SUBCOOLED AND LOW QUALITY FILM BOILING OF WATER ...

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NUREG/CR-2461 NUREG/CR-2461

ANL-81-78 ANL-81-78

SUBCOOLED AND LOW QUALITY FILM BOILING OF WATER

IN VERTICAL FLOW AT ATMOSPHERIC PRESSURE

by

K.K. Fung

t'h>ARGONNE NATIONAL LABORATORY, ARGONNE, ILLINOIS

Prepared for the Office of Nuclear Regulatory Research

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sagiy4;jaoroaro nont DOE 40-550-75CR-24b1 R ppH

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NUREG/CR-2461ANL-81 .78

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ARGONNE NATIONAL LABORATORY.9700 South Cass AvenueArgonne Illinois 60439

SUBC00 LED AND LOW QUALITY FILM BOILING 0F WATERIN VERTICAL FLOW AT ATMOSPHERIC PRESSURE

by

K. K. Fung *

Reactor. Analysis and Safety Division1

Based on a thesis submitted aspartial fulfillment of the requirementsfor the degree of Doctor of Philosophy

at the University of Ottawa

i

August 1981

,

Sponsored by theDivision of Reactor Sr'aty Research

,

U. S. Nuclear Regulatory Commission -

Washington, D. C. 20555Under Interagency Agreement DOE 40-550-75

NRC FIN No. A2014 -

* University of Ottawa, Quebec.

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SUBC00 LED AND LOW QUALITY FILM BOILING OF WATERIN VERTICAL FLOW AT ATMOSPHERIC PRESSURE

by

K. V.. Fung

ABSTRACT

Subcooled and icw quality film boiling is usually encountered

in safety analyses of nuclear reactors. In most of the previous

subcooled film boiling studies, cryogenic fluids were used either in a

stagnant pool or a forced convective set-up. These data cannot be

applied to reactor safety analysis without excessive conservatism or

skepticism.

In this study, a unique method is used to establish flow film

boiling of water in a vertical tube at atmospheric pressure. The data

cover a mass flux range of 50-500 kg.m-2 -1,3 and an inlet subcooling

range of 5-70 C. It is found that the heat transfer coefficient depends

on the mass flux, inlet subcooling and the axial distance from the

point where film boiling first starts.

A physical model is developed to predict the wall temperature

of a tube during inverted annular film boiling. It considers the thennal

boundary layers in the subcooled liquid core and in the superheated vapour

film. The predicted wall temperatures and void fractions compare well

with the measurements.

NRC FIN # Title

A2014 Heat Transfer Coordination for LOCA Research Programs

11

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TABLE OF CONTENTS

P89%8

'NOMENCLATURE xii

!.0 INTRODUCTION 1,

II.0 TWO-PHASE FLOW AND HEAT TRANSFER 3

II.1 B' oiling Curve in Pool Boiling Systems 3II.2 Forced Convection Boiling 5

III.0 LITERATURE REVIEW 17

III.1 General 17III.2 Review of Experimental Studies 17III.3 Breakup of Liquid Core in Inverted 24

Annular FlowIII.4 Review of Theoretical Analyses 26III.5 Correlations for Low Quality and 35

*

Subcooled Film Boiling

IV.0 EXPERIMENTAL DESIGN 37

IV.1 Background 37IV.2 Test Section 39IV.3 Flow Loop 42IV.4 Experimental procedure 44IV.5 Data Acquisition 45

i IV.6 Measurement of Void Fraction 47'

V.0 DATA REDUCTION 50

| V.1 General 50V.2 Tabulation of Heat Transfer Data 50

! V.3 Void Fraction Data 51

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--a-m'---wit 17-'+ C - ee w-w- = w * -- - e'mi' T- --''t*-- ' r -"'v'u -

- _ - . - __ -

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VI.0 DISCUSSION OF RESULTS 52

VI.1 Dryout Location Inside the Hot Patch 52VI.2 Effect of Hot Patch Temperature 57VI.3 Reproducibility of Results 59VI.4 Temperature Profile Along Film Boiling 65

SectionVI.5 Effect of Axial Location 68

' VI.6 Effect of Inlet Subcooling 71VI.7 Effect of Mass Flux 73VI.8 Effect of Quality on the Convection 75,

Heat Transfer ComponentVI.9 Minimum Film Boiling Temperature 83VI.10 Void Fraction Measurements 84

VII.0 THYSICAL MODEL OF INVERTED ANNULAR FILM B0ILING 95

VII.1 Appruach and Assumptions 95VII.2 Momentum Equations 98VII.3 Energy Equations 104VII.4 Constitutive Equations 108VII.5 Solution of the Momenttsn and Energy 111

EquationsVII.6 Prediction of Wall Temperature 117

VIII.0 DISCUSSION OF THEORETICAL MODEL 119

VIII.1 General 119VIII.2 Effect of Mass Flux 119VIII.3 Effect of Local Equilibrium Quality 120VIII 4 Effect of Inlet Subcooling 126VIII.5 Actual Quality in Subcooled Film Boiling 133VIII.6 Slip Velocity 137VIII.7 Effect of System Pressure 141VIII.8 Effect of Tube Size 141VIII.9 Sensitivity of Model 144

IX.0 COMPARISON WITH EXPERIMENTAL RESULTS 152;

I IX.1 General 152. IX.2 Prediction of Wall Temperature 153'

IX.3 Comparison with Other Correlations 168IX.4 Prediction of Void Fraction 170

4

X.0 CONCLUSION AND RECOMMENDATIONS 176

X.1 Summary of Contributions 176X.2 Recommendation for Future Work 177

iv

_ - . _.

_ - _ _ _ . . . _ _ _ _ _ _ . -__ -. ,. - . - - - -

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Table 1 Low Quality Film Boiling Experiments 178

Table 2 Low Quality and Subcooled Film Boiling Correlations 179

! Table 3 Dimensions of Test Sections 180

Table 4 Range of Test Conditions 181

Table 5 Minimum Film Boiling Temperature 182

1 Appendix A Gamma Densitometer 183'

Appendix B Correladons Used in Data Reduction 194

Appendix C Data Tabulation 21 2

Appendix 0 Boiling Curves 236,

Appendix E Void Fraction Data 258

Appendix F CHF Correlations Used to Compare Hot Patch Power 264

Appendix G Void Fraction Correlation 266

Appendix H Radiation Coeponent 269

Appendix I Solution of the Mathematical Model. for InvertedAnnular Film Boiling 277

Appendix J Sample Calculations 296,

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! References 299i

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LIST OF FIGURES

Figure Page,

1 Pool Boiling Curve 4

2 Flow Regimes in a Directly Heated Tube 6

3 Schematic Flow Regime Map 9

4 Forced Convection Boiling Curve in a Directly Heated 10Tube

5 Effect of Quality on Boiling Curve 12

Temperature Distrigution Along 0.408" Tube for ({}6 14= 14,500 Btu /hr-ft From Dougall and Rohsenow(1963)

7 Schematic Representation of the Effect of Quality on 15Forced Convective Boiling Curve

8 Range of Available Film Boiling Data 18

9 Spreading of Dry Patch Downstream of Hot Patch from 38Groeneveld (1974)

10 Schematic of Test Section 40

11 Dimensions of Hot Patch 41

12 Boiling Flow Loop 43

13 Typical Trace of TS Thermocouples during Test 46

14 Schematic of Test Section and Y-Densitometer 48

15 Block Diagram of Y-Densitometer 49

16 Prediction of Hot Patch Heat Flux using CHF Correlations 54

17 The Effect of (Inlet) Subcooling on CHF From Fung 55(1976)

18 Heat Flux at the Hot Patch for Test Sections C&E 56

19 Comparison of Data at Two Different Block Temperaturep. 58Nominal Block Power = 500W, Mass Flux = 100 kg.m-2.s-'Inlet Subcooling = 100C

vi

20 Heat Flux at the Hot Patch for Test Sections A&C 60

21Mass Flux = 50 kg.m-gion during Film Boiling.s,I, Inlet Subcooling = 700CTemperature Distribu 61

22 Temperature Distribution during Film Boiling 62Mass Flux = 100 kg.m-2.s-1, Inlet Subcooling = 700C

Temperature Distribution puring Film Boiling 6323Mass Flux = 150 kg.m-2,3 , Inlet Subcooling = 200C

24 Temperature Distribution 64Mass Flux = 200 kg.m-2,s puring Film Boiling, Inlet Subcooling = 100C

25 Temperature Distribution during F 66Nominal Mass Flux = 200 kg.m-2.s,jlm Boiling

26 Temperature Distribution during Film Boiling 67Nominal Inlet Subcooling = 200C

27 Typical Results at Two Mass Fluxes 69

28 Variation of Heat Transfer Coefficient along Heated 70Length

29 Effect of Inlet Subcooling 72

30 Effect of Mass Flux at Inlet Subcooling of 200C 74

31Nusselt Number ag s Function of Equilibrium Quality

a 76,

for G = 50 kg.m .i

32 Nusselt Number as a Fynction of Equilibrium Quality 77for G = 100 kg.m-2.s-'

33 Nusselt Number as a Function of Equilibrium Quality 78,

for G = 150 kg.m-2.s-l

34 Nusselt Number as a Function of Equilibrium Quality 79for G = 200 kg.m-2.s-l

35 Nusselt Number as a Fynction of Equilibrium Quality 80for G = 300 kg.m-2.s-



38 Minimum Film Boiling Temperature 85

39 Axial Variation of Void Fraction, 87Mass Flux = 500 kg.m-2.s-l

vii

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40 Axial Variation of Vojd Fraction 88Mass Flux = 300 kg.m .s-l

41 Void Fraction as a Fu 89Mass Flux = 500 kg.m getgn of Equilibrium Quality.s

42 Void Fraction as a Funct1 90Mass Flux = 300 kg.m-2.s 9n of Equilibrium Quality8

43 Void Fraction as a Function of Equilibrium Quality 91

44 Effect of Mass Flux on Void Fraction 92

45 Comparison of Measured Void with Predictions 94

46 Development of Vapor Film 96

47 Laminar Vapor Film 99

49 Energy Balance for Vapor Control Volume 105,

50 Predicted Wall Temperature 1 21

51 Definition of Thennal Laminar Sublayer 122

52 Heat Transfer Coefficient as a Function of 123Equilibrium Quality

53 Convective Component of Heat Transfer Coefficient 124'

54 Predicted Vapor Film Velocity 125

55 Effect of Equilibrium Quality on the Heat Transfer 127Coefficient

!

i 56 Effect of Equilibrium Quality on the Heat Transfer 128i Coefficient

57 Effect of Inlet Subcooling on the Heat Transfer 129Coefficient

58 Development of the Temperature Profile in the Liquid 130Core

59 Fraction of the Total Heat Flux Transferred to the 131Liquid Core

60 Predicted Actual Quality at G = 200 kg.m-2 -I.s 134

61 Predicted Actual Quality at G = 400 kg.m-2 -I.s 135

62 Percentage of the Total Heat Flux Transferred to the 136Liquid Core

viii

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_ _ _ _ . - . -,

_ _ - - - _ _ _ - .- ..

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63 Predicted Liquid Core Velocity 138

64 Predicted Vapor Film Thickness 139

65 Predicted Slip 140

66 The Effect of Pressure on the Heat Transfer 142Coefficient

67 Effect of Tube Of ameter on the Film Boiling 143Heat Transfer Coefficient

68 Effect of the Transitional Point Between Laminar 145and Turbulent Vapor Film on the Predicted Wall-

Temperatures

'.69 Effect of Increase in Interfacial Friction on the 146

Predicted Wall Temperature

70 Effect of Increase in Turbulent Thermal Diffusivity 148of the Liquid Core on the Predicted Wall Temperatures

71 Effect of Different Assumptions of the Average Vapor 150Temperature on the Predicted Wall Temperature

72 Predicted Void Fraction with Different Assumed Vapor 151Temperatures

73 Comparison of Predicted and Measured Wall Temperatures, 154| G = 100 kg.m-2.s-l. T = 600Cin

74 Comparison of Pwdicted and Measured Wall Temperatures, 155G = 100 kg.m-2.s I, T = 800Cin

75 Comparison of Preficted and Measured Wall Temperatures, 156G = 150 kg.m .s ,T = 700Cin

.

76 Comparison of Pr 157G = 200 kg.m-2.sgicted and Measured Wall Temperatures,,T =9 Cin77 Comparison of Predicted and Measured Wall Temperatures, 158

G = 200 kg.m-2.s-l, T = 800Cin78 Comparison of Prepicted and Measured Wall Temperatures, 159

G = 250 kg.m .s , T = 700Cin79 Comparison of Pre 160G = 300 kg.m-2.s picted and Measured Wall Temperatures,,T = 900Cin80 Comparison of Predicted and Measured Wall Temperatures, 161

G = 300 kg.m-2.s , T = 80.5%,

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in

ix

_ , . . - _ _ _ - - .

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81 Comparison of Pre 162G=400kg.m-2.spictedandMeasuredWallTemperatures,,T = 800Cin

82 Comparison of Predicted and Measured Wall Temperatures, 163G = 500 kg.m .s-1, T = 950Cin

83 Comparison of Pre 164G = 500 kg.m-2.s ficted and Measured Wall Temperatures,, Tin = 800C

84 Comparison of Pre 165G = 200 kg.m .s picted and Measured Wall Temperatures,

, Tin = 900C

85 Comparison of Pre 166G = 396 kg.m .s picted and Measured Wall Temperatures,,T = 840Cin

86 Comparison of Pre 167G = 495 kg.m-2.s picted and Measured Wall Temperatures,,T = 900Cin

87 Comparison of Predicted and Measured Void Fraction, 171G = 100 kg.m-2.s-I

88G=200kg.m-2,spictedandMeasuredVoidFraction,Comparison of Pre 172

i 89 Comparison of Predicted and Measured Void Fraction, 173'

G = 300 kg.m-2.s-l

90 Comparison of Predicted and Measured Void Fraction, 174G = 400 kg.m-2.s-l

91G = 500 kg.m-2.s picted and Measured Void Fraction,Comparison of Pre 175

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NOMENCLATURE

xi

_ _ _

N0fENCLATURE

NOTE: The less frequently referenced symbols are defined when they

appear in the text

A Area Re Reynolds nun 6er

Cp Specific heat t Time; thickness

D Tube diameter T Temperature

De Hydraulic equivalent diameter, tube diameter U,u Veloci*y

E Energy flux V Velocity

f Friction factor y Velocity normal to solid surfacet

F View factor X Quality

3 Acceleration due to gravity y Distance perpendicular to well

G Mass flux Z Axial coordinate

h Enthalpy, heat transfer coefficient

i Enthalpy

I Count rate

J Radfasity

k Thermal conductivity

L Length

m Wave number

M Momenttan flux ,

Nu Nusselt number (h D/k)

P Pressure

Pr Prandt1 number (u Cp/k)

q Heat transfer rate

r Radial distance

R Radius

xii

__= _ _ -- - _ - _ - - ___ __

. _ _ _ _ _ _ _ _ _ _

J

NOMENCL

_ _ _ _

GREEK Subscripts

cm Void fraction, thennal diffusivity a Actual value

r Evaporation mass flux c Core, convective, condensation

6 Film thickness CHF Critical heat fluxc Eddy diffusivity, emissivity Cond Conduction

n Dimensionless distance conv Convection

e Angle, dimensionless temperature e Equilibrium value, evaporation

u Dynamic viscosity E Empty

v Kinematic viscosity f Saturated liquid

e Surface heat flux F Full(

p Density fg Difference between saturated vapor and saturated liquid value

a Surface tension g Saturated vapor

o$g Stefan-Bohn constant H Heat transfer

t Shear stress i Initial, interface, inner

r, Transmissivity in InletAH Local subcooling, h -h

3 1 Liquida[n Evaporation mass flux

o Outer

ai Local subcooling. T -T3 rad Radiation

s Saturation

sat Saturation

T Turbulent

v Vapour

w Wall

xiv

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!i -1-j|| 1.0 INTRODUCTION

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During the normal operation of water-cooled reactors, the fuel

sheath temperature is maintained at near saturation temperature. How-|'

ever, during a postulated loss-of-coolant accident (LOCA), the fuel

cladding may experience a rapid increase in temperature as a result of

a change in heat transfer mode from nucleate boiling to the less effi-d

cient film boiling. The subsequent injection of emergency core coolant

will result in a return to moderate sheath temperatures by changing

the heat transfer mode from film boiling via transition boiling back tor

nucleate boiling and forced convection. It is during this phase that

film boiling in an inverted annular flow regime is most likely to be

encountered.,

'

In most of the previous subcooled film boiling studies, cryo-

genic fluids were used (see review reports by Groeneveld & Gardiner

(1977) and Fung (1978)). One of the reasons why water has not been'

used more frequently in the investigations is the high heater wall

temperature required to maintain film boiling at low qualities in

water.

Previously derived correlations are based on data obtained in

cryogenic fluids and therefore lack the necessary data base to

substantiate their applicability to water reactor safety analysis. At|

present, one of the correlations recomended to be used in this region ,

is that by Hsu (1977), which is a modified Bromley equation for pool

film boiling.

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-. . .- _-. --- _ _ - . . - - __ -. . --

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In this investigation, a unique technique is used to establish

steady-state film boiling for water during forced convection inside a'

directly heated tube. It permits the measurement of film boiling at,

I

low qualities in which the flow pattern is inverted annular. This has

not been possible in the previously reported film boiling studies using

a directly heated channel.

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II.0 TWO-PHASE FLOW AND HEAT TRANSFER

II.1 Boiling Curve in Pool Boiling Systems

When a heated surface is maintained at a temperature above the

saturation temperature of the liquid close to it, a change in phase

from liquid to vapour may occur. If we plot the heat flux and the cor-

responding surface temperature (or mcre frequently the wall superheat,

T -T ), we obtain what is referred to as the boiling curve. Aw s

typical boiling curve in a pool boiling system is shown in Fig.1.

If the surface is heat flux controlled, e.g., by resistance

heating, the boiling curve follows a path similar to A B C D F G on in-

creasing heat flux and G F E C B A on decreasing heat flux. Initially,

up to point B, heat transfer is by forced convection. At point B, the

surface attains the minimum superheat for nucleation, and vapour bub-

bles are formed on the surface. The region from B to D is called

nucleate boiling. One of the main characteristics of this mode of

boiling which distinguishes it from the others is the continuous

contact between the liquid and the heated surface.

At point D, the maximum limit of nucleate boiling is reached.

The heat flux at this point is called critical heat flux (CHF). Var-

ious other names, such as dryout, burnout, first boiling crisis and

departure from nucleate boiling (DNB) are also used.

If the heat flux is increased beyond the CHF, the curve will

follow path DF. The transient is usually very fast so that no stable

operation can be maintained along 0F. Point F lies on the film boiling

.

- -

- _ . _. _ -.-.

-5-

region of the boiling curve.

At heat flux above F, the film boiling curve follows path FG.

On decreasing the heat flux, the curve traces out GFE instead of

returning to 0. This hysteresis is typical of pool film boiling. It

is due to the fact that once a vapour tilm is established, it does not

collapse suddenly but instead decreases in thickness as the heat flux

is lowered.

The minimum wall temperature required to maintain stable film

boiling is called the Leidenfrost temperature. It corresponds to

point E in Fig. 1. On further reduction in heat flux, the surface

temperature will rapidly drop to that corresponding to point C, which

is in the nucleate boiling region.

The above discussion applies to a heat flux controlled system.

In a temperature controlled system, such as one in which heat is

supplied by a condensing vapour, the continuous curve shown in Fig.1

is obtained. The region DE, which is not attainable in a heat flux

controlled system, is called transition or partial film boiling. In

this boiling mode, liquid-surface contact is intermittent.

II.2 Forced Convection Boiling

II.2.1 Flow Regimes

Forced convection boiling is commonly investigated in electri-

cally heated channels. The mass quality increases along the heated

channel, with corresponding changes in the flow regime. Fig. 2 shows

|

,

, _ _ ~ - __

_ _ _

-6-

Y W

SINGLE PHASE FLOW SINGLE PHASE" VAPOURo ,"*

*-,

,'.

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DISPERSED FLOW DISPERSED FLOW*,.

*.'..,

, ' ..,**o*'

, ,

,t " , , .

,'. . I o .*** "

.a ogANNULAR FLOW "*

** h,.f INVERTED ANNULAR9.

FLOW' SLUG OR FROTH FLOW

p;3 qhg - -

BUBBLY FLOW BUBBLY FLOW

,Jj i(: u

': nL

SINGLE PHASE LIQUID : SINGLE PHASE LIQUID

% %

(a) Low heat flux (b) High heat flux

FIGURE 2 FLOW REGIMES IN A DIRECTLY HEATED TUBE

. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

-

-7-

two possible configurations of the flow structures inside a heated

channel. Fig. 2(a) is most comonly encountered under low heat flux

conditions while Fig. 2(b) is typical of high heat flux conditions.

The flow regime and heat transfer modes along the channel

length can be divided into the following regions:

(a) Single-phase forced convection

(b) Bubbly flow

Here the heated surface attains the minimum superheat for

nucleation. Bubbles are formed and remain in the wall region, since

the bulk of the liquid is still subcooled.

Further downstream, the liquid becomes saturated. In this

fully developed bubbly flow bubbles exist in the core region and

coalesce to form larger bubbles.

(c) Slug flow or froth flow

Here nucleate boiling is still the dominating mode of heat

transfer. However, the vapour volume is comparable to the liquid

volume. At low mass flux, the flow is intermittent, with vapour plugs

followed by liquid plugs. At high velocity, the flow structure tends

to be the more homogeneously mixed froth flow.

(d) Annular flow

The vapour now occupies the central part of the channel, with

liquid droplets entrained in it. A liquid film exists around the

heated surface. Often, nucleation is suppressed and heat transfer is

-8-

by forced convection across the liquid film and then by evaporation at

the vapour-liquid interface.

(e) Dispersed flow

This is the post-CHF region. Heat transfer is by forced

convection to vapour, evaporation at the droplet surface and droplet-

thermal boundary layer interaction.

(f) Inverted annular flow

Fig. 2(b) represents the situation in which the heat flux is

sufficiently high that boiling crisis is encountered by bubble clouding

or subcooled dryout (Tong (1972)). Here the post-CHF flow regime is

inverted annular, with a vapour film separating the liquid core and the

heated surface. It is usually encountered at void fraction below 30%.

The transition between this flow regime and the previously discussed

annular flow can be represented schematically by a flow regime map, as

shown in Fi g. 3.

11.2.2 Forced Convection Boiling Curves

A typical boiling curve for forced flow inside a directly

heated tube is shown schematically in Fig. 4. The same curve will be

traced out on increasing or decreasing heat flux. This is different

from the hysteresis behaviour in the film boiling region in the case of

pool boiling. The reason why the heat flux does not fall below the CHF

along the film boiling section of the boiling curve is that a rewetting

front propagates along the channel as the heat flux is reduced.

I_ ____ __

-9_

ji

Wen

5 '

dn

$ LIQUIDSLUG DISPERSED*

INVERTEDANNULAR

I I ,

VDID FRACTION a

FIGURE 3 SCHEMATIC FLOW REGIME MAP

.

- - - - - - - - _ _

- 10 -

Ee

D+

W

INCREASING CRITICAL

CHF QUALITY

sa

O& k

5LaI B

=

N55=

WALL SUPERHEAT

FIGURE 4

FORCED CONVECTION BOILING CURVE IN A DIRECTLY HEATED TUBE

. . - - _ _ _ _ _ _ _ _ _ - _ _ _ _ _

. _ _ - _ .___ - -__- -____ - _ __.

- 11 -

,

However, if the system is temperature controlled, the film boil-

ing heat flux can fall below the CHF, as in the case of pool boiling.

A heater with a high thermal capacity, such as a nuclear fuel bundle,

will resemble a temperature controlled system. In most of the previous

film boiling studies in directly heated channels, this part of the film

boiling curve could not be measured. In this study, a unique technique

is used to enable the measurement of this portion of film boiling in a

directly heated tube. Instead of approaching film boiling from CHF

(which will result in excessive wall temperature), it is established by,

;.

,

using a hot patch to stop the propagation of a rewetting front.

It trust be mentioned that a temperature controlled system is

not generally used because of its relatively high cost and technical

difficulties. With water as the test fluid, the ch'oice of the second-

ary heating liquid is very limited. Only liquid metal will possess the

high heat transfer coefficient required in such applications.

Ij II.2.3 Effect of Quality on the Boiling Curvei

The forced convection boiling curve depends, among other fac-

tors, on the mass flux and the quality. Consider a boiling curvei

plotted with quality as a parameter. In Fig. 5, a set of curves of

different quality in the film boiling region is shown. The film boil-

ing curve has also been extended to below the CHF, since this is the

region considered in this study.

i

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- -,_m_ _.. __ , , . . , _ . - - _ _ . . - . . _ _ _-

_ _ _ _ _ _ - . . .- .-_

.

- 12 -

CONSTANT QUALITY

_ C' E' _ D'5d s'/ /

s' / /r CONSTANT HEAT FLUX - '

OI A B[EC D

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Z

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o6

Se

0 0

00 De

f }

h9 /Ct FIGURE S EFFECT OF QUALITY

"OH BOILING CURVE

A T,

FLOW

_

- 13 -

Consider a point near the beginning of the heated length where

inverted annular film boiling has been established. This point is

shown as point C on Fig. 5. The wall temperature has been observed to

increase initially and then decrease towards the downstream end.

Fig. 6 shows an axial temperature variation 'taken from the report of

Dougall and Rohsenow (1963). The same form of temperature distribution

has been observed in the present study.

Therefore, point D in Fig. 5 will lie to the right of point C- and point E to the left of point D. At higher heat fluxes, all points(

will be shifted upwards to C', D' and E'. The dotted lines such as CC'

will therefore represent a segment of the boiling curve for a fixed

I point on the heated channel.

Since point E is downstream of point D, the quality is higher at

E. Therefore, the constant quality line through E represents a higher

quality than that through D. In addition, since the temperature at E

is lower than that at D at the same heat flux, the heat transfer coef-

ficient is also higher at E. This improvement in heat transfer is

| due to the increase in vapour velocity in the dispersed flow regime.

The effect of quality on the boiling curve has been studied by

Groeneveld and Fung (1976). The parameter trend is shown schematically

in Fig. 7. The lines corresponding to low quality and subcooled film

boiling, where inverted annular flow prevails, have been included in

this figure. Note the opposite trend of quality in the two different

_

600| | | | | | 1

''[__---0 ~~

D *s-

o+0500 - --

/ -+ ' Ng,+

of/$ '

400,- ' -

cL '(

# 300 DATA FOR FREON' - fj3 INSIDE A 0.408" 1.D. TUBE- -

I .

(f)7 = 14,500 Btu /hr-f t G = 4.92 x 10 lbm/nr-f t2 5 2o RUN 2 j,

%200 - -

( )T2 5 2= 14,400 Btu /hr-ft G = 4.92 x 10 lbm/ hr-f 1+ RUN 3 ,

,

SOLID LINE - THEORY ,

100 BROKEN LINE - EXPERIMENTc --

,

I l | | | | 1

0 5 10 15 20 25 30 35

i {:'

'

0.408" I.D. TUBE FOR ( = 14,500FIGURE 6 TEMPERATURE DISTRIBUTION. ALONG .

Bt u/ hr - fI 2 FROM DOUGALL AND ROHSENOA (1963)

. z.%.

%

% ' '

4

. _ - _ _ - _ _ . - _ _ - _. _ - _______ ___ __ _ _

.- - - __ . _ _ - - - _

r

'

- 15 -

1

i,

i

/Q |

'

% !

s'& t

D'h

x% 't

x //| //

\

f / )

,Q .D4

~I 'W SAT j

FIGURE 7

SCHEMATIC REPRESENTATION OF THE EFFECT OF QUAllTY ON

FORCED CONVECTIVE BOILING CURVE|

|<

$

- .. - - -.. -_ - . . . . . ... - . - - - - . _ - . - . - . - . _ _ .

!

:

!

- 16 -,

.

d

|

) flow regimes. The changeover point would be a suitable criterion fori'

demarcation of inverted annular and dispersed flow. Obviously, the'

i quality at this changeover point will depend on the mass flux.

!i<

I1

i

i

I

i

|

!i

!

i

f1

|r

;

!

!

.i

!i

-

: :

i

k

Ii

__ _____ _ . _ _ _ . . ._. _ - _ _ _ . - _ _ -__ _ _ _ _ - , - - - -

|

- 17 - '

III.0 LITERATURE REVIEW

III.1 General

Post-CHF heat transfer has been the subject of several review

reports (Collier (1975), Groeneveld & Gardiner (1977), Fung (1978) and

Mayinger & Langner (1978)). In the present study, we are primarily

concerned with convective film coiling of water in low quality and sub-

cooled conditions. Therefore, pool boiling studies will only be men-

tioned briefly if they are pertinent.

The experimental conditions covered by the available film boil-

ing data have been examined by Groeneveld & Gardiner (1977) and the

result is shown schematically in Fig. 8. Most of the previous investi-

gations have concentrated on the high pressure, high mass flux and high

quality conditions, which are relevant to boiler tube design. The

areas marked " Data available but questionable" are relevant to water

i reactor accident conditions and interest in it has been enhanced in the

last ten years. The present study is part of an integral research

program aimed at a better understanding of the post-CHF heat transfer

under reactor accident conditions.

III.2 Review of Experimental Studies

III.2.1 Experimental Work

As mentioned earlier, cost of the previous studies of low

quality film boiling have been carried out with cryogenics and

refrigerants, mainly because of the relatively low wall temperatures

1

.

_ _ _ _ _ . _ _ _ _ _ _

_

- 18 -

5-

AVAILABLE7 4- DATA"

NO -

h 3-OATA

x

c5 2-

DATA AVAILABLE \lD NO

BUT GUESTIONABLE DATA

I i 0 ^ l I I I\.17\ 0. 2 0.4 0.6 0.8-0,2 -0.1 '

SUBC00 LING OUALITY \"x OATA

uNO 5

OATA ya 10- AVAILABLE

DATA

ANFIGURE 8 RANGE OF AVAILABLE FILH BOILING DATA

i

- 19 -

required. Recently, because of the interest in LOCA heat transfer,

some of these studies have also been carried out in water. Table 1,

taken from Groeneveld & Gardiner (1977), sumarizes the studies

reported in the literature.

The last four entries in Table 1 represent a new approach to

obtaining post-CHF heat transfer data in the past five years. It was

mentioned earlier that the post-CHF region below the CHF is unattain-

able for water in a directly heated thin-wall tube due to the propaga-

tion of the rewetting front. To circumvent this, these authors used

thick copper cylinders in which the flow was passed through a central

bore. Heat was either supplied by cartridge heaters embedded in the

copper block (Fung, Cheng & Ng, Smith) or by wound-on heaters

(Newbold). Due to the high thermal conductivity and thermal capacity

of the copper cylinder, this type of test section approaches the situa-

tion of a temperature controlled system.

The experiments of Fung and Cheng & Ng were primarily designed

to obtain transition boiling data. The limitation of this type of test

section in obtaining film boiling data is the melting point of copper.

Practically, it is limited to below 750*C. Therefore, the obtainable

temperature range in film boiling is relatively narrow. The

experiments of Newbold covered the temperature range below 700*C.

- _

- 20 -.

The extension to this technique is to incorporate the copper

block in a tube. . Smith (1976) adopted this method to obtain data in

the inverted annular and dispersed flow film boiling regimes, but he

was unable to maintain film boiling in his test section using a single

block. Instead, he had several copper blocks clamped to the tube.

Each of the blocks is independently heated by cartridge heaters and

acted as heat source of the tube. Most of the data were obtained in

the dispersed flow regime.

III.2.2 Parametric Trends Observed

The overall film boiling heat flux is affected by the mass

flux, system pressure, liquid subcooling and some other secondary

factors. The parametric trends observed in experimental studies are

discussed below:

(a) Effect of mass flux

Bromley (1953) studied film boiling of liquid in upward flow

over horizontal tubes. He observed that an increase in mass flux

increased the heat transfer coefficient. Kalinin (1969) observed

a similar trend in cryogenics. On the other hand, Dougall &

Rohsenow (1963), Ellion (1954) and Rankin (1961) found no signifi-

! cant effect of mass flux on the film boiling heat transfer coeffi-;

cient measured in vertical channels. It is possible that, beyond a

certain mass flux, the vapour velocity is more dependent on the

!

_ .____________ _ _ _ _ - _ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ _ - _ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

,

- 21 -

.

vapour content of the flow than on the velocity of the flow prior

to vapour generation.

At low flow, situations are similar to pool boiling and buoy-

ancy force becomes important. For upward flow over horizontal

tubes, Bromley (1953) used the Froude number as a criterion for

transition between free and convective film boiling. The

criterion is as follows:

U/gD < 1.0 free convection

U/gD > 2.0 forced convection

No similar criterion has been proposed for flow inside tube.

Newbold et al. (1975) studied the effect of flow direction on

film boiling in internal flow in a short test section using a

transient quenching technique. They found that the heat

flux increased with increasing mass velocity, although for upflow

2the increase is small above 400 kg/m s. The higher rate of in-

crease near the low flow region is probably due to the relatively

higher increase of the inertial . force over the buoyancy force.

For conditions of higher inlet subcooling and pressure, they

observed no significant difference between upflow and downflow.

However, with low subcooling the heat transfer in downward flow is

significantly lower than for upward flow. They attributed this to

the buoyancy force acting in the opposite direction to the inter-'

facial shear, thus increasing the vapour fiim thickness and

reducing the rate of heat transfer.

-. . . - . _-

- 22 -

(b) Effect of subcooling

The effect of subcooling on the film boiling heat transfer

coefficient has been investigated experimentally and analytically.

Increased heat transfer with increasing subcooling was observed in

the pool boiling experiments of Tachibana & Fukui (1960) and Hein

(1975), in quenching experiments with spheres by Farahat (1977)

and Dhir (1978), and in transient cooling of short cylindrical

blocks under internal forced flow conditions by Fung (1976), Cheng

(1976) and Lauer & Hufschmidt (1976). Ilnder steady-state condi-

tions, inverted annular film boiling has only been observed in

cryogenics and refrigerants. Kalinin's data (1969) for downward

flow of nitrogen in a vertical tube and Bromley's data (1953) for;

vertical cross flow over a horizontal tube showed enhancement of

heat transfer with increased subcooling.

Theoretical analyses of forced convective film boiling on

flat surfaces were carried out by Cess & Sparrow (1961) and

Sparrow & Cess (1962), using a laminar boundary layer approach.i

They found that the surface heat flux increased with subcooling,

and the rate of increase was higher at a lower temperature level.

(c) Effect of pressure

No systematic experimental studies on the effect of pressure

have been carried out. Qualitatively, pressure affects the film

boiling heat transfer primarily through the changes in fluid

- 23 -

properties. In his theoretical analysis of laminar film boiling,

Ellion (1954) derived the following expression for the film

thickness

I kp I4 y y

6 = 12 AT Z(hfg g yo p s

The conductivity and viscosity increase with pressure, where-

as the latent heat and liquid density decrease with pressure.

Together, they tend to increase the film thickness. However, the

most significant change is in the vapour density. For example,

for ATs = 5000C and Z = 0.1 m, the calculated film thickness at

0.1 MPa and 10 MPa are 1.6 and 0.42 m respectively. A thinner

film will have a stabilizing effect on film boiling, since the

onset of turbulence will be delayed.

With a thinner vapour film, the heat transfer coefficient

will be increased. Therefore, at the same wall superheat, the

total heat flux will be higher at higher pressure.

(d) Effect of surface property

In film boiling there is no direct liquid-surface contact.%

Therefore surface roughness and oxide layers are unlikely to have

an effect on film boiling if the roughness height is less than

the laminar sublayer thickness. If the roughness peaks penetrate

|'

. - _ _ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ _ _ _ _ _ - _ _ - _ - -

i

l

- 24 -

through the sublayer the film boiling heat transfer coefficient

will be increased due to the existence of eddies near the~

asperities.

III.3 Breakup of Liquid Core in Inverted Annular Flow

In experiments carried out inside heated tubes, the mass quality

increases along the flow direction. Therefore, starting from ani

| inverted annular flow near the inlet, the liquid core will break up as

quality increases until eventually a dispersed flow is formed. Tong

(1965) considers the inverted annular flou as a liquid jet discharging

into a moving vapour stream. The stability criterion under such condi-

tions is

m.c + > U -Uy c

,

where m is the wave number. Rankin (1965) iy;c:luded in his analy. *s a

vapour thrust force which exerted on the vapour-liquid interface during'

I dry wall collision and derived the following stability criterion

'

2 2 -- -

am 2k AT p r 12+ 1+ mt > U -U

3 y y cO *P Pc c V g y- -

P 'v --

i

|

I

.

.-

,,

- 25 -

where AT = T, - Tsat

~ -2Cp AT

h)g=hfg h1 + 0.4

g

f Cp AT Zt =y Y33Pr hyg

f = friction factor

Z = distance from dryout location

Both stability criteria depend on the wave number m, which is

unknown.

j The breakup of the liquid core will result in a slug flow ini-

tially. This will increase the vapour-liquid interfacial area and

therefore will increase the local heat transfer. Such a situation may

lead to rewetting of the surface. If no rewetting takes place, the

flow will gradually change to a dispersed flow. If there is a rewet-

ting front, the heated surface will gradually be quenched.

Dougall & Rohsenow (1963) used a different correlation for the

heat transfer coefficient in dispersed flow and in inverted annular

flow. They suggested that as the vapour film thickened due to increase

in quality, the heat transfer coefficient, and hence the Nusselt

number, decreased accordingly. The minimum Nusselt number then

, _ . - . _ . _ _

_ _ _ _ ________________ _____ _ _________

- 26 -

corresponds to the transition between inverted annular and dispersed

flow. Smith (1976) used the same criterion to compare his data.

Various authors use the void fraction as a criterion for transi-

tion from inverted annular flow to dispersed flow. For example, HughesI

(1976) uses 0.2 and Kaufman (1976) uses 0.4 as the critical void frac-.

tion. This type of criterion is convenient to use but does not des-

cribe the mechanism of the transition. In addition, other effects,

such as mass flux and pressure, also influence the transition. This is

evident from the various flow regime maps suggested (see Collier

(1972)).

III.4 Review of Theoretical Analyses

Several theoretical analyses relevant to the present investiga-

tion are discussed in this section. In all analyses, the purpose is to

predict the wall heat flux, or the temperature, depending on which

boundary conditiod is known. The most commonly adopted strategy is to

solve the boundary layer equation for the vapour and/or the liquid.o

Most of the uncertainties lie in the interface and many assumptions are

made for the interfacial shear stress and heat transfer.

111.4.1 Ellion (1953)

In his analysis, Ellion makes the following assumptions

(i) the liquid is at the saturation temperature;

(ii) the interfacial velocity is virtually zero;

- 27 -

(iii) all the heat leaving the tube is used to vaporize the liquid.

This assumption also implies that there is no superheat in

the vapour and that the rate of vapour generation is constant

along the tube;

(iv) the total pressure gradient is equal to the hydrostatic

pressure gradient.

By assumptions (i) and (iii) the interaction between the '

liquid and the vapour is known and consequently no analysis of the

liquid core is required.

For the vapour film, Ellion makes the following assumptions

(v) the vapour film flow is laminar, and

,(vi) the vapour properties are independent of temperature.

,

! With some additional assumptions which are comon to boundary

I layer analysis, the momentum equation for the vapour film is reduced toi

2du og

2+-=0dy p

subject to the boundary conditions of

u = 0 at y = 6 and y = 0

Once the velocity distribution across the film is known, the

total vapour flow can be obtained by integrating across the vapour

film. By assumption (iii), the vapour flow at any location can also be

calculated from the power. These two equations for the vapour flow can

be solved to give the vapour film thickness. The film boiling heat

_ _ _ _ _ _ _ _ _ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

- 28 -

transfer coefficient is then calculated by assuming pure conduction

across the vapour film and is found to be

3N-I P2. P kfg v y,

12 AT Z_ sat y_

It is noted that hFB does not contain any subcooling nor

mass flux effect. This is the result of assumptions (i) and (ii).

III.4.2 Rankin (1961)

Rankin considers the separate cases of laminar and turbulent

vapour film flow. The analysis of the laminar film is similar to that

of Ellion, except that the interfacial velocity and the pressure gradi-

ent (which is assumed constant) are unspecified and appeared as param-

eters in the solution.

For the turbulent film, Rankin assumes that the interfacial

velocity is zero. The vapour flow is then considered to be equivalent

to that between two parallel stationary plates. The interfacial shear

stress is taken to be the seme as the wall shear stress. By neglecting

the momentum change of the gas, a force balance gives

buoyancy force = frictional force2

or 6g(p - p ) = 2 x ifp U .g y yy.

_ _ _ _ _ _ _ _ _ - -

.

- 29 -

By neglecting the body force of the vapour the film thickness

is expressed as

2p* U

*6=f 4.2.1P 92

The momentum and heat transfer at the wall are mlated by the

Colburn 'j' factor as follows:

f h 27Pr ,d " 7 " Cp p Uy y y

By assuming that all the heat goes into vapour generation,

the change in vapour flow can be related to the vapour generation at

the interface as follows:

I 4Ip d(U 6) = dZ

(h.y y

f)9

h ATdZ=

h.fg

G AT fU" V.'. d(U 6) = dZ

2 hy Pr/3'y

g

_ _ - . - -

- 30 -4

Substituting 6 from Eq. 4.2.1 the following equation is

obtained

dU Cp g AT, pgy30

dZ " 2h)gv p Py r

Direct integration gives the vapour velocity. The final

expression for the heat transfer coefficient is

3 Z'bf 'g Cp AT,pg yp* 26 Pr hy_ g ,

III.4.3 Dougall & Rohsenow (1963)

In this analysis the liquid core is assumed to be at satura-

tion. Rather than solving the boundary layer equation for the vapour

film, it is assumed that the universal turbulent velocity distribution

holds from the wall to the point of maximum velocity in the film, and

its mirror image for the other half of the film. Close to the inter-|

face, they consider the following possible situations

(i) the turbulent layer exists up to the interface,

(ii) a buffer zone exists next to the interface, and

(iii) both the laminar sublayer and the buffer layer exist.

!

I,

- - - - - . - . - -

- 31 -

The total shear stress and heat flux are then assumed to be

given by'

I ' duV +Ef* Ey 4.3.1y m ...y i

and =- a +E 4.3.2p y H ...

It is further assumed that the Reynolds analogy is valid,

i.e., cm "CH. The eddy diffusivity is then calculated from

Eq. 4.3.1 by using the universal velocity profile. For the case of

constant heat flux, Eq. 4.3.2 is integrated to yield the temperature. A7

typical comparison between the theory and their data is shown in Fig. 6 (p.14).

III.4.4 Kalinin et al. (1977).

This is a one-dimensional flow model of subcooled film

boiling. The liquid is assumed to move in the form of a turbulent jet,

which is separated from the pipe wall by a turbulent film. Both vapour

and liquid are assumed to move at the same velocity. In addition, the

heat transfer from the wall to the vapour and from the vapour into the

liquid are assumed to follow the equations:

mNuy = A Re 4,4,1,,,

Nu =BRej 4.4.2g ...

_ _ _

- 32 -

Since the velocity distribution has been assumed uniform, the

momentum equation is not required in the analysis. Then, by combining

the continuity and energy equations, a differential equation for the

dimensionless vapour film thickness b = hisobtainedandisshown

below

h-$)(b)-E $ (b) = 0 4.4.32

...

H is a function of Nu and the axial wall temperaturey

distribution. E is a function of Nu and Nug. Qualitatively, Hy

contains the prior history of the flow and E is proportional to the

fraction of heat that goes into the liquid.

The solution of Eq. 4.4.3 requires that A and B in Eqs. 4.4.1

and 4.4.2 be specified. Once b has been found; the heat flux is

obtained from Eq. 4.4.1.,

1

The most questionable assumption is that of the equal and

uniform velocity in both liquid core and vapour film. The usual

assumption in a one-dimensional two-phase separated flow analysis is to

assume a different velocity for each phase. The assumption of equal

velocity in the two phases seems more realistic in dispersed flow than

in inverted annular flow.

|l

,

- 33 -

111.4.5 Barnes (1973)

In this analysis, similarity transformation is applied to the

governing equations. Analogous to laminar flow, a similarity var-

iable n, a stream function $ and a dimensionless energy function & are

defined. Since the analysis is applicable to turbulent flow, n and $

are functions of the eddy diffusivity. The latter is assumed to follow

an exponential relationship with n, as well as being dependent on the

fluid velocity. The general form is

e = c ,(1 - exp(-bn)) + v(exp(-bn))

where c, is the free stream eddy diffusivity and b is a shape factor.,

|

| The momentum and energy equation are then reduced to a set ofr

j simultaneous ordinary differential equations of the transformed veloc-

ity and temperature. For the solution, the entire fluid domain is

divided into three regions, namely the vapour film, the mixture layer

| and the subcooled liquid. Numerical method is used to solve the

coupled ordinary differential equations, using experimentally measured

wall temperatures to evaluate the vapour and liquid thermophysical

properties.

III.4.6 General Coments

Most of the analyses assume that the liquid is at saturation.

This enables the vapour generation rate to be calculated. By further

_ _ __ _

- _ _

- 34 -

assuming-the velocity at the vapour-liquid interface, the liquid layer

or core is decoupled from the vapour film. This seems to be an over-

simplification compared with the case of subcooled film boiling where

significant thernal non-equilibrium exists.

In most of the analyses (except Barnes'), the vapour-liquid

interface has been assumed to be smooth. In the actual situation, it

has been observed (Dougall & Rohsenow) to be wavy. The quantitative

effect of waves on the interfacial shear and evaporation rate is not

amenable to theoretical analysis at the present time, since the predic-

tion of the same quantities in the case of a smooth interface still

depends on some correlation parameters based on experimental data.

Assumptions must therefore be made concerning the momentum and energy

transfer at the interface.

In addition, in the case of a subcooled liquid core, part of

the heat transferred at the interface eventually goes into heating up

the liquid. This portion of heat transfer diminishes as the liquid jj

temperature approaches the saturation temperature. The situation is

quite similar to a thermal-entry-length problem (Kays (1966)) in which

the heat transfer coefficient assumes a very high value at the

entrance, and gradually decreases along the length. No analysis of

film boiling has so far considered this developing thermal boundary

layer in the liquid core. This is because in saturated flow the liould,

remains at a constant temperature.,

_

.

- 35 -

III.5 Correlations for Low Quality and Subcooled Film Boiling

Most of the correlations are developed based on cryogenic or

refrigerant film boiling data. Their applicability to water has not

been established through comparison with water data. The available

correlations have been examined by Groeneveld & Gardiner (1977) and are

tabulated in Table 2. Basically, they fall into three categories

(i) Correlations based on laminar flow of the vapour film. These

correlations have the form

N-3 P (P P } f9V V E Vhc=A +h-

E 0 'v sat -

where A depends on the vapour-liquid interfacial velocity and L

is a characteristic length.

(ii) Dittus-Boelter type correlations

These correlations assume that all resistance to heat transfer

lies in the vapour film. By suitably defining a Reynolds number

the correlations have the formbNu = a Re ppcp

.

where F is incorporated to fit particular sets of data.

_--_ _ -- - _ ---- _ - _ ----- --- - ----

. _ . . _ . - _ - __ ,

- 36 -

,

1

(iii) Phenomenological correlation.

These are developed, b; sed on physical phenomena, with some

empirically correlated parameters. They are less cumbersome to<

apply than mechanistic models, which require the solution of,

j several simultaneous differential equations..

I

a

f

l

s

t

,

!l

.

!

(

!;

!

!

|i

i

t

)'i

- . - _ _ - -- . _ . - ._ __..._ _. _ _ _ __ _,-

_ ..

- 37 -

IV.0 EXPERIMENTAL DESIGN

IV.1 Background

The most widely used system in forced convective boiling studies

is a heat flux controlled system where the heat output of an electri-

cally heated element is independently controlled. However, as men-

tioned previously, such a system does not permit the measurement of

flow film boiling data for subcooled water due to the advancing of the

rewetting front..

The method used in the present study is the so-called hot patch

technique originally used by Groeneveld (1974) to study the effect of

flux spikes on CHF in Freon. Fig. 9, taken from the original report by

Groeneveld, shows schematically the flow structure downstream of the

hot patch. It was noticed that, after steady film boiling was estab-

lished at the hot patch and the test section power increased gradually,

the drypatch propagated downstream into the heated tube. This method

was subsequently used in a water flow loop by Groeneveld & Gardiner

(1978) to establish stable film boiling.

In the case of water, the vapour film established inside the

bore of the hot patch cannot be propagated downstream into the test

tube by raising the power. This is due to the much higher latent heat

of water compared with Freon. Therefore, both the hot patch and the

test tube have to be preheated to above the Leidenfrost temperature

before the injection of water. The role of the hot patch is then to

- ______________ _ __ - _______-________ __-_____________________________

- _ - _ _ _ _ _

- 38 -

N

/{\ /() <p-WETTED WALL

/a //'

//// T, >> T,a,///

CORE CONTAINING //TWO PHASE MIXTUE ---*-

OR SINGLE PHASE LIQU10 M-p-o/-

I /'

VAPOR ANNULUS

[{''//

--,

/\ N/ at

| /\ \y/\ N'

'\ \/

Amx i/'

/.x.qg'1

FIGURE 9 SPREADING 0F DRY PATCH DOWNSTREAM 0F HOT PATCH

FROM GROENEVELD (1974)

_ _ _ ___.

- 39 -

prevent the rewetting front from propagating into the tube. This

slight difference in procedure is imperative in the successful

establishment of stable film boiling.

IV.2 Test Section

The test section is shown schematically in Fig.10. Details of

the hot patch clamp are shown in Fig.11. It was machined from oxygen-

free copper and equipped with eight cartridge heaters. The support

ring was welded to the test tube and the copper block was then held to

it by three bolts prior to brazing.

The copper block was brazed to the Inconel tube in an inert

atmosphere using a high temperature silver solder (melting

point = 780"C). Two types of test section were made: one with a

6.25 cm long and one with a 2.54 cm long hot patch. A total of four

test sections was used in the study. Details of the dimensions of the

test sections are presented in Table 3.

The optimum length of the test section was determined experimen-,

tally during the initial trial runs. It was desirable to have the

he ted length as long as possible, so that the effect of length on the

heat transfer could be examined. The actual length was finally deter-

mined by using the following criteria:'

(i) the maximum wall temperature must be below the melting point of

the tube, and

.- . _ _ .

- 40 -

INCONEL TUBE

/ I D = ll 94 unOD = 13 08 an

10 P

9?

8P

7P

DC6r SUPPLY

5P

4P3e2PIHOT

PA T C H

CARTRIDGEHEATERS

ELE'N (CM)T/C FROM TOP OFn

HOT PATCHI 32 6

COOLANT 3 94 12

5 18FIGURE 106 24

SCH EMATIC OF TEST S E CTIO N 7 358 45

9. 50

10 55

_ _ _ _ _. .

- 41 -

89=

25.4__ _

_

|

8 8

3.21

/ / /s n , p

/ || / /

/ / /[' 63 5i ,

2 4

1 / // /

'

/ / / /. v.

8 i,

8 holes 95 3 dia. equispaced 13.1~ ~on 63 5 p.c.d.

25.4 (tight fit in copper block)

I : ; 1

i . -i

I13.1-

31.8

MATERI AL OXYGEN-FREE COPPER (BLOCK), 304 STAINLESS STEEL (SUPPORT RING)I

All dimensions in mm

FIGURE 11 DIMENSIONS OF HOT PATCH

_ _ ___

- _ _ _ ____ _ . . _ _ _ _ .

I

|

- 42 -

(ii) at least a 20% variation in total test section power between

the runs at maximum and minimum (before quenching) wall

temperature.

The least variation in test section power ur:: ally occurred at the

lowest ficw because the axial temperature variation was then the

largest.

Chromel-Alumel thermocouples were spot-welded to the outside

surface of the tube at various axial locations to monitor the tempera-

ture during the test. All thermocouples were located on the same

vertical plane. The locations of the thermecouples are tabulated in

Table 3. The entire test section was insulated with a high temperature

ceramic fiber and covered on the outside with aluminum tape to reduce

radiative heat loss.

IV.3 Flow Loop

The experiments were carried out in a low pressure water loop

unofficially referred to as the SOUP LOOP (Subcooled Or Unsaturated.

Post-CHF). A schematic diagram of the loop is shown in Fig.12. Part

of the flow through the preheater was bypassed back to the tank in

order to obtain a more stable control over the coolant temperature at

the inlet to the test section. The flow rate was measured by a bank of

rotameters, each covering a different range. The bypass leg downstream3

-. - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ -

- 43 -

5;c"S 5

2*

a m a S_

e L.s u -- -

1 rod ba

8x = m

b a| 35mm

==

, ,> 4

m.-

O,- gB

S 5 --

m m :

g

2 5 i: =>x <

-

a* E

g = __=w

ei E

YW(

9- -a 'eam" x8-

II1*

1* Vf(nRQ1 ~n|

| 1 m

i eA' g N

o =5 eT _E E

-O wOm

M @

Ao r "55 08:i G83

_

..

- 44

of the flow meter was set to the same flow resistance as the test

section leg. During the heat-up of the test section, the flow was

directed through the bypass leg to establish the required flow rate and

subcooling.

IV.4 Experimental Procedure

The test section formed part of the flow loop through which

water was circulated at atmospheric pressure at the test section exit.

Initially the test section and hot patch were heated separately, while

the desired flow condition was established in the bypass leg. The hot

patch thermocouple and selected test section thermocouple signals were

displayed on an 8-channel chart recorder. When the hot patch tempera-

ture exceeded the minimum temperature required to maintain film boiling

(usually in the range of 350-550*C, as determined in previous calibra-

tion tests), the test section power was increased to raise the tube

temperature rapidly to the 500-800*C range. The flow was then diverted

from the bypass to the test section. As soon as the first test section1

thermocouple showed a decrease in temperature (indicating that water

had arrived at that location), the test section power was increased to

prevent the test section from being quenched. The presence of the hot

patch at the entrance created a vapour annulus around the circum-,

1

ference, which could be maintained downstream in the heated tube under

appropriate conditions.

- _ _ - _ _ _ - _ _ _ _ _ _ _ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

- 45 -

When steady film boiling was established, the test section power

was increased and/or decreased in steps of approximately 10%. After

each change, conditions were allowed to reach a steady state before the

data were recorded. A typical sequence of event is shown in Fig.13.

For tests at high mass fluxes (typically G > 300 kg m-2's )it,

was necessary to, start with a lower mass flux in order to prevent the

test section from being quenched when the coolant first reached it.

The mass flux and the heat flux were then gradually increased to reach

the desired flow rate.

IV.5 Data Acquisition

The test section power and thermocouple signals were scanned at

regular intervals (generally set at 20 seconds) and converted to engi-

neering units by means of an on-line computer. The hot patch power was

measured separately by a wattmeter and the flow by a rotameter. These

values were keyed into the computer through a teletype console by the

operator. All information was displayed on a CRT. When conditions

were steady, data were recorded on magnetic tape. Further processing.

of the data was carried But on a CDC Cyber 170.

For each inlet condition, data were recorded at several power

level s. The test section power was limited so that the maximum tube

wall temperature did not exceed 1100*C. The test was terminated when

the tube started to rewet as the heat flux was reduced. The test

matrix is given in Table 4.

- 46 -

}. f. , .I.LJ.1 If ; .L. [.2yi . . [. . .. .d!_{_ L___ ! .!_lai? .. l[ ._..$'

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- 47 -

IV.6 Measurement of Void Fraction

A Y-densitometer was used to measure the void fraction for test

sections 0 and E. The relative positions of the radioactive source,

test section and detector are shown in Fig.14. A schematic design of

the electronic circuit is shown in Fig.15. The source (50 mci

Cesium 137) is encased in a collimator which, together with the detect-

or, can be moved parallel to the tube axis to measure the attenuation

at different axial locations.

The gama beam was collimated so that it was just slightly wider

than the tube diameter at the measurement region. This ensured e max-

imum sensitivity and ease of alignment. It is desirable to have a high

counting rate in order to reduce the random error for a fixed sampling

time. This can be achieved by increasing the height of the collimator

windows. For practical reasons, the collimator windows in the present

setup were adjusted to give approximately a counting rate of 12,500 c/s

and a sampling time of 200 s was used.

The void fraction is deduced from the attenuation of the Y-beam.

A detailed description of the calibration and error estimate is given

in Appendix A.

- . .

_ _ _ _ . .. .. ...

. . .. .

* -*

RELACS

; :

RLE REZ E

' NY MONL I

I AA T"HN

$CA

;

RE

ITE

G MOI 0L TA^ I

N SA N

ED-I YFO

MARGAI

DPM KA C

OLB;

:

51

.

GI

FPMAERPN

OI ;TEACI RDUAORS

O uP8EdG5M

EG Y|

ARLTEPLWPOOUVPS

, |

- 50 -

V.0 DATA REDUCTION

V.1 General

For each run the test section current, voltage and thermocouple

signals were recorded. The test section power was then calculated

based on the current and resistance of the test section. This was used

to calculate the local heat flux and heat transfer coefficient (based

on wall superheat), taking into account the conduction along and across

the tube wall and heat loss to the surroundings. The various

corrections are discussed in Appendix B.

V.2 Tabulation of Heat Transfer Data

The thermocouple signals and the test section power fluctuated

slightly about their mean values at each steady-state condition. To

reduce this random error, the data for each run were recorded on

magnetic tape for three times, with approximately 10 seconds between

each recording. The total amount of raw data is therefore too large to

be tabulated completely. It is decided to examine the data and pick

out only one set from each group of three recordings. The criterion

used is that the data of the chosen set are approximately the mean

among the group of three.

The reason why averaging is not used is that the heat transfer

coefficient averaged over the three sets will not be equal to.the

average heat flux divided by the average wall superheat.

. - . -

,

N l

i1

i - 51 -

.

ie

The reduced data are tabulated in Appendix C and shown as

) boiling curves in Appendix 0..

V.3 Void Fraction Data,

l The densitometer signals were processed as explained in;

i Appendix A. The void fraction data are tabulated and plotted ini Appendix E.

!I

I

,

!

ii

I

!

1

6

1

>

1

!

-- - - - - . _ -. ... . - - - _ - ._. . .. _ - - - . . . - .- ., -

-52-

VI.0 DISCUSSION OF RESULTS

The results are shown graphically as heat flux versus wall

superheat in Appendix 0. To this author's knowledge, these are the first

non-transient data on subcooled flow film boiling of water obtained over

a wide range of mass flux and inlet subcooling. Furthemore, the test

conditions are comparable to those that are expected to exist inside

reactor core during the initial stage of emergency ccre coolant injection

after a LOCA. The only similar data available are the few points

obtained by Ellion (1954) in a short test section (7.6 cm). The

reflooding tests of the FLECHT program (Cadek,1971) also contain some

infomation, but it is difficult to interpret the data due to the

transient conditions.

Observations from the data and the parametric effects on the boiling

curve are discussed in this chapter. Test section C contains the widest

test range and will be used as the reference when comparisons are made

with other test sections.

VI.1 Dryout Location Inside the Hot Patch

The location where film boiling starts inside the hot patch was not

measured during the experiment. However, such infomation is important

in the theoretical analysis. If it can be ascertained that dryout occurs

over a short distance compared to the length of the hot patch, then it i

can be assumed that the entire block is in film boiling. :

1

To analyze this, the total power over a unifomly heated tube in which |,

CHF occurs at the end is calculated using two correlations: the local |,

conditions fom of the Macbeth correlation (1963) applicable to low

I

- _ - _ . - _ . -

-53-

pressure and short length and the Menegus (1959) correlation. The

correlations are shown in Appendix F. The results are shown as critical

heat flux versus critical quality in Fig.16.

Another independent check can be made from the CHF data reported

by Fung (1976) in his transition boiling study. The copper cylinder used

in that study was similar to the hot patch of the present experiment.

The CHF's deduced from the transient cooling curve are reproduced here in

Fig. 17.

For comparison, the average heat flux over the length of the hot

patch during film boiling tests for test sections C and E are shown in

Fig. 18. It can be seen that the average heat flux at the hot patch is

much lower than any of the predicted CHF's. This means that film boiling

with a lower heat transfer coefficient exists over most of the hot

patch. Dryout actually occurs near the inlet and the difference in

temperature between the wet and the dry side of the surface will cause

significant axial conduction along the copper cylinder. This also

suggests that the average heat flux of the hot patch is always higher

than if pure film boiling occurs on the entire surface.

..

_.|

-54-

5.5

G 500 kg.m-2 -I.sn " n n ,5.s

O Macbeth Correlation6 Menegus Correlation

4.5

m

N4.s

ee

E\ 3.53Ev n 100n n n3.s ,

D__JL1_ 2.s

l-CL1J 2.s

500

*

100;

,

i.s>

.,4,e.., . 3 e. 1 .2 e-si ise-si s.

CRITICAL QUALITY

FIGURE 16 PREDICTION OF HOT PATCH HEAT FLUX USINGCHF CORRELATIONS

,

k

_ _ _ .

-55-9.

|

I20 G = 67.5 kg/m s/

a = 135e. /+ = 405j

x = 675 //i = 1350

j

MACBETH CORR. / -

*' G= 1350

/

//

I '+/ -6. f ,-

N / s'E / e'E +'

f,

E \ / /\ \s / /3 S. s f fE N -

/ /. _ pM+ N 405x' \ \ / /\k -

'\ /g7I l

\\ /u4s \_ _ _ _ _ _ _ _ _ ;t _ _

,

/\ /\ /g'

% /3. ' ' - +/

kO Mg 67.5,/

''

' - - 'K 'x p____ -B- ,/e\ v- ~Q________g''

Ag 1

- E ~ ~ ~ ~ ~ ~ _.RI.

0. 10. 20. 30. 40. 50. 60. 70. 80.

ATSUBi (C)

FIGURE 17 THE EFFECT OF (INLET) SUBC00 LING ON CHF FROMFUNG (1976)

. _ .

-56-

| | | | |*

(kg.m .s ) open symbols - Test Section Cclosed symbols - Test Section E_

x 100+ 2000 300V 3000.8 -

a s00

X -

-0.7

0.6 -

-

0.5 N5 x

5d 0.4 -

|24

0.3 '

/-

/

0.2 -

4

I~

0.1

0 -

10 20 30 40 50 60 70

INLET SUBC00 LING (OC)

FIGURE 18 HEAT FLUX AT THE HOT PATCH FOR TEST SECTIONS C&E

_______________ _ ______________ _ _

-57-

For the same inlet conditions, test section E has a higher average

heat flux. This is because of the larger fraction of its length over which

CHF occurs. The increase in the heat flux near the region of saturated

inlet condition for test section E can be regarded as the behaviour of

CHF around this region. This phenomenon has been observed by Fung (1976)

in a transition boiling study and is attributed to the higher turbulence

due to vapour generation.

In Fig.18 the heat fluxes at G = 50 kg.m-2 -I.s do not follow the

same trend as those at higher mass fluxes. It is conceivable that at

such low mass flux, the void fraction increases rapidly over a short

distance into the hot patch. In other words, inverted annular flow does

not exist over any appreciable length at such low flow in the present setup.

VI.2 Effect of Hot Patch Temperature

Fig.19 shows two sets of results at virtually identical conditions,

except for different hot patch temperatures. No significant effect on

the film boiling curve was observed. It was also observed that a change

0in hot patch temperature of the order of 100 C did not significantly

change the power required to keep the block temperature constant. This

is in agreement with the results of Cheng et al (1978) and Fung (1976),

who observed a constant film boiling heat flux in the region just beyond

the minimum film boiling temperature. In both studies, the test sections

used were very similar to the hot patch of the present test section. The

above observations were valid for all conditions, except at the lowest-2 -Iflow of 50 kg.m .s where the post-CHF flow regime and the vapour

superheat might have changed significantly inside the hot patch.

j

_ _ _ _ _ _ _ _ _ _ _ ,

-58-

9 __

O _._

7 -D T C- 2

TEST SECTION C -

T c- 4s . _ _ A+ T C- 6

Block Temp.s - x T c- B _

4 _Continuous line 450*C

_

Symbol 385*C3 _

_

m

N2 _

_gN3Yv

210 --

X s _TC2 I4 6 8J s _

_d 7g __

G -

D -

g 5 -_

LJT 4 -

-

3 ___

I

I7 _

_ |

|

10'200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0 1000. 1100. 1200.

TW-TSAT (DEG C)

FIGURE 19 COMPARIS0N OF DATA AT TN0 DIFFERENT BLOCKTEMPERATURES. NOMINAL BLOCK POWER = 500W,MASS FLUX = 100 kg.m-2.s-lINLET SUBC00 LING = 1000

_ - _ _ - _ . . _ _

____ - .

-59-

.

VI.3 Reproducibility of Results

A total of four test sections were used in the experiments. Each

differed in some aspects from the others. The exact dimensions of each

test section are shown in Table 3.

In addition to the difference in dimensions, there was structural

difference in the region of soldering between the hot patch and the

tube. Such difference was not intended purposely, but was the result of

imperfections in the soldering process. In particular, if the gap

between the central core of the hot patch and the outside surface of the

tube was not completely filled with solder, then some spots would be

rewetted more easily. In such case, a higher average heat flux would be

required. A comparison of the average heat flux in the hot patch in test

sections A and C is shown in Fig. 20. The difference in power in the hot

patch affects the heat transfer coefficient in the film boiling section,

because the local enthalpy of the fluid will then be different.

A more suitable basis to compare the data from different test

section is the wall temperature versus thermodynamic equilibrium quality.

This is not totally satisfactory, because nonequilibrium exists due to

vapour superheat. Nevertheless, it is informative to examine such plots.

In each of Figs. 21-24, data from test sections A and C are compared.

The nominal inlet subcooling, flow and average heat flux are kept close

between the two test sections. For sensitivity comparison, two runs

using the same test section are shown for two power levels on each of the

g raphs .

Fig. 21 shows data at the lowest mass flux and inlet temperature.

The wall temperature is most sensitive to changes in inlet temperature under

. . . . . . . . - _ . . _ _ _ .

... .

- - . -

-60-

1.0-2 -1G (kg.m ,3 ) o en symbols - Test Section C

closed symbolc - Test Section AO 50

0.9 >< 10 0-

+ 2000300v400A5000.8 -

0.7 -

0.6 -

[-

_ 0.5- x5a

'O.4 -

$#

~

0.3 I

~

0.2

-

0.1!

I I I I I I I*

10 20 30 40 50 60 70

Tin ( )

FIGURE 20 HEAT FLUX AT THE HOT PATCH FOR TEST SECTIONS A&C

-61-

|200.

TEST SECTION AVERAGE HEAT FLUXItse. 2kW/m

+ A 53ines* A C 54

O C 47__

inse. a

noon.

o^ass.

wtr

-

] Off. A-I-C1 ess.go_E

| w 8. .

I- +

--I 7ss.._1C3 7ss.

ess. +

|

bii.

see.

585..4E-fi .2E-81 s. .2E-51 .4E-51 .eE-SI . E- f l

GUALITY

FIGURE 21 TEMPERATURE DISTRIBUTION DURING FILM BOILING0MASS FLUX = 50 kg.m-2.s-l, INLET SUBC00 LING = 70 C

_ _ _ _ _ - - _ _ _ _ _ _ ..

-6?-t 2sq.

! I

i15s. TEST SECTION AVERAGE HEAT FLUXkW/m2

O A 74liss. + C 73

A C 70

1s58.

issa.

O965.

LLIED ess.F- +GE ass.yG-EW ess.F-

-J 75s..JE3 7ss.

ess.

ess.

555.

Ses..4E-si .2E-st s. .2E-51 .4E-si .6E-si .8E si

QUALITYFIGURE 22 TEMPERATUREDISTRIBUTj0NgURINGFILMBOILING

MASS FLUX = 100 kg.m- .s- ,

INLET SU8C00 LING = 70 0C

_ _ _ _ _ _ _ - __-_

_ . _ _ _ _ _ _ _ _ _ . _ _ _ _ _

-63-|2ss.

It58. TEST SECTION AVERAGEHEgTFLUXkW/m

O A 1358'88* A A 124

+ C 130

tess.

issa.

O

W a,

oss.

C(r ess.yQ_EW ses.|-

J 7ss._JC3 7ss.

est.

ess.

55s.

ses. ;.4E-si .2E-s! s. .2E-81 .4E-81 .eE-St .GE-81

QUALITY

MASSFLUX=150kg.m-2.sgURINGTHEBOILINGTEMPERATURE DISTRIBUTIONFIGURE 23

,

INLET SUBC00 LING = 200 C

- _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ .

-64-

1285.

:58. TEST SECTION AVERAGE HEAT FLUXkW/m2

O A 135tiff. + C 135

A C 121X C 125*

1858.0* INLET SUBC00 LING = 20 C

IBBS.

O958.

W '

988.. .

858.gQ.T'W off.

1--

d 758.JE

788.

e55.

688. .

I

565.

585..4E-fi .2E-Bi 5. .2E-fi .4E-fi .0E-fi .9E-81

* QUALITY

TEMPERATUREDISTRIBUT{0N{URINGFILMBOILING |FIGURE 24MASS FLUX = 200 kg.m- .s- ,

INLET SUBC00 LING = 100C

_ _ _ _ _ _ _ _ _ _

- _ _ _ _ _ _ _ _ _ _

-65-

these conditions. Fig. 22 shows very close resemblance of the data from

the two test sections. It is noted that tiiese runs have the closest

matching conditions, including the power at the hot patch. Fig. 23 and'

24 present some further comparison.

In drawing any conclusion from these comparisons, it should be

reiterated that the basis of comparison (i.e. Tw vs. Xe) is not totally

sati sfactory. This will be reconsidered after the development of the

theoretical model. The slight difference in the construction of the

test sections makes it difficult to compare statistically the systematic

error in the measured temperature in different test sections. However,

qualitative comparisons show that the results are quite reproducible.

VI.4 Temperature Profile Along Film Boiling Section

Fig. 25 and 26 show the temperature profile at a fixed mass flux

and a fixed inlet subcooling respectively. It can be seen that at low

subcoolings (close to saturation), the axial temperature gradient is

rather moderate. A steeper increase is observed for inlet subcooling0greater than 20 C. At low inlet subcoolings, the vapour generation

rate is high. The corresponding increase in vapour velocity will tend to

lower the temperature increase. The heat transfer mechanism can be4

regarded as convection dominating. At high inlet subcooling, most of the

heat input goes to heating up the liquid core. The vapour film gradually

thickens and increases in velocity. Most of the resistance to heat

transfer lies in the vapour film. The wall temperature is therefore more

sensitive to an increase in film thickness.

..

-66-

12ss.

INLET SUBC00 LING (OC)Itse.

O 5+ 10

ties. x 20| 30

Aisse.

Atess.

^Oxses.

T AD ess.I-GE ess.W QO_ %u

E 'c. iW ses. r

H $lui

$|758-

E $|'2 a|7ss.

?|$ !

mIess.

E|wees. i

$|23,I55s.

I

Isu.ts. 2s. se. 4s. s. m. m. .s.,

AXIAL DISTANCE CM

FIGURE 25TEMPERATUREDISTRIBUTIONDURINGF{LMBOILINGNOMINAL MASS FLUX = 200 kg.m-2.s-

I

_ __ . _ _ _ _ _ _ _ _ __ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

-67-12ss.

MASS FLUX kg.m-2.s-lItse.

O 50A 100

ties. + 200x 300

1 400185s. - 500

1 -_

tees.

+ess.

LL)'

TD ess.F-E a*T

85s.gQ_E i

iIW ess. ._,

H siu|l 7ss. m

_] wE El3

"|ves. ,.8~lg|ess.

aless. O,

l

!|858-

|ses. .

s. 1s. 2s. as. 48. 5s. es. 7s. es.

AXIAL DISTANCE CM

FIGURE 26 TEMPERATURE DISTRIBUTION DURING FILM BOILING0NOMINAL INLET SUBC00 LING = 20 C

.

'

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . _ _ . . . _ _ . . _ _ _ _ _ . _ _ _ _

-68-

It can also be observed from these figures that the wall

temperature reaches a maximum-at locations quite far away from the power

clamp. The subsequent drop in temperature is therefore not likely to

be due to end effects. Most likely, it is due to the increase in vapour

velocity and liquid-vapour interfacial velocity, which together will

improve the heat transfer.

VI.5 Effect of Axial Location

Fig. 27 shows that the axial location has a strong effect on the

film boiling temperature. This effect may be more readily seen when the

heat transfer coefficient is plotted as a function of axial location.

Fig. 28 indicates that the heat transfer coefficient decreases initially,

but levels off towards the end and, under certain conditions, it may

increase. The observed axial location effect can be attributed to the

following:'

(a) Entrance offect

Analogous to a single-phase themal-entry-length problem, the heat

transfer coefficient near the entrance is much highar than at a " fully-,

developed" downstream region. In addition, in the present situatit,..,

the creation of a vapour blanket at the hot patch considerably disturbs

the flow. The resulting higher heat transfer coefficient at the

beginning of the dry patch decays with distance downstream. Fully |

developed conditions cannot be attained for our two-phase conditions i

Ibecause of the continuous vapour production; however, in analogy with

single phase heat transfer in an entrance region, most of the change in

heat transfer coefficient due to developing boundary layer would have

taken place within 25 diameters from the hot patch.

. - .

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

-69-

9 _ '

-

B-

' -

P = 100 kPa TEST SECT!0fl C~

-

E _-

AT 0C=g

_ in _

4_

-

3 _-

G = 250

kg m-2 3-1 _2 _ p 5

-a - - - . ,Y a # MO"

7"/A' A'-

_TC2yC42 #}O- TC6 TC8,9 - -X e

2 8 --

-

_J 7 --

g_

TC2 TC4 TC6 G = 50_TC8,9

g s Jih I # kg m-2 s-I _

sf g ,+ [ 4 -

[LJX < - P 4 ,

3 _-

2 - -

10'200.0 300.0 400.0 500.0 GO O .0 700.0 800.D 900.0 1D00, 1100. 1200.

TW-TSAT COEG C)

FIGURE 27 TYPICAL RESULTS AT TWO MASS FLUXES

- . . .

______.

-70-

/

0*C & t *I **A AT =in ave

5*C & co AT 54 W m|

= =in ave

P = 100 kPa i

|G = 300 kg m- 2,3-1

TEST SECTION C

Note: H is based on T,- Tsat

350 _e64E

o*300 _

if a

o oO

n0 _, ,&wE 6

oe afoh 2# _

V oo8 0 o oz

ISO I I I I I

O 10 20 30 40 50 60

AX1 AL LOCATION (C M)

FIGURE 28 VARIATION OF HEAT TRANSFER COEFFICIENTALONG HEATED LENGTH

_ _ _

_ .- .

-71-

I

(b) Change in Vapour Film Thickness

At axial locations near the beginning of the vapour blanket the

vapour film thickness is least due to the higher local subcooling and

shorter film boiling length. Here heat transfer across the vapour film

is mainly by conduction. At downstream locations the vapour film

thickness increases because of increased vapour flow rate. Initially

this results in an increased thermal resistance, thus lowering the heat

transfer coefficient. At locations further downstream the vapour velocity

also increases, especially when the bulk of the coolant reaches

saturation. This increases the convective component and the overall heat

transfer coefficient will then increase.

(c) Increase in Vapour Superheat

Near the end of the heated length, high void fractions are encountered

during the low flow or low subcooling tests. As has been observed in

other post-dryout studies, a significant degree of vapour superheat can

be expected for these conditions. For such cases, the wall superheat,

(T -Tsat), is no 1 nger a representative temperature difference. Thew

heat transfer coefficient based on this difference, instead of (T,,-T ),y

will result in an artifically lower value.

VI.6 Effect of Inlet Subcooling

Fig. 29 coinpares data at different 10% ; secoolings. It can be

seen that the effect of inlet subcooli-: : u.itive for short filmboiling lengths: the film boiling hea- flu ? TC 2 (L/D=5) increases

with an increase in inlet subcooling. However, at TC 8 (L/D=38), the

effect is much reduced and the trend is reversed.

.

-72-

9 _ _

8 -

7 - Numbers on curves refer to inlet -

s _subcooling in C

_

S _ Broken lines indicate TC2 _

4 - Continuous lines indicate TC8 -

TEST SECTION C3 _ _

10 5y20 / 7 10 G = 500^20m / / /2 _ - / f -2 -I ~E p p

|/~ kg.m s

\ / /

3 / '7x

v

N 50 510 -_ ,9,,/2 5 G = 100

3o / so --

X S // /

J_ /, / / -

kg m- 2, s-1 -_

~3 e _

7 p /la. ./ /

--

6 _ / _

& / 5,30g s _ j _

tJ3- 4 _ _ !

TC8,4

3 - 45 m -

1

I2 _ TCl- _y

T ll l

6 cm

10'200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0 1000. 1100. 1200.

TW-TSAT (CEG C)

FIGURE 29 EFFECT OF INLET SUBC00 LING

_ _ _ _ _ _ _

-73-

The effect of inlet subcooling can be understood by considering

the local subcooling or quality effect. Near the hot patch the void

fraction is low and the flow regime is inverted annular. Heat transfer

is primarily by conduction through the vapour film. The vapour film

thickness depends strongly on the bulk coolant temperature and hence

the inlet subcooling, with the thinnest films occurring at the highest

subcoolings (Groeneveld and Gardiner (1977)).

It has already been shown that at locations well downstream

from the hot patch convective heat transfer becomes more important. This

is especially true for lower inlet subcoolings, due to the corresponding

higher local void fractions. Consequently, as can be deduced from

Fig. 29, the film boiling heat flux is higher for lower inlet subcoolings

at TC 8.

VI.7 Effect of Mass Flux

The effect of mass flux at a fixed axial location and inlet subcooling

is shown in Fig. 30. The film boiling heat flux increases with mass flux

for all axial locations.

In inverted annular flow the vapour layer thins appreciably a t high

mass velocities, thus decreasing the resistance to heat transfer across

the film. This thinning of the film may be sufficient to allow liquid

contact with surface asperities and may explain the observed higher

rewetting temperatures at higher mass velocities.

- _ . _ __-_

,

-74-

9 _ _

8 _ _

7 - Broken lines indicate TC2 _

6 -

Continuous lines indicate TC8-

.

6 - TEST SECTION C -

4 _ _

3 _ _

5 00,,. 500 -2 -l^

'M3 kg.m s" -350 ~2 ~

s ' ,,.E 250\ /

3 / ,'', '

- 250y / -

'v - - 150

/,'$0/710 _ _ ' __

x 9 _ - _

] 8 _ _'

? _ _g6 - 750 -

H / 50CI s - / -

w '-,

1 4 _ / _

TC8,3 _ _

i 45 cm

i 2 __

TCl- ' _'

q

T 3L6 cm

10'200.0 300.0 400.0 S00.0 600.0 700.0 800.0 900.0 1000. 1100. 1200.

TW-TSAT (CEG C)

0FIGURE 30 EFFECT OF MASS FLUX AT INLET SUBC00 LING OF 20 C

-75-

VI.8 Effects of Quality on the Convection Heat Transfer Component

The total heat flux can be considered to consist of a radiation

and a convection component. To examine the convection component, the

radiation component is first subtracted from the total heat flux. The

radiation component is calculated according to Appendix H together with

the void fraction correlation of Appendix G. The convection component

is then expressed in terms of a Nusselt number. Thus,

I - Rad O'TOT

N" CON*

(T, - T ) ks y

The Nusselt number thus calculated is plotted against the local

equilibrium quality in Figs. 31-37. It can be seen that for mass fluxes-2 -Iabove 100 kg.m .s , Nu decreases initially with increasing Xe. It

CON

passes through a minimum before increasing again. To the left of the

minimun, the flow regime is primarily inverted annular and a significant

fraction of the heat transferred from the wall is needed to heat up the

subcooled liquid core. To the right of the minimum, the flow regime is

dispersed liquid droplet flow and the wall heat flux is primarily used

for evaporation. The increase in Nusselt number with increase in

equilibrium quality in this region can be attributed to the increase in

vapour velocity.

-2.s'I),It can be seen from Fig. 31 that at the lowest flow (50 kg.m

there is no prominent component for heating up the liquid core. This is

because at such low flow, the liquid core is broken up into slugs and

droplets . It is also evident that the Nusselt number decreases with

axial location. This again suggests a high vapour superheat at this low

flow.

-76-

203.

RUNAME=C 50 . Te-2,ga. _

O TE-4

a TE-6

A TE-9*

+ TE-10

140.

n20.

H.__]Ld

100.gCDDZ

es.

00.

40.

a @ B23. 3 eDOg Qgn

cot h DA * * ' ' '* 4e .4 4 Ag' g,. 9 @+ , a + + 4^^ + *f ++ +

'

.e4 82 0.0a .02 .e4 .00 .20 10 .12 .14

XE

FIGURE 31 NUSSELT NUMBER AS A FUNCQUALITYFORG=50kg.m-}IONOFEQUILIBRIUM.s-

.

-77-

200.

RUNAME=Cl00 . TE-2io,.

O TE-4a TE-0

A TE-9lea.

+ TE-10

140.

120.

I--JL1J

100.g(ODZ

02.

D"

00.

"O%a a

on "40. 'bo O

i%2a-

. e Q y y = ,. . . .

2..04 .02 0.58 .52 .84 .00 26 -10 .12 .14

XE

FIGURE 32NUSSELTNUMBERASAFUNCT{0NOFEQUILIBRIUMQUALITY FOR G = 100 kg.m- .s-l

-

-78-

200.

RUNAME=C150 . TE-2,g,,

O TE-4

a TE-6

a TE-e,g,,

+ TE- 10

140.

D

120."

F---"

_JLiJ 103.g ,

WDZ

es. Ol

Ooes.

O@

a%40. ,

C l0

2s. , , . + +

|

*s.

.04 .02 2.28 .82 .04 .se .se .is .32 .34 |

XE ;

FIGURE 33 NUSSELT NUMBER AS A FUNCTION OF EQUILIBRIUMQUALITY FOR G = 150 kg.m-2.s-l

. _ _ _ _ _ _ _ _ _

|

||

-79-

200.

RUNAME=C200 o TE-2,g,,

O TE-+4 TE-8

100.~

+ TE-19

140.

122.|

l-'

_) o olij 8a o

108.g OW

op3Z

00.

DDO02. o

Oo

O42. _ g

T>syss" " .e2,.

'2.

84 82 0.86 .82 .24 .56 90 18 .12 .14

XE|

l FIGURE 34 NUSSELT NUMBER AS.A FUNCTION OF EQUILIBRIUMi QUALITY FOR G = 200 kg.m-2.s-l

_ _ _ _ - _ . ___ -_-_ -__ _ - _ - __ _ _ _ _ - _ _ _ _ _ .

-80-

208.

RUNAME=C300 . Tu-2,,,,

O TE-44 TE-8

188.~

+ TE-10

148.

128. a

H_ILIJ

iss. agC'DZ a0

08. ,

a

O

60. J

O

040. 0

? ,e+**

|.

a2..

s..04 .82 0.88 .02 .84 .20 08 .lg 12 .14

XE

QUALITYFORG=300kg.m-2.sgFEQUILIBRIUMNUSSELT NUMBER AS A FUNCTIONFIGURE 35

!

|1

- -- - _ _ _ . - - _ _ _ _ _ -

.

-

-81-

238.

RUNAME=C400 . TE.2ig,,

O TE-4

4 TE-6~

100. -

+ TE-IB

149.o

o

120._

l- a

__]DLLI o

100-gCD D

aD o

Zes. O

Ooa

00. o,

E* Qo D

",@48. s++

2s. ,

s.84 .02 F. 09 .92 .84 .38 .90 18 .12 14

XE

FIGURE 36 NUSSELT NUMBER AS A FUNCTQUALITYFORG=400kg.m-{0NgFEQUILIBRIUM.s-

_ _ _ _ _ _ . -

. .

-82-

229.

RUNAME=C500 . TE-2,g,,

O TE-44 TE-6

A TE-8100.

+ TE-10

a148. o

Oo

L28. D

f-- "-Jll.) o

tes.gD a

'D CP

Z ^es. O

*OoD

a

es. ,

0oa8 O

20j48.a

28.

's.84 .02 s.Es .02 .s4 .ge ,gg ,g ,g

XE| |

|

QUALITYFORG=500kg.mj0NgFEQUILIBRIUMNUSSELT NUMBER AS A FUNCTFIGURE 37 *.s-|

||

- _ _

-83-

VI.9 Minimum Film Boiling Temperature

In some of the tests, as the power was reduced, the test section

started to quench. In most cases quenching started either at the start of

the film boiling section, or at the power clamp at the downstream end.

This kind of rewetting is Cue to conduction along the heated surface,

and is sometimes referred to as " conduction-controlled rewetting". The

measured temperature at the moment of rewetting is not a genuine minimum

film boiling temperature, since it is influenced by the conditions nearby.

Ari samination of the strip chart records of the thennocouple

traces shows that in some cases the first thermocouple that indicated

rewetting was TC3. Such rewetting occurs when the liquid penetrates

through the vapour thermal boundary layer next to the wall. At heat fluxes

above the minimum film boiling temperature, the vapour generated at the

vapour-liquid interface will create a vapour thrust force. If the force

is large enough, it will push the liquid away from the wall, thus

preventing liquid-wall contact. If the liquid momentum in the direction

towards the wall is greater than this thrust force, the liquid will contact

the wall momentarily. Whether the surface will rewet after liquid-wall

contact has been initiated will depend on the thermal capacities of_ the

wall and the liquid in the region around the point of contact. Henry (1974)

has considered the minimum film boiling temperature to be determined by

this transient cooling, mechanism. This kind of instantaneous rewetting

has also been termed " impulse cooling rewet" (Tan & Griffith (1976)).

Table 5 shows the minimum film boiling temperature extracted from

the strip chart records of the present experiments. Since rewetting occurred

. .

-84-

during a reduction in test section power, no record of the exact power at

rewetting was available. The equilibrium quality shown on this table was

calculated at a steady state film boiling condition just prior to the

change in power. Usually the change in power was a reduction of 5-10%.

A better way of estimating the exact power is to extrapolate the heat

flux versus the wall temperature at location 3 to the minimum film

boiling temperature. In view of the somewhat qualitative nature of this

discussion, we will not pursue it here.

Fig. 38 shows the minimum film boiling temperature plotted against ,

the local equilibrium quality. It shows the asymptotic trend of the

minimum film boiling temperature. At high qualities, the minimum film

boiling temperature is close to the saturation temperature. This can also

be seen from Fig. 4, where the high quality boiling curve changes gradually

from CHF to film boiling. At negative qualities (corresponding to

subcooling), the minimum film boiling temperature increases rapidly..

VI.10 Void Fraction Measurements

Details of the void fraction measurements are discussed in Appendix A. i

The 95% confidence interval for the measured void fraction ranges from

13% to 16%, depending on whether the void fraction is near the upper

(100%) or lower ends (0%). This level of accuracy has been optimized

with respect to the sampling time, the radioactive source strength and*

\

the information obtained. Furthennore, this interval only contains the

random error due to radioactive decay. Other measurement errors have not

been included.

- - - - -

-85-

sea.

O A 58.885*8' a A 58.257

+ AISS.287X CISS.857

,,

i CISS.ISD

- C186.285

> CI5S.188446.9 C255.2s5< 25s. assO \ 0o C358.295425. p

ko Z c588 286W \ ET 4ss.D \F \C

\g ass.L2_1 \Q_

\E see.

H \j 346. \J \ - 1

E \1 32s.

\ I>\ass. 3

' ' ~ x'%

286.

28s. e

.3E-WI .2E-5t .IE-Si f. .IE-Gi .2E-Ei .3E-51

QUALITY

FIGURE 38 MINIMUM FILM BOILING TEMPERATURE

_ _ _ _ _ _ _ -

-86-

Figs. 39 and 40 show some of the data plotted as measured void

fraction versus axial location. It can be seen that the void fraction

at a fixed axial location increases with a decrease in inlet subcooling

and an increase in heat flux. The increase in void fraction with increase

in equilibrium quality is small in the high subcooling region (large

negative quality). This implies that most of the heat input goes to

heating up the liquid core.

The same data as shown on Figs. 39 and 40 are replotted against the

equilibrium quality based on local pressure in Figs. 41 and 42. It can

be seen that the data fall fairly closely on the same line. This

suggests that the local phase and velocity distribution is primarily a

function of equilibrium quality.1

Fig. 43 shows all data from test section E plotted together. The

mass flux effect is more evident in Fig. 44 in which only the best curve

through the data of each mass flux is shown. It can be seen that an

increase in mass velocity increases the void fraction at a given

equilibrium quality.

Inside the hot patch the flow regime will be inverted annular when

film boiling is first established. The distance over which this flow

regime is maintained depends on the void fraction. Plumer (1974) suggested

that at a void fraction of approximately 40% the inverted annular flow,

|

regime changes into dispersed flow. It can be seen from Fig. 43 that the

void fraction is in the neighbourhood of 50% at zero equilibrium

quality. This would suggest that most of the data in the net equilibrium

quality region are in the dispersed flow regime.

- _ - - - - - - -

-87-

.

i.e.

D

RUNAME=D500 o,,* aTEST SECTION D @

MASS FLUX = 500 kg.m-2.s-lO

AT (O )C88. in O

a 5A 20

78.

^ZUO e,,

F--OE Oss.Q'LL

Q=

4s.-

O>

38.

26.

is.

s.s. is. 2s. as. 4a. se, es.

AXIAL DISTANCE (CM)

FIGURE 39 AXIAL VARIATION OF VOID FRACTION,MASS FLUX = 500 kg.m-2.s-l

- _ ________________ _ _ _ _ _ _ _ _ _ _ _

__ ..

-88-

tea.

::

RUNAME=E300 hOs. O A

:es. :

A

7s.

ZO e,,

HO O

E Jss.y

LL

C A4''~TEST SECTION EMASS FLUX = 300 kg.m-2.s-l

ATbn( C)3s.

O I -

a 10'

+ 17,

t 2s. --

1s.

s.s. ts. 2s. 3s. 4e. se. es.

AXIAL DISTANCE (CM)

FIGURE 40 AXIAL VARIATION OF VOID FRACTIONMASS FLUX = 300 kg.m-2.s-l

l

- - - - - - - - - - - - - -

-89-

in.03

RUNAME=D500 O.. e

TEST SECTION DO D500 05 OgATin( C)o D500 20 -- Q3

8s* O O 5AA A 20A

*78.

/*

Z S AOO ,,,

Ap a &

O

h 5s.00 A

1 O

O 44s._.

O A>

3s.

A

a. gA

ts.AAA A

s.. E-81 .z-s t s. .g.g l . E-si .eE-si

GUAL.ITY

FIGURE 41 VOID FRACTION AS A FUNCTION OF EQUILIBRIUM QUALITYMASS FLUX = 500 kg.m-2.s-1

_ _ _ _ - _ ._ _ _ _ _ - _ _ _

-90-

100.,

I o a^o%RUNAME=E300,

O Esco 01a E3co -to + .

80. * E 3 00 17 TEST SECTION E _

AT in

n. A U Ia 10 _

+ + 17Z3 so. +

HU M

h 2. 00 -

L

0 40. A

a aD

30.

|

+20.

+

k10. _

+

0.

.4C-01 .Z-01 0. .E-01 .4C-01 .EiC-01

OURLITY'

!

| FIGURE 42MASSFLUX=300kg.m-2.sgNOFEQUILIBRIUMQUALITYV010 FRACTI1N AS A FUNCTI

_ _ _ _ _ _ _

- - - _ _ _ - _ _ - _

-91-

1es.

t ' 'COG2 o88'

tOMIt/AL M S U X* N 6' Oa

(kg.m-2.s-lyagO 100 $ $a88. a 200

kX 400

I g gI 50078. XX g

X2

hQ '

3O iG' .. g

58.Q p+k x6oo +

I;'

Q i+a> XX b

; l'30. OXI

X28. ;

'ly%

is.

QNx

s.' * E'Il .g- g g *E'81 . g.g g E-81

QUALITY

FIGURE 43 V0ID FRACTION As A FUNCTION OF EQUILIBRIUM QUALITY

_. . _ _ _ _ _ _ _ _ . a

__

-92-

100. -

h

so , . TEST SECTION E og

80.

7D.

ZO so,-

HUCI 50.yk

O40._,

OD s

1

30. 7

5

20. o

Re$

10.

O.

.4C-01 - E-01 C. .E-01 .4E-01 .GE-01.

CUALITY !FIGURE 44 EFFECT OF F \SS FLUX ON V0ID FRACTION

-. -- _

_ - _ _ - _ . -_

-93-

In Fig. 45 the calculated void fraction, a = X X + p /o .S.(1-X) ,y

based on constant slip and either saturated or film temperature for the

vapour, is compared to the data measured during a low inlet subcooling

test. In calculating the void fraction it is assumed that the liquid has

reached saturation before the first measurement station. This assumption

is reasonable since the coolant is only slightly subcooled at the test

section inlet. The predictions show that high slip ratios tend to lower

the predicted void fraction, while higher vapour superheats can affect

the void fraction either way through a decrease both in quality and

vapour density (the cross section average enthalpy remains constant).

For the condition of Fig. 45 an increase in vapour superheat increases

! the void fraction.1

Fig. 45 shows that a no slip assumption results in a large over-

prediction of the void fraction just downstream fren the hot patch. Hence

high slip velocities must have developed inside the hot patch. Also,

near the start of the film boiling region the measured axial gradient in

void fraction is lower than any of the predictions. This may be

explained by (i) an increase in slip ratio, or (ii) a combination of

increase in slip ratio and decrease in vapour superheat.

Further downstream the axial void fraction gradient becomes larger

than any of the predictions. This is ') be expected as the flow regime

will be dispersed and the liquid corr will have broken up into small

droplets . It has been shown (Groeneveld & Delorme (1976)) that smaller.

|

| droplets are more likely to be accelerated, thus reducing the slip ratio|

ar>d hence increasing the axial void fraction gradient above the constant

slip ratio predictions of Fig. 45.

An empirical correlation of the void fraction data is developed in

Appendix G.

_

-94-,

100 -

- *

, ~ .- - - - a

#

./,O

90 -

//,-.

I O /80 - /

//

70 -

v

g P = 100 kPa60 - ~ ~

G = 495 kg m sNw

AT = 5.1*Ca gs

O A f = 231 m m50 ave

A Measured void fraction

40 - / Calculated void fraction :

S = 10, T = 0. 5 (T +Tsat), Tliq sat/ =Tvap v

S = 10, T =T =T |----

-[, vap liq sat;

| 30 S =1T =T =T. _ . -vap liq sat,

I

:' x Measured wall temperature

20m

1000 - .O" , - - - ,, --a

_

-

10 --

-900 _._

:s1 I I I I I I

*' O

O 10 20 30 40 50 60 701

AXIAL DISTANCE DOWSTREAM FROM THE HOT PATQi (ca)

FIGURE 45 COMPARISON OF MEASURED VOID WITH PREDICTIONS I

|!;,

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

-95

VII.0 PHYSICAL MODEL OF INVERTED ANNULAR FILM BOILING

VII.1 Aporoach and Assumotions

The present analysis uses the one-dimensional integral technique

similar to the analysis of single-phase boundary layers. A similar method

was used by Dougall and Rohsenow (1963) to analyse film boiling. The

present analysis differs from the analysis of Dougall & Rohsenow in that

it considers the initial stage of the vapor film to be in laminar flow.

As the vapor film thickens, it changes into turbulent flow. In- addition, a

subcooled rather than saturated liquid core is considered in the analysis.

A sketch of the laminar and turbulent regions are shown in Fig. 46. Note

that the designation of the laminar sublayer is slightly different from

the usual convention in the laminar film region. The reason for this will

be explained later.

In the analysis, the acceleration of the vapor is taken into account.

Heat transfer to the subcooled liquid core is analysed by solving the energy

equation. The main feature is that the enhancement of the heat transfer

coefficient at the start of film boiling (equivalent to the entrance region

in single phase flow) is considered in the analysis.

The wall temperature is calculated by assuming pure conduction

through the laminar sublayer adjacent to the wall. The radiation component

is calculated by assuming radiation exchange between two cylindrical

surfaces.

The following assumptions are made in order to obtain a simplified

form of the momentum and energy ec,uations:

(1) The change of vapor momentum is negligible in laminar flow. Consequently,

the viscous force is balanced by the buoyancy force.

-

-96-

Vapor

fH | Loc o'Laminar Sublayer = g m Veloci G/ I

/ |

/ \lj | |Turbulent Film

/ | Velocity Profile

/ |

/ $

Il/I

/4

|#}n

/

/ '

?'/

ILaminar Film y

/

/Liquid

|

|

Fig. 46 DEVELOPMENT OF VAPOR FILM'

__

- _ _ _ _ _ _ _ _ _ _ _ _ _ _

-97-

(2) Axial conduction is negligible in the fluid.

(3) Kinetic energy and viscous dissipation are negligible. This is true

at low Mach number.

(4) The vapor density is much less than the liquid density, so that (o -o )t'g y

(5) The liquid density is constant.

(6) The liquid core is assumed to be in turbulent flow. This is justified-2 -Isince the Reynolds number at G = 100 kg.m s is 2100. At low

Reynolds number, there are two possible situations. The first is that

the liquid volume is large, and conditions approach that of pool

boiling. The second is that the liquid volume is small, so that

transition from inverted annular flow to slug flow takes place over

a short distance. The present analysis does not consider these

boundary cases.

(7) The velocity profile of the liquid core is assumed uniform. This is

reasonable as a consequence of the previous assumption.

(8) The flow is steady.

The momentum and energy equations will be developed based on these

assumptions. Further assumptions will be made and justified in order

to reduce the complexity of the mathematics.

After the solution of the governing equations has been obtained, a

sample calculation is performed to check the validity of the assumptions.

This is shown in Appendi.x J.

. .. . _ _ . -

,

-98- >

+

VII.2 Momentum Equetions-

VII.2.1 Laminar vapor film

In the laminar film region, the vapor film is thin. . Therefore,

we can use Cartesian instead of cylindrical coordinates. A sketch of the

region under consideration is shown in Fig. 47. According to assumptions'

(1) and (4), the momentum equation is simply

p = -90gy

This is identical to the derivation of Ellion (1953). The

solution is-90 2

gh+C)yEu =2y y

v

At y=0, u=0'

,

.. C =0 [*

2

Assuming an interfacial velocity equal to the. liquid average velocity

U , the constant C) can be evaluated.. The result isg'\

99 y6 'g

(1- ) + Ugf,

u =2,,vy g

This can be substituted into the definition of the vapor masss

flux:6

fup'dy-

2nRG =

i y 2 yynR

2~90 y6

t

(1- ) + U f dy=g p 1

0- 2p gv

V

Assuming that the vapor properties .can be evaluated at the film

temperature, we cbtain

3 -

2 ~9P 6 U6-t g

Ov"R D *v 12u 2 .y

..

7

-

-99-

Vapor

#Locus of"|Paximum Velocity /

|

/ \#Velocity Profile =

/ \

/ \

/\ |#

Heated Wall &

/,

/

h

/

/ y

//

Liquid

FIG. 47 LAMINAR VAPOR FILM

|

-

____ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

-100-

The energy equation will give another relation for G . This willy

be considered later. Differentiating the above equation with respect todu

the axial direction Z, and assuming that is negligible in the laminardz

film region, we obtain.

2- -

.dG 2p gp 6 Uy y g t d6

7Idz R 4u +T dz* -----

- v _

This equation will be used in later analysis.

I VII.2.2 Turbulent vapor film

The liquid and vapor control volumes under consideration are shown'

in Fig. 48. The forces in the Z direction acting on per unit length of

the vapor control volume are

(i) Wall shear -2nRT,=

(ii)interfacialshear - T (2n)(R-6)(cose)=

4 Az

- 2n(R-6 t= f g

(iii) pressure force at Z and Z+aZ~

2=-[ p n(R -(R-6)2)y

= - n 6(2R-6)dz -2np(R-6)hy

dp

= - n6(2R-6)dz - 2n(R-6)p taney

(iv) pressure force at liquid-vapor interface2 2n(R-6)az sine

E"

t cose Az

2n(R-6)p tane=g

(v) Gravity force - this is assumed to be negligible.

|

l1

, _ , _ .__ -_ . _ . - -

-101-

dp g

P + dzg *0

U \/ hg

\ Azrp g

\\g i a# | 4

9P

t

(i) Liquid Control Volume

1

dp gyp + dz Azy >

4 \\

|\

W i y

\J

L L .\<

P Uy y

1

(ii) Vapor Control Volume

FIG. 48 CONTROL VOLUME IN INVERTED ANNULAR FLOW

-102-

The changes in momentum are:

(vi) Change in momentum between I and z+6z

RI 2d=g 2no u rdryy

R-6

(vii) momentum crossing the liquid vapor interface

U 2nrp u drag yy

R-6

The pressures p and p at the two sides of the interface arey g

not equal, due to the presence of a vapor thrust force. If the evaporating

mass yndergoes a change in normal velocity from V to V across theAN p

interface, then the difference in pressure is

p -p *f5(V "0 )y g y gn" (N v~

It is shown in Appendix J that this effect is small. Therefore, we assume

that the pressures on both sides of the interface are equal. Under this

assumption, the second term in (iii) can be equated to (iv). Combining

(i) through (vii), the resulting equation is

rR R

gh,2nroudrh 2np u rdr - Uyy yy

J R-6R-6

- 2nT (R-6) - n (h) 6 (2R-6) - - - - -= - 2nRT 7.2j

Similarly, for the liquid core, the forces in the direction of

flow are

(i) Interfacial shear = 2n(R-6)Tj

-103-,

(ii) pressura force at z and z+Az'

=-h p u(R-6)2g

d

= - n(R-6)2 + 2np (R-6) hdz g

= - n(R-6)2 2np (R-6) tan 0g

(iii) Pressure force at liquid-vapor interface

= -2n(R-6)p taney

(iv) Gravity force = - ga n(R-6)g

(v) Change in momentum,.R -6

d 22nU rdr=ogg g

0

Conbining (1) thmugh (v) and again assuming p = p , the liquid momentumy g

equation isR-6

g(R-6)2oh 2n U rdr = 2nT (R-6) - n(R-6)2 ( )-g2 n -----7.3g g j

"O

Equating the pressure gradient in eqs. 7.2 and 7.3 we obtain*R

2RT 2r$(R-6) rdr6(2R- ) * 6(2R-6) + n6(2R-6) "P "v~*

v

R R-6Ut d

- - 2nro u drn6(2R-6) dz . yy

"R-6.R-6

7.4+ 2no U rdr= go ------

n( R-6)2g gg

0

This equation can be regarded as the relative motion equation

since it describes the difference between the rates at which the two

phases are gaining kinetic energy.

* In the more advanced two-fluid models currently being developed, the pressuregradients in *Se two phases are considered to te different. This will resultin a cross flow component in addition to the evaporation / condensation,In the present analysis, the evaporation component is dominating.

.. .

_

-104-

The vapor mass flux can be defined as

Rrj2nrp u dr.G =

y 2 yynR s R-6

Similarly we define the momentum fluxes to be

,R2

M 2*P u rdr=y 2 vy

"R-6

,R-6

=h 2and M 2no u rdrg ggnR

"O

Equation 7.4 can then be written as

dM dG6(2R-6) dM1 w 2ri ,go (2R-6)62r

1V V-U 7*5- -----

dz 1 dz(R-6)2 dz

R R-6 2R

!

VII.3 Energy Equation

VII.3.1 Vapor film

Consider the vapor control volume as shown in Fig. 49. We assume

that all condensation and evaporation take place in a narrow region at the

liquid-vapor interface. This region has been combined into the vapor

control vol',me under consideration. Therefore, the condensation and evapor-

ation mass fluxes r and r, (based on interfacial area) will have enthalpyc

h associated with them. According to assumptions (2) and (4), the energy if'

equation for the vapor control volume can be written asR

nD$ = 2no (R-6)(cose) + 2" "v v h rdrPTOT g y

R-6

+ 2n(r -r,)h (R-6)(cose) 7.6-------

c f

.- .

-

-105-

BuV

u + 3g AZy

L,e 0

,

\j NI': c

\ 6e

* TOT "

N4 g g

N l'en

pig, 49 ENEggy gALANCE FOR yAPOR CONTROL VOLUME

!

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ._ _

-106-

Here 4 is the net heat flux based on interfacial area going intog

the liquid core. If the liquid core is saturated,this component is zero.

The total energy transferred away from the heated wall goes to increasing

the total vapor enthalpy.

A mass balance at the interface givesR

n(D-26)(r #e)cose " ~ rdryyc R-6d6

2 v= -nRdz

Similar to the definition of the momentum flux, we define the energy

flux asR ,

=hE 2no u h rdry yyynR

_3

Eq. 7.6 can then be written as

dE dG2

2RcTOT - 2(R-6)$1 cose dz f dz=R h-

It is more convenient to consider $g based on the projection of the

interfacial area on the tube wall. Using this modified definition of

$g, the above equation becomes

dE dG2 v 2

2RO (R-6)t =R -Rh 7.7------

TOT g dz f dz

VII.3.2 Liquid Core

So far, the liquid velocity has been assumed uniform across a

cross-section. However, the liquid average velocity changes due to

evaporation and increases in vapor film thickness. Consider the 2-dimensional1continuity and energy equations written for the liquid core: '

. _ _ . ..-

_. . . .

-107-

Bu

hhyg + 37E=0 7.8-----

~

3T ST 13 BTgg+vg p = p (c D)'W 7.9and u

H -----

Since we assume that u is uniform, we can replace the partialg

derivative w.r.t. I with the ordinary derivative. Thus eq. 7.8 becomesdU

V * 20t dz

"Defining r* = R 6 .the above equation can be written as

3r*vg* ddU

1 g=0(R-6)r* Br* z

Integrating over the cross-section of the core, we obtain

vg = f - (R-6) fE

Since v,is finite at r* = 0, f(z) = 0.

.~. v = -(R-6) fE

g

At the liquid-vapor interface, this velocity v can be consideredg

as the rate at which the radius of the liquid core is diminishing.

Substituting v into equation 7.9 and again using the dimensionless radialg

distance r*, we obtain

r* dug gg30 1 1 3

)303r* * 2 dz ar*E Y (R-6)2 F Br* r*(cH

T -T(r*,z)Swhere 0= 7.10-------

T -Ts in -

It is interesting to consider a limiting case of eq. 7.10. Ifdu g

6 = constant and = 0. Eq. 7.10 reduces to a familiar thermal-entry-lengthdz

problem (Kays (1966)). However, since c depends on r* as well as z, theg

_

-108-

i

reduced equation cannot be treated as a Sturm-Liouville problem. Never-

; theless, it can be seen that qualitatively the heat transfer at the boundary

! of the liquid core behaves similarly: an enhanced coefficient near the

| entrance, gradually diminishing to the steady state value.1

The solution of eq. 7.10 will give the temperature gradient.I The heat flux $g at +.he vapor-liquid interface is thereforei

BT

(c e )Tr&g -p cg pi gr=R-6

(T -Tin)30 3

| g p (c e)F (R-6)7 11=oe -------

gI p*al

I

However, equation 7.10 cannot be solved independently, because:

j the vapor film thickness is ur.known. It is coupled with the momentum

equations 7.1 or 7.5 and the vapor energy equation 7.7.

:

VII.4 Constitutive Equations

In order to obtain a complete set of equations, the constitutive

j equations for the interfacial shear and eddy diffusivity has to be specified.

VIII.4.1 Interfacial Shear^

The general approach of estimating the interfacial shear in two

phase flow with vapor generation is to determine a correction factor for,

the non-evaporating /non-condensing friction factor. For adiabatic two phase1

flow, there are at least two approaches of estimating the friction factor.

; One approach is to express the equivalent roughness of the interface in

terms of other system parameters such as phase velocities, void fraction

and surface tension. This approach has been used by Cohen & Hanratty (1968),

!

, _- -__ _-,. . - - - - . - -,,_m.. - - - , . _ _ _ _ _ _ _ _ _ . - . . , _ . __ - - _ _ _ . - ,

-109-

and Kordyban (1974) with some success. However, as comented by Hughes

et al (1976), "the general applicability of the approach has yet to be

established due in part to the large number of degrees of freedom

associated with the interface flow regime."

Another approach is to use empirical correlation. Wallis (1970)

found that, in gas-liquid annular flow, the interfacial friction factor

can be correlated by

f4 = 0.005 [l+300 6/D] 7.12-----

Moeck (1970) also proposed a similar correlation. Eq. 7.12 has

been suggested to be used in inverted annular flow by Hughes et al (1976).

In such case, 6 is the vapor film thickness. Until more satisfactory

correlations become available, eq. 7.12 will be used in the present

analysis.

A simple approach to correct the effect of evaporation is to use

the Reynolds flux concept. This has been developed by Silver and

Wallis (1965). The correction is expressed as follows6

i- -------=e 7.13f

where f * and f are respectively the friction factors with and9 g

without phase changes;

2r&= _

P "vv

and the phase change mass flux is

| dG2 V'

nRd2

r = 2n(R-6)

2R dG,

,

2(R-6) dz

._

-110-

Various similar expressions have been proposed by Mickley (1954)

and Spalding (1963), among others. In view of the empirical nature of

these expressions, there is no strong reason to reject either one. For

the present analysis, we will use eqs. 7.12 and 7.13, together with a factor

C) to be adjusted later. Thus the interfacial friction factor is. -

* If = C)0.005 1+300 e - - - - - - - - 7.14j

- -

and the interfacial shear is1

f ju-uj(u-u)j =yj pTy y y

VII.4.2 Eddy Diffusivity in the Liquid Core

The turbulent thennal diffusivity is required in equation 7.10

and also in evaluating $g from the temperature gradient at the vapor-liquid

interface. Two possible sources of information available from literature

are pipe flow and jet flow. The absence of a rigid wall in the present

situation seems to suggest that conditions are closer to jet flow than

pipe flow. In free turbulent jet flow, it has been found that a constant

value of c, across the main part of the jet is a reasonable assumption

(see Hinze (1975)). ,Using Prandtl mixing length hypothesis, the eddy

diffusivity for momentum transfer is given by

= C)b(G ~bmin}em mx

where b is the mixing zone and C) is a proportionality constant.

We intend to use this expression in the present analysis. Since

the turbulence in the vapor-interface, and consequently within the liquid

core, is expected to increase as the vapor film thickens,, the mixing zone

will be defined to be half of the vapor film thickness (i.e. from the point

of maximum velocity to the interface). In addition, the velocity difference

-111-

is assumed to be proportional to the average liquid velocity. Therefore,

c ,= CUg (h 6 )T

By assuming that Reynold's analogy is valid, the constitutive equation

for the eddy diffusity of heat transfer in the cresent analysis is

cg cu,, (y 6 ) - - - - - - - 7.15T,

VIF.5 Solution of the Momentum and Energy Ecuations

To carry out the analysis further, information about the velocity

profile has to be specified in the turbulent film region. We first express

G,M and E in tenns of dimensionless quantitiesy y y

* *Define u =

v

and n= yV

We further stipulate that all vapor properties to be evaluated at the film

temperature i .e. at T =hT,+T). From the definition of Gy 3 y

R1 f 2* O u rdrG *

y 2 vynR J

R-6

and on replacing r by (R-y), we obtain.

~ .5 .62

G =7 p u Rdy - pupdyy yyR

'O 'O.

*+-

f*f.

2.

# +* dn - u r,dnRu u= g yy y

'O-_

__

-112-

6+

*

Define 9) u dn=7.16-----y

0

f* 6 *+and 8 " Ud"* 7 I72 'i

~~-~~

)0-

2 -

..G =2 0#"'

Ru S) S 7.18- -----y y 2'W

Dv

.R

1 2My =q 2np u rdryy

7R ,

R-6

6

2 2( R-y)dy=7 ouyyR s

0 + +P +2 lf

q+u *22 U UdU= pv dn - vyy yi. 0 0 J

e6+*2Define 9

3"U "v dU 7 I9~~~~~

O1

'6' |+2and 6 u nda 1=

7.204 -----|

'O

F T,'

-

12 2 Cv 1.

M =7ov R e3-04 7.21-

y -----y y yR v

I. _

||

|

|

_ _ _ - - - _ - - _ - -

- _ _ _ _ _ _ _ .

-113-

Re

2:o u h rdrE =y y y y

R-6

1 @y 2 * #v"vrdrh x=

xR jR-6

7.22hG= -----y y

R-6

2M 2#0 u rdr=g 2 tg

sR

0

2 (R-6)2*}2*tLU 2

-2-

* II 6)2G g

KAt o (1-a)g" '

2-2'

2 (G-G)R= (1 g) j3 y

(R-6)2L~

-2~

~( G-G ) Rj y=- 7.23-----

R-6 -

L-

VII.5.1 Laminar Vapor Film

Substituting eq. 7.22 into eq. 7.7, we obtain

2R (h -h )G = 2R* TOT - 2(R-6),g 7.24-------

y f y

The liquid heat flux s is given by eq. 7.11. Using a numericalg

method to integrate step by step in the z direction, we can assume that at

each step (h -h ) is independent of 2. Therefore, eq. 7.24 becomesy f

dG2R (h -h ) V = 2R* TOT - 2(R-6)$g.y f

Combining this equation with eq. 7.1, the vapor film thickness

can be obtained upon integration.

.

_ _ _ _ _ _ _ . _ - . _ _

.

-114-

VII.5.2 Turbulent Vapor Film

i

For the turbulent vapor film, we assume that it consists of a

laminar sublayer and a turbulent layer. This two-layer approach has been

used by Hsu (1960) to analyse film boiling on vertical surface. In the

present analysis, we assume that in the laminar region

+u =n 0<n<n

s

and in the turbulent region the velocity profile is symetrical

about the line of maximum velocity. In each half of this turbulent region,

the 1/n-th power law is assumed to hold, i.e.1

+= Cn* n>nu

s

| where m=7

C = 8.74

Similar approach was used by Dougall and Rohensow to analyse the

vapor film. They used the universal velocity distribution law.

The thickness of the dimensionless laminar sublayer n has to bes

specified. Its lower limit is equal to 5, as has been found in pipe flow.

The buffer zone usually lies between 5 < n < 26. However, similar to the

analysis of Hsu (1960), the buffer zone is omitted and the laminar and

turbulent layers are both extrapolated so that they coincide at the

transition point. Hsu chose a value of 10. Therefore, for the presentanalysis

5<n < 26s

Having specified the velocity profile, eqs. 7.16, 7.17, 7.19,

7.20 can be expressed in terms of n.

- _ _ _ _ _ _ _ _ _ _ _ _

___

-115-

6+I

Oj= Cn" dn*

..

0

U Ucs p j_.

ndo + 2 Cn'dn= ',

0 ns

m+1~

2 ( m+1)n , ,+ 2 mci n -n |5

7.25= 3 ----2 m+1 ( /

Similarly2el bl.

02* U +2 0 ~U L----- 7.26s,

3 2+m 2+m3m0 = + 7.27(n

_U3------

s j2+2m 2+2m4 2[ )n m m

4=j+mC i n -n 10 7.28s ------

Integrating the energy equation 7.24, we obtainz

2R (h -h )G [2R$ - 2(R-6)4 ]dz=

y f y TOT g~

0

and af ter rearranging

I

[2R&by " R (h -h ) d

TOT -2(R-6)$g]dz2y f 0

From eq. 7.18 -

2 -

P "v2 vG =7 Rp e) 0-y y 2

-

V -

_ _

-116-

.zv

*

R O) - 0 [2R$ TOT - 2(R-6)$g]dz* 7.29.. -----

2 2p (h -h )y y f s

_E. O

Pv

The integral on the R.H.S. can be evaluated since $ isTOT

specified and o is given by eq. 7.11. Let us denote the R.H.S. by Q andg

substitute eq. 7.25 and 7.26 into 7.29. The resul t, after rearrangement,*is m+1 -

2 5 RQ-n +2 men, ,+1,

3 6 - - 7.30n =

( 2m+1 )R- (m+1)- _ (m+1)(2m+1)

-

.

If we assume that inverted annular flow exists at void fraction

below 40%, then (2m+1) R n [6(m+1) in most situations. The laminar

sublayer thickness is also negligible compared to R. Thus eq. 7.30 can

be simplified tom

m+1 m+1

-/ " ~

(Q - n 2 R + 2mcR n m+1)s Y sU *

2mCR

Differentiating w.r.t. z, we obtain _j#I

m+1

2 f + 2mcR n m+1)da 3 Q-n 3 ,+ )" m ,+1 Hdz 2mCRj

. .

(m+1-1 S

"2CR U /

-t

f n." g 7.31--

h is, except for a factor, equal to the integrand of eq. 7.29.

_. - ____ _ _____________ _

-117-

We can now substitute M and M into the momentum equation 7.5.y g

Details of the mathematics are given in Appendix I. The result is an

ordinary differential equation in the form

jh=f +# 7.32f -----

2 3

where the f's are functions of n, n , 6, G and fluid properties. hiss

given by eq. 7.31. Thus eq. 7.32 can be integrated numerically.

At each step of the numerical integration, the film temperature

is updated. This is equivalent to evaluating the wall temperature and is

considered in the next section.

VII.6 Prediction of Wall Temperature

Assume that the total heat transfer rate at the wall can be divided

into a conduction and a radiation component, i .e.

9 TOT " 9 cond* 9 rad

It has been shown by Nicholas (1965) that such assumption is

justified for optically thin * gas.

(1) Conduction Component

For the conduction component, we assume that most of the resistance

to heat transfer lies in the laminar sublayer adjacent to the heated wall.

If we denote this sublayer thickness by 6 , then the conduction components

is given byk (T,-T )y s

9cond = 2nR 6OZ

s

* An optically thin gas means either self-absorntfer. is negligible or thedimension of the gas body is small. See Atpendix H for detaileddiscussions.

- _ _ _ _ _ _ _ _ .

-118-

In the laminar vapor film region, 6 is taken to be the sums

of the distances from the wall to the point of maximum velocity and

one half from the point of maximum velocity to the interface. The

reason is that from the point of maximum velocity to the vapor-liquid

interface, the turbulence induced by the irregularity of the interface

results in less resistance for heat transfer. Such assumption also gives,

i a smoother transition from the laminar region to the turbulent region.

From section VII.2.1, the maximum velocity occurs at,

(1 A)+ =0b= "dy 2p 6 6y

'

2u U '~

or y = f V1+2'

90 6g. -

! Close to the pointwhere the vapor film is first created, y may

be greater than 6. In this case, the maximum velocity occurs at the inter-

face and therefore y = 6.

In the turbulent region a is given bys

* - n$ 6 I6s 2n-n |s

!

(ii) Radiation Component'

lThe radiation component is evaluated by assuming radiation exchange 1

between the tube wall and the liquid core, with an absorbing medium in )

j between. Details of the calculation are given in Appendix H.|

i

.

__ __ _______ __ _____ _ . _ _ _ _ _ _ _ _ _ _

-119-

VIII.0 DISCUSSION OF THEORETICAL MODEL

VIII.1 General

The theoretical model developed in the last chapter incorporates the

effects of (i) mass flux, (ii) inlet subcooling, (iii) axial location,

and (iv) system pressure to predict the surface temperature at a given

heat flux and an inlet condition. Most of the uncertainties lie in the

value of the thermal diffusivity of the liquid core and the thickness of

the dimensionless laminar sublayer n . In this chapter, the parametrics

effects on the predicted wall temperature and heat transfer coefficient

are discussed. Unless otherwise stated, the standard run has the

following values for the parameters:

D = 0.01194 me

n * = 10

P = 105 kPa

C) = 1.5 (This is a factor to adjust the interfacial

friction factor, see equation 7.14.)

VIII.2 Effect of Mass Flux

At the same inlet condition, the quality at any axial location

depends on the mass and heat fluxes. Therefore, to compare the effect of

mass flux on a local condition basis, the inlet temperature is kept

identical and the heat flux is set proportional to the mass flux, so that

the local equilibrium quality is identical.

- _ - _ _ _ _ _ - _ _ - _ _ _ _ _ _ _ - _ _ _ _ _ _ - _ _ _ _ _ _ _

-120-

The wall temperatures predicted by the model are shown in Figure

50. The discontinuities in the curves are due to the transition from

laminar to turbulent film flow. In the theoretical model, laminar flow

is assumed to change to turbulent flow when the local Reynolds number

n is greater than n*. The neglect of a transition region results ins

some discontinuity and shows up in the predicted temperature profile.

Figure 51 shows schematically the thennal sublayer where all the

resistance to heat transfer is assumed to concentrate. By appropriately

choosing n*, the discontinuity in d at the transition point can beTh

minimized. This will be considered in the next chapter.

The overall heat transfer coefficients (based on T -T ) are showng 3

in Figure 52. This 'overall' HTC includes the contribution by thennal0radiation, which ranges approximately from 10% at T,=600 C to 40% at

T =10000C. The ' convective' component of the HTC is shown in Figure 53.y

It can be seen from these two figures that both the overall and

convective heat transfer coefficients increase with mass flux. The

increase is greater at higher mass flux.

The increase in the convective component is due to the increase

in the vapour film velocity. The predicted vapour velocity is shown in

Figure 54. It indicates that the vapour velocity increases with mass

fl ux. The vapour velocity reaches a maximum at around 45-50 cm. This will

suqqest that the actual flow regime will change into a dispersed flow.

VIII.3 Effect of Local Equilibrium Quality

The predicted heat transfer coefficient at the same inlet

temperature and mass flux are compared in Figure 55. It can be seen that,

_ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

-121-

1200.

2G (kg.m-2.s-1) 4(kW/m ),,,

O 100 50A 200 100+ 400 200 -

less.

+P = 105 kPa #T = 800C +,, p,3 , , in +0 -

+m

688.tt! +TDp 7ss. +

'

G "Of 3W Ag + 3 4ees. _

I- ALtJ

^F sea.

J h

_J +D DE 4se. o o 3-

3 3

0388. O

A O

288.O

100.E. 5 IS. 15. 28. 25. SS. 36. 48. 46. 58.

AXIAL DISTANCE (CM)

FIGURE 50 PREDICTED WALL TEMPERATURE

_ . . _ . _ _ _.

-122-

'

|

, -

1

I

/# LOCUS OF MAXIMUM // VELOCITYg

/ \ , o- bg/ \ r

\ /DIMENSIONLESS/ -6E / e '/ VELOCITY/ Th ! j / s PROFILE

t / /

I / /j'# ! / / VAPOR - LIQUID

/ VAPOR - LIQUID / INTERFACEINTERFACE /j /

/ / -

/ !,

//s '

(i) LAf11NAR VAPOR FILM (ii) TURBULENT VAPOR FILM

FIGURE 51 DEFINITION OF THERMAL LAMINAR SUBLAYER

|

\

'

|

'

|

|

- - - = -_ .

_ _ - _

-123-

.

000.

I IO

G (kg.m-2,3-1) 4(kW/m )2

7ss. O 100 50A 200 100+ 400 200

P = 105 kPa^.... T - 80 C

in

TOTh = T -T

, w s

rses.-

N.O

EN2 4... +m

I o+

300.4

s +D +

+ +^ +a +

2... 3, ,D O

D D

ISS.-4.9 -3.5 -3.3 -2.5 -2.3 -1 5 -1.3 .5 e.8

EQUILIBRIUM QUALITY (2)

FIGURE 52 HEAT TRANSFER COEFFICIENT AS A FUNCTION OFEQUILIBRIUM QUALITY

I

-124-

800.

G (kg.m-2 .s'I) 4(kW/m )2o700. ~

O 100 50~

A 200 100+ 400 200

| oes. -

3 P= 105 kPaT = 800C+ in

~TOT"* rad^

h_% 500. (T -T ) _g 3

N..

G 488.__

3v +

Z po 388- -

O +I

* +200. _

.__

a +

} $ P oS4

100.

8.

-4.0 -3.5 -3.0 -2.5 -2.0 -15 -1.0 -5 0.0.

EQUILIBRIUM QUALITY (%)

FIGURE 53 CONVECTIVE COMPONENT OF HEAT TRANSFER COEFFICIENT

--

. _ _ _ - - _ - _ -

.

________

-125-

24.2G (kg.m-2.s-I) 4(kW/m ) -

50'

O 100 100A 200 200 ++ 400g,,

+m 18 P = 105 kPa_

(D T = 800Cing

{ te.a3

+>- a

14. ,

O oO 12.

a_JW o> is.

Of +O e.g

h a D

6 +0

a04. +

bh2. -

we. D

B. 5. 18. 16. 28. 25. 36. 36. 48. 46. 58.

AXIAL DISTANCE (CM)

FIGURE 54 PREDICTED VAPOR FILM VELOCITY

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

-126-

at the same inlet condition, the overall heat transfer coefficient

decreases with the local equilibrium quality. -Figure 56 shows the same-2 -Ieffect at a mass flux of 400 kg.m .s .

The increase in the heat transfer coefficient in the subcooled

region is due primarily to the increased portion of the ' total heat flux

transferred to the. liquid core. This results in less vapour generation

and consequently a thinner vapour film.

VIII.4 Effect of Inlet Subcooling

The effect on the heat transfer coefficient of an increase in inlet~

subcooling at a fixed mass flux is shown in Figure 57. At higher inlet

subcoolings, higher heat fluxes are required to maintain the tube in film

boiling. Therefore, the comparison shown in this figure is based on the

same outlet equilibrium quality at the end of a 60 cm tube. The improved

heat transfer is most noticeable near the entrance. Further downstream

the heat transfer coefficients for different inlet subcooling approach,

l

each other.

The improved heat transfer is due to the higher heat transfer to,

Ithe liquid core near the entrance. The development of the temperature1

profile of the liquid core for one typical case is shown in Figure 58.{

It shows a high temperature gradient at the entrance, which subsequently4

diminishes as the thermal boundary layer develops. Figure 59 shows the

percentage of th_e total heat flux transferred to the liquid core. At

higher inlet subcooling, a higher percentage is used to heat up the

liquid core. The increase in the percentage of liquid core heat flux at

-127-

ees.

see.(kW/m ) G = 200 kg.m-2'.s-I

P = 105 kPa0 200 T. = 800C

'"See. A 150

+ 100

|| 4ss.

m

M 4s8..

N.

h "'' 3+\3 O" ass.

I.+A O268.

o O

+3b+ b280. ,

+ + ++ s+

A150.

188.-4.s -3.s -3.s -2.5 -2.s -1.5 -1.s .s s.g ,s g,g

EQUILIBRIUM QUALITY (%)|

FIGURE 55 EFFECT OF EQUILIBRIUM QUALITY ON THE HEAT TRANSFERCOEFFICIENT

_

-128-

Oss.

1

2 -I& (kW/m ) G = 400 kg.m-2 .s

P = 105 kPa;gg, 3T " = 800CA 150

ID 200

e88. O

m

588. A

m.S

488. O#2

v

D

388.

O1 AO

A O O,A 0

288. 4

108.

8.i -4.0 -3.5 -3.3 -2.5 -2.0 -1.5 -1.6 .5 8.3

1EQUILIBRIUr1 QUALITY (%)

FIGURE 56 EFFECT OF EQUILIBRIUM QUALITY ON THE HEAT TRANSFERCOEFFICIENT

.

-129-

ees.

G = 200 kg.m-2 -I550.in(OC) 4(kW/m2).s T

P = 105 kPa+ 70 192O 80 150See. a 90 108

es.

+

m

M 4se. __

.

ea.* see.EN3 +v see.

IO

250. A4e +

D+gO2es. A

faAao

'

158. aA

I S S . --

-B. -5. -4. -3. -2. -1. 2. 1. 2.


FIGURE 57EFFECT OF INLET SUBC00 LING ON THE HEAT TRANSFERCOEFFICIENT

. ----

-130-

kfI

I0 - 4

50 cm0.2 -

40

, 0.4c -

T T 30

E 0.6 -

205 10

0.8 -

1

I1 -

i

I

k i l I i1 0.8 0.6 0.4 0.2 0

r

R-6

G = 200 kg.m-2 .s-lT = 800CinR = 0.00597mNUMBERS ON CURVES INDICATE AXIAL DISTANCE

FIGURE 58 DEVELOPMENT OF THE TEMPERATURE PROFILE IN THELIQUID CORE

_ _ _ _ _ __

-131-

Iss.

La

G = 200 kg.m-2 -I.sos.A P = 105 kPa

2ATin ( C) $(kW/m )

_es.

A 70 192 ,

O'90 108| 0

78.

i N ^j v

es. OgOl-

{ Es. O

ctN A

O48.

O_.]-

1 Elo_ se.

O

28. g

18.

| E.

s. 5. Is. 16. 28. 26. 36. 36 48. 46. 58.

AXIAL DISTANCE (CM)

FIGURE 59 FRACTION OF THE TOTAL HEAT FLUX TRANSFERREDTO THE LIQUID CORE

__.

-132-

around 25 cm is due to the increase in the turbulent thermal diffusivity.

Had this diffusivity remain constant along the boiling length, the heat

flux to the liquid core would have decreased monotonically. However,

in the model the diffusivity is assumed to increase with the vapour

rilm thickness. Physically, this can be interpreted as the increase

in the waviness of the vapour-liquid interface. Consequentty thei

resistance to heat transfer is reduced.

Based on the results of Figure 57, an entrance length * of 6 L/D's

is obtained. This is much shorter than in tube flow and can be attributed

to the increase in turbulence in the liquid core due to the creation of

the vapour film.

In the experimental program of this study, a hot patch was used

to establish film boiling at the entrance. As discussed earlier, CHF

occurred initially and developed into inverted annular film boiling. It

| is not certain how the radial temperature profile of the liquid core

adjusts under such circumstance. In the theoretical analysis, the liquid

core is assumed to be at a uniform temperature at the entrance. This

results in the very distinct entrance effect. Referring buk to

| Figures 31-37 where the convective Nusselt number is plotted against the <

equilibrium quality, the " scatter" can be attributed to this entrance

effect, although at a reduced magnitude.

* When a fluid flows into a heated channel, the temperature at the boundary

changes instantaneously from its temperature prior to entrance to the

temperature of the channel . The heat transfer coefficient is therefore

infinite at the entrance. As the thermal boundary layer develops, the

heat transfer coefficient diminishes. An entrance length is the length

over which the heat transfer coefficient approaches the " fully-developed" HTC.

_-

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _

-133-

VIII.5 Actual Quality in Subcooled Film Boiling

In low quality inverted annular film boiling, the vapour is

superheated above the saturation temperature, while the liquid remains

subcooled. Therefore, the equilibrium quality can be negative. The

actual quality, however, is always between 0 and 1. Its value depends

on the actual vapour temperature and the slip velocity. In the

theoretical model, the vapour film is assumed to be at the " film"

temperature (=0.5*(T +Tsat)). The velocity distribution across the filmy

is either calculated (for laminar film) or assumed to follow the law

of the wall (turbulent film). The vapour mass flux is obtained by

integrating across the film. The calculated actual quality (=G /G) isy

shown in Figures 60 and 61 for two mass fluxes. It can be seen that,

at the same mass flux, the actual quality increases with the heat flux

level. This can be explained as follows.

The heat conducted to the liquid core is limited by the temperature

gridient in the liquid thermal boundary layer next to the vapour-liquid

interface. Figure 62 shows the percentage of the liquid core heat flux

(based on tube diameter) at the vapour-liquid interface. At the same mass

flux, the thermal boundary layer develops approximately at the same rate

(neglecting for the time being the influence due to evaporation). Any

increase in the total heat flux will therefore be used for evaporation.

Figures 60 and 61 show that the actual quality depends not only

on the equilibrium quality, as is usually observed in higher quality

(dispersed flow) film boiling, but also on the heat flux.

.. __

!-134-

i

2.s

i.e P = 105 kPaG = 200 kg.m-2 .s-I

21.e c(kW/m )A 1000 150 U

|m 1.4 . .

Nv

>- 1.21-- O._.

!

E t.s

D0

-J .e U aCD}-O eg A

O\

.4 A

OO

2O a

OOAO as.s

^-4.s -3.5 -3.s -2.5 -2.s -1.5 -1.s .5 s.s .5 1.s


FIGURE 60 PREDICTED ACTUAL QUALITY AT G = 200 kg.m-2 -I.s

__ . _ _ _ _

-. . .. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ _ _ - _ _ _ _ _ _ _ _ _ _ _ _ _ .

-135-

.as

!.2e

P= 105 kPaG = 400 kg.m-2 -l

~

.s

4 ( kW/n$2).24

D 150.22 A 200

m an .2av

>- .ie

t--

g .ie

E14e

J .t2C 3D -

F- .ie

OC

.se A

.seA

.s4

A D

.se D3 g

cp os.es n o D 43t,

-4.s -s s -s.s -e.s -2.s -i.s -i.s .s s.s

EQUILIBRIUM GUALITY (%)

FIGURE 61 PREDICTED ACTUAL QUALITY AT G = 4 00 kg.m-2 -I.s

_.

-136-

tee,

b

D'8'

P = 105 kPa3G = 200 kg.m-2 -I.s

A2se. D 4(kW/m )

AA 100

O A O 1507,,

NOv 3

es.gOl--

'._. a

68.3o_ oNO

4s._.

d A !O*--. O

I AQ_ Gs.

O

r es.|

| || 1

is. I

s.-4.8 -3.5 -3.5 -2.5 -2.8 -1.5 -1.8 .S 8.8 .5 1.8


FIGURE 62 PERCENTAGE OF THE TOTAL HEAT FLUX TRANSFERRED |TO THE LIQUID CORE

|1

_ . _ _ _ _ _ _ . _ _ _ _ ___

_ _ _ _ _ _ _

|

-137-

VIII.6 Slip Velocity

In the theoretical model, the velocity of the liquid core is

assumed uni form. Nevertheless, it can still vary with the axial distance.

The development of the liquid core velocity is shown in Figure 63. The

increase is higher for higher initial velocity, i.e., for higher mass

fl ux. This can be explained by considering the hydrodynamics of the

vapour film.

The vapour velocities (averaged over the cross-section of the

vapour annulus) corresponding to the same conditions as those in Figure 63

have been shown in Figure 54. The vapour film thicknesses are shown in

Figure 64 The vapour velocity and the vapour film thickness are

primarily determined by the balance between the buoyancy force and the

sum of the wall and the interfacial shears. Figure 64 shows that the

vapour film is thicker at higher mass flux. A thicker vapour film

results in less flow area for the liquid core, and consequently the

liquid velocity will increase. -

Figure 65 shows the slip under the same system conditions. It

indicates that there is less slip at higher mass flux. For higher mass

fluxes, the slip reaches a maximum and then gradually decreases. This

suggests that the flow is changing to the more homogeneous dispersed flow.

It may seem doubtful whether some of the high slips as shown in

this figure actual can exist in inverted annular flow. The high value

is actually fictitious. It is due to the neglect of the velocity gradient

in both the vapour film and the liquid core in calculating the slip.

. . . . . . . ___

. . .

-138-

1.s

I

G (kg.m-2 -I) &(kW/m2)9 .s

0 100 50A 200 100

.0 + 400 200.

P = 105 kPa.7

.6 +m

N +E .5 + +" ++

+J ++) .4

3a

g 6 O O

1 O O O O O O O O O 3

Ia.e !

3. S. t8. 15. 28. 25. 30. 35. 40. 45. 52..

AXIAL DISTANCE (CM)|!1

FIGURE 63 PREDICTED LIQUID CORE VELOCITY|

|,

_ _ _ _ _ _ _

-139-

1.8

.

-2 2' ~

G (kg.m .s~) O(kW/m )

0 100 50a 200 100

'{ 1 4 - + 400 200Ev P= 105 kPa

(g 1.2 _

CDLLJZ +'I i.eO

_

aIF-

.e ___

E_] +~

All- .c ____+

0' +O 3 3

AQ- +E 4 _

3 OO O

> + A OUA+ D

.2 D+4)

__

s06.0

e. s. te. 15. 2s. 25. 32. 35 40. 45. 50-

AXIAL DISTANCE (CM)

FIGURE 64 PREDICTED VAPOR FILM THICKNESSs

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

i

-140-

120.

|

-2 2'

G (kg .m .s') 4(kW/m )-

O

O 100 50I"' --

A 200 100+ 400 200

00. _

P = 105 kPa O

80.

70.

O_~ e0. 4J O a0) a

50.

O 3

40.O

+ ,30.

O +O.

20. _A

O +A

10.O A +

A~~

3 --

8. n

0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50.

AXIAL DISTANCE (CN)|

|

FIGURE 65 PREDICTED SLIP

|

_ _ _ _ _ _ _ _ _ _ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . _

-141-

VIII.7 Effect of System Pressure

Although there is no data obtained at elevated pressure in the

present study, the theoretical model is derived for more general situation.

Recently, some low quality film boiling data are reported by Groeneveld

et al (1981). Figure 66 is reproduced here from this reference. The

predictions by the present model are shown for comparison. It seems that

there is a shift in the equilibrium quality between the predictions and

the experimental data. This is probably due to the exclusion of the " hot

patch" region in the theoretical model.

The increases in the heat transfer coefficients with increases

in system pressure is due primarily to the increase in vapour density,

which results in a thinner vapour film.

VIII.8 Effect of Tube Size

The effect of an increase in the hydraulic diameter on the heat

transfer coefficient is shown in Figure 67. It can be seen that, away

from the entrance region, the heat transfer coefricients increase with

the hydraulic diameter.

Near the entrance, the trend is different. This behaviour can best

be explained by considering the vapour film thickness. A thin film means

that the heat transfer resistance from the wall to the vapour-liquid

interface is small . On the other hand, the eddy diffusivity of the

liquid core is directly proportional to the vapour film thickness.

Therefore, as the vapour film grows, the resistance to heat transfer

increases in the vapour film but decreases in the liquid core. Depending

on which resistance is dominating, the heat transfer coefficient may

reach a maximum shortly downstream of the entrance.f-

-142-

DATA FROM GROENEVE_D ET.AL. (1981)1.400G = 0.377 Mg.m-2.s-1 PRESSURE

Ilin = 100 kJ kg-l x 8 MPa

HEAT FLUX USED IN PREDICTION + 6 MPa1.200 PRESSURE FLUX A 4 MPa28 MPa 240 kW/m

6 MPa 215 kW/m2 a 3 MPa23 MPa 180 kW/m

1.000 PREDI CTIIHL

W8 MPa bMg k.800

# fg6'MPa

% M4 +%Mk

t h'Wp

% %.c00 + + +Wn .pf

% # % y3 MPa b

.400 Nd #4 g gya

#a e **

.200

0.000.100 .060 .020 .020 .060 .100 .140 .180 .220 .260 .300 .340

|

QUALITY

FIGURE 66THE EFFECT OF PRESSURE ON THE HEAT TRANSFER COEFFICIENT

.

_ _ _

_ _ _ _ _ _ _ _ _

-143-

508.

P = 105 kPa2450. - 4 = 150 kW/m

Tin = 800C

De(m)408. - 0 0.008

A 0.05+ 0.10x 0.01194m

y ase.

oJ.

&

E see. __y3m

1. < _

ese. _

I a'

?286.

ISh. .- -

iee.-4.0 -3.G -3.8 -2.5 -2.8 -1.5 -1.0 .5 0.0 .5 1.E


FIGURE 67 EFFECT OF TUBE DIAMETER ON THE FILM B0ILINGHEAT TRANSFER COEFFICIENT

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . _ _ _ _ _ _ _ ..

-144-

In a uniformly heated channel, the increase in equilibrium quality

is inversely proportional to the hydraulic diameter. In inverted annular

film boiling, the heat flux which can be applied to a tube in film

boiling is usually limited by the wall temperature at the downstream

region where the quality is highest. Therefore, for the same length of

tube, the one with a larger diameter can sustain a higher heat flux.

VIII.9 Sensitivity of Model

Most of the uncertainties of the theoretical model lie in thetra ns . 1en the laminar and turbulent film flow, and in the constitutive

equations for she interfacial shear and the liquid eddy diffusivity.

Most of the dependence of these quantities on the flow field has been

incorporated into the model, leaving only a few factors to be adjusted

later. The effects of changes in these factors are discussed in this section.

VIII.9.1 Transition point between laminar and turbulent film

Figure 68 shows the effect of the changeover point from laminar to

turbulent. It shows that n = 12 gives a smooth transition. A larger n

also means that the vapour film is in laminar flow for a longer distance.

!! VIII.9.2 Interfacial shear

Figure 69 shows the effect of interfacial shear on the predicted

ter:perature. A higher interfacial shear means that the vapour experiences

a greater retarding force. This results in a thicker vapour film and

consequently higher wall temperatures.

|.

_ _ _ -_ _ . . -~

_ _ _ - _ _ _ _ _ _ _

-145-

1200.

P= 105.KFH G= 200.KG/r1**2.3 FLUX = 150.KLuth e 2

il00- - O ETAG= 7 --

A ETAS= t 0

+ ETAG= 12X CTA3=L5

1000. _

,

+ o900. ___. 3

4 ox800. _ g ___

h"

3 100. -

9h f a

600. ___

$500. _

400.

300.

O. 5. 10. 15. 20. 25 30. 35. 40. 45. 50. 55.

AXIAL DISTANCE (CN)

FIGURE 68 EFFECT OF THE TRANSITIONAL POINT BETWEEN LAMINARAND TURBilLENT VAPOR FILM ON THE PREDir.TEDWALL TEM ERATURES

-146-

1200.

P tas.KPA 0= 208.K0/No*2.8 FLUX = 1ss.Kt4Eis *21100. O Cet0.7 .

a cct.1. 0+ cc t . l . 5X CC1 2. 0

g

AO

900.O O

+

X

800. + b U^

C g*

t O

$"

3 ,ee.

@

eaa.

S 1

sea. |

423.|

I

s80.a. s. te. is. ee. 2s. se. es. 4e. 4s. sa. ss. |

AXIAL DISTANCE (CM)'

FIGURE 69 EFFECT OF INCREASE IN INTERFACIAL FRICTIONON THE PREDICTED WALL TEMPERATURE

|

_ _ _ _ _ _ _ _ _ _ _ - .

-147-,

VIII.9.3 Liquid eddy diffusivity

Figure 70 shows the effect of the liquid eddy diffusivity. A

higher diffusivity results in a greater fraction of the heat flux conducted

to the liquid core and therefore less for vapour generation. Consequently,

the vapour film is thinner and the wall temperature is lower.

VIII.9.4 Vapour density

In the development of the model, all of the vapour properties are

evaluated at the film temperature. The properties that affect the

prediction most significantly are the thermal conductivity and the density.

The fonner affects the prediction of the wall temperatures through its

effect on the resistance to heat transfer in the laminar sublayer next to

the wall. The latter affects both the predicted wall temperature and void

fraction.

Since the laminar sublayer is close to the wall, it seems reasonable

to assume that the thermal conductivity can be represented by its value at

the film temperature.

The portion of the vapour film beyond the laminar sublayer is assumed

not to offer any resistance to heat transfer. This implies that the density

of this portion should be evaluated at the saturation temperature. However,

in models that contain some simplifying assumptions, it is often found

necessary to distort a certain aspect to compensate for it. The ultimate

criterion lies in the comparison with the experimental data.

In order to show how the vapour density affects the predicted wall

temperature and void fraction, two runs are compared: one with the vapour

_ _. ._

-148-

|

1200.

| |,

P= 105.KPA G= 200.KG/M**2.S FLUX. ISO.lW N'*21100. O T(RB.O. 05

A TtRB.O.08+ TtRB 9.15

1000.a

+|

O900.

+ |

O

800. O A^

O & + +4

Ov 3

3 700.F-- A

O +

600.

AS00.

_

400. +

1

|

|'

300.0* 5. 10. 15. 20. 25 30. 35. 40. 45. 50'. 55.

AXIAL DISTANCE (CM)

FIGURE 70 EFFECT OF INCREASE IN TURBULENT THERMALDIFFUSIVITY OF THE LIQUID CORE ON THEPREDICTED WALL TEMPERATURES

..

- _ _ _ - _ _ _ _ _ _ _ .

_ _ _ _ _ _

-149-

density evaluated at the film temperature and the other at the saturation

temperature. The results are shown in Figures 71 and 72. It can be seen

that the predicted wall temperatures are increased by approximately0100 C when the density is evaluated at the film temperature. The void

fraction is also increased.

The measured data favour a higher wall temperatures and void

fractions. Therefore, all vapour properties in the model are evaluated at

the film temperature.

-______ _ _ _ _ _ _ _

-150-

| \

1209.

l ! !

P= L35.MPA G= 233.KG/Mo*2.S R.UX. ISO.KW Me*2ILSO. O Tv= 0.S* (TW+TS)

A TV TS

1900. O

O|

980. O

AO^ 800. O

Q O A

Om

] 730. A Oa

F- O .

A

633. _

O

S03. a_

400.

320.a. s. te. is. 2.. 2s. s. as. 4. 4s. se. ss.

AXIAL DISTANCE (CM)

FIGURE 71 EFFECT OF DIFFERENT ASSUMPTIONS OF THE AVERAGEVAPOR TEMPERATURE ON THE PREDICTED WALLTEMPERATURE

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _____________- _ ______-________________________ ____ ____. . _ .. ..

_ _ _ _ _ _ _ _ _ _ _ _ _ _

|

-151-

100.

P. 105.KPA G. 200.KG/Me*2.S FLUX- 150.KtuM* *298" O TV 0.S*(TW'TS)

A TV.TS

es.

70. __

hN- O

"''ZO a-.

F-. 50.gEcr O aw

4e.

o- ao> se. O

b

28. OO

O O

OO10.b

28.

-4- -3- -2. -1. g- 1. 2.


FIGURE 72 PREDICTED VOID FRACTION WITH DIFFERENTASSUMED VAPOR TEMPERATURES

_ _ _ _ _ _ _ _ _ _ _ . . _ _ _ _ _ _ _ _ _ _ _ _

-152-

IX.0 COMPARISON WITH EXPERIMENTAL RESULTS

IX.1 General

The theoretical model is used to predict the experimentally

measured wall temperatures. The parameters which are left indeterminate in

the model are then chosen to fit the data. To recall briefly, the most

sensitive parameters are the interfacial friction and the eddy diffusivity

in the liquid core. With the wide range of data in hand, it is possible

to use a least squares fit methodology to calculate the best estimates for

these parameters. In fact, a preliminary version of FORSIMOPT (Selander

1980)) is available. This computer code will solve the coupled PDE/0DE's

and adjust the parameters in the equations to fit the input data. However,

the computing time would be prohibitively large, since at present it

requires about 100 s computing time on the CDC Cyber 170 to integrate a

tube length of 50 cm. In addition, some of the constitutive equations used

are only best estimates for the real situation. It does not seem to justify

a more precise (in the statistical sense) evaluation of the parameters. |

The final method used is to choose various sets of data and compare

them with the predictions graphically. For each set of data, the average

of the pressures at the inlet and outlet of the film boiling section as!

calculated in the data reduction is used as the system pressure. In |

addition, the average heat flux is used as the boundary condition. The

initial conditions are the mass flux and the inlet temperature.

The value of the inlet temperature requires some explanation. The

theoretical model does not take into consideration the inception of CHF

at the start of the test section. It is not certain how this would

affect the radial temperature profile of the liquid core. Therefore, in |

|

,- - - - _ _ . - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - .

.

-153-

the theoretical prediction, two initial conditions are considered: in

one case the inlet temperature is set equal to the measured inlet

temperature, while in the other case an "artifidial" inlet temperature is

used so as to match the equilibrium quality at the first thermocouple

location. For those runs with high mass fluxes, these two values are not

sery different and the predictions should bound the measured data.

Such " eyeball" fitting produces the following estimates for the

parameters

(i) Transition between laminar and turbulent vapour film, n = 12s

(ii) Proportionality constant for interfacial shear, C) = 1. 5

(iii) Proportionality constant for eddy diffusivity, TURB = 0.08

in addition, the theoretical model is only written for inverted

annular flow. As the thickness of the vapor film increases, some of the

assumptions may break down and will cause the predicted temperature to blow

up. There are also uncertainties in the transport properties of superheated

steam (Groeneveld et al (1981)). Therefore the computer solution is

allowed to proceed up to a film thickness of 0.5R (n = 75%), or a wall0temperature of 1600 C, whichever occurs first.

IX.2 Prediction of Wall Temperature

The predictions are shown in Figures 73-86. It can be seen that, in

general, the trend and the magnitude of the wall temperatures are correctly

predicted. The model performs best at high subcoolings, when it can

safely be ascertained that the flow is inverted annular. At close to

saturated inlet conditions, the generated vapour will probably be entrained

in the liquid core. This effect has not been modelled.

.-

f

-154-

12ss.

P= 181.HPA G= 180.KG/No*2.8 FLUX = 02.HWNe a2

1198. O CtBS.481 measured

A TIN:48.4 C{,

+ TIFkO6 4 C J predicted

1988.,

O088* O

O088. +^

U O

v +O2 7ee. ^

+O s

+ aees. 0 a4

A

OO5e8.

A

|A4es.

.1

see.;

s. 5. 18. 15. 2s. 25 Gs. 05. 4a. 45. 58. ,5 .5.

AXIAL DISTANCE (CM)\

FIGURE 73 COMPARIS0N OF PREDICTED AND MEASURED WALL ITEMPERATURES, G = 100 kg.m-2.s-l,T. 600C=in I

-155-

1200. __

P= le4.KPA O= 100.K0/Ne*2.8 FLUX = 71.KWNe * 2

1100. O Cise.20 measured-

A T!Ns08.8 C{+ TIN:93.7 CJ

l800.

9J8.

O

O+^ Ben.O--

Oh

U] 706.

#O ,

D Aees. +

+ aa

sea. ,

a

488. g

.

Gee.s. 5. 18. 15 28. 25. Gs. G6. 48. 45. 58. 55.

AXIAL DISTANCE (CM)

FIGURE 74 COMPARIS0N OF PREDICTED AND MEASURED WALLTEMPERATURES, G = 100 kg.m-2.s-l,T 800C=

in

- - _ _ _ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ _- .. - -

. _ _ _._ . . .

-156-

1290.

P= 197. 6 G= 158.K0/Me*2.8 FLUX = 137.M4me s2

1t98. O CtSS.Get measureda TIN =70.3 C

predicted+ TIN =85.5 C

1999. O

O

900.A

U+^ 899.O +

3

" + A

2 7se. O

F-

AOeso.

dsee.

408.

see.8. 5. te. 15. 2s. 25. 3s. 35. 40. 45. se. 55.

AXIAL DISTANCE (CM)

COMPARIS0NOFPREDICTEDANDMEAjuREDWALL iFIGURE 75TEMPERATURES, G = 150 kg.m-2.s- '

,

00CT =

in

.. _ _ - _ _ _ _ _ _ _ .

_ . . .

-157-

12se.

P= 113.MPA O. 20s.K0/Mee2.8 Ft.UX* 187.HWN'*2time. O c2es.sSt measured

A TIN 95.9 01+ TIN:15p.3 C J predicted

HSU (10600C)tees.

see.

d O o+ g^ ees. o

O A+

HSU (7100C)v

f'

ELLION

ees.

See.

4se.

Oss.e, s. te. 15. 26, 25. 3s. G6. 4a. 46. se. SS.

AXIAL DISTANCE (CM)

FIGURE 76 COMPARISON OF PREDICTED AND MEATEMPERATURES,G=200kg.m-2.s-{UREDWALL

,

T 950C=in

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - ____

-158-

1298.

P. 1IG.KPA O= 200.K0/Ne*2.8 FLUX = 143.HWNe *2

iies. O C2ss.2el measureda TIH=05 8 C predicted+ TIN =0s. 8 C

;&S8. -

4 no o

oss. o ---

+ aO

^^ 088. ELLION

~ .O A

] 780.H + oa

HSU (6120C) |ees.

AO

580.|

4es.|

| see.|| s. s. is. is. 2s. 2s. os. os. 4s. 4s. ss. ss.

| AXIAL DISTANCE (CM)|

|

I FIGURE 77 COMPARIS0N OF PREDICTED AND MEA URED WALLTEMPERATURES, G = 200 kg.m-2.s- ,

800CT =in

!

_ _. ________ _ _ . _ _ . _ _ _ _. _ _ _ _ _ _ _ _ _

-159-

1288.

P= le7.KPA 0= 250.K0/Nes2.8 FLUX = 173.KWNe s2

atse. O c25s.381 maasured

a Tir>78.7 C [ predicted+ TIN *00 8 C)

tese. AO

bO000. ._

,

+ a^ Gee. OO +

^v

] 788. + 3}--

Oees.

O

OSee. AO

O

4se.

Gee.8. 5. t8. 15. 20. 25. 30. 36. 48. 45. 58. 66.

AXIAL DISTANCE (CM)

FIGURE 78 COMPARISON OF PREDICTED AND MEATEMPERATURES,G=250kg.m-2.syVREDWALL

,

T 00C=in

- . _ _ _ ._ -.-

-160-

,

1200.

P= t l8.HRA 0= 3SS.KG/Mee2.8 FLUX = L58.HWM+ e2

1t98. O C398. l91 measuredA TIN =00.9 C k predicted+ TIN-93 9 C)

1998. . _ _

O

A U90s.

o+ 4g

898. O _ _ _ , ,

b+v

) 3 7ss.3

F--

ees.

500.

4se.

,

see.s. 5. te. 15 28. 25. 30. 35 40. 45. se. 55.

AXI AL D L8TANCE (CM)

; FIGURE 79 COMPARIS0N OF PREDICTED AND MEASURED WALLI TEMPERATURES, G = 300 kg.m-2.s-l,

T. 900C=

| in

_ ._.

-161-

12se.

P= 112.HPA 0= 308.K0/N'*2.8 FLUX = 109.HWN'*2

ties. O C3se.201 measured

A TIN 88 5 Clpredicted+ Tit &8e,5 CJ

tese.

O

+ 0a

988. -

+ 0 3

+ 3^ 088.C)

-

+ A

"00

] 788. + 0

00

000.

GL

SEE.

4es.

SSS.

s. 5. 18. 15. es. 25. Gs. GS. 48. 46. 58. SS.

AXIAL DISTANCE (CM)

FIGURE 80 COMPARIS0N OF PREDICTED AND MEASURED WALLTEMPERATURES, G = 300 kg.m-2.s-l,

0.50CT =

in

I. ._ -

-162-

12ee.

p= 184. 6 O= 489.KG41* *2.8 FLUX = 172.MWMe *2

1ies. O C4ss.2s1 measuredA TIN =89.9 C predicted+ TIN =84 7 C

tees.

+AO

999. +A

+AD

^ Aees.U + mv

A3 7es. +

gF-A

ees. + 0LP

O

see. OA

4es.

308.s. 5. te. 15. 2s. 25. 38. 35 48. 45. 5s. 55.

AXIAL OISTANCE (CM)

FIGURE 81COMPARIS0NOFPREDICTEDANDMEAyUREDWALLTEMPERATURES, G = 400 kg.m-2.s- ,

T = 800Cin

- . . - _ _

..

-163-

12se.

P= 135.HPA 0- 5ss.K0/Me*2 8 FLUX l93.lWMe *2ties. O c5ss.s51 measured

a m w.s c predicted+ Temn.2 c

less.

AO O

O b AO

ess. O + 00 A

+ A^ ass.

A+v

3 7se.>- a

ess.

see.

Abs.

see.s. 5 ts. 15. 2s. 25 as. 35 4s. 46 58. 55

AXIAL DISTANCE (CM)

FIGURE 82 COMPARISON OF PREDICTED AND MEATEMPERATURES,G=500kg.m-2.s-{UREDWALL

,

T 950C=in

_ _ - . -

- _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

-164-

l120s.

P= tes.HPA 0= See.KG/Ne*2.8 Fl.UX= 21e.MWNe e2

11ee. O CSee.2e2 measured

c) predictedA TIN =0s.eCf+ TIN =84 3

Otees. +

b

b+ 0

ese. ^

+ 30

ese. + ^^

OO^v

+2 7s8.I- ^

O

O

ees. O AO

See. ^

<

488.

I

Gee.s. 5. t8. 16. 28. 25. 38. 35. 48. 45. 58. 55

AXIAL DISTANCE (CM)

FIGURE 83 COMPARISON OF PREDICTED AND MEASURED WALLTEMPERATURES, G = 500 kg.m-2.s-l,T = 800Cin

. _ _ _ _ _ _ _ _ _ _

_ _ _ .

-165-

1208.

P= 113.KPA O= 268.KG/Nu 2.8 FLUX = 132.HWNu2

11es. O E288.191 measured

A TIN =08.8 Cpredicted

+ TIN-04.B C

1888.

O

088.O

+ A

^ 888.O + a

O+ Av g

3 7se. - 0 AF- +

ese.

see.

/

4se.

Ges.8. 5. 18. 15. 28. 26. 30. G6. 46. 46. 58. 55.

AXIAL DISTANCE (CM)

FIGURE 84 COMPARIS0N OF PREDICTED AND MEASURED WALLTEMPERATURES, G = 200 kg.m-2.s-1,T = 900Cin

_ _ _ - _ - . .-

-166-

12ss.

P= l15.HPA G= 396.KG/No*2.8 FLUX- 186.HWMe *2iies. O E4ss.16l measured

a mm.s c predicted+ TIN =87.2 C

tees.

O A+

Aess. _

+

+-A

4- O

] 8ss. A

+_ 0 A

3 7ss.

f-A+ 0

Uess.O

O A |

See. 4

| 4se. A

!

ass.s. 5. ts. 15. 2s. 25. 3s. 35. 4s. 45. Se. m.

AXIAL DISTANCE (CM) |

|FIGURE 85

COMPARIS0NOFPREDICTEDANDMEAyVREDWALL'

TEMPERATURES, G = 396 kg.m-2.s- i,

T 84 C=in

-_ _ _ _ _ _

-167-

l??f-1 .

Ij

k*

P. I'<.C. KPf) 0- 495.KGM1* *2.3 FLUX- 105.KWNa *21te0. - O ECel.121 measured

_

b Tif1 90.0 C{+ TIN =tc.4 CJ

1020. . A4 O O

4+

900. .-

4 O

A

hA

^ 8P7.f. )

.-

O + 3O_,

O3 7ee. .

3 ---

p +

eee. .

3 -

+5e0. _

A

400. - ---

| 300.! 0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50. 55.

| AXIRL DISTANCE (CN)

FIGURE 86 COMPARISON OF PREDICTED AND MEATEMPERATURES,G=495kg.m-2.sjUREDWALL

,

T = 900C

-168-

Figure 74 snom that, at low flow, the predictions are best

obtained by using the " equivalent equilibrium temperatures" at the start

of the film boiling section (see discussion in last section). This

indicates that the temperatures of the liquid core is fairly unifonn.

Figures 32 and 86 snow greater discrepancies in the predictions.

This is probably due to the overestimate of the system pressures. As

explained in Appendix 88, the pressure in the film boiling section is

estimated by using appropriate correlations. The pressures estimated for

the runs shown in Figure 82 and 86 are 135 and 126 kPa respectively. It

is felt that these may err on the high side. A higher pressure will result

in a higher vapour density and a higher saturation temperature. Both-of

these will contribute to an underprediction of the wall temperatures.

IX.3 Comparison With Other Correlations

As mentioned earlier, there is no appropriate correlation available

to predict the film boiling heat flux under the conditions investigated in

this program. None of the correlations considers the effect of subcooling

in the liquid core. For saturated conditions, the correlations of!

Ellion (1954) and Hsu (1977) are the most applicable, although they still

lack the effect of mass flux. The correlation of Ellion (1954) was)

derived analytically based on a laminar flow assumption. The local heat

transfer coefficient is expressed in terms of two components:

h=b +hc rad

1.s I

oB (T * } ~ IIsat+273)4kghfoDtg. x

S w

12aT 2 S +S -1 ATsat "g_ w t sat

:

i

|_ . _ _ _ _ _ _ _ _

..

__ __ _ _ _ _ _ _ _ _ _ _ _ _ _

-169-

The data shown in Figure 76 were obtained under very low inlet0subcooling (T =95C). Under these conditions, the correlation ofin

Ellion underpredicts the wall temperatures. Figure 77 compares the

correlation with the data at a higher inlet subcooling. It can be

concluded that the lack of agreement is due to the omission of the

subcooling and mass flux effects.

The predictions according to the correlation recommended by Hsu

(1977) are also shown on these two figures. This correlation was derived

based on post-CHF FLECHT (Cadek,1971) reflood data. The correlation is

reproduced here in SI units:

h=hHsu,TB + hmod.Bromley

= 281.5 p .558 exp (-0.00481 p .1733 ATsat)0 0

- 0.25- -0.25 - 3k o (o -py)hfggt , g q g+ 0.62

2x)9(P-"g) "g satoTt

- - - -

(P is in kPa)

Since the correlation consists of the sum of two components, it

predicts two different wall temperatures at the same heat flux. The lower

temperature corresponds to the region of transition boiling and the

higher temperature the film boiling region. It appears that this

correlation does not predict adequately the^ trend of the data. The

discrepancy can also be attributed to the lack of mass flux and subcooling

effects in the correlation.

_ ...

-170-

IX.4 Prediction of Void Fraction

The void fractions calculated in the theoretical model for the

runs shown in Figures 73-86 are compared with the measured values in

Figures 87-91. The calculated values correspond to inlet conditions that

give the best agreement between the measured and predicted wall

tempera tures . The solid curves are the same ones as those shown in Figure 44,

which are the eye-ball fit to the measurements in Test Section E.

It can be seen that the measured void fractions are usually higher

than the predicted value. This can be due to entrainment of the vapour

in the liquid core. The result of entrainment is a thinner vapour film

and an increase in the effective eddy diffusivity in the liquid core. The

latter was probably included in the proportionality constant for the eddy

diffusivity used in the model.

|

-171-

tee.

P T 493. _.

(kPa) (f")2(kW/m )

C

O101 86.4 92ea. _. A104 93.7 71

7s. __ #4' n

m

00. -

o-

I- se. AgEcrLL

4e.

9 a

oy se.

s28.

D ao n

1B. -- O a

8.-2.0 -1.5 -1.0 .5 8.8 .5 1.E 1.5 2.0

EQUILIBRIUM QUALITY (%)|

i

FIGURE 87 COMPARIS0N OF PREDICTED AND MEASURED VOIDFRACTION, G = 100 kg.m-2.s-l

_

-172-

see.

De- P T 42(kPa) (o (kW/m )

0113 100.3 107es. A 115 85 143

+ 113 94 132

Je.

Nv

"'~Z OO._,

H se.gg, +cr

fr %

& ^ D<e.+

.--.

O a> 38. _

.

|1

|2e. y + D

D3 + 0

Ole. + |3

e.

! -4. -3. -2. -1. O. 1. 2.

EQUILIBRIUM QUALITY (%)1

FIGURE 88 COMPARIS0N OF PREDICTED AND MEASURED VOIDFRACTION, G = 200 kg.m-2.s-1 |

|

||

||

_ . . __ _. ._

-173-

tea.

os. P T c2(kPa) (o (kW/m )

O 118 90 15888*A 112 80.5 169

79. .

N"

.

es.yo._,

b 5..

E(rLL

4e. O_O~ ao> 38. O

s29. O

A

19. O

o4Is.

-4.9 -3.5 -3.0 -2.5 -2.0 -L.5 -L.8 .5 9.9


FI6dRE 89 COMPARISON OF PREDICTED AND MEASURED VotDFRACTION, G = 300 kg.m 2.s-l

. . _

-174-

tee.

|

Os. P T 42(kPa) (o (kW/m )

es. 0104 80 1720 115 87.2 185

70.

Nv

O.

o._.

F-Sm.o

C&L1-

40.

o-.

O> se. +%

0

28. DoO 4

18. O

a

be.-4.0 -3.5 -3 0 -2.5 -2.0 -1.5 -1.0 .5 0.0


FIGURE 90 COMPARISON OF PREDICTED AND MEASURED V0IDFRACTION, G = 400 kg.m-2.s-l

_ ,_ .~

-175-

Lee.

08- - P T 4 _

2(kW/m )(kPa) (o

es. .__ D135 97.2 193a 108 80 210t126 90 195

,e. --. _

Mv

88-| z

o-

rS8- -O

E c.

Q' T,#u.

4e. +o b D~

O> se. - a+a

b D

28. + 0g

4 O3

te. + 0

l

O

e.-4. -3. -2. -1. e. 1. 2.


FIGURE 91 COMPARIS0N OF PREDICTED AND MEASURED VOIDFRACTION, G = 500 kg.m-2.s-l

-176-

X.0 C0f1CLUSI0flS AND REC 0fNENDATIONS

X.1 Suninary of Contributions

Subcooled film boiling data were obtained for water under forced

flow condition inside a tube. They covered a wide range of mass flux and

inlet subcooling similar to those expected to exist in a nuclear fuel

channel under accident conditions. Similar data were nonexistent prior

to this study. The use of a unique technique to initiate film boiling

was instrumental in the success of the experiment.

A theoretical model was developed to predict the surface temperature

of a tube during inverted annular film boiling. It incorporates the effects

of mass flux, inlet subcooling, pressure, thennal entrance length and

hydraulic diameter. The predictions compared well with the exoerimental data.

Both the theoretical and experimental results indicate that the

overall heat transfer coefficient based on (T, - Tsat) is a function of themass flux and local equilibrium quality, except in the thermal entrance

region. This region is estimated to be in the order of 6 L/D's from the

start of the heated length (which coincides with the start of the film

boiling length in the theoretical model). Over this length, the heat

transfer coefficient is enhanced by increasing the inlet subcooling. The

fully developed heat transfer coefficient decreases with increasing

equilibrium quality, until the flow regime changes to dispersed flow.

Heat transfer by thennal radiation is an important component at high wall

temperature. It may account for about 40% of the overall heat transfer.

The theoretical prediction indicates that the heat transfer

coefficient increases with pressure. This is due primarily to the increase

in vapour density and consequently decrease in the film thickness, and to

the increase in the thermal conductivity.

_ _ _

|

l

-177-

X.2 Recommendation for Future Work

The theoretical model developed in this thesis gives a skeleton of a

mechanistic model of inverted annular film boiling. Most of the uncertainties

lie in the interfacial shear and eddy diffusivity. The constitutive equations

for them have been under development constantly. It is suggested that such

projects be pursued with greater emphasis. This is also consistent with the

requirements of the current two-fluid models development.

One of the simplifications adopted in the development of the theoretical

model is an assumed uniform velocity in the liquid core. In other words, the

velocity boundary layer of the liquid core at the liquid-vapor interface is

neglected. This effect can be included in the further development of the model.

The eddy diffusivities of momentum and heat transfer in the liquid core

have been assumed equal. There are situations where this assumption is not valid.

Therefore, potential users of the model should note this limitation. This is

also an area for further investigation.

The theoretical model still needs some data at different pressures and

hydraulic diameter to substantiate its applicability. As a theoretical model of

such complexity is almost always impossible to put into a large computer code,

these additional data will be useful for developing an empirical correlation. The

theoretical model may be used to identify the important parameters.

The theoretical model only considers inverted annular film boiling.

Each phase is considered to flow separately, without any entrainment of the

other phase. The region of low quality dispersed flow has not been included.

Such data were almost always obtained in the experimental program. Several

models for dispersed flow film boiling at high quality are available in

the literature. It would be of academic and practical interests to

consider the low quality region.

|-

.

. _ _ _ _ _ _ _ _ _ . _ _ _ _ _ _ _ _ _ _ _ _ _ _

I 178,

1

1TABLE 1. LOW QUALITY FILM BOILING EXPERIMENTS I

(from Groeneveld & Gardiner (1977))-

allTh0S CtGIEETif FLIAf COND1710ms sanIn 045tavaT10ml

nr .ie,, taa.,. variac.i flew ..ho... CC1., rer vi.g e 1.0, h, se swe,ende.t .f ,elocit,.a nebber. eve, a herisental be....e. . h,i

.,il ie. !c n.1 wer. v.ed. .a f or ei45 > 2.0, he i. ,r.,ortio- i t. u%.0.D. - . . . . ... fi.nea of 0 t.

1D53 - 12.6 m= 6.56=105 kge-r 1a- 16.2 mm were tested.

Bher phy , garaude, Morigental plate Messne, besseme, Noted a moderate increase la the heat transferand Zahrednik and methanon were coefficient with velocity.weed. nass tn.sesof 0.67= 101 to

1970 6.04 103 6 .-2.-!3were te.ted.

Rankin vertical tube Freen-113 and No ef f ect of velocity in film boiling reglen.1.D. - 24.9 am methanol at mes

1961 length - 1.22 e it aee of 0.368 1 3to 4.04 810* has"9 *Is

Douse 11 and vertical tube 0.4) = 103 to me ef f ect of velocity on inte belling,koheeww t.D. - Ic.2 mm 1.11 = 103 kge-2 -1

- 4. 6 .m Fr.on-als I

1903 tength - sa6 .a

I kge-2,-1 Subcooltag increased the heat transfer coef.Mutta and Verti4 al flow 0 6.56= 10Steeley over horisontal of N-hesone, CC14 ficient by as much as a f actor of 4 The

cylindere bemeene, and ethyl ef f ect of subcoolitig increased with increased0. D . - 9.4 asm alcohut with velocity.1957 = 12.6 mm subcooltage of

- 16.2 sun 0 - 4%*C

Kallein Downward flow na 36.32 30 - 18.9 * 103 Tenfold increase in heat transfer at highest4 m*2 -1 of nitrogen subcooling over saturated conditione.a vertical tube 3 a

I .D. - 4 as with 0 - 38'C sub- Threefold increase la heat fles ever mese flea1960 to 20 mm coo 11 ass la range t ested.

tength - 100 em transient tests.

Ellion Upward flow in a 0. ))6 = 101 to 1.13 No ef f ect of subcooling er velocity on filmver tical annulue * 103 kge*I *I of belling heat transfer.s0.D. - 63.5 mm water with 20*C to

1954 a .D. - 6.4 em 36*C sobc olang.length - 76.2 am

Fung Upward flow in a 0.068* 101 to I.)$ Heat fles increased fourfold at highest sub-vertical tube = 10I kge*2 -1 of cooling over esturated conditions. Heat flessI .D. - 12. 7 me watar with 0 - 74*C increases tenf old at highest flow.1970 length - 101,6 .a . bcooling, in

transient tests.

Cheng and kg Upward flow in a 0.19 = 103 kge-Ie-1 Heat fles increased stafo1J for highest sub-ver: Scal t ut.e of water at sub. coo!!ng. Noted a sudden die sa the teillag

1978 I.D. - 12.7 am cooling of 0 - 26*C curve near the etniews (11a belling tempera-length . 37.15 mm in transient teste. ture at 26*C subcooling.

Newbeld Westical tube 0. 03 6 = 101 order of magattuJe increase la well heat flestoupward and down. 1. 24 5 = 10I kge-2 -l over mass fles range tested.ward flow. of water la t ran-

IO7G I .D. - 10 mm stent tests.tength = 77 em

seith vertical tube 0.025= 101 to Too f ew data potats at the same avbcoelinge1970 1.D. - 12.5 mm 0.152 = 103 kg.-2e-1 and pres. ore to draw ar coactuetons on

tenath - 1.22 m of water, velocity effect.

_ - _ - _. - . . _ ..-. - - - _ _ - .. . _ _ _ _ . . - _ . - _ .

!!

| 179

TABLE 2.LOW QUAllTY AtlD SUBC00 LED FILM BOILING CORRELAT11NS

(from Groeneveld & Gardiner (1977)),

.m u .u,,i ""' *"- .,,m. -, ===.

f:'I - ,. ... .m

., ...,,.u..,,,u. .

. . . . . .. u. .. . . ..a,

. u.

u.. .

,

;. , . . . ., e. . i. - a 1aiw e..

.a.. ...~ ..

. . . . . . . . .u,.. -. |j 1oaa..

. ..

... , , , . . , . , , . . , , . , . , .a ,...: .

! l.1 * /. - il ".i .'

.,,..,,I....c,..,*t . , ,

,

u ~.y, . qs .a u. . .,.*. . * a. - *. a,u , ~'~, .

1901, , * . , , , ,'

I *. ". i i.. . i.

. . . . . .,,,,,..,.g .. ,s a . in u. .. c ....iu-. . ,<, .gf . . , . o uw...

. ...

.wu uj ,, . . u i ; ., . .i .4 . u.....

, ,, ., ,

.,. .. .. .

| 1970 s . * " ! *' B' h *J " " ' ' '.

.. . 2'id'

.

'

. . <' { 2- ra a ,, - ,,>* . l ' *. . . . . . . . , . , , * t.,, u. s. . ca .u o,. , ,

| . . . . . u....f

.'. u n . c, . . u'

197G . . . . . .* " . '' . " . . " . * . ' . " . ". .;;;;';;;;*m,..

unr .s . .u.. . u ..u a n. . s in a c.u c

. . * .d. . 8.'' g'g. ; . .,,, sa . i.2.ggyu . .u s

.

t .u .. .

.. 3 .

..l......4. .) 8 ... 6.. p l. .L.. d . ..! . 3. 8 5t. u.

1. 3 . 3.*Cg.3 .l.d 4...que.#4 .. .

18s4. .s.1 ..83 4 .. 4

. I tie..

~ . . uj .. * '[ . , . n ...t . . . . g. . .. ..

..ls.*1000 "* ~~*-a'a.'** * * ' - '|

(s y 2...en.ea 1.- , , , ,( .

,

..[....................].

. . "s ". | y. }..,...) 81 4

.. . .. . a[. . . .. r . . ., p]. .

u. . ., , :... . . .. .

..,,..'\L**

.,

i r. us vouai. .. i.: (e n tus u uu1__,,

| * . . , *. .u . i.. . as ama. .

. no.. .

| 1903 4 5" * -

i . e. t*.

-

.

.

.

* , - _ . _ _ _ ~ . _ _ _ _ _ . . _ . . , , _ _ . - , - - _.-. _ _ _ _ _ - - . . - , _ _ _ . - _ - _ - _ . .. - , - _ . , . , _ _ - . ,, _ , - ,

-180-

TABLE 3

DIMENSIONS OF TEST SECTIONS

Test SectionA C D E

ID (mm) 11.81 11.94

OD (m) 12.70 13.08Heated length between

power clamps (cm) 77.26 75.25 76.73 76.14Heated length before

hot patch (cm) 3.81 3.81 3.786 3.35Heated length beyond last -

j TS thermocouple 1.39 8.44 9.946 9.79! Hot patch OD (cm) 8.89 "

! Hot patch length 6.35 2.54= =

Cartridge heaters 8 x 250W 8 x 150W= =

Thermocouple location

Elevation (cm) from'

top of hot patchj TSO O'

TS1 2.46 3

TS2 5.08 6

i TS3 7.38 9i

j TS4 10.28 12

| TS5 15.08 = 18 =1

! TS6 25.48 24 !I '

iTS7 35.48 35

TS8 45.88 45

TS9 55.64 50!

TS10 70.27 55 |

TS11 Bottom of downstream power clamp

i

!:

4

2

- . . _ _ _

_ 181.

TABLE 4 RANGES OF TEST CONDITIONS

Nominal Mass Flux (kg.m-2.s-1)T (OC)in

(nominal) 50 100 150 200 250 300 350 400 500

1 C E E E E E

5 A A,C A,C C A,C C,E C C C

10 A A,C,E A,C A,C,E C C,E C,D C,D,E C,E

20 C A,C,E A,C C,E C C,E C,D C,D,E C,D,

30 A,C C C C C

40 C C C

50 C C

60 A,C C

70 A,C

, Entries in table indicate test section.

. . . . . . . -. _. ._ _. . - . _

,

-182-

TABLE 5 MINIMUM FILM BOILING TEMPERATURE

'

.

| |8,

iRUN NAME T-MIN (C) QUALITY .s''A 50.605 365. .154800E-02

'

A 50.104 302. .218500E-01A100.207 313. .390580E-03 c

C100.057 293. .252773E-01 i

'

C100.10D 335. .923900E-02 v~

C100.205 335. . l e.7 310 E-0 2 i'

C150.106 310. .994483E-03C250.205 419. .134619E-01 '

C250.303 414. 214547E-01C350.205 427. .180936E-01

'

C500.206 405. .143072E-01C500.206 405. .143072E-01

' ' '

\ ..,.'

i i

h. -'A

,*

.

4'

V

.

s -

',

s -

e

\

s

e -- --- -- e- . - - - - - . - . - . , - - , , * , - ,,.m,, . ,.. - . . , si:

-183-

|

APPENDIX A: GAMMA DENSIT0 METER

A .1- INTRODUCTION

The radiation attenuation method of density determination is

bastu on the absorption of gamma rays by the intervening material.

Theoretically, the attenuation can be predicted and related to the void

fraction of the measurement region. In practi:e, a calibration curve

is obtained in a mock-up structure. The methpd is a well established|

| technique. A comprehensive review of the methodology can be found in

Lassahn et al. [1979]. In what follows, the features specific to the

present investigation are described.

A.2 DESCRIPTION OF Y-DENSIT0 METER ARRANGEMENT AND CALIBRATION

The densitometer consisted of a 50 mci Cesium 137 source encased

in a collimator, a detector and electronic counting devices.

The dimensions of tne collimctor and detector window are shown

in Fig. A.1. The relative positions with respect to the tube are shown

in Fig. A.2.

Calibration was carried out by measuring the attenuation through

Lucite plugs of various ' diameter inserted into a mock-up tube to1

simulate inverted annular flow. The % attenuation is defined as,

I -It E'

A= g .yE F

-,

!

__ _ _ _ _ _ -.

AMERSHAM SS TYPE 31 CAPSULE

\ .62 x7 4.06

E *

' 4.06 50.8a

ws h NNNNNN-

139.7 *- 76. 2= -

VERTICAL SECT ION THROUGH MID - PLANE OF COLLIMATOR COLL IMATOR (FRONT VIEW) $;?

LEAD SHIELDING

25.4 THICK'

__/ /,

/ [ t 4,

-d h23.4 - 30.56

- 20.24

DETECTOR WINDOW (FRONT VIEW)

( ALL DIMENSIONS IN mm)FIGURE A.1 DIHENSIONS OF COLLlilATORS

_ _ _ _ _ .

1I ' | || 1

h m'

m:- _

*- _.

- -- -

,

-

,

RO

RTOAT MCI 9ELTL - 5EO 1

DC 2

TNEMEGNARE

- IB NUT R

ETEM0TI

t St

a f

ED

[-F0

- , WEI

V- N

AR 4 LO . PT 2A 5M 1

I 2LL m AO mC

- .n GE iI

C FR sU nO oS i

snc ei

n

_i

d-

-__ l

l

; A_

e__

-

--

_

_

,

| li | .| l

-186- j

!1

where IE and Ip are the empty and full count rate respectively.

It can be shown (Chan [1980]) that if I /IE F is close tounity, then A = 1 - a.

The above method of relating the attenuation to the void frac-

tion is applicable in an arrangement in which the beam is at least as

wide as the tube cross section. In such case the void fraction is the

average void fraction over the cross section. In order to investigate

the effect of preferential void distribution on the measured attenua-

tion, a computer program was written to calculate the attenuation,

assuming that the flow was homogeneous in one case and inverted annular

in the other. The results are shown in Fig. A.3. It can be seen that

the present arrangement does not differentiate between flow regimes.

A.3 ERROR ANALYSIS

Let Ip and IE be the full and empty count rate averaged over

a period of T s. Since radioactive decay follows Poisson statistics,

the associated standard deviations are /I /T and /I /Ip E

respectively. If during the calibration runs n repeats are taken, then

we have

average full count rate = IF 1 K /Ip/nr

dverage empty Count rate = lE 1 K /I /nTE

where K is a function of the confidence level. -

For 95*. confidence, K = 1.96.

,

..

-187-

1001HOMO. MODEL-

O INVERTED ANNULAR MODEL

O CALIBRATION POINT

90 -

80 -

70 -

SOURCE TUBE-"" i {60 - - - --f >; -

3-

,

D ---- 31.1 cm - -

S-W

~

48 cm -=

eda

f 4G -

=

30 -

20 -

10 -

1 I I I'

0 20 40 60 80 100

VOID (%)

FIGURE A.3 GAMMA DENSITOMETER CALIBRATION CURVE

. .

-188-

If we assume that I /IF is close to 1, then the full scaleE

difference can be written as

F.S. = (I - I ) + 2KY I /"TE F _ E

If a count rate of I is obtained over the same period i during

the experiment, then |

IE-I% attenuation "a" = I -IE F

OII - I) O(IE-I)aa E FT* IE-I IE-IF

A(IE- } O(IE~I)F63 " +a

g ,ggEtF E F_ -

I I IK E E

" U Y/ EIE-IF .

( + }+IE-IF

For the densitometer used in this investigation,

|

I E - I * 0.05 IE jFi

| NE = 2,500,000 for T = 200 s;

,

. --_- . - __ ---. _ _ . _ . - _ . .- . . -

, . _

-189-

*

IE = 12,500..

For 95% confidence level'..

'

aa = 1.96 (1+'/5)+2a 0.05x12,500i20b 5,

= (3.236 + 2a) x 0.01108

.

Since a=1-a

* aa = 62..

The following table shows the variation of an at different void

fractions

a a% aa = ta %

0 100 3.6

0.5 50 4.7

1 0 5.8

The lower oa at large void fraction is due to the slight increase in

count rate.

. _ _ . _ . .

-190-,

A.4 EFFECT OF THERMAL EXPANSION OF TUBE ON THE ATTENUATION

Consider a small element AL along the tube with mass equal to am.

4amDensity of tube wall = p2

. nD al

dp_ d(AL),_

{..,

p (AL)

!

The linear absorption coefficient is given by'

p = Kp(T);

where K is a coefficient calculated from the absorption cross section

of individual atoms of the tube matericl.

The change in the absorption coefficient is therefore

1

au = K h AT;

= - K ALE a(AL) ATdT

=-Kpa AT

=-pa ATL

,

~ u = po(1 - a (T - T ))- o!

|

i

I

*

!

I.-. --_ - _ _ . - . . _ _ _ . _ _ _ .

-191-

For the water inside the tube, the change in temperature is too

small to affect the absorption correlation.

A computer program was written to calculate the change in count

rate due to expansion of the tube. The results are shown in

Table A.1(a).

Measurements were taken by heating up the tube. The results are

shown in Table A.1(b). The higher ratio in the measurement was partly

due to the lower gas density at higher temperature, and other

| uncertainties in the insulating material surrounding the tube. It was

therefore decided to use the theoretical prediction to correct for the

measured count rate. The correction curve is shown in Fig. A.4.

!

. - _ .

-192-

20 -

18 -

16 -

14 -

$$ 12 -

2

c

8oe 10 -

k -2.76 + 0.13924*T2y8 -

6 -

N -3.66 + 0.0976*T4 -

575 !

2 I I I "I I I I200 300 400 500 600 700 800 300

Wall Temperature (*C)|

FIG. A.4 TEMPERATURE CORRECTION FOR COUNT RATE

|

||

, _ _ . .

_ _ _ _ _ _ __

-193-

TABLE A.1

TENERATURE CORRECTION FOR EMPTY COUNT RATE

(a) Theoretical

Empty count rate ratio

T ('C) I/I o

20 1

200 1.0003218

300 1.0005145

400 1.0007084

500 1.0009096 -

600 1.0011290

700 1.0013760

800 1.0016540

900 1.0019530

1000 1.0022470

1100 1.0024830

(b) Experimental

Counts in Average over

T (*C) 200 s No. of samplings I/I o

20 2664271 7 1

400 2665365 8 1.0004106

600 2669760 10 1.0020600

970 2671814 7 1.0028310

-

-194-

APPENDIX B: CORRECTIONS USED IN DATA REDUCTION.

B.1 Temperatures on the Inside Surface of the Tube

Since the thermocouples were spot-welded on the outside of the

tube, the temperature of the inside surface has to be estimated by

solving the steady-state heat conduction equation for the tube wall.

If the axial temperature gradient is assumed to be much smaller

than the radial gradient, then the governing equation is

2dT,1g,f=0dr r dr k

where q"' is the heat generation rate in W/m.

The solution in terms of the inside surface heat flux Q is

2--

4R R R 1

Tj=T -g 9 9A"o 2 2

k R -R R 2__g $ $

For a typical value of & = 200 kW/m2 and k = 20 W/m2 ( C/m),

To - Tj = 2.9 C.

B.2 Electrical Resistivity

For each test section, the electrical resistance per metre of

the tube was calculated from the voltage and current measured during a

heat balance run in which the maximum wall temperature was about 90 C.

._ ___._.

- - _ _ - _ .

-195-

The resistance at other temperatures was then corrected by using the

information from the supplier. Table B.1 tabulates the thermal proper-

ties of Inconel 718. Assigning a value of 1 to the resistivity at

90 C, the resistivity at other temperatures is given by the following

polynomial derived by fitting the data:

tRESIST = 0.98372 + (0.162122E-3)*T

- (0.11893E-6)*T2

+ (0.307604E-10)*T3

The total power along the tube is calculated using the resistance and

the measured current. This is then compared with the product of the

measured current and voltage. The difference is generally less than

1%.

B.3 Thermal Conductivity

The normalized thermal conductivity is given by the following

correlation:

#COND = 0.875672 + (0.120865-E-2)*T

+ (0.174172E-6)*T2

- (0.102129E-9)*T3

tT is in *C

_ _ . .

-196-

B.4 Heat loss by Axial Conduction

Consider the heat balance for an elemental length AZ

0 1

h

Q rLoss _g

W J

00

.

QFB + QLOSS + Q1 " Qo + 9"'AV B4I-

where the Q's represent the heat losses from different faces and q"'is

the rate of heat generation per unit volume.

In terms of the temperature gradient, Eq. B.4.1 becomes

>Rg ( \7-2nRkaZh -2nRkaZh - 2nR k+ AZ +g g

i o 'R1

|I

R i'

) 7

h 2nRk h dR + q"' n 2AZ dR = - R -R AZ

'R$

. ..

_ _ _ _ _ _ _ _ _ _ _ - _ . _ _ _ _ _ _ _ _ _ _

-197-

,

.

Dividing each side by 2nAZR , we obtainj

>R

BT R BT R 3 ISTI 'R2 2'-RO I'U-k- -k k - dR = q"'--

3R R BRR $ R 'R I Ij o

$

BT BT

*FB =-k- $ =-k-**

LOSS.

3R R, 3rR Ri o

R2

R k 3 T(R,Z) 'R2 2'-R I'. ~ . $pg - $ LOSS 2

R dR = q"'R, R R BZ 2R

$ $ $,

R$

Since the wall is thin

2 23T 3T2 (R,Z) = independent of R = q (R ,Z)g

3Z BZ

Rf-Rf_ LOSS Rf+Rf$ 0, 9,,,.

kFB

**22R R R 2R BZi o i i

Power

q.u ,

2 2n(R, - R ) x HL

2 2 2Power P. r -r 3T= - - - + k 2 (R,,Z)9 -$ LOSS

FB 2nR, x HL R, R 2r BZ$ j

d

. . . . . . . . . . - . . .

-198- |;

The second spatial derivative is approximated by first fitting a

second order polynomial through three adjacent points and then differ-

entiating the resulting polynomial. Thus, if the temperatures at Z , Zj 2

and Z3 are T), T2 3 respectively, then the Lagrange interpolationand T

is

'Z - Z Z-Z ~ -

2 3 1, 3T(R ,Z) = T) + , ,T, ,, ,,

2I Z -ZII-Z2, , Z) - Z3 2 1.32 -Z3,

Z-Z ,Z-22,j,

*

Z - Z). Z - 23 3

'

and

3T{R,Z)=2~

T T T'

I 2 3+ ,,j g 2 ,, , , - ,

31 _, Z) - Z2, , 1 - Z3, \2Z - Z), Z-Z ,Z - Zj, 3 2,-Z-Z

2 3, 3

For calculating the gradients at the first and last test section

thermocouple locations the "zeroth" and "n+1"st thermocouples were used

if they were installed in the test section. Otherwise, the temperature

at the mid-plane of the hot patch was taken as the "zeroth" tempera-

ture; the n+1st temperature was taken to be 450*C below the nth temper-

ature and the location was at the bottom edge of the downstream power

clamp. The value 450*C was chosen after an extensive survey of the

data in which the n+1st thermocouple was installed.

- -

_ _ _ _ _ _ _ _ _ _ _ _ _ - _ - .

-199-

.

B.5 Test Section Radial Heat Loss Through Insulations

During the comissioning of a new test section, a dry test was

carried out in which the test section was heated up with a slow flow of

nitrogen through it instead of water. The heat flux, after being cor-

rected for axial loss, is the radial heat loss. Near the start and end

of the heated length, axial conduction was greater than the radial heat

loss under dry conditions. Therefore, it was decided to take only the

heat loss at the location of zero axial temperature gradient and derive

a correlation of heat loss vs. wall temperature. This cerelation was

then used at other axial locations. For test section A. ~e correla-

tion is

4 LOSS = 1.23974 - (0.381818E-2)*T2+ (G.129243E-4)*T

3 2+ (0.194641E-9)*T kW/m of tube inside

surface

Test sections C, D and E were similar and the heat loss

calibration was carried out for test section C only. The correlation

is

* LOSS = 0.22619319El - (0.16666573E-1)*T2+ (0.46720548E-4)*T

3- (0.21880536E-7)*T

_ _ . . ..

_ _ ._

-200-

1

Another independent estimate can be made by considering the

conduction through the insulation.

Consider the situation shown in the following sketch

600PC,,; -

e $0 C

|

d59h~ 6.5 cm =

a

The thermal conductivity of the insulation is assumed to be

0.035 W m-1 C*-1*

Heat loss by conduction through insulation.,

k AT2

g An (r /r$)r g

0.035 x (600 - 50)3

4 0.0059 x In (0.065/0.00596)

1351 W/m2 of tube inside surface.=

Using the second correlation expression, the result is24355 W/m ,

,

.

..-- + n - ,- , _ . . ~ ~ - , --, -

- - . _ . - - - - - - - - - - - - -.

_ _ _ _ _ _ _ _ _ _ _ _ _ .

-201-,

The discrepancies cay be due to the following factors;

(1) the insulation close to the tube had been observed to heat up to

a glowing red condition. This had the effect of reducing the

insulation thickness;

(ii) for a power supply rated at 50 V and 1000 A, it was not possible

to measure accurately the actual power at such low levels (a

2heat flux of 4355 W/m was equivalent to a total power of

114 W for the test section). The empirical correlation will

tend to correct for any zero offset.

B.6 Heat loss in Hot Patch

Dry test similar to those described in B.5 were carried out for

the hot patch. The correlation derived is

Q = 155.344 - (0.437261)*T2+ (0.487421E-3)*T y

This correction was applied to all test sections. It would

appear that test section E should have a different heat loss since the

hot patch was shorter. However, the difference was found to be small.

This was because the heat loss was more dependent on the area of the

exposed insulation surface.

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ -_ _

-202--

B. 7 Linear Expansion

The coefficient of linear expansion is given in Table B.1. The

correlation deriveu by means of least squares fit to these data is

a = (12.87195823 + (0.3832677818E-2)*T

+ (0.5699454545E-6)*T2 + (0.3313636364E-9)*T )*10-63

m/m/ C

All dimensions of the Inconel tube are corrected for thermal

expansion.

B.8 Pressure Drop Along the Heated Length

The test section pressure at the inlet was measured by_ means of

a pressure gauge. The actual value was not recorded because of its

fluctuating nature. Instead, the pressure along the heated length is

calculated by using two-phase pressure drop correlations. The calcula-

tions are carried out backward from the point where the flow discharges

into the reservoir at atmospheric pressure. From this point to the end

of the heated length, the Claxton#(1972) correlation for two-phasei

frictional pressure drop for adiabatic round tubes is used. The

calculations are as follows. |

* Strictly speaking, this correlation should only be applied to ?$ flowin which the wall is wetted. Since there is no correlation available forpredicting the pressure drop in inverted annular flow, we will use it as q

a first approximation. I,n addition, the correlations quoted in this !

section are taken from some proprietory report. Therefore, the {references will not be given. Luckily, those who work in this field will j

have no difficulties identifying the sources.

l

-. _ _ _ _ _ _

-203-

Re = G.D9 u g

G(I - x)0Re =L u,

Calculate f and f from the expression

16/Re if Re 12000f,

0.0014 + 0.125/Re .32 if Re > 20000

2 21 2f G x y'dp g 9*,Ej9 0

2f 62 (1 -x)2Tg' gy

g,

,dZ,g

bf dPX=dZ ;Hi jg g

_

bf (G) = 28 - 0. 3G1

.

fp 10.2p

A= 1 _.d.

("9 )L

. 2-9 f + 2.51

T; exp= _

. 2.4 -10-4 (G) _

C = - 2 + f)(G) T)

2 2O = 1 + C/X + 1/Xg

G'ORe =

20 pg

_ .. . ..

. _ . - . - - . -. .

'

i

1 -204-

Calculate f from Re as before.g9 g

IL

T2 " (I ~ *) #ot,

2

&fo= $f*T42

2 2If 4 <1 put $o,j

2 2'dP 2f G Vg $go )gg

M"!D

Along the heated length, Rouhani's (1969) modification of'

Becker's (1962) correlation is used. The two-phase multiplier.is

defined as

2, (dP/dZ)two-phase'

9

(dP/dZ) single-phaseliquid.

= 1 + 2850 (X/P) - 8150 (X/P)2:

,

where X is the vapour weight quality:

P is the pressure in bars. ;4

'i

This correlation is applicable in the range 0.001 < X/P < 0.1.

However, it is extrapolated to beyond the upper range for the following

reasons:

i

- . - . - . . , .

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

.

-205-

(a) the length over which the quality is greater than 0.1 in the

present study is relatively short, if exists at all.

(b) The overall pressure along the heated length is small.

Therefore, errors introduced due to extrapolation will not

significantly affect the pressure drop estimate.

For qualities below 0.001, the value at 0.001 is used.

B.9 Program Listing

A listing of the computer program incorporating all the '

corrections can be found in the following pages.

. .

.

-206-

TABLE B.1

THERMAL PROPERTIES OF INCONEL 718*

(Ann. 1800*F/h)

Temperature Thermal Electrical Mean Linear

Conductivity Resistivity Expansion

*F Btu /sq.ft/ht/*F/in. Ohm / circ. mil /ft. in/in/ F x 10-6

200 86 762 7.2

400 98 772 7.8

600 111 775 8.0

800 123 784 8.1

1000 135 798 8.2

1200 147 805 8.6

1400 160 802 9.1

1600 173 799 9.5 |1800 185 801 9.8

2000 196 811 10.0

,

*From International Nickel Company of Canada Ltd. (INCO)

CONVERSION FACTORS

1 Btu /sq.ft/h/F/in. = 0.14423 W/m2 (*C/m);

h

1 Ohm / circ. mil /ft = 1.662 x 10-3 Ohm / nun /m2

in/in/*F = 5/9 m/m/*C !

l!

l

|l

1

1_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . .

1

-207-

PeMaam Goth 74/14 0FTel */ FTW 4.S*494 0949-43 11.39.39 PaES &

L D"'mam 30gL f teF9T.OlrTPUT.FJEC3.TAFel, TAFStaGIFT' PVT.* TAFe?*PJ4CE . FtAT)

C

5_. -g..

CC paucaM G0tt CairyLATus teacht staf FLc5 FWDe FitJe totL850C eAW OAT 4. vastuos CossaCT!0pm as4 !acomprimaT59. SSSC AF9EMOtB B OF T9EE18 St E . 5. FUteGC

te smmmu m-w .mu cccmmuum- - .- CCSEAL Sfoes '18. Les .8?OeT9 4 54.103 .8199re t10. 686RAAL s o t a t . 0La&T t 13 t .Tzaf t 131, pp | LIIDaemJII/m&TES/ tt e l l tISTBCSS SUtlAsIS,FSL

L1 t erTEGES easesaak Twt na).Tt133,3t n 39 Fet i g13,3i4 38,ett|113,ef t L31asAL TT f 66. 8IlbSEAL SNf 54.141.5?OSF fl0. LSBtaAL TSSIS4b .P4IO q191

34 eTimeetal/L130/s e 143Coseeries / States e ung e, amAs . f tCE. F BBC AL. Ytels. Tue&B,TICT. FTSCA&creeens FossesT/st, gh. FT!?LS ( 4) . FluSTITi t s . FEAETIT E 41. FVasTIT f Si e

11s5Tvtag .4794fe e te p

'TJueWm e tTMEAses fsICOS418 6. FSTm31 f II , Fe testa t al .FSTMB 3 4 31.FETue4 E 35,31 IF9Tset 8 3 5. FSTsIB4 ( 31. F1 Tie 7121.FStueS t il .F5fMB9 8 3i .FSTiele s ti

1 .S TBIEJX I 5 V1614C TCD80eois /TITLK/ Tj ft4E E. T tTTACf SUSl4CE .SfJBf.KY. EAsLOCE, s a m tm,

1 rasLoca.vaal4CTOATA F& ARA /l . h t 9 814G S-4/

le cafa ET/g . 3. .g . 9. 6 3. 10. 34. 39. el. 54. .St. 4 8./DATA E/1.1.11. 14. 4 3. 3 3./QATA Jo971.J0FT3 8.5/OATA 90131L/9./

h&Th J06Fute/6/lt DATA stPTITL/1/

CC Joeres=0 esams 40 Psierage CAacsC J0FT1*K leo 9 TDTC OEFhcLT 15 etT9 flat ame utTEGUT fuergse CAallS

48 CC eMITLa-o asASS NO FtET GRAOteGCC DEFAULT IS WETE NEAstleGC

og ep?tTL=4

JCFT L * LC

- "CCCCCCCCCf'CCCCCCCCCCCCCCC

Se C Test SpCTIOe CarsasTt?C

- MCCCCCCCC- -CC

49 CC WF5 IS TUTEL astmera 0F T T atuUs cessC amo is stat 7stCAL asalgiassCs paa stTsa

Pat 14AAA DOEL T 4 /14 OtTet EDUNDe*=*/ FTW 4.8*499 64-6943 11.39.39 Ped 38 3

C Er.ausp IS 354T30 Laum2TB BEfques f.ast T TC stJT IS esATED LSearf11 GEFOSS BGT F41t'B

se C met te s mo**1.et uti /et seTsaC resta la SEATED PttingTttC #EII * 11 0 , of ADetT1oenAL sesastrDED PotirTO

C Os * TUDR ISC 1350 * 7t'tB M)

6% C FOUT * OUTLET PsassuntC AOLEleG e 4Dt ASATIC L230GTW DtzteTRAAA 0F BEATES LEleGTWC AleGT * VERTICAL DISTAseC198191E55 O!SCEAES ASIS tiroC CF NEAftO LENGTEC

?S CCCCCCCCCCC- - mwwwwCCCC . .CC CEANGS *WS FO!J4W!sG vhetaaLES Fot DIFFERENT T337 $3CTIONSC A4AG C94au.S DATA CAaD FOR REATED LEIAET9

?S C CEa#48 FUleCT1up TS IM68CCC Test sectrois a

WT3* L i84 EI00=4. 99 9115

ELEWO*G. 89 79SL87*G.0 3 39wso* 4De~e 81194

0% D000-6,8130 0CC sun samr.asCCCLmm mm mwsm

90 C---

souTe L e t . 3AOLamo=e . 61ArmuTe n . 4 %ptSteem t. n e t e *et

99 Eme t m 3. 8 ( D000*1buco-Og e bt e /DSC B00189 IE USED ts cotasCT!on OF WALI. TElertaaTUSSC

m2We*4.1 *De* t 41./ t t . s pe/Dopot H al l * 4 LOG 100D0/CS) -4. 51CALL Phoe s t9 Ef t

Lee C anno F est emmt19 Coeff tIII;W

t sot

IC=439 asas et.lth acmasas.t.reL

ISS IP t hw r ts . st , e . s 9 Top * nocaT&*

ff se 30 0.e Go TO leease.ermassaEC* tCe LFe*Ff4h4T f FSLB

114 maAO 89.931 ITtJI .Jol WTE1.?TW. TOUT.79T.TRL.88.891F 1559 . eE. 96 BRAS 19.531 (TTi t h . I*L.mSU)

CC ISSNTIFT F188T ase LAst TgtauenCOUPLaBC

_ _ _ _ . _ _ _ _. _ _ _ _ _ _ _ _ _ _ _ __.

-208-.

PEDeessa 90th 74/T4 cPT*1 segut****/ PTe 4.8*e90 46-49-43 11.39.39 Pam 3

119 If (WWW .aQ. 69 GO TO 34T f il *TT t 3 BTS*TT f itfl134*TTf 364D TO 144

138 le (*UIrf tput

Trib TakT t 13'1 *T111 B *4St .T13 *1 Lt .ST1*STtil

13S ef f tle-I.18364 CONTrutit

CC CAI4ULATs taLST auT9ALPT

* C138 C

PTIs**Ttuel . el * 3 3.PS81T*FOUT/S . 89 48CALL POLAA8 4 4. .PTIS. 8. 8.11S te*G d 31 * 2 3 34.

135 C CALCUL&Tg PLets Penettu AT GUTts? e ATnt PmUngC

CALL 70tA&WIP90tf.4. 8.R,11SPee s a t ti-sq 3t l * 3 3 36.ELdNFT*B t II * 2 3 34.

L ee C CALCULAft 90utt AmD SirfBALPT UP 10 EMD QP GOT PATCEC

Pt*Pe-eloo s ITattPrAs* STAT * RitO*3 I ** 3POW 1*PG* POW

149 4 t il *EIW ePaul/G/ P AAAAC

CC PILA 30tLIe6 SECTrom

150 C

C CAkWT3 AADI AL ESAT PLUEC

0011 t*3.irr$199 PeIDeaso*es tST ITi t i n *t!** 3/PSAln/11. * 3. *ERPas tT t til *T t til

CCC CAE4ULAft Ast&L esAT CaucocTrou

IPf13*TI t*L D / t IETII*11-STt E-19 9 8 (BTtI*11-STitilllet 1 *Ti tl/t f 0Titt-WTit* Lit *tBTe tt-STit 19 5)

3 *Ti t-L I / I istit-Li-et t Ill * tWTit-11-ef t!*1b t )31 CouTTeUR

ARPE IoOTt1*TCOne tT f Il B *tDalC

Les C Copaact nActAL maAT PtAI POS ALL I4IBGEBC

PW t i ti e t Pe t S* AEPWI-TO LGBe f fill l *1904. 0 / 81. *EIPAs tT t 31 l *Ti tl lI P t ufU . 50. al STi t l * efteLeef s ti-sta t-il

174 Pau'PUu*EL* psi t t l *f . 01*PfB1m* E L . *8EP ASITt ti l *Ti ti t **3C

PauwmAs soil 74/Te OPT *1 */ PTu 4.0*490 00-49-45 11.39.39 Pasig a

CC cceaacTas WALL Tempta& Tuss

171 C

CTutII*Tq ti Petill'ecoup/TConDif ell)

C1st C

Cet fleetsetton* Pal / tG*PARRAsSe t -1

11 Coirftsvatal C

C CALfDLAft TOTAL TEST SecTtou P0utaTWTS*T r trrtellPONC*Fols* AM RSTITUTEl 'ELfue*40*tI*tI* ti. *SIPas tfuTRl *TWTEl

1 / t 1. * 3. * fEF AS q TWTT. *TWTWOL 90 EQUT* elm *Fout/GsPhaEA

Potes*S t * ttCmm - -c

195 C WTERATE PSEGUta 0807C

smw.miCCCMC' -

n0UT*foafT-eloutt /uPG200 CALL 70tJantPSSIT.S.,1.RA.1)

CC cat 4Tt.6TS PhaBEURS DEDP ALGIIG ADaA&& TIC SBCTICW 00uusTSAMC OF BRATM LENGTRC

365 CALL A0peac tG. 50UT.DE.DPDilPole-OP0t * ADLRieGPGetT1* Pea **P91,6.3940

CALL POLA Ad . Pta tT1.8. . S . 88.11SPGlet GB 4 85 -R813)l * 2 324.

Ils E L I* d erAFP- em f a l * 3 336 t f E PG1EAIMie t E13 * 50UTl *t .1IP t EAW4 . 63. 4. ) EAWO*4.v4WG* e a mili *as t li n 't. 9' t I .-tavGl *tAWG*4. 5*it t fl * As til l l/ _e.814P 13* POUT + POL - AcacT*t . 81/vavG/1000.

319 WTB*13 *P13PP' RTE *1)*E13tie

CC CALCULATE passaUt3 Da>F ALouG BEAftp LAmeTEC

338 . WFt1*trft*100 AT J83.uft1LA*errt e l-JLL1 *LL* 1CALL SE P9eo f 6.If LLit .PP f LLil .DS.0POSI

335 PP t LLi o PP r LL15 -SPOR * t eTILL15 -ST t LL) l *G. 011 +0.01 * t BTILL11-ef t LL) l *9.81/1994./ t (Ba t tl e t t.= 5 t LLil l *2 RAITS * EILLill/16.018)

PettT3*Pt s LLl / 6.st et

- _ _ _ _ . _

-209-

PEDseNs AOIL '4/14 OSTE 1 */ PTB 4.8*4se 64-09*43 13.39.39 Dem 1

CALL potJAu # Pt3173,s. .E 83,11Ile s tLLe e +s* LLt /3136.-ass 3B sis satel astal)

Tsaf f Lha s e 88 t 131 - 33. 5 /1. 0et.daf r LLe sea t 3) * 3 338.

17 COWPf eUS#F I PF t 1 B .LT. PGPfl Pf l1| eplEFT

839 CC DATA ThaU1Af t0WC

IP s t e87 . 80 18) CALL PAGE 11837%WRITS it.47) e'IsAstS , TI S . 4, pt .TML, pGee, PGsC

344 wasts s e ,44 6 PP f t g . PFi138,518.etti.TBAf t10,5Le&Tt13es tTT a e .491) (Tuf t1B ,1 te t.w?TlW83 73 8 4. 6 9 31 < PWI IIII ,3 tet. eT33WAITS #4. 4 91) (B f III ,31e2.sTE)FB8?*f t tT*1

349 WT11eerTS-1CC stoas DATh Poe Pt.0TC

00 la Jo1.pTE1Alt STesTt IC.J1 *Tw f J+1 b

STOSP f IC.J1 *Mi lJ 811STOePSI EC JisFFiJet tSToes IIC.J t *e 4 J+16STODTt i tC .Ji eT5 Af tJ*11

399 MI t.f *14 ePet IJ+1t *S.44113 CwTreus

IF d JC97UB . 30. e s r,o TO 31wetTg a 7,113 ssfusass3.Tf e,6,7tt,P5,811)ut!TE T 1,13 ) etw tJi .Jel. rPul .Tul10.Ttte WT9*13

344 wtIts 4 9.73) ( Pe t f J I .Je a . WT33QO TO 39

C34 CowTtuus

IF t JOSPUB . 83. 0) aretT3 47,74) 4Bel C

tr i JorT1 .3Q.11 00 TO 19C Plot caTa

use re=200. 9 InAze1300.Tief s*19. S finAt=1 53

370 TIClete.s Leg . 4 S TL*4 . 6

IP + uoTITL B0 OS GO TO 3 3PTTTL511 b ette 1 * SUS ('or t.SFTITLaillet eeS FttJe a'ct

319 PTITLS I 31 e143 LJHG *)90 TO 31

33 CoorfievtPTITLR e ll e 10 E l * 'l

33 CowTtet eIGG F9 UBTT T t t ) =100 4 * #UmastS**

P9veTIT t Il e sed. Alt apSTvaaselCALL CE AD t 4, N AME. ? ,1.STVka$ 11))PEART!?t ti *1OW t **'W-TB AT

84 5 FRAATET 4 3 t e135 (CSG Cl *)

Ph.ERAn 30th 74r94 CPTel h]UWike**/ PTB 4.8+490 54-##e43 11.39.39 FeG8 4

PTast! Title 109 t*tlEAT FtBPTuttT Ile10eg e tw/ neeFTasTIT f 3 5 e1053 t * pS T4LME.G .11

390 S TitLaXT*4.1SU5b*1Ca e l .53st4CT*4 1P5Yte t (11 *108 (*T 'C* l'1

P5 fte 3 (1) *10E t *T/C. 3 *)PSTns3 d l e =10E q *T 'C-

399 3*lPSYMB4 41 b =108 4 *T T*P9 YtW9116 e10E l *T/C

4 *)1 *)

PSTime t tl e10d l *T/C+ 6 *)P5The1411 e 10s t *T 'C- T *)

300 P9fsISO li p *195 I"? C 8")PSTust elletce t*T T. t*)Pt vise 10 i n g e195 ("T/C-18 *3JoeoP'* LWTt 3 *e873- 3

109 80 14 4tet,arTt3,300 13 Je1. ICTes tJ) eeTotTIJ. Ill -SfotTS lJ,IIIMia f 21 stfons f 4,III /1804.

13 CDwTtsDR310 sTtuccol t ri e s t

CALL G IsWLT IJt20PT. T03. M t s . IC. S . II,39JoaopTo 3

14 CONT!WUEIInt

319 SYtectsiIIIeleDO 11 Jet. ICTBS (J) eetost f J. Ill *100.FE IS q Jn es?Otf (J,3 81/1988.

Il CairTleCE339 CA14 51mPLTIJCDOPT,TS3, M ts . IC. 0,18,33

eo TO 3951 poema? (48,4 33.r19,3,Ilt)53 FGeseAf t 15.8P1.39to Poen&T (18.145,16513.43

33$ 47 fogant 112. Ale,48.19813.56691 POetAT 135.*T*,35.10213.4 bE93 Fonseaf 13 5, * F* , 3 E.10 B13. G lsol Posstat t as,*S*,3E.19813 6,/l

71 foesnaf (&4. P9. 3. 3FG ,0, P1.4.319. 3333e 73 poematt13PS.al

il FeasiaTE10PS.1174 roesia? (148,Pt.el

suo

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uS E TS #4.40566 MelekT * *1*)

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& FUNCTim MIST IT1C *C macTt3 CAL AdutSTITITT or IECUusk 718 agenLIsasC

S se tste g e t S. 34?t04 B+10 *T-4.1109 348*e t *T*4.16 313 39-3 b M*4. 903?IleSTUWme

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4 PURCTIOW TBIASe tT) $' * .O

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*g

f9LLSeef I f *. 310044148 f *T*. 44 T309480-el *T=.164449%f 38-1)*To I$ * .3J4191199015

. i, 'TF ET .LT.300.5 191488*4. N'*asTUW 'g

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S..W,..- . . , , . . _ . . , ,,. . 0...e - . . , ii . 3. 3, . .

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9 Capoten./ es t t n * Le .36 8

ERJGe n . < Ba t 11846, e L SflBCLees e 36 og ,agg 3

vtsCreets t 9 6 8 8. 408 3L8 17 e ss .GT. 0. 3 GG TO 31

19109tmlGD TO 13

il 950=e*tB*05/WfSCGSELac* e t . - E39 805/WISCL

19 Ft.*F9't"? t SSLt74*F9 E CT r asras

CDre54 3. *fG* t I6*1ti e *33 fggf amog

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s*4GsT f D90st/tret41Fl o a t . -e . 3 *too? 'G8

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18 IF 4 E 0 . LS. 9. 6 GO TO 3 3tie e t 4. -stl **II *FL, FLO

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3 3 Doet*-3. *FtO*G'.G*TurWho, tuot,08Il DFetentOS et S60esTUnethS

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L SUBEDUTtWS DIP 0eo 86.38 p.08.0F056Causeum 'h&Tsas em t L31

CC TWO-FEASB Ft|CTIONAL FOSSSUBS Cede POW EEAftB BOUWB GUCTS

5 C eautast 'S asetFICATICE OF 38C5ta*0 COta3LAT10mCC ISPUT FSSSSGBR 18 IB EPAC

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19 IF I EA , GT. $. 4 740 40 9 EAet.1748 48EPBASeE A/ i f 1180. I148tWLe t . * 199 0 *t P942-0110. * EFS498 EP448BRLaG*et'VISCLFt*FBICT r aabe

3G DF9E *- 3 * PL *G'4*T4fDf L/RBOL/SSP90S *0 FOS /199 0.29TUmseEb

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_ _ _ _ _ _ _ _ _ _ _ ._ _ . _ _ _ _ _ . _ _ _ _ _ _ _ __ ___.___

_ . - . - .. . - - - .. .. - - - . . . . . .

-212-

| APPENDIX C : DATA TABULAf!ON1

.!'

1i

; Each run is identified by an eight character alphanumeric run name. The

legend is shown below:

1 A 100 10 1

N'! test sequence

Test Section mass fluxi

(kg.m-2.s"1) inlet subcooling

(OC)

e

Note that the mass flux and inlet subcooling in the run names are nominal

values and should not be used in any actual calculations.

" Each set of data consists of five lines. The format is:

ii

First line

,

Run name;

Inlet Temperature (OC)

Mass flux (kg.m-2,3-1)

Hot patch power (W)

Hot patch temperature (OC)

Measured power (W),

Corrected power (W)i|

'.

_ _ _ _ _ _ _ _ _ _ _ _ _ _

-213-

Second line

Calculated pressure at end of hot patch (kPa)

' Calculated pressure at end of heated length (kPa)

Inlet enthalpy (J/kg)

Enthalpy at end of hot patch (J/kg)

' Saturation temperature at end of hot patch (OC)

' Saturated liquid enthalpy at end of hot patch (J/kg)

Third line

Corrected wall temperature (OC) at 10 locations.

Fourth line

2Heat flux (W/m ) at 10 locations.

FJfth line

Enthalpy (J/kg) at 10 locations.

* In the case where this calculated value is below the atmospheric

preseure, the atmospheric pressure is tabulated in its place. The

saturation temperature and liquid enthalpy still correspond to the

calculated pressure.

|

_ _ _ _ _ _ _ _ _ _ _ .-

_ ____

-214-

The distances in cm of the ten thermocouple locations measured from the

top of the hot patch are:

TEST SECTION

A C, D and E

TS1 2.46 3

TS2 5.08 6

TS3 7 38 9

TS4 10.28 12

TSS 15.08 18

TS6 25 48 24

TS7 35.u8 35

TS8 45.88 45

TS9 55.64 50

ITS10 70.27 55

.. .

. _ _ _ _ _ _ _ . _

,, 91. r* ,,6 J

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? e3 .994130soet .101110E*44 1847 39 t*44 .999429E*G3 .9979 35t*e3 .811029ttel .0 30021B*S3P .191514eese .19e241g*D4 .13999 3 3*e4 .1693428*44 .180981t*6e .18890 33*45 ,109??23*04 .19134 3R*04 .193 33*Reet .193901B*048 434139B*06 .4 39115sete .4440e9t*44 . 4499 9 4 8*e4 .419911E*c4 .440091t*e4 .4071328*e4 .103?titees .5131448*e4 .920112Bece

B480.012 .98?t04t*02 .364048t*e3 .419990R*e 3 .323410E*03 .1414113 44 .19410 0 E +44.142269t*el .13?t99 test .411624t*64 . 4 306 40E *e4 .199'90s*0 3 . sees tesote

T .943394t*43 .941470E*el .144319t*e4 .10 34 51E *e4 .lete?2E*e4 .18 4140t *44 .901270t*43 .911839t*e3 .493410B*e3 .019??9t*e3P . 21139 4 B+ D4 .3163943 66 . 3108 46 8+44 . 24 9 304 t *46 . 30 *04 3t*04 .2e9 2028*e4 . 210 394t*$4 .2121313*06 .313033t+04 . 213 349 e*eet .4341738*0s .4416e9t*S4 .4473015*04 .4127028*64 . 44 34 91R*44 . 4 7 44 4 48+ D4 . 4 949 3 3 8 *e4 .S13441t*et .l22t ,48*04 .S319314*04

515140t' e3 .482911see4 . e sa494 s *44Beet. ell .te113es*e2 .364840s*e3 .400332E*e3 *

.139402R*63 ,131741t*43 . 4136 24 teeg .429%448* . 5 0414e 5 *4 3 . e14 316 t*06.e415est*05 .t0014?B*e3 .917101B+43 .9542023*43 .9???0 38 C6? e3 .911840E*e3 .934411s*e3 . 0 94 409t *e 3 .4312308*e3 .51?e99t*e3

P .1749400*64 .17 3191t*66 .113 3018606 .173190gede ,1t314et*46 .112115 B eet .1739338 04 .1?4 311B*04 .1711elteet .1?S149E*043 .434123 tees .4344418*04 .44 5193 tote .44??11t*44 .454?338 04 .4451168*e4 .442341S*64 .491104t*6e .9811048*46 .9137233*64

_ _ _ _ _ . _ _ . _ _ _ _ . _ . . . _

-235-

3444.414 . 98t l#48*43 3944 60See l 4804614 45 1131998 45 .4254698*44 . 404 94tB*G4.11 L 9195 *4 5 . ;9 a litee t 44 2*4 9 t*44 4 4 0 2 4 4 8 *e4 19 7914 3*4 B . 494 G l o g ote

T 99 3149 tee l e %8 3 % e t ee l . e t119etee l e *0 ? le t*G l e 6 '1548*4 3 6384'84*99 . 94L % A 49 *e l 8444 518*4 9 .e049958 01 . ?t4 3 31 B+4 99 19 9 f 96 0 e*G 1449 4 28*c4 11409 e sc ,4 3 . 54 e 8 *04 . 813141 t *44 .19 294 38+oe 19 B444E *04 .19 4441 E *44 .11* 40 9 t *44 . li t t e l t eesG 4333948*44 .4341238*94 . 444 3418%d A* 4 312 0 *06 412 3% ?t*e4 .4403193*44 4794138*94 .4004468 44 .49%3393*o4 . 5 41916 E*46

3444.101 . e 96940 t*e 3 l'9 0'etee n . 54 2964 8*e4 . 919414 tee s. 346 0448*e ' . ele ne ttee l.12 59 99 t *e t. 3 T64 f't*04 19 t e60s*44 . 04 9 919 ee l .4447393 041 s * * se 004 B

9 9946 89t*41 . 8 L 12 41 S*4 ) .4194468*44 .e44 3 288e0 3 9913098*01 9441913*9) . 6L040E*4e .98?Se9 teel 9942448*e 3 . 9 33299t*43F 1994548*ee 1969635*e4 . 941098*4e 199 99". E*ee 194 5 3 4t*04 .1944946*04 19 4491 t *e4 .19e g g e g*et .19 9 t e %3 *c4 .19174 %ge 's

8 .1944 9 38*e6 . %el t193*e4 . eetteste.< e . 41212 t t*04 4 2 3 3 e ?g*c4 . 4 3 34et E*c4 , eg120 38*44 . 44 0 3 31g est 4 74 pe g g,44 .4353393*44

1494.103 . e t t;e 9 8*4 3 .194e as t u n . 441e 4 33*41 4e ntletee n .s994e lseet . 4 44 4a le*4 41 J 44 J 38*4 8 .13 2 f lJ a ee t . l? 9120t*44 .194 L 44 Beet .le190 2 t*4 3 e e lte 4E *e4

f .?293188*41 .?%14444*43 . 'e lte 9 8+el . ?9464 9 E*4 3 44 913 J t * . 91194 3 8 *s t . 9 ? ? )2 4 5 *4 ) ,9961195*49 . 9414 29 t*41 .9149660*45.1020448.93P 1849 BLO*ee 1044 4 95*04 .104171S*44 . L e leitteet 44 . L e092 33*e4 .17 9 413 8 *e4 1'44618644 .198 31'f ee4 .10 0 0 % ? teet

t .3949428*44 .199 f 29t*e4 . e 64 % 12 t eet .4491038*44 . 41811' t*e4 420 3 31B**4 . 4 4 9 44 e E *e4 . 4411 ? t Seet . 4 6 941't *06 . ef 49L18*46

3444.14) .999 9 64E*43 3 94 8 48 t *4 3 . 444 98 9 E.41 .19 e ? 40t*4 3 .440'940*44 . 4 319 418*e 4. ! J 3 290 Ret t .119 319t*4 5 . 3 ?%91c E*e4 J S 9 7 01E *G4 .10% )%4a*4 8 . 4 414 * ? t*44

? . 64 4 241 t*4 3 68096 & B+41 . f eeleetee n .7 4 4419 t*4 ) 50 %e99 t *41 Ste?1JE*4 9 . 914884e*43 . 9 311398*e l . 9941448 *4 3 . 9 9 3 317 E ** )P .16 ?? 3 E Beet .14 9 393 B+44 .164 9 7't*44 .14 416 9 e*44 .16 4 90 e t *e4 . it lT1't *e4 .16 714et*44 .1623318*04 .14 3 749se64 1414 4 e3 *443 .3944028*44 . 3 98 3 '1t*04 . 6e a teeteet . 46 7 012 3 *e4 . 4199 9 L A*44 .4341295*04 .4394615*44 .4910345*44 . 468910E *04 .4484098*44

8404.104 .009440S*43 .3944446*41 .4104418*4 1 .994 Mote #3 .4419118*et . a#9810E*44.114 4 99E*e l .1193418*41 3124918*44 .5042148 *6 10 'e t 2 E *4 3 419 4 0 9 t*64

7 9914248*49 4S le2 ? tee t 4142148*41.1 %e 194 5.4 56 6419 't* 739981S*01 5049448*43 8441 ?tS*e 3 0 4 9 09 'R * G 1 . S?4 250s*4 3 0498998 41

F .1930095*44 . iSI 413 0*44 1911140004 44 .1490428*44 1 s t 9 94 5 *e4 .4441428*44 .146'228*44 .1444345*04 .1441454*445 3991745*e4 . 3940648+et . 29'9e flect .eG L O P DS*e4 409441 sees . 4 L'3*es*44 . 4 31. 1S *c4 . 4 419148 *e4 .4945433*44 . 4904 S it ece

3444.109 .te2930t*43 . 594640 tee l . 415112 9*41 91e9449 eel . 364199s*44 .149 2 2 0 t *e4.1 L 419 8 8*4 9 .114141S*41 31'49 2 8 *e4 1912440* Le llettoe t . 413 54e 8*d4

0946023.44f .'129440*43 .14 341'8+e l . te9 340s*4 3 4 %419 5 8*e 3*

43 .?llfeate4) . 6099 e l e *4 3 . 819440E*4 3 . 0 3 399 35*4 5 . 4 2299e t ee lP .13?1988*44 .114 419E *44 .11984 ?t*04 .13 4 6 86 t e44 . 4 2e 28*04 1114423*04 .1319 2 7 t**6 . !!! aecteet .111 s tog *64 .1914415*448 394?? 35*e0 .194 4 0 8 8 *e4 .4417948*04 . 409 AS I5*44 412 3 3 48*e4 . 4 L 9 Ll't*44 .4319 41E *e4 . s e 314 ? t *44 .4400148*04 . 494 56 7E *04

8400.141 .0 39 300E*42 . 3 94e46 tee t . 514417 t*41 12 9 44 0 E*01 . 4 94 4 61E *4 4 . e f te690*44.11418 t E*4 5 .1149 91E*e l .1914 21a*44 1496423 04 .1G le 'et *e l .4314128*44

? . 59 8 3 39 5*e 5 . 584e9 9t*4 5 . 400 011 t*41 . 4139 ?va*4 5 7 344 41t*4 3 0334145*43 . 91014 e 5 *4 5 . 9 7 412 e 3 *4 3 ,94eeettoe t .94129et*43

P .1448440*44 .166392teee .199994t*44 .18%1148*04 1914s tE*64 181431S*06 . 4 7 9699 t**G .119414 9*e4 .17 915 L t*44 . i ttie 't eetE . 314 4114*e4 . 3 79 3 ? e s*M . 34 44 9e t a44 . seee69seet 19e ssts*04 . ee ts 4 2 8 *06 . e 314 21t *44 . 4 44604t+ee 4 4 8 400 t *44 .4141918*04

B484.14 3 .54&1868*43 .1940 4c'*4 9 ,513912 e*6 9 Ste4eeE*e 3 9567 41 t*e 4 .9 390 2 4t+44.12 0 919 E *4 ) .119 3 L 4S+ 39 31 t 99.e4 3 713148 *e g . iget t br ee l . 4 ;9600 seet

.44 P*944*el .e944 818 elT .4404 3 L B+4 9 43 7 91394 0*e t S ite ts te e t . 94 49 3 9 t *4 3 .104 2 L ? s*4 4 .10 6 514 t*e 4 . 99499 28*e 1 . 97999 9t*41P . 3004 B le*44 . 3 9012 % 8*e4 . 34 '9 99 5 e4 . 3 04 40 9 E*04 , 29 44 9 ' t *44 . 2e19 94 t *M .2409493 44 .30644 st+0e . 26ees ts*44 . 29 le t t*44E . 3 ?t e t ' 5*04 . 36 L 79 28*46 . 3G T16 De*e4 . 3 0 212'8 *e0 e e l! 44 B + 44 . 413 5 0 9 3*M .4131938*04 .4%e6 40$*44 .4944 30s+44 . e4 5 3 2 30 *e4

8908.811 .9049448*41 . 499 29dE*4 ) .4992945*41 1121e e t ee l . 440 2 0st*44 . 4 2e e49E*44.11944 3E+e l .1349 . 4 L 41s lE*44 .4249448*04 .168 399t*41 4943?23*44.e 30491B+0 3 .e9 9'e'es*e li4*41 . 48 89 L 1 E *01 9849 3 2E *e l . 0 00 eis t * 41 . 9 9 444 % E*e l . 0 9 94 %e t *e l . 3 28 314t*4 5 9 0 2017t*4 5T . 50294 93*e 5

9 .14 l ? 9 e 8*04 .1490444*44 .16 24 "t *44 .1619 2 9 t *44 1614 7 4 8 *e4 .161 % 4 9E *e4 .161 ? t 2 t +M .16 24 b1B+44 .14119 2 3 *e4 .16 4101 B *04B . 4 J 991 '9* D4 .4353898*44 . 414 ? 17t *44 . e ett ee E*ee . 4448 71 B+44 .4114 148*04 . 44 40 44 8 *G4 . 417 )e ts *44 .44 34e9E*e4 . 44814 7t+44

8980.813 . 98 9199t*42 . 49 9 3 9d f oe l . 40 9 44 0 t*41 1115645*41 19919 2 W ee t . 4G L 4 24t*44. 4 lt ? )' E*e l .13 014 0 S *e 1 . 4114 3 4E+44 e 2644 2s*44 10904 'f *4 3 . 4% ? 2 41 t*44

? . 06 913e t*41 . 98 L L 9 2E*e 3 . teile t E *e ! . 9144 f l 8 *a l .4449428*43 . 94 4'449*e l .9434145*43 . 44404%B +4 B 04410 3B+0 3. 8 29412 t.ee lF .1011990*04 .143411S*44 .1022'45*06 .18 L41'E*e4 . i el le 7teet 10192'tece 10144 tE*04 . i e l te t teet .10 :410E*e4 1343448 04

t .4341145*04 .4343248*06 . 4 80 L 445*M .4419925*04 4 4 9119t * 04 . 4 % ? 19 0 S *e4 4711188*04 . 4818 9 20 *e4 .098243S*44 .4966445*46

3184.813 .te?lteteet .4912948*4 9 .444910e*41 .5394 368*19 .544501t*44 . 5117e 1t *44116 94 J E *t t .14 2 24 5 8*e 3 . 4114 3 4 t -e4 . 4 3 ? t 9%3+04 .1004423*41 .49 % %33*44

9 . 949804 8*4 8 . 994 20e s*4 5 .9919918*4 3 . 98 te49t*41 9999 L 13*4 3 .16e e t it*e 4 .94 289 ? t*4 3 . 914111 E*4 3 015101 B 41 .e 49 2 0 0 E *0 3F .2443548*44 . 2e lt lltede .2431308*04 . 29 344 7t*44 . 362111t*e4 101914 E *44 .2424448*04 . 2040 3 33*e4 . 2 3 4914 t *e4 . 3 01111 E *e4A .4 314 9eE*44 . 4 39 712 E+G4 . 4 3999 98+ee . 444 3 44 t*44 e l2 ? e t t*04 . 4612 J1t *04 . 4 ?t t i le *M . 4 91419 E *04 .4903165*64 . 54134 9 E *os

3984.161 ,904'0eE*43 . 494 290 0*41 . 913 4 4 9 t*4 ) 1419000 03 .llee245*44 .4182948*04.1261110*01 . 4 3% 2 71 tee l . 31121s t*44

.e4 9ee4E *e l .9010'es*e t .9460148.06391014t*44 .106 39 4 9*e 3 . 4 444415*

0 3 .514840s*01.916106t*4) .teltetteel? .114 344t*43 146 4 2 4 t*01 . ?*199 t S*4 3 .0112e e*4 3F .1944 418*e4 .1944J40*4e .1991448 04 .1944115 04 .1917 4 0 t*M .1929939*64 .19 L 19 e E*04 .1949945*44 .19 L l14 B+04 .19 L 4 4 9E * 440 . 39%130 E*e4 . }9919 7 t*e4 . 44124 4 R ve . 48 7 3 3 2 5 *44 4114 L e S* 04 .4334'93*94 .4141818*44 .4119493*44 . e 95 319t*45 . 44 4 911 t *04

tiet . n e l .8913448 43 .491396 tee t . lll3 71S*4 9 14 4800 t ee) . 9 9 2 04 4 t *e 4 94e69et*ee. 4 30 900E*4 3 .1284995*01 . 3 7 4941t *04 .14 9 2 0 *E *04 .19 t 20c e*4 5 . 4 4 9 411t*94

? .??4 3 4e 8*e l .3J00949 65 .g lie fiesel . 848 9 01 t*4 3 9 5 419 48 * 01 . 910 04 9 3*e 3 .19 3 4 9 2E *e 4 .19189 4 8 *G 4 94 4 71t E * . . 9419115 *0 3

P . 21 ?914 B e64 . 314 79 2E*e4 . 316 9 4 % E *G4 .3114458*44 . 314998 t*64 . 211914 3* e4 . 312 2 44 5 * M .312%S95 04 .3111448* se .213911E*eeE .5936128*44 390 394E+G4 . 44 2 G ?9 B *04 .4811848*44 41418 't*44 . 421516 8 *0e . e e L1)la*e4 . 4 H e l t E *04 . 44 416er* *4 . 4119 t et eet

5%80.103 . 944 ?SO E*0 2 . 499 390E*4 3 .11604 ?seel 1111405*03 .e499945*44 . 4494 36 E*441219485*43 . 4 20191t*4 5 . 371114t*44 3 94 9 41 t *04 . t e 4 2 'S t *41 . 4 4 L 14't * 44

9 .64 314 5B+99 .44349es*43 7043148 4 3 .718)t1Beel *e99 31E*4 3 82TT13 tee n .9400523*41 .9L?ti' t+4 4 .telletteel .941341t*01F . L 14164s eet 1117 99 E*e4 .17 39 ? 5teet .1120st E*ee 17121't *M .1714 59 8+ee . lessecteot .1441113 46 .1487495*44 .4444498 468 . 3941448*e4 .3917945*44 . e 413 418*44 . 44 4 6 300 *e4 412094 t *e4 . 4191444*4e . 4 3113 40*44 . 4 418 61t *44 . 4 49 ? ? 9 8*G4 . 4114 4 4 3 *te

8904.144 .943948t*02 . 491290 teel .90944 78 4 5 111140 t *0 3 .4204198*44 4 0 014 9 E *4 4.119491 Bee t .11 P4 39 A *41 .3199918 94 1940 2 4 8 *e4 .1941193*45 . 4 38194t*e4

? 4169198*48 .6893398*43 .614701S*43 794411t*41 190 le t a * 6 3 . ?s t t 19 5 +0 3 .44 314 tE*41 . 0 9 29 9 2f *41 . 06126 f t*4 3 . 0 749185*4 3F .1993325*04 .199 2438*0e .14e e'Ot*04 .111491E *64 .194e 23f-48 19 93 31R*44 .154 2 44E *ee .11344 ? E*e4 191892 t*44 .1146214*045 .39e140s 46 . 39 *e14 E*oe . 404490 S*44 .4819688 44 4164425*04 . 415 9 7 5t*04 . 42e ?9 3B+44 .4399065*44 . 4 4 464 9 t ee6 .4443318*44

GBWigu * ret)CMa teG Costyt3TS

||||

|

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ __

- _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

-236-

APPENDEX D : BOILING CURVES

The boiling curves shown in thi3 appendix correspond to the data

tabulated in Appendix C.

1

J

f

_ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . _ _ _ _ _ _ _ _ _ _ _

_ _ - _ _ - - _

-237-

'' l i I I I i i I i l i i l i I I I I l - *- I i I i i i l i l i I I i l I i i i l-o ,4. RUNAME*A 50.50 - *,. o ,4. , RUNAME=A 53.05 ~.. a ss. .

. ._ . sn. ._. sn. e,_ . ss . . _ ,_

. w. . _

. sn. e.. .ss..

. ._ . ,s . ._

._._

__

N.

__

N.. ,.. . ,_

_cN c

NE E-

10 *;- :- 10 ' +t- -

:-x .u.

o ,_ ox .3 S o

_

dd .' *._, , , .

.. * .. _ ._ a. ,,. ... ,o, .

,.._ ,. .. _

_ . .a: . . = * _

y _

a._

.. _

y _ . .. o.e . . ,,. ..._

_ .__

s._ s_

_

10 IIIIIIIIIIIIfIIIIII 1Ii1| | fIIIIIIIIIIII10m. .. . . . =. . ...n. .. . . . . m. . i =. n . ..Tu-TSAT (DEG C) Tu-TSAT (DEG C)

-

-l| I I I I I I I I I I I I I I I i l-'

v o es. a RUNAME=A 50.10 . __g y;jy) ) | [ ttI| 1Ii| 1~

._ +w*_

, o,.. PUNAME=A 50.10 ~._ aw.

=v4.. _,- ,,4..

_

.,4..._.. *m*

-, , , . ,

-

,_ . w. , _

.

g_ -

*- .-

. n. 8

k. - *

,

E_ _

! EiO ; :- -

x ._ _10.__

D 1. = *- r*J - _

7k *- o. -

d- a ~

. .._ 0 .a ...

,-_

,, .g

._ .._ g

_o ,. ,.- , , ,

g g

.. _ Y_

o. .. .. .'

._ _

N- -

,, _

gg. k!k!! !

t 0 ,,g | | gj| | | | | | | | | | | | | |. . . . . .. . . . . . . . . .

Tu-TSAT (DEG C) ,, ,, , , , , , , , , m. , , , , , , , . , , , , , , , , . . , , , .

Tu-TSAT (DEG C)

_ _ . . _ _ _ - - - - -' _ _ _ _ . .

- ._

-238-,

&IilIIIiiiiiii i *IIi- git | 11i| | | | | 1 3, ! ~ |TIj-i'h a e4. s RUNAME=A 50.20 _~ RUNAME=.9 S).35.-,_ o,43.- a v4 .

. ,_ ,,4..

,_ *s4.. , , , , . . ,-,4. . - ,-

, ,a. ..

.- 4 '4 .. ._ .,4..

_

.s8- .

,3_ _ ', '

.

N N* e* 3-

.

k_ ~e t

h

2 H'

.~ *

w

t O'g- _. g8.. ,,,- 4 w

D 7"

O-

,G - g - O a ,_J '- 0" *

'.". ,j / * -La- *, =. - 4 O * **a * *'- o . . " * - - , , . ,

* o =.,_

Y_

o*. ,

= . ' ' _

W_

o,*c + "*C *. =, ,

" * -

4-2-

. 3. - , .

.

.. ~

-_ _ _

1

9

10' IIIIIIIII!IIl!III!!I ! I '...' m' .! I I I I I I I I I I I I ' .10' ;

m. . . . .. ... . . . m. .. .. ... ii . im. m. . . . . ... = .. . .. :... ..TW-TSAT (CEG C) Tu-TSAT (CEG C)

Le.

< >,.

*IIIIIIIi1Ii| i l i l i l 1- A| | | | | | | | | 1 i l i i i l i i 1-.~RUNAME=A 50.70 : ',: a va. :,_ o ra. : RUNArT * A 50. 60 ,

.- a '4 *_ ._ s ,4. . .

.+'4. .,4.. ,.-m ,4.. - s_ ,,4..

- *.- ,,

y.- *'4.. ._ .,4.. - <

_- _ ,_ _ J

E.N*

C- _ e ,_ -

C.5 \

$ b~ '10*;- _-- 1g*_ --

x .- x .s-

_ o-

D '

d- o 3 ,_

" . . -

' *- d * + * **- ,o.. , , ._ o .

-.,.- o* , , " * _

E_

o". *a ++ . n . _,_

Y_

* ++ *c =. , = , . _ ,._

W_

.. , ,

-- _-

.,_.

#- _ .. _

ici llIIIIII!I1!IIIiiIi II III!IIIIIIIilIIi1 ..,g,m. =. ... . m. . ...,,.s. . ,. . .. . m. . . . . . . . .

TW-TSAT (CEG C) Tu-TSAT (CEG C)

_

_ _ _ _ _ _ _ . . _ _ . _ _ _ . _ _ _ . . _ _ _ _ _ _ _ __ _ . _ _ _ _ _ _ _ _ _ _ _ _ _

__

-239-

-l i I I I I i 1 I I I I I I I I I I l- -11 I i i l I i l i l i l I i i i i l-. ** : a r4 3 Rtf#t1E = A 100. 05 : , o ,4. RbNAr1E*A100.10 :,

._ . sn. ._ .. . sn. .

_

.-. ,n . . .r.,4.. - .- . ,n. .

-4..._ . sn. .

_ ._ . sn. ._

a-_ _

s__,

N. N.. > -

r_-

_

C_

_

Egs

r_ _

10 *;t- o. _-- 10 *;,- .* + ..* = --*

. o.,o . . -x , ,, x ,_ o.. .. - -,.

_2 ,-

,.3, ,_ . ... .a._ o. _ w .-

,* .+g ..

.- o. . . . . - - o. *. . = * .

. .,

_o.cy _ o. + s. .

c. . . ..W _

._ % + = . --_

.__

_ _ __

igi liiIIIIIIIIIII IIIII 10' IIIIIIIIIIIIIIIIIII...... . . . . . ii . i=. m. m. =. ==. = . . =. . . .. . . . ..

TW-TSAT (CEG C) TW-T3AT (CEG C)

.'.! I I I I I | i i I i i l l I I I I:*: QUNAr1E = A104.35 - $-

1 I I I I I I I I I i l i l i I 1 |-_|.

a ,4. , RUNAr1E = A100.20 :, ,4. s

.. . sn. . -,- .,,a..

.

+ ,n- *-

. n. .,-.,4..

-.- . , , . .

.- . rn. . -._ . ,a. . -

._ -

._ --

_N NI5

- -| |, -a

E.2 1O-

10 *;-' * :-

~* iga. . + . . --

: - |_o

* *-

9 . . . , " -x . o.d

_ - 3.o , + -,

,_ - -e : o.._ -

.,,..-. , .

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TW-TSAT (CEG C) TW-TSAT (CEG C)

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Tu-TSAT (DEG Cl Tu-TSAT (DEG C)

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Tu-TSAT (CEG C) Tu-TSAT (DCG C)

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,. _ _ _ _ _ _ _ .. . . _ - -

l,

i-258- i

a

11

APPENDIX E : VOID FRACTION DATA

L

The void fraction data are tabulated in the following format:

First line

Run name (see Appendix C)

Equilibrium quality at void fraction measurement locations

Second line

Void fraction at five locations

The void fraction measurement locations are at 1.5, 15, 30, 4 3 and

52 cm downstream from the top of the hot patch. Note that not all

five locations were measured. The equilibrium quality is calculated

by interpolation between the thermocouple locations (See Appendix C).

The data are shown graphically following the tabulation.

J

1

< - , , . - , . _ . . ~ . . . ... _ _ . , , . . _ _ _ _ _ _ . , , _ , , , _ _ _ _ . , _ __ ______.__ _ _m_

- _ _ _ _ _ _ _ _ _ _ _

| -259-|

D350.101 .106E-01 .380E-02 .394E-02 .107E-01 .155E-0158. 59. 76. 83. 92.

D350.102 .ll7E-01 .393E-02 .493E-02 .128E-01 .183E-0158. 65. 79. 84. 90.

D350.103 .ll8E-01 .318E-02 .651E-02 .153E-01 .214E-0157. 63. 79. 85. 94.

D350.104 .123E-01 .272E-02 .791E-02 .178E-01 .247E-0158. 68. 81. 89. 92.

D350.105 .157E-01 .425E-02 .919E-02 .201E-01 .278E-0157. 69. 81. 86. 96.

D350.201 .214E-01 .108E-01 .811E-03 .ll2E-01 .184E-019, 28, 69. 86. 93.

D350.202 .231E-01 .110E-01 .219E-02 .140E-01 .223E-0110. 35. 73. 86. 95.

D250.203 .251E-01 .118E-01 .279E-02 .158E-01 .249E-0110. 38. 74. 90. 96.

D350.204 .223E-01 .ll4E-01 .565E-03 .ll2E-01 .186E-016. 28. 67. 83. 94.

D350.205 .202E-01 .108E-01 .518E-03 .862E-02 .150E-013. 22. 56. 78. 92.

D350.206 .202E-01 .lllE-01 .121E-02 .757E-02 .137E-013. 14. 45. 72. 85.

D400.101 .157E-01 .684E-02 .376E-02 .124E-01 .187E-0150. 60. 84. 91. 97.

D400.102 .168E-01 .626E-02 .566E-02 .162E-01 .236E-0147. 61. 88. 93. 98.

D400.103 .178E-01 .533E-02 .742E-0245. 64. 91.

D400.104 .110E-01 .353E-02 .225E-02 .102E-01 .155E-0148. 57. 79. 87. 96.

D400.201 .240E-01 .135E-01 .199E-02 .821E-02 .154E-0110. 29. 68. 86. 93.

D400.202 .256E-01 .138E-01 .733E-03 .108E-01 .189E-0111. 36. 75. 89. 93.

D400.203 .268E-01 .140E-01 .491E-04 .125E-01 .213E-0112. 39. 79. 89. 95.

D40C.204 .212E-01 .128E-01 .236E-02 .684E-02 .133E-018. 22. 58. 77. 87.

D400.205 .216E-01 .130E-01 .344E-02 .511E-02 .109E-015. 18, 52. 75. 88.

D400.206 .184E-01 .107E-01 .216E-02 .542E-02 .107E-014. 18. 44. 68. 84.

D500.051 .142E-01 .667E-02 .151E-02 .915E-02 .145E-0158. 62. 84. 87. 92.

D500.052 .150E-01 .637E-02 .260E-02 .ll5E-01 .177E-0151. 65. 83. 93. 97.

D500.053 .172E-01 .752E-02 .285E-02 .127E-01 .195E-0151. 64. 83. 90. 97.

D500.054 .131E-01 .654E-02 .673E-03 .735E-02 .120E-0147. 58. 80. 85. 89.

D500.201 .265E-01 .171E-01 .679E-02 .220E-02 .862E-027. 21. 52. 73. 83.

D500.202 .293E-01 .186E-01 .688E-02 .335E-02 .107E-019. 27. 59. 78. 87.

D500.203 .246E-01 .161E-01 .685E-02 .125E-02 .702E-027. 20. 43. 66. 79.

D500.204 .211E-01 .134E-01 .501E-02 .230E-02 .758E-027. 17. 38. 60. 76.

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . ._. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ _ __-

-260-E100.021 .1913-01 .3848-01 . 412 8-O L

74 94. 57E100.422 .1948-01 5348-OL .S312-01

13. 92. 91.2140.023 .1588-01 343E-0L .5878-01

14,.

93. 99,2100.024 .1508-01 .2788-81 .4108-01

79. 49. 91.3100.625 .1438-01 .2415-41 .3112-01

10. 91. 91.C100.026 .1488-0L .2318-01 .3208-01

10. 87. 94.B140.101 .4498-02 .1493-41 .3918-01 .S355-41

42. 80. 99 97E704 102 . 3478-02 .1928-01 .4428-01 .45 a t-01

44 . 43. 91. 90.B100.103 .4128-02 .1218-01 .2978-01 .4488-01

40. 78. 92. 94.3100.L04 .2798-02 .1938-01 .2 44 E-01 .3478-01

37. 73. 99. 91.5100.101 .4708-42 .4428-02 .1928-01 . 29e t-O L

30. 44. 87 93.5100.201 .9418-02 .133E-01 .3418-01 .5578-41 .5938-01

50. 74. 94. 91. 974104.202 .7328-02 .L128-01 . 31S B-O L .4898-01 .4098-01

45. 79. 92. 94 91.E100.203 . 7298-02 .9728-02 .2818-01 .4385-01 .S472-01

49. 77 94. 94. 95.5200.811 .230E-02 .1475-41 .2863-01

61. 87. 93.8200.012 .3028-02 .160s-01 .3162-01

65. 97. 92.E200.013 .170 E -0 2 . L 74 8-01 .3548-01

42. 37 93.5200.014 . 2 $ 08 -0 2 .1348-04 .2458-01

41. 87 92.1200.011 .3098-02 .1312-01 .2438-6L

63. St. 91.E200.014 .3458-02 . L 273-e1 2778-41

44. 86. 90.8200.0L7 .3328-42 .1138-01 .2 fit-01

41. 43. 11.E200.101 13 78-01 . 3 4 3E-0 3 .1415-0 L .2748-01

20. 17. 83. 92.I200.162 .1498-01 .4 79 E-0 3 .1818-01 . 3 3 2 E-01

Lt. 63. 89. 96.

E200.103 1353-01 1473-41 .1198-01 .2353-01LO. 13. 82. 90.

E200,104 .1343-01 a.2448-02 .9323-02 .1918-6110. 43. 79. 49.

8200.201 .24 38-O L .072E-02 .1048-01 .274E-017 34. ?S. 98.

E300.011 . 4943-02 .5053-01 .1448-01 2648-0143. 82. 89. 99.

3300.012 .S128-02 .53St-42 .1763-41 .2848-6114. 41. 87 94.

3300.013 .6428-02 .1928-02 .198E-01 .3213-01S0. St. St. 97.

5300.014 .4798-02 .4453-02 . 210 E -01 .3128-0112. 19. St. D4 .

R300.019 .3518-02 4498-02 .1478-41 .3358-0110 40 87 92.

B300.101 .203E-01 .7328-02 .7218-02 .1998-0114. St. 42, 94

3300.102 .2002-01 =.7818-02 .SO 4 8-0 2 .1773-OL17 49. 79. 90.

5300.103 .1718-01 .6L18-02 .373t-01 .148t-OL14 42. 18. 97.

3300.104 .1648-01 .7048-02 .3093-03 .1278-0115. le. 72. 64 .

3300.171 .271E-01 .1473-41 .1148-0 3 .1168-019. 24 47 St.

3300.172 .2773-01 .1598-0L .30 38-0 2 .6248-027 Lt. St. 54.

3400.011 .1133-01 .133E-02 .1018-01 . 20 2E-0146. 72. SS. 94

3400.013 .101E-01 .971E-03 .9433-02 .1848-0144. 73. 45. 90.

3400.014 .9198-02 .1138-42 .8128-02 .1633-4149. 73 49. 91.

E400.101 .2358-01 .1318-01 .1703-42 . 8 4 3 t -0212- IS. 70. 85.

2404.102 =.230E-01 .1348-01 .2748-02 .6418-0211. 33. 66. 02.

3400,103 .2228-OL .1358-01 .3743-02 .4978-029. 28. 41. 76.

4400.104 .2108-01 .1328-01 .4748-02 .1148-024. 23. 54. 77.

1400,105 .1798-01 . 1098-01 .3443-02 . 2 90 E-0 212, 19. 93. 12.

E400.361 .2028-01 =.1648-01 8092-02 .102E-029 13. 43. 10.

8400.142 .3958-01 .1878-OL .6912-02 .3268-025 18. 54. 60.

B100.011 .1173-01 .4901-02 .180s-02 .892E-02 .1345-4130. 40. 77 44. 93.

3300.012 .1248-O L .5198-02 .3158-01 .1028-01 .L578-0141. 60. 74. 87. 93.

BS40.013 .118E-01 . 3 312-0 2 .2432-02 .1148-41 .1778-4148. 42. 74. 90. 94.

3500.141 .2378-6L .1198-01 .4498-02 .1313-02 .4928-0217 27 55. 14. St.

3500.102 a.2398-4L .1448-01 .477E-02 .1958-02 .4178-0214 29. St. ft. 49

8500,103 . 218E-01 .144 8-01 .4712-02 .3328-03 .1148-4216. 30. 13. 71. 79.

3100.104 . 2053-01 .1383-01 .6448 02 .2953-03 .4338-0221. 31. 47. 68. 79.

_ _ _ _ _ _ _ _ _ _ _ _

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.

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. . . . . . ..z... .. .x . . . . . . . . . . . . . . . . .z . .. .x.., .. . ....,

QUALITY OUR !TY

i. i.

I I I I I I laal I I I I I I I I I I I I^b o

m. - RUNCr1E=E300 - SB. _ RUNAME=E400 oogg3 g -

O (soO os o O E *00 osa E 300 iG * a E+00.io E

es. _ * t .co 'i ? - es. _ + (400 4 .a -

%e

n. _^ - m. .s$ -

.. . a

$ .. _+ - $ .. _ a -

wW T W gg. ,o o - g m. _ g --

t L o*

9 .. _^ - 9 .. _ -

3 9^aa

m. - - m. _ -

a

am. _

,- m. -

,-

,

%.

*a6 -i.. - i..* s*

I I I I 1 I I I I I ( 1 | ! I I ( I i 1.. ... . . . ..r... .. .a-.i . . . . . .. . . . . ..x . .. . . , . . , .

CUALITY OUA.ITY

_ _ _ _ _ _ - _ _ - _ - _ - - - - - _ _ _ _ - - _ - _ _ _ - _ - _ _ _ _ _ _ - - _ _ _ _ - _ _ - _ _ - - _ _ .

_ _ _ _ _ _ _ _ _ _ _ _ _ _ . _ _ _

-263-

..I I I I I I I I I I

_ plt m E=E500 [_,,

= tsos u 9a g 30s - 80 a

a_

se_

o.an. n a _

t a,

8 o_ .. _ o _

ha de

W ^

g .. __ _

w a

9 . _ e _

o.>

.. _ j _

a

m. _ a _

%a

I.. _ _

! I I I I I i I i I..

. . < . . .x... .. . x ... .. . ....

CUC:LI TY

I ________ _ _ _ _ _ _ _ _ _ _ _ _ . _ . _ _ _ _ _ _ _ . . . _ _ _ _ _ _ _ . _ _ _ _ _ _ ._ - . . _ . _ . . - - . _ . . . _ _ _ .

-264-

APPENDIX F

CHF CORRELATIONS USED TO COMPARE HOT PATCH POWER

To be applicable to the present situations, the CHF

corrections must be based on data at low pressure, low quality and4

short heated length. The Macbeth correlation chosen is based on the'

local conditions hypothesis, Ate: has generally been found to be valid;,

) at low quality boiling crisis ONB). The form of the correlation is

De x G x i f9d CHF= At-Cy 4 X,

*-

1 (G x 10-6)Y2A =y De1 o

4 (G x 10-6)YSC = y3 De7

yo = 1.12

y1 -0.211

y2 = 0.324

y3 = 0.0010

j y4 = -1.4

y5 = -1.05,

t

!

!

1

__- . - . _ . . - . _ . _ _. _ _ - . .. , -,- . _ . _. , . . . .

-

-265-

_

_

2 2(6 in Btu /ft h, De in ft., G in lb/ft h, i fg in Btu /lbm)

The Menegus correlation is

II I fI

6] )3)TYy

6=a 1+ST+ 1+YV+ 1*c 0 9 0 Dk lk I ( )

2(4 in MW/ft , T in 'C, D in ft, V in ft/s)

T is the difference between the saturation temperature and the

bulk temperature of the liquid (water).

The Greek letter parameters are all pressure dependent.

At 14.7 psia the values are as follows:

a = 0.1107

S = 21.20 x 10-3o

i = 1.244 x 10-4S

Y = 1.113 x 10-3o

i = 19.542 x 10-4Y

1 = 8.04 x 10-36

_ _ _ _ _ _ _ _ _ . -- __

_ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ _ _ _ - _ _ _ . _ _ _ - _

-266-

APPENDIX G VOID FRACTION CORRELATION

An empirical correlation is development for the measured void

fraction data based on local equilibrium quality and mass flux. The form

of the correlation is:

Y = 0) * (100) * #2

a=p3+ II'0 * an p4 (X, - Y)*3

Void fraction data measured in test sections D and E are used for

the least squares fitting procedure. The estimates for the parameters are:

p) = -0.001867

p2 = 0.000777

p3 = 0.47146

p4 = 101.2|

The standard deviation o is 0.076. The predicted void fraction is

plotted against the measured void fraction in Figure G.l. A scatter plot

! is shown in Figure G.2

As this empirical vorrelation is based on data at one pressure,

extrapolation to other conditions are not recomended.

_.

-267-

i.e

C5p.-

.7 @CO O

Sg%ao

m g % gaw

a_

of mEB'*

@ a o O

8.s%5 o a

[ 0o aU

SY .4 o O

b O O O O O OO

0% o O Do [$Oa,3 g

a a 9OO O C

k @DDO

.2 2 9

00 mO

1 00

e.2B.O .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0

MEASUREJ V0ID FRACTION

FIGURE G.1 PREDICTED VOID FRACTION

.

_

-

_ _ . . . _ _ _ _ _ _ _ _ . . . _

-268-

t

.38

.25 0

O.29 DO

O.l5 000

O O Ob S'''

OnO O O

Ef' h OO 330

0- .35 g 6 O

> O% % O0Oc88 cp O Oa.ae OO w a cw oo

080O k.a o "E O O OO oo b

~ ' 'O gO 00-

Og OOO .ie abo Olt) O UZ O

.i5

O

0.200

.25 00

.38

O

.3E-0t ..IE-9i .IE-st .3E-0t .5E-81 .7E-0t

XE

FIGURE G.2 SCATTER PLOT OF PREDICTION ERRORS

.

_

. _ . .

-269-

APPENDIX H

Radiation Component

H.1 Network Diagram

We assume that all surfaces are gray and that the vapor film

is optically thin. The latter assumption will be justified later. This

is equivalent to saying that there is no self absorption within the vapor.

We can then consider the vapor as a single node in the radiatior.-network

method of estimating the heat transfer between surfaces. We further

assume that the liquid core is smooth and concentric with the tube.

Referring to sketch below, the energy leaving surface (1) and

arriving at (2) is given by

U A)F12 v912 * T1

O aV

The energy leaving surface (2) and arriving at (1) is given

by q21 JAF I=2 2 21 v

' Net exchange between wall and liquid core..

=q = J)A)Fg 12 v - J ^2 21 vT f T2

A)F12 2 21=AF*.

.

J)-J2q,g = (J)-J }A Fl 12 v

1

'

T *..

2

A) Fl2(1'Ev)

__ _ _ _ _ _ _ _ _ _

________ - _____________________________-_______ _.______

-270-

|

Since we consider the vapor film as one single node, the energy

leaving the vapor is given by

i

dv"EEv bv

Portion of J,. reaching surface (1) = A F Jyg y

*Af CEv v1 v bvPortion of energy leaving surface (1) and reaching the vapor

J)A)F)y y= a

U A)F)y y" cl

." q J)A)F),c - A F )c Ey bv=wy y yy

(J) -Ebv)*1

AF cj jy y

*

41 net 9 -P. * S*<

w w-v

dl-d2 J1-Ebv, +1 1

1

AF12(I ~Cv) A)F)y y |cj

J) -J J1-Ebv Ebw - U12* rad, net 1 I I

* "

1

F12(I ~Cv) F c cjy y w

1-c,

H.1-------

The network diagram can then be represented by the following

sketch.

- . _ _ - _ _ _ _ .

_.

-271-

,

1

E g A) F12(l-C I _g v J2 c

btC O^

1-c '

I -C2

cA ^1 lv y cAE C"I g2

1

f C2 2v v

-

'bv

H.2 Vapor Emissivity

The vapor emissivity can be approximated by

Lyyc = 1-ey

where a is the mean absorption coefficient and L is they y

mean path length given by

2E[0 -(D-6 )2)aA

L =4 1=4 tuba 4X*v A n0+n(D-26) 2n(D-6)

_ 2(D-6)2626=-

2(D-6)

_. -_____ _ _ . _ _ _ __ ___ . .

__ _ _ _ _ _ _ _ _ _ _ _ _ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

272--

To estimate the mean absorption coefficient > we use the Planck

mean absorption coefficient K for optically thin gas. The functionalp

relationship between K and the absolute temperature is shown in Fig. H.l.p

An approximate fit to the curve is

T-1000)700K = 5.4e T in R '

p

and K in (atm-ft)~1p

,(1.8T-1000)700 Uor K = 17.72e T in K andp

K in (atm-m)~)p

To simplify the calculation, K is evaluated at hT, T ). It isp s

felt that since the method which we have adopted in estimating the radiation

component is only approximate, it does not warrant to calculate K at somep

other average temperature,,

! |

We can check at this point whether the assumption of optically;

I thin gas is justified. The optical thickness is defined as

T = KLo y

where K = extinction coeff.

= absorption coeff.

= 26(10 m-I) = 206 m-I*

T.. g

4For void fraction a = 0.3, 6 = 9.75 x 10 m

T = 0.02 < < i*.. g

.

__m_ . _ _ _ . _ _ _ _ _ _ . .

-. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

-273-

o

EIS i s i a 4 i e

g _ -

E _ _

Il2 - -

y _ _

E _ % _

s9 -

g _ _

;:

_ go-

A_

y6$ - t

5 -

ys - -

x _

y _%_

_

tO t I '' "

1000 2000 3000 4000 5000TEMPERATURE , *R

FIG. H.1 PLANCK MEAN ABSORPTION COEFFICIENTS FOR WATER VAP0k

(From Tien (1968))

_ _ _ _ _ _ _

- __________________ _ _ _

-274-

|H.3 View Factor

Since surface 2 sees surface 1 only, we have F21 * I

<

AFj12=AF2 21 = A2'

..

A

12 " { * 0-26 " (I T)2 26

F0

To find Fjy, we consider the following:

Energy leaving surface 1 and reaching vapor = J A F)y yajj

Energy leaving surface 1 and reaching surface 2 = J) A Fj 12 vT

Energy leaving surface 1 and reaching surface 1 = J)A Fj jj yt

Total energy leaving 1 unit area of surface 1*

..

=Jj = J (Fjy y 12 v + F))Ty)a +F Tj

F)y y + F12(I-*v) + Fjj(1-c ) = 1'

c.. y

26F12 * I T

F =jj

.'. F jy y + l-c =1c y!<

F =1jy

Similarly

21 v)J * J (F2v"v + F T2 2

'

F +F T *I2v"/ 21 v..

. . .

.

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ - _ _ _ _ _ _ _ -

- - ._. . _ _ _ _ _ _ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

-275-

Assuming that the vapor is aonreflecting and that Xirchaff's

identity applies, then >

c +T =1a +t =y y y y

1-F21(I'Ev)I *

2v cy

H.4 Net Radiation Heat Transfer

Referring to the network diagram, we can calculate J) and J2

as follows:

1-c( g *)J) = oT -op

W

C A,t(J)-J )A Fj 12(I'Ev) * IJ -Ebv)A F2 2v v + (J -Ebt) 1-c 0C =

2 2 2L

C ^2 bfE ^2t t

2 ' 2 2v*v - A) F12(I-*v) * l-c'

d F ~JAF1 j 12(I-C )*l-c vg g

bv^2 2v v+E F C

J and Jj 2 can be substituted into eq. H.1 and $r can be

evaluated. The parametric effect of film thickness and wall temperature

is shown in Fig. H.2.

-_

_ _ _ _

-276-

.*

.

FIG. H.2 PADIATION COMPONENT,

0 .31 0.56 0.75 0.89 0.97gas,.

p

a 700*Ctes. a 800

+ 900X1000

14s. I1100__

L2s.

N

{ tas.

N3v-C 88.

XD_Ju_ es.

4s.

. w" m2s.-

l...50E-63 . IE- 92 .:::E- 82 .35E-62 .4E-62

FlLN THICKNESS (M)

.

_ _ . _ _ . . _ _ . . . _ . . _ _ _

_ _ - _ .

-277-APPENDIX I SOLUTION OF THE MATHEMATICAL MODEL FOR

INVERTED ANNULAR FILM BOILING

I.1 LAMINAR VAPOR FILM%

Combining eqs. 7.1 and 7.24(a) we obtain

2'2p gog6 Uy 1 d6 1- -

2R$ - 2(P-6)c*R 4u T dl * R2(h -h f)*_ T0T g

y y _

- -

This is the governing equation for the vapor film

thickness 6. It is coupled to the liquid core through $g.#The criterion of transition from laminar to turbulent film is the

local Reynolds number. For laminar flow, the wall shear is given

bydU

yT *w v dy

-gog6 U'g"v 2u +6"

_ v .

*The friction velocity is Therefore, the local Reynolds *.

v

number is

Emax

U *s v ,,

where 6,,x is the distance from the wall to the point of |

maximum velocity in the vapor film. I f n >n *, then the flow iss

turbulent.

.

_.

|

__

-278-

I.2 TURBULENT VAPOR FILM

The governing equation for the vapor film is eq. 7.5,

which is snown here again for convenience: J.

*.s

d M,,

- 1 az - 6(2R-6) dM1a

dG 2~ 2i 901(2R-d)6vI.2

'*~ * --

R-6)2 az R - R-d 2dzq

.

Each of the terms in this equation is expressed interms of 6 and n.

From eq. 7.21

-

_9-

VM pv R 9=y y y 3 - 0 4y

(2+2m)P "Y ( ) (UT)

* P V O +v v s 36 v

s

R 1-

x ,

+2m(2+m) 6# 3.

2) s(2+2m-

Pv"v (2mc6s( +m) 2 2m

,

"v _ -2 2 4 Rd6

2 2 ( 2 +2m) R d6'

s+pP v"v (2mc 2)Ds (2+m)P dZpPv"v 's 2 dZdZ36

3s

(2+m) _

+ P v (2mc 2)( )(n) Ry y

6 '2 2m bT dZ(2+m) 6

_~

(2+2m)-

-

* I d6. d6 h22(2mc )(n)2 R lpP V

- T+6)\2dZ#y y /6 \2 dZ j

(2+m)3

.

_ _ _ _ _ _ _ _ _ . _ _ _ _ . _ _ _ _ _ _ _ _ _ _ _ _ _ _ . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . _ . _ _ . _ _ . _ _ _ . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

. _ _ . ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . _ _ _

-279-

.

[d6-2 2 4

Pv"y Us=

z3R6 s

- 2+2m 2+2m -

2 U d6- 4mc 2 n s s

OV"Y 6 2-

2 dZ(2+m)R (T 6

_f63) _

s

2+m -

2 * l R 1 2+2m d34mc E+ p y fq) i

- 2+2mj2 vv m dZg (6 3(2+m)

'

_

h d6n[ R T2 BT~7-

((2+m)([6 +S)) -s

From eq. 7.18

2 vG = R0 - ! S

I 2v 2py

= RO) - 02

s

s32 S /R- 2m+1/

*

(R 3)-s mc n=U 2 s m+1s

(m+1- 6 -

T6m, p +p g

+ 2mc n m+1 -3

2m+1-

-

_ _ _ _ _ _ _ _ _ _ __ . _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ - - - - -_

.

-280-

6Assuming that 6 <<R ar.d (2m+1) h>>(m+1)(6 + ), the above equation

s s ,can be simplified to |

\

+1 m+1* *R R 2 2mc . + 2mcRG

* 7 's (m+1) s (m+1)Uv 2p

y

Differentiating with respect to Z we obtain:

1

2 dG iii dn= 2RC n2p d2y

h= 2RC n

From eg. 7.23

1 (G-G )R _ 2_

y

"I *{ , ,R-6

[g= ( -6) (G-G )y

Differentiacing we obtain

2 _ [t# d6 dG1 ) p y

~[(R-6)dZ - (R"-T) dZ dZ~2yMg

. . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

i-281-

By definition

T( 6-[6])

*Uvy

f "v U\T *Dw vI 6

N0~

To calculate the interfacial friction, the Reynolds flux $ is given by

2 dGR y

* * o U (R-6) dZy

2~

R 2RC dn_ p U (R-6) 2 v) U Hy y g

Substituting h from eq. 7.31, we obtain

U v d34 = J-6)p U dZ -yy

hand hasfollows:S

Finally, d and can be related to

n n5 s

6 =6x 6=s

2n-n 2n-ns s

6 = 6-6T s

3 U 6ns d6 s s da( M}dZ * W 2n-n (2n-n )s s

U 26ns d6 s do

*2n-n ~ (2n-n )2 Es

s

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

____. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - -

.

-282-

d6 d6 \T d6 3

' F " 3Z F .

d6 's 26ns dn

" dZ ( ~ 2n-n } * (2n-n )2Hs

s

2(n-n ) d6 + 25n5 s da2n n dZ (2n-n )2 dZs

s

All the tenns in eq. I.2 can now be expressed in the dependent variable

handtheindependentvariableh(whichinturnisrelatedtoh).

Collecting all tenns containing h to the L.R.S. and all terms containinq

h- to the R.H.S. , the result is eq. 7.32, viz

f) h=f *I I.32 3

.

I.3 Liquid Core

The governing equation of the liquid core is eq. 7.10, which isdU*

as 1 1 a * as r t asU i- *" I'H + *+TFt H " (R-6)2*

r 3r ar 3r.

This partial differential equation is discretized in space to yield

a set of simultaneous equations, which are then coupled to either eq. I.1

or I.3, depending on the flow regime.

1________ - __ _

_ _ _ _ _ _ _ _ _ _ __ _ _ _ .

I.4 Solution Scheme

The coupled ODE /PDE is solved by using a simulation package FORSIM*

on a CDC Cyber 170. A listing of the program FISHY [ilm Superheated

b pothesis) is shown on the following pages. An output from a typical

run is also included.

* Carver M.B. , et al . , The Forsim VI Simulation Package for the Automated

Solution of Arbitrarily Defined Partial and/or Ordinary Differential

Equation Systems, Atomic Enercy of Canada Limited report, AECL-5821,1978.

__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - __.___ _

-E84-

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CCl*P90fbST505&LITY CDe$Thir? POS INTERFEIAL SERAA

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-285-

r, .. ,Foa,. ,.,,4 .Me1 - *,MK, m..4., .1-4 2, . . . 1. ,4 . 3

11 l CrJu iTwee . 7831iTLee. 96riase13. 4f.asse . 48

16e CC Le1. 9C

E' A42= MAS *WTaeCCCCCCCs"CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCwwwwumw

-CLAS C SaCT!as 2

CC $PIECIPT Tute la ns .taLET *W4PEAAFJRE 1C) .Penamatta r EpasC maa5 FMIR i EGw in** 3. 6eC

11e Os*4.81194f tW8el.Fe ll t .

G*3 39.C <

139 a*e. Sees32*980

C tergsmation fysmituaTIon CaittatonC SpeCIpT mAalsuute Ft;Je TWiegutSS. mad. TURS LEBrrt. thaA e4LL fBtfC 7F%Es Ttasstaarrtout Cat?tagon AAS SULLT In

144 COttJtAsee . S*E8 Fle=e. 6

CMetha =1606.

Lt1 CC SPECIFT GUTrot Fam3uWWCT, CSf alfLT TS EVTM 9.14C

930UT=4.81C

154 C GG To teseotr?IBS FLUE TO SPECIFT TWS GRAT FLSECwwwwmi. -

CC SECTton 3

ISS CC safena"un LJ3 pts peDessTtssC

PFFWe9 e 6. 49 46CALL scLease4FFPE.4. 0tausT,as.1,

16e euBOL*1., as e 16 *14.418

SLeas t i t * 3 3 2 4.apan s 33 o gg ag g 35 t e33 26.vt ACLeast31*1.4443Cous0Leam a e i *1. 7 le

14% CPL *et i f l * 4194. 6

Tsafe ' as ( 131.s CON 0f.FRLeCFL*TIBC1

8 3. t /1. 0amou'ela at t79*14.618S E *USL=Ol*0L/ BIBOL/ CPL

110 C,

C CAkh INLET EurfRALFT

streaMrTIng great 3 t o /14 OPT *1 SOUWDe***/ TRAC $ FTW 4.6*4 3 3 81-49 37 17.81.34 54GS 4

Cf tBref te*1.8*13.CALL POI.aas ( 4. . T157.0095t7. 85. 21

179 SIW*As t 2 9 ellle.OTimeftat-Tis ,Ele = # 3tle-el.a /WPr3

CC IBITI AL GUESS OF #4LL TEEPERATUSE

184 CTWoT5Af *10.

CCC ssT UP 1stTI AL CosIDITtates

149 C31 CowTiteUS

IF tB .GT. 4.5 Go TO 13Del.see .90mER=4.

190 goe .Uvoe .0011 l*1.Gr1

11 g ( Ii =1.

S t WFOIlrfl *G .19% C

C BLEne433 COI651TIONSC EdesBS kJUh04AfC

3C l ! .1. a s =1.300 Sc a l. a. li et .

K I 3.1. ll ee .K 4 4.1.11 e4.

CC VPtta GupisOAST

205 CRC 11. 3.1 L *4.EC ( 2. 3.13 *1.WC i 5. 3.16 ed .eC e 4.3.1 se

310 - . - -CC DTsamIC $5CTtomaC

all cc tsCTtoes 4C CEKE PbMe REGIstSC

33 CowTtout3 JS esL?ame*est a

C ALL Paast? t eFDE.870 TWT 9.OI.01.gastC CAirUtATE FLGIS P90FT9?IESCC

335 C Cairt. Aft tourEMitt tasED ces pagvices TALat OF TWC Tus LaCattan or surCE uas et cue stTess SIDs or TGR CUtsas? sip?amCE 3C TWIS stLL FoeCE TER rersGRaf tsG 30EFTIME TO Tass sinagA33 STEFS.8SFilmier.C Cu TER toLanAleCE cu strasse-?ns11

.

__________

_ _ _ _ _ _ _

-286-a

,

SUtuMF?tst 870473 74/14 oFTe g */ taAC3 ryg 4.6*433 44-41 31 17.04.34 pass S

C338 TP= 4 7WefG AT1 *9. 5

TFhSRS* fF* 4 S*l3.I Ft teucT . 30. + L sGG TO 3 3I P i &&$ 4 fF ears-TA8& D . Lt. S p 30 TO le

13 COWTINUS339 YMisfF4dR5

CALL FOL4Aas i PF95.YFM83.042 WIT . RS.18asairet. . as t Li * L4.819Twe*2 L 28 *1134.V15CVaps t 16 * L. 4043

344 C040*=as t 4 i *1. 7 3 g

C Pweta a ll *4144. OCCCCCCC CCCCCCC-- sC fus AweesTse sTheft is Tus LM1maa Pfue aestoesC At EVast STED TBS LOCAL ESTWOL.DG WOBqSta IS C5aCEO

349 C IF ET4 IS CLc45 10 8745. 75E3 AT TBA uSET COWWRaGED STSPc CouTaDL IB swifCERD GVER 10 TBS TumagtarT FIWe 93GICsCC

24 Coerttwut250 :P s isuuT .8Q. *1l 40 TO IS

IFi t? .33. 4 p GO TO 341F v CSLf 4 . LS . 0. 6 10 TO 35Fa tCTVeegmT e l. S t a sm0L*0tLT&/ 3./ as0V*G/ts0L/OS '.th*Vt SCV/SSOW1T ot . S * i k . *Gs EBOL/Baod * 3. *VISCV/ 9. 01/Dthtb DELTAp

215 IP 'T . 4T . L. D Ten.

fThensov'OELT4*Fa tCTV/Vt aCW'.TEF t i ff h-ETAS) . 7F. 9. MOL ICUuv EQ. LJ 30 TO 34

39 Coerttuus264 C

rNrvv'CCCCCCCCCCCCCCCCCCC fCC 3ECTton 1

365 I.AMIRAa FIUB SEG805C

1?=4CC CA&CUL&fS VAPOS mA38 Ftml & LIQUIS COSS Olnw* VRW3CITT

270 CGV* 3. /n*esov* t t. 81 *asoL*gsLf4*es&fA/13. /VisCV

k *Gs RBOL/ 3. B *SELTAULa te-qev /RSOL/ (E-OELThe / r A-OELf44 892

L F LGV . GT. L . 5- 5 4 OveGv/ R50V/t L.-(1.-08Lf4 E) **3D379 C

C SET UP Javtum1pG EQuaTIcie FOR LIQUID ConsC

DU1AE*0BB i 8.UL.S. L 6OULos*4 Lim!00LDE.-& . 24. b

394 If tOELTA . LE. 4.5 '.*L.

DSLT47*0 ELTA * 11. = f IDELT AseCEL Th-Cttf A f *4.1SDDT*4PIDE' T&f t30 12 teL.arottr?

26S w ill efDet*Rs t Il 'U R t I n

\

SUse0071WE Opga?S 14/?4 0FT*1 nGUuDe*** / TRACS FTE 4.6*4 53 S L-4 5-27 17.81.24 94G3 6

L2 CONTrautCALL PUPIle.De 55.mCUPL,880tET.sSQoS300 L 3 1 * 3. apo t u?

USIElae De L 1I / As i II / I S.081,74) / I S-OSLTA) *0. S*ta i II *DcLD8'US III i/UL290 13 CouTINUS

CALL PAAFIS r WFOS. sPOI1rf.O.BS.98.UERICCC Cas4ULAft osat fanasFtB TO LIQUID Cons

291 CPW ILa-03 e spo tsp?) *tDOT*tAOL* CPL *UTIE

If t rBIL .Lt. 4. 4 Pe t Lee.CA&& FLUM (MITOT,5)

C300 FutelFIIIfot*n rette *2.

I P q 941 . LT. 0.1 ru l ee .

PWIt**WE / 3. /WISCV/ IET-EL)1F iM13 . L7. G. I #911ee.IF tg .LT. 9.1 Q*4.

30 5 CC CAIEUIATE * Buses !W OtPFt98WTI AL EQUA?IDE Foe Th70e Ftue

Demost & *4. 91 * * SOL *DSkTA *DELT A/VISCV / 4.casuoenaer2. amoL/ 2.OswMu= A . *geam0V * i sv-EL# ' logan)sel *ctieust2 3

318 an Saru tQD 90 34

le CourttsetC

315 CC SaCTtcet 4C Temaut.strF FILseC

n= 7

| 339 Cee . 74 (' C 2*C *C g

| IP t !T .55. 91 40 TO 3?EDOT JeEDOTstift i6.416 Q . M I I . DELf t.0ELTAs .TW, E . 08

331 41 FeastAf 1// ,15 * A7 ftASSITIOS F3000 LAstE5Aa TO futacLANT F1MI.1 fE3 UA4 utt OF C. FBIS, BSLTA. DELTA 4. TW. S. OS hAS*./3 .54.8513.1'

3? CturttuutC

336 CCC

WT&f* '0-ffas38t/ 3. *28aeC*t/ +m*13 * stas ** t im*Li/en )L / <n * *DE k * 'O* L t

335 tr 1staf . La . 6. ) watts (6.62 0.Pe t t.JELTh.Dattas.8?a?.fu S.08C

42 rommeAf t//.LE.*tT415 eSGAf!VB. v4 LOBS OF 9,PEIS. DELTA.BSLTAS.& f7 Af. ful. 8,01 AR2* . #,18. es13. SsIP t et4T . kf . s.6 9. TO et

144 EThef?Af ** fe/ 'm*1) 9IPirTh . LE . ETAan STafefThe

C

_ _ _ _ _ . _ _ _ _ _ _ . _ _ _ _ _ _ _ _ . . _ .__ . _ _ . _ _ _. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . _ _ _ _ _ _ _ _ _

.g8].n

SGSMUTt40 UPDAf3 74/14 SFT*1 - */ TRACS PTW 4.6*433 41-46*31 17,3& 24 pag 3 7

f?41*2. *gTA-ET ASSTAJegT AA *ff AA

149 C

3v*Q'so. *Wl acV / R3 , tTAL3

:4LTA ctLTA * STASDSLT AT*GELFA-05LT AS

Clie UL* i4-Gvi, bact/ 4 5-CELTal / f B-OELf 46 se2

SEILDE*cta e 1.UL.S. . a sDLL&S*ALin eDUL45.-a. 26. AUvoqv/ts0w/ t 1.-4 4. -oaLTA/ Sa ** 31

4 ALP *4. POELt&T3SS EDOT 1*AP i aALFi

CC 5091 DIF"SIVtf? IS C30Sf5 TO 95 *IIS aEATTS Of fBR CURRENTC C A&cDLATTa VALLs Ame 7tAf AT TRamSitt3se Peon LAmtmAA TOC P'J esutENT

164 EDDT*Am45&iED011.8DCf 2D

SET UP 90UAft''Me8 POS LIQUIS CCES

Dia L 4 t el , ePC 11p?14 9 W e ! P *SDDT*ts t 14 *UE lIl

14 C0erTIeM?SCALL PUPE (W.Dw aS.aCUPL.pPJtWT.st; DEILe L1 1*2.apolufT

JE s I e * dOwe Is/ RS t 1)/ 't-OSkTAs / 43-ORLFAs *4. SatSill *00La8 *Ta t 13 t /UI.3 ?S L1 CourtInU5

CALL PAAriu l uPOB. mV *WT.G . GI . OR.gma nC

MIL *-1B q uPOINTI *fDOVotsch* CPL *Uf13IP' NIL _Lt. 4. i M I L*4.

371 CALL Ft.4X t reIM. 8)C

P9 tes NITO"*t-pu tLI 8 3.

NT B*te E/ 2. /V!SCV/ 'WW-uLaIP t N13 . if . 3.I MII*e.

348 CC

81*fW.S a , WHOL*1 A/ ia-OSLTO * tG=GV) 3 **3C

0540ste*WI SCT*TT SCT< 4 SOW *1./ 3./ t*f?AS2 * RTAS 3*fT ASitT AL385 L calf ASf og:.ths

DEss en&et* 2*w*C 2.'E/ ines t a f ( 4 ETal ** t . 3*3*est ,WI eL / s ttLT AT A5a / 'DSLT&f/ I*csLTAsa *rTA2 - t F'' As * * i ' 2 * len'3 *csLT, DuifASe DELTAstt /gg 6

3 *tytP v*vtsCv/asowl/gTAA190 C

CDEurstJo2* v Dg-catran *cgLf4*seasesusk/ t r e-CELTAl **31

C199 C

DEamJa*- a Lawasse*0gnease1 *osom2CCC

SUSeE3DTISE DFDATE 10/94 ctf*& notDDe***/ ? TACE P?W 4.6*4 33 91-45-27 17.61.24 pace S

osee DsDEet i37M ** t-L/Bt D / tC*Sb 'F513*9. S

Caselevt 3CT*vitCT *ps0V* 3. /1.*fTad3* stas 2/trA2

1 < t/ DELTAS /DELFAS*0 ELTA *TTAS*3MS2*e etWOW*TISCVev1Sdritets*C2/ 4 2**I5 8

444 1 t t IT&** t t 3 *2*40 /t:3 | /1 (BELTAT/ 200ELTAsa **Il1 -4 5tas** d ( 3*2*It. /46 i d DELT AS/ DSLTASa "DELT&*f?ASiYTAA

Cens l*4 /9 *TisCV*vi SCV/BA0V*EI*C3 * ETA *H I 3 * set /ml

& * a 3 DELT AS* e 2*2*et 14/ 4 2 *ess -6, e-ST&*t? A$e WT A2410 1 *t*oaLT A/1 ( DELTAT/ 2 *0E LT&38 ** 31/ ( 2*Ett I

CCC

RES4*4*sost 'Eseret, as0L) *0tLTA* f DS-0ELTU / I IS-OELTAs * 3 pe419 & *v t acT*C* t igT Al ** t le h8 6

/*Cnast*2* , eTAs * eto / * rvisCv'T18Cv/asortt

L / ( f 40ELT A*CELFAal *@. ll **31DG*GT, R509/ t DB-OSLTAI /DSLTA* 1G=GV) /RBot./ t t-DELf 41/ ( 3-OELFO

429 asTFUE * 2. ov f SCV/ # t-OSLTAs , R$0V/UT*PW18Pe tcT1*e . 00 l* I I . 'see . *DELT A< De t# B ICTSmCC l*FBICT! * ' RE7 4 -afTFUE/ 3 /Ft fCT t s -aETFU1/ Fe tCT16ABS **P5 tCTS *tadiresJ ot2 *oG* AS$ 4 0GB / L B-OELTAa

C429 amS7*4. stetSOL* E OS-0ELTAs * CELT &/R2

Re fe=PW I I * 3 *vlaCV/ 3 2 *ULC

Ese* d-EBS t-as t2-Em S 3-RSSel 'OueOS- eu B5-esS4 *eBS ? *eBSGIT*&

438 C,

C I

la C0er?Tutt Ig

e ll C DIFFRAENTI AL R3 POS DSLTA IC g

Deaf 43*ess /Utmest/BIP iDELTAS .LT. e. b DeLfAJ'4.

e40 Cs-me - sCC SECTIGB ?C *p

441 C EEAT fnAmerta saCT10eC

tr iDELTA .LE . e. p G3 TO 39CC

494 C CCeIDJCTim ContriaissTC

MICop*COSUW/CBLT AS * (Tie-TSST)CC AActation Costrohrpf

GSS Cftas>TSAt* 2e .

. _ _ _ . _ _ _

_______ _ _ _ _ _ _ _ _ _ _ _

-288-

SURROOTINE UPDATE 74/74 OPT *1 ROUND*+ */ TRACE FTM 4.6+433 81-05-27 L7.01.24 PAGE 9

IF (TW . GT . TRAD ) C ALL PH E RAD ( PH RAD , D ELTA . TW , TS AT , ENITW , D4 ITL. P)C

PHRAD=PHRAD*1.E3460 PHICAL*PHICON+PHRAD

CCC ITERATION SECTIONC

465 CC COMPARE ACTUAL AND CA!4ULATED HEAT FLUXC

PMIDIFoPHITOT-PHICALPERDIF=A3StPHIDIF/PHITOT)

470 TW=(FRICON+0.5*PHIDIF)* DELTAS /CONDV+TSATIF ( N .LT.TSAT)TW=TSAT+500.

C39 CONTINUE

C475 CCCCCCCCCC 7 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

CC SECTION &CC CBECK CP ?!*'.E REMAINING

480 CC

TLEFT*KTIME(2)CALL FINISH (5..TLEFT,5HTLEFT)IF (IMOUT .NE. 1) GO TO 40

485 CC STORE UP !NFORMATIONC

IS=IS*1STOR2AtIS)*GV/G

490 STORAD(IS)=PHRAD/PHITOTPOWaPOWER*3.1416*DESTCRXE(IS)=XIN+ POW /G/3.1416/R2/EFGSTORA3(IS)=1.-(1.-DELTA /B)**1STORTM ( IS) *TW

495 STORPL(IS)*PHIL/(R-DELTA)STOR3(IS)*3STOPCM(IS)*PHICON/(TW-TSAT)*DE/CONDV

CCALL RITER(RS,10HCOORDINATE)

500 CALL RITER(0,10HTEMPERATUR)CALL RITER(UR,10HGRADIENT )CALL RITE (2, DELTA, DELTAS. ETA,W,Q)CALL RITE (DENOMO.DENOM1,DENOM2,DEMOM.DULD1, DELTA 2)CALL RITE (RHS1,RES2,RES),RBS4,RESS.R536)

505 CALL RITE (RHS7,RESS,RES,UV,CL.PHIL)40 CONTINUE

CC CHECK WRETHER TERMINTION CRITERION HAVE BEEN REACHEDC

510 CALL FINISH (~1.E-10, ETAT,4HETAT)CALL FINISH (3,IFIN,6HLENGTH)CALL FINISH (DELTA.DE!J4Ax,5HDELTA)CALL FINIS 3(5.,TLEFT,5HTLEFT)

SUBROUTINE UPDATE 74/74 OPT =1 ROUND=+ */ TRACE FTN 4.6+433 81-05-27 17.01.24 PAGE 10

CALL FINISH (TW ,TWMAX,2H N)515 RETUPN

41 CONT 14CECC END OF PROBLIN SOLUTIONC POST-TERMINATION HA,4DLING

520 CWRITE [6,661) TURS,G, TIN.CC1,PHITOTWRITE (6,662) P.TSAT,ETASWRITE (6,663)DO 111 J 1,IS

525 WRITE ( 6,6 66 ) STOR1(J) ,STORTN (J) STORIE (J) ,STORXA(J) ,1 STC RAH ( Jl , STORQN (J) ,STORPL(J) ,STORAD(J)

111 CONTINUE662 FORMAT (II,*P= *,F7.1,* EPA *,5X,*TSAT= *,F7.1,'C *

,

I *ETAS* *,FS.2)530 661 FORMAT (*1*,1X,*TU7B = *,F6.3,*G = *,F6.0,* TIN = *,F7.2,

1 *CCl* *,F7.2.1X,*PHITOT**,E9.3)666 PORMAT(11,F4.2.11.F6.0,1X.2(E9.3,12),F4.2,12,F5.0,11

1 .E10.4.1X,F4.2)663 FORMAT ( 1X , * 1* ,41, * TW" ,5 X , ' XE* ,81, *1 A* ,8x , * VOID * ,12,'NUSS * ,

535 1 3X , * P9IL * ,6x, * PHRAD/PHITOT* ,/)RETURNEND

,

_ _ _ _ _ _ _ _ _ _ _

.

-289-.

$Usm00?IME FLUX 74/74 0FT*1 30UW0****/ TRACE FTW 4.6*433 81 4S*27 17.01.24 FAGE 1

1 $Uta00TIus FtKI(M ITOT.IlCC SPECIFT EEAT FLUE ALouG TER TUSS *N W/4'*2C

$ MITOT*L46.E3RETUsaUD

%

Sota0UTIEE FsIRAD 74/74 0FT*1 300no****/ TRACE FTW 4.6*433 81 4S*17 17.01.24 Faci 1

1 $Ut n0UTINE NIEAD I FLUI . DELTA.TW. TS . In tTW. INITL. F lCC ESTIMATE EADIATION EEAT TRANSFER COMPONENTC

$ C FLDA IN EW/R**2C

Capeton/LENGTE/DE

C CRANGE ALL TEMPERAT'JRES M ABS TEMFC

10 TWE*Twe273.TSE*TS*273.TVE *0. 5 * t TWE*TSE D |

C TIEW FACTORS g

C g

15 VF1V*1. |V71182* DELTA /DE g

V712*1.*vF11 g

A1A2*(DE/(DE*2.*DELTall**2C

20 C TAFOR OLISSIVITTC

FIAJeL 8 *17. 72 *EIF i * ( 1. 8 *TVE * 1000 .1/700. )EMITV*1. * B EF l * 2. * DELTA * Ft.AsCE * F /10 0. )VF 2V* ( 1*V712 * (1* EMITV) ) / EMITV

29 CCleTF 2V* EMITT* A1A2 *VF12 * ( 1* ERITV) * EM ITL/ ( 1* ENITL)

CRe s1 * ( 1* A1A2 *V712 * ( 1* ENI TV) /C11 'V712 * ( 1* ENITFl *TWE ** 4meS2* t EnITV*VF2V' ITVE **4) *IstITL/(1*ERITLI * (TSE * *41 )

30 1 *V712 * ( 1* EM ITT) /C1RES )* EMITV*TF 1T * (TWE * * 4*TVE * * 4 )RES*( S. 464E* 11) * (RES1*RE S2 *tES31DENOM* 1 *vF19'MITV/EMIW* f 1* EMITW ) * ( 1*ERI TU) /EM ETW'(1

1 A1A2 *VF 12 * ( 1. * EMITV) /C1) *VF12 * ( 1* ERI TV)11 FLUE *EES/ DEMON

RET 3RENO

m

_ _ _ _ _ _ _ . _ _ . _ _ _ _ _ _ _ _ _ _ _ _ . _

. ,

-290-

...........................................................

* F0R$Ie VI *

* FORTRAs Ca!IrrtD DISTRIBUTED SYSTEM SINULATION PACIAGS ** POR PARTI AL AND 08 ORDINAAT OIFFERENTIM SWATIONS *

* UStaa nAmuaL - ATOMIC ENanGT OF CAAADA REPORT AKL $421 *

* A4TTBORS m B CAAYta AND D G STTWANT , C . R. N. L./ A, t.C. L. *

e.e.....e.e................................................

FIIJE SOILING ANALTSIS

TER FOLI4 WING VARIABLES WILL 83 OUTPUT AT Ttt TIME INTERVALS SPECIFIED BT DTOUT

CEEE TRAT TER VARIASLES IN TER CALLS TO RITE AA2 IM ft!S ORDER

2 DELTA DELTAA ITA TW Q DEMOMO DENOM1 DEMON 2 DEMON DULD$ DELTASRA41 RES2 RE S ) RES 4 mE S S RES6 ass 7 maSe Ras UV DL PBIL

PARTIAL DIFFERENTIAL EQUATION 1 WILL St DISC &ET!SID USING 3 POINT PORMUIAS AND 31 SQUAL SFATIAL DIVISIONS

CASE NUM882 1 USING GaAAS STIFF (VARIA8La STIF) INTEGRATION AI40RITEM MITH FULL JACOBIAN ANALYSISINTEGAATING 34 EQUATIONS

TIMS 0. STEP .10003-02 CPTIMS 3.477

VAa!ABLS S P A T I A L T A R I A T I O E

COORDI1Att 9. .1000t+00 .2000 .3000 .4000 .5000 .6000 .7000 .8000 .9000 1.000TDFERATUR 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 6.GRADIENT 0. O. G. 4. O. 9. 6. 3. 9. G. -45.00

1 = 0. DELTA = G. DELTAS = 0. ETA e -a Tw = 112.6 Q = 0.Druon0 * 0. DINOM1 = 0. DENon2 e .1047 DENOM e 1871. DULDS * 9. DELTA 1 * S3.95RESL e 0. RSS 2 = 0. RES 3 e G. ras e * 0. RaSS = 0. RaSS e 0.RES 7 * 9. asse e G. Eg 8 e 602.8 UV = 0. UL = .3093 FW IL e $70.2

Tint .10003-41 STty .54273-04 CFTInt 12.83

VARIASLt S P A T I & L V A B I A T I O W

C00aDINAft 9. .1000t*00 .2000 .3000 .4000 .5000 .6000 .7000 .8000 .9000 1.000TIMFt3ATUR 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .9994 .9777 0.

.91923-12 .64183-09 .24988-06 .1.000GRAD!srt O. G. 4. 9. .0944s-04 .14273-01 -1.481 -27.27

8 e .10003-41 DELTA e .41323-04 DELTAS e .37823-04 ITA e 7696 Tw = 271.9 0 * . 23 2 ?t-0 2Dtuono * 0. DEMORE e .3098 DENon2 = .1047 DENon * 6532. DULDS = .1099 DELTAS e .2445as S1 = 0. RES2 = 0. 9553 = 0. Esse = 0. RaSS e 9. Aza6 = 0.Rs S7 * 0. RESS * 0. RMS * 9.534 UV = .3047 UL = .212S Ft!L = 866.9

-

TIME .20003-01 STEP .11693-03 CPTInt 19.27

VARIABLE 3 P A T I A L T A R I A T I O N

C00RDINAft 4. .1000t*00 .2000 .3000 .4000 .5000 .6600 .7000 .a000 .9000 1.000T mFERATUR 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .9999 .9944 .9074 6.GaAD!str 3. O. G. .14448-10 .24333-08 . 39148-06 . 38853-04 . 32473-02 1393 -3.195 -15.42

.20003-01 DELTA = 74432-04 DELTAS * .58348-04 ETA = 1.126 tw e 337.2 3 = .58538-02I e

DENnMO e G. DEMcM1 e .7691 DEMOM2 = .1047 DEMON * .13172 0$ OcLas = .3069 ctLTAE = .70230. Rus 2 a 0. Ras) = 0. Ess4 o G. EsSS = 0. RES6 * 9.RaSt =

Ra87 e 9. nsse * 0. Esa = $$.24 UV e .3633 UL e .2146 Pu!L = 844.0

?!ns .30003-01 STtF .11758-03 CPTIME 19.87

VARI ASLE S P 4 ? I A L V & R I A T I O N

COORD!naft 6. .1000 t *00 .2000 .3000 .4000 .5000 .6000 7000 .4000 .9000 1.000Tp FERATUS 1.000 1.000 1.000 1.000 1.000 1.000 1.000 .9767 .7167 4.

.26418-11 219 38-0 9 .16428-0 7 .94$12-04 .50188-04 .17643-02 . 9960GRADItwr 0. .41653-01 .6403 -4.791 -7.713

I e .30003-01 DELTA * .13618-03 DELTAS e .10368-03 ETA = 2.039 Tw e 447.0 Q = .20938-41DENone = 0. DENon! = 2.253 DENon2 = .1047 DENon e .33238*01 DCLOS = .4276 DELTAS * 1.134RSS L = 0. as S2 a 0 Ras) = 0. ras 4 * 8. AsSS e 6. ems 6 * 0.ras 7 e 0. Rase * 4. RES * 225.0 UV = .0657 UL = .3192 FEIL * 719.1

?!nz *0003-01 STEP .22908-03 CPT!n2 20.79

V4a!ASta S P A T I A L V A I A T I O s

COORD!uaTE 0. .1000t+00 .2000 .3000 .4000 .5000 .6000 7000 .8000 .9000 1.000TpFE RA TUR 1.000 1.000 1.000 1.000 1 000 .9992 .9191 .6024 4.

.35703-08 .11198-06 .1.000 ! .23498-01 . 9902GRADIEw? 0. .3 36 3E-05 .414st-04 .16348- t .2470 -1.571 -4.970 -S.696

8 = .40003-01 DELTA = .18773-03 DELTAS = .14203-03 ETA * 2.904 Tw S19.0 0 * .42691-01=

DENOMO e G. DENont = 4.J02 DINOR2 * .1047 DENOM e . S $ 90 t *0 5 DULDI e .3312 DELTAS e .6106RS S 1 = 0. RES2 = 0. RES3 * 0. anSe = 0. RaSS * 9. nsse = 0.

*BES7 * 9. Ra Se = 0. Esa e 206.S UV = 1.462 OL = .2231 FWIL = 760.4

_ _ _ _ _ _ _ _ _ . _ _ _ _ _ _ _ _ _ _

-291-

fists . 5440 s.e t STEP .44995 43 C Pf!ME 21. 9e

was2Asts 3 7 4 7 t a L T 4 a I 4 7 I O e

CODRDIsaft 8. .130e E *e8 .2000 .3006 .4444 . idte .6464 7904 .8400 . teet 1.448TeasteATUs 1.044 1.040 1.400 L. Get L . 044

.4 4615-44 . FI L2E-01 .108 73-4 3 .134 3 8-02 . 9994.13973-41 . 9944. 96a4.8421 .4999 4.

c.aAct EWT 0. 1881 . tee s = 2.19e 4.534 4.780

t e , Sect s.41 OsLTA * .22933 43 DeLths * .16638 43 ffa e 3.446 vs a 157.1 0 e .41123-64DEasosee * 9. DE3eAe1 e 1.280 CEMIAe2 * .1947 DEN 04 . .71948*49 00LDS * 2412 DEL?ns . .5264es S 1 e 3. ENS A e 4. Es S S e 8. ans4 * 8. aust . 9. asse e 6.tas ? - 3. asse e 4. ans e 226.1 UV * 1. 90 2 UL e .2254 pa t h e 154.4

?!as .64005 01 GTEP .47474-43 CP?tas 22.19

wass asLa S P & T I & L W 4 3 I & T 1 O e

C'30mp t saft 4. .10003 00 .2004 . leet .4004 .S400 .6404 7004 . Sese . Sete 1.044TEntraATUs 1.000 L . 400 1.400 .9996 .9972 .9839 9204 .7621 .4243 9.

.11948-4 4 a .12185-0 3 .1.005GaAD I ENT 3. 11128-42 . 36168-4 2 . 54973-4 L .2720 . 9013 2.504 4.04& 4.010

t e . 64608 0 L DSLT4 * 21103 43 ORLThe e .18943-43 ETA e 4.211 *w e S99.1 0 e .82425-41DEm>stG e 1. DEmmel * 4.797 Ot*As2 * .1047 Otmus * .etecteel DOLDS * .2SL9 DEL 7at e .5281as s t e 3. assa e 3. ass 3 * 9. umse e 4. asst * 9. ans6 e 4.pas F e G. nese e 4. aus e 282.5 UV * 2.391 UL * .228L Pu tt e 734.4

Tint .70609-91 STE P .43453-43 CPT1ms 22. J S

was2 Asts $ 7 4 7 I a L T 4 a I a T I 3 e

Coue0f aAft e. . Lete t *ee .2006 .3000 . 40e4 .5000 . Seet 7044 .8849 .9944 1.000YtmetaafUn 1.000 1.000 1.040 .9997 .9952

.13925-4 3 . 94153 43 . 3444E-02 *. 31493-4 L . 99L1 . 9431.3799 .6471 .3644 8.

GaAeltu? 4. .1386 . asst . L. 34 9 -2.198 -3.421 *3 442

1 o .70005 41 DELfA e . 2 4 2 ? E-4 3 DELTAS * .31349 93 ET& a 4.514 fu e 613.2 Q e .1974DEJeosee e G. Dammt1 e 3.249 Druoss2 o .1447 camms e .18 7 0seet cotas e .2347 ost?As o .5495an s k e D. amS2 e 9. 3333 * 6. ease e e. ans) e s. asse = 0.mas? * 4. asse e 4. ass e 324.4 uw e 2. 914 UL e .2181 Pe tt e 79 7.4

fles .60008-44 $7E P .44323 43 CPTing 22.43

VAalssLS S P 4 ? I 4 L W 4 8 8 & T I O e

CnDaOlsaft 3. .10043*06 .2000 .3000 . 4 Gee . $ 300 .6044 7004 .4496 .9004 1.644TEMPTS ATU R 1.004

. 9999 . 43518 02 . 19668-91 . 99467093t-O L . 9 894 . 32174..9997 .9997 .9345 4254 .4241

Geas t tw? 9. .43358-43 .2618 .7142 *L.SJS - 2.570 3.239 -3.815

I e .9400s.01 DELt4 e .31223 43 cettas a .23105 43 sta . 9.131 es e 441.1 g m .1347Dg erAes e S. Druuna e 9.428 'JEssnes2 e .1847 Otwas e .12 7 7E *44 DULD S * .2318 DEL?As * .4809ans t e 4. ass 2 = 0. ass 3 e S. esse * 9. ess) e 3. anse e G.nes t * 4. Russ e 9. ass e 364.7 UV e 3. 4 7 L UL * .2329 Felt o 484.3

TINS . 9004 8-e l STE P .11713-42 CPTIns 22.64

wast asta S P 4 ? 2 & L W 4 S I & T I O e

C00aDIWAff 4. .1000 E *44 .2006 .3064 .4400 .5009 .6064 .7044 .Seet .9000 1.000ftmetaafus .9999 .9997 .9990 .9963 .9412 .0444 .7794 .5429 .2560 8.

*.33648-02 . 13728 41 .44443 41 . 9873Gaac1Erf S. .1929 a . 48 71 . 9L34 *1.471 + 2. 4 70 -2.914 -2.479

2 e . sett e-e t DELT4 e .34420 43 DELTAS e .31195-43 ETA * 1.477 fu e 467.3 0 * .1439Deseosto e 3. Dammel e LL.67 DE38L8sJ * .1947 OtmEs e .14 74 E *e4 Octos e .2704 OELTas o .4564au s t e 9. am 8J e 8. aus) e 9. asse e 8. ness e 9. anse . 9.ass ? e e. an6s e 9. ass e 442.3 av e 4.842 UL e .2352 PmIL e 679.5

fins .1044 ATTP .19958-42 CPTImA 22.9%

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CCE) tit eAf t G. .10GO B +44 .3004 .3000 . 444e .5400 .6400 7064 .0446 .9000 1.000tuppv** *. .9994

. 9994.94258 02 . 32603-41 . 9913.99318-41 . 9715.9971 .9373 .Stel 7182 .Stal 3349 8.

GaAs t ant 4. .2473 . 1144 -1.071 +1. 7 3 7 2.317 -2.438 2.447

1 e . LOGG 08L74 * . 3649E.4 3 DELTES * .I7198-43 ETA e 6.173 fu e 645.4 0 * .1991DEwasee e 8. Demuel e 13.se Osam2 * . L84 7 Denase e .164 73 *e 4 outas e .3127 otLTas o .4379mes t e S. enes2 e 3. sus t e 9. asse = 0. ass 5 e 9. asse a 8.ass 7 e 4. anse o 0, aus e 431.4 gv e 4.6e6 UL e .2374 Pe tt e 653.7

Tras .1144 STtp .40028-43 CPTIssE 24.21

unaI&aLg S P & T I a L T & a t A T I O e

Cooma t eart 9. .1440E *C8 .2944 .3006 .4040 .5006 .6404 7000 . Seet .9006 1.000ftmPsaATUp 993L .9971GaAe g aart 8. a.24929 41...63&73-81 . 9034 . 95e9 . 9053 . 4696

9913 .8144 .4637 .332s 8..1576 .3514 . ease *1.184 1.713 + 2. 2 2 4 -2. 4e 3 -2.103

5 e .1106 DELT4 * .39243-43 Dettns * .29943-43 BTh a 4.494 fu = 103.2 0 e .2279Ot*Ee9 e 4. PTII0 gel e 14.73 Dtajst) e .1947 pasase e .18 713 *e4 DULg3 e .2694 DSLTag e .4184Em S1 * 4. Ms2 * 5. amS 3 e 9. asse * 9. satt e 4. ASS 4 e G.ass t . 4. assa . G. ass e 447.4 gy e 3.241 gg e .2395 Pet t e 637.9

t tata .5240 STE P .13198-42 CPTimE 24.71

wast asLa a P a ? I 4 L V 4 8 I A T t O e

CODantn4TE 9. .1884 E *00 .2000 .3606 . e448 .5006 .6444 .7004 .Seet 9944 1. tee?EleFSRAfUS .9912

. 993438428 41 . 9447 . 9700 . 6170 . 0784 . 7742.6237 .4309 .3121 S.

GaADI ENT 4. .1844 . 2 34 L .4112 .8016 1.238 -1.7 34 -2.499 -2.291 *1.993

8 e 1204 DELf 4 * . 4 L 6't-0 3 DELfts * . 313 2 E.4 3 ETA e 7.146 Tu * 716.0 0 . .2624DemAe8 e 6. Dtmast e 16.39 Osastm2 o .1847 Daarse e . 26183 *e 6 OctJ3 * .1912 DSLT&s e .4449am e n e 8. as s 2 e 4. ass 3 e 6. asse e S. ens) e G. pase e 9.ass ? * 4. es se e 4. ans e 497.4 Uw e S.776 U1, e .2416 P91L * 422.9

*

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7tma .1388 sTIP .10S13-42 CPTras 27.49

vasI&sta 3 p a ? I a L T a a 1 4 T I O e

COceD rea?S 4. .18448*44 ,2000 .3000 4444 .5404 .4448 7068 . Seet .9404 1.340TEnttaAFC B .9999 .9874 .9740

. . 9143 . 9129.44LS .7320 .5420 3976 1949 8.

GaADI 5w? 4. * 46338-41 .1538 .3052 .5543 . 0961 -1.299 +1. 6 94 * L . 944 -2.429 + 1. 8 34

>

8 e 1340 DELTA e .44043 43 DELfka e .33993 43 RTa e 7.630 Ms e 731.4 g . .2903Lam.est = 0. DeeJetl = 10.10 DeaJe2* .1047 Dsurdt * .22593*06 DULD3 e 1936 otL?as * .3997an el o 3. anS2 e 4. Ess 3 * 9. asse e d. ansS e 8. asse e 3.nsa ? e s. asse e 3. asa e 126.9 UV e 4.373 OL e .2436 Pg I L * 644.2 -

?!as .1404 STEP .13445-42 CP*Ina 27.97

wastaaLa S P 4 7 1 & L v 4 B t 4 1 3 0 sxpotsatt 3. .19808*40 .2000 .300s .4400 .5400 .6030 7004 .0006 .9000 L.006te.staA?va .9s13 . 977 L .9431 .9344 .8849 .4050 .4927 .3439 .3479 .1795 0.2aAC L ENT 4. *.84498 41 . 2046 .3794 . 6323 . 95S4 -1.316 + 1. 64 2 -L.440 -1.a f a .L.68S

*3 . .1444 csL?& a .46323-03 DELTAS . .34083 43 rTa e 3.1Je =w e 743.6 2 e .331438teuse e 4. DEMOm1 e 19.42 Deucas2 e . L8 4 7 Druon * .24405*44 3CLDS e .1999 ott?&2 e .3797

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4331 e 3. ass 2 e 9. ass 3 e 3. ass 4 * 5. Eas t e 3. asse = 3ass ? e 3. ma ss e 3. aus * 157.4 UV e 6.941 UL e .24S6 PE !L e 'e' S

+

?Im3 .1980 STEP 13448-82 CPTins 28.41

vaaf asL3 3 7 a T I a L v & 8 1 *. T I O 9

CODac tuaft 8. .1448 t *4 0 .2000 .3000 .400s .5400 .6400 7000 .4000 9900 1.040TtpPlaATUS .9444 4433

. 94%4 . 7494.6%42 .5002 .3414 .1444 6..9100 .814%

GaAD t s.s? 6. .1129 .2547 a.4496 . 4901 .9992 * L. 314 -L. SS L 1.731 1.932 1.158

3 e .1940 t sLT A = .48145 03 CELTas e .36478-43 eta o 8.620 ''w = ?17.2 3 * .37e5DENQue * 9. DEmme! * 21.4% 051sDm2 * .1447 CENDR e . 26 81S *$ 6 DULDS e .1940 CELTAS * .J446am S L e 9. as s2 o 4. RMS 3 e 8. 3384 * 4. asst e 9. asse e 8.as s ? * e. mast * 4. nas e 144.4 UV * 7.632 UL e .2471 PWIL e 170.2

?!MS .1640 5717 .13448-42 CPTims 26.86

vaala3LS s P 4 ? ! A L T 4 E I A T t 0 s

C00RD1satt 8. .1000s*40 .2006 .3006 .4444 .5004 .6848 .7000 .0004 .9490 1.803TEMP t aATUR 9124 .94%3

. 9239.4841. . 3223

.733%. 6176

.4717 3L76 1539 3.'iaA0 ! EIPP B. * .13 ? S 2994 * Salt 1444 -1.324 L.297 -1.113 .L.e29 -1.649 +1.446

8 e 1440 SELTa e .5470s.03 ctLfas e .38305-43 f?4 e 9.872 Tw e 747.6 3 e .4144ggwosse e g. cmersal e 23.40 ggnan2 e .1847 Dam.NE * . 2 0 0 4 E =4 6 OCLD8 * .1471 :P*.Ta8 e .3573ha s t a 1. RE S 2 e 5. Ass) e 4. pas 4 e 8. BASS e 3. p.ase a 8.Ess ? e 8. sess e 3. ass e 911.2 7v e 0.25S DL * .2494 Pg t h = 444.3

Tins .1708 STEP .15 4 2 t+4 2 CPflat 20.16

vaa2AsL3 s P a ? I & L y 4 0 1 4 T t 0 e

CODRD!satt 9. .10 60 g*00 .200-3 .3004 .4800 .1400 .6000 7000 .5000 .9440 L.040TButtaATCH .9311 .9232 .5905 .8149 . f e91 .4979 .5820 .4418 .2941 .1432 3.GAADI STT 3. *.1508 *.3373 . 5443 *.7926 .L.0 34 *L.269 *1.447 -1.531 1.496 -1.346

8 e 1 ?O G DELTA e .32913 43 StL?aJ * .39468-43 ETA * 9.903 ew a ???.3 Q e .4961DENEeG * 3. DeMusil * 21.14 DEW [de2 * .1047 OEM[ES * . 3 0 84 B *0 6 DULD5 * .1024 SELTA2 = .3491RE S L * 8. aE S2 e 8. DRA3 o 0. asse e 4. ass t e 9. ange e 9.as47 e 8. aass e S. ass e 642.7 UV e 3.876 gb o .3113 Pg I L e $10. 3

TIME .1834 STEP .23095-42 CPTina 2s.21

v&A23.aLa 5 P & F t a L T & a I & T I O e

CooaDinaTE 4 .1440sede .2006 .3000 4004 4000 . tede 7000 .0390 .9000 L . 344f tpPtaATUS .9041

. 8974 . 8706 . 4234 7151 .4430 .3494 .4101 .2746 .1335 3.GaAc t rw? 4. .1799 .3441 .5713 . 0624 -1.438 -1.233 *L.3?9 -L. 4 34 -1.399 *1.294

0 e 1400 DEL T A = .54863 43 DELT&A * .41203-43 t*4 e 9.919 Tu * 701.1 @ * .4999Dfwlse0 a 4. Oceant a 24.0? Destwe2 * .1447 DEMME * .32025*46 DOLDE o 1774 ctL?a5 e .3422am S1 = 0. Ad S 2 e 8. as s3 e 3. Esse e 3. agg S e 0. asst e 6.au s ? e 8. nase * 9. 9as * 670.4 '5v e 6.497 7L e .2931 Pu tL e S34.4

TI4E .1960 STE P .24154-82 CPTIm3 28.32

was t asta O P & T I A L T 4 a 3 4 ? ! O e

CocaD t eart 8. 14043*e4 .2000 .3 ace .4044 .5000 .5004 .7006 .3300 .9000 1.000TEMPlaATV3 .5777 .5403 .0396

. 7947.7246 .5390 .1105 .3924 .2547 .1247 4.

GaADisas? 9. 1983 .3443 .S944 .4494 -1.416 -L.191 -1,311 .L. 3 52 *L. 304 -1.174

8 e 1940 Detta e .56045-43 DEL?ns * . 4 2 ?C S-e l E*4 e 18.97 TW e 794.6 Q = .3439CEMJeed * 4, Ogucual e 28.49 CEmum2 e .1447 DEEMe e .34948*44 DULDS e .8643 9tLT&8 * .3344ad S L * 8. AE S 2 o 9. 2s83 e 0. anse e G. eell * S. Sase * 8.as S T e S. ames * 8. ass * 497.3 ;;'t * 14.15 UL e .2549 9. L * 122.9

tima .3040 STEP .16 2'B -4 2 CPTims 29.11 evaal&&La s P & T I & L T a a 3 A T I O eCootDImatt 8. .10308*48 .2004 .3000 .4440 .S436 .6404 7814 . B844 .9000 1.089TEstPlaATUS .4462 .4364 .8049 7%47 .6864 .1999 .4009 .3401 .2422 .1164 8.GaAc t sw? e. . 1943 . 3942 .4429 . . e4 5 9 . 9920 - L .141 +1.244 - L. 2 71 *L.222 - L.0 90

8 e .2443 CELTA e .18038-03 csLTas e .44175-43 ETs e 18. e 4 Tu e 803.7 o e .5494Demusee e S. Dews 1* 30.57 cessan2 e 1347 'w== * .37148*46 BULDS * 1704 stLTA8 e .3268AS 5 L * 8. eBS 2 e 8. ass 3 e 4. tage e G. aest e 8. meSe e 8.AB E T e 8. asse e 8. ass * 724.7 gv e 10.03 75 e .2547 PW I L e $4 9. 3

_ _ _ _ _ _ _ _ _ . _

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.

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8 e .3120 del?A e .54768-03 DELTAS e .41423-43 f?k e 61.24 74 e 411.3 0 e .4344 *

Damme e 0. Dammt e 12.39 otson2 e . Leet DPgras e 39243*44 ocLDS e .1745 DSLTAS e .3200as t L e 9. as sJ e 8. ans) e G. asse e 0, asst e 9. asse e 9.ma s t o 0. Base * 9. ma s e 711.4 UV e 11.49 UL e .318$ Pg IL e 499.9

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8 e .2206 DELf4 * .62818-49 OtLias * .47018-03 ETA e 11.49 74 e 019.4 3 e .6049Dramt e 8. Dtumi e 34.23 camm3 = .1947 Damm e .41348*06 DULDS * .17L9 DELt&S * .3159as s k e 6. Essa e S. Rest e 9. asse e 4. asSS * 9. am84 * 9.ans t e 0. asse e 4. aus e 779.0 uw e 12.1% UL e .2402 re t L e 482.1

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Tins .2306 STEP ?2145-43 CPTIns 36.44

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. 4442.3051 .1992 .95103-01 4.

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8 e .3100 Dat?A e .64448-43 DELfas e .11373-43 RTA o 14.02 Tu o 040.3 2 e 7339DBam60 e .14160*01 Otw et! * .16248*01 Otanse2 e $l30. DtuCEs * e. 3434 teel DULD8 e .9003 DELT&$ e 1.614an s! e 1.247 assa e .2814 ans) e 7.844 anst * .34248-4L anst e 401.4 asse e 1844.as s? e 2032. asse e 2.17L nas e -314.0 UV e 11.44 UL e .2454 pu t L e 435.9

TINS .2400 STEP .61494*44 CPT1Ra 41.20

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. 1774.02308-01 9. .

GaADERWT 0. . 1990 .3944 ..S796 .7371 . 0409 a.9413 *1.00$ .9402 .099L .7344

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T!ma .2500 STEP .16735-43 CPT7R3 41.36

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. 3490 . 2550 . 1617.74613-41 4.

GRADIgwy 4. .1944 . 3083 .1444 . 7100 . 8393 .9183 .9451 .9114 . 4160 .649L

8 e .2500 DELTA e .44198-03 DELTAS * .17688 03 Sta e 14.44 Tu e 089.4 Q e .8120 | gDe ersee . .11228*0% Duermt e .1747s*05 Danan2 e 1822. Esale e . 34 518 *e l DULD5 = .0410 delta 8 e 1.309 m"a s eea s t = 1 079 ass a e .3312 mas) e 0.301 * e .44348 41 aast e 342.7 asse e 2009. Inas? e 2444. Ruse e 2,770 ass = .305.3 OV e 15.42 UL e .2422 en t h a 341.0

flus .2600 STEP .$1798-43 CPT!ns 47.21

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. 6210 . 48304073 .3231 .3141 .1477.192) .5454

. 67723 019.

GaAstswT 4. .1935 .3404 .55J1 .ette *.3014 .4716 *.0011 *.0422 7454 .6019

8 e .2400 DELTA e .92348-63 DELTAS = .40108-03 RTa e 15.22 Tu e 878.3 0 e .9177Damone e .19 81E *0 5 Dem)sta e .1820t*01 Otmona e 914). Devon e . 37488*05 DULDS e .0019 08tTAS = 1.442pa s t * 1.013 ass 2 * .3184 ans) e 0.741 asse e .54423 41 assl e 323.7 asse e 2241.ses 7 e 2674. anst = 3.034 ass . 32s.1 UV e 10.56 UL * 900 ru t L e 344.4

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VA3!A3La ...-..-. 3 P 4 ? ! A L y a a ! A T I o e

CT RDruAft G. 1000t*00 .2000 .3000 .4000 .3000 .6000 7000 .4000 .9000 L.000f tnPR AATUR .1912 .5974 . Stat .4914 .3707 .2984 .3192 .1344 .41148 0 L 0..10648*12 . 3897GaADItwT .L899 .3717 *.5390 . 6713 *.??04 *.0249 +-8202 *.7770 *.6708 5302

I e .2700 CELTA e .10103-02 CELTAs a .42?ls-03 sTA * 15.66 m e 091.0 Q e .9456Dawwo e 4971. CerA1 e .1844t*01 canon 2 e 10 8 0 t *0 5 Dtwost * . 3 8 21t *0 5 00' D 8 e 1.032 CELTAE e 1.448SESL e .9519 mall * .3732 ma 93 e 0.946 pag e e .44115 01 Rasl e 300.9 aas t e 2471.aB 57 e 2903, as se e 3.313 pas e 330.3 UV e 15.53 UL e .3009 Pe t L * 274.3

?!ms .2004 3T87 . 6 2J 15-0 3 CPTsus 14.42VAA! ASLS $ 9 A T I A L T A a I A T t 0 eCJotDruATS J. .1000t*00 .2000 .3000 .4000 .5000 .6000 ,?000 .4000 .9000 1.000ftMPtaATUa .5404 .1911 1237 .4793 .4200

. 3111.27%0

. 1947 .1219. 94848-01

8.GRADisr? 9. ..Lall . 3621 ..$187 . 6443 7357 =.7793 7733 . 7L62 .6147 .4740

3 e .2900 DELTA e .110S8-03 CELTAs a .45103-03 tTA e 16.12 W e 943.0 Q e 1.056Canoh0 * 8037 Druoset . .14718*05 Cenon2 e 11843*05 Damon e *.3949t*01 DULD8 e 1.233 CtLTA8 e i.431Ra s t e .079% Ra 52 e .3091 As s 3 e 9.293 Ms4 * .70313-01 Bas 5 e 303.0 Rast e 2759.wa s 7 e 3140. asse e 3.606 ass . 344.1 UV e 19.61 UL e .3125 Patt o 243.0

?!ns .2900 STEP .64148-03 CPT!n3 St.JGVAAIA3Le 3 P A T ! A L V 4 R I A T t 0 g

conRDIunTS 0. .1000s*30 .3000 .3000 .4000 .5000 .6000 7000 .3000 .9000 1.000TfmptRATUS .3361 .5172 .4904 .4474 .3911. 3244 . 1790

.3123 .1990. 48048-01 8.GaADIfw? 4. *.1004 . 3522 . 1017 . 6209 7007 . 7345 7190 .4567 .5533 .4197

I e 2900 DELTA e .1210s.02 CELTAs a .60538-03 ETA e 14.59 *if . e 914.4 Q e 1.130DEmone e 1233 Druoni e .19738*05 Druon2 e 15565*05 DRuon * *.4149t*05 DULDS e 1.450 DELTA 2 e 1.799As si e .8227 Ra s 2 e 4017 Ra s l * 9.542 asse . . 9 3 a nt-01 nast o 264.9 paso e 3041.Ra s 7 e 3414. Rasa e 3.483 ass . 444.0 UV e 19.64 UL e .3262 PSIL e 220.5

?!mt .3000 STEP .52815-41 CPTINS 62.62VARIASLR S P A T I A L T A E I A T I O eCODantuATE 0. .1000 t *00 .2C00 .3000 .4000 .5000 .4000 7000 .0000 .9000 1.000f thPE RATUR .4000 .440L

. 4149 .4133 .3594 .2949 .2296 .lett .90003 0L .43913-01 4.GRAD (INT 4 .- *.1736 .3373 .4703 a.1477 . 4$74 e.6818 .6602 . 5911 a.4973 .3790

2 * .3J00 DELTA = .13218-02 DELTAS a .71968-03 ETA e 17.05 T4 = 914.1 Q e 1.302Dtmone e 6437 DEMoni * .19 91E *01 Dem0M2 e 190$8*05 Denom * *.4399t*02 001.08 = 1.704 DELTA 3 e 1.919Re s t e .7721 pa s 2 e 4101 ass 3 e 9.035 use e .1137 asst a 2r+.0 Rase e 3469.Ras ? e 3700, mast e 3.932 ams e el14.6 UV e LS.43 'JL e .3425 Ps1L = 240.5

Tlat .3100 STEP .74903-04 CPT!n3 66.63

VARIASLE -- 3 P & T I A L Y A a 1 A T I O NC00RDIuAft 4. .100v8 00 .3000 .3000 .4000 .5000 .6000 7000 .4000 .9000 1.000ftMPB AAT'Ja .4657 .4371 .4134 .3749 .32SL .2471 .2056 1444 .0796R-01 .39095-41 8.GAADIENT 4. .1635 .3164 .444S . 9446 .8037 =.4197 . 5939 . 5306 . 4414 .3404

8 e .3100 CELTA = .144?S-02 OELTAs e 75678-03 ITA = 17.49 TW e 141.2 Q 1.271eDruono a 500s. Denomt . .18238*05 CEN042 * .23538*05 Demon = .4757t*05 DULDS e 1.474 DELTAS e 2.213Ra s t * .7325 pa s 2 e 4193 aa s) e 10.12 asse e .1390 anst . 231.s asse e 3911.Ras? * 1999 Rase e 3.897 pas . 620.3 UV e 15.67 UL * .3612 PEIL e 219.6

?!at .3200 STEP .31168-03 LPT!nt 70.72

V Aa t a4LE 5 P A T I A L V 4 A I A T 1 0 eCCoac t uett 6. .10008*00 .2000 .3000 .4000 .5000 .6000 7009 .8000 .9000 1.000TEMPERATUS .3979 .3904 .3445 .3336 .2887 .2364

. 1822.1201 70358-01 .35098-0L .4.GRADitWT 0. .1409 . 2974 .4034 . 4490 . 5304 .1490 . 5218 .4600 .3931 .3040

8 e .3200 DELTA = .15408-02 DELTAA e .79578-03 ITA * 17.91 *if * 995.8 Q * L.337DENNe0 e 5191. DEnomi e .1760t*J5 CENCeel * . 2 9 2 7E *0 5 Otwost * *.S206t*05 DCLDS e 2.311 DELTA 3 e 2.241Re s t * .6484 Ra s 2 e .4201 4a3 3 e 10.23 Rase e .1730 RESS e 214.4 has 6 e 4322.Ras ? e 4304 Rase e 4.072 ass . 694.4 UV e 15.55 CL * .3827 Ps!L = 280.0

fina .3300 STEP .38135-03 CPT!nt 74.84

UAAIASLS a P 4 T I A L T A E I A T I O uCm RD! waft O. .1000t*00 .2000 .3000 .4000 .5000 .6000 7000 .8000 .9000 1.000TEM PtaATUa .3467 .3402 3210 .2904 .2513 .2062 .1500 1120 .68813-01 .33018 01 0.GaADitWT 6. e.1307 . 2519 . 3525 . 4257 *.4471 *.4714 *.4I30 *.4074 .3452 .2744

8 e .2300 DELTA e .17118-02 DELTAS a .03928-03 rTA e 1s.31 Tw e 948.7 0 e 1.399DEMOMO e 4709. Crucul e .171 J E *01 DruenJ e 36 e is *05 ctwose = .58468*05 CULDI e 2.507 DELTA 3 * 2.420kBS1 e .6%43 R35 2 e .4238 Ra3 3 e 10.42 ang e e .2096 ras 5 e 201.2 ass 6 = 4810.Ra s 7 e 4412 asse e 4.102 Ras e 847.3 UV e 15.14 UL * .4073 PWIL e 297.9/// Till8 RUM TERMINAft0 BBCAUsf Tt3 CUR 1 TNT VALUE OF TW 13 1937.9

. _ _ _ _ _____ - _ _ _ . ._. - - - - - - - - - - -

_ _ _ _ _ _ _ _ _ _

-295-

| T'J33 o . 00 04 e 190. 719 e O S . 3 SCC l * 1.14 Pet TGT* .1448*44Pe 115.4874 T9afo 6 0 3 bC rT AS* L2. 2 03 *is as sa #0f D #989 7511. Pt eAS/ 79 t?OT

4.04 L14. . . J 4 88-01 0, 9.56 4. 09 t Ja *01 3. 04.21 473. . 3 3 t-41 .9440-4% .33 114. .14 4 3s eet .03.82 137 . 3278-36 .2385-44 .8J 201. .14323*44 .84.et 447 . . J L 48-O L .8448 44 .99 115. . & 34 & t *44 .J0.9 %29. *. le sta e & .1948-48 .44 04. 13 2 9t *04 .11.81 117 a . 2 940-41 .3018-43 af 73. . L3 t et ec t .12.44 1 09 . . 2 8 3R-0 L .3943-44 .00 43. .12 ? ?s >44 .15.97 4.6. ..J72a-e& . 32 es.4 3 .49 14. .13 44g +0 4 .17.98 641. . 24& B-S L . 44 J s-4 3 48 11. .12178044 .19.49 447 . 2 %e t -41 . 41 L S.4 3 .44 47 .&&lts*l4 .24.10 484. =. 2 315-41 .4088-93 .L3 43. 146 ?E *04 .22.&& 703. . 3273-41 .LJ 28-43 .13 44. .1 L eet wee 23

&& 717 . 2180-41 .1428-43 .18 18. . &l2 3 R *44 .25.L3 131. *.2070 .1415-43 14 14. .1109t *44 .24

f ee. .. a tes 41.&4 4L , a a t s-4 2 19 19. .19745*44 27.1% 717 = .19 53-01 .2121-42 .14 33. .1014 t *4 4 .3914 144. . . & 7 4 8-4 L .2378-43 14 31. .14828 44 .30

.17 777 .. 415 4& . 24 J 5-4 2 .17 30. .14143*44 3440 79%. 1925-41 .2885-03 19 29. .94948*49 .3219 791. . 14 4 8 -e & .31?s.43 .15 20.

.9401s*4945 32

23 Se e. . ; 1.E.41 .3445-42 19 37 .9461s .33.21 811. * A.)t-4& . 3 7 ea-e 2 .19 36. . t a ess eet .34.33 Ste. .d 50 -41 .4104-42 .24 2%. .94333*45 .39.23 644. . e40 0-9 2 .5411-02 .22 al. . 03 %c e *41 .37.34 017. ..s718-42 .4118-42 .34 II. . f t 9st eet .39.2% G4 9 ..?415-43 .4475-43 .24 21. . 6 41%g *41 .44.d4 379. . 4%45-02 .7335-42 .29 24. .40 f ts *4 5 43

37 492. . 14 % d-e 3 7803 42 .31 49. . 51448 *e % 42.JG 941. . 4548-42 .8443-42 .34 LS. 49919 4% .44.29 941. . 3273-43 .9 Log-42 .34 17. .44323*0% 41.30 92 0. . . J 19E-4 2 .9778-42 .39 L P. .5170s*01 44.31 941. . 1105-42 .19% B-e t .48 14. . 5141E *4 9 .47.33 tit. . 9112-49 .1118-41 .44 15. .63943 09 48.33 949. .1688-42 . i& 84 -0 & .49 14. . tge23 04 9 .e9

mit moS/st 1. 3 arLgA2te99 01-4%.2517. 61.17. P I SE f tG PtseSf0/%l17. 41.17. 3 P dg002434 copo s - P t LS t u PUT . DC 0 417. 41. 31.P185T. E 3 31-Pf t. f 204.1311.17.41.21. Pv4G 4.5. JR Eles!)17. 01. 21. LI BRARf l LI B4 3 4)17. 01. 3 3. PT4 6 Pee . T)17.03.20. L.174 CP SEC00so$ CusiP!L& TION f tNE17.03.24. ROUTE. PL JT , T t D* AA. DE P.! ? . 4 3.18. 4TTEu , Pne s us .17. 0 5.16. P P4 IS! ? . 3 3. L S . Poes t u17.03.19.77 CYCLS m0. * 000& ?.0 3.19 ATTES I POLS 44. IDestCasI50erl!?.0 3.19. PP4 ISL t.0 3.19. POLSAW17.41. 4 3. GO.17.24.04.PP CYCLE 80. * 30417. 2 4. 0 4..*0PYv 0.GO. PUR3 f e.1G3. POLAAm .17. 2 4.12. afTUts . Poe s t m . t/.0.! ?. 24.13. ATT ACE. Pot &In.CT*1.& ? . 3 4.12. P P4 IS! ?. 34.12. Poes taL T . J S .15. LI taAm t . Poe S 14.L! b4 34.L ? . 3 4.17. MAP toPPI17.36.10.30.! ? . 2 6. 44. WE P4f &L La".,ADES BRitOSS =17. 24. 44.IIOW-SIISTtw? LIBRASY GIVTu .IIrwPATA2, Lcaaga gascas .37910E f. 24. 4 9.L P. 36.4 9.TEIRD TO LOAD INTO 3L003 BELJe ORIGIN .L F . 2 4. 4 9 - GET . BT

17.24.49.LAST Ptir. mass etAD . s. 3QB f.24.4 9.LAST FI:J ncrtsSED= LI B4 3 4! ? . 2 7.84. sus-Patah Loacta sam 0R5 -L?.3 7.04.TEIED TO LwAO f wTo sta)CR SELcw Oc tGiu .17. 2 7.34. Gr r . t?17.3 ?.80.LAST PnoGnan peac . v.sg!?. 27.90.LAST P1La acctasED- LIB 4 34& ?. 31.14. S TOP17.31.14. 141.44 ? CP 18CQesOS * REC 12TICII ?!ME17.11.14 EE!T.S.17.31.14.09 00008130 wraps PtLa OUTPUT . Dc to! ? . 35. le . m3 21144 eneDs ( 114484 RAS US EDI17. 31. h t .CP4 111.04 2 3tc. 43.384 LDJ .19.3%.15.20 14.454 s ec . . 00 0 kaJ .17. 3 9.19.cn 112??.1Je EDS. 14.911 437.17.3% 11.31 100.00017. 3 5.11. P P 8 7.14 % SRC. DATE 0&-41-2 7! ? . 3 5.15. tJ tuo Or Jow. 1%

_ _ _ _ _ _ _ _ _

- _.

-296-

APPE.tiDIX J: SAMPLE CALCULATIONS

The governing equation for the turbulent vapor film is eq. 7.5,

which ,is shown here again for convenience:

dM dG dM 2T., 2:, * go (2R-6)6y v (2R-6) ~ g t~

dZ 1F' (R-6)2 dZ R R-6 2~

g

| (a) (b) (c) (d) (e) (f)

The physical significance of each term is:

(a) rate of change of vapor momentum

(b) rate cf increase of vapor momentum due to evaporation

(c) rate of change of liquid momentum

(d) wall friction

(e) interfacial friction( f) buoyancy force

The magnitude of each of these six terms are evaluated here, using

the results from the theoretical solution. The system parameters are:

G = 200 kg.m-2.s-I

o = 115 kPa

T = 85 Cin

t = 146 kW/m2

n * = 12

. _ _ _ _ _ - _ _ _ _ _ _ _ _ _ _

- _ _ _ _ _ _ _ _ _ . . _ _

-297-

A listing of the theoretical solutions can be found in Appendix I.

Immediately af ter the transition from laminar to turbulent vapor film

(at Z = 23 cm), the predicted conditions are:

6 = .6866E-3 m

.5137E-3 m5 =3

n = 14.02

T = 840.30C

U = 15.46 m/sy

.2658 m/sU =g

-I

h(h) = 1.654 m-I

h=35.13m

[d5 = 5.137x10-3 gj,

4.737x10~3 m/m=

Using the equations developed in Appendix I, the value of each of3the six terms is (unit in N/m );

dM

(a) [ = -116.7 - 26.8 + 17.38= 30.3

d(b) U *g = 2.17

(2R-6(c) = 53.95(R-6)

2r(d) = 405.6

R

_ _ _ _ _ _ _ _ _ . - _ _ _ . -_

-298-

2r

(e) R- = 1656.

.

9( f) 2(2R-6)6

2032=

R

L.H.S. = 30.3 - 2.17 - 53.95 = -25.82

R.H.S. = 405.6 - 1656 + 2032 = -29.6

The difference between the LHS and RHS is due to the slight difference

in the fluid properties used in the hand calculations. It can be seen that the

rates of change in vapor and liquid momenta are comparable in magnitude.

In the development of the model, it was assumed that the difference

between the pressures p and o is negligible. The equation for the difference isy 2

= Ni 2( )p -o -y g y

|

| whereAE1 is the rate of evaporation per unit area of interface.y

|

dG2 y )

.

0* = nRv dZ * 2n(R-6)

= 2.67x10-2 kg.m .s'I-2

= 2.46x10-3 Pap -oy g1

1

The pressure difference ccross the interface is therefore negligible.

- _ _ _ _ _ _ _ _ _ _ _ _ - _ _ _ _ _ . _ _ _ _ _ __ _ ._

-299-

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.

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______

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|

|

,,

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I - 303 -

Distribution for NUREG/CR-2461 (ANL-81-78)

Internal:E. S. BeckjordC. E. TillR. AveryB. R. T. Frost dP. A. Lottes (37)ANL Contract FileANL Libraries (2)TIS Files (6)

External:NRC, for distribution per R2 (325)DOE-TIC, Oak Ridge (2)Manager, Chicago Operations Office, DOEChief, Office of Patent Counsel, DOE-CHPresident, Argonne Universities Association, Argonne, Ill.

,

Reactor Analysis and Safety Division Review Connittee:S. Baron, Burns and Roe, Inc., 700 Kinderkamack Rd., Oradell, N. J. 07049W. P. Chernock, Combustion Engineering, Inc.,1000 Prospect Hill Road,

Windsor, Conn. 06095L. C. Hebel, Xerox Corp., 3333 Coyote Hill Road, Palo Alto, Calif. 94304

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W. Kerr, Dept. of Nuclear Engineering, University of Michigan, Ann Arbor,Mich. 48105

S. Levine, NUS Corp., 4 Research Place, Rockville, Md. 20850S. Levy, S. Levy, Inc. ,1901 S. Bascom Ave. , Campbell, Calif. 95008T. H. Pigford, College of Engineering, University of California, Berkeley,

Calif. 94720

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SUBCOOLED AND LOW QUALITY FILM BOILING OF WATER ...

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