simple-process factor SU = separative power G = rate of feed to unit

By taking larger cuts it is possible to increase indefinitely the changein concentrationof the residue (column 3 of Table 1). However, the separativepower remains finite. If it is supposed that the element continually separates G moles/sec of charge into 0G moles/sec of distillate and (1 - 6)G moles/sec of residue, then by Eq. 1.42,

The value of 6U is found in column 4. More fundamental than 6V is 6 V / 6 ~ which , gives the separative power per unit amount distilled (column 5). It is seen that this term

153

155

APPENDIX E

provided

Appendix E PROPERTIES OF CONCURRENT TWO-PHASE ELEMENTS A type of element which occurs inthe case of concurrent contacting of two separate phases, and in many other instances, e.g., the concurrent centrifuge, is illustrated in Fig. 1. Figure 2 shows a simple cascade of such units. The element has entrances for two separate

in the equilibrium state. Here k > 0 is a transfer coefficient and z is a parameter expressing the size of the unit. F o r example, in a catalytic-exchange process, z would be the volume of catalyst and k would be proportional to the surface a r e a per unit volume of catalyst. These equations a r e consistent with the equations of conservation L N(z) + 1 n(z) = constant = L N(0)

+ 1 n(0)

(2)

and solving Eqs. 1 and 2 together it is found that, after some r e arranging, N(Z)- N(0) = (1 - 9) [n(0) - ~ ( 0 +) ( f f - 1)N(1 - N)] (1 - e-bz) n(z) - n(0)

=-9

(3)

[n(O) - N(0) + (a - 1)N(1 - N)] (1- e-^)

where

Fig. 1 -A

concurrent two-phase element.

When z streams, and unlike the element of Fig. 1.1 (Chap. I ) , there is no particular necessity for the s t r e a m s to be of the same composition. This extra degree of freedom gives considerable flexibility and r e sults in more efficient cascades. There will be exhibited phenomena associated with the degree of equilibrium attained in each unit, and the dependence on cut characteristic of Rayleigh distillation elements (Appendix D) is absent. The equations governing the transfer of material from one phase to another a r e dN L -= -k [(N - n) - ( a - 1 ) ~ ( 1 N)] dz

-

W,

the last t e r m in Eq. 3 approaches 1, s o that

is the percentage of equilibrium attained in the unit. The separative power of the element is

Expanding the value functions about V [ N ( ~ )and ] collecting,

+

and substituting from Eq. 3,

y1

[ d o ) - n(z)]

1

156

THE THEORY OF ISOTOPE SEPARATION

APPENDIX E

STH OR TOP STAGE

It is seen that fiU is a function of (3 and N(0) - n(0). Taking N(0) = n(O), which corresponds to the no-mixing case for single-entrance elements, the following is obtained:

(s

and from Eq. 4, the separative power per unit length is -5U - -

z

k(ff - l y , p 2 - ln(1 - 6 )

+ 11TH STAGE

(no-mixing case)

which has a maximum at (3 = 0.715.

5U

-=

z

0.40724

2

sTH STAGE

(no-mixing case)

*

However, this choice of N(0) - n(0) is not the most efficient. Differentiating 6V from Eq. 7 with respect to N(0) - n(0), it is found to be a maximum when " ( a N(O)- n(0) = 2-6

- DNO

- N)

(s - W

H STAGE

which is

and a relative maximum with respect to 6 when 6 = 0, namely,

-6U z

-

k(a-If 4

f ST STAGE

(optimum concentration relation)

By comparing Eq. 12 with Eq. 9 it i s seen that the optimum concentration relation permits an increase in separative power of 19.6 per cent over the no-mixing case.

Fig. 2-A

cascade of concurrent double-entrant units (rectifying section).

158

THE THEORY OF ISOTOPE SEPARATION

It i s further evident from Eqs. 4, 8, and 11 that 0 = % i s the most efficient cut, because this value r e s u l t s in a minimum value of ( L + Z) for a given 6. It is instructive to compare these two c a s e s (no-mixing and optimum-concentration relation) by considering the net transport for the cascade of Fig. 2 with 9 = ?h. This is

APPENDIX E

159

Total volume = - (2 -13) In (1 - 6 ) dl - q ) 6 Total pumping =

1

- q)

2-/3 (3

Number of stages = 2 - 5 (1 - q)(3 On the other hand, f o r a given flow Ls the maximum possible t r a n s port through the sth stage is Ls ( - 1)Ns+l(l - Ns+l) ff

which occurs when -- vs

1-vs

-

1

Ns+1

a l-Ns+l

The percentage transport ( r e f e r r e d t o the maximum) is therefore

When N(0) - n(0) a r e related by Eq. 10 and 9 = ?h,q turns out to be %; when they a r e equal, q = (3/2. Thus in the no-mixing c a s e the net transport is related to the degree of equilibrium. The r e s u l t s just obtained may be generalized by introducing the parameter q directly into the formula (Eq. 7) for <5U.It i s then found that (for 0 = lh)

Fig. 3-Factors for total pumping,volume, and number of stages; a comparison of nomixing cascade and "constant per cent transport" cascade (q = 0.5).

F r o m Eqs. 14 and 15 expressions a r e obtained for the total volume of a cascade, the total pumping, and the total number of stages. These a r e , for the constant-transport case,

The volume and pumping a r e both minimum with respect to q for q = ?h. Figure 3 gives the three factors a s a function of fi for the two choices q = 0.5 (minimum volume) and q = (3/2 (no mixing). The minimum volume for q = 0.5 i s a t 6 = 0 and i s 19 p e r cent l e s s than the

160

THE THEORY OF ISOTOPE SEPARATION

APPENDIX E

minimum volume for the no-mixing case which occurs at 0 = 0.715. Unfortunately both the pumping and the number of stages a r e infinite

It is obvious that the more general "constant per cent transport" cascade offers greater opportunities than does the no-mixing case which might a priori be considered the most efficient. A few interesting possibilities a r e noted which illustrate the flexibility of the constant-transport cascades. The factor for minimum volume in the no-mixing case is 9.82 and corresponds to a 71.5 per cent approach to equilibrium. In the constant-transport case this volume factor corresponds to many choices of q and 0;among others, to q = 0.5 and?! , = 0.82. In this case, however, the total pumping is 24 per cent l e s s than in the no-mixing case, whereas the number of stages is only 3 per cent higher. It is thus possible to save a considerable amount of pumping while losing only slightly on the number of stages. Another constant-transport possibility with the volume 9.82 is the choice q = 0.4 and ft = 0.77. The pumping is 15 per cent l e s s and the number of stages 4 per cent l e s s than the no-mixing values for ft = 0.715. The no-mixing case thus represents no extremal. A good balance between volume, pumping, and stages will be struck by the choice q = 0.40 and = 0.80. But if one of these three factors overweighs in importance the others, different values should be chosen. Thus if it is especially desirable to reduce the volume to a minimum, the choice might be q = 0.5 and (3 = 0.5. If it is desirable to reduce to the limit both the pumping and the number of stages, the choice should be q = 0.4 and 0 = 0.90, o r even 0 = 0.95. If the number of stages is not important, then q = 0.5 would be chosen.

REFERENCES

I

Fig. 4-Factors and fi.

----

NO MIXING

for total pumping, volume, and number of stages as functions of q

at 0 = 0. It is seen, therefore, that the significant region will lie between ft = 0.95 and ft = 0.50. Figure 4 gives the factors for volume, pumping, and number of stages for q = 0.3, 0.4, 0.5, and 0/2 within this range of 0.

Cohen, K., and I. Kaplan, Report A-302, Sept. 19, 1942. Kaplan, I., and K. Cohen, Report A-301, Oct. 29, 1942.

161

INDEX

Aston, F. W . , 103

Bardeen, J., 131 Beams, J. W., 105, 113 Boundary conditions, rectifier, 87 stripper, 87-88 Bramley, A., 105, 119, 124, 131 Brewer, A. K., 105

Campbell, G. A., 50 Cascade, 3, 5 combined, 78 cut in, 5 design of, 11 efficiency of, 6 equilibrium time of, 24-27, 38, 82 flow in, 13-15 total, 9, 77-78 ideal, 5-10 stages of, 5 ideal and nonideal, 10-12 losses in, 67-74 per stage, 69 production rate of, 12, 94-102 rectifier of, 21 relaxation time of, 25 square, 30-44 equilibrium time of, 39, 144-146 production loss of, 101 production rate of, 94, 96 disturbance function of, 96 steady state of, 31, 62, 65 transient phenomena of, 62 -64 Type Ioperation of, 40-43, 47, 54

Cascade, square, Type I1 operation of, 41 -44, 47-53 squared-off, of countercurrent elements, 34-35 rapid approximate solution for, 3538 scheme for, 38 stages of, 3-5 number of, 3, 12 size of, 9 stripping sections of, 21-24, 77-78 Centrifuge, concurrent flow, 105, 114 countercurrent flow, 105, 119, 121-122 evaporative, 104-105, 111-114 separative power of, 109-1 11 Centrihge separation, design of, 105-106 differential equation of, 106, 109 history of, 103-106 maximum separative power of, 109111 theory of, 105 Chapman, S., 131 Cohen, K., 29, 38, 61, 78, 124-125, 136, 143, 149, 153, 161 Column, countercurrent, 31 number of stages, 31 -32 fractionating, 30 relaxation phenomena for, 54-61 thermal diffusion, 30 Combination of separation plants, 74-78 Concurrent contacting, 131 Concurrent two-phase elements, 154-161 Control, hydrodynamic, 72-102 Control problem, 79-102 Countercurrent centrifuge, 30 Countercurrent column, 31 number of stages, 31 -32 Countercurrent contacting, 129 Countercurrent devices, 30-3 1 Cowling, T. G., 131 Cut in cascade, 5

THE T m O R Y O F ISOTOPE SEPARATION

Debye, P., 131 Deuterium, separation of, 126, 132 Deuterium separation, catalytic exchange method, 132-136 Diffusion, thermal, 131-132 Dirac, P, A. M., 29, 109, 125 Distillation, Rayleigh, 150-1 53 separation in, 152 separative power in, 152 Doetsch, G., 94 Dunning, J. R., 29, 38

Elements, number of, 14, 19 Equilibrium fractionation, 67 Equilibrium time, 24-27, 38, 82 of square cascades, 39, 144-146

Feynman, R. P., 67 Flow, total, 9, 15, 23 Fluctuations, 80 -90 nonperiodic, 90 periodic, 90 Foster, R. M., 50 Fuchs, K., 29, 84, 91, 102 Furry, H,, 119, 131

Joly, J., 104 Jones, R. C., 119, 131

Kaplan, I., 29, 38, 61, 78, 106, 125, 136, 143, 149, 161 Keith, P. C., 136 Kuhn, W., 105, 119 Kurti, N., 29

Laplace transform, 84, 89, 93 Lefkowitz, H., 78, 149 Lindemann, F. A., 103

Martin, H., 105, 119 Mayer, H., 78, 84, 149, 161 Mixing, 6, 1 5 Mole fraction, 5 Mulliken, R. S., 104 Murphree, E. V., 125

,

Net transport, 21, 26

Gevrey, M., 145

Onsager, L., 119, 131

Holdup function, 25, 77-78, 147-149 in column, 59 net desired material (N.D.M,H.), 25-27 in top reservoir, 59 Humphreys, R. F., 105, 113 Hutchison, C. A., 61, 136

Peierls, R., 29 Plants, separation, combination of, 74-78 Poole, J. H. J., 104 Process factor, centrifuge, 105 infinitesimal, 9 large, 1 5 simple, 5 Production rate, variations in, 94-102

Isotope separation, 27-29 by chemical exchange, 126-127 by thermal diffusion, 131 uranium, 15

Rayleigh, J. W. S., 150, 153 Relaxation phenomena, 54-61

INDEX

Schwartz, B., 78, 149 Separation, isotope (see Isotope separation) two-phase, 126-131 Separation plants, combination of, 74 -78 Separative power, 13-15, 19-21, 37-38 (See also Centrifuge; Distillation) Separative work, 72 Simon, F., 29 Skarstrom, C., 105-106, 113 Square cascade (see Cascade, square) Stages of cascade (see Cascade)

Thermal diffusion, 131-132 Thompson, W. I., 62, 78, 136 Transcendental equation, roots of, 139- 143

Transport, loss of, 90-91 maximum, 33

Urey, H. C., 105-106, 125, 133

Value function, 13-14 elementary, 15, 17

Waldman, L., 131 Waste material, 18 Wirtz, K., 131 Work function, 64