12
,1 ;¡II 1 '1 I!! : .1 ¡ti '11 ~.! Chapter 8 CENTRIFUGES 1. IDSTORICAL INTRODUCTION The effect of the earth's gravltattonal fleld on the eomposttíon of the gases In the atmosphere at varlous helghts has long been recog- nized. For, If PI ls the partíal pressure of a gas of molecular welght MI, accordlng to the Boltzmann dlstrlbutton law (6.1) where h ts the helght; g ls thegravltatlonal constant; R ls the gas constant, 8.314 x lO" ergs¡OK/mole; and T ls the absolute tempera- ture. Then for two gases of dlfferent molecular welghts M 1 and MI (6.2) The suggestlon that gravltational or centrlfugal !lelds are suitable for lsotope separation was made by Llndemann and Asten' In 1919. In fact, they proposed that samples of neon taken at ground level and at 100,000 ft would show a sensible shlft In tbe ísotoptc abundance ratio. In a centrlfuge, vastly more powerful Helds are avallable. The energy per mole at radius r in a centrifuge rotating wlth angular ve- locity w ls ~ Mi(wr)2. This expresslon for the energy replaces - MIgh in Eq. 6.1, givlng () MI(wr)2 PI r = PI(O) exp 2RT (6.3) I Phll. Mag., 37: 530 (1919). 103 104 THE THEORY OF ISOTOPE SEPARATION Between the perlphery r = r a and axis r = O, for two dlfferent gases, from Eq. 6.3, (6.4) where wr 2 ls the veloclty of the inslde wall of the rotor, the lnner perlpheral veloclty. In a blnary system, lf N ts the mole fractton of the ísotope havlng mass MI' PI + PI = p. N = pJp 1-N = palp (6.5) and p - p(r)ls the total pressure. Then Eq. 6.4 becomes ( --L) . (--!L) ex (MI - M2) (wrl)· 1 - N 1- N P . 2RT r. o (6.6) It ls usual to take MI > MI' and accordingly (6.7) \ so that Eq. 8.6 takes the form ) where Qo ts the (equllibrium) simple-process factorof the separation. The value of the simple-process factor for a peripheral veloclty of 300 meters/sec, T = 300 o K, and a mass dlfference of one unlt ts, by Eq. 6.6, Qo = 1.0183. Shortly after the suggestlon of Llndemann and Aston was made 1 , at- tempts were made, In rather crude apparatus, by Joly and Poole and by Mulllken2 to reallze this separation factor, but these were unsue - cessful. The Idea of the evaporative centrifuge was introduce~ a~ t~e time by Mulliken. In this method a small amount of ltqutd is same h .h Dur introduced lnto the centrifuge, forming a layer at t e per ip ery. - 1 Phll. Mag., 39: 372 (1920). •J. Am.Chem. soc., 44: 1033 (1922).

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Page 1: Cohen Capítulo de Centrífugas

,1;¡II

1

'1I!!:

.1

¡ti

'11~.!

Chapter 8

CENTRIFUGES

1. IDSTORICAL INTRODUCTION

The effect of the earth's gravltattonal fleld on the eomposttíon ofthe gases In the atmosphere at varlous helghts has long been recog-nized. For, If PI ls the partíal pressure of a gas of molecular welghtMI, accordlng to the Boltzmann dlstrlbutton law

(6.1)

where h ts the helght; g ls thegravltatlonal constant; R ls the gasconstant, 8.314 x lO" ergs¡OK/mole; and T ls the absolute tempera-ture. Then for two gases of dlfferent molecular welghts M1 and MI

(6.2)

The suggestlon that gravltational or centrlfugal !lelds are suitablefor lsotope separation was made by Llndemann and Asten' In 1919.In fact, they proposed that samples of neon taken at ground level andat 100,000 ft would show a sensible shlft In tbe ísotoptc abundanceratio.

In a centrlfuge, vastly more powerful Helds are avallable. Theenergy per mole at radius r in a centrifuge rotating wlth angular ve-locity w ls ~ Mi(wr)2. This expresslon for the energy replaces - MIghin Eq. 6.1, givlng

( ) MI(wr)2PI r = PI(O) exp 2RT (6.3)

I Phll. Mag., 37: 530 (1919).

103

104 THE THEORY OF ISOTOPE SEPARATION

Between the perlphery r = ra and axis r = O, for two dlfferent gases,from Eq. 6.3,

(6.4)

where wr2 ls the veloclty of the inslde wall of the rotor, the lnnerperlpheral veloclty.

In a blnary system, lf N ts the mole fractton of the ísotope havlng

mass MI'PI + PI = p.

N = pJp

1 - N = palp

(6.5)

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

(--L) . (--!L) ex (MI - M2) (wrl)·1 - N 1 - N P . 2RTr. o

(6.6)

It ls usual to take MI > MI' and accordingly

(6.7)

\so that Eq. 8.6 takes the form

)

where Qo ts the (equllibrium) simple-process factorof the separation.The value of the simple-process factor for a peripheral veloclty of

300 meters/sec, T = 300oK, and a mass dlfference of one unlt ts, byEq. 6.6, Qo = 1.0183.

Shortly after the suggestlon of Llndemann and Aston was made1

, at-tempts were made, In rather crude apparatus, by Joly and Poole andby Mulllken2 to reallze this separation factor, but these were unsue -cessful. The Idea of the evaporative centrifuge was introduce~ a~ t~e

time by Mulliken. In this method a small amount of ltqutd issame h . h Durintroduced lnto the centrifuge, forming a layer at t e per ip ery. -

1 Phll. Mag., 39: 372 (1920).• J. Am.Chem. soc., 44: 1033 (1922).

Page 2: Cohen Capítulo de Centrífugas

__ "t1uO;;-

__ rn vacuum-chamber centrifuges, which_ _. nuraIlon and thermally lsolated to eliminate convec-

__ -"".n:urrents, flnal1y permitted successful centrlfugalisotope separa-tton to be obtalned In 1939. Beams and Skarstrom2 at the Unlversityof Virginia, usíng the evaporaUve centrl!uge method on CCI..,reporteda 13 per ctmt change in the Cl35/Cl37 ratio. Shortly thereafter Hum-phreys,3 using the same technique on ethyl bromlde, altered theBr'78/Br81 abundance ratio by llper cent.

Thesimple-process factor of centrifuga} separatlon depends on1yon the mass difference. Unlike other methods (e.g., diffusion throughbarriers, where a varles as AM/M) tt ts no more dIfficult for heavyelements than for líght, In addítton, ít ls posslble to calculate a priorl.tbe separatíon to be expected from centrtíugal apparatus wltbout anyundetermtned constants or tbeoretical uncertaíntíes.

The simple-process "flow-through centrl!uge," as the concurrentflow type was called, ts an applicaUon to gaseous flow oí the conUnu-ous cream separator, As originally conceived, a single stream oí~as enters one end oí a rotor through a hollow shaft, and two streams,re taken off the other end, one Irom the periphery and the other near'le axis. This method produces a small change in conceiltratlon perachine.The countercurrent flow type was designed to atta in considerableoaratton in a single centr1fuge, thus reduc1ng the number of stagesiuíred and the amount of material circulated between stages, As~inally proposed by Urey," círculatton was to be established byinuous distillation of UF8 from the bottom cap of the rotor. Ther .was to be condensed on the top cap. 'The heavy liquid wouldbe forced out to the perlphery and would flow down the walls toottom cap, countercurrent to the vapor flow, and complete the

Bramley and Brewer5 and Marttn and Kuhn' in Germany pro-to establish countercurrent flow by thermal convectlon.

Modern Phys., 10: 245 (1938); Rev. Sel. Instruments, 9: 413 (1938).Rev., 56: 266 (19391.Rev., 56: 684 (l939).s on Progress in Physies, VI: 72 (1939).1, 92: 427 (1940).Ik. Chem., A189: 219 (1940).

_~ ",--.,,--'''feXpOse the theory of centrifugal sepa-_ ~.-YUiOU8 tYpes of centrl!uges. The important questlon of

- meehanical deslgn wll1 be dealt wlth only In so lar as lt ls essentlalto the understandlng 01 the process deslgn. The partlal differentlalequatton 01 centrifugal aeparatlon la derlved 'In sec .• 2. In sec, 3 thetheoretlcal maximum separative power 01 a centrifuge i8 found. Thesolutlon for thé evaporaUve centrl!uge ls glven In Seco 4, that lorthe concurrent centrifuge In see, 5, and that lor the countercurrentcentrifuge in Seco 6. Sectlon 6 investigates at length the properUes 01the countercurrent centrifuge.

The wrlter was lntroduced to the art of eentrifugatlon by C. Skar-strom, formerly of tbe UnlveraUy 01 Virginia and later wlth the Stand-ard OU Development Campany. Harold C. Urey parUcipated actively1nthe early work, until bis other responsibutUes became too heavy,contr1buUng ideas and physlcal interpretaUons oí mathematlcal re-sults. Irving Kaplan shared in 1.11 the developments subsequent toDecember 1941, notably the work on cascades oí centrl!uges (in-corporated largely in Chaps. 1 to 5) and the design problem •

Z. PARTIAL DlFFERENTIAL EQUATlON OF CENTRIFUOE

As a preliminary to the derivaUon of the dl!ferenttal equatlon, con-slder íor a moment a centrifuge contalning a single pure gas. Thegas ts supposed to rotate uniformly at the same speed as the rotor,t.e., w radlans/sec; to have no other motíons; and to be lsothermal.Choose a cylindrlcal coordlnate system, rotattng wlth the gas, withthe origin at the axes oCrotatlon. The eUect of the rotatton ts to setup a centrifugal force w2r per un1t mass. Unde~ the lnfluence óí thisforce, a pressure ¡radient isestablished according to the hydrody-namíe equatlon

(6.8)

Here pis the denslty of the gas mixture in grams per cubic centime-ter. The quantUles p and pare related by the gas law

RTpp"--M

Combinlng Eqs. 8.8 a"" It ,.

Page 3: Cohen Capítulo de Centrífugas

The denstty dlstrlbutlon (Eq. 6.10), co!tsldered from a molecularpolnt of vlew, 18 a dynamlc equtllbrlum between the effect of randommotlon of the molecules (dlffuslon), whlch tends to erase densttygradlents, and the effect of the centrlfugal fleld, whlch tries to pile a11the molecules at the perlphery. The diffuslon counter to the densitygradlent glves a current densUy (in grams per seeond per squarecenUmeter)

ap Mw.r-O-IO'-Op-ar RT

Rere O Is the coefficlent of seU-difluslon. The outward current persquare centlnieter caused by theeentrlfugal force, which balancestMs Inward current, must therefore be

(6.11)

In mixtures, Eq. 6.11 holds for each component separately

(6.12)

01 Is the diffuslon eoeffielent for nioleeules of type 1tbrough the wholemixture. For Isotopie mixtures the exaet meanlng of the diffusloncoefficlent Is of no slgnlf1cance, and henceforth the subscrlpt w1ll bedropped. EquaUon 6.12 ts the deslred expresslon for the flux of mole-cules caused bya centrlfugal fleld, and U ís now posslble to derivethe dlfferenUal equatíon,

Conslder then a centrtíuge contalnlng a perfect blnary gas mixture,Isothermal and rotaUng uniformly. In order to accompl1sh separa-tíon, the gas ts made to flow radla11y and axia11y. In terms of thecylindrical coordinate system chosen as befare, the motion of the gasas a whole (drift velocUy) is given by the compónents

(Radial) . u = u(r ,z)

(Azlmuthal) v = O

(Axial) w = w(r ,z)

Because of thls motion of the gas, the equUibrlum dlstrlbutlon (Eq.6.3 or 6.10) Is conUnua11y upset. The system moves material In aneffort to reestabllsh equ1l1brlum¡ In thls way a transport of the de-slred material 18 set up,

107 108 THE THEORY OF 1S0TOPE SEPARATION

Imagine a small annulaz reglon In the rotor, bounded by the cyl1n-ders r = ro and r •• ro + 6r and by the planes z = Zoand z = Zo+ 6z•. Ina short Ume Interval 6t there flows across the cyl1ndrlcal face r lO roan amount of deslred Isotope (denoted by the subscrtpt 1),

(6.13)

In thls equaUon P1u ts the flow caused by the drift velocity, PlUe. Isthe flow Impressed by the centrifugal fleld, -o apJar ts the con-trlbuUorf from dlCfuslon agalnst the densUy gradlent, and 2lTro6z Isthe area of the cyl1ndrlcal face. Llkewlse the flow across the planeface z = Zols

(6.14)

The change In content of deslred material In the volume element tstherefore

Introduclng Eq8. 6.13 and 6.14 tnto 6.15 and makíng use of the rela-tlon of Eq. 6.12 for uc, (dropplng the subscrlpt O),

1•• ( M w1r e» ) &(P1w) O &I~l~ " r Dp ~ - D ~ + P1u - + &at • - r ir 1 RT &r az. z(6.16)

Now

MI' MI NPI = RT PI = RT P (6.17)

and

(6.18)

SubsUtuUng Eq. 6.17 In Eq. 6.16, ustng Eq. 6.18 and notlctng that Op= constant, tt 18 found that

Page 4: Cohen Capítulo de Centrífugas

CENTRIFUGES

~ = .QE. -!. [r ~ + (MI - M,)wlra N(l - N)]P at r élr ar' RT

1 él él . al- r ar (Npru) - az (Npw) + D aza (.Np) (6.19)

whtch ís the destred parUal differenUal equatíon of centrifugal sepa-ratíon,

The three ktnds of centrifuge descrtbed In Sec. 1 are governed byspeclal cases of Eq. 6.19. In the evaporaUve centrifuge w = O andru :: constant¡ ln the concurrent and countercurrent centrifuges u = O,and w descrlbes the respecUve types of flow. The solutlon of Eq.6.19 for these cases ts the subject of the subsequent secttons devotedto the particular centrlfuge types.

3. MAXIMUM SEPARATIVE POWER OF A CENTRIFUGE

In thls section the maxtmum separative work that can be performedby a centrifuge' wUl be determined. It wUl be used as a standard toevaluate the efftclency of the vartous centrifuge designa,

Uslng the result of Seco 6 (Eq. 1.60),

(T- NP) (N' - N)6U = N2 (1 - N)2

The net transport oi desired material (in moles per second) across asuriace element do ís, accordlng to the analysls given ín a precedlngsectíon,

(6.20)

where ro ts a unit vector along a radlus through do••uel ts the vectorvelocity created by the centrifugal field directed along ro and w1thmagnitude glven by Eq. 6.12, and the symbol V ts the vector gradlentoperator.

Thus the separative power for a small slab of tblckness ds, withfaces do normal to VN and sldes parallel to VN, ts

dU ~ DpRT (6.21)

I Followlng Dlrac.

109 110 THE THEORY OF rsOTOPE SEPARATION

The symbol • lndlcates the scalar productoTaklng VN as a variable and da ds and everythlng el se constant,

Eq. 6.21 ts a maxlmum when

1 w1r ).•VN • T (MI - Ma) RT N(l - N ro

that ts, along the dlrecUon of the centrifugal neld, and one-half tbeequtllbrlum value. Then

(8.22)

(dU) Dp [(Ma - M,)wlr]1 do drmax - RT 2RT

IntegraUng over the whole centrifuge, of length Z and radlus ra'

(6U)m&ll _ J (dU)m&ll _ ~~ [(Ma ;~,)Wl r ir,rl 2u Z dr

Dp [(Ma - M,) (wrl>,]1 .Z (moles/see)-1ff 2RT 2

•[(Ma - M,) (wr.).] .Z (grams/see)

-pD 2RT 2

(6.23)

Note that the expresslon In brackets ls the exponent of ao (Eq. 6.7)iThe mt.nlmum number of centrifuges oi length Z and perlphera

veloctty wr requlred to perform a glven separaUon ls, by Eqs. 1.44and 6.23, AU/(6U)max' The value of (6U)maxdependa on the perlPhera~veloctty of the fourth power, whlch puts a premlum on hlgh perlpheral veloclties. The separative power ls a~so proportlonal 'lo thelength. But for constant peripheral veloctty, the separation ls In-

orwl,.. • .!.

p

where T la lbe atreaa in lbe wall and p ta Ita denstty. The rotor wUl break when T •pwl rI la greater tban the tenaUe etrength of a metal.

Page 5: Cohen Capítulo de Centrífugas

• CENTRIFUGES

dependent of the radtus. The radlus affects the holdup and in tcurrent centrifuges the flow lnto the rotors, but ls o~herwls:~~e~:=vant. It ts usually chosen by mechanlcal conslderaUons.

SURFACE dc:r

\\\\\ /t\ Im\'t'/

\ IV

(0.)

(b)

Pig. lU-Stresses ia a th1 taU 'dlagram. a ro ng cyUnder. <a)Stress~ la a wallsector. (b)Vector

4. THE EVAPORATIVE CENTRIFUGE

In the evaporative centrifuge N ts lnd dslderlng the steady state, aN/ at = O andeEpeqn6

en1t9'of d

Z'and w '" O. Con-• • re uces to

~ :r {DP [r ~~ + 2AAr N(l - N)] - NPru} ;: O

\ where

(6.24)

(6.25)

111

112 THE THEORY OF ISOTOPE SEPARATION

8uppose vapor ls removed from the axts at arate L moles/seco

Then- :T 2wrZu = L (8.26)

Multtplylng Eq. 6.24 by the constant 2,..Z/ RT and íntegrattng,

(6.27)

where

" • 27rZ ~~ (moles/sec) (6.28)

The lntegraUon constant was evaluated by noUng that the left slde oíEq. 6.27 vanlshes at r '" O. '

Replaclng N by R as dependent variable,

(aR ') l 1l R -R(O)l" r ar + 2AAr R = L R(O) - R(r) 1 + 1 + R(O)

(6.29)

R -R(O) .Now 1 + R(O)' lE - AAra N(O), whtch can be neglected with respect to

untty. After thus slmplifylng Eq. 6.29, the solution ts readUy found to

be

and o~ lntroduclng Eq. 6.7, whlch In terms of AA ts\

(6.30)

tMs' expresslon may be wrltten

R(O) = R(r2)Oo(2::L) (6.31)

Equation 8.31 glves the relation for mole fractton ofproduct N(O)In terms of the composlUon of the charge N(ra) and the rate of pro-

Page 6: Cohen Capítulo de Centrífugas

'~I,.~1

1]

CENTRIFUGES 113

duetton,' If tt la aaaumed that the production la only a small Iracttonof the charge, or that successlve batches o{ product are kept separate,the separative power of the centrlfuge Is (ct. Appendlx D)

(6.32)

The value o{ 6U ís a maxlmum when L = 2'Y:

(6.33)

whlch ts the same as the theoreUcal maxlmum separative power of acentrlfuge (Eq. 6.23).

In the development [ust glven the gas was assumed to be Isothermal.Thls -pícture Is far too simple. When vapor ts wlthdrawn at the axis,the gas cools by' expanslon and a temperature gradlent ts Induced.The gradlent ts determlned by the balance between the rate of cooltngof the gas and thermal conductlvlty. As L Increases, the gradíentincreases until the gas reaches the dew potnt or the adiabaUc gra-díent, The mean effective temperature T eff ts therefore reduced as afunction of L.

An attempt may l1e made to account for this effect by writing inplace of Eq. 6.31

(6.34)

Slnce T/T eff Increases wlth L, the ssparatíon wll1 not {al1 off quite sofast with L In practice as Eq. 6.31 would tndícate,

A unlt welght oígas rotatlng near the perlphery with angular ve-'loe lty w has the angular momentum r~w; at the 'axis tt has none. Whengas Is moved from the periphery toward the axis, means must beprovided to absorb this excess angular momentum, which otherwisetends to Increasethe angular velocity of the gas at smaller radlt,This is accomplished by radial baffles2 (lookíng down along the axis,a star) which prevent the gas from rotating at angular velocities díf -ferent from that of the rotor. .

1 Thls result was found by Humphreys, Phys. Rev., 56: 684 (1939).a Beams and Skarstrom, Phys. Rev., 56: 266 (1939).

114 THE THEORY OF ISOTOPE SEPARATION

Although the evaporattve centrlfuge~ wtth L = 2", delivers 100 percent of the theorettcal separatlve power, lt ts not easlly adapted tocontinuouS operaUon.

5. THE CONCURRENT CENTRIFUGE

Llke the evaporattve centrlfuge, the concurrent centrifuge ls aslmple-process machlne slnce the enrlchment factor per machlne tslhnlted to ao'

In thls devlce axial flow ls used to obtaln an axial as wel1 as a ra-dial concentratton gradlent. The gas enters (Flg. 6.2) In two streamsat one end of the rotor and flows axially to the other end, where thestreams are removed separately. Durlng the passage through thecentrlfuge the streams tend to assume the (radial) equtlibrlum dts-trlbution (Eq. 6.6). The flow pattern adopted {or the concurrent cen-trifuge conslsts oí two thln cyllndrlcal streams located at radll riand r (perlphery) andflowlng parallel to the z axis. (The opUmumpositton ior the tnner stream ts not at the axis, as wll1 be seen sub-sequently.) There ts no radial mass flow.

A rotor wlth a double entrance (so that entering streams of twodifferent concentrattons can be used) will be considered; prevlouslythe literature has covered only single-entrance machines. The double-entrance centrifuge ts more flexible than the tatter. Cascades ofthese elements, whlch are more general than the element of Chaps. 1to 5, are studled In detall In Appendlx E. Among the more lnterestingfeatureq of thls study ts the result that the no-mixing case (bothentering streams of the same composition) is not an opttmum.

The solution given here follows roughly the demonstraUon by Mur-phree, whlch for the case considered ts superior to the more generalmethod that was originally used. The value of u is zero, and the flowsare gene rally so large that ba(:k-diffusion ts negligible. Then Eq. 6.19reduces for the steady state to

O :: Dp .!- (r ~ + 2AAr N{l - N)] - -aa (Npw)r ar ar z

(6.35)

In the reglon between r = ri

and r = ra, w = O, so that Eq. 6~35 inte-

grates to

_-ºP.. 2lT[r ~ + 2AArl N{l - N)] • !(Z)RT Sr

(6.36)

Page 7: Cohen Capítulo de Centrífugas

GAS EXIT

1

J11 CAP

Illi

z o z

1 t\

11

1

~..

ROTOR WALL 'f

1I '4

CENTRlFUGES

III"'--SHAFT

115

zoO

rll. 1I.2-Slmpl. madelof a ¡asICua concurrent c.ntrltu, ••

118 THE THEORY O, ISOTOPE SEPARATION

where f (z) ti the tranlport 01 dellred tlOtOpe acrol. a cyllndrlcalelement 01 unlt length between the two atream ••

But also, tf L(r.) 11 the flow ln the Itream at r • r.,

(6.3'7)

and

L(r.) N(r.,z) + L(rl)\N(rllz) • L(r.) N(r.,O) + L(rl) N(rllO)• B (a constant> (6.38)

Integratlng Eq. 6.36 between rl and r., constdertng AA N(1 - N)constant to terma 01 the order 01 (AA)',

_ ~ III(N(r.,I) - N(r"l) + AA,(rl-l1) N(l - N») • f (1) In t (8.39)

and substttutlng lor f (z) from Eq. 6.3'7 and for N(r"z) from Eq. 8.38,

~~ 211[ ~. - N(rl) (1 + i:-) + AA(ra - 11) NU - N)]

d r= L1

-d N(rl) ln.:.J.z rl

IntegraUng thts ordtnary dtfferentlal equaUon for N(ruz) and re-arranglng the result,

N(ruz) - N(rllO) = (1 - 9) (N(r.,O) - N(rl,O)+ AA(r: - r~) NU - N)] (l - e-ba)

N(r"z) - N(r.,O) • - 9 [N(r.,O) - N(r¡,O)+ AA(r: - r~) NU - N)] U - e-ba)

(6.40)

where

(6.41)

Page 8: Cohen Capítulo de Centrífugas

CENTRlFUGES

Th, ~eparatlv, power when N(r.,O) and N(ruO) ar, molt favorablyr,lat,d 11, accordln, lo Bq. 11 oí Appendlx B,

6U • [L(r.) + L(r.)) 8(1- 8) (1 - eobl

) (ln a )' (1 - 4.)12 <t + eobS) o ~

, 1

and the separat~ve power per unlt length ls

!!!...ZODP(ln l' (1- ••••) (¡-jrz RT ~ bzb + e061) , \ln"it

whleh ls a maxlmum wlth res t lo /ra/r •• 0.534, pee r. r, for any value of bz when

!!!.. 21r Dp • (1 - ,obl)z RT (ln a o) bz(1 + ,.6.) 0.40724

and a relatlve maxlmum wlth respeet tobz when bz lO O, namely,

('ZU}mu •. 0.4072411' ~~ (tn (0)1 (6.42)

If thls result ls eomp d lthare w Eq. 6.23, tt ls seen that thls ls 81 45per cent of the theoretlcal efflelency. •_ For a single entranee centrlfuge, lt ls neeessary te take N r Odl~(~l'O). The separatlve power ls then, aeeordlng to Eq. 8 of A;;n~

,

6U •• [L(r.' + L(r.)] 9(12

-,9) (1 - eobl)' (ln ao)' (1 - *rand per unít length

• 'N( O) ( e·kr., - N rilO) • - ¡-;e:b& AA(r: - t1) NU - N)

117 118 THE THEORY OF ISOTOPE SEPARATION

Tbe maxlmum wlth respect to r/r, ls the same as before. The ex-presslon1 (1 - eoba )I/bz has a true maxlmum at bz lO 1.25643 and thus

(.!!!.) • (0.40724)' 27rDp(In (Jo)'Z ma. RT

(6.43)

. whlch has an effleleney of 66.34 per eent. The eondltlon bz •• 1.25643glv;,:=ia deftnlte relatlon between the length of tbe rotor and the flow,namely,

47rDp zL. + L••• 1.5786RT 9'<t - 8)

(6.44)

if lt ls remembered that the separative power IS the product of theconcentratlon difference and the transport, the maximum of 6U wlthrespect to r¡/r, is easily understandable. The concentratlon differ-ence ts the maximum when r/r, = O, but the transport (through theeyUnder of radtus zero) is zero. The transport ls a maxlmum whenrl/r ••• 1, but the coneentraUon differenee is zero. The maxlmum ofthe product thus Ues somewhere In between.. A fuller discusslon of the significanee of the dependence of the re-sults on 1 _ e-bl, the degree of equUibrium attained, ts given in Ap-pendix E. The phenomenon Is a quite general one.

NoUce that b contains the factor 9(1 - 9) (L1 + La)' Decreasing9(1 _ 9) and lncreaslng L

I+ L, leaves thls factor unchanged, but small

total flows are deslrable and 9(1 - 9) should obvlously be made aslarge as posslble, which requires 9 = Ya (cf.Eq. 6.44).

Further properttes" of the coneurrent centrifuges may be obtainedfrom the equatlonsglven.

1Note that the correspondence e-k - t1/r: maltes thls talte the lorm

UtIn "it

1

.For example, lor any r¡!r. the dependence on z/(L + Lj ts the same, and themaxlmum lor the stngle entrance centrlfuge la at the sa~e place, bz • 1.256 [wtth theapproprlate value of ln (r./r,) In b]. But for z/(L1 + L.) preasslgned (and not bz) themaxlmum wtth respect to r¡!r. varles.

Page 9: Cohen Capítulo de Centrífugas

CENTRIFUGES

8. THE COUNTERCURRENT CENTRIFUOE

Countercurrent separatlng devlces In general have the property oímulttplylng the equUlbrlum slmple-process factor many Umes in oneuntt. Because of the nature of the flow, a concentratlon gradlent whoseslze ts Umtted only by back-dlffuslonls estabUshed in the dlrecttonof the flow.

Slnce a large separatton ls obtalnedln one untt, much of the recy-clíng between untts whlch ls otherwlse necessary to multlply the stm-ple-process factor ls avolded. Furthermore, the problem of cascadeoperatton ls very eonsíderably slmpUfled slnce the number of unttsIn series requlred to effectuate a given fracttonatton decreases enor-mously. The cascade becomes broad lnstead of long and ts easUybroken down into lndependent parallel sections. Countercurrent flowIs also efficient from a process standpoint because it is possible tomaintain maxhnum separative power through the entire unít," For allthese reasons, which have long been appreciated by chemical engi-neers, countercurrent processes are preferred if at aH possible.

The dlfferenttal expresslon of Eq. 6.19 becomes for the counter-current centrifuge

aN Dp a [ aN ] aN a2Np - = - - r - + 2~Ar2 N(l - N) - pw(r) - + Dp --st . r ar ar az az2(6,45)

The solution of Eq. 6.45 may be íound by foHowlng tbe standard meth-ods for solutíon of partial differentlal equatlons, Thls method wasutilized In lune and luly of 1940. However, Eq. 6.45 ís particularlysimple stnce the gradients of the concentratlon In all dlrectlons aresmall. Thus the equatíon for a countercurrent centr ííugal column canalso be solved by the method used by Furry, Jones, and Onsager" forthe thermal-diffusion column. This pos~ibiltty was first pointed out,and partially worked out, by Bramley." A sim'Uar procedure wasemployed by Martin and Kuhn."

The results of the two methods are of course ídentícal, The secondmethod will be demonstrated here because tt ís easier and because it

'Incontrast to .concurrent devlces, In which aN/ ar cbanges w1th z, 80 that íf theseparaUve power ts an opUmum In one place 1t Is necessarUy less In every otherplace.

a Phys. Rev., 55: 1083 (1939).a Sclence, 92: 427 (1940).·Z. physlk. Chem., A189: 219 (1940).

119

The integratton constant (a function of z and t) was e~aluated ~y re-membering that r( aN/ ar) - O as r - O.

The second integration wtth respect to r requires a little more.artiflce in order to get a form ln whlch the lntegration constant canbe evaluated. The boundary condltlons are

Equatton 6.48 may be tranSformed by lntegrating by parts the flrstterm on the left-hand side,

( it. pw (t. aN dr itPRw

Tr dr)

271 :z N(r2,z) o RT r dr - Jo ar . o

_ 2 .!- ¡t. Dp ~ r dr - 211It.L aN r dr (6.49)- 11 az o RT az o RT at .

Equatlon' 6.49, although lt does not glve the lntegra:tlon constant thatwould permlt the integratlon of aN/ ar ltself, tells how to lntegrate

120 THE THEORY OF ISOTOPE SEPARATION

permlts the case in whlch the mole fractlon of the product ls large lobe lncluded wlthout added compUcatlons.

The method depends essentlally on the fact that aN/ar ls of theorder of N(l - N)2AAr, whlch ls a small quantlty. Thus the varlatlonof N with respect to r, as compared to the varlatlon of r, p, or pw,may be neglected. For example

f~ pw r dr - aN f pw r dr + terms of the order of AA2

az az

Consequently Eq. 6.45 may be integrated with respect to r, The flrstlntegration gives directly, dropplng alN/ azl and aN/ at, which are oísecond order.

aN 1 aN It pw_ = -2AAr N(l - N) + - - - r drar r az o Dp(6.46)

.(6.47)

Page 10: Cohen Capítulo de Centrífugas

CENTRIFUGES!I

I. .

élNl' pw r dr, whlch Is [ust as useful for preaent purposes. SubsU-élr otutlng Eq. 6.46 In Eq. 6.49, lt la found that

(6.50)

where

271 (M; - M1)w2 1" l'Cl = RT RT o r dr o pw r dr

211' 1 1" dr (I' )1Ca • - -. - pw r drRT Dp o r o (6.51)

211' Lc. = RT o· p r dr = holdup per unlt lengtb (moles/cm)

Equatlon 6.50 has the form of Eq. 2.1, the typlcal equatlon for tso-tope separatlon In a square cascade or eolumn, The propertles ofsuch columns, lndlvldually and In cascades, for fixed values óf c

l, c

s,

and c. were studled In Chaps. 2, 3, and 4. The followlng dlscussloncovers the investlgatlon of how these coefficlents vary as a result ofchanges In operatlng condítíons, The material 111the earlier chapterswill be referred to constantly.

In order to examine propertles of the countercurrent centrifuge,attentlon will be confined for the moment to a single centrifuge. Forconvenience it will also be assumedthat N<l, althou~h this is by nomeans essential to the argumento Then the steady-state solutton ofEq. 6.50 is (Eq. 2.14)

N(Z) 1 + r/J exp [2EZ(l + lP)] _ (1 + lP)e2cS(l+ljI)N(O) = 1 + lP exp [2EZ(l + lP)] - 1 + lPe2cS(l+ojI)

121122 THE THEORY Ol' mOTOPE BEPARATlUN

•where

P PRT

lP• e; • 21r (M - M ) .wl Ir. r dr i' pw r dr

I 1 RT O O

(M - M ) w2 1" r dr Ir pw r dr

I 1 RT O O

2E = ~ = -1 -l-f'· dr (I' pw r dr) 2Dp 2 + Dp o. ro'

(6.52)

. d th f cUon pw whlch measuresThe quantlties cl and ca depen on e u~ d o~ the flow pattern,the up-and-down flow, In two ways: They epen n the absoluteor the locatlons and relative slze oí the streams, and o b the post-magnltude oí the flow. The pattern ts butlt lnto( the :oto~e)y oí baffles.tlon of oriflces In the end caps and posttton or a sen be ad-

But the magnltude of the flow ts an oper~t~~~:::~:b~::~~t~~~al con-justed from outslde. Thls lndepednden~eif ge gives lt more flexlbilitystrucUon In a properly conceive cen r uthan the otherwlse analogous thermal-diffuslon column,

Therefore a new quantity L ts lntroduced.

L = ;; ['. [pw] r dr (6.53)

whlch ts the flow up the column plus the flow down the column, Then

Irpw r drO L

does not depend on L, and Eq. 6.52 becomes

PlP=-alL

_ a1L2E - e + a La

2 a

/L d • c,/L1 do not vary wlth L.where al • Clan a, f L and ./. For any value

N(Z)/N(O) 'la now a functlon o '1"h the rate of production ts zero.N(Z)/N(O) ls greatest w en

N(Z) = exp ( a1L z)N(O) ca + aaL2

(6.54)

of L,Then

(6.55)

Page 11: Cohen Capítulo de Centrífugas

CENTRIFUGES 123and the maxlmum posslble valu f ()/centrifuge and í 1 t e o N Z N(O) íor a glven length oí

or a g ven ype oí countercurrent !low ls

(6.56)

",here 2Eols the maxlmum oí a,LcI + a,L1 wlth respect to L and la

(6.57)

The correspondlng value oí L ts

(DP) IrlLo = &. = 27Tr2 RT o Ipw I r dr

Va; J2 Irldr (Ir )2O r O pw r dr

(6.58)

The exlstence oí a maxímum In E ts henon oí back-diffuslon . Th P yslca11y due to the phenom-E. When L, the total fl~w in~h:o~ce~t~atlon gradient ls proportlonal totíon gradlent Is sma11, back-díüusí uge, is large and the concentra-E varies lnversely as L But on ts negliglble (e, <: a,LI) and

. as soon as L ts small a d dcentratlon gradient are largeb k d n E an the con-flna11Y.limltlng the attainable'fr;~tl~n~:~:~~n plays an lmportant role,

It wtll be found convenient to measure 11 flTherefore set a ows In terms of Lo.

Lm=-Lo (6.59)

Substituting in E,

N(Z) (l + I/J) exp [2EoZ 2m (_ _ +__~:.:!+:..!r:!!.a:...:..l::..+.:...:I/J)lN(O) =

1 + I/J exp 2EoZ 1~mm2 (1 + I/J) ]

In terms oí the dlmensionless parameters m and .1. P 1 giY", S ven as

P = ml/JPo

(6.60)

(6.61)

124 THE THEORY OF ISOTOPE SEPARATION

where

P 4 2 Dp0= 7TEor-

2 RT(6.62)

EquaUons 6.57 to 6.62 constltute a complete descriptlon oí the be-havior oí the countercurrent centrifuge. When the type oí now pw isspecifled, Eo, Lo, and Po are determined. L and Pare the conditlonslmposed and are measured on flowmeters by the experlmenter. ByEqs. 6.59 and 6.61, m and r/J are thereforeknown. And finally, sub-stituting Eo, m, and r/J In Eq. 6.60, the fractlonation of the column 18

predictable.The maximum separaUon factor occurs at L = Lo,but thls ls clearly

not the most efficient circulation rateirom the standpoint oí separa-Uve power, slnce the parasltlc phenomenon of back-dUíusion ts verylmportant at thls ñow, The relaUve weight of back-dlffusion is re-duced by lncreasing L (thus decreaslng the concentration gradíent),These qualltative conslderations are conítr med by examining cm.From Eq. 2.33,'

su = 5...Z = a~L2Z I -.5- Z (~) - 5- Z (~) (6.63)4c, 4(cI + a,L ) - 4a, 1 + mI - 4c, 1 + mI

whose largest value ts reached asymptotically as m - ".It should be noticed that the llmlting value oi 6U ts approached very

rapldly: When m = 3 the separative power ts 90 per cent, and whenm = 5 lt ls 96 per cent oi c~Z/ 4c,. At these values of m the concentra-tlon gradient ls 60 and 38 per cent, respectively, oí the maxtmum.!Thus lt ls not necessary to dlscard the advantages oí large separa-tion ln one unlt, or to lncrease the flows lnordlnately, to get adequateseparation efflclency.

REFERENCES

Bramley, A., The Centrlfugal Separator, Oct. 7, 1940.Cohen, K., Report A-54, october 1940; Report A-52, Jan. 21, 1941; Report A-50, Jan.

30, 1941.

'Conslstent wlth lhe earller sectlonsof lhls chapler,lhe symbol 6U ls b.lng used forthe total separatlv. pow.r, and not the separatlvepower per unlt lenglh as In Chapo 2,Sec.3.

"Th. factor 182m/U + mi).

Page 12: Cohen Capítulo de Centrífugas

CENTRIFUGES 125

Cahen, K., memorandum lo H. C. Urey, May 14, 1940.Cahen, K., and l. Kaplan, Report A-I0l, lan. 28, 1942.Cahen, K., and C. Skarstrom, Columbia Ser. No. 4R-X138, Feb. 8, 1941.Dlrac, P. A. M., Brltlsb MS, 1941.Ka~lan, t., and K. Cohen, Report A-195, lune 17, 1942.Murphree, E. V., Separatlan of Gases by Dlffuslonal Metbods, February 1942•

..