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AlAA PAPERNO. 6-183

JET PUMP D ESIGN AN D PERFORIUNCE A N A L Y S I S

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b yD A V I D R. CROFTS h e f f i e l d P o l y t e c h n i cS h e f f i e l d , E n g la n dandD A V i D G . LILLEYU n i v e r s i t y o f Ar izonaTucson, Ar izona

WASHINGTQN, D.C. / J

Fo r permission to copy or republish. contact th e American Instituti: o f Aeronautics a nd Astronautics,1290Avenueof the Americas. New York, N . Y . 10019.

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David R. Croft*Sheffield Polytechnic, Sheff ie ld , hglandand

university of Arizona. TWSM. ArizonaDavid G. Lilley**Abstract

The design of je t punps i s normally bawd onw r h t ather than theory, and, what theorythere i s , has been developed from smle energyand m t w n balances.exhaustive analysis, based on a new primit ivevariable f in i te d i f ference procedure , n d f i e d topredict two-dixensional axisymretric j e t p mpflcws.gfred gr id system fo r ax i a l a d r ad i a l ve loc i t i e s ,a line relaxation procedure fo r ef f ic ient so lu t ionof the equations aid a two-equationk--E turbulenced e l . The analysis pan i t s the invest igation ofje t prmp paramters and their effect on the over-a l l perfomawe of the device, and i n so doingeqlains th e ncchanism of mixing bfedeen theprirrary and se an da ry flu id s. Conputed values fo rboth tk intexial flow characteristics and theovc ral l p z r f o m c e of va rious je t pmp configura-t ions are presented together w i t h eqxr i r renta ldat a which vali dat es tk a c m a c y of the m u t e rrrpdel

This paper presents a mre

The nwe rica l techniqce inmlves a s tag-

NcmnclatuRCDdHI ,J.JkLeesMNPQRSu,v

aEil611P

omstantmixer tube d i m t e rprirnvy j e t nozzle d i m t e rfl ui d headm s h p o i ntturbulent flux vectorki ne ti c energy of turbulencemixer tube lengthturbulence length scaleprimary nozzle to mixsr tubelength spacinge n t r a i m m t r a t i ohead r a t i otiE-man pressureflcw ratearea r a t i osource termtim-nran vel oci ty cOmpOnents i nz , r direct ionsaxial, radial coordinatesturbulent eXd...anqe a f f i c i m tman-flau ra te of st ra in tensorturbulence energy dissipationrate = k3I2/2ef f ic iencydiffuser half-angle

turbulent viscositytkremean dais i ty

a Pran dtl-Sch idt n h rT turbulent stress tensor+ depndent variableD 'Rctor d i f f e r en t i a l owra to rm r s c r i p t s* preliminary u,v and p fieldbased on e s t i m t e d pressuref i e l d p*

get U,V,Pmrrection value to u*,v*,p* to

Subscripts .DJ re la t ing t o pr+ je tm axial s ta t ionn ,s ,e ,w north, south, east, west facesof cell0P,N,S,E.W p i n t , north, south, east,S re la t ing to semndary flodLl re la t ing to turbulent v ismsi t yf o d a

position a t exit from diffusermaximum value a t a par t icular

value a t primary nozzle exi twest neightors

1. Introduction?he Phenmnonind us tri es . Thfy provide a mans whereby fluidsmy be plmpea without an y rmving parts, d?d areused with awbard fluids (e.g. slurry) and/or ins i tua t ions where mnmntional ptn'ivs are d i f f i c u l tt o place, as w e l l as to p r i m other p m s . Cpera-t i o n of a j e t p~ng3i s by the 'primary' fluid en-t ra in ing the 'secondary' fl ui d through mixing withit: t h a t i s , by energy transfer between the fluids.c l ea r ly the pcrfomkmcc of such a p q s ther r foredepxden t on tb manner of mi;cing of the two fluids,which i s inflilfnced by the geowtry of the surfacesover which the fluids flcw, as w e l l as the p e s -sures, veloci t ies , and o t k r p r o pe rt ie s of t kfluids.Perfonnance Prediction

Ejector je t p q s are widely used i n thr process

In t he pas t , j e t p m esign has been largelybased on empirical forrrrulas and constants, n o d i -f ie d with appmpriate cofff icients , w h i c h r e l a t eth e gxxEtric and i l ai d propert ies to Lk pres-sures and velocit ies of the f luids'- ' .

The authors acknmledge w i t h muiy tk&s the helpful assistance of Nan Cliar les~or thi n discussions and preliminary xork w i t h the c q u t e r proc&xn as reported i n Ref. 18.*Principal Lecturer, kp ar tn rn t of ~b%?chanicalnd Prcduction Engineering. . .

**Visiting Associate Profcssor, kp ar tn rn t of Chcmical Engineering. Wxrber hvIA.

1

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&-tl~'-~mre advanced approaches, includingone- and t w o - k n s i o n a l computer nodels have beendeveloped for use in j e t p w prediction, givingguides to pir fon mc e but with a l imitcd range ofapplication.The technique discussed here simulates two-dirrcnsional d s y r m t r i c j e t p q s clirect1.y i nterms of a t ~ - ~ q u a t i o n-E tu rbulenc e m de l, in-mrporated in the governing e l l ip t i c par t ial dif-fe re nt ia l equations, and solution is direct ly by af i ni te difference relaxation techniqm.prowxent and use w i l l SiFi f icantPy reduce thecost and t h eq ui re d f or j e t p m design. Solu-t i on my be via the str ean function-vorticity orprim itive pressure -velocity approach. Whereas thef o m r approach, used i n t he 1968 computer programfrom In fx ri al College for exanpie', reduces by onethe nmher of equations to be solved and e1imi.-nates the troublesOP(e presswe ( a t the eqxnse oftrouble with the vor t ic i ty equat ion) , the pre-ferred approach nm is S1MTI.E (mxnonic for zemi-*licit ~ ~ 3 dor gressue l i k e d - q t i o n s)&ich focuses attention directly on the la t tervar iables 'o .mr k here i s developd irurediately on this newtechnique, the basic ideas of which had been e mbodied in to the 1 9 7 4 I w r i a l C ollege TFACl(reaching e l l i p t i c -&synmttric charac ter i s t ics-eur is t ica l ly) c w u t e r program'".

Its i r -r

I t psseses many advantaFs and the

outline of the paperThe paper presents recent work i n t he f i n i t edifferenoe solution, via a new primitive pressure-wlo ci~ ty ariable f in i te d i f ference code, of two-dirrensional axisimr;etric j e t p q s . A n associatedexp?rinmtal study i s touched on only obliquelyand describfd b ri ef ly i n section 2 .

are based on the s a m system:presented and mulded into a ccmnn fom i nsection 3, bu t the simulation of turb ulence,which has been discussed a t lenqth quite re-c e n t l ~ ~ ' - ' ~ ,akes a minor role here.general flags, the equations are e l l i p t i c i ncharacter and together with this simulation prcb-la i s the necessity to solve the equations: alenqthy n m r i c a l r r l a x t i o n r r e t h d is a w mpriate.

,- Theoretical mdeli ng and solu tion techniquebasic equations are

Governing

The pr&iction precedure i s dea l t w i t h br i e f lyin section 4 , w i t h th e details 1-eleqatedto th ereferences. l%e mnputer program solves di r e d l yfo r the primitive pressure ard wloci ty var iables ,unlike previous mthods" "' which obtain these byway of stream function and vor t ic i ty . In ad-dition, the u and v velocities xe p s i t i o n e dbetwem the mdes where p and oLher variables arestored and tk &ination of staggered grid and al i ne re laxat ion nethod leads to rapid solution.S q l e carputations are discussed i n section 5 ;the fi na l conclusions slmPnarize the a c h i e m t s .2 . The S y s E

The Sincjle Nozzle Jet P qA sing le nozzle j e t pump is mnsidered, ass h n Fig. 1. The primary (driving) fluidf lms, w i t h re la t ive ly high wlocity and pressure,

thruu+ the prinury nozzle and entrains th es m d a r y (entrained) fluid, w i t h re la t ive ly lo wvelocity and pressure, so as to fonn the output.Flow rate quantities QJ, Cr, and 4 = Qs + QJ andheads F ~ J , IS and Hg are s t m i n the f igure, to-gether with prinury nozzle d i w t e r d , mxer tubediaeter D , d i f f s e r angle 0 , nozzle to mixfrtube spacing & and mixer tube lenqth L.Design Pa rme t e r s

The mjor p a r m t e r s of interest i n the p i r f o r(i)

m c e and therefore d e s i 9 of j e t p w s are:the ra t io of entra ined f lu id ra te to pr i -m r y f lu id rate, k nm as the entrairmatr a t i o M,the r a t i o of pressure head gained by theent rdn ed f lu id to the pressure head lo stby the prirmry fluid, hm as the headr a t i o N.

(ii)

1. PriFaty drivin g flmid.2 . Sernndaq entrained f lu id.3 . Nozzle.4. S u c t i o n c h x k r .5. Mixer tube .6. Diffuser.Fig. 1. The single nozzle j e t p q(iii) the area r a t i o R , w h i c h i s the r a t i o be-

tween the driving nozzle and mixer tube-s,the ef f ic iency n, which is the ra t io h-tween enerqy transfer t o the entrainedflu id and the enerq-y transfer to thedriving f luid.(iv)

For t h e system just descriked these quant i t ies are:M = Qs/QJN = (HD - H ~ ) / ( H J - HD)R = d'/D2 (1)n = MN = (Hg - Hs) (Ib - HD) 1

%er gccnrtr ic cii ta affect ing pzrfommce +e th omlxcir tube l e n g t i i / ~ ~ ~ . ~ ! ~ t e ra t i o (L/D) , thc nozzleto mixer tubc spacing (Cs) and t k nc lded ang leof the dif fuser 0 . Typical pcrfommce curves off o r exarrple n against N and 1," against M fordi f ferent d /D values m y be fomd h tk litera-ture'.

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Expcrinmtal workTo -1-t the avai lable e q e r h t a l mrk,an expfrinrntal j e t pwp r i g using water fo r bothdriving and entrained flu id has been constructed asshum in Fig. 1. P r a c t i c a l results obtained f r mthis provide mqxirisons w i t h the mnputer mde.1predict ions given later.dThc pimp is a single central nozzle type, w i t ha mixer tube dimter D of 0.0254 m and a suctionch& d i m t e r of 0.1016 m. These parts are madeof pe~spex o that the mixing p m s s can be o bs e m d , pa r t i cu la r ly a t the onset of cav i ta t ion.The nozzles are interchangeable and the Kozzle tomixer tubf spacing can be varied.corded tests , the p r k y f lu id was delivered by a7.5 kW centrifuged pmp t o a nozzle of diamter0.0127 rn.

For the re-Thearea rati o R w a s therefore 0.25.

,

I n s t m t a t i o n on the m i r r e n t a l r i g permitsthe r e a s u r m t o f pressure and veloc i ty p ro f i l e sa t s ta t ions from the nozzle to the end of the dif-fuser . Fur the r ins tmen ta t ion i s being developedto enable nrasuremznts to be made using laseribppler anemmetry,3 . Analysis

Gomming EQua t ionsThe turb ulen t f lux (Reynolds) equations of mn-servation of mss, m m t m , tuxtmlence energy andturbulence dissipatia? rate cpvem the two-diren-sional steady tubulent fl5.4. These transportequations are a l l similar and mntain terms for theconvection and diffusion (via t u h u l e n t f l u x terms)and source Sb of a general variable $ (which con-tains t e r n describing the generation (creat ion)and consmption (dissipation) of $ ) , Dfferringtheir presentation u n t i l they are pu t in a ocnmonform, FQ. (6), f they are to be solved for tire-man pressure p and velocity, then the turbulentf lux colknowns ( turbulent S t T e s s tensor T and turbulence fl ux vectors J@ fo r $ equal to k and E )must be specified prior to solution. I t i s mn-venient nmJ to consider the problem of closure ofthe equations.

Flux Iaws

wheret in r -man f l o w rate of s t r a in tensor.is the turbulent v iscosi ty and b i s the

?he exchange coefficients il and r$ are connectedt o th e other f lu id proper t ies svch as dcnsity andturbulence characteristics by a varie ty of al-gEbraic relations, and Prandtl-Schmidt n h r s re-late other exchange coeffici ents t o the turbulentv i s m s i t y u: these are defined byu+ = P f l $ (4 )

and often given valua near unity.the turbulent transport the two-equation k--E tu?bulence mxf l I is used, wherrby the t u h u l e n tv i scos i ty is calculated fromTo describe

11 = c pk2/E (5 )Pand two different ia l equat ions are solved for theh.io turbulence quantities k (kinetic energy of tur-bulence) and i (turbulenco energy dissipationr a t e ) . I t m y be noted that turbulence lengthscale i s re la ted to these via E = k 3 / e . Con-stants appearing her e and i n Table 1are givenv a l ~ seconmended in Ref. 10 :1.00, C I = 1 . 4 4 , c 2 = 1 . 9 2 , uk = Y.oO,and gE =1.21.

c - 0 09, cD =camrian Form of the Governing F e t i o n s

A l l these linkages provide a high degree of non-

H e r e i t i s nrrelylinearity and simultaneity in the problem.tional i n f o m t i o n about boundary wndiitions needsto be suppl ied and inmq orate d.

Mdi-noted that the s imi la r i tv between the differential

~s p a t i o n s and their d i f f k i o n r e la t io n s allows themall to be pu t in the wmmn form

for 0 = u,v,k and E .() are gim in Table 1.Thf f o r m f o r the source tflmk b u l e n t f lux wctors J fo r transported f lui dscalar properties @ m y be &scribed by thefoliagin g diffusion-flux law: Table 1. The form of the source tennin the general equation fo r$, Eq . ( 6 )= - r v + ( 2 )0 0

0 Shere the direction of J$ is the e xact o w s i t e oftha t o f Q$.llormal to th i s d i rec t ion , it n u s t be expressed asI f there is a diffusional comp3nent

The turbulent transport of m t m s via thei n so sinple a m e r a3 the transport of C$by dif-fusion; or ra the r it can but f i r s t the whole e xpression must be considered and only then transferthose unwant& mp e n t s , which do not vanish i nv iew of the continu ity equation, into th e s o w e

turbulent stress tensor T which cannot he expcpmssed Vk

tern. The approxim te expression is E

3

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where

au av '](-- +z)a r4 . Prediction Prccedun?

The TechniqueI n t e r l i r k a F s beb!een the 0 ' s present d i f f i -culties to solution of the gowming equations.

%hosebetween the axia l and radial velocity cawp n e n t s are of a peculiar kind, each containing anunknown pressure gr adi ent and the c m p n e n t s arelinked add itio nall y by another equation, t ha t ofmss conservation, in which pressure does notappear. E xar rp le s of research papers mst relevantto he resent work are l i s t e d in the references.1 0 - 1 5 - 1 ' The wo rk here incorporates the following:

a f i ni te difference procedure i s used inwhich the dependent vari abl es are the velo-c i t y mrrp0nent-s and pressure:the pressure correction p' i s deduced f r man equation w h i c h i s obtained by the combination of the continuity equation and theucmnta equatiaqs (yielding a nfw f o m ofwhat i s !am in t he l i t e r a tu re as thePoisson equation for pressure);the idea i s present a t each i tera t io n of af i r s t approximation u:v:p* to +,e solutionfolla\'ed by a SucceeClFng correction(FU* + u v=v* + v' and p=p*+ p 7 (u'and v' are re la ted t o p') ;the prcedun? inmrporates displamd gridsfor the axial and radial velocit ies u andv, which arc placed beween the nodes herepressurr p and other variables ar e s t o r d ;andan %lici t li ne- by -li ne r e l a t i o n tech-nique is fnployed i n the solution pmedure(requiring a tricliagonal matrix to be in -verted. in otder to @ate a variable a t a l lp i n t s along a cohmm) .

(i)

(ii)

(iii)

( iv)

(v)

here is qim only t o the m a i n pints.The Staqqered Grid and Nota-

Fig. 2 shajs sone of the rectangular computa-ti on al rresh. W.e intersections of the so l id l i nesm k the grid nodes where a ll variables except theu and v velocity mmpanents are stored.la t ter are stored at points which are denoted bythe arrows and lcc at ed midway between th e gr idintersecti ons, and the taonrrrlng-shapd envelopesenclose a t r i ad of points denoted by a s inglelet ter P = ( 1 , J ) . Similar remarks apply also t othe four n e i y h r s N = ( I , J + l ) , = ( I , J - l ) , E =(I+l,J) and W = (1-1,J) . Thus, for exmple,UI,J = U ( I , J ) i s the axial velocity a t refercmcelocation (1,J) e m hou$i it actual ly rcpresfntst k elocity positioned a t ( I -$ , J ) .a x r a n m i i t has b+o special rrerits: f i r s t l y , itp1.aces the u and v vflcci.ties bctwcen thepres sws which driw than and it is oasy to cal-culate the pressure gradients which affect thm:and secondly, these veloci t ies are direc t ly

The

This gr id

4

available fo r the calcula t ion of the convectivefluxes across the koundaries of the olntrolvoluies surrounding the gr id nodes where p, k andE are stored.

NI T -7 1

1 A I3Jf - 5zhree grids: f u r p etc ( 0 )fo r u velocity (+)for v vrlocity (1)

Fig. 2 . Staggered g rid and notation

Thuse cwputer storage loc atio ns which appearto be external o t he d m i n and i t s boundary, seeFig. 3 fo r a sohemtic of the qrid system used,,rpayhe used to store the w a l l valws of the appropriatevariables. Boundaries are located midiay betweenthe nrsh lines so that normal velocities are lo-cated dire ct ly on the bunda ries ; for exarrple u i slocated on a vertical boundary, v on a horizontalone. N o t i c e t ha t there is no necessity fur thedistancf hetween successive grid p i n t s to be uni-forn; f m p e n t l y a gradually eqand ing rectangu hrm sh system i s used on the grounds of accuracyand ooonony.

NJ

RU I

_.J = lI s 1 2 NI

ZI I l

Fig. 3 . Grid specification - an -le_-olution P m d E"," thz cp+?minq partial cliffercntiaL equationsl i k e Pq. (GI ill13 r e d u a d t o a se t of f i n i t e dif-f e n m a api t i .ons for values of the variable @ , a tp i n t s OF the grid system covering the solutiondamin.wnd it ion s, cunsti tute a system of strongly-coupledsir ult me ou s al +x xi c equations. Though theya p p f a r linear thcy are not since the cofff icientsand sourm t c m s axe thcmsolvr?s functions of so1113of t i ie v&riables, cm.niithc w l c c i t y e q u t i on s ar estmq1.y l inked tiuowjh the pressure.

rq the t edmiqw descriw i n the

l?Iese, toqcther w i t h appropriate boundary

The solution

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proceeds by the cycl ic repEti t ion of the fo l lmingsteps:Guess the v a l u c s of a ll variables includingp*. Hence calc ulat e auxi liar y variablesl ik e density, viscosi ty etc.Solve the axial and radial mxentum equa-tions to ob ta in f i r s t estimates u* and +.Solve the pressure correction equation toCalculate the pressure p ard the mrrectedveloci t ies fmn u and u=u* + u v=+ + v'and p=p* + p' (u - and v' are re la ted t o p ') .%lve the equations for the other variablesk and E successively.T n a t the n m v a l l r s of the variables asinproved gw s s e s and return to s tep (i).& p a t the process u n t i l mnvxqaice.

obtain p'.

In t he solution pro ced m algebraic equationsconnecting a $-value w i t h i t s four nei-rinqvalws, are solved "any tixes, coefficie nt andsource p3 at in g being carried out pr io r to eachoccasion.of the w e l l - W m +diagonal F t r i x &gorithm(TDMA) whereby a se t of equations, each w i t hexactly three dcncxms in a p a r t i d a r order ex-cept the f i r s t and l as t w h i c h have exactly two un-! a m s m y Se solved sequentially . In the two-dim nsi ona l problem one considers the val.ms a tgrimints along a ver t ica l gridline t o he vnknown(valuss a t P, N and S for each p i n t PI , but takeas h m M , mSt recent valms being used, theva lms a t each E and W neighbor. The 1D t is t k napplied t o this ver t i ca l gr id l ine . In t h i s m e rone can traverse along a l l th e l i ne s i n the ver-tical diir ebi on seqyuentially from le f t to r i gh tof the integrat ion &nuin.

The practice used hfre is to mke use

A t each such i t e r a t i on it i s necess- to awto take a wei&ted average of L'e nwly ca lcula tedvalw and theprevious valw a t each point. Finalconvergence i s decided by way of a residual-sourcecri terion, bhich nrasures the departure frcmexactness fo r the variable Q a t the point P.

.~ oy s m deqree of m d e r r e l w a t i o n , f o r exaple

5 . Results and Discussionconputer program has been se t up to nukepredict ions fo r the saw c o n f i w a t i o n as studiede x p e r k t a l l y and shown i n Figs. 1and 3. 'Ibepredict ions shavn relate to an area r a t i o of 0.25for different primary j e t velocities and nozzleaxial positirms.

As exanples of the currant predictive capabil-a ty of the proqram, several sanple mrputa t ionsa m presented to illustrate:(i) the in ternal behavior of the j e t p u p ,W i c u l a r l y i n the region of mixing be-tween the primuy and secondaq flcws, and

the ef fec t of nozzle axia l ps i t ion;the preclicted overall prfonrance of theje t p q nd a canparison w i t h e2pxirmnt.dresults.(ii)

conputations w e r e generally mde with a 15 x 1 5variable size gr id , w h i c h alladed the solution ofa twical pr&lem t o be obtained in about 100-150ite&ion s; and taki ng the &valen t of a'mut 1-2m i n of ccC6600 6 im .

The ccnputer w k l predic ts valu3s of axial andradinl vfl .oci t ies u and v, turbulence ki ne ti cenergy k, tub ule nce enerqy dissi[aLion i, urbu-lence lm iq th scalethroughout the fla qfi eld . Discdsscd f i r s t is aconfig uration with prinruy nozzle t o mixer tu&,spacing &/o = 2.25 and in l e t primary j e t velocityand s tat ic press'xe p

= 22 m-1.Figure 4 shows the tu%ulence energy k andlenqth scale 2 dis txibut ions within the flow.

energy c o n t o w reveal that a t the p i n t of " im -pact" of tAe primuy j e t on the entrained flowthere i s a high rate of energy generation.high valucs of lenqth scale i n this region quidclydissipate the enerqy into th e ent rai ned flow. 'Therapid decrease in the damstream lev el of lenqthscale results in a v e q gradual reduction inenergy level i n the mixing tube section. This in-dicates that the mixing ti& lenqth is p?rhaps tccmat sin= minim1 mixinq i s actual ly occurring,a useful result i n usinq the c n q u t e r &el tooptimize a desicp.

The

The

Fig. 4 . Pr ed ic t ed turbulenceenergy and lenqth scaledis t r ibut ions(D = 2.25, % = 22 ms-')

Fig. 5. Predicted a ~ . a lelocity distr ibutions i nmixing regio n(t& = 2.25, = 22 m-' )'5

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As a further i l l u s t r a t i on o f w h a t is actual lyh a p w i n g i n the mixing region it i s revealing toeMmine the axial and radia l wlo ci ty prof i les .Figure 5shows the increase i n axial velocity asthe entrained fluid nears the p r i m q j e t , andFig. 6 the variation of radial velocity, a t a s e et i an j u s t dms t r eam of the nozzle exit. col-l i s i on between the radial and axial flows resultsi n a violent m i x i n y p m e s s and the consequentgeneration of turbulenc e energy. Figures 4.5 and6 also show a rapid decay in turbulence energy,axial velocity and radial velocity away from thei n i t i a l mixing region, particularly in the radia ld i rec t ion outwar& t a a r d the suction ch- wall:a fa ct which suggests th at the suction chanber istm large, and that a s im il ar p r f o m c e would Iravailable w i t h W l e r equiprent.

The positio n of the nozzle rel ati ve to themuth of the mixing tube i s another design featurewhich the d e l can eas i ly simulat.e.and 7 portray the tuloulence energy k and lengthscale 8 d i s t r i b u t i m s f o r two sys t em:a primKy j e t velocity of 22 E-' ut varying innozzle-mixing tube spacing !D.w h c r e the nozzle is adjacent to the nnuth of *demidng tubs (Fig. 7 w i t h LJD = 0.5) the turbu-lence energy mntours show that there i s less dis-s ipa t ion into the secondary fluid and consequentlyless entrainrent .i l lus t ra tes the in tens ive local mixing activity a tthe nozzle exit; the rapid decay i n the axial di-rection minimizes the further decay of turbulencei n the mixing tube and again revEals t ha t thelength of the miwing tube i s excessive. In tamsOf predicted entraitwent ratio M, the configura-tion i n Fig. 4 generates a d u e of 0.58 and forthat in Fig. 7 a Value of 0.48. Inte-diatevalues of t h ozzle to mixing tube spacing can bet r i e d to optimize the entraitwent ratio.

Figures 4both with

In the case

The length scale dis t r ibut ion

'-O 1.i *--..-- ' m * ,,,*