ML20237H424

Evaluation of Revised LOTIC-3 Drain-Flow Heat-Transfer Models

ML20237H424
Person / Time
Site:	Sequoyah
Issue date:	05/31/1986
From:	Gido R, Koestel A LOS ALAMOS NATIONAL LABORATORY
To:	NRC
Shared Package
ML082700172	List: ML20237H424
References
CON-FIN-A-7286 LA-UR-86-2053, NUDOCS 8708250012
Download: ML20237H424 (61)

Text

4 enc L O 5 U RE l

k. A-U R 2053 to. *.. ~., e-.

t.....so.,

..:.e e, ie u,.v.. 3 e c.o...~. vo, iee u~i.e si... o.c.n-.,i e se,,, on.. c.,,..c, w.,es.e ~c. x TITLE EYAlATICI; 0F EI'i EC LOTIO-3 DFAH;-FLOW F.IAT-TEA!;5FIE CI;5 Avinossi A, E e s*,e ;

?. G. Gid?

SUB M* E O T O

El 8,.ccept. ace of tms..t.c6e tne cuensne, r.coomaes in.? the U S Gov.<nm.nt e.t. ns. nonencews ve roy.ny-fr.e hcen to ow h.w or r.oroesc.

the pwDbsm.c,0.m O' tm.S Coat.tDuOR. Or to.460m Oth..$ 10 00 80. for u $ Gov.rnm.nl purpos.6 1,.to....-o,~....u........,,,e.,,,e......,.....,,.s,,.,........,,,,m.......,~........,v.,..,,ee...,

i l

O $

A $"_ _ O $ L sal m sNationalLaboratory Los Alamos,New Mexico 87545

,d.-) (I 7'7 I N h [ k

-(f xA l

Foaw no su a4

/~

~-

i, e n.,,,

EVALUATION OT REVISED LOTIC-3 DRAIN-FLOW HEAT-TRANSTEk kk)DELS (TIN A7266 Draft NUREG/CR.

by A. Koeste] and K. G. Gidc Mai 195t Ererg) Divisier les Alaces N;tient] Laborat(ty Ur.iversity of Califerr.ia Los Alates, fN 6754f

(

1 ABSTRAC7 Tia $ repert provides the NRC with at ardependert assesseert of the acceptability of (al revised Westinghouse LOTIC-3 drain flow beat-trarsfer codels. ard (b) the codel ut ilization of arbier.t air drair-flov test results.

It additiet. correctier facters are esticated to quar.tifv 6pecific deficiencies f the prepesed drair flos heat trar.sfer models and the utilizattor cf atbi e nt -a i r d r a i r -f l ew test results to represer.1 the actual sce-conder.ser lower-c orpo r t ee r.t hi gh-t eepe r a t ur e -st e st e r.vir or. cert.

1 INTRODUCTION Ar result of recert charges it the Westirgheuse rethods for c octu t i r.;

rai src crer;> rele.se data fer postulatec n.a:n-stear-lite-break (MSLE'

s.. :ce: t s. ke st irrhees e hu. f ound it rece ssary te mode l more rechar.:st ically the therca? respctse of the lower comparteert ir. ace-conderser-certaiteert plants.

l 3 te WCAP-5345(P), Westirrhouse has proposed char.gir.g the currett,

l Ir Scyplerert staff approved versier of the LOTIC-3 cott aineert response code WCAP-5354(P).

Stpplecert 2.

The ebjective cf the proposed char.ge is to accourt cere realisti-celiy fcr there2) processes occurrity it the lower comparteert durir.g a pc'ste nted MSLE.

It particular, kef. 1 describes the revised codels deveicted St bestirghcuse fcr estimatiet ef the crain flow heat receval.

Use of the tevised LOTIC-3 code for sce-cordenser MSLB atalvs:s appears tc restP tr cerrartrett temperatures that are sagrificar.tly less that stat the c6Trer*1s ac c eptable ver sior. of the LOTIC-3 code veuld predict.

If accepted by the NFC staff. the revised LOTIC-3 code coald provide a basis for resolutier of M5LE cercerr as at relates te the qualificatier cf'dalety-related equiptert f

1 irside certaittert.

Tne purrese of this repert is to provide the NRC with the following" 1.

Ar independent assesseert of the acceptability of the Ref. 1 revised kestitt ouse LOTIC

.t h

A.

drair-flev heat-trar.sfer models, and E.

the rodel util:2atter. ef ambient-air drair.-flow test results.

j u w 2.

Esticated correctier factors to quantify specific d. deficiencies of A.

the proposed drair-flow heat transfer models, and E.

the utilizatier. of anbiett-air drair-flow tesi results to represet the actual ice conderser lover-compartment high-temperature-stear e rvi r onne r.t.

i 9

.. _ {

.y

{

]o-

.)

>V o

n1 o,

g.

}

I e

3.

\\ (j

. /

)

.(

/

.g.

l

\\

);

s;i

'a a

/

l.,

IL DISCUSSION revised Westinghouse i.0 TIC-3 drain-flow beat-transfer model deccribed '

fht t

in ses

'l relies beavily or; utilization of ambi e r.t -a i r drain-flow-simulation e

s t e s t s..

As a result, representation of heat transfer to the drain flow in a h

results rduires high-temperature steae environment using the ambient-air test l

based cr. understanding and application of the ecchanics involved.

e xt r a ppla t ico The understanding of the mechanics is enhanced considerably by the experimental l eessurecents and observa'. ions of the ambient-air tests.,

u gnsfer in Ref. 1 is calculated from the product of ibrae Drain-flow heat teres ree Nocer.slature, t

(1)

[,A ST th u expressier. can be developed into a fore that sicplifies under-I r. a d d 2 t i e r...

standing effects of specific paraceters and phenomena, e.g., drop size, drop velocity and entraineer.t.

drain Proce: ding',weexpresstyeliquidsurfaceareaofthedropsin-the fic> as nar40, where n is the nueber of drops per unit voluee and'6 is the drop 2

d'atater.

Also,

?

bf' = c

r. u,A where s = liquid flow rate at tbc drain outlet, cV' = fractions splashed (V* = 1 for freely falling drops).

uj = absolute drop velocity (uj = U+u ), and g

A = cross-sectional area of flow.

The beat transfer per unit voluer is 1

l

- b n r 6'(T-Tj) sbete b = best transfer coefficient and (T-Tj) - temperature difference.

Also, beat transfer to the drain flow drops Q6 I'

. O3 h r. r E*(T-Tj)A 2 where 2 is vertical distance. Coebining the equations above results in d 6 h 2 V' (2) o 0

AT T 6o 6(u +U) 0 f

g T-Tj where * =

and 4T is the maxieue possible temperature difference.

o o

T c s

Equetion 2 will be used to evaluate specific aspects of the revised LOTIC-3 drain-flow beat-transfer models in Sec. A below.

In particular. Sec. A 1.

discusses the ecchanics involved in calculating beat transfer to the drain flow and the utilization of the ambient-air drain-flow test results.

procedures suggested by Ref. 1 for calculating beat transfer 2.

describes to the drain flew and the utilization of the.achient-air drain-flow test results.,and 3.

establishes.

if practical....;roxicate correction factors to quantify differences between the procedures suggested in Ref. 1 and what we believe is more appropriate for the calculation of drain-flow heat transfer in the actual ice-condenser lower-compartment high-temperature stese environeert.

A.

General Evaluation In this section, soce of the cechanics involved in calculating beat transfer to the drain flow and the utilization of the ambient-sir drain-flow test results are discussed in general. Also. fundamental differences froe the factors approach suggested in Ref. 1 are established and approximate correction to quantify differences between the approach discussed in Ref. 1 and what we believe is more appropriate are estiented. Details of the developments are con-tained in the appendizes.

4 Tr e e lv-Fa llint-Dror Ve locity - The absolute velocity of the drop is 1

drop rela-22 vet by uj-l4u. as used in Eq. 2, where, U is the velocity of the g

is the velocity of the en-tive to the gas entrained by the falling drops and ug is caused by cocentue transfer froe the drops to trained gas. The entraineett and U for air and stene the surrounding gas. Appendia A esticates values for ug Table 1.

Note that the relative ateosp'heres with the results sucutarized in velocity and the Eas velocity require different air-to-stene correction factors, which are based solely on hydrodynamic' considerations, Reference 1 propeses esticating the absolute velocities in stese by ceasured in air by the relative (tereinal) codifyirg the absolute velocities velocity correction, which is ((p,/#,) V8al.22]. This approach appears to be the absolute velocity is equal to the tereinal bases. or the assutption that veio: sty.

the steat-ateospbere values for the relative drep velocity Ve believe that entrained-gas velocity should be based on separately-corrected values.

anc the of This would require determination of the entrained-Eas velocity as a function flov rate for ar air and a sicae ateosphere, i.e., u, and u,. respective-drain sicplified approach used in App. A could be used for guidance in caking ly.

The the detercitatior..

Then. (a) the relative velocity in air U, is obtained by velocities censured in the ac:bi e nt -a i r subtracting the u, free the absolute (b) U, is obtained by caking the air-stene correction of U,, and (c) the tests.

The values in abselete velocity it stene is obtained from U,+u,, see App. A.

would be 1.16 Table 1 indicate that the resulting absolute velocities in steac:

tires the absolute values it air. Although this value is quite close to the 1.22 factor recoceended by Ref. 1 the principles involved arc different.

Freelv-Fallint-Drop Size - Drop size forcation by a free fall is dis-2 cussed it. App. B and drop size forention at sheet breakup is discussed in the freely-falling-drop diaceter in air is equal App. C.

Appendix B shows that As a result, drop sizes ecasured in approximately to the diateter in steam.

tests do not need correction for application to a stene steosphere.

acbient-air In addition, we believe the drop size is controlled by the free fall.

Reference 1 discusses the use of the achient-air-test drop sizes for a stene atmosphere, which we believe is appropriate. Ilowever, Ref. I discusses also the detereinaticn of the drop size based on the disceter being equal to tbc drain-flov sheet thickness just before sheet breakup, which we believe is inappropriate.

1

1 F

t 4

1 If tbc drop size is determined by the sheet thickness at sheet breakup.

Eq. 9 in App. C shows that the beat transfer could increase by tbc ratio j

6,7=(5F, ) v i,

- 1.023 s

for 9, = 0.075 and 9, = 0.050 lb /ft'.

However, if the aerodynacie effect on g

the sheet is reduced by a decrease in the ambient-gas density, the liquid sheet may breakup by perforations, see Ref. 2.

This results in an increase in drop Fire esticated to be e sperferated sheet) p 6 (w2 vy she et J l

based ce Fig.15 of Ref. 2 for a sheet in a reduced ambient-gas density equal to

]

th: stete density. This would result in a beat-tran.fer reduction by the factor

)

1/;.2' = 0.75.'

l

,3.

Terr.crature Difference fer Falline Drops - The temperature difference (f

f c' r en array of falling drops can deviates from that for a single drop f allin in ar. infinite environment. For the array, the drops interact priearily with tbc firite quantity of the bulk atmosphere in the immediate surroundings of the drops, i.e., the atmosphere entrained by comentue transfer. This results in an available temperature difference that is less than the carimum available, See which is the diff erence between the atmosphere bulk and drop temperatures.

/

App. D.

Reference 1 proposes use of the full temperature potential.

To account for the reduction in effective AT. temperature-difference cor-rection f actor. T f ri Eq. 2. is estimated in App. D.

A value of Te nt = 0. %4 results for a drain flow rate of 400 gp: and a fall height of 32 ft.

Appendix C shows the effect of a steam atmosphere on the sheet size before breakup. This may be of-value in estienting beat transfer to the sheet.

Specifically.

Eq. C-8 of App. C shows that the li i length (r*) can in-creaseinasteamatmospherebythefactor(p,/p,)gudsheet

/' = 1.2 for 9, = 0.075 and C, = 0.050 lb /ft' g

1

t

. - The beat-transfer coefficient (b) corree-4.

Heat-Transfer Coefficient stear-tion is affected by the cocbined effects of the drop-size, velocity, and prayerty correction factors.

5.

Splashinn - Equipeent interference with the free fall of the drain

)

a file and splashed

)

fic< results in separation of the drain flow liquid into drcps., H0 wever, the drain-flow trajectory and icpingecent angle are affected by the gas (air or steac) environment.

As a result. the splashing fractions see App. F.

observed in the nebier.t-air tests cust be adjusted for use in stent, of the splashing fractions ecasured in Reference 1 propeses direct use 837 of losing a portion of the drain flow to file flow is accounted TLe effect i t. -

E;. 2 by the correctier factor V*.

The achient-air drain-flow test ir.clude censurecents of V".

However, the angle of rrr.

3: 5:vssed it Fef. 1 fraction splashed, depends on the resistance of

t; +;e sr; which effects the the reduc-Apper. dix F esatir.es this phenomenot ar.d esticates t i r a r). c r i g a s.

titt i s. V when scaling froe air to sicae to be Y'tsica

, 0.55 - 0.766

)

U*I t#(a3r1 for a drain flow rate of 403 gpe icpinging on a sicae generator.

6.

Heat Transfer to Dispersed Drops - The drain flow may interact with equipeent (e.g..

cable " ays and flat surfaces) to create a complicated ateosphere of falling and suspended drops. As a result, calculating the beat For the surrounding stecsphere to the drain flow is complicated.

trarsfer frot exacple. the t empe ra t ur e difference potential is scaller for drops in the interict of tbc dispersior.. A possible approach, discussed in App. E. is to as-energy is transferred by turbulent eixing f roe the surrounding stene succ that inte at array of dispersed drops.

As a result, a model for esticating the ef-fective beat transfer to drops in suspension is provided.

for beat transfer to drain flow that has interacted Reference 1 accounts in the with the cable trays and flat surf aces by using-(a) velocities observed ambient-cir tests and (b) the maxieue possible AT.

We believe that this ap-preach is inappropriate because the effect of the dispersion on reducing the l

energy transfer is ignored.

^

t lL _ - ___ - ___ _ _ _ - _ _.

7 a pproxima t t er.. App. E use s a codel based or the assurption that As at crergs is tra sferred by turbulert eixarg from the surreurding steam inte the array of dispe r sed drops. The resultirg Eq. 2 heat-transfer correction factor T

is 0.945 fer a dicat flow rate of 400 gpt and a fall height of 32 ft.

prep Heat Trarsfe-to the Sheet - The heat transfer to the sheet should be based er the ir. terre! resist.r.cc cf the liquid beir.g the ca.nor resistar.ce. Th =

results free the licuid-int erf ace resistance beirt cirical ar. a stear-rich erv:-

r e r.re r t.

Leference 1 preposes the use of a heat transfer correlation where the ca s e r h e a t - t r a r.s f e r r e s : s t a r.c e is it the liquid. which is appropriate.

I 1))

APPLICATION LIM]TATich5 AJylicatter of the certertier facters and util:2atter cf the codels devel-

pe.

this repert e.st be d:.e $sth a good understard:r; ef the corstratris fr-tre code! develeptert.

Ir particula: our analyses 1.

are cursory with the primary purpose best; the ider.tificatter ef redelling areas th:t eight r.c e d additscral ard core cocplete I

trsestigatter ther. su;pested in Ref.

1.

utilize r.a.i e r s irp l i f y s r.; assumptior.s. such as, a cor.stant area (fer

{

t'r.e crair flew e r.t r a a r.r e r t aralysis). on e -d ime r.s i e r.a l geometry, at:

t urifere d:stributier of dreps, 3.

base. l i g. i t e d scope because most of the calculators are orl) fer 400 rp. and 4.

r.as ret have 2dertified all techar. ses requiring further nr.vestigatter..

1 l

4-It.

CONCLUSION AtC Rf.C04ENDAT10N5 preserted based or our Tne fc:les.rt cerclusters ard recorrer.datiers are

e ;e.

tre LOT]C-3 'lef 1 p r r c e t.i r e t for calet)atir; beat trar.sfer tc the ice-cerderser lower compartmer.t.

drait. flew att ar

• Sere ci the LOTIC 3 tr ecedure s we re f our.t te everestir. ate the heat transfer tr the dic:- flow, o Tb precedures fcr caltuistir.g heat trans:er te the drait flow should be ted;;ied w::: the redels cf this report es guidelar.es.

Ir particular. the precedures of tras repcrt shcule be expanded te account for (a) tor c r e d i te r.s i c r.c. and ner.hercrerecus effects. (b: differer.t dreir. flow rates, and (c1 seriatier of drer sizes.

e er red:f:ca icr.

LOTIC-3 c COMPAEL could be used to deterr. ire the

fe;-

r' tbt red: f i c a t iers c.r ice-corderser tiersphere temperature.

ACKN0'4 LED *.1ENT S Tre authir-wei. d like te thant Mr. Y. J.11uarg a r.d

k. L. Palla, Jr.

of 11 - NF-

- n ; a t.b : e d i s c u s s i er. r e g a r d i r.g t h i s r e pe r t.

1 1

l l

_m._--__

m_______.-___________m_

9 N7'DCLATURE

- Area such as ibat for beat transf er and flow A

)

B

- Paraceter grouping used in App. D 12 b/(c 6 p g) j C

- Drag coefficier.t d

c

- Specific Leat I.

- Drag force: Base diateter for a flatten drop, see App. B Diaceter at*

top of truncated cone in Ap;. C, see Fig. C.2 s.

i 1

- Differentia; operator, e.g., dV is a differer.tial volute J

)

c

- k' ave atplitude ratio used in App. C, which is assumed usually to be con-E stant at -12 1

- Fractior. used ir. Ary. C see Fig. C.1: Integration factor used in App. D g

- Gravitational onstant t

- Heat-transfer coeffi;ient t'

- Sheet thickness at breakup used in App. C K

- Paraceter of nozzle used in App. C with K =r b o

o k

- Thereal conductivity

/

- Mixing length used in App. E t;

- Nueber of drops.

l l

. - Nutber of drops per unit volume c

Mass such as, that at the drain outlet (d ) and a general liquid 5

- Flow rate, o

flow rate (Ej)

App. D differential equation parameter B + 1/ug P

Heat transfer such as that to tbc drops (Q ): Drain sheet flow in 6

Q App. C: App. D differer.tial equation paraceter 1/ug e

- Heet transf er rate per unit area used in App. E i

- Radius c'f

• bag-code-breakup bag oper.ir.g described in App. E Radius of

• bag"-mode-breakup cylindrical-ring ligament-described in r

App. B; Ligacent radius in App. C.

i.e.. rj; Radius of cone base used ir.

App. C

- Drain flow sheet radius free source ic App. C

-I 7

- 7eeperature l'

- Relative velocity U

- Liquid-sheet velocity used in App. C g

i

- velocity such as liquid velocity (uj) and gas velocity (u )

g L

V

- Volume V

- Traction splashed such as froc a stean generator i

e

. Weber number parameter grouping used in App. C V

Vertical-fall distance 2

Greek F

Correlating coefficient used in App. A where u -oU g

o 4

- Dissipation. per unit volute used in App E ar

- Nezzle pressure drop a~

- Tenpe:ature difference Drop diaeeter I

- Turbuler.t eddy diffusivity used in App. E 7

- Half angle in App. C, see Fig. C.2 I

Growtb-wave length i

Viscosity v

Density c

- Surface ter.sion Temperature difference 0

Temperature-difference correction factor 9/9 =(T-Tj)/(T -Tj) used in o

o 7

App. D Heat-trar.sfer effectiveness used in App. E n

i

32 Subscripts, a

- Air g

- Gas

/

- Liquid: Li g ace r.t

- Original: Beginning; Overall: Atcosphere o

o:1

- 0;ticue 5

- Stect i

I l

I l

1 I

t

-.- y

\\

. f l

TABLE 1 FALLING-DROP-FLO't-F] ELD VELOCITIES (ft/s) AT 32 ft Air or Drain Flow Rate, gpe Steam / Air y,jo'i'i'5 Stets 400 1200 Ratio e

Air 19.5 19.5 1.22 U,lative to ictediateand U,.

drop velocity Stese 23.6 23.8 re surroundings, (u, & u,), velocity of Air 11.2 13.9 1.12 u

i ediate surroundings.

Steae 12.5 15.5 Air 30.7 33.4 1.15 uj - U + ug drop absolute velocity.

Stenc 36.3 39.3 i

• See N cen:Jature.

1

-14 APPENDIX A FALLING-DROP-FLOW-FIELD VELOCITY ANALYSIS Consider ar. array of drops falling because of gravity as shown in Fig. A.1.

The drop array exerts a drag force on the surrounding gas, which is entrained and fails at the gas velocity u.

For simplicity, a constant flow area for the g

fallirg drops and er. trained gas will be assueed.

Actually, a more complete analysis would yield a variable flow area that would increase with falling distar.ce.

Cer.servation of comentue with the entrained gas assumed to have zero ini-tia:

rertue ir, the vertical (z) direction results in L-CY-d i s. F

)

A r 6 /6 and D = p g e 6'/6 which assumes equilib-3 l's a ng dV = A dz Ej = f r. uj y

j rice with the drag = weight we get i E dZ l

= u du ujt F g f Using U = uj - t = cor.stant, which applies at equilibriu=. produces

~

g E 5 c2

/

U u du +u#du 2AF g g g

g E

= 0 to 2 = 2 and u

=u gives 1rdegration between the licits z = 0 and ug g

g ijg 1 u'

u*

(A-1)

U1 + 1 a

2AF 2

3.

G

O.

. = a U to give To f acilitate the solution of Eq. A-1. let ug sjg 2 s

(A'2) 2 0 V' G

which is plotted in Fig. A.2.

Equation A-2 can be used in conjunction with the and drain-flow test results of Ref. I to esticate the entrairied gas velocity ug absolute drop velocity uj = U + u.

Table I succarizes the results for drain g

flow rates of 400 and 1200 gpe falling in air and steam.

Figure A.3 coc: pares the estimated results based on this analysis with the censured values in Ref. It.

s Calculatier for Air at 7 - 32 ft To selve Eq. A-2, the following will be used:

U = U, = 19.5 ft/s for a deforced drop, see App. C.

dj = 55.5 lb /s (m 400 gpe).

e

• A = 227 ft' = r D*/4, where D a 17 ft from Fig. 5-38, pg. 5-49 of Ref.

1, which is based el the assucption that the drain flow at 32 f t has the shape of a circle.

e c = t, = 0.075 lb /s.

g g

Substituting values yields

"/E Z 55.5 32.2 32

~

2 A P,U, '

2 227 0.075 19.53 a U, = 0.575 x 19.5 = 11.2 ft/s and uj - 19.5 + 11.2 for which a = 0.0575, o

=

g

= 30.7 ft/s.

Sicilarly, for 1200 gpc (dj = 166.5 lb /s). D = 23 ft, A - 415.5 ft r,,,o,73, e

= 13.9 ft/s and uj = 13.9 + 19.5 = 33.4 ft/s.

u g

l

\\

l l

l l 1

Calculatier for Steer at 2-32 ft For e steac atr:osphere c, = 0.050 lb /f t', U, = 23.6 f t/s ( App. C) and the f

e jet size is assuced to be the snee as that in air.

1 i

Fer 400 gp (55.5 lb /s),

e l

l

\\

F A2 ff.5 32.2 32 2 A P 0, 2 227 0.050 23,6'_ = 0.257

=

3

- o U, = 0. 527 x 23.8 = 12.5 ft/s and uj = U, + u p

for vbich a = 0.527 u g 23.5 + 12.5 - 36.3 ft/s.

52cilarly for 1200 gp: (dj - 166.5 lb /s) in steac. A = 415,5 f t, o -

g

0. e f '-

t.g 15.5 ft/s er.d uj - 23.5 + 15.5 = 39.3 ft/s.

l l

1 l

i L__________._.___.______________

1 4

t A

~

l l

O e

O i

o V

y

/

v O

o1+

ed;/

i f o fo l

D ge 3 y i g q) f i=

y e4pd }b I

e o

I o' i o

-i 1

l y

l h

U 0

yQ i

VO G

90 Q l "l 7

I

).

9 I

v if Y

4

'l I

}

1 Fig. A.)

Fa.lir.g drop-flow-field showing entraineent free surroundings a r.d char.ge i r.

velocity of gas flowing in imediate surrour.dir.g of drops.

1

)

i I

-1S-O.f f

1

/

~

05-

/,/

a lr

/

s-N g' 'e

},'

s 0,t -

.-[ M.

C o.1 <

05

$4-07 u

Fig. A.2

= c U.

Equatior A.2 used to detercice correlating coefficient a where uE l

_______________________.___________________.___m.

.. i X

8 50.

/

/

(j 2s.

/,.

s

/

'~

b

/

,2e.

!~.

8J 15.

W Ey le.

r i

[ 5.

8.

O.

5.

10.

15.

. 20.

25.

50.

a ELEVATION BENEATH DRAIN P]PE trT) cf7 Fig. A.3 Mean liquid velocity uj (f t/s) vs drain-flow f all height 2 (ft). The

• are eensured values free Ref. 1 for the range of test drain flow rates (200 to 1200 gpe) and tbc

• and *x are calculated in this appendix for 400 and 1200 spe, respec-as tively.

Note that the minimuc: velocity ecasured is close to the free-fall equilibrium velocity (19.5 ft/s).

j

a 20-APPEND 1X B DROP SIZES sheet near the Drops are forced initially by the breat-up of the liquid drain.

The controlling instability is sinuous (flag flutter). which involves aerody'namic forces. Prop sizes forced in this canner are esticated in App. C.

During the fall, the drops cay grow in size by agglomeration or they esy break-up due to instability.

These two opposing processes wi result ultimately in a dynacic equilibrium of small drops growing and large drops breaking up. The breaking up of large drops will create small drops and, if l this process is fastr.7 than that of agglomeration. the resulting array is coc-posed of (a) drops forced by the fragmentation of the unstable drops and (b) drops that are of maxieut stable size. This appendix addresses the drop-breakup pheno =ena. A search of the literature produced the following drop-diaceter (6) values for drops falling in atmospheric air e Reference 2 states the drops will react according to drop diaceter. Specifically. (a) 6 - 4 c= (0.158 in) drops will begin to defore. (b) 6 = 6 e= (0.237 in) drops will flatten markedly and (c) 6 - 6.5 me (0.257 in) drops will begin to breakup by forcing an open-side down

• bag."

see Tig. B.1.

• Reference 3 states that drops of diaecter greater than 8.5 ne (0.336 in) will breakup iccediately because of surface instability caused by a cocbi-Drops diameters between 4.5 ne nation of capillary and gravity waves.

(0.176 in) and 8.5 me (0.336 in) require time to breakup with some breaking presented in teres of up sooner than others. Therefore, results are-average time to breakup, halflife of tbc drop, and probability of breakup. The data indicates that for 6 less than 7.5 se the average fall distance before breakup is greater than 101 m (330 ft), which exceeds the drain flow fall distance. Reference 4 gives a critical diameter of 8 ne (0.316 in).

. e Reference 5 gives a critical disceter of 8 cc (0.316 in) in still air and l 5 cc (0.195 in) in turbulent air. Referer.ces 6 and 7 assuee that raindrops breakup incediately after they attair. a diaceter of 6 c= (0.237 in). 'Assucing that a drop is stable when the ir. terr.al pressure is equal to the exterr.al pressure results ir. I 2 2

1. U 60U f

=-f E B (B-1) or = 2 b o v A spherical drop will f all at a tereir.al relative velocity (U) deterc2ined f'oe r 2 aU g plr 2 f l' = Cgr6 I an' U (B-2) 3t C g d w Eliminating U froc Eqs. B-1 and B-2 gives 6oC 6=[ d}'/2 (B-3) PE A Sub'stituting the values o = 49.9 x 10~' lb /ft (68 F), ty = 62.4 lb /ft', f e 2 8 t - 32.2 ft/s and Cd = 0.5 (10' > Re6 > 10 ) results in 6=(649.9x 10*'O.5)'/2 12 = 0.186 in. (4.71 ac) I 62.4 32.2 32.2 1 ______m 1

4 fb ;, value agrees reasonably well with the experimental values stated in the at ove refsrences. The following analysis relates the original unstable drop size (6,) to.the breakup by the " bag" mode. see drop size after fragmentation (6) based on balanced Fig. B.1. Assumir.g a quasistatic breakup and that the dynamic bead is by the t s.' surfac< nsion files results in p U' R p U' E I (x R )= 2 r R 2 e or =8 2 c Eq.a t ir.g the veluee of the resulting circular-rir.g ligatent to the original s'.:r r g:ves rb' t 2Rr7 6 acc"rding to Rayleigh, drops forced free ligneents result in r._ br 3.7E r Elicinatir.g i and R results in -- - 0.215,l G U }i: (B-4) s e7 t o e To evaluate 6/6, f ree Eq. B-4, the tereinal velocity (U) of an unstable drop of size l cust be determined. To do this we will assume the velocity to e be that of a flattened drop and that the velocity remains constant during the " bag

• foreation.

In addition a beelspberical shape for the flattened drop, as shown in Tig. B.2. is assuced so that

\\ j { i r6* (B-5) 2 (r Ds I. } and D = 2'/86 0 = 6 6 Equating the weight to the drag, with a drag coefficient Cd = 1.'we get i E E(3'/*b *) and plbrb*='4 = g 0 2 4 c 2 1 5 (B-6) 14 0 U"I F 323/. r l Sutut31utari this expression for U into Eq. B-4 results in

1. I4 1

(B-7) o (E 3 p )'/2 6 - = 0. 215 t O g = 32.3 lb *ft/(Ib +s'), and a = 49.9 x 10-' lb /ft. j = 62.4 lb /ft', f with e f e e 6 (B-6)- -- = 2 2. 3 6 0 t o Sicilar values substituted irto Eq. B-6 results in i U=(32.2 0. b^5 )=150f'/* (B-9) o 'l The eueber of drops forced b,r the " bag" breakup is estimated to be J (B-10) N = ( b )' ..-m_ -_.--_m___

1 l . assucing the resulting drops are about the same size. the draft Based on the preceding, the expected equilibrine drop sizes of The smallest drop that will breakup immediately (Ref. 3) flow can be estiented. has a diaceter of 8.5 cc (0.336 in). Note that this size is of the order e x pe c t,e d ' free the sheet breakup, see App. C. The drop size resultlng from the breakup (Eq. B-5) is 1 l '1 l f - 22.3 f ' = 22.3 f. 6): - 0.0175 ft (0.21 in) l e I the aueber of drops forced per breakup (Eq. B-10) is j 0 1 4 336p 3 3 g, 0.21, the tercital velocity before breakup (Eq. B-9) is [ U = 150,0.336.'/: = 25.1 ft/s , and J 12 the terrir.al velocity af ter breakup for a defore:ed drop (Eq. B-9) is U=150,0.21)W = 19.5 ft/s Ir. a turbulent e nvi r onee r.t. such as might be created by the drain-flow This would re-spray, soec of the " break-up" drops (0.21 in) will also breakup. sult ir the follovir.g quantities (based on Ref s. 3 and 4). From Eq. B-S 6-22.3(0.21): - 0.00653 ft (0.082 in) 12 For this size drop, the tereinal velocity can be determined from Eq. B-2 because Therefore, for Cd

• V-the drops are expected to be spberical, see Ref. 2.

3 62.4 32.2 0.052 = 22.1 ft/s U. 30.075 0.5 12 1 L

. To suecarize, this appendix shows the following:

• Drain-flow drop sizes should be between 0.21 and 0.082 in.

This range of values is confirmed by the test observations presented in Figs. 5.10 and 5.11 of,Ref. 1.

• Tercinal velocities should be between 19.8 and 22.1 ft/s. This velocity j

I range is confirmed approximately by the Fig. 5.33 of Ref. 1 free-fall velocity near. the drain-flow discharge, which is where the entrained air velocity is a citieue. }' Drop sizes do not change appreciably in a steam environment except for effect$ of surface tension, liquid density, and C. These are expected to d be sr.all, see Eqs. B-3 and B-7. The terminal velocity will be effected mainly by the steam density in the exp;ession, U, = U,(r,/t,)'/2, neglecting the effect of scall changes in C. g Drops forced by sheet breakup (see App. C, Eq. C-9) are expected to very by l the f actor. L, - 6,(r,/t,)'/" which is a small ef f ect. 1 1 1 1 w: o t

r A sudoC4, w.~ c.6 k. fa .Ur Fs4 T Fig. B.1. falling drop just before the bag B a g c or.f i g u r a t i e r. of a breaks to fore a circular-ring ligament of radius R and cross-sectier radius 7. Note that there are two surface tensier files that balar.ce the dynamic pressure. J l a s. -a-1 Fig. B.2. Geometry for a drop of diameter 6, changing to a flattened drop shaped like a becispbere with base diaceter D. i

. APPENDIX C DROP SIZE FROS! AN AER0DYNo11CALLY-UNSlABLE lhTISCID LIQUID SilEET Two types of oscillations can-produce instability, i.e., sinuous _ (flag l nee ected because its j l fluttgr) and dilational. Dilational instability can be growth rate is always less than that.for sinuous (Ref. 6). Vben aerodynamic f forces are reduced.

• sheet tearing" will determine the drop size.

In a low den-sity sicae environment this " sheet tearing

• may be controlling.

To obtain the drop size that results from sinuous instability, the follow-ing equations froc Ref. 6 vill be used; (C-1) 'or

/

gives tbc ligament radius .r gives the taxi =ur-growth-wave length (C-2) Sept = po U' :- J ) L - 3. 7 5 r y gives the drop size, and (C-4) V er gives the sheet thickness at break-up I i h;= Pj l g FjU h c with W= 2o K = Parareter of nozzle = r h , and' o E = frM, the wave aeplitude ratio, assumed usually to be 12. yo Equation C-4 can be solved for h as follows b -E o(F V2 e ) /2 FK / o$ i b,/. and i pjl I J ________________________________Q

. (p,E o,:fs(P U2e)n fK fn (C-5) i r; sc thrt Ecs. C-1 through C-5 can be solved to produce , ' A l'o" 2 r ) ' / 2, {2 e h )'/ ,,4 e c h _E pU-S. 7b g go 2 e l'/2(#p o 3'/3(# D2e)'/* f /o and l - 3.76[p U p, E ' go 2 e s'/ (I o 1,'/* (C-6)

2. 7 5 t,L.-

L)'/3 (p e o g can use the nozzle pressure drop in edditict., we

,\\-[

to psve bl e (p,'/3(I ,be s ' / 2, p /) /t (C-7) 3 4 7b 7; 2 TL:s expression shows that drop size for a fixed nozzle depends only on the pressure drep (AP) and the density ratio becaLse K - K (AP) for a fjapper o o valve. The ratic of drop size to file thickness at breaku: is obtained by dividir.g Eq. C-6 by Eq. C-5, i.e., E 3.75(G U ')'/3(E (p1)'/* 2e o '/2 g jo g and -= b N( O I/> No '/3 E 2c tj I i -. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _. _ _ - - - _ _ _ _ _ - _ _ _ - - - ~. _ _ _ _ _ _. _. _. - - _ _ _ _ - - _ - _ _ _ _ _ _ _ _ _ _.. _ - - _ - - _ _ _ _ _ _ _ _ - - - - _

~_ 6 ., E ( r/ )S / 6 (C-8) . = 3.76f-[-)'/3 (c,U,-)2/3 2p 6 'Lc y 1 of, in teres of AP across the no:zle.

3. 76 (po )' / 2 (gp )2 / 3 (-1)5/*

(C-9) E e 6 g b a

.\\

is a constant u and Equations C 6 throurb C 9 apply to water sheets where Uc gravity plays ne role. The velocity ir. the drain-flow sheet is quite depender.t grasi.y durint its 5 ft e x t e r.s i on. However. epproxieste calculations will be or cade w: t b Ec s. C-6 ttrough C-9 to see if the tecbanics of sheet instability car be used to esticate drop sizes resalting free dreir-flow sheets. C.I. Cerrariser of Theerv vith Ref. 11 F:rst, Ref. 11 will be used to evaluat e the equatier.s developed because it f ) l -a! used as a basis fer esticat stg drop diaceters resulting f ree a sheet. i r. produced by a l Ref-

1. vbich we are reviewing.

Note that the Ref. 11. sheet was l spray rozzle. see Tig. C.1. Free the geocetry of fig. C.1. 1 I Q

• 4 U

ard l Q-2ri b U = 2 r r sino b U se thai 1 p-(C-10) rb =K = e 6 sir.1 1 To obtain the parameter grouping used in Ref.11. i.e., 6/D and AP D. Eq. C-7 is rearranged and Eq. C-10 substituted for K to give o I

t \\ E sir.7 F)'/3(E ' / 2 ( ~# J'/' l D: r 1 t - 3 75'. E .'4. 7 5, 5 s i r.? E }'/ 3( AP D)'/ 3(#s )'/s 0 / 6 1 - = I' E Subis t Li ang the value s ? - 30. E - 12, c - 4.92 x 10 lb /ft, #j/p = (62.4 0 f g lb..t')/(0.075 lb /f t') - $32 and converting AP D (Ib l.t) to AP'D' (Ib /ir.), f f y e i.e. t7 D = AP'D' x 12 produces a .t.92 x 10*) .'/3 '/s } ( f b sit.3D 12 AP'D' 12 which a plotted in Fig. C.2. The agreerrer.t of the 6/D in Fig. C.2 is witbin an ordet ( ~ ea g t. i t ud e. which is as good as could be expected free the theoretical deri m,on. M:re specifically, the theory predicts values of 6/D that are axi.ately a factor of two higher than the experseer.1 values.a a p' s icila r ear.nc r. Eq. C-9 car. be arranged to correspond to Ref.11 pa-I r. a

t ti c ~. t i as fo}}owf l

p A --- - 3. ' 5 f E 6 s i r.7 ) / 3 ( ) )3 ( J)Sf s 1 b g l l

- 3.76(12 5 = 0.5) /3(4.92 x 10* * )* / 3(832) /8 2 0 ' ~<

(C-12) i 5 t ,p.g. g (AP'D'),,, I h 1 Eqantion C-12 is plotted in Fig. C.3 for cocparison with the results presented 6 n Re f. 11. The Ref. 11 curve is obtained by combining Figs. 11 e r.d 12 f ree j Ref. 11 J alt is interesting to note that Eq. C-11 results ir. 6 va rying with AP"'/2 and Eq. 12 of Ref. 11 ir.dicates 6 varying with AP/'.

. is within an order of magnitude and 6/b' is too high Again the agreceent oy a factor of 2 to 3. C.ll. Application of Theorv to Drain Floy The theory developed will now be applied to the drain-flow tests of Ref.1. drain-flow sheet is depicted i r. Fi g. C. 4. Observation of ge,tetty of the Tbc o the sheet geocetry at 400 spe (55.5 lb /s) indicates a sheet spread angle of g 100# and r of 5 ft. As a result. (C-13) Q-2rrfbU One iar reathitt simplification cust be made in the case for drain flow. Ir.acely, that U - (2 g t')'/ Sin:e the sheet velocity is created essentially by the of 5 feet. U itercases f rom the drain pipe to the break-up fall ever a distance point. This is a deviation free the theoretical development where the velocity was assumed to be a constant. theoretical d e ve l o pee r.t. Therefore. the esticates that follow cust be considered approximate. Ussr.g the expression for U gives f O O L =y b C 2 e F.' 2 r F(2 g t')'/3 Pjl(2 g I')'/#} and AP = 2 Substituting values gives 55.5 lb /s I y = 0.0281 and K 62.4 2r (2 E 5)'/*

s . [(2 32.2 O 5 ft)]' lb 62.4 f ft-s' AP = 312 ft* ft ib" 2 32.2 -s*1b 7 ) 'I t f = 3)2 y %'i, h i - 12. L q. C-7 c a.r. be s o l ve d t o g i ve -0.02 tis'/3 4.02 x 10*'3'/2 - 62. 4 ) ' / ~ 3,,3 j 12 312 O.075 0.03be ft (m 0.46.4 i t. ) ard Eq. C-9 1 12 10~3 62.4 sS/' 2.<5'. I'/3(4.92 x )(0.075 = 4.63 2" O.0261' 312 7bese, sults car be applied to the observations of Ref. I by reducing the values by a f actor of 2 to 3 as indicated by the cocparison with measurement s f ror Re f. 11. This results in 6

• 0.2 in and 6/b' = 2.

The drop diameter is cor.f ireed approxir.ately by the drain-flow test measureme nt s ar.d the 6/b' value indic.ates the drop diameter will be greater thar. the sheet thickness. C.111. Sheet Breakur Letrth r The observed value for t in Ref.~1 is 5 ft. This can be verified eppsosientely by using Eq. 24 free Ref. 8 c(jU'by'l' gK p p n E = Pjy : 2e 'b' for far.-spray nozzles. For the drain-flow whe e N =t o sheet 1

( l l 1 l l l b' = and O' = (2 g r')'/* i 2 r r F U* i f 0 h r'F g(2 t')'/; { I Coebining to eliminate b and U and using the simplification that U, = U' results i r. } ( Q (C-14,.)s 5 2 g,;('F,r T}r!f

1. * / f f'

= 3 s .c Substitutir.g tbc values for 400 gpt 55.5 lb /s f,3 y . - 0.650 0 = 62.4 It /ft-s r ! = j or.g = 0.25. E = 12 w lb lb .1 j = 6 2. 4 -,#. o = 4. 92 > 10 f t f ft-results 2 r. g.92 x 10O.88932.2)2/ = 3.95 ft 4 0.075)/' 62.4 1 ,= = 12 (2 32.2)'/3' 62.4 r 0.25 which is confireed approximately by the observed value of 5 ft. C.IV. Theoretieel Dro-Site can be combined with Eq. C-6 to provide an expression for Equation C-14 the fan angle is 100' drop size based on only the esperip otal observation that and, therefore. F = 100'/360' = 0.25. Starting with Eq. C-6

. N 1 (C-6) ? - 3.75'2 c oi'/3(#;g)'/* IjEl'c (2 g r*)'/ ard b - Q/(2 r T r U) results in .: r.d sub' :ttting F.D -rL,U p (C-15) 6 E ' '( 2 g ) ' ' ' ( f'jr T# 0 d' (f J'/ j Subsiit.titt the values for 400 gpe used in Eq. C-14 gives 3.1 ,4.92 x 10-3 0.5$4 '/* 0.075 of,. '# *(2 32.2l 1.94 e 0.25

62. [

0. 4 34 I t (0.521 it) crJy 131 highe r ther. the value obtained in Sec. C. II.

whic. 2. C.V. Theoretical katic f/t' The theoretical ratie of drop size to break-up flic thickness car be coe-potcJ 1s substituting 1 0 and AP = P g r F., j ( l 2 e f(2 g r,), I { arte Ec C-9 to give \\ es:/2(# 1,5/5 l [

3. 7[. 2, E e T ' / 3 (p/

r p 1 b' 2 g s' 9 g 1 Using Eq. C-14 for r produces i 6 3.75 2.D fe (#o )S /'(r T )* /'(# //" (C-16) J i

1. g U

b" (2 E ) ' 1 l Us att t he 400 gpr value s give s i i

. ife(4.92 x 10*8 ) / ' ( r 0. 2 5 )' / * ( 0. 075 ) ' / ' ' i 62.4 6 3.75 2 12 0.889 - =y' (2 32.2) 1.94 = 5. 4 4 C.11. vbich is or.ly 131 bipher thar. the value obtained in Sec. I f l l I 1 I i 1 l l l

u._ ~36-l [ o tp t Nh / r* Fig. C.) Spray-sheet peometry used to estimate drop sizes af ter sheet break;p based or. Ref. 11. D is the spray-nozzle diameter, r* is the sheet length ler.gth h' is the sheet thickness, r is the sheet radius at r *, and 7 is the sheet conical ar.gi:. ) I I J

o. as. ,[,_k._.--_ 1 d 1 I ( i \\ \\ NlS 'Dnn{ st ( N I. \\ J" i 1 1 1

• A M

b e d 7, _~\\\\ 8 i 5/o Fig. C.2 Comparison of Ref. Il ecasured spray-sheet drop diace-l (6) with values predicted by Eq. C-11 of this analysis. ters D is the nozzle diae:ter and AP is the nozzle pressure drop. The primes refer to a change in units, see derivatiot of Eq. C-11. i

o 2 - r __*1 .. ~ .v e ,a ..g m. . 4-t ~,,.. . $:.:...i M - M%K != - _~:iV-l-,- ,n,r, - e.'w.. ,W~ ?- ;5.a' llC ~- a at ji+ --M.9-i..-

s. _n.

jp.-.i _f.= M ; '.; ? -'. g "f [ ~[h-[ pth ~j j.I }l!I '.- ] j..h ID ; x.. .i x.. C - i... 1 5_ 3 a 4ifi 2 iA 45 ~.U i-i vi i_-,.;.. t -t. . +..+ - [#.L 1.}.!! jh4 - h ~ l ! rXi 8 X 5-g-3.i t.iai i eN ..;..=. i w; ..a.uiw*=m v.a..

e. a.

-,x _. ; rim 5_ 1 l... - 233EE Rd. s-C. t f M i. ,N. y. 1 gj g N-p5 ~ e 4 i-t ,; :1 4

• i -:

Ti. a r - r a 4-1 i 64str - i 'w' .r d '.' & -l Wig- ) 5 W"}: '. h S h i.E. T)

t. e ? i r -

.e 4 - .i t,.. i .% w i _i '

t. g j-t v 4 1--

, f. ' r )' 6' 4 \\ \\ ~;D1 '- l t 1 i,. .d-1-\\;1dh W %P =-

• b-i--+ !. =

' i. l

.j ' l F, _' H. l'I ._.E._.-_ 5..'j k. L',J %~ ~~' i. }, .t..- .i 1 Jo 200 AP e' (4/Q' Fi I C. 3 L' of dror 5 i2* f to sheet thickness h vs nozzle pressure drop AP' and diameter D' based on Ref.13 measurements and th. analysis of this report.

. ) i / / i / y ; t i o,- s \\. %, -- iCP t b+ 1 i y -,7 __j l 1 s Eig. C.A' Spray-sheet geometry used to estieate drop sizes af ter sheet breakup based et Ref. 1. 7' is the sheet leegth and h* is the sheet thickness. f i

, APPENDIX D IIEAT TRANSTER FROM ENTRAINED ATMOSPilERE TO SPRAY in temperature prpose of this appendix is to estimate the reduction The differerte potential for beat transfer to the spray because the beat transfer is the the spray and its ietediate surroundings of atmosphere entrained as l betweer is, the temperature difference potential is not the differ-spray falls. That ence between the atmosphere temperature and the drop temperature (T -Tj). In-o effective temperature difference potential is (T-Tj) where T is the stead, the As the drops and temperature of the gas entrained and falling with the drops. entrained gas fell.-the gas is cooled so that T < T and (T-Tj) < (T -Tj). o o dx u shovr in Fig. D.1, the gas-phase beat balance over the increcent

ves e e 'l + c T de =06

• C e T + d(e e T) and g

o g g g l c(T -T)de = Q * ' M dT e g 6 g ( ) Intr this we can substitute 2#u du E r b ~f f Q6 - t, n s O (T-T )dV and n-d5 j /I 6 Tbc latter results froc a cocentue balance on the incremental voluce assucing l The resulting expressior. is l the drag force equals the drop weight, see App. A. l l 1 b (T-Tj)du2 + dT (T -T) = 0 m cbpjg g i To sierlify, let e - (T-Tj), 6, = (T -Tj), vbich results in g

n . - e)b = 12 h ec 6 p,g 8 + de du (e O u Using T - e/e, as a censure of the decrease in potential for beat transfer gives' (1. T)) = -12 b T du + dT u cbsg S g g Rretscrgirg for integration we get 12 b )T= and

• (

+ l l N+PT=Q 't where P=B4 1 B= , and Q=1 12 h u cbpg u i g g that can be This is a linear first order nonbocioteneous diff erential equation integrated with the integration factor F JP du ~Bu F=c G=ue E g Solving for T with the assistance of the integrating factor results in -B u ) (D-1) I 8 T = B u (1 - e g Calculated Values

. To deter-calcu eted for 0.20 ir. diaceter drops falling in air. sill be ri :.e the hea1 trar.sfer coefficient b, the Rar.z-Marshall correlation with a Frandii t.utber of one will be used, i.e., (D-2) n=b 2+0.6 ) ) 6 v g use of this equation icplies the beat transfer is priearily ser.sible Note that c re r g> - 2 r, - 27,9 x 10~* ft /s. For air at 260 F and

ate.

g. - 0 0:944 Btu /(br+ft 0F) and U, - 14.5 ft/s (see App. B).

Substituting ir. Ec. D-2 gives b, - 26.2 Btu /(br ft #**F). fn: stea at 300'T and 7 psig, r, - 1.9 x IO*' ft /s. a r.d U, - 23.6 ft/s (see App. B). Substitutir.g into s, = 0.016 Btu /(br ft**F) Eg. D-2 give s b, - 25. 2 Btu /(br *f t2**F). B>.g is r.eeded also to solve Eq. D-1. For air, c, - 0.24 Btu /(Ibe OF) and u - 11.2 ft/s (see Table 1 for 400 gpe), which results in (B u ), = 0.122. g p Fr ee E:;. D-1. 7, = 0. 94. g 12.5 ft/s (see Table 1 for c, - 0.475 Btu /(!be**F) and u For stent, 4.40 pn 1. which results in (B u ), = 0.0739. Free Eq, D-1. T, = 0.96. g t I l

c yl ,.Y A s.W

• >a$

e Licatomed 'f/*,**.,*,.{[ % cx3 I h \\ m, cTtclGncT) e Fig. D.1. Teres used in erergy balance for an array of falling drops and gas entrained free a surrounding atmosphere at i t ecpe r ature T,. T is the temperature of the entrained gas and Q is the energy transferred to the drops. Note that 6 T(T because the entrained gas is cooled as it falls. o l 1 l 1 l i i i .J

1 . APPENDIX E TURBULENT-DIFFUSION HEAT TRANSFEP TO A UNIFORL1 D15PER510N OF DROPS The purpose of this appendix is to esticate the eff ectiveness (n) of beat a dispersion of drops such as eight be createc by drair.-fle.' spray trarsier te interaction with cible trays. The heat transfer is assueed to be cor. trolled by turbulert propagation so that di (E-1) t cdx c. e; plies. See Fig. E.1. r -Servatier of crergy over the increcer.t dx sbovr ir. Fig. E.1 results i t. cg- = Og x + {-4 # c.dTg- + d r dT }),! dT r d ig- -t f

• bt:t froduces 1

7 GE l e C t 61 Assutir; a ur.ifere distribution of drop heat sinks gives i A h(T-Tj) j Vg = y wbere V = 2aL* Width Height. Note that Aj and V can be obtained from test obser-vation ir. Ref. 1. Also. Ict e - T-Tj which gives uY/dx = de/dt. Substitution results in g:o AjbP (E-2) -0 ds' \\. <0c t--_

1 , Differential Eaustiet Selutie:. Solutier. of Eq. E-2 as a liccar homogeneous second order differential e-quation with constant c. coefficients results in (E-3) B - C r* * + C e- , where j 2 Ab i c - (V e F c) f: f Applyirg the bour.dary conditsens A l d6 a r.d (f ),,t = 6 Ig7,3 1 = 0 0 results it. p ,c(L-x) ,-c(L-x) (E-4) f ,L e ,-L e c which is equivaler.t te w f_, costic(L-x)) (E-5) 6 cosb(t L) 0 Effectiveness of Heat Trar.sfer to Serav Drops The heat trar.sf er f rom the surroundings at temperature T to the disper-o siot of drops is, froe Eq. E-1, dT de (9 ) -0 " "' # C Igi )x=0 " "' # C { 2 g ) =0 t Free Eq. E-5, (de ),,p = -c 6 tanb(c L) 7-l so that, with the expressior. for e and dropping the x-0 connotation, 1 L.________________._____________

= a Ajra rc Ajb it (E-6) q.-(---, jf tanb{(\\. 9 c) j L) ) o e Nate that this espression is similar to that for an extended surface with length L, const nt area, base temperature T and insulated at L. e.g., see Ref. 13. o The maxieuc beat transfer occurs when the easieuc temperature potential available (f ) applies everywhere so that e b Ajlf (E-7) e (qt)ct, y Tt e a. trarsfe effectiveness is then Ajb s tanb[LI'Vefc)ir) 41 1,Ve e c't; =, n = ( q1 )e. 3 L Ajb or. ir t.cres of c. tar.b(L c' (E-6) n= Le .Ajb wherec=(Yapc)f: i l to detercine c. the turbulent eddy diffusivity e is needed and will be obtained by usir.g the following froc Ref. 14 1/3f8/3 (E-10) with Pr = 0.7 t e = Pr t where ( is the dissipation per unit volume of the potential energy of the falling spray

l % (E-11) d'

• F V/

Af 5 E \\ i and the eixing length / = 0.15 D/2. an expression used often for a free l t u r b e l'e r.t jet. Values for Heat-Transfer Effectiveness n in air and stese Values for n will be esticated f or drain flow f a!!ing vitt 6 = 0.20 ir,. dj = 55.5 lb /s (400 gpe) and A = 227 f t' f or a spray diame ter y of 17 ft esticated free photographs in Ref. 1. In addition, the drop velocity to approximate the terminal relative velocity calculated in App. B. is assumed U,

• 20 ft/s and U, = 23.6 ft/s.

i.e. j Fcr air. the dissipation froc Eq. E-11 is l bE $5.5 32.2 ft / a"Ag " 227 o,073 7 a ar2d. free Eq. E-10, with the eixing ler.pth / = 0.15 D/2 = 0.15x17 ft/2= 1.25 ft g,'/'/'/* 105'/8 1.25 = 9.37 ft l = ,a. Pr 0.7 6 t Determining e with h, = 26.2 Btu /(br ft 2 *F) from App. D and I Aj djr 6 6 $1 ~ F 6 ujA Y

1. r6UA j

6 6 6f. 5 = 0.0707 ft =- 62.4 V.20 20 227 Substituting p, = 0.075 lb /ft and c, = 0.24 Btu /(lb,'F). 8 y Ah ifr 0.0707 26.2 y'/ = 0.0552 ft / a Veac 9.37 0.075 0.24 3600 s/hr t

l. l l i l 1 l l l ac that, with L = 17/2 = 5.5 ft. c,L = 0.4692 and, froc Eq. E-8 I tanh 0.4692 0.4376 = 0.933 = na 0.4692 0.4692 i For steac with t, = 0.05 lb /ft8 g, - 155 ft'/s' and <, = 10.7 ft /s. g Using b, - 25. 2 Btu /(br ef t * **F). U, - 23.8 f t/s and c, 0.475 Btu /(lb *0F) g I e, - 0.0429 c,L = 0.364 and n, = 0.945 i s l l l 1 i l l l l l l

le l. \\ . l l / (> J o p Q (pu,e+ i.*n b, 'P f a i s p l s'. p* s f9 % T"q,v -eecdjb - fLj e '? ~$Ph ?'5 'f .ll

• O &\\)

Of C ao y .C' G UVW j;g. e. c *o cs .o.. 0 .Qp l l _ _C lE hy ' ')

• *xt J

G ' lL = J,, f. ',N * **a E r; v, ' ;' o w J ' i 0

• )00 c, g

9 '.4; ' LL' V i .f. ( 4 Fig. E.1. Turbulent-diffusion heat transfer to a one-dimensional slab of drops distributed uniformly. The slab thickness is 2+L and the ateosphere temperature on either side of the slab is T. c 4,.

1. I l ) - i APPEhDIX F SPLASil ANALYSIS This apperdix examines tbc ecchanics of splasbicg to estimate the impor-tance ' of the e r.vi r onee tt a l gas. In particular, equations for the fraction splashed are developed. Also, the analysis should yield estimates of the size cf the ** splashed drops. The analysis is developed free several primitive l equatior.s i n Re f. 15. For e xample. the periphery of contact between the drop and the top of the file is. 1 r = 2 r(2 R U t -Ut #)'/* d wher. is tite free iritial iepact, see Fig. F.1. For the splashed file, 3 R U '/; d Ut and v = n! ) , where o - 1.19 3 h=2 1 Tne rate of voluce flow obtaired from the precedict equations is I 3RU'/:(2r(2 rut - U t )) d 3 + ) d V=hVP=70 t n! Usit; dieresionless tier as suggested in Ref. 16, i a r.d dt* = b t results in t' = I k d d I U y, f. dV U r a R t*( t R )'/:(2Rt*Rd - t":Rd d 3'#t d di R dt. d d (F-?) b=roRd -(3)'/ (2 - t')'/ t = dt' i Ir.t e g r a t i on f r oc V = 0 a r.d t ' = 0 t o V a rd t

• yi e l d s l

___.__m____._______m_ __m .m___m.. _ _

. f [16(2)'/2 -($+6t*)((2-t*)')' ] " "I3) The fraction splashed where V R* is V' = y'd r o p drop " d V' = l'I9I'I'/ j16(2)'/* - (8 + t, t*)((2 - t')') ] (F-2) V' = 0 when t' = 0 and dV'/dt * = 0 when t' = 2, which are physica18-I,o i e that correct. The latter condition ccans that

• splashing"terminateswhenthebottot(

I of the arps: ting drop contacts the wall and the " water-barrer* effect doeinates. TMs causes horizontal spreading and the forcation of a cylindrical cavity. Also, any drops forced due to wave action during spreading will be neglected, see Ref. 17. Fre: tbc above, the "splashtice" is, t - b/U. If the impacting drop approaches the file at an angle o, the " splash time" can be approximated by b t = U sino , see F Q. T.2. in teres of dimensior.less time t*, b (F-3) t' = R 53E' d To coepute the fraction splashed V' from Eq. C-2 using this expression, the film thickness b cust be determined. Assuee a vertical surface being splashed by an impacting array of drops of j I total mass flow rate 5, and forming a surface film of width W. The file flow rate becomes. (F-4) d - (1 - V*)E n 1 1

a . I The thickness of a turbulent vertical file b as a function of the file flow i rate $/w was derived ir. Ref. 16. The reduced equation for the water file is I 10~ 3 (f.)1/1 r (ft)=3.33x10[(1-V')[]7/1r j b - 3.,33 x 1 and, in teres of the dicensionless

• splash tiec,*

(F 5) B1-V')b} 3' 33

• IV t' =

s i r.c kd ir. te res of t' and V' that Equatices F-2 and F-5 are two sicaltar.cous equations car. be solved graphically for V* and co= pared with test results. F.1 Corrariser Vith Test Results The f ollowing ir.forcation free Ref.1 vill be used for the splashing of 400 l (55.5 lb /s) of drair. flow free a stene generator: sp e indicated by Fig. F.3, which was copied fro =

• At icpact ar.gle o of 23 as Ve!. 1.

A censured salue for (1-V*)e,3 = 0.36 from Fig. 5.42, pg. 5 55, of Ref. 1. steae generator diaeeter is 12.5 f t and, as an approximation, the file The flow width (w) is esticated to be equal to one-half of the circumference = 2. 6 3, " V-f(r12.5)=19.6ft and = The maxieue breakup drop size is assumed to impact the surface of the sicac' gen-erator. i.e., 6 = 0.21 in (see App. B). Equation f-5 yields,

~ e e % t' = !(1 - V')2.83)'/'2 = 1.78(1 - V*)'/ (F-6) I 0.21 0.391 2 71e solutier. of Eqs. F-2 and F-6 is shown in Fig. F. 3 with V* = 0.70 and the

l file flow fraction (1 - 0.7) = 0.30. which compa re s reasonably well with the

{ I reasured value of 0.36. f F.11 Splashirr Fraction in Stear Assu=ir.g that the mechanistic *sp' sb* codel is correct. the only quantity l 3rvolved that car. be changed by the gaseous environment is the angle of sepact n. This results fro: the effect of drag on the drain water trajectory. l 1 l lo e s t ica t e the effect of a different gaseous e nv i r onee r.t. the angle of icpingecent without drag is estimated from the trajectory equation E *'. (F-7) 4 y = x tar.f 0 2 V cosf l g g l where 6 is the initial drain flow trajectory angle (-20' f rom photographs in 0 Ref. 1). V, is the initial velocity (* 15 f t/s ). y is the vertical coordinate a a r.d x is the horizontal displacement. Substituting values and solving for x after a vertical fall of 32 ft (y = -32) gives x = 17.7 f t. This can be coe-pared with the value of 10 f t ceasured in Ref. 1. In additfor.. Eq. F-7 solved for the zero-drag icpact nr.gle gives a = 47. Assucang the icpact argle in stene is the average of that in air and that with zero drag gives ,, 23 + A7 - 35" 2

• For 400 gps.

W 4 . Eqt:s t i ci t + with a new sina - sin 35" - 0.574 becomes t;=1.75 (3, y;),jir, 3,33 (3 y;)t/tr (y.g) 0 The graphed solution in Fig. F.4 gives V' = 0.55 and the file flow fraction 1 - 0.55 - 0.45. This indicates that the fraction of the impacted drain flow going into file flow could increase significantly in a stcae ateosphere, i.e., 0.45/0.30 - 1.5. The size of the splashed drops 6, can be estimated by using the equation l, - 1.5o b = 1,59 U t/2 = 0.95 U t f roc Re f.15. I r. teres of

• splash time,'

2, = 0.?$ t*Fd = 0.95 t'6,/2 and 6,/6, = 0.475 t'. With t - 0. 21 i n, t j = 0. 88 and t ' = 0. 78 ( f r oc F i g. F. 4 ) o (6,), = 0.056 ir. and (6,), = 0.076 in Note that these are maxieue drop sizes created near the end of the splash. These values indicate that the difference between splashed flow' drop sizes in air and stene is scall, i.e.. the ratio 0.076/0.085 = 0.89 is near unity. Comparing with the censured drop sizes for splashed flow from the steam generator (Fi. 5.31 of Ref.1) indicates f ( 6,),( c a l c ula t e d )_ 0.055 - 0.59 ( 6, ),( ce a s u r e d ) 0.15 L.-.---.--_-__-...-_-_.-_-.

\\ -$S-1 ) l gV K c g_( fv / Eh i / fb l -M

L_

. L, j i<fs/ 7/// - - /isss a l Tig. F.1 Parameters used it splash analysis of a drop icpacting a f i l e.. R is the drop radius. U is the drop velocity, r is d the radius of the periphery vbere the drop contacts the top of the film, V is the velocity of the

• splashed" drops, and 6, as the diaceter of the " splashed
• drops.

l l l l l M 46 t i. i t y. ia i s ap l Fig. T.2 Geometry of a drop with velocity U approaching a file of thickness b st the angle a from the film surface.

l-o, ~ . i f e

• ; *.p*,

1. w-p.

,,y.g.!Y '*E,' I h [g4' ~ I@, hh' A I d ' ' _ Qg g ( I \\ 3 y; es y..- ~ ~ 7 4: im J 1.- [ '.j {:l1l 4 1[i(p \\ (.~,l RCP Y1 9 k L i l i 2 4f 1 i .. i. .i d1 fgg b--44: n I Il ! i, [ \\ n \\ \\\\ \\\\ ' ~ ' wn> cm qa.; @ g Fig. F.3 Drain flow of 400 gpc impacting the siculated reactor-coolant 0 puep (RCP) at an ar.gle of a = 23. This obse-vat ion wa s assumed to apply to a stcae generator (SG) because the RCP same vertical and and 50 a re located at approximately the borizor.tal distances from the drains.

v l . I.D - N i u. F4 N s v. 1: 8 '.. yac c 7 >. t * = 055 s V 4 k g,g 5 Shum ? I 4 e h Equ,C.-2 c C' S~ l* O f.y. c% e e s, an te ss #sp Is.sk &e. " Fig. F.4 Fraction splashed V* vs the dieension less splash time ta for Egs. C-2, C-7 and C-E. The curve intersections show tbc solutions of simultaneous equation for air and steam. 1 .---____.__________.______._________o

w w o . REFERENCES 1. L. J. Dovies. L. L. Hochreiter. V. M. Kavalkovich. J. A. Kolare h. Lec. J. C. Feck. S. 5. Tsai, "Ic e Condense r Dra ir. Te st kesults. D;ta Atalssis. and Developmer t of Drain Flow Models fc7 the LnTic-Ill ice Cerderset Code.' Ve s t i r.phou s e Electric Corporatic. Fr err a. ers rerert VCAP-1096cP ( he vetoe r. 1965.. 2. L. Pr rdt;. Essertiels cf fluid Dvrarics with Arrlicatiers. pg. 325 10c.t r e t e refererae tc L. H oc h s c hm e r.u c t bassertattet. Hejdelberg. (1'1' H:f:rer Pvt. Ct. NY (19f2.. 3. t becabasss2. T. Genca. and L. 15 cr.c. " Life Tiee of Water Drops befcre breekirt and 5ize Distribution of frag =ert Droplets." Jourt. l Met Soc. Jurer. Vo;. 42 ht-5 (1964:. I 4 1. Lerard. "Uber Keger, Meteorologic Zeitscript 21-FT 246-202

lo av L.

C. B!crch;rd. "Tne behavior of Vater Dreps ir Tereiral Velocity it l A:- Tr rs. Acer Geephys. U r s c r.. 31 pp E3t-542. c. 1. L:rgr.ir. "The Productiet of Ra st by a Chait Reactier.in Cuculus C! cues et Temperatures Above Freezing.' J.

Meteor, 5,

pp 175-192. 1 I \\144b. ~ I. H. L u r.d l a r. "The Production of Showers by the Coalescence of Cloud brer.ett. V. art. J. Kes. Meteer. Soc. 77, pp 402-417 (1951). E. F. Eiserklar. N. Doebrewsk i. D. Ha s s er.. " Drop Forcatier free kapidly Mav:rg Liquid Sheets.' leperic) Ce lege Report JRL ho. 44 (May 1959). h. L etrevski V. K. Jehrs. *The Aerodynamic Instability and Disir-terratier c.f Visceus Liquid Sheets.* Chetical Err. Science. Vel. 1E. pr. 203-214 Pe rrator Press (1963 ). It. J. C. P. H at;. "The Break-up of Axisymmetric Liquid Sheets ' J. Iluid Hech. \\c; 4 3. pa r t 2. pr. 305-314 (1470..

11. 5.

Veirnerg. *Hett Tratsfer to Lov Pressure Sprav of V;ter in a 5 tent Air stnere.'

1rst, cf Me c t..

Er.p.. Proceedirts L-Fr. 240-2fb ( 19: 2-fi 1;. T. Garsber;. 'Liccid Jet Br ea k-ur Cha rac t e r izat iot with Applicatier. tc Melt-kater M.xtry. Proceedarf. ltt. AhS/EN5 Mtg. Sat Diege. CA. (Februts 2-t. 19 6 t. 1.' V. 5. Arpac: cerec: tier Hett Trorsfer. Addison-Wesley Publishing Cc. f 1 46t i. 14 5. t. Pet e rk;r - A. Pollard. A. L. Sitghal. 5. P. Vanka. Editers. heretic;; Prec:ctier e fles. Heat Trarsfer. Turbulence and Combustier. Perpacer Press. Ni 119b3. If. h. G. Cne rg-Leerg ( Pr o.ie c t advisor Pref. Shi-C Yao),

• Droplet Gerer-atier by 5 p l a s h i r.r.

M. S. Pr o.ie c t keport, Dept. of Mech. Erg. l Ca rr.e g i e -Me ll er 07 a ve r s 1 v ( Apr il 20. 1975). te V. C. Mack'ar. 6. J. Metaxes. *$plashirt of Drops er Liq.ic Lasers.' 1

~ 50. 17 O. G. Erge]. " Water Drop Cellisior,s with Solid Surf aces.* Journal of j kesearch of the hatiotal Bureau of Standards. Vol. 54, No. f. Re-J searci pa pe r 250) (May 19553,

15. l..

(i. U i cie a r d A. Koestel. T or.t a i nee r.t Cendensing lleat Transfer. Secord I r t e r r.i t t er.e l Topical Meetirp or Nuclear Thermal Hydraulics. S;rta B.reart. (A (Januar3 11-14 1963). i 1 l l 1 ( l 1 4 I l l l __._________.__________a}}