UNITED STATES EPARTMENT Of IOMMERCE

UBUCATION NBS TECHNICAL NOTE 608

U.S.

)EPARTMENT

OF

COMMERCE

National

Bureau

of Standards

Steam-Water, Critical

Flow in a Venturi
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mMi S<TM Of ST^SAKs

- 10 o UNITED STATES DEPARTMENT OF COMMERCE

Maurice H. Stans, Secretary

\J.% , NATIONAL BUREAU OF STANDARDS • Lewis M. Branscomb, Director

NBS TECHNICAL NOTE 608

ISSUED JULY 1971

Nat. Bur. Stand. (U.S.), Tech. Note 608, 25 pages (July 1971) CODEN: NBTNA

Steam-Water, Critical Flow in a Yenturi

R. V.Smith

Cryogenics Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

^ "' c.

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide

a means for making available scientific data that are

of transient or limited interest. Technical Notes may

be listed or referred to in the open literature.

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. (Order by SD Catalog No. C 13.46:608), Price 35 cents Stock Number 0303-0882

20402

Contents

Page

1. Introduction 1

2. Experimental 4

3. Qualitative Observations 4

4. Analytical Procedure 8

5. Results and Discussion 14

6. Conclusions 16

7. Acknowledgments 17

8. References 18

9. Nomenclature 20

in
List of Figures

Page

Figure 1. Flow circuit of the steam-water loop. ... 5

Figure 2. Pressure versus distance along

the venturi showing the effect

of raising the downstream pressure .... 6

Figure 3. Pressure versus distance along

the venturi. Curves are from the

analytical program and points from

the experimental program. 12

Figure 4. Pressure versus distance along

the venturi. Curves are from

the analytical program and points

from the experimental program 13

IV
STEAM-WATER, CRITICAL FLOW IN A VENTURI

R. V. Smith

This paper is the second part of an analytical and experimental investigation, in which the primary object was to test

the hypothesis that the flow of the gas phase controls critical

and near critical two-phase flow for cases where the gas and

liquid flow essentially in separate streams. In the first part

of the investigation, a two -component system (air -water) was used. The results presented here substantiate the hypothesis. The analytical results also indicate the use of one dimensional flow equations with reasonably accurate estimates for droplet size and for the drag and heat transfer

coefficients (without consideration of mass transfer—vaporization or condensation) describe critical and near-critical

flow reasonably well. This indicates that mass transfer may

be a secondary effect for these flow conditions.

Keywords: Critical flow; pressure profile; steam;

venturi; water.

1. Introduction

This study was the second part of a two -phase critical flow program. In the first part, (Smith 1968, 1971) a two -component, twophase system (water and air) was employed, and the experimental program was quite extensive. This second part was an extension of that

work, and was conducted primarily to compare two and one component

behavior. The analytical program was essentially the same as that

used for the two component system. The hypothesis to be tested for

both studies was that the gas behavior primarily controls the critical

(choking) two-phase flow for higher qualities where essentially all the
gas flows in separated and continuous streams. A more detailed description of previous work and objectives of this program may be

found in Smith (1968, 1971).

This study differs from many of the previous works in that it follows the flow from an upstream point and it defines critical flow

as a state that occurs when the gas, rather than the mixture, reaches

critical conditions. In the largest number of studies the primary interest has been in critical flow at the geometric point where critical

flow occurs. This restriction of the study is useful, but it imposes

several disadvantages. One is that the fluid conditions at the critical

point are difficult to determine. The pressure gradient in that region

is very steep and experimental measurements from conventional wall

taps in a converging or straight section must be extrapolated to the

point of critical flow. Also, in one component flow the quality or gasliquid distribution cannot be directly measured and must be calculated

by various equilibrium or phase-rate-of-change assumptions. In almost all works at the critical point, thermal equilibrium has been

assumed. The first phase of this study and other works, such as Fauske (1965) and Carofano and McManus (1968), indicate that this is not a valid assumption.

Further, the velocity of each phase cannot be determined ac- curately because experimental void (gas -volume fraction) measurements are so difficult to obtain that such velocity data must be evaluated

with the use of uncertain quality data. These disadvantages make data

for the point of critical flow of limited use, for example, for designers

of flow systems.

Nevertheless, work at the point of critical flow has resulted in

fairly reliable expressions for the critical flow rate, for example,

Fauske (1961), Moody (1965), Zivi (1964) and Cruver and Moulton (1966).
These works, although they have the same general range of reliability

of prediction, vary widely in assumptions regarding equilibrium and the

gas and liquid velocities. The assumptions made for the gas -liquid

velocity ratio are, in almost all cases, substantially higher than ex- perimental measurements reported by Fauske (1965) and Klingbiel (1964).

In the program described here, an attempt was made to minimize

and resolve some of these difficulties by studying the flow from points

upstream where fluid conditions are known more reliably. Thus, estimates regarding mass, momentum and energy transport can be made

throughout the region of rapid acceleration to critical flow. These

estimates can then be checked by comparison of analytical and experimental pressure profiles. This procedure of analysis from an upstream

point is also followed in a more recent work by Henry and Fauske (1971)

and that approach together with the interface analysis procedure re- ported here is very similar to that reported by Carofano and McManus

(1969). This work and that of Carofano and McManus were developed

independently at about the same time.

The second difference between this study and most previous

works lies in the analytical conditions prescribed for critical flow. For

the most part, in previous work the fluid has been treated as a mixture

and critical flow has been related to mixture properties. In this treatment, dealing with cases where the flow is essentially separated it has

been assumed that critical flow occurs when the gas reaches critical

flow. This assumption for critical condition is the same as earlier

works using the "vapor choking" model at the point of critical flow,

for example, Ryley ( 1 9 61 ).
2. Experimental Procedure

The experimental apparatus was similar to, but smaller than,

that used in the first part and is shown in figure 1. The two-phase

mixture comes from the heater and flows through the annulus of the

venturi. The rack and pinion permitted the venturi core to be moved

axially, which allowed axial pressure measurements to be closely

spaced near the throat where the gradient was very steep. The support

for the venturi core was rigid with concentricity thoroughly checked

before installation. Any possible variations could be observed visually

and by examination of motion pictures and still photographs taken during

the runs. The quantity of liquid film flow at the tube wall was measured

by extracting the film. The ratio of liquid at the 'wall to total liquid

flow (m /m ) is shown in figures 3 and 4. The mixture quality at the

venturi entrance was determined by calculations which took into consideration the energy input and in which thermal equilibrium was assumed.

Critical flow at the throat was verified by the presence of a shock wave

in the divergent portion of the venturi. If the shock were very near the

throat, the readings in that region could have been disturbed, so only

data from runs where the shock was greater than 0. 05 inches downstream of the throat are reported. Liquid flow was measured by calibrated rotameters between the condenser and the heaters. Pressure

was measured at the entrance and near the throat by transducers.

Estimated 3 a error in the flowmeter reading was 3% and in the pressure

readings was 4%.

3. Qualitative Observations

As in the case for air-water flow (Smith, 1968, 1971), there

were some qualitative observations which support the hypothesis of gascontrolled flow. As shown in figure 2, after critical flow was estab-
Venturi core with

axial travel

Rack 8

pinion

HEATER

Pump

Figure 1. Flow circuit of the steam-water loop
DISTANCE DOWNSTREAM OF VENTUR I INLET, cm

1.0 1.0 2.0

26

Hydrodynamic throat of venturi

.

a.

UJ

QCCO

CO

LlI tr

a.

UJo

CO

CD<

Downstream valve gradually closed

-!6xKf

14

10

UJ

QC

Z>CO

CO

UJ

DC

Q_

UlO

CO

CD<

ud

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CD

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co

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ft

<du

CD

o

CD -M

CD

ft

CM

CD

CD

o

0.8 DISTANCE DOWNSTREAM OF VENTURI INLET, in
lished, the downstream pressure was raised from its minimal value

and this moved the shock wave toward the throat. When the shock

reached the throat, the pressure was approximately that which would

be expected for ideal steam flow alone.

Also, when the standing shock wave reached the throat, the

changes in the downstream pressure were transmitted upstream. If

the gas were not controlling the critical flow behavior, one might expect pressure signals to be transmitted upstream at different velocities.

For example, some pressure signals might be transmitted through a continuous liquid phase at a high velocity and have some influence on the

upstream pressure throughout the range of downstream pressure changes.

Since the upstream pressure was not noticeably changed until the shock

reached the throat, it would appear that the vapor flow characteristics

were controlling.

The accuracy of the experimental critical pressure ratios (p /p ) was estimated to be within ± 0. 03. This was primarily a result of the

uncertainty in the location of the hydrodynamic venturi throat. The

location was fixed by studying runs where the back pressure was raised

to produce subcritical flow, as shown in figure 2. The hydrodynamic

throat was identified as the minimum pressure point for the sub-critical

runs, where downstream pressure changes changed the upstream pressure.

Smith (19 68) showed that with a similar venturi the geometric throat,

the hydrodynamic throat as defined here and the throat indicated by

pressure measurements with critical, all gas flow all fell within an

axial range of 0.020 inches. It was concluded that the relationship

would be approximately the same for the case reported here. As

studies following this general procedure are continued, further detailed

examination of the critical flow at the throat is indicated. These

critical pressure ratios varied from 0. 5 to 0. 6, considering the
experimental uncertainty, in the general range indicated for isentropic to

isothermal flow of the vapor alone, which is 0. 55 to 0. 61. Of course,

vaporization or condensation and momentum transport would change

the value of p /p from that for ideal gas alone. It would appear, howc o ever, that the general agreement, between the experimental data and

the expected behavior for the ideal gas flow indicates that the characteristics of the gas flow was the primary factor in controlling the critical

flow behavior for the conditions reported. Critical pressure ratios in

this range were also reported by Carofano and McManus (1969) for a steam-water system with similar flow conditions.

4. Analytical Procedure

The analytical model was the same as that used for the air -water

portion of the program. Thus, the rates of vaporization or condensation were considered zero or negligible. It was recognized that the

vaporization and condensation effects could be significant. However, it seemed useful to compare data processed using this "frozen flow"

model with the experimental data.

Briefly, the analytical model consisted of a liquid film flowing

at the tube wall and a gas core containing entrained liquid droplets.

As mentioned previously, the distribution of liquid between film and

droplets was experimentally determined by extracting and measuring

the rate of the liquid film flow. Assuming one dimensional flow,

working expressions were derived from the conservation equations.

Frictional pressure drop, due to forces at the tube wall, was considered

negligible compared to the momentum pressure drop in equations (1,2).

Potential energy terms such as changes in elevation, and pressure

changes were neglected in the energy expression (6).
The momentum equation for the mixture was written as: m (du /dl) + m (du. /d£) + mrr (du /dl)

-**/&,= -*—* ^_*: *—

• (1)

Fluid velocities from momentum and continuity equations were:

- (d A /dt) u

g g for gas velocity du /dl = , (2)

A [1 - u

2 /(dp/dp )] g g g for liquid velocity

•A

£f(dpM) - PfAffg+2^r

fg

f£gPg [(u

g -u

ff

) /2]

du„/dt =

(3)

*ff TMf£

and for liquid droplet velocity

,2

p r

( vUfe)

-u c

dur /dt = —-i-i ? J d dr • (4) fe' 4 3 I

TR

d

P

f U

fe

Energy transport and fluid temperatures from energy and heat

transfer equations were:

for interface energy transport

q = h A, (T - T,) /c , c fgv g f , (5)
for gas temperature

u (du /di) + (m, /m ) u, (du. /di)

, , g g fe g fe fe dT /di = -a = fi g c 8

Pg

(m /m ) u (du /di) + 1 /m (dq/di)

. a a

c Pg

(6)

and for liquid temperature

dT/di= Ja/di_ . (7)

f mfc

f

Gas density from the ideal gas equation of state was

dp/di - Rp (dT /di)

dp /di = ——

o

& . (8) g RT v y

Effective gas flow area, empirically adjusted to account for waves and

droplet distribution, was:

A ^ = A - Ar - C, Ar , . (9) gE t f str If str 1 w '

After considerable study, the droplet diameter, the values or expressions for drag coefficients, and the interface heat transfer coefficients were determined. These were selected from data reported in

the literature for similar cases and, from among that group, were primarily the values or expressions which best fitted the experimental,

critical pressure-ratio data for air-water flow. It was found that variations in the assigned values of these terms by factors of two to four

produced computed critical pressure ratio changes no greater than five

percent, except in the case of droplet radii, where the variations were

10
as high as ten percent. Therefore, the final results were not particularly sensitive to the assigned values for these terms. Droplet

radii, R , data were from Wicks et al. (1966) and Ryley (1961).

Droplet drag coefficients, C , involved three studies, Ingebo (1956),

Lappel (1950), and Rabin et al. (I960), with the last of these chosen as best describing the data. The liquid-gas interface drag coefficient,

f , was set at the equivalent fluid-solid interface value for complete

turbulence with rough tubes, using conventional friction factor curves. The convective heat transfer coefficient, h , for the droplets was of the ce Ranz and Marshall (1952) type for laminar flow. The interface heat

transfer coefficient for the liquid film, h r , was the value for a fluidcf

solid system in turbulent flow, Dittus and Boelter (1930), and this value

was partially confirmed by temperature measurements of the liquid

film.

The effective gas flow area in (9) was determined by first subtracting a computed liquid flow area, which would occur for the liquid

flowing in a smooth stream, and then subtracting the final term in the

equation, which is a blockage factor, to account for the reduced gas

flow area caused by waves and by droplet distributions. The value of

C , which is a multiplier for the venturi-inlet, stream liquid flow areas, was empirically determined so that the analytical program would

achieve critical flow at the venturi throat. Although the value of this

coefficient did empirically fix the location of critical flow for the analytical program by adjusting the effective gas flow area, it did not

influence the relationship in equations (1-8), which determined the

pressure distribution that was used to compare analytical and experimental results. The value of C varied between 1. 5 and 3. for the

range of experimental data reported. Although the higher values of

C may seem large, one must remember that this is an adjustment

11
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DISTANCE DOWNSTREAM of VENTURI INLET , cm

I I I 2

34

30

10

mt = 0.069 lb m /s =(0.0315 kg /s)

x, =0.35

m ff /mf =0-58

m,=0.069 lb m /s

= (0.0315 kg /s)

x, =0.47

m ff /m«=0.45

1

o

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o

m t =0.0493 lb m /s =(0.0224 kg /s)_

x, =0.59

m ff /mf =0

(liquid film removed

at entrance

)

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> <U 0)

<U rci <u ^ X! S mi CO CD

CD > P »^o *h D l

Ph u £

XF

(1) 3

GO

0.4 0.4 04 0.8

DISTANCE DOWNSTREAM of VENTURI INLET , in

13
from a liquid -flow model which is far from realistic. The multiplier C

is applied to the stream flow area, which at these high qualities, is very small compared to the total area. Also, estimates of wave behavior and droplet distribution indicate that these blockage values

are in a reasonable range, (Smith, 1968).

The computational procedure was to numerically integrate the

flow equations from the entrance point of the venturi to the point of

critical flow, using the Runge-Kutta (1962) method. The axial length

of step between computation points •was determined by,

Ai , = c_ [1 -JTlcA. (io)

step 2 i

The constants C and C were adjusted to produce the required accuracy

in the final numerical results. This demanded that C be approximately

the value of the length of the converging section in the venturi, to in- sure a very short step length near the throat where the pressure gradient

is very steep. The point of critical flow was detected in the numerical

system by examining the behavior of (2). If the gas flow is primarily

controlling, the mixture flow will be critical when the gas flow reaches

the critical condition. For that case, the denominator of (2) becomes

zero. The maximum uncertainty for this computational procedure was

estimated at plus or minus 0.01, for example, for the critical pressure

ratio.

5. Results and Discussion

The analytical and experimental results are shown in figures

3 and 4. These data are for different total mass flow rates, all at relatively high quality.

14
Both experimental and analytical data show a. typical, gas -flow

profile. The agreement between the analytical and experimental data

is quite good, especially considering the "frozen" flow (no vaporization or condensation) model used. The agreement appears to be somewhat poorer than that for the air-water data. The steeper slope (tendency for the experimental pressures to be lower) which appears to prevail for the experimental data may indicate the influence of the

vaporization or condensation omitted in the analytical procedure. One

should note, however, that in these figures the slope of the pressure curve

is very steep and this tends to make the agreement between the experimental and analytical data appear somewhat better than it actually is.

In comparing these data with air-water data from Smith (1968,

1971), the difference in the agreement between the experimental and

analytical profiles appears significant but not great. This rather

strongly indicates that the gas behavior exerts primary control. It may also indicate that the influence of condensation and vaporization,

while noticeable, is of a secondary nature for these flow conditions.

Carofano and McManus (1969), using very similar analytical and

experimental procedures, found what appears to be about the same

level of agreement with the pressure profiles. In their analytical model,

they took into consideration condensation and vaporization rates while

employing a fog flow (liquid totally entrained) model. Rough comparison of critical mass flow rates on runs with similar entrance fluid conditions showed agreement at least within ± 5% between the experimental

critical flow rate data reported here and that of Carofano and McManus

(1969).

Calculation of the critical mass flow rate, G, using the equation

of Cruver and Moulton (1966),

15
mt -< u

g/"f> -1/2

CA1 (dvg/dp) s + CA 1 (dx/dp) s + CA

3 (dv

f /dp)

£ 11)

where

and

and

C 1 = xj"i + jV^fH rx( 3u/uf+ i)-

L 3 u

g/u£ L g

C

'A2

A3

U

g

[

2 + 2x (Ug/uf - l)j + Uf ( Ug/af

) C 1-2X) ( Ug/uf

- lj

- -4»

(u /u

f

) (1-x) 1 + x (u /uf

- 1) (1 + a /3u )

1/3 with (u /u_) = (p-/p )

' , predicted mass flow rates about double of those

g f f g reported here. This is not unexpected , as in that type of equation,

thermal equilibrium to the throat is assumed. For most of the runs reported here, this assumption would predict superheated steam at

the throat. Therefore, it can only be concluded that the Cruver and

Moulton type of equation will substantially overpredict the flow rate

if the entrance quality is used; since liquid was always observed at

the throat, the equilibrium assumption is known to be incorrect.

Cruver and Moulton (1966) and Fauske (1965) have suggested that this

type of equation may use a slip ratio higher than the actual ratio to compensate for the error in the equilibrium assumption. These data

would tend to support that suggestion.

6. Conclusions

This work, following a more comprehensive air -water study,

further substantiates the concept that gas flow behavior is the primary factor in controlling critical, two-phase flow for higher void or

16
quality flow conditions where the phases flow separately. The analytical model for one-dimensional flow, with estimates for interface

phenomena of momentum and energy transport, but neglecting mass

transport (vaporization or condensation), showed good agreement with

the pressure profiles just upstream of and at the critical point. The

critical pressure ratios were in general agreement with those expected

for ideal gas flow.

7. Acknowledgments

The experimental work for this program was carried out during

a period when the author was attached to UKARE, Harwell England.

Mr. L. B. Cousins and Dr. G. F. Hewitt made substantial contributions to the program.

17
8. References

Carofano, G. C. and McManus, H. N. (19 69), "An analytical and

experimental study of the flow of air -water and steam water

mixtures in a converging-diverging nozzle", Progress in Heat

and Mass Transfer, Vol, 2 (Pergamon Press).

Cruver, J. E. , and R. W. Moulton, (1966), "Metastable critical flow

of steam -water mixtures, " AIChE, Fundamentals of Fluid

Mechanics Symposium, Detroit.

Dittus, F. W. , and L. M. K. Boelter, (1930), Univ. of Calif. Pubs.

Eng. 2, 443.

Fauske, H. K. , (1965), "Two-phase two -and -one -component critical

flow, " Proc. Symp. on two -phase flow, U. of Exeter, England.

Fauske, H. , (1961), "Critical, two-phase, steam-water flows," Proc.

19 61 Heat Transfer and Fluid Mech. , Inst. Stanford Univ.

Press, 79.

Henry, R. E. and Fauske, H. K. (1971), "The two-phase critical flow

of one -component mixtures in nozzles, orifices and short tubes, " ASME paper No. 70-WA/HT-5, to be published in ASME transactions, J. of Heat Transfer.

Ingebo, R. D. , (1956), "Drag coefficients for droplets and solid spheres

in clouds accelerating in airstreams, " National Advisory Committee for Aeronautics, Washington, D. C. Tech. Note 3762.

Klingbiel, W. J., (19 64), "Critical flow slip ratios of steam -water

mixtures," Ph.D. Thesis, University of "Washington, Seattle.

Lappel, C. E. , (1950), "Dust and mist collection, " in Chem. Eng . Handbook, 3rd ed. (McGraw Hill Publishing Co. ).

Moody, F. J., (1965), "Maximum flow rate of a single component,

two-phase mixture," Trans. ASME, Series C, 87, 134.

18
Rabin, E. A., Schallenmuller, R. , and R. B. Lawhead, (I960),

"Displacement and shattering of propellant droplets, " Final

Summary Report, Rocketdyne, A Division of North American

Aviation, Inc.

Ranz, W. E. , and W. R. Marshall, (1952), "Forced convection heat

transfer from a single sphere, " Chem. Engr. Prog. , 48 .

Runge-Kutta Method as summarized, for example, in Hildebrand,

F. B. (1962) Advanced Calculus for Applications (Prentice-Hall).

Ryley, D. J., (1961), "The dispersion of water globules in steam,"

The Engineer.

Smith, R. V., (1968), "Two-phase two component critical flow in a venturi, " D. Phil. Thesis, Oxford, England, and Smith, R. V.,

Cousins, L. B. , and Hewitt, G. F. , (1968), " Two -phase two

component critical flow in a venturi, " UKAEA Report

AERE-R5736.

Smith, R. V. (1971), "Two-phase, two-component critical flow in a venturi," ASME paper No. 71-FE-4, to be published in

ASME Transactions, J. of Basic Engr.

Wicks, Moye III and A. E. Dukler, (1966), "In Situ Measurements of

drop size distribution in two -phase flow; a new method of

electrically conducting liquids," 3rd Intl. Heat Transfer Conf. , ASME-AIChE.

Zivi, S. M. , (1964), "Estimation of steady state steam void fraction

by means of the principle of minimum energy production, " Trans. ASME, Series C, 86, 247.

19
9. Nomenclature

A = Area, cross -sectional, flow area when single subscripted.

2 Interface surface area when double subscripted L

2 2

c = specific heat L /t T

C = coefficient

D = tube diameter, L

ir - interface drag coefficient, dimensionless

fg

2 g

= gravitational acceleration, L/t

h = convection heat transfer coefficient, M/t T

k = thermal conductivity, ML/t T

jJ = length along venturi, L

m = mass flow rate per unit time, M/t

p

= pressure, M/Lt

q

= energy (heat) transport at interface ML/t

R = gas constant, ML2 /t 2 T

Rj = droplet radius, L

t = time, t T := temperature, T

u = velocity, L/t

3 V = volume, L

x = mixture quality, dimensionless

20
Greek Letters

3

Subscripts

density, M/L,"

c = convection (first subscript)

= critical when subscripted to p or T

ct = total convection

d = droplet

dr = drag

e = entrained

f = liquid

fe = entrained liquid

ff = liquid film

fg = liquid-gas

f str = equivalent liquid stream flow

g

= ga-s

= stagnation conditions (first subscript)

p

= constant pressure

t = total

1 = venturi entrance

= coefficient identification

2, 3 = coefficient identification

21

form NBS-114A (1-71) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEETS

1. PUBLICATION OR REPORT NO. NBS-TN-6082. Gov t Accession No.3. Recipient's Accession No.

4. TITLE AND SUBTITLESteam -Water, Critical Flow in a Venturi

5. Publication Date July 1971

6. Performing Organization Code

7. AUTHOR(S) R. V. Smith

8. Performing Organization

10. Project/Task/Work Unit No. 275C478

9. PERFORMING ORGANIZATION NAME AND ADDRESS NATIONAL BUREAU OF STANDARDS DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

11. Contract/Grant No. NA

12. Sponsoring Organization Name and Address 13. Type of Report & Period Covered NA

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant bibliography or literature survey, mention it here.) This paper is the second part of an analytical and experimental investigation

in which the primary object was to test the hypothesis that the flow of the gas

phase controls critical and near critical two-phase flow for cases where the gas

and liquid flow essentially in separate streams. In the first part of the investigation, a two-component system (air-water) was used. The results presented here

substantiate the hypothesis. The analytical results also indicate the use of one dimensional flow equations with reasonably accurate estimates for droplet size

and for the drag and heat transfer coefficients (without consideration of mass transfer

--vaporization or condensation) describe critical and near-critical flow reasonably

well. This indicates that mass transfer may be a secondary effect for these flow

conditions.

17. KEY WORDS (Alphabetical order, separated by semicolons) Critical flow; pressure profile; steam; venturi; water.

18. AVAILABILITY STATEMENT

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