A study of nonequilibrium flashing of water in a converging diverging nozzle volume 2 modeling

Pressure distributions, photographic observations, diamet rical averaged centerline void fraction distributions, detailed transverse distri-butions of the chordal averaged void fractions

B.J.e Wu N Abuaf and P Saha

Date Published: June 1981

EXPERIMENTAL MODELING GROUP

UPTON, NEW YORK 11973

Prepared for the U,S, Nuclear Regulatory Commission Office of Nuclear Regulatory Research Washington, D,C 20555

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NRC FORM 335 1 HI 1'0fn NlJMBE R (AIJ,,,n"rlIJv DOC!

NUREG/CR-1864, Vol 2

BIBLIOGRAPHIC DATA SHEET BNL -NUREG-51317, Vol.2 I

4 TITLE AND SUBTITLE (Add Volume No., If apprO(Jflare' '] IL ,."." hlan.' of J

A STUDY OF NONEQUILIBRIUM FLASHING OF WATER

IN A CONVERGING-DIVERGING NOZZLE 3 RECIPIENT'S ACCESSION NO

VOLUME 2 - MODELING

9 PERFORMING ORGANIZATION NAME AND MAILING ADDRESS (Include Z,p Code) DATE REPORT ISSUED

Brookhaven National Laboratory

8 (Leave blank'

12 SPONSORING ORGANIZATiON NAME ANO MAl LING ADDRESS (Include Z,p Code)

10 PROJECT/TASK/WORK UNIT NO

U.S Nuclear Regulatory Commission

Office of Nuclear Regulatory Research 11 CONTRACT NO

13 TYPE OF REPORT I PE R I OD COV ERE D (InclUSive dares)

16 ABSTRACT (200 words or less)

A steady water loop with well controlled flow and thermodynamic conditions was designed, built, and made operational for the measurement of net vapor

generation rates under nonequilibrium conditions The test section consists of

a converging-diverging nozzle with 49 pressure taps and two observation windows

at the exit Pressure distributions, photographic observations, diamet rical

averaged centerline void fraction distributions, detailed transverse

distri-butions of the chordal averaged void fractions at 27 axial locations, and area

averaged void fraction distributions along the nozzle were recorded under

the phase velocities was recorded during the present experiments, the

calcula-tion of vapor generacalcula-tion rates from the available experimental data involved the

assumption of a slip model between the two phases

17 KEY WORDS AND DOCUMENT ANALYSIS 17a DESCRIPTORS

17b IDENTIFIERS/OPEN·ENDED TERMS

18 AVAILABILITY STATEMENT 19 SE5UR~TY Cl /'r~ (Tef'! report) 21 NO OF PAGES

NUREG/CR·1864 BNL·NUREG·51317

N Abuaf, B.J.C Wu, G.A Zimmer, and P Saha

Manuscript Completed: December 19aO

Date Published: June 19a1

Prepared for the UNITED STATES NUCLEAR REGULATORY COMMISSION

OFFICE OF NUCLEAR REGULATORY RESEARCH CONTRACT NO DE·AC02·76CH00016

FIN NO A-3045

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States Government Neither the United States Government nor any agency thereof nor any of their employees, nor any of their contractors subcontractors or their employees, makes any warranty, express or implied or assumes any legal liability or responsibility for the accuracy completeness, or usefulness of any information, apparatus, product or process disclosed or represents that its use would not infringe privately owned rights Reference herein to any specific com- mercial product, process or service by trade name, trademark, manufacturer or otherwise, does not necessarily constitute or imply its endorsement recommenda- tion, or favoring by the United States Government or any agency, contractor or subcontractor thereof The views and opinions of authors expressed.herein do not necessarily state or reflect those of the United States Government or any agency

FOREWORD

the program and the original principal investigator who conceptualized the search and experiment, supervised the design and construction of the flow loop and defined the measurement techniques to obtain the necessary data He also put the Experimental Modeling and the Systems, Control and Data Acquisition

their appreciation to Dr Jones, Jr for his technical contributions to the project both in the experiment and the analytical modeling areas and for fruit-ful discussions that the authors had wi th him during his involvement wi th the program

The final series of experimental runs (Runs 254-397), the required data duction and analysis, and the development of the general void growth model were

organization and content of this final report

During this research several reports and papers have been published and a

BNL-NUREG-26003 and BNL-NUREG-27138 have been transmitted as preliminary data

the analytical work performed

ABUAF, N., JONES, O C., Jr., ZIMMER, G A., LEONHARDT, W J., and SARA, P.,

"BNL Flashing Experiments: Test Facility and Measurement Techniques,"

BNL-NUREG-24336, 1978; also Proc of the CSMI Spec Meeting on Transient

Two-Phase Flow, Paris, France, June, 1978 (in press)

Optical Probes," ASME paper 78-WA/HT-3, presented at the Winter Annual

Meeting, San Francisco, Calif., December 1978

ABUAF, N., ZIMMER, G A., and JONES, O.C., Jr., "BNL Instrumentation Research Program," BNL-NUREG-2605l, April 1979; also NUREG/CP/0006, May 1979

ABUAF, N., FEIERABEND, T P., ZIMMER, G A., and JONES, O C., Jr., "Radio

Frequency (R-F) Probe for Bubble Size and Velocity Measurements,"

NUREG/CR-0769, BNL-NUREG-50997, March 1979; also Rev Sci Instrum., 50(10),

pp 1260-1263, 1979

ABUAF, N., WV, B J C., ZIMMER, G A., and JONES, O C., Jr., "Preliminary Data

ABUAF, N., JONES, O C., Jr., and WU, B J C., "Critical Flashing Flows in Nozzles with Subcooled Inlet Conditions," BNL-NUREG-275l2, March 1980; also

"Polyphase Flow and Transport Technology, R A Bajura ed., ASME, Aug 1980 ABUAF, N., WV, B J C., ZIMMER, G A., SARA, P., and JONES, O C., Jr.,

"Nonequilibrium Vapor Generation Rates of Flashing Water Flows," Proc of the ANS/ASME International Topical Meeting on Nuclear Reactor Thermal Hydraulics, Oct 1980

JONES, O C., Jr., and SAHA, P., "Nonequilibrium Aspects of Water Reactor

Safety," BNL-NUREG-23143, July 1977; also in Symp on the Thermal and

JONES, o C., Jr., "Inception and Development of Voids in Flashing Liquids," BNL-NUREG-26464, June 1979; also presented at the Joint U.S.-Japan Inf Exch

on Two-Phase Flow Dynamics, Japan 1979

JONES, O C., Jr., "Flashing Inception in Flowing Liquids," BNL-NUREG-26134,

1979

SARA, P., "A Review of Two-Phase Steam-Water Critical Flow Models with Emphasis

on Thermal Nonequilibrium," BNL-NUREG-50907, Sept 1978

Model of Vapor Generation in Steady Flashing Flow," BNL-NUREG-25709, March 1979; also ANS Trans., 32, pp 490-491, 1979

ZIMMER, G A., WU, B J C., LEONHARD, W J., ABUAF, N., and JONES, O C., Jr.,

"Pressure and Void Distributions in a Converging-Diverging Nozzle with

Nonequilibrium Water Vapor Generation," BNL-NUREG-26003, April 1979

ZIMMER, G A., WU, B J C., LEONHARD, W J., ABUAF, N., and JONES, O C., Jr.,

"Experimental Investigations of Nonequilibrium Flashing of Water in a

Converging-Diverging Nozzle," BNL-NUREG-2S716 , Aug 1979; also "Nonequilibrium Interfacial Transport Processes," J C Chen and S G Bankoff eds., ASME,

1979

Water Reactor Safety Research Division,

ABSTRACT

A steady water loop with well controlled flow and thermodynamic conditions was designed, built, and made operational for the measurement of net vapor

a converging-diverging nozzle with 49 pressure taps and two observation windows

at the exit Pressure distributions, photographic observations, diametrical

distri-butions of the chordal averaged void fractions at 27 axial locations, and area averaged void fraction distributions along the nozzle were recorded under var-

the phase velocities was recorded during the present experiments, the tion of vapor generation rates from the available experimental data involved the assumption of a slip model between the two phases

calcula-The development of voids in nonequilibrium flashing flows is shown through the Oswatitsch integral to be dependent on three major factors of the void

inception point which determines the initial and subsequent liquid superheats and must be accurately described; of the interfacial mass transfer rates, which depend on the local superheat and must be specified; and the local interfacial

Alamgir and Lienhard (1979) was extended to predict flashing inception in pipe and nozzle flows with subcooled inlet conditions A void development model for bubbly flows (a< 0.30) was based on a simple concept for interfacial area de-

this model, bubbly flow, bubbly-slug flow, a transitional flow comprising the annular and annular-mist regimes and finally fully dispersed droplet flow were assumed to occur at successively higher void fraction ranges

On the basis that flashing inception occurred at the throat in nozzle flows with subcooled inlet conditions, and that the pressure undershoot can be

calculated from the Alamgir-Lienhard correlation, a method of calculating the

existing data

Comparison of the BNL experiments with TRAC-PIA predictions revealed that, although the code gave a good qualitative description of the flow, it was

significant quantitative discrepancies in the predicted and measured flow

4.3.1 Low Activi ty Single-Beam Densitometer for Axial

5

4.3.2 Multibeam y-Densitometer for Transverse and

4.3.3 High Activity Single-Beam y-Densitometer for

Transverse and Axial Void Distributions ••

EXPERIMENTAL RESULTS • • •

5.1.1 Single-Phase Pressure Calibration • • • •

5.1.2 Pressure Distributions Under Flashing Conditions 27

Diametrical Averaged Void Fraction by a Low Activity

Single-38

45

5.2.1 Gamma Densitometer Calibration

5.2.2 Axial Distributions of the Centerline Diametrical

Averaged Void Fractions for Flashing Close to

45

5.2.3 Axial Distributions of the Centerline Diametrical

Averaged Void Fractions for Flashing Upstream from the Throat • • • • • •

Obtained by Means of the Five-Beam Gamma Densitometer • •

5.3.1 Calibrations • •

5.3.2 Transverse Void Distributions and Area Averaged

Void Fractions • •

Obtained by Means of the High Activity Single Beam

5.4.1 Cal ibrations · · ·

5.4.2 Transverse Void Distributions and Area Averaged

and Area Averaged Void Profiles

COMPARISON OF TRAC-PIA PREDICTIONS WITH EXPERIMENTAL DATA •

7.1 Comparison of TRAC-PIA Predictions with Experimental Data

Consisting of Pressure and Diametrical Averaged Centerline

7.2.2 Comparison with Runs Performed at a Nominal

7.2.3 Comparison with Runs Performed at a Nominal

Conditions • • • • • • •

Fraction Distributions Obtained wi th the Low Activity Single Beam Densitometer • • • • • • • •

Void Fraction Di stributions Obtained wi th the l1ulti Beam Gamma Densitometer • • • • • •

Void Fr action Di stributions Obtained wi th the High Activity Single Beam Gamma Densitometer • • •

LIST OF FIGURES

3.1 Schematic of BNL Heat Transfer Facility

3.2 Inside Dimensions of TS-2

3.3 Deviation From Design of TS-2 Inside Dimensions

3.4 Schematic Representation of the Test Section

4.1 Graphical Representation of the Data Acquisition System

4.2 Schematic Representation of y-Densitometer

4.3 Photograph of the Five Beam Gamma Densitometer

4.4 Horizontal and Vertical Cross Sections of the Source Holder

4.5 Schematic Representation of the Detector Holder

5.1 Typical Pressure Distributions Along TS-2 for the Single-Phase Flow

Hydrodynamic Calibration Runs

5.2 Dimensionless Pressure Distribution for TS-2 Data are Averaged for

All the Hydrodynamic Calibration Runs Performed

5.3 Typical Representation of an Isothermal Flashing Experiment in the p-T

Diagram

5.4 Pressure Distributions Under Flashing and Nonflashing Conditions in

TS-2

5.5 Dimensionless Pressure Distributions in TS-2 Under Flashing Conditions

Compared With Single-Phase Hydrodynamic Calibration Data

5.6 Comparison of Pressure Distribution in Two Experiments to Show the

Reproducibility of the Results at Low Mass Fluxes, G = 3.03 Mg/m2 s 5.7 Comparison of Pressure Distributions in Two Experiments to Show

Reproducibility of the Results at High Mass Flux, G = 4.45 Mg/m2 s 5.8 Pressure Distributions Showing the Effect of Condensing Tank Back

Pressure for Identical Nozzle Inlet Conditions

5.9 Photographic Observations for the Experimental Conditions Presented in

both the front and rear windows is 50 mm

Inlet Conditions Which are Close to Those For Onset of Flashing in the Test Section

in Fig 5.10

Fig 5.12

on the Pressure Distribution in the Test Section

Fig 5.14

on the Pressure Distribution in the Nozzle

on the Pressure Distribution in the Nozzle

on the Pressure Distribution in the Nozzle

Section While the Flashing Onset is Upstream From the Nozzle Throat Calibration of the test Section Both Empty (Air) and Full of Water as

a Function of Axial Distance

Pressure and axial void fraction distributions in the test section Plot of the difference between the dimensionless measured pressure drop and the nondimensiona1 pressure drop measured in the single phase

Pressure and axial void fraction distributions in the test section Plot of the difference between the dimensionless measured pressure drop and the nondimensiona1 pressure drop measured in the single phase

Pressure and axial void fraction distributions in the test section Plot of the difference between the dimensionless measured pressure drop and the nondimensiona1 pressure drop measured in the single phase

Pressure and axial void fraction distributions in the test section Plot of the difference between the dimensionless measured pressure drop and the nondimensiona1 pressure drop measured in the single phase

Pressure and axial void fraction distributions in the test section Plot of the difference between the dimensionless measured pressure drop and the nondimensiona1 pressure drop measured in the single phase

Pressure and axial void fraction distributions in the test section Plot of the difference between the dimensionless measured pressure drop and the nondimensiona1 pressure drop measured in the single phase

Pressure and axial void fraction distributions in the test section • P1ot of the difference between the dimensionless measured pressure drop and the nondimensiona1 pressure drop measured in the single phase

difference between the dimensionless measured pressure drop and the nondimensiona1 fressure drop measured in the single phase calibration,

5.34 Pressure and Axial Void Fraction Distributions in the Test Section

Difference Between the Dimensionless Measured Pressure Drop and the Nondimensional Pressure Drop Measured in the Single Phase Calibration,

5.47 Radial Distributions of the Chordal Averaged Void Fractions at Various

Axial Locations Obtained by the High Activity Single Beam Gamma

Densitometer for Run 310

Axial Locations Obtained by the High Activity Single Beam Gamma

Densitometer for Run 310

Axial Locations Obtained, by the High Activity Single Beam Gamma

Densitometer for Run 310

Axial Locations Obtained by the High Activity Single Beam Gamma

Densitometer for Run 310

Axial Locations Obtained by the High Activity Single Beam Gamma

Densitometer for Run 310

Axial Locations Obtained by the High Activity Single Beam Gamma

Densitometer for Run 310

diametrical void fractions (A) and of the pressure drop (B) in the

and at a mass flow rate of 8.8 kg/so

(B) for Runs Performed at an Inlet Temperature of-149°C and Several Inlet Mass Fluxes

(B) for Runs Performed at an Inlet Temperature of 149°C, a Constant Inlet Mass Flux of 4300 kg/m2s, and Decreasing Nozzle Exit or

Distributions Along the Test Section

Distributions Along the Test Section

Axial Locations Obtained by the High Activity Single Beam Densitometer for Run 306

Comparison of Centerline Diametrical Averaged Void Fraction

Distributions with the Area Averaged Void Fractions at the Same Axial Locations for Runs 306 and 310

Axial Distributions of Pressure (A) and Area Averaged Void Fraction (B) for Five Runs Performed at an Inlet Temperature of 121°C and Increasing Inlet Mass Flux

Comparison of Axial Distributions of Pressure (A) and Area Averaged Void Fraction (B) for Two Runs Performed Under "Identical" Conditions Data in Runs 133-136 Were Recorded with the Five-Beam Gamma

Densitometer with the High Activity Single Beam Densitometer

for Runs Performed at an Inlet Temperature of 121°C, a Constant Inlet Mass Flux, and Decreasing Nozzle Exit or Condensing Tank Pressure

Three Dimensional Representation of the Chordal Averaged Void Fraction Distributions Along the Test Section

Three Dimensional Representation of the Chordal Averaged Void Fraction Distributions Along the Test Section

Three Dimensional Representation of the Chordal Averaged Void Fraction Distributions Along the Test Section

Three Dimensional Representation of the Chordal Averaged Void Fraction Distributions Along the Test Section

Three Dimensional Representation of the Chordal Averaged Void Fraction Distributions Along the Test Section

Axial Distributions of Pressure (A) and Area Averaged Void Fraction (B) for Runs 149-153, Performed at an Inlet Temperature of 121°C with MgO Particulates Present in the Water Loop

Axial Distributions of Pressure (A) and Area Averaged Void Fraction (B) for Five Runs Performed at an Inlet Temperature of 100°C and Increasing Inlet Mass Fluxes (Five Beam Gamma Densitometer)

Three Dimensional Representation of the Chordal Averaged Void Fraction Distributions Along the Test Section

Three Dimensional Representation of the Chordal Averaged Void Fraction Distributions Along the Test Section

Distributions Along the Test Section

Axial Distributions of Pressure (A) and Area Averaged Void Fraction (B) for Two Runs Performed at Similar Inlet Conditions but Varying

Condensing Tank Pressure

Axial Distributions of Pressure (A) and Area Averaged Void Fraction (B) for Two Runs Performed at an Inlet Temperature of 121°C, Two-Phase Inlet Conditions, and at Two Inlet Mass Fluxes

Converging Part of the Test Section in Runs 82/821 and the Least Square

Based on the Least Square Fit to the a and p Data

and Void Fraction Distributions;cr, Measured; Cubic Spline Fit;

of Vapor Generation rv for Three Values of the Distribution Parame.ter Co·

5.82 Results of Calculations on Data of Runs 133/134 Pin = 350 kPa,

Pct = 233 kPa, Tct = 121.7 C

Pct = 234 kPa, Tct = 121.6 C

Alamgir-Lienhard Correlation

by the Alamgir-Lienhard Correlation (1979)

Seynhaeve, et al (1976) with the Theory Developed by Jones (1979) Using the Approximate Static Flashing Overexpansion Value of 18 kPa for the Computation Jones (1979)

50°C Subcooled Inlet Conditions (Brown, 1961)

Data in a Converging-Diverging Nozzle with Subcooled Liquid Inlet

Lienhard (1979) (solid line) with the Locus of the Liquid

Depressurization History (circles connected by dashed line) in Brown's Nozzle (1961)

Lienhard (1979) (solid line) with the Locus of the Depressurization History in BNL's Nozzle (Runs 76 to 79)

(1979) (solid line) with the Locus of the Depressurization History in Reocreux' Pipe Experiments (1974)

1955) Heat Transfer Coefficient with Measured Instantaneous Heat

6.12 Variation of a with ?r and ZNVG in Present Model and Comparison

with Experimental Data of Reocreux

6.13 Void Fraction Distribution in a Constant Area Channel as a Function

6.14 Void Fraction Distribution in a Constant Area Channel as a Function of

6.15 Void Fraction Distribution in a Constant Area Channel as a Function of

6.16 Effect of Expansion Rate on the Pressure at the Point of Net Vapor

Generation

6.17 Variation of Point of Net Vapor Generation ZNVG with Mass Flux G and

6.18 Variation of Cr with Mass Flux G and Initial Temperature Tin in

Data

6.19 The Flow Regimes Map for the General Model

6.20 Comparison of Correlation of Table 6.2 with Experiment

6.21 Nomenclature of Bubbly-Slug Flow

6.22 Surface to Volume Ratios for Cylinders and Spherical Caps

6.23 Heat and Mass Transfer Rates for Spheres

6.24 Comparison Between the Void Fraction Data for Run 353 and The

"Best-Fi t" Calculation Using the General Model

6.25 Comparison Between the Void Fraction Data for Run 358 and the

"Best-Fit" Calculation Using the General Model

6.26 Comparison Between the Void Fraction Data for Run 362 and the

"Best-Fit" Calculation Using the General Model

6.27 Comparison Between the Void Fraction Data for Run 145 and the

"Best-Fit" Calculation Using the General Model

6.28 Comparison Between the Void Fraction Data for Run 137 and the

"Best-Fit" Calculation Using the General Mode

Comparison Between the Void Fraction Data for Run 137 and the

"Best-Fit" Calculation Using the General Model

Comparison Between the Void Fraction Data for Run 344 and the

"Best-Fit" Calculation Using the General Model

Comparison Between the Void Fraction Data for Run 291 and the

"Best-Fit" Calculation Using the General Model

Comparison Between the Void Fraction Data for Run 284 and the

"Best-Fit" Calculation Using the General Model

Comparison Between the Void Fraction Data for Run 273 and the

"Best-Fit" Calculation Using the General Model

Comparison Between the Void Fraction Data for Run 278 and the

"Best-Fi t" Calculation Using the General Model

Comparison Between the Void Fraction Data for Run 296 and the

"Best-Fit" Calculation Using the General Model

Comparison Between the Void Fraction Data for Run 268 and the

"Best-Fit" Calculation Using the General Model

Comparison Between the Void Fraction Data for Run 304 and the

"Best-Fit" Calculation Using the General Model

Comparison Between the Void Fraction Data for Run 309 and the

"Best-Fit" Calculation Using the General Model

The Optimum Bubble Number Density at the Inception Point vs

the Liquid Superheat at the Inception Point

The Interfacial Area Density at the Inception Point vs the

Liquid Superheat at the Inception Point

7.4 Comparison of TRAC Predictions and Homogeneous Equilibrium Calculations

Pin = 395 kPa, Tin = 99.3°C

Pressure Distributions and Area Averaged Void Profiles

Tct = 100.4 C (mexp =4.6 kg/s, IIlTRAC = 4.8 kg/s)

Data for Pin = 143

7.8

7.9

Tet = 100.7 C (mexp = 6.1 kg/s, mTRAC = 6.0 ,kg/s)

Comparison of TRAC-PlA Predictions With BNL Experimental

Pressure Distributions and Area Averaged Void Profiles

Comparison of TRAC-PIA Predictions With BNL Experimental

Pressure Distributions and Area Averaged Void Profiles

kPa, Tin = 100 C, Gin = 4520 kg/m2s, Pet = 113 kPa, and

Tct = 100 C

Data for Pin = 248

Data for Pin = 247

Averaged Void Profiles, as well as Vapor Generation Rates Calculated

Pressure Distributions and Area Averaged Void Profiles.' Pin = 305 kPa, Tin = 121.2 C, Gin = 3.7 Mg/m2s, Pct = 234 kPa, and

Tct= 121.7 C (mexp = 7.5 kg/s, mTRAC = 7.0 kg/s)

With TRAC-PIA Predictions for Run 145 Presented in Figure 7.11

Tct = 121.7 C (mexp = 8.9 kg/s, mTRAC = 8.4 kg/s)

7.14 Comparison of Vapor Generation Rates Calculated From Experimental Data

With TRAC-PIA Predictions for Run 133 Presented in Figure 7.13

Averaged Void Profiles as well as Vapor Generation Rates Calculated

Pressure Distributions and Area Averaged Void Profiles for Runs

Pressure Distributions and Area Averaged Void Profiles for Runs

Pressure Distributions and Area Averaged Void Profiles for Runs

Pressure Distributions and Area Averaged Void Profiles for Runs

Pressure Distributions and Area Averaged Void Profiles for Runs

Pressure Distributions and Area Averaged Void Profiles for Runs

Comparison of Experimental Mass Flux Gexp and Critical Mass Flux

Gcr at C-J Point in BNL Experiment Run 77

Comparison of Experimental Mass Flux Gexp and Critical Mass Flux

Gcr at C-J Point in Run 56 of Schrock, et a1., (1977)

Comparison of Critical Throat Mass Fluxes Measured by Powell (1961) with Those Calculated by the Present Method for Different Nozzle Inlet Pressures and Temperatures

Comparison of Calculated with Measured Critical Throat Mass Fluxes (Powell, 1961) for Various Nozzle Inlet Conditions

LIST OF TABLES

Pressure Calibrations

Measurements Under Flashing Conditions

Fraction Data with Single Beam Gamma Densitometer

of Water, IF

Gamma Densitometer)

Source Gamma Densitometer)

Gamma Densitometer)

5.10 Summary of Experimental Conditions for Flashing Experiments (Single

Source Gamma Densitometer)

7.2 Comparison of 121 C Inlet Temperature Runs with TRAC-P1A Predictions

(Run 423)(1974)

specific heat at constant pressure diameter

pressure differential between the test section inlet (Tap 1) and a

include any gravitational head effects.) DP/l/2 Uo ' dimensionless pressure differential DPm"'- D~

inverse of cavitation number friction coefficient

mass flux (inlet) mass flow rate of vapor expression defined in Table 6.2 gravitational acceleration

mass flow rate number density Nusse1t number pressure Peclet number interfacial heat flux radial coordinate, bubble radius radius

surface temperature time

turbulence intensity test section inlet velocity free stream velocity

velocity speci fic volume

volume flow quali ty axial coordinate

void fraction

volumetric rate of vapor mass generation chordal area subs tended by the gamma beam thickness perimeter

dummy variable of integration

Y attentuation coefficient, viscosity mass density

surface tension depressurization rate half subs tended angle for spherical caps equivalent sphere radius

bubbles single phase calibration

saturated vapor interface

test section inlet liquid

measured, or mixture point of net vapor generation initial point, stagnation condition reduced by the critical value

bubbly-slug flow regime saturation

steady state throat, Taylor bubble vapor

dimensionless throat

area averaged

area averaged quantity

6 ANALYTICAL MODELING AND COMPARISON WITH EXPERIMENTS 6.1 Introduction

In order to predict the critical flow rates of a vapor-liquid two-phase ture through pipes discharging into a low pressure environment, it is necessary

trans-ition from single-phase (liquid) to two-phase flow, due to rapid tion of the liquid in the flow causing the liquid to vaporize or flash, is par-ticularly important, because the results of the flashing process may be taken as the initial conditions for the ensuing two-phase flow

depressuriza-In addition to those presented in this report, flashing experiments have

vapor generation model developed at BNL will be discussed, and also its cation to the evaluation of experimental data obtained at BNL and elsewhere

appli-In analogy to the process of nonequilibrium condensation of a supersaturated vapor during rapid isentropic expansion (see, e.g., Wegener and Wu 1977) and subcooled boiling (see, e.g., Jones and Saha 1977), the transition from liquid

to vapor-liquid two-phase flow by flashing may be visualized as taking place in

flowing liquid may be sufficient to bring the liquid to a saturated state, since the temperature change in adiabatic liquid flow is generally negligible Vaporization does not occur upon reaching the saturation state, because a finite

the liquid becomes superheated as the pressure decreases further and falls below the saturation value

Second, the bubble nucleation, either homogeneously or heterogeneously, gins after an "induction" period when a certain liquid superheat has been at-

given flow system may depend on the flow conditions and the time rate of

ther-modynamic state of the parent phase, small changes in the liquid superheat may

rates will rise quickly to a peak value after the liquid superheat increases

stable vapor nuclei and the accompanying loss of latent heat by the liquid to the vapor is expected to decrease the liquid superheat slightly so as to cause

nucle-ation zone is a function of the physical properties of the liquid and vapor,

the flow velocity, the depressurization rate, and other flow variables In the case of condensation in supersonic flows, it has been found (e.g., Wegener and

Wu 1977), that the nucleation zone is only a few millimeters wide, and the

rela-tively long overall vaporization zone (of the order of 0.1 m) and the

relative-ly low flow speed (of the order of 10 m/s) in observed experiments, nucleation zones a few cm thick (transport time of a few msecs) may still be considered

narrow one

In addition, Jones and Zuber (1978) found that for the early phases of ble growth in a variable pressure field, where the pressure varies with time ac-

growth period is shorter, and the growth rate is always much lower than for

loca-tion downstream from the nuclealoca-tion zone is dominated by the bubbles generated

the nucleation zone, a certain population of bubbles with an average size mined by the nucleation mechanism is present, and that additional bubble gener-

earlier research (Edwards, 1968)

Third, vaporization of the liquid, which is still superheated at the

ef-fective at a void fraction of about 30% in substantially altering the bubble population and size distribution, leading eventually to bubbly-slug or annular-mist flow

Unlike the nonequilibrium vapor condensation in supersonic nozzle flow, in which homogeneous nucleation dominates, flashing as a result of rapid depres-surization of liquid flows in commercial pipes is most likely initiated by het-erogeneous nucleation of vapor bubbles in the bulk liquid and/or at crevices or

Oswatitsch's (1942) treatment of condensation in supersonic nozzles, Zuber et

al (1966) proposed an expression for the calculation of the mass flow rate of

1

Z

where ~h is the perimeter of the duct, J( Q is the nucleation rate per unit wall area at point s along the pipe, m(Z,s) is the mass at Z of a vapor bubble

integration effectively sums the vapor mass of all the bubbles nucleated before

con-densation in high speed flows (Wegener and Wu 1977; Wu 1977) with remarkable

mainly because of the lack of understanding of the heterogeneous nucleation

Rohatgi and Reshotko (1975 a,b) carried out such a calculation for flashing flow of liquid nitrogen and compared their results with the experimental results

unit volume, and the other being a function of the contact angle between liquid

parameters by "best fits" to the experimental data, but they did not apply the analysis to any steam-water data

To circumvent this difficulty, it was decided to treat nucleation and bubble

the end of the nucleation zone, or the so-called "net vapor generation" point ZNVG' while focusing attention on the calculation of vapor generation rate in the bubble growth region and beyond

As shown in the following sections, the local vapor generation rate in the initial bubble growth region can be calculated from conduction limited bubble growth equations developed by Forster and Zuber (1954) and by P1esset and Zwick

which are unknown a priori:

nonequi1ibrium conditions and constitutes the origin of the time

scale t for the bubble growth history;

the inception point

The conduction controlled vapor generation rate proposed was first applied

to the experimental data of Reocreux (1974), and the values of the three

specified, which determines the onset location ZNVG' the critical bubble

ZNVG' can be fixed from a semi-empirical onset correlation developed by

below the saturation pressure (at the initial temperature) at the inception point to the reduced initial temperature and depressurization rate, and can be

determined, the critical radius at the onset pressure can be calculated, and following the reasoning of Alamgir and Lienhard, the only free parameter left is

was followed for the BNL nozzle experimental results presented in this report

A second approach to the solution of the same problem involves a study of

reasons for seeking such a relationship can be summarized as follows:

The flashing of a superheated liquid under nonequilibrium conditions peared to be a process analogous to chemical reactions or condensation in high

modeled successfully by an equilibrium flow down to a transition point (termed

"sudden freezing point" by Bray) followed by a frozen flow, i.e., no change in

frozen (supersaturated) down to the inception point, where a sudden transition

to equilibrium occurred, and the flow was considered to be in equilibrium and

is applied successfully except when the flow is too slow compared with the

characteristic time of condensation

It was desirable to find out whether an approach similar to the latter could

considered as a frozen flow (superheated liquid) upstream from the inception

the jump conditions are applied across the transition zone at inception, does not give the details of the transition zone but correlates the upstream and

requires the additional implicit assumption that the transition region is

nar-6.2 Flashing Inception

The importance of correctly predicting the flashing inception point under

usual-ly under varying degrees of nonequilibrium conditions, strongusual-ly affects the

the-rmodynamic conditions at this inception point determine the local inception superheat, i.e., the local pressure and thus the critical bubble radius at on-set The inception point also constitutes the origin of the time scale for the bubble growth history, and thus forms the initial point for all the void de-velopment integration schemes and also fixes the initial superheat or driving

calcula-tions including TRAC-PlA have a bubble growth model, the results predicted

point is clarified in detail in Section 7, where the present experimental sults are compared with the TRAC-PlA predictions

re-In this section, it is shown that the correlation developed by Alamgir and Lienhard (1979) relating the static pressure undershoot below the saturation pressure (at inlet temperature) at the inception point to the reduced initial temperature and local depressurization rate can be applied both to pipe flows (Jones 1979) and to nozzle flows to determine the flashing inception point 6.2.1 Static Depressurization Results

Alamgir and Lienhard (1979) developed a correlation to predict- the pressure undershoot at flashing onset below the saturation pressure during a rapid de-

pipes) heterogeneous nucleation occurs at the surface imperfections of the pipe walls, they introduced a heterogeneity factor to be determined by experimental correlation which reduces the critical work of formation of a critical bubble

An expression for the nucleation rate per unit area and time was presented, and its integration from the time the local pressure reached the saturation pressure

to the inception time when the depressurization stopped yielded a total density

of nuclei per unit area which was equated to a critical packing density when an entire surface is covered by bubble nuclei This derivation provided an expres-sion for the pressure undershoot as a function of local properties, the hetero-

was assumed to depend on two independent variables, the reduced initial perature and the depressurization rate, and the functional dependance on each was evaluated by using published data for rapid depressurization blowdown ex-

The final correlation proposed by Alamgir and Lienhard, 1979 for the sure undershoot is as follows:

the depressurization rate in Mbar/s

The ranges of the correlation were

while assuming that the mass flow rate (or depressurization rate) plays only a

depen-dance on depressurization rate and lacks a direct comparison with experimental data, and since its adequacy was checked with comparisons of critical pressures