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COLLEGE OF ENGINEERING

DEPARTMENT OF NUCLEAR ENGINEERING LABORATORY FOR FLUID FLOW AND HEAT TRANSPORT PHENOMENA

Technical Report No. 18

CHOKED FLOW ANALOGY FOR VERY L O W

QUALITY TWO-PHASE FLOWS

by :

Frederick G . Hamitt M . John Robinson

Under contract with:

National Aeronautics and Space Administration Grant No. NsG-39-60 Washington 25 , D . C .

Office of Research Administration Ann Arbor

March, 1966

k

https://ntrs.nasa.gov/search.jsp?R=19660017369 2018-05-24T09:13:32+00:00Z

ABSTRACT

Two t h e o r e t i c a l models t o pred ic t a x i a l pressure d i s t r ibu t ion ,

void fract ion, iliid velocity iil a cav i t a t ing ven tu r i are applied. The

theo re t i ca l p red ic t ions a r e compared wi th experimental da ta from cold

water and mercury tests, and good agreement f o r the pressure p r o f i l e s is

found. The predicted void f rac t ions a r e found t o be too high, probably

because the models assume zero s l i p o r negative s l i p between the vapor

and l iqu id phases.

The analogy between the cav i t a t ing ven tu r i and o the r choked flow

regimes i s explored. One of the t h e o r e t i c a l models used is based on the

assumption t h a t the cavi ta t ing v e n t u r i is e s s e n t i a l l y e n t i r e l y analogous

t o a deLaval nozzle operating i n a choked flow regime with a compres-

s i b l e gas .

The cav i t a t ing ventur i i s an example of an extremely low qua l i ty

choked two-phase flow device. The present study i s thus somewhat appl i -

cable t o the study of liquid-cooled nuclear reac tor pressure ves se l o r

piping ruptures , which have received considerable a t t e n t i o n i n recent

years . However, the qua l i t i e s encountered i n the present cav i t a t ion

case are an order of magnitude lower than those usual ly considered f o r

the reac tor s a fe ty analyses, so t h a t t he present study is a l imi t ing

case f o r these .

ii

ACKNOWLEDGMENTS

The authors would l i ke t o acknowledge the work of D r . Willy

Smith and M r , I an E . 3. Lauchlan, former s tudents at the 'u'niversity of

Michigan, whose void f r ac t ion da ta w a s used f o r t h e comparisons wi th

the t h e o r e t i c a l l y derived numbers i n t h i s i nves t iga t ion , and Captain

David M. Ericson Jr., USAF, doctora l candidate a t t he University of

Michigan, whose experimental p ressure p r o f i l e da ta w a s used f o r t h i s

comparison wi th ca l cu la t ed pressure p r o f i l e data.

.

iii

TABLE O F CONTENTS

Page

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . iii

NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . V

L I S T OF TABLES v i . . . . . . . . . . . . . . . . . . . . . . . . . . L I S T OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . v i i

I . INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . 1

. . . . . . . . . . . . . . . . 11. BACKGROUND FOR PRESENT STUDY 5

111. CALCULATING PROCEDURES . . . . . . . . . . . . . . . . . . . 10

A. H o m o g e n o u s F l o w M o d e l B . H y d r a u l i c Jump M o d e l

I V . RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . A. Theoret ical Models B . C o m p a r i s o n w i t h E x p e r i m e n t a l D a t a

V. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . V I . A P P E N D I X . . . . . . . . . . . . . . . . . . . . . . . . . .

A . Sonic V e l o c i t y i n Low Q u a l i t y T w o - P h a s e M i x t u r e s . . . . . . . . . . . . . . . . . . . . . . .

B. C o n d e n s a t i o n ShockWave Analysis . . . . . . . . . . . C . H y d r a u l i c J u m p M o d e l . . . . . . . . . . . . . . . . .

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . .

24

35

36

36

37

38

39

Y

i v

.

~

i 1 I

Symbol

g,,g

h

J

k

L

M,&

P

R

t ,T

V

v

X

P z

NOMENCLATURE

Meaning

sonic ve loc i ty - area

r a t i o of vapor t o l i qu id volume

f l u i d (subscript) - force

g rav i t a t iona l accelerat ion constant , g - vapor (subscr ipt)

enthalpy

conversion constant - mechanical t o heat energy un i t s

t herma 1 conductivity

l i qu id (subscript)

Mach number, mass flow r a t e

pressure

universal gas constant

ven tu r i th roa t (subscr ipt) - t o t a l (subscr ipt)

spec i f i c volume

ve loc i ty

qua l i t y

parameter defined eq. (3) of Appendix

densi ty

a x i a l dis tance from throa t discharge i n ven tu r i s

V

LIST OF TABLES

Table

I. Calculated Parameters from Shock Wave and Hydraulic J m p Y ! d e ? s . . . . . . . . . . . . . . . . . . . . . . .

Page

18

v i

LIST OF FIGURES

i Figure Page

1. Basic ven tu r i flow path dimensions . . . . . . . . . . . . 6

2. Sketch of cav i t a t ing ventur i hydraul ic jump flow analogy . . . . . . . . . . . . . . . . . . . . . . . . 9

3 . Calculated sonic ve loc i ty v s . void f r a c t i o n f o r homogenous mixtures . . . . . . . . . . . . . . . . . . 11

4 . Calculated qua l i t y v s . void f r a c t i o n . . . . . . . . . . . 12

5. T-S diagram of the cavi ta t ing ven tu r i process . . . . . . 14

6 . Mach number r a t i o f o r both models vs . void f r a c t i o n f o r both shock wave and hydraulic jump models . . . . . . . 19

7 . Calculated pressure jump vs . void f r a c t i o n f o r shock wave model . . . . . . . . . . . . . . . . . . . . . . . 20

8. Normalized pressure and void f r a c t i o n v s . a x i a l pos i t i on f o r water, experimental v s . ca lcu la ted comparison f o r standard cav i t a t ion . . . . . . . . . . . . . . . . . . 2 1

9. Normalized pressure and void f r a c t i o n v s . a x i a l pos i t i on f o r water, experimental vs . ca lcu la ted comparison f o r f i r s t mark cavi ta t ion . . . . . . . . . . . . . . . . . 22

10. Normalizsd pressure and void f r a c t i o n vs . a x i a l pos i t i on €or mercury, experimental vs . ca lcu la ted comparison f o r standard cav i t a t ion . . . . . . . . . . . . . . . . 27

11. Void f r x t i o i l vs. radius f o r w a t e r a t severa l a x i a l locat ions f o r standard cav i t a t ion . . . . . . . . . . . 29

12 . Void f r a c t i o a v s . radius f o r water a t severa l a x i a l locat ions f o r f i r s t mark cav i t a t ion . . . . . . . . . . 30

13. Vold f r a c t i o n p ro f i l e s f o r standard and f i r s t mark cav i t a t ion conditions i n mercury . . . . . . . . . . . . 31

v i i

Figure Page

14. Average venturi void fraction vs . axial distance for mercury . . . . . . . . . . . . . . . . . . . . . . . . 32

. 15. Velocity prof i les as function of radial position and

cavitation condition, observer at tap position G , cold weter, 1/2 inch t e s t section . . . . . . . . . . . 34

v i i i

I. INTRODUCTION

Very low qua l i ty two-phase "choked" flows are important today i n

many appl ica t ions . That which has received the greatesc ac ten t ion i n

recent years i s the postulated nuclear reac tor accident wherein a rup-

t u r e of a high-temperature high-pressure l iquid-carrying pipe i s

assumed. Another r eac to r sa fe ty problem a l s o involving somewhat s i m i l a r

flow considerat ions i s the occurrence of a rapid f u e l temperature t ran-

s i e n t i n a liquid-cooled core. I n t h i s case the evaluat ion of the maxi-

mum ve loc i ty of t ranspor t of t h e liquid-vapor mixture from the region

becomes important.

Other appl ica t ions of such low q u a l i t y choked flows which have

been of importance f o r many years are cav i t a t ing flow regimes such as

e x i s t i n pumps, valves , marine propel lors , hydraulic tu rb ines , ventur i s

and o r i f i c e s , e t c . The present da t a and analyses a re motivated by these

l a t t e r appl icat ions and hence involve extremely low q u a l i t i e s as com-

pared t o the usual reac tor problem, s ince they deal with f l u i d s such as

low temperature water with i t s very low vapor t o l iqu id dens i ty r a t i o .

However, the void f r ac t ions are of a magnitude s i m i l a r t o those con-

s idered f o r the reac tor cases . Hence the present study examines i n

d e t a i l the extremely low qual i ty end of the spectrum from the viewpoint

1

2

of the reac tor s a fe ty problems, which i s , nevertheless , the primary area

of concern i n the usual cav i t a t ion s i t u a t i o n .

The e a r l i e s t s tud ies which, t o our knowledge, have considered

the analogy between low-quality two-phase flows and the conventional

compressible choked flows dea l with cav i t a t ion (L,2). I n both cases

cav i t a t ing ventur i s were considered, as i n the present study. I n the

paper by Hunsaker (1) it i s noted t h a t the maximum pressure rise occurs

a t the v i s u a l downstream termination point of the cav i t a t ion cloud. The

Randall (2) paper discusses the use of the choked flow fea ture of a cav-

i t a t i n g ven tu r i f o r flow-rate regulation and notes t h a t the flow region

downstream of the throa t e x i t cons is t s of a high speed j e t surrounded by

vapor, a f a c t a l s o noted by Nowotny (2). However, none of these pre-

s en t s a quan t i t a t ive ana lys i s of t he flow such a s the analogy t o the

con-rentional choked flow s i tua t ions presented here .

The exis tence of a close analogy between cav i t a t ing flow phenom-

ena i n cen t r i fuga l o r a x i a l pumps and compressible flow phenomena i n

s imi l a r compressors has been real ized f o r some t i m e , including the f a c t s

t h a t the occurrence of cavi ta t ion i n a pump i n l e t w a s somehow very s i m i -

l a r t o the occurrence of Mach 1 condi t ions i n a compressor i n l e t , and

t h a t both r e su l t ed i n a choked flow condi t ion. More quant i ta t ive

approaches have been made i n recent papers on cav i t a t ing flow i n cen t r i -

fugal or a x i a l flow pumps i n order t o explain some of the p e c u l i a r i t i e s

i n the e f f e c t s of the cavi ta t ion on the pump performance (A,>).

A recent: excel lent quant i ta t ive study of the flow of l o w qua l i t y

a i r -water mixtures through a converging-diverging nozzle, ( i . e . , a

3

ventur i" from the viewpoint of cav i t a t ing (flows), has been made by 11

Muir and Eichhorn ( 6 ) . This reports on the exis tence of a compression

shock when the flow i s underexpanded, but presents no ana lys i s of t h i s

shock,

is reported by Starkman, e t a1 (1). I n t h i s p a r t i c u l a r case the minimum

q u a l i t i e s considered are i n the 1 t o 2 percent range, which i s consider-

ably above the range considered i n the present paper. The exis tence of

compression shocks under su i tab le conditions i s mentioned, but no quan-

t i t a t i v e data thereon i s presented o ther than the observation t h a t the

condensation shock i s much less abrupt than the conventional normal

shock i n a compressible gas flow. F ina l ly , a very recent experimental

and theo re t i ca l study of water-nitrogen mixtures a t about 50 percent

void f r ac t ion by Eddington (E) deals with shock phenomena i n t h i s type

of mixture.

A study of low-quality flow of water through a s i m i l a r geometry

The above s tud ie s , which have only been se lec ted as t y p i c a l ,

s ince many more e x i s t i n the l i t e r a t u r e (see bibliography of reference

5, e .g . ) a r e motivated by conventional flow appl ica t ions r a the r than

nuclear reac tor s a f e t y problems, However, var ious s tud ie s motivated by

reac tor s a fe ty a l s o e x i s t i n the l i t e r a t u r e .

Xoody (10) and Min, Fauske, and P e t r i c k (11) are t y p i c a l . Generally

these dea l with s i m p l e converging ( r a the r than converging-diverging)

passages, s ince a converging opening b e t t e r models a poss ib le rupture of

a pipe o r pressure v e s s e l .

i n the throa t of the converging sec t ion , but do not concern themselves

with shock waves which a r e only involved i n the diverging

Studies by Levy (21,

Hence they consider the choked flow analogy

4

("supersonic") por t ion . Generally the q u a l i t i e s considered a r e g rea t ly

i n excess of those of the present paper.

I n addi t ion t o a l l the above, there has a l s o been some recent

interest (12) i n the q u i t e analogous s i t u a t i o n of high-quality choked

flows such as are encountered with sa tura ted vapor expanded through a

nozzle. I n t h i s case it has long been known t h a t condensation shocks

sometimes occur a f t e r the flow has a t t a ined a non-equilibrium sub-cooled

condition due t o the r ap id i ty of the expansion. This type of flow s i t u -

a t ion is primari ly appl icable to tu rb ines operating with sa tura ted o r

w e t steam i n l e t condi t ions.

11. BACKGROUND FOR PRESENT STUDY

Considerable experimental da ta has been gathered i n the authors '

laboratory on cav i t a t ing flow regimes i n ven tu r i s both with water and

mercury a s test f l u i d . This data includes measurements of a x i a l pres-

sure p r o f i l e s , ve loc i ty p ro f i l e s using a micro-pitot tube, and void

f r ac t ion d i s t r i b u t i o n by gama-ray densitometer, a s wel l a s high speed

motion p i c tu re s tud ies of the flow. A port ion of the da ta r e l a t i n g t o

the void f r ac t ion and p i t o t tube measurements a l ready appears i n the

open l i t e r a t u r e (13,14), while addi t iona l void f r ac t ion (15) and pres-

sure p r o f i l e da ta (16) has not y e t been published. A f u l l descr ip t ion

of the closed loop ven tu r i tunnel f a c i l i t i e s both f o r water and mercury

has a l ready been given (17).

ven tu r i s used i s shown f o r convenience i n Fig. 1. A 1/2" diameter

c y l i n d r i c a l th roa t of 2.35" length i s located between nozzle and d i f -

fuser sec t ions , esch with 6" included angle f o r the port ion adjacent t o

the t h r o a t . The d i f fuse r continues a t t h i s cone angle out t o almost the

f u l l pipe diameter.

However, the basic flow path design of the

With a wealth of experimental data i n hand f o r t h i s cav i t a t ing

flow geometry i t i s des i rab le to develop an ana ly t i ca l model capable of

pred ic t ing a p r i o r i the pressure p r o f i l e s , v e l o c i t i e s , temperatures, and

void f r ac t ions which would be encountered once the required independent

5

5

- 0 0 3-25 v)(D

t

. m C 0 -d m E

-4 a E" c 4J a a 0

VI

= 3 u 2 rl

I -4

4J

cu k 1 3 U C

0 8

6

m

a 4

7

var i ab le s had been set .

cav i t a t ing ventur i case, it also has some appl ica t ion t o o ther c a v i t a t -

While such a model appl ies pr imari ly t o the

ing flows such a s a r e encountered i n turbomachinery, e t c . , and a l s o t o

reac tor s a fe ty problems, even though these nay normally involve somewhat

higher q u a l i t i e s than those considered f o r the present case (maximum of

about

As already mentioned, e a r l i e r i nves t iga to r s (&3J have noted

t h a t a cav i t a t ing v e n t u r i flow can o f t en be considered approximately a s

a high speed c e n t r a l j e t issuing from the th roa t and with a diameter and

ve loc i ty approximately equal t o those a t the t h r o a t . Our own invest iga-

t i o n s with a p i t o t tube i n water (s), by v i s u a l observation i n mercury

where the gap between the transparent w a l l and the c e n t r a l mercury col-

umn could e a s i l y be seen (l3), and by gamma-ray densitometer void f rac-

t i o n measurement i n mercury (13,14), tended t o confirm the s u i t a b i l i t y

of t h i s model. However, l a t e r void f r a c t i o n measurements i n water (IS),

disagreed i n showing a predominance of void f r ac t ion near the ax i s . The

reason f o r the disagreement between the p i t o t tube and gamma-ray densi-

tometer data in water i s not known. However, the average void f r ac t ions

presented a s a function of ax ia l pos i t ion i n F igs . 8 and 9 a r e taken

from the garnayray densitometer da t a .

Two possible theo re t i ca l models f o r the flow were inves t iga ted .

Neither matches the observations on the flow regime completely, but

hopefully should give r e s u l t s of reasonable engineering accuracy.

i> HDmogeneous flow mode1 wherein it i s assumed tha t the vapor i s

dispersed 9rzifcrrr.ly thmughe-t t he f l u i d fer el!. p lanes normal tc t h e

8

a x i s , t h a t the two phases always have the same ve loc i ty , and t h a t

thermal equilibrium e x i s t s . The flow i s assumed adiaba t ic everywhere,

and i sen t ropic except f o r t he condensation shock wave which i s assumed

t o e x i s t a t the termination of the cav i t a t ing region.

stream of the shock i s assumed to be zero. Conservation of momentum

m a s s , and energy are s a t i s f i e d f o r a l l cross-sect ions.

The qua l i t y down-

i i ) Assuming the c e n t r a l j e t model with diameter and ve loc i ty equal

t o those a t the th roa t , i t is assumed t h a t the c a v i t a t i o n termination

region is analogous t o a hydraulic jump. The region around the c e n t r a l

j e t i s assumed f i l l e d with stagnant vapor a t a pressure equal t o the

sa tu ra t ion pressure ex i s t ing a t t h e th roa t discharge. While the "height"

of the jump i s terminated by the sidewalls of the ven tu r i a t t h e i r loca-

t i o n , the pressure a t the w a l l i s assumed equal t o t h a t a t t h e appropri-

a te submergence i f a n ac tua l hydraulic jump exis ted (see sketch of

F ig . 2 ) .

g rav i ty is neglected so t h a t the wal l pressure a t t he jump i s assumed t o

apply over the e n t i r e cross-sect ion. A s i n the conventional hydraulic

jump analysis, momentum and mass a re considered t o be conserved across

the jump. Af te r the jump t h e l i qu id f i l l s the e n t i r e cross-sect ion, so

t h a t the qua l i t y here i s zero as i n the previous model. Note t h a t the

void f r a c t i o n upstream of t h e jump i s simply the r a t i o of a rea around

the j e t t o t o t a l a rea f o r any cross-sect ion.

I n t h i s model as applied t o the conical ven tu r i d i f fuse r ,

+

f

I

9

i y”

t I

3 0

U E

$ M E 4 U cd U 4

c)

VI 0

%

c r)

t I

N

bo 4 h .

h 2

111. CALCULATING PROCEDURES

The ca lcu la t ing procedures followed f o r the two models a r e sum-

mari ze d be low :

A . Homogeneous Flow Model

1. Throat Discharge t o Cavi ta t ion Terminat ion Point

From the throa t discharge t o the cav i t a t ion termination point

t he flow is assumed t o be isentropic-adiabat ic . The throa t ve loc i ty and

w a t e r temperature a t the throat can be considered independent var iab les ,

and hence known f o r a given case. Assuming t h a t the flow i s analogous

t o a compressible gas choked flow i n a deLaval nozzle, Mach 1 must exist

a t the th roa t discharge where it i s assumed t h a t the cav i t a t ion w i l l

begin.

e x i s t f o r a cy l ind r i ca l th roa t due t o the e f f e c t s of f r i c t i o n . 111 the

present i dea l flow ana lys is of course the pressure would remain constant

along the throa t . A s shown i n F i g . 3, sonic v e l o c i t i e s i n l o w qua l i t y

water-steam mixtures can be extremely low (see Appendix f o r appropriate

r e l a t i o n s ) , and hence of the order of reasonable throa t v e l o c i t i e s f o r

cav i t a t ing ventur i s , f o r void f r ac t ions i n excess of Then, as

shown i n Fig. 4, the qua l i t y is of the order of 10 t o 10 , and i t s

ne t e f f e c t upon the thermodynamic mixture proper t ies except f o r sonic

Here, i n a real case, the minimum pressure would be expected t o

-6 -8

10

*.

11

0 -

LL Ir. 0 L L O oolc! O O n N W Y ) II n n

w w w l - l - k -

4 0 0 0

I l l I I I I I I I I I I I I I I 1 1 1 1 1 I I I Y)

I

k 0 w c 0 4 +, 0 cd k w U 4 0 > Im rn k Q) > A +, .r( 0 0 r( Q) > 0 4 a 0

U 0, +J

cd r( 1 0 r( td u m

M

R

rn

4

12

. I- /

/

I- /

O TEMP=ZOO*F

0 TEMP=60*F

V TEMP =53.5*F

/

I- // 1860

I I I I 1 l l l l I I I I I l l l l I I I 1 l l l l l I I 1 1 1 1

10'2 to4 I

VOID FRACTION

Fig. 4 Calculated quality versus void fraction

13

ve loc i ty is negl ig ib le .

considered pure l iqu id .

Hence the flow issuing from the th roa t can be

The computation of conditions f o r any point downstream of the

th roa t discharge and within the cav i t a t ion region is e n t i r e l y s imi l a r t o

t h a t normally employed for the computation of condi t ions i n a nozzle

handling w e t steam. Conventional steam tab le s (19) can be employed.

The assumed process an a T-S diagram is sketched i n Fig. 5, the region

between

shock but i n the cav i t a t ing region), being the por t ion under present

discussion.

( th roa t discharge), and 1 (a point immediately upstream of the

Using the steam tab le s (19), it i s necessary t o f ind by t r i a l

and e r r o r t he temperature a t point 1 such t h a t the flow w i l l remain

i s en t rop ic , and the conditions of energy and mass conservation w i l l be

maintained (see Appendix f o r appropriate r e l a t i o n s ) . For the cold water

example computed ( throa t ve loc i ty of 6 4 . 6 f t . / s e c . ) , the temperature

drop t o the point of cav i ta t ion termination i s only a very s m a l l f rac-

t i o n of a degree. This i s typ ica l of the cav i t a t ion case. Even so,

t h i s temperature decrease must be considered s ince the s p e c i f i c volume

and entropy of the vapor a r e extremely sens i t i ve t o temperature i n t h i s

region. Also, the ve loc i ty of the mixture i s qu i t e constant from the

th roa t discharge t o the cav i t a t ion termination poin t , with the void

f r a c t i o n increasing appropriately t o counter the increase i n cross-

s ec t iona l area of the d i f fuse r . The t h e o r e t i c a l model f o r homogenized

flow i s thus cons is ten t with the hypothesized c e n t r a l j e t flow regime i n

t h a t both pred ic t r e l a t i v e l y constant ve loc i ty through the cav i t a t ion

region.

I-

w 2 I- Q L1L w n 2 w I-

C

a

14

) SATURATION LINE

DIFFUSION AFTER SHOCK

d EXPANSION FROM INLET TO THROAT IN CONVERGING ( SUBSONIC 1, PORTION 4

SHOCK /v \EXPANSION FROM THROAT

TO SHOCK IN DIVERGING, (SUPERSON IC)rPORTION.

1861 ENTROPY, s

Fig. 5.--T-S diagram of the cavitating venturi process.

15

2. Cavi ta t ion Termination Region (Shock Wave)

The conditions immediately upstream of the pos tu la ted shock wave

are as given from t h e ca l cu la t ion previously discussed. I f t h e v e l o c i t y

of sound is computed f o r t h e mixture a t t h i s po in t , i t i s found t h a t t h e

flow i s highly supersonic as would be expected from t h e analogy with

compressible gas flow. It i s required t h a t immediately a f t e r t he shock,

which is assumed t o requi re zero a x i a l ex ten t , a l l t he vapor has con-

densed (Fig. 5), so t h a t t he qua l i t y and void f r a c t i o n are zero. Thus

it is clear t h a t a f t e r t he shock the flow i s highly subsonic s ince t h e

v e l o c i t y of sound i n a l i q u i d i s t y p i c a l l y of t h e order of 10 t o 10

f e e t per second.

3 4

Afte r the shock the f l u i d may have a pressure i n excess of vapor

pressure f o r the new temperature, which i s presumably s l i g h t l y above the

temperature upstream of the shock by v i r t u e of t h e l a t e n t heat re leased

by the condensing vapor. I n fac t i t is clear that t h e enthalpy a f t e r

the shock must be g r e a t e r than t h a t a t th roa t discharge, s ince the

k i n e t i c head has decreased. In general , then, the temperature must also

be higher since, while t he re was a s l i g h t q u a l i t y a t t h roa t discharge,

t h e f l u i d i s assumed vapor f r ee a f t e r t h e shock.

The ca lcua t ion of conditions a f t e r t he shock from the upstream

conditions requi res an appl ica t ion of conservation of momentum and mass.

The only quant i ty of i n t e r e s t t o be found from conservation of energy i s

t h e temperature a f t e r t h e shock. However, sample ca l cu la t ions show t h a t

t h i s quant i ty does not change appreciably over the shock. The conserva-

t i o n of mass and momentum equations can easily be arranged t o the form

16

of eq. (6) and (7) i n Appendix B y with the assumption t h a t 6-4. Hence

t h e v a l i d i t y of the theo re t i ca l r e s u l t s is l imited t o xC0.01. Using

these equations i t i s possible t o make the ca l cu la t ion . However, i n

order t o explore the analogy with compressible gas flow it w a s des i red

t o express the r e l a t i o n s i n terms of Mach number. This can be accom-

p l i shed as shown i n Appendix A following t o some exten t t h e procedure of

Jakobsen (9 . The opera t ive equations are now eq. (12) , (13) , and (14),

and from these the conditions a f t e r the shock can be computed.

3 . Cavi ta t ion Termination t o Venturi Discharge

Since the flow i n t h i s region is vapor f r e e , the computation i s

e n t i r e l y straightforward and assumes f r i c t i o n l e s s flow. I n terms of t h e

T-S diagram (Fig. 5), t h i s portion is an i s en t rop ic compression from 2

t o cl, where t h e l i q u i d i s subcooled.

B. Hydraulic Jump Model

1. Throat Discharge to Cavitation Termination

I n t h i s region a cen t r a l l i qu id j e t (zero qua l i ty ) is assumed

surrounded by stagnant vapor at t h e vapor pressure corresponding t o

th roa t discharge temperature.

out.

t he j e t ve loc i ty remains equal t o the th roa t ve loc i ty .

t he void f r a c t i o n a t any a x i a l pos i t i on wi th in t h e c a v i t a t i o n region is

simply t h e r a t i o of t h e c ross -sec t iona l a r ea around t h e j e t t o the t o t a l

c ross -sec t iona l a rea . This i s a l s o approximately t r u e f o r t he shock

wave model s ince , a t l e a s t f o r t he examples computed, the ve loc i ty

upstream of the shock is about equal t o t h e th roa t ve loc i ty .

Temperature i s assumed constant through-

Since t h e j e t diameter i s assumed equal t o the th roa t diameter,

For t h i s model

17

2. Cavi ta t ion Termination Region (Hydraulic Jump)

Conservation of mass and momentum i s appl ied across the cavi ta -

t i o n termination region as i n the conventional hydraulic jump ana lys i s

(20) except t h a t t he pressure i s assumed uniform across a plane normal

t o axis immediately upstream of t h e jump, i.e., g rav i ty is neglected.

The j u q , as the shock wave, is = s a m e 3 t o reqrrire zerc a x i a l ex ten t .

The analogy t o a hydraulic jump s t e m s from considerat ion of a two-

dimensional case wherein the ventur i w a l l would be f i c t i t i o u s l y removed,

and the l i q u i d allowed t o assume the height downstream of the jump which

it would a t t a i n i f the flow were hor izonta l , and under the influence of

g rav i ty (Fig. 2) .

The appl icable equations a r e shown i n Appendix C , the operat ive

equations now being eq. (15) and (17). It i s noted t h a t t he momentum

equation, (16), is i d e n t i c a l t o t he comparable momentum equation from

the shock wave ana lys i s , ( 7 ) . Thus the models w i l l give s imilar r e s u l t s

i f V i s the same i n the two cases. I n the cases f o r water flow which

have been invest igated, t h i s is e s s e n t i a l l y t r u e , even f o r the 200°F

1

water (Table I and Figs . 6 and 7 ) . Since the pressure rise predict ions

from the two models are almost i d e n t i c a l f o r the cases examined, only

one t h e o r e t i c a l curve is shown i n Figs . 8 and 9, where the experimental

r e s u l t s and theory a r e compared. This equa l i ty of r e s u l t s between the

models is a l s o the case f o r void f r ac t ion , so t h a t again only one

t h e o r e t i c a l curve is shown.

0000000

. t m c o o h m \ D m o O o o

. e . . . . I

E: 0 d U (d U d

% c)

a & a a

U m $4 0 VI

a, d

d 0 d L, rl a E: 0 c)

m 4 s m Q

9

I I I l l I 000000 d d d d d d

rn

c?d b B 00 d d

m m x x c o a o m o o o d m o o o h N O O O O O O

o e m o bh1000000

~ . . . . . . . . 0 0 0 0 ~ ~ 0 ~

. * D O * .

0 0 0 0 d 0

0 0 N

b

19

\\

0 0 0 0 0 0

2 w4 y"1

(u UI PD

LL

In r! n

% w I- D

1 1 1 1 I I I I 1 1 1 1 I I I I I

0 2 w4 y"1

Y 0 0

9

k 0 w a 0 d +, 0 0 k R a 0 * m

k Q) * 0 d c, (d Er: k Q

d

lm

3 z .Eo d 16

b 20

d

ZL

I I I I 1 I I

.+-

+-

t-

22

.. 4

W w - r

I I I I I I I

a a* b u> Lo * m cu - - 0

23

3 . Cavitation Termination t o Venturi Discharge

The calculation f o r t h i s region i s identical t o that already

discussed for the shock wave model.

I V . ReSULTS

A. Theoret ical Models

As previously discussed, ca lcu la t ions have been made f o r the

ventur i shown i n Fig. 1 f o r water a t 53.5”F, 60.0°F, and 200°F. The

lower temperature matches experimental water data with 64.6 f t . / s e c .

t h roa t ve loc i ty f o r void f r ac t ion and pressure p r o f i l e s (15,16) taken

f o r two d i f f e r e n t ex ten ts of the cav i t a t ing region:

(standard cav i t a t ion ) , and 1.75 inches ( f i r s t mark cav i t a t ion ) , down-

stream from the throa t e x i t .

0.786 inches

The ca lcu la t ing procedures have already been described, and the

equations involved a re shown i n the Appendix. An examination of these

w i l l show t h a t t he calculat ions a r e independent of th roa t ve loc i ty

unless ac tua l magnitudes of pressure r i s e and Mach number a re required.

The f l u i d propert ies , however, a r e required immediately, so tha t i t is

necessary t o specify the f lu id and i t s temperature. Assuming, then, t h a t

water is the f l u i d a t the three spec i f ied temperatures and assuming var-

ious values f o r t he r a t i o of cross-sect ional a reas between cav i t a t ion

termination and th roa t , i t is possible t o compute qua l i ty , void f rac-

t i o n , and sonic ve loc i ty immediately upstream of the shock. For a given

water temperature, there i s of course a unique r e l a t i o n between void

fraction and sonic T!elncity> end alsn v n i d f r l c t i n n lrzd flj121-ii-V rant4 1””’J ¶ ----

24

25

these a re shown i n Figs . 3 and 4 , respect ively, and l i s t e d i n Table I.

It i s a l so possible , without specifying ve loc i ty , t o compute t h e

r a t i o of Mach numbers (M1/M2) across the shock and the r a t i o of shock

pressure rise t o throa t k ine t ic pressure u t i l i z i n g e i t h e r of the calcu-

l a t i o n a l models.

t o conventional compressible gas cases, increasing from the order of 10

t o the order of 10 as the void f r a c t i o n increases from the order of

A s noted, the Mach number r a t i o is very large compared

3

t o 10". Even la rger r a t i o s occur as void f r ac t ion approaches

uni ty . However, these may not be meaningful s ince, as previously men-

t ioned, the assumptions made r e s t r i c t the v a l i d i t y of the ca lcu la t ion t o

the r e l a t i v e l y low qual i ty range.

decrease markedly a s void f r ac t ion i s decreased, and become uni ty f o r

zero void f r ac t ion , s ince a condensation shock i s not possible i n t h a t

case.

2 A l s o both M1/M2 and (p2/pv)/,oVt

To obtain e i t h e r actual Mach numbers o r pressure rise, it i s

necessary t o specify the ac tua l th roa t ve loc i ty .

B. Comparison with Experimental Data

1. Axial Pressure P ro f i l e s

Experimental a x i a l pressure p r o f i l e s have been measured (16) f o r

t h e ven tu r i of Fig. 1 f o r water a t 53.5OF, th roa t ve loc i ty of 64.6 f t . /

sec., and f o r the cav i t a t ion terminating a t 0.786" and 1.75", respec-

t i v e l y , from the throa t e x i t . For comparison with the theo re t i ca l

models, a normalized suppression pressure i s formed by subtract ing vapor

pressure and dividing the r e s u l t by the throa t k ine t i c pressure.

26

According t o classical conceptions of c a v i t a t i o n , t h i s number should

always be zero a t t h r o a t discharge.

now w e l l recognized (21, e.g.).

i c a l models are such t h a t t h e normalized suppression pressure must be

zero a t t h i s po in t . For e i t h e r t h e o r e t i c a l model the pressure is then

e s s e n t i a l l y constant t o the end of t h e c a v i t a t i o n region, while i n the

r e a l case it f a l l s t o a minimum downstream of the th roa t e x i t (Figs. 8

and 9 ) .

The f a c t t h a t it normally is not i s

However, t he assumptions of t h e theore t -

Similar pressure p r o f i l e da ta is ava i l ab le f o r mercury (22) a t a

v e l o c i t y of 33.1 f t . / s e c . and 115"F, and f o r t he c a v i t a t i o n terminating

a t 0.786'' from the t h r o a t e x i t . This da ta w a s normalized as previously

described.

shown i n Fig. 10.

The r e s u l t i n g comparison t o the t h e o r e t i c a l p r o f i l e s i s

A t t h e end of the c a v i t a t i o n zone t h e t h e o r e t i c a l models each

p red ic t a s t e p rise i n pressure, which goes through a maximum f o r

increas ing ex ten t of the cav i t a t ion region. It can be shown from

eq. (17) of Appendix C , which app l i e s t o the hydraulic jump model, t h a t

t h i s maximurn occurs f o r a value of v e n t u r i rad ius , r , equal t o (2) 1/2 rt,

where rt i s the v e n t u r i throat rad ius .

show a s t e p r i s e i n pressure a t t he end of t h e c a v i t a t i o n region (which

i s marked i n Figs. 8, 9, and 10 by t h e s t e p r i s e of predicted pressure) ,

but they do show a s u b s t a n t i a l g rad ien t of about t he co r rec t magnitude

t o match the pred ic t ion from the models.

The experimental p r o f i l e s do not

Af t e r the c a v i t a t i o n termination region t h e pressure continues

t o r i s e more gradually f o r both experimental and t h e o r e t i c a l p r o f i l e s as

27

I

i

0 E El -1

a

I 0

3 P 3 a W

8

kf 3 4

0

a a 0 W

-1 3

I

W -1

5 1 E l

a 5

B

I-

E w - W

I

0

+-

I 06z'zch)b 1 - 3tJnSS3tld 03ZIlW'lrVtlON ( 'd-d

I

v) W I 0 z I t- w J z - 5i 0 E I t-

3 t- z W > r 0 E LL w 0 z v)

A

X

E

F a 5 a

o a l ? 0

o u a m

28

expected f o r a s ing le phase d i f fuse r .

mental a x i a l pressure p r o f i l e s match reasonably we l l .

Thus the theo re t i ca l and experi-

2 . Void Frac t ion P r o f i l e s

Local void f r ac t ions for the flow regimes coiisidered here w a s

measured as a funct ion of radius and a x i a l pos i t i on with a gamma-ray

densitometer (13,15).

w a t e r and Fig. 13 f o r mercury. For the water tests the void f r ac t ion i s

g rea t e s t i n regions along the ven tu r i ax i s , contrary t o previous expec-

t a t i o n s from s imi l a r measurements and v i s u a l observations i n mercury as

w e l l a s e a r l i e r p i t o t tube measurements i n w a t e r ( t o be discussed

l a t e r ) .

very high along the ven tu r i wall and small along the ax i s , ind ica t ing

the s u i t a b i l i t y of the hydraulic jump analogy f o r t h i s case .

From p r o f i l e s of t h i s type it i s possible t o compute average

F igs . 11 and 12 show the r e su l t i ng p r o f i l e s f o r

For the mercury t e s t s it w a s found t h a t the void f r a c t i o n w a s

void f r ac t ion as a funct ion of axial dis tance.

f o r water and mercury f o r standard cav i t a t ion in. Fig. 14. This , plus

add i t iona l data f o r f i r s t m a r k cav i t a t ion , i s rep lo t ted i n F igs . 8, 9,

and 10 t o show the comparison with ca lcu la ted values f o r the two flow

models. While the loca l void f r a c t i o n d i s t r i b u t i o n s between water and

mercury d i f f e r widely, it i s noted t h a t the averaged d i s t r i b u t i o n s i n

both cases are considerably l o w e r than the ca lcu la ted models would ind i -

c a t e near the end of the cavi ta t ion termination region. It is noted

t h a t the void f r ac t ion d i s t r ibu t ion i n mercury follows t h a t ca lcu la ted

qu i t e c lose ly almost t o the point of the jump, ind ica t ing t h a t the

Such p r o f i l e s a r e shown

hydrsulic J--r ; r r m n *---*- nnslnui i O J in chis case mnrr -J be quite g c d . 9c'n'cv2r, the

29

30

25

20 s z 0

0 15

a a

5

t 0 5 .IO .20 -25 .30 DISTANCE FROM INCHES

Fig. 11.--Void fraction versus radius for water at several axial locations for standard cavitation.

30

30

25

$ * 20 Q

2 I-

15 D 0 > -

10

5

DISTANCE FROM c-INCHES

Fig. 12.--Void fraction versus radius for water at several ax ia l locations for f i r s t mark cavitation.

31

z

r .3 .2 . I

It- END OF

CAVITATION

1.25

1.00 I I

I I

I .75

I 30

STANDARD FIRST MARK

Fig. 13.--Voia traction profiles for standard and first mark cavitation conditions in mercury.

32

z 0

P)

m U m d a d m d X m

2

: rn k W > C 0 4 U c) m k v(

a d 0 ? d k 3 U C s

: P 2

W bo m k

I

. . b o h 4 k k 3

c) k

8 k 0 w

33

averaged void f r ac t ion near the jump i s qui te a b i t higher i n both

models than the experimental values. This is p a r t i a l l y due t o the f a c t

t h a t these models assume i n one case t h a t the ve loc i ty of the vapor i s

zero and i n the o ther t h a t it is equal t o l iqu id ve loc i ty , whereas i n

a c t u a l i t y there may be s igni f icant "slip" so t h a t the vapor ve loc i ty i s

considerably g rea t e r than tha t of the l iqu id .

the void f r a c t i o n f o r a given qua l i ty would be less, and thus somewhat

nearer t he experimental values.

If t h i s were the case,

3. P i t o t Tube Measurements

Fig. 15 shows micropitot tube ve loc i ty p r o f i l e s f o r water

(21,13) across a ven tu r i s imilar t o tha t used i n these t e s t s .

tube is located upstream of the cav i t a t ion termination point f o r two of

t h e c a v i t a t i o n condi t ions shown (except "no cavi ta t ion" and f i r s t mark),

a t a point about 2 3/8" downstream from the th roa t e x i t .

t h a t i n t h i s case a well-defined c e n t r a l l iqu id j e t ex is ted with a

ve loc i ty of 90 t o 95% of throat ve loc i ty .

t h e order of 90% t h a t corresponding t o throa t ve loc i ty ind ica tes t h a t

t he f l u i d densi ty must have been close t o t h a t of pure l iqu id .

The p i t o t

It i s c l e a r

That the impact pressure was

34

r s a s " aaaa0 00602 o a x o

0 0 0 0 0 0 0 0 rc)

0 0 0 u2 Ir! * n

1 0 - 0

AI

'NI '33NWlS10 l V l 0 W t l

V . CONCLUS IONS

Major conclusions from t h i s work follow:

1) E i the r the homogeneous flow, thermal equilibrium model o r

the hydraulic jump model are capable of pred ic t ing axial pressure d is -

t r i b u t i o n i n a cav i t a t ing ventur i qu i t e c lose ly , even including the

behavior of the condensation shock marking the downstream termination of

the cav i t a t ion zone. This has been v e r i f i e d i n water and mercury.

2) Both models predict void f r ac t ions i n both f l u i d s , averaged

over the cross-sect ion a t f ixed a x i a l pos i t ions , t h a t are considerably

higher than those measured by gamma-ray densitometry. This may be due

t o the f a c t t h a t ne i ther model allows a vapor ve loc i ty g rea t e r than the

l i qu id ve loc i ty , whereas i n a c t u a l i t y such a pos i t ive " s l i p " probably

does e x i s t .

35

VI. APPENDIX

A . Sonic Velocity in Low Quality Two Phase Mixtures

The sonic velocities were computed following the methods of

Jakobsen (3. The operative equations are as follows:

where, v u v B = Y = 2 Void Fraction (= V.F.)

V;otal vL 2 P L ~ L = (Vg/Vf) 2 R a’ r - ‘ v v

a V = (gokRT)1/2 = 59.9(T)ll2 for water ( 4 )

Equation (1) is derived in reference 4 starting with the basic

relation :

and assuming constant temperature with no evaporation or condensation

during the sonic disturbance. The first assumption tends to give higher

sonic velocities than the alternative assumption of thermal equilibrium:

36

37

while the second assumption tends i n the opposite d i r ec t ion , but prob-

ab ly t o a lesser ex ten t .

B. Condensation Shock Wave Analysis (Assuming x = 0 a f t e r shock and p = pv before shock)

1. Basic Conservation Relations - - - 1-1 = v /v v1 I Vf (1-x)+v x 2 f

A g Mass :

b

2 ho = h x + hl (1-X) + V1/2goJ

Ig f Energy :

n L -

- h2f + v2/2goJ

2. Assumptions and Rearrangement t o Working Form

m

x = m v / 5 = m v / y , for low qua l i ty (x <.01)

z For the cav i t a t ion problem, fi<1, so 1-x = 1

then (6) becomes :

V,/(v +v B) = V2/vf = V 1 f /v (Bt-1) f f

o r

Vl/V2 = B + 1

38

Introduce Mach number:

M1 = Vl /a l , V1 = M a , e t c . 1 1

then (10) becomes :

Subs t it u t ing

And momentum

o r

And from (1)

equation (7) becomes :

2 2 P f P 1 = P2’Pv = (MIMplaL - M2aL)/goVf

a = aL(l+B) / (l+Bvf/vg) (1+Br) = aL (l+B)/

(MI /M2) ( 14) 1

C . Hydraulic Jump Model (Assuming space around j e t t o be f i l l e d with stagnant vapor and j e t s t a t i c pressure = pv)

Momentum:

BIBLIOGRAPHY

I -

b

1. Hunsaker, J . C . , "Cavitation Research, A Progress Report on Work a t t he Massachusetts I n s t i t u t e of Technology," Mechanical Engineering, Apr i l , 1935, pp. 211-216.

2. Randall, L. N., "Rocket Applications of the Cavi ta t ing Venturi," ARS Journal , January-February, 1952, pp. 28-31.

3. Nowotny, H., "Destruction of Mater ia ls by Cavitation," VDI-Verlag, Berl in , Germany, 84, 1942. Reprinted by Edwards Brothers, Inc. , Ann Arbor, Michigan, 1946. English language t r a n s l a t i o n a s ORA I n t e r n a l Report No. 03424-15-1, Department of Nuclear Engineering, The University of Michigan, 1962.

4 . Jakobsen, J . K. , "On the Mechanism of Head Breakdown i n Cavi ta t ing Inducers," Trans. ASME, J. Basic Eng., June 1964, pp. 291-305.

5. Spraker, W . A . , "Two-Phase Compressibil i ty E f fec t s on Pump Cavita- t ion," Cavi ta t ion i n Fluid Machinery, ASME, 1965, pp. 162-171.

6 . Muir, J . F., and Eichorn, R. , "Compressible Flow of an Air-Water Mixture Through a Ver t ica l Two-Dirnensional Converging-Diverging Nozzle," Proc. 1963 Heat Transfer and Flu id Mechanics I n s t i t u t e , pp. 183-204.

7 . Starkman, E . S., Schrock, V . E . , Meusen, K. F. , Maneely, D . J . , "Expansion of a Very Low-Quality Txo-Phcse Fluid Through a Con- vergent-Divergent Nozzle," J. Basic Eng., Trans. ASME, June, 1964, pp. 247-264.

8. Eddington, R. B . , " Invest igat ions of Shock Phenomena i n a Super- sonic Two-Phase Tunnel," AIAA Peper No. 66-87, 3rd Aerospace Sciences Meeting, New York, Jsnuary, 1966.

9. Levy, S., "Prediction of Two-Phase C r i t i c a l Flow Rate," ASME Paper 64-HT-8, J . Heat Transfer , Trans. ASME, May, 1960, p. 113.

10. Moody, F. J . , "Paximum Flow Rate of a Single Component, Two-Phase Mixture," ASME Paper 64-€E-35, ASHE Trans., J . Heat Transfer.

39

40

11. Min, T. C . , Fauske, H. K., Pe t r i ck , M., "Effect of Flow Passages on Two-Phase Cr i t ica l Flow," I&EC Fundamentals, Vol. 5, No. 1, Febru- a ry , 1966, pp. 50-55.

12 . Deich, M. E . , Stepanchuk, V. F., Saltanov, G. A., "Calculating Com- pression Shocks i n the Wet Steam Region," Teploenergetika, 1965, 12 (4) : pp. 81-84.

13. Smith, W., Atkinson, G. L., H a m m i t t , F. G., "Void Frac t ion Measure- ments i n a Cavi ta t ing Venturi," Trans. ASME, J. Basic Eng., June, 1964, pp. 265-274.

14. H a m m i t t , F. G . , Smith, W., Lauchlan, I., Ivany, R., and Robinson, M. J . , "Void Fract ion Measurements i n Cavi ta t ing Mercury," E Trans., Vol. 7 , No. 1, pp. 189-190; a l s o Nuclear Applications, February, 1965, pp. 62-68.

15. Lauchlan, I. E. B., "Void Frac t ion Measurements i n Water," Term Paper, NE-690, Nuclear Engineering Department, The University of Michigan, Ann Arbor, Michigan, August , 1963.

16. Ericson, D. M. , Jr. , Capt. USAF, unpublished data, Thesis underway, Nuclear Engineering Department, The University of Michigan, 1964.

17. H a m m i t t , F. G., "Cavitation Damage and Performance Research F a c i l i t i e s ,I1 Symposium on Cavitat ion Research F a c i l i t i e s and Techniques, pp. 175-184, ASME Flu ids Engineering Division, May, 1964.

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