The Pennsylvania State University

The Graduate School

Department of Mechanical and Nuclear Engineering

SCALED EXPERIMENT ON GRAVITY DRIVEN EXCHANGE

FLOW FOR THE VERY HIGH TEMPERATURE REACTOR

A Thesis in

Mechanical Engineering

by

Suchismita Sarangi

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Master of Science

May 2010

ii

The thesis of Suchismita Sarangi was reviewed and approved* by the following:

Seungjin Kim

Assistant Professor of Mechanical and Nuclear Engineering

Thesis Advisor

Fan-Bill Cheung

Professor of Mechanical and Nuclear Engineering

Karen Thole

Professor of Mechanical Engineering

Head of the Department of Mechanical and Nuclear Engineering

*Signatures are on file in the Graduate School

iii

ABSTRACT

The process of lock-exchange and gravity driven exchange flow for fluids of

differing densities is of particular interest in the postulated Depressurized Loss of Forced

Convection (D-LOFC) for the Very High Temperature Gas-Cooled Reactor (VHTR).

This event involves the gravity driven ingress of air into the helium filled reactor vessel,

ultimately leading to a possible oxidation of graphite components in the vessel. The

present study performs a scoping experiment using water and brine as simulant fluids to

study the exchange phenomena. To design the test apparatus, a scaling analysis is

performed to maintain the exchange time ratio to be unity with the Gas Turbine Modular

Helium Reactor (GT-MHR) reference system for the vertical standpipe break. The

apparatus consists of two rectangular acrylic compartments connected by pipes and is

designed to investigate the effects of the break angle and break length. The break angle

is varied from horizontal to vertical at every 15 degrees for L/D = 0.63, 3.0 and 5.0. The

volumetric exchange rate is obtained by measuring the time rate of change of mixture

density using a hydrometer. A flow visualization study is performed to gain physical

understanding of the phenomenon. In general, the present results show similar

characteristic phenomena to those found in previous studies for the initial stage of

ingress, where the mixture density changes linearly with time. As the ingress progresses,

however, it is found that the mixing phenomena inside the compartment and the

compartment geometry make significant impacts on the ingress rate. Unlike the previous

iv

studies, the present results show that the average exchange rate for the entire ingress

event can be up to 70% lower than that obtained from the initial stage alone.

v

TABLE OF CONTENTS

LIST OF FIGURES……………………………………………………………...… .. vii

LIST OF TABLES…………………………………………………………………... ix

NOMENCLATURE…………………………………………………………………. x

ACKNOWLEDGEMENTS…………………………………………………………. xii

CHAPTER 1 Introduction............................................................................................ 1

1.1 Research background and motivation ............................................................. 2

1.2 Literature review ............................................................................................. 4

1.2.1 Gravity driven lock exchange flows…………………………………… 4

1.2.2 Exchange flow rate estimation...……………………………………….. 10

1.3 Objectives of present study ............................................................................. 16

CHAPTER 2 Scaling Analysis ................................................................................... 18

2.1 Identification of scaling parameters ................................................................ 19

2.2 Scaling methodology ...................................................................................... 25

2.3 Design of test facility ...................................................................................... 30

CHAPTER 3 Effects of geometric parameters on exchange flow .............................. 32

3.1 Experimental setup ......................................................................................... 33

3.1.1 Test apparatus ....................................................................................... 33

3.1.2 Experimental procedure ........................................................................ 37

3.2 Experimental results ....................................................................................... 40

3.2.1 Flow visualization study ....................................................................... 41

3.2.2 Density measurement ........................................................................... 49

3.3 Data Analysis .................................................................................................. 51

3.3.1 Initial region ......................................................................................... 52

3.3.2 Non-linear region .................................................................................. 61

3.3.3 Mathematical treatment for final region ............................................... 72

3.3.4 Volumetric flow rate and Froude number analysis .............................. 76

vi

CHAPTER 4 Conclusions and recommendations for future Work ............................ 88

4.1 Conclusions ……………………...………………………………………….. 89

4.2 Future work …………………………………………………………………. 91

REFERENCES………………………………………………………………………. 94

APPENDIX A1 Derivation of equations…………………………………………..... 98

APPENDIX A2 Exchange Volume Calculations………………………………….... 103

APPENDIX A3 Experimental data and correlation for L/D = 0.63……………........ 112

APPENDIX A4 Experimental data and correlation for L/D = 3.0……….................. 120

APPENDIX A5 Experimental data and correlation for L/D = 5.0………………….. 128

APPENDIX A6 Volumetric flow rate and Froude number for L/D = 0.63…………. 126

APPENDIX A7 Volumetric flow rate and Froude number for L/D = 3.0…………... 134

APPENDIX A8 Volumetric flow rate and Froude number for L/D = 0.63………… 142

vii

LIST OF FIGURES

Figure 1-1: Schematic diagram of the two categories of lock-exchange flow ............ 9

Figure 1-2: Effect of inclination angle on Froude number ......................................... 15

Figure 1-3: Effect of mixing length on Froude number .............................................. 15

Figure 2-1: Schematic of VHTR ................................................................................. 21

Figure 2-2: Schematic diagram of inclined test facility .............................................. 27

Figure 3-1: Schematic diagram of test facility ............................................................ 33

Figure 3-2: Complete experimental test facility ......................................................... 35

Figure 3-3: Components of test facility ...................................................................... 36

Figure 3-4: Interface at the break for L/D = 5, 00 .................................................................................

42

Figure 3-5: Global mixing phenomena for L/D = 5, 00 ....................................................................

44

Figure 3-6: Mixing interface at the break for inclined and vertical pipe .................... 46

Figure 3-7: Global mixing phenomena for vertical and inclined break ...................... 48

Figure 3-8: Time rate of change of density for L/D = 3, 60 0 .......................................................

50

Figure 3-9: Data and correlation in initial region for L/D = 3, 60 0 .........................................

53

Figure 3-10: Effect of Break Angle and Length on Ci ...................................................................... 55

Figure 3-11: Effect of Break Angle on ttr,i for L/D = 3 ............................................... 56

Figure 3-12: Effect of Break Angle and Length on ρ * tr,i ............................................................... 58

Figure 3-13: Volumetric exchange rate in initial region for L/D = 3, 600 ...........................

60

Figure 3-14: Experimental data and correlation for L/D = 3, 600 on regular axes ..... 62

Figure 3-15: Experimental data and correlation for L/D = 3, 600 on log axes ........... 63

Figure 3-16: Effect of Break Angle and Length on Ctr ..................................................................... 64

viii

Figure 3-17: Normalized density and time in non-linear region for L/D = 3 ............. 65

Figure 3-18: Comparison of experimental data with correlation for L/D = 3, 900 .........

67

Figure 3-19: Variation of density with time with multiple time steps for L/D = 5, vertical

break .............................................................................................................. 69

Figure 3-20: Experimental data and correlation for θ ≤ 300 ........................................ 70

Figure 3-21: Experimental data and correlation for θ ≥ 450 ........................................ 71

Figure 3-22: Effect of Break Length and Angle on ρ* tr,f ............................................ 74

Figure 3-23: Volumetric exchange rate for L/D = 3, 900 ............................................. 77

Figure 3-24: Froude number for L/D = 3, 900 ............................................................. 78

Figure 3-25: Effect of break angle on volumetric exchange rate for L/D = 3 ............. 79

Figure 3-26: Effect of break angle on Froude number for L/D = 3 ............................. 80

Figure 3-27: Effect of Break Length and Break Angle on Froude number of entire mixing

period ............................................................................................................. 82

Figure 3-28: Comparison of average Froude number with previous studies ............... 84

Figure 3-29: Comparison of initial region Froude number .......................................... 85

Figure 3-30: Effect of Break Length on Froude Number ............................................ 87

Figure 4-1: Conceptual design of slit break ................................................................. 92

ix

LIST OF TABLES

Table 2-1: Reynolds and Froude numbers for the horizontal lock exchange flow ...... 30

Table 2-2: Scaling parameter ratios in vertical break for various break lengths ......... 31

Table 2-3: Scaling parameter ratios for various break angles...................................... 31

Table 3-1: Test Matrix for current study…………………………………………….. 40

x

NOMENCLATURE

A = area of cross section of break pipe

C = constant for slope

CD = discharge coefficient

D = diameter of break pipe

Dh = hydraulic diameter of pipe

Fr = Froude number

g = gravitational constant

L = length of break pipe

Q = volumetric flow rate

Re = Reynolds number

t = time

j = local superficial exchange velocity

V = Mixing volume

Greek Letters

ρ = density of fluid

µ = dynamic viscosity of fluid

γ = density ratio

θ = Angle of inclination above horizontal

Δ = difference (in density)

xi

Subscripts/Superscripts

f = final condition

H = Higher density fluid (brine)

i = initial condition

k = fluid

L = Lower density fluid (water)

HA = Helium-Air

R = Ratio

tr = transition

WB = Water-brine

xii

ACKNOWLEDGEMENTS

The author would like to express gratitude to her advisor, Professor Seungjin Kim

for his valuable guidance and encouragement. Prof. Kim not only imparted indispensable

technical knowledge but also continuous inspiration throughout the period of research

work. Sincere thanks to all the students in the Advanced Multiphase Flow Laboratory for

their help and cooperation during author’s period of research.

The author would like to acknowledge Dr. Fan-Bill Cheung for reviewing this

work. The research work presented is supported by the U.S Nuclear Regulatory

Commission.

Finally, the author would like to express deep indebtedness to her family

members and friends for their continued support, motivation and encouragement. Sincere

gratefulness is extended to the author’s parents, to whom this thesis is dedicated.

1

CHAPTER 1

Introduction

2

1.1 Research background and motivation

The Depressurized Loss of Forced Circulation (D-LOFC) is one of the most

important design based events for the Very High Temperature Gas-Cooled Reactor

(VHTR). This event is caused due to the accidental rupture of the primary pipe or

maintenance standpipe in a VHTR, leading to a depressurization of the reactor vessel.

Once the vessel pressure reaches atmospheric pressure, air may ingress into the reactor

with a simultaneous outflow of helium by different mechanisms depending on the

location of the break.

For the horizontal primary pipe break connected to the lower section of the vessel,

after the initial depressurization, the density difference between helium and air may cause

the ingress of air into the reactor by lock exchange phenomenon as described by Reyes et

al. (2007). In this process, cold air enters through the horizontal pipe break and fills the

lower plenum of the vessel while hot helium simultaneously escapes through the break.

As the heavy air settles at the bottom of the vessel, molecular diffusion process promotes

further ingress of air. This causes the density of the gas mixture to increase and after a

sufficiently long time of about 150 – 200 hrs (Oh et al., 2006) when the temperature

difference of the gas mixture is large enough, natural circulation is initiated. An abrupt

increase in the rate of ingress is observed at the onset of natural circulation (Takeda and

Hishida, 1992). On the other hand, for the vertical maintenance standpipe at the top of

the reactor vessel, following the initial depressurization, the unstable layer of heavy air-

helium mixture above the light helium gas triggers immediate gravity driven air ingress

3

into the vessel. Once the reactor vessel is filled with the gas mixture, the temperature

difference of the gas mixture causes the onset of natural circulation (Hishida et al., 1993).

Since the VHTR core design consists of fuel located within a graphite matrix, the

oxidation from an air-ingress event could compromise the structural integrity of the

graphite, leading to a possible release of fission products. The bulk natural circulation

causes extensive graphite oxidation and generates a large amount of heat which increases

the core temperature (Oh et al., 2006). Hence, it is essential to determine the time

available for mitigative actions before the onset of natural circulation. The present study

focuses on the gravity-driven exchange stage since it is an essential mechanism of air

ingress before the onset of natural circulation.

The air ingress following the initial depressurization could be caused due to

molecular diffusion as well as gravity driven exchange flow due to the difference in

density between the helium and air. The most dominant mechanism causing this ingress

phenomenon has been widely debated in the past. Earlier studies (Takeda, 1997, Takeda

& Hishida, 1996 and Oh et al., 2006) focused on molecular diffusion ignoring the effects

of gravity driven exchange flow. Similarly, No et al. (2007) performed numerical

simulations for the assessment of the GAs Multi-component Mixture Analysis

(GAMMA) code to analyze the proposed molecular diffusion following a guillotine pipe

break. The onset of natural circulation was predicted to be at 150 – 200 hours after the

rupture of the primary pipe. The importance of gravity driven flow as a dominant ingress

mechanism was described by Oh et al (2008) by performing CFD analysis using both

FLUENT and GAMMA codes to take into account the effect of density difference driven

stratified flow and molecular diffusion respectively. It was observed that the ingress of

4

air due to the stratified flow assumption occurred very rapidly and the entire bottom part

of the vessel was filled with air within about 60 – 90 seconds, after which the flow

stabilized and molecular diffusion predominated. Based on this analysis, the onset of

natural circulation was predicted to be at ~160 seconds, which is significantly earlier than

the predicted time for molecular diffusion. Furthermore, in a recent study conducted by

Oh et al. (2009), it has been suggested by computational simulation as well as

mathematical formulation using species transport equations, that the time scale for

density driven exchange is nearly 800 times longer than that of molecular diffusion.

Therefore, the present study focuses on the gravity-driven exchange stage since it is

crucial in determining the rate of air ingress before the onset of natural circulation.

1.2 Literature Review

1.2.1 Gravity driven lock exchange flows

The density gradient driven flow discussed above has been widely studied in the

past due to its varied applications like waste water discharge into rivers, oil spills, smoke

movement, indoor air flow etc. apart from the proposed accident scenario. Several

experimental studies have been performed to investigate the flow pattern caused by the

intrusion of a heavy fluid mass into a fluid of lower density on a horizontal plane,

referred to as gravity driven lock exchange flow.

5

Front shape and speed

The earliest theory for predicting the speed and shape of the propagating front

was developed by von Kármán (1940). This theory was developed by considering energy

conserving heavier current propagating in an ambient lighter fluid of infinite depth. The

flow was considered relative to a coordinate system moving with the heavier fluid. The

speed of the front was predicted as a function of the depth of the current and the density

ratio between the mixing fluids. Furthermore, the slope of the front at the point of

stagnation was predicted to be 600.

Benjamin (1968) developed a theory for the propagation of the front using

momentum and mass conservation in a frame of reference moving with the current for

inviscid flow. This theory suggested various possible solutions for the speed of the front

depending on the depth of the current. It was also shown that the depth of the current

would be half the depth of the channel for an energy conserving flow. The front speed

and the shape of the front were found to agree with those obtained by von Karman (1940)

for the energy conserving assumption. It was shown that if the gravity current occupied

less than half the depth of the channel, the current is not energy conserving and the

maximum energy flux is reached when h = 0.347 H, where h is the depth of the current

and H is the depth of the channel.

Interface shape

The shape of the interface between the two layers of fluid flow has also been

studied extensively in the past. A mathematical formulation was developed by Yih and

Guha (1955) to predict the interface between the two fluid layers. It was shown that fluid

6

motion in such stratified flows results in the formation of a hydraulic jump described by a

change in depth of the propagating front. Mathematical formulations were developed

utilizing conservation of momentum for the two layers of fluids to determine the jump

conditions and the depth downstream from the jump. The interfacial shear was neglected

and the pressure distribution was assumed to be hydrostatic in order to model the change

in depth of each layer due to the occurrence of the aforementioned jump. The predictions

were subsequently validated with an oil and water mixing experiment.

Further studies on the horizontal exchange flow were performed by Keller &

Chyou (1991) covering the complete range of density ratios between 0 and 1. The

hydraulic theory was formulated using mass and momentum conservation for the two

layers and two models were predicted depending on the density ratio γ between the

mixing fluids. The assumption of hydrostatic pressure distribution in this case was

replaced with the assumption that the static pressure difference and the stagnation

pressure difference upstream and downstream of the hydraulic jump are proportional,

with the constant of proportionality limited between 0 and 1. For small density

differences (γ ≥ 0.281), both the gravity currents were assumed to be energy conserving

and connected by a long expansion wave and an internal hydraulic jump, whereas for

larger density differences (0 < γ ≤ 0.281), the heavier current was assumed to be

dissipative in nature and the two currents were connected only by a long expansion wave.

Experiments were also performed using several different fluid pairs, covering a density

ratio range of 0.001 to 0.9 to validate the results obtained from the proposed model by

flow visualization studies. However, some discrepancies between the model predictions

and the experiments were found due to the effects of viscosity and surface tension.

7

Lowe et al. (2005) performed experiments and developed models to investigate

the shape of the front as well as the interface for fluids having small density differences,

i.e. Boussinesq lock exchange, as well as large density differences, i.e. non-Boussinesq

lock exchange. It was seen that for the Boussinesq case, the speed of both the currents

are constant and nearly equal and the flow is symmetrical about the centerline such that

the front of each current occupies half the depth of the channel as described earlier by

Benjamin (1968). For the non-Boussinesq case, it was seen that the heavier current

travels at a higher speed than the lighter current although both the currents still move at

constant speed. The flow is not symmetric as was seen for the Boussinesq case and the

depth of the heavier current is found to be slightly smaller than half the depth of the

channel. In order to model the shape of the interface, the theory proposed by Keller &

Chyou (1991) was derived again in this study and experiments were performed using

water and brine to cover a density ratio range of 0.6 to 1. The two proposed flow

configurations as suggested by Keller & Chyou (1991) are shown in Fig. 1 (Lowe et al.,

2005). Fig. 1-1 (a) shows the flow configuration for 0.281 < γ ≤ 1 and Fig. 1-1 (b) shows

the flow configuration for 0 < γ ≤ 0.281 as discussed earlier.

This study was extended to propose a second possible solution considering only

an expansion wave connecting the two gravity currents as shown in Fig. 1-1 (b) for the

entire range of density ratios for the non-Boussinesq exchange flow. The data obtained

from the experiments were compared with the results from computational simulations by

Birman et al. (2005) and the results from the two proposed models with and without the

internal hydraulic jump for the entire range of density ratios. It was found that the theory

in which the two gravity currents are connected only by a long expansion wave, with the

8

heavy current being dissipative in nature was found to be most representative of the non-

Boussinesq lock exchange flow. This contradicted the theory proposed by Keller &

Chyou (1991) for the range of density ratios 0.281 < γ ≤ 1 which assumed both the

currents to be energy conserving and connected by both an expansion wave and an

internal hydraulic jump.

9

Figure 1-1: Schematic diagram of the two categories of lock exchange flow: (a) Both

currents energy conserving, connected by an expansion wave and an internal hydraulic

jump and (b) light current energy conserving and heavy current dissipative, connected by

a long expansion wave (Lowe et al., 2005).

10

1.2.2 Exchange flow rate estimation

In order to understand the effects of geometric parameters on the general flow

characteristics resulting from gravity driven exchange, several previous studies have been

performed using water and brine as simulant fluids. Leach and Thompson (1975)

performed experiments using both water-brine and carbon dioxide-air as simulant fluids

to investigate the accident scenario for a Magnox reactor. These experiments were

performed for the horizontal pipe, covering a break development length-to-diameter ratio

(L/D) range of 0.5 to 20. The water-brine experiments were performed using a sealed

box containing brine connected by a pipe to a tank containing water and recording the

weight of the brine box with time by using a force transducer. The carbon dioxide-air

experiments were performed using a pipe with a quick release valve at one end to start

the ingress of air, connected to a box containing carbon dioxide and measuring the

change in concentration of gas sample in the box. A dimensional analysis was also

performed in order to estimate the volumetric flow rate in terms of the measurable

parameters as discussed above. It was suggested that after the start of the ingress, the

reactor vessel can be purged with an external supply of carbon dioxide (coolant gas) in

order to prevent further ingress of air. The purging flow rate required to replace the air in

the vessel and prevent further ingress was also determined. The results were expressed in

terms of a non dimensional parameter referred to as the discharge coefficient of the flow

which scales the effect of inertia force to buoyancy force and is essentially similar to the

Froude number that has been used in several studies later. This discharge coefficient was

given as (Leach & Thompson, 1975):

11

(1-1)

where Q is the measured volumetric exchange rate, D is the diameter of the pipe, Δρ is

the density difference between the two and g is the acceleration due to gravity. Due to

the very small density difference, ρ can be the density of either fluid, without making an

effect on the calculated value. It was seen that for the case of horizontal pipes, the

exchange flow is not significantly affected by the L/D ratio and the discharge coefficient

is constant.

To investigate the effect of inclination of tubes on gravity driven flows, Mercer &

Thompson (1975) performed experiments using ducts inclined at various angles between

the horizontal and vertical for a range of L/D = 3.5 to 18. The test apparatus consisted of

a sealed tank containing brine connected to a duct. The open end of the duct was

immersed in an open tank of water and a quick release seal at the end of the duct was

employed to start the ingress. Two experiments were performed for various orientations

of the inclined break, namely, tilting the entire system with the pipe connected normal to

the compartments, and tilting only the break with the pipe connected to the compartments

at the orientation angle. The exchange flow rate was determined by measuring the

change in weight of one compartment with time. The results obtained were expressed in

terms of a non-dimensional volumetric flow rate q* similar to the discharge coefficient

described in Eq (1-1). The value of q*

was given by Mercer & Thompson (1975) as:

12

(1-2)

where ρH is the density of the heavier fluid. It was observed that the value of q* obtained

for the horizontal ducts was independent of the L/D ratio and was found to agree with the

value of discharge coefficient obtained in previous studies (Leach & Thompson 1975).

Furthermore, it was seen that q* increases as the duct angle is increased above the

horizontal until it reaches a maximum value at a peak angle dependent on L/D.

Thereafter, q* starts decreasing as the duct angle is increased until it reaches a minimum

value for the vertical duct. The q* value in general was also found to decrease with

increasing L/D.

To further investigate the effect of break length in vertical tubes Epstein (1988)

conducted experiments using tubes having L/D range of 0.01 to 10 connecting two

compartments containing water and brine respectively. A rubber stopper at the end of the

tube was used to start or stop the ingress as desired. The volumetric flow rate was

calculated by measuring the density of the brine filled compartment at regular intervals of

time (~ 2 min). It was observed that the density of the fluid in upper compartment

decreased linearly with time. The results of these experiments were presented in terms of

the non-dimensional Froude number which scales the effect of inertia force to buoyancy

force and is similar to the non-dimensional parameters used in previous studies. The

Froude number was defined as:

(1-3)

13

where is the mean density of the two fluids and A is the area of cross section of the

pipe. The area of cross section of the pipe is given by:

(1-4)

and the mean density is given by:

(1-5)

It was found that the Froude number increases with L/D until it reaches a

maximum value at L/D = 0.6 and thereafter decreases with increase in L/D. Based on

L/D, four exchange regimes were observed, namely the oscillatory flow regime for small

L/D described by Taylor wave theory (I), the Bernoulli flow regime described by an

inviscid counter current exchange flow model by the application of Bernoulli equation

(Brown, 1962), the turbulent diffusion flow regime for very high L/D described by

chaotic mixing of the two fluids leading to slower exchange between the compartments

and the intermediate flow regime between II and IV described by Bernoulli type flow at

the ends of the tube and turbulent diffusion at the center of the tube (III). The maximum

exchange flow rate was found to occur in regime III. Finally, a suitable formulation was

developed to predict the Froude number in each condition and this was found to compare

well with that obtained from the experimental data.

14

To study the exchange flow parameters for fluids having higher density

differences, various studies have also been performed in the past utilizing two gases as

simulant fluids. An experimental study on the effect of break length and angle using

helium and air was performed by Hishida et al. (1993) in order to understand the air

ingress processes during the primary pipe and stand pipe rupture accidents of the High

Temperature Engineering Test Reactor (HTTR). The test apparatus for the stand pipe

rupture experiment consisted of a pipe connected to a box containing helium, with a

cover plate that could be removed to start the ingress of air. These experiments covered a

wide range of inclination angles from horizontal to vertical for L/D = 0.05 and L/D = 10.

The effect of break length on the exchange rate was also investigated for the vertical

break. The exchange flow rate was obtained by measuring the change in weight of the

helium compartment during the exchange by using an electronic weight balance and was

expressed in terms of Froude number. The variation of Froude number with L/D was

found to agree well with previous data (Mercer et al., 1975 and Epstein, 1988) as shown

in Figs 1-2 and 1-3. It was observed that the Froude number is constant for L/D < 0.1,

increases with increasing break length upto L/D = 0.6 and decreases thereafter. However,

although the overall trend of variation of Froude number with pipe angle was similar to

what was observed by Mercer et al., the peak Froude number occurred at much higher

angles as shown in Fig.1-2. The peak Froude number for L/D = 10 was found to occur at

600

for the helium-air exchange, whereas that for the water-brine experiment from

previous studies occurred at angles less than 150 for L/D = 8 and 13.

15

Figure 1-2: Effect of inclination angle on Froude number (Hishida et al., 1993).

Figure 1-3: Effect of mixing length on Froude number (Hishida et al., 1993).

16

Furthermore, the differences in exchange flow pattern and flow rate for different

fluid pairs including air-helium, Ar-air and SF6-air were studied by Tanaka et al. (2002)

using similar experimental procedure as Hishida et al. (1993). Here, the effects of

inclination and fluid properties on exchange flow parameters in a rectangular channel

were investigated. The axial velocity profile was obtained using Laser Doppler

velocimeter and flow visualization was performed to analyze the flow pattern for

different break angles varying from 150

to 900. The exchange flow rate was obtained by

measuring the change in weight of one compartment with time and found to depend on

both of the break angle and initial density difference between the mixing fluids. It was

found that the exchange flow rate increases with the increase in density difference

between mixing fluids. Furthermore, it was also observed that the exchange flow

increases with an increase in pipe angle, reaching a peak value at about 750

and decreases

thereafter.

1.3 Objectives of present study

For all pairs of mixing fluids it has been reported that the exchange rate strongly

depends on both the L/D and angle. Furthermore, the Froude number is identified as the

major non-dimensional parameter to account for to account for these effects. Based upon

the previous works, the present study seeks to perform an adiabatic scoping study using a

scaled water-brine tests to observe the two dimensional mixing phenomena and

investigate the effect of geometric parameters on the exchange rate during the postulated

D-LOFC in VHTR. The exchange rate and Froude number are calculated by measuring

density of heavier fluid for each L/D and angle and the results obtained are analyzed in

17

comparison with previous studies. The results of the scoping study performed are crucial

to understand the exchange mechanism for gravity driven exchange flow in order to assist

in the design of a heated helium-air test facility to aid in the future licensing of helium

cooled gas reactors.

18

CHAPTER 2

Scaling Analysis

19

2.1 Identification of scaling parameters

The present study aims at performing a scoping experiment to establish database

and provide understanding of the gravity driven exchange flow phenomena occurring

during the postulated accident scenario. The experiments are performed using water and

brine as simulant fluids in a scaled test apparatus. The design of the water-brine test

apparatus is developed based on the scaling analysis and other practical considerations.

The scaling analysis described below has been performed in a previous study by Kim &

Talley (2009). Since the water-brine test is a scoping experiment, the test apparatus is

designed taking the following factors into consideration:

To be able to perform scaled experiments over a wide range of geometric

parameters as given in previous studies,

To be able to perform detailed flow visualization studies on a local as well as

global scale,

To be able to observe the two dimensional mixing phenomena during the process

of ingress of heavier fluid

To be simple to implement different pipe lengths,

To be light in weight so as to easily perform experiments for the inclined

conditions.

20

The scaling of the water-brine test apparatus is based on the General Atomics Gas

Turbine Modular Helium Reactor as given in INEEL/EXT-03-00870 (2003) design. As

shown in Fig. 2-1, this reactor vessel consists of two pipelines connected to the vessel;

the horizontal coaxial primary core inlet/exit duct, and the vertical refueling standpipe.

For initiation of air-ingress into the reactor vessel, these entry points of different size and

orientation are considered.

21

Figure 2-1: Schematic of VHTR (INEEL/EXT-03-00870, 2003).

Primary

Pipe

Standpipe

22

In the scaling of adiabatic gravity-driven exchange phenomenon, the previous

studies by Turner (1973) and Epstein (1988) suggest that Froude number is the

dimensionless number that accounts for the effect of both the break length and

orientation. The Froude number scales the effect of the inertia force of the ingressing

fluid with respect to the buoyancy force induced by the fluid density difference and is

given by:

(2-1)

where k denotes the fluid (k = H for heavier fluid and k = L for lighter fluid), D denotes

the break diameter, g denotes the acceleration due to gravity, Δρ denotes the density

difference between the two fluids and denotes the mean density of the two fluids as

given by Eq (1-5). Here, jk is the superficial fluid exchange velocity given by:

(2-2)

where u denotes the fluid exchange velocity and α is the fraction of the total cross-

sectional area occupied by the fluid k, at the break, which is given by:

(2-3)

23

The Froude number can also be rewritten in terms of the volumetric exchange flow rate

given by:

(2-4)

where Q and A are volumetric exchange rate and break area, respectively. Here, the

volumetric flow rate can be measured in the experiment by monitoring the density change

with respect to time as given by Epstein (1988):

(2-5)

where VH and VL are the volumes of heavy and light fluids, respectively, and ρH is the

density of the heavier fluid at a given time, t. The subscript i denotes the initial time, i.e.,

t=0. This expression for Q is obtained by mass balance of the two mixing compartments

as given in Appendix A1-1.

Based on the Froude Number and volumetric flow relations, the following ratios

are identified as key scaling parameters for the water-brine test:

: Froude Number Ratio (2-6)

: Global Exchange Ratio (2-7)

24

: Local Exchange Ratio (2-8)

: Exchange Time Ratio (2-9)

where the subscripts WB and HA denote water-brine and helium-air, respectively. The

fluid exchange rate has been taken into consideration through two scaling parameters,

namely a global scale, , and a local scale, .

The volumetric exchange rate ratio for each fluid-pair can be deduced from Eq (2-

4) by:

(2-10)

As such, can vary based on the Fr number and break size. Similarly, the local

exchange rate ratio is obtained from Eq (2-1) by:

(2-11)

25

2.2 Scaling Methodology

In order to calculate the scaling parameters, the Froude number for a given

simulant fluid pair for a particular case is estimated from previous studies based on the

effect of break angle from Fig. 1-2 and break development length from Fig. 1-3. To take

into account the effect of the break angle, the data for air-helium is taken from the L/D =

10 experiment by Hishida et al. (1993) while the data for water-brine experiment is taken

from the L/D = 8 experiment by Mercer & Thompson (1975). The break angles are

varied at 15° intervals between horizontal and vertical. To take into account the effect of

break length, Froude number is obtained only for the vertical pipes over a range of break

lengths between L/D = 0 to L/D = 10. To determine the break size, the pipe diameter for

water-brine test apparatus is varied over a range of L/D ratios comparable with previous

experiments for both the break angle and break length effects. The pipe size for helium-

air is taken to be a prototypic VHTR hot-leg diameter value of approximately 1.5 meters

for the break angle effect, and a standpipe diameter of approximately 0.75 meters for the

break length effect.

Hence, the global and local exchange rate ratios are calculated by substituting the

Froude number ratio and break size obtained as described above, in Eqs (2-10) and (2-

11).

The exchange time ratio that scales the total time required for complete exchange

of the fluids for a given exchange volume is given by:

26

(2-12)

where VHA and QHA are the volume and the volumetric flow rate based on the reference

GT-MHR dimensions. Based on the two entry locations in the GT-MHR vessel, different

amounts of fluids are exchanged, depending upon the volume below the break location.

In the vertical standpipe case, a complete volume exchange is assumed, approximately

351,000 liters, such that the heavier air in containment will completely displace the

lighter helium in the reactor vessel. The volume of the core is excluded from the vessel

volume to obtain the available volume of the annular region between the vessel and core.

In the primary break case, the air is assumed to displace any helium below the plane

parallel to the top of the primary pipe break, approximately 94,000 liters or 27% of the

total vessel volume. The volume of the lower plenum which includes the shutdown

cooling system has been neglected for calculating the exchange volume. Furthermore,

the support structures under the core and any internals in the upper hemisphere are also

neglected in the volume calculations. The detailed volume calculations are described in

Appendix A2. To estimate the exchange time for water-brine experiment, it is noted that

the volume of fluid to be exchanged also varies with the break angle. In the case of the

vertical break the entire volume will mix, whereas in the horizontal break approximately

half of the volume will be exchanged. The volume that will remain unmixed, for a break

angle θ, is obtained based upon the planes that are perpendicular to the direction of

gravity and touch the highest and lowest point of the breaks as shown in Fig. 2-2. The

27

volume not located within this region will always be exchanged. The detailed volume

calculation for the present study is included in Appendix A2.

Figure 2-2: Schematic diagram of inclined test facility (showing unmixed volume for

inclined break).

Unmixed

water

Water-brine mixture

in upper compartment

Unmixed

brine

Water-brine mixture

in lower compartment

28

It should be noted that the exchange time ratio thus calculated by Eq (2-12)

represents the time ratio for complete exchange of fluids between the two compartments

under adiabatic conditions as opposed to a local exchange time ratio based on the local

front velocity at the break. The global timescale is chosen as the primary scaling

parameter due to its importance in determining the timescale for mitigative actions.

Here, the prototypic exchange time can be estimated as a function of break angle and L/D

based on the previous adiabatic Froude number measurements. As such, the parameters

of test apparatus volume and break size are varied to make the global time scale, , of

order unity.

In view of scaling the inertia force of the exchange fluid with respect to the

viscous force, the Reynolds number is also examined. It is noted, however, that both

Epstein (1988) and Hishida (1993) showed the flow structure and exchange mechanisms

of the exchange fluid are primarily determined by the L/D ratio in a vertical break.

Hence, only the lock-exchange flow in the horizontal break is considered in examining

the Reynolds number. The Reynolds number for a fluid is given by:

(2-13)

where Dh denotes the hydraulic diameter and µk denotes the viscosity of the fluid. Since

each fluid occupies half the depth of the channel as discussed by Benjamin (1968) and

Lowe et al. (2005), the hydraulic diameter for the front of a fluid is given by:

29

(2-14)

Based on the available Froude number data from previous studies, the fluid superficial

velocity can be calculated from Eq (2-1). Hence, the Reynolds number can be obtained

as:

(2-15)

Assuming that each lock-exchange fluid will occupy exactly half of the pipe

cross-section, the length scale in the Reynolds number is chosen as the hydraulic

diameter of half of the pipe cross-section. The Reynolds number obtained for water-brine

and helium-air is shown in Table 2-1. The subscript H refers to the heavier fluid while L

refers to lighter fluid. As can be found in the Table, while the Reynolds number ratios

are far from unity, the exchange flow is laminar for all the cases. Hence, the flow

structures and exchange mechanisms can be scaled reasonably well in the adiabatic

water-brine scoping tests ensuring that the exchange time ratio is near unity. Fluid

temperatures for the heated condition are chosen as 9500C and 43

0C for helium and air,

respectively. For unheated conditions, the fluid temperatures are chosen as 200C for all

the fluids.

30

Table 2-1: Reynolds and Froude numbers for the horizontal (θ=0°) lock-exchange flow.

Condition Fr D [m] ReH ReL

Heated Helium-Air 0.01 1.50 1312 18

Unheated Helium-Air 0.01 1.50 1450 183

Water-Brine 0.15 0.01905 53 55

2.3 Design of test facility

Based on the scaling approach described above, the dimensions for the test

apparatus are determined. From the calculated scaling ratios, a break size of 19.05 mm is

selected in order to maintain the exchange time ratio between the water-brine scoping

apparatus and the prototypic VHTR, close to unity. The scaling ratios for this break size

are given in Tables 2-2 and 2-3 for the various tilt angles and L/D values, respectively.

From the scaling tables it is clear that the exchange time can vary significantly based on

the tilt angle and L/D value. It can also be found that for a fixed L/D the exchange rate

increases as the break is tilted more towards horizontal. In the vertical standpipe break,

the exchange time increases as the mixing length increases. Since the L/D ratio captures

the break size effects, the separate-effects scoping studies have been performed with one

break size. The height of the tank is chosen to be the same as the width for simplicity.

Based on the dimensions determined from the scaling analysis the water-brine test facility

is designed as described in Section 3.1.

31

Table 2-2: Scaling parameter ratios in vertical break for various break lengths

Pipe Size [mm] L/D Fr]R Q]R j]R tG]R

19.05 0.06 1.00 1.409E-05 0.022 0.99

-- 0.13 1.00 1.409E-05 0.022 0.99

-- 0.5 1.00 1.409E-05 0.022 0.99

-- 0.75 1.00 1.409E-05 0.022 0.99

-- 1 1.00 1.409E-05 0.022 0.99

-- 5 1.00 1.409E-05 0.022 0.99

-- 10 1.00 1.409E-05 0.022 0.99

Table 2-2: Scaling parameter ratios for various break angles

Pipe Size [mm] Tilt Angle [deg] Fr]R Q]R j]R tG]R

19.05 0 (horizontal) 15.00 3.736E-05 0.232 0.70

-- 30 1.75 4.359E-06 0.027 9.80

-- 45 1.00 2.491E-06 0.015 18.80

-- 60 0.62 1.533E-06 0.010 32.09

-- 75 0.71 1.779E-06 0.011 28.63

-- 90 (vertical) 1.00 2.491E-06 0.015 21.05

32

CHAPTER 3

Effects of geometric parameters on exchange flow

33

3.1 Experimental Setup

3.1.1 Test Apparatus

The test apparatus for the water-brine experiments is designed as shown in Fig. 3-1. The

apparatus consists of two narrow rectangular acrylic tanks of dimensions 25.4 x 25.4 x

3.81 cm where the narrow rectangular geometry is chosen to highlight the two-

dimensional mixing and enhance flow visualization studies.

Figure 3-1: Schematic diagram of test facility. All dimensions are in cm.

34

The complete test facility is shown in Fig. 3-2. In order to investigate the effect

of break angle, the test apparatus is attached to a pivoting support, which can be locked at

any position between vertical and horizontal as shown in Fig. 3-2 (b). The tilt angles are

monitored by a digital level, accurate to ±0.1°, attached to the test apparatus.

The compartments are connected by break pipes of various lengths as shown in

Fig. 3-3 (a). To minimize optical distortion, the break pipes are machined from solid

rectangular acrylic blocks by boring holes of the break size into the pieces. As such, the

present test section provides an ideal condition for detailed flow visualization study. To

initiate the ingress at the beginning of a test, a careful consideration is made in designing

a manual shutter mechanism. Unlike most of the previous studies as described by

Epstein (1988), Cholemari & Arakeri (2005) and Kuhn et al (2001) where a simple

rubber plug has been employed as a stopper, the present study employs a sliding gate

mechanism. The gate employed is shown in Fig 3-3 (b). The method employed in

previous studies may induce unwanted agitation in the tank that can affect the initial stage

of the ingress pattern. Therefore, the sliding gate shutter mechanism reduces the

disturbance to the flow at the beginning of ingress.

35

(a) Experimental setup, vertical (b) Experimental setup, inclined

Figure 3-2: Complete experimental test facility

36

(a) Pipe break component (L/D = 5) (b) Manual sliding shutter

Figure 3-3: Components of test facility

37

3.1.2 Experimental Procedure

The experiments are performed separately for flow visualization and data

collection. In determining the density difference between the fluids, values in the range

of = 0.025-0.030 are chosen to be consistent with the previous studies of Mercer &

Thompson (1975) and Epstein (1988). Since the density difference between the mixing

fluids is small (γ = ρL/ρH = 0.97), the flow can be categorized as Boussinesq exchange.

However, it may be noted that the exchange flow for the prototypic condition, using air

and helium is non-Boussinesq due to the larger density difference (γ = ρL/ρH = 0.15). The

implication of this density difference is discussed in Section 4.1.

The detailed setup procedure is as follows:

a. The break piece of interest is installed between the two compartments.

b. The test apparatus is cleaned and tested for leakage.

c. The density of brine is measured using hydrometer and the brine is dyed a darker

color using food dye.

d. The bottom compartment is filled with water and the shutter is closed.

e. The top compartment is filled with dyed brine and the top lid is installed to close

the top compartment.

f. The setup is inclined to the desired angle accurate to ±0.10 indicated by the digital

level.

Thereafter, the experiment is performed in two steps, i.e flow visualization and data

collection, as described below:

38

Flow Visualization

In order to perform the flow visualization, the setup is prepared by following steps

a through f above. The experiment is then performed as follows:

g. The video camera is installed in position depending on whether the local or global

mixing phenomenon is being investigated. Both a high-speed and regular speed

movie camera is used to capture the local and global mixing phenomena in the

break and the compartments, respectively. For the high-speed movies, a capture

rate of 125 frames per second is employed to allow observation of the flow

structure in detail.

h. The camera is turned on, the shutter is opened and the stopwatch is started

simultaneously.

i. After the ingress is completed as seen by visual observation, the shutter is closed.

j. The test facility is set back into vertical position, a sample is collected from the

upper compartment and the final density is measure using hydrometer. The time

on the stopwatch is recorded.

k. The video is investigated to record the time at which the mixture interface reaches

the break height.

Data Collection

After the flow visualization is performed, the data collection experiment is

conducted to obtain the rate of change of density in the upper compartment as and hence

evaluate the volumetric exchange rate as described in Sections 3.2.2 and 3.3. The test

39

facility is setup as described in steps a through f and then the experiment is conducted as

follows:

l. The shutter is opened to start the ingress and the stopwatch is also started

simultaneously.

m. After a fixed time determined from flow visualization, the shutter is closed and

the test facility is set back to vertical position. The time intervals are very close

initially and then spread out as the ingress progresses.

n. A sample is taken from the upper compartment and the density is measured with

the hydrometer. The time interval and density are recorded.

o. The sample is returned into the top compartment and the fluid is stirred

thoroughly.

p. The top lid is closed and the facility is set back into the inclined position.

q. A wait time is allowed for the interface to be stable again at the inclined position

and then the steps l through p is repeated.

r. After the completion of the ingress, the final density is recorded and the graph is

plotted to obtain the rate of change of density with time.

In order to investigate the ingress phenomena for the break length and angle

effect, the test matrix shown in Table 3-1 is developed. The break length is varied with

L/D = 0.63, 3.0 and 5.0 to account for the prototypic reactor condition where the length

of the primary pipe between the GT-MHR reactor vessel and the power conversion unit is

approximately 4.5 meters or 3 diameters. In addition, this range allows comparison to the

40

previous experiments. For each of the three L/D ratios, the break angle is adjusted from

vertical to horizontal in 15° increments to observe the effect of break angle.

Table 3-1: Test Matrix for Current Study

L/D Degree from horizontal

0.63 0, 15, 30, 45, 60, 75, 90

3.00 0, 15, 30, 45, 60, 75, 90

5.00 0, 15, 30, 45, 60, 75, 90

3.2 Experimental Results

The experimental results are divided into two categories. The first set of results

consists of observation of the flow phenomena and exchange mechanism from flow

visualization, for different break lengths and angles. The second set consists of the

results obtained from density measurement experiments for all conditions of the test

matrix.

41

3.2.1 Flow Visualization Study

The flow visualization study is performed in two ways, i.e using a high speed

camera to capture the local mixing phenomena at the break and using a regular speed

camera to capture the global mixing phenomena in the compartment.

It is observed that the mixing phenomenon depends greatly on the angle of

inclination. For the case of 0° (horizontal break) when the gate is opened, the fluid

propagates from the brine compartment to the water compartment, with a well defined

front or nose as shown in Fig. 3-4 (a). The slope of this nose at the stagnation point is

found to be approximately 60° which is in agreement with the shape of the front as

predicted by von Karman (1940) and Benjamin (1968). Following the nose, a stable and

well defined interface between the brine and the water is observed as shown in Fig. 3-4

(b). It is seen that the propagation fronts of brine and water are symmetric about the

middle of the pipe with each fluid layer occupying approximately half the depth of the

pipe. This observation is consistent with the predicted interface characteristics for a

Boussinesq lock exchange flow as described by Wilkinson (1986) and Lowe et al (2005).

Furthermore, it is also seen that the interface is sloping downwards as it reaches the end

of the break. This phenomenon can be described by considering an expansion wave

typical of the lock exchange process, as described by Keller & Chyou (1991) and Lowe et

al (2005).

42

(a) Front shape at start of ingress (b) Interface at a later stage of ingress

Figure 3-4: Interface at the break for L/D = 5, 00: (a) front shape and (b) mixing interface

Flow direction of brine Flow direction of brine

θ ~ 600

43

As the brine ingresses into the lower compartment it forms a traveling layer along

the bottom as shown in Fig. 3-5 (a). This traveling layer is seen to have a greater

thickness at the front as compared to the trailing edge. This can be considered as an

internal hydraulic jump that is typically formed when a high density fluid ingresses into a

channel containing low density fluid as discussed by Yih & Guha (1954) and Keller &

Chyou (1991). Once the brine reaches the opposite wall, it is forced upward by the

ingressing brine as shown in Fig. 3-5 (b). Since denser brine is continuously filling

below the nose, the brine that is forced upward cannot fall back down and begins to

propagate back across the compartment towards the break. This characteristic wavy

mixing phenomenon as shown in Fig. 3-5 (c) occurs until the denser fluid reaches the

height of the break.

44

(a) initial ingress stage (b) increase in height of brine (c) backward wave propagation.

Figure 3-5: Global mixing phenomena for L/D = 5, 00.

45

For the case of inclined angles, the interface is no longer stable at the break as

seen in Fig. 3-6 (a). From the figure, several layers can be seen corresponding to a

counter-current flow. The ingress rate is generally higher compared to the horizontal

condition due to the force of gravity enhancing the exchange as explained in previous

studies by Hishida et al. (1993). It is also seen that the region of interfacial mixing

increases as the angle of inclination is increased from the horizontal as discussed by

Tanaka et al (2002). Due to this interfacial mixing the symmetry of the propagation

fronts seen for the horizontal case is absent. However, in the case of the vertical pipe

break, it is observed from Fig. 3-6 (b) that there is no clear stable interface between the

fluids and the mixing phenomena is highly three-dimensional. The upward water stream

and downward brine stream interact strongly and fluctuate irregularly similar to what was

reported for air and helium by Hishida et al. (1993). These flow characteristics for the

vertical pipe with L/D = 5 are found to be similar to the characteristics of the turbulent

diffusion flow regime (IV) described by Epstein (1988). Moreover, this three-

dimensional mixing causes a reduction in the ingress rate.

46

(a) Interface at the break: L/D = 5, 450

(b) Interface at the break: L/D = 5, 450

Figure 3-6: Mixing interface at the break for inclined and vertical pipe.

Brine flow Water flow

47

Similar mechanisms are observed in the mixing phenomena for the inclined and

vertical break cases based on global compartment observations. As the brine ingresses

into the lower compartment, the plume spreads and the outer region is slowed by the

shear flow with the stagnant water. This slower region is then displaced outward by the

continuously ingressing plume as seen in Figs. 3-7 (a) and 3-7 (b), and the brine spreads

throughout the compartment rapidly. Additionally, when the plume reaches a wall, the

slower fluid at the wall is displaced upwards by the ingressing brine, after which it cannot

move downward due to the presence of the heavier fluid below. It is noted that such

mixing of the brine throughout the compartment occurs rather quickly on the order of 1 to

2 minutes depending on the break length and angle. In all cases, it is observed that the

ingress rate decreases after the brine has spread throughout the compartment.

48

(a) Global mixing for L/D = 0.63, 45 0

(b) Global mixing for L/D = 0.63, vertical

Figure 3-7: Global mixing phenomena for vertical and inclined break.

Brine flow Brine flow

49

The flow visualization study also allows for determination of density

measurement time based on the rate and pattern of ingress observed. The time at which

the mixture interface reaches the break is recorded from the flow visualization. Then the

time intervals at which density measurements need to be recorded are determined from

the flow visualization study to obtain reliable measurements within the resolution of the

hydrometer and also obtain sufficient number of data points to be able to capture the

nature of ingress. As shown in Section 3.2.2, the measurement frequency is initially

higher when the ingress is faster, but lower as the ingress slows down.

3.2.2 Density Measurement

As described in Section 3.1.2, the density change in the upper compartment is

measured throughout the exchange using a hydrometer by stopping and restarting the

ingress. An example of the density measurement is shown in Fig. 3-8 for L/D=3, 60°.

From the figure it can be seen that the data can be clearly divided into two regions. In the

initial region the change of density is linear with respect to time, while in the subsequent

region this density change becomes non-linear. It is found from the flow visualization

study that initiation of this transition corresponds to the time when the brine reaches the

break height. After the transition occurs, the ingress rate slows as the density difference

between the fluids in the two compartments decreases.

50

Figure 3-8: Time rate of change of density for L/D = 3, 600.

1010

1015

1020

1025

1030

0 150 300 450 600 750 900 1050 1200 1350 1500

ρH

[kg/m

3]

Time [sec]

Measured Data

Time brine reaches break

Theoretical final density

51

This result of two characteristic regions is different from the findings in the

previous water-brine studies, which reported the density change to be linear throughout.

It is speculated that such differences in mixing characteristics are attributed to the

difference in mixing compartment volumes. In the previous studies by Mercer &

Thompson (1975) and Epstein (1988), the lower compartment volumes were much larger

than the upper compartment, such that the mixing phenomena may have had little effect

on the overall ingress characteristics. It is speculated that due to the larger volume of the

lower compartment, the density of the lower compartment is not affected by the ingress

of heavier fluid. As a result, ρL in the denominator of Eq (2-4) can be treated as a

constant. In the present study, the mixing volume is much smaller and the results

highlight the effects of mixing of the fluids on the rate of ingress. It is important to note

that the present study is more pertinent to the reactor condition, because the reactor vessel

volume is finite and smaller compared to the containment building volume.

3.3 Data Analysis

Based on the two characteristic regions observed during the exchange, each

region is treated separately for analysis of volumetric flow rate and the Froude number.

52

3.3.1 Initial Region (0 < t ≤ ttr,i)

In the initial portion of the exchange, a constant rate of density decrease in the

upper compartment is observed. For this region the density data can be fit using an

equation of the form:

(3-1)

where Ci is the slope of the initial region and ρH,i is the intercept based on the initial

density measurement. For each break length and angle, the values of Ci are varied to

obtain the best fit to the experimental data. The data for the initial region is shown in Fig.

3-9 for L/D = 3, 600 where the straight line represents Eq (3-1) for this condition.

53

Figure 3-9: Data and correlation in initial region for L/D = 3, 600. (Error bars are ±0.1%)

1017

1022

1027

0 200 400 600

ρH

[kg/m

3]

Time [sec]

Measured Data

Linear (Initial Correlation)

Time brine reaches break

ρ = 1028 - 0.0350t

ρtr,i

54

The value of Ci found for various break angles and lengths is shown in Fig. 3-10.

From the figure, it can be seen that Ci increases, i.e. ingress rate increases in general, as

the break angle increases from the horizontal up to a maximum angle for a given break

length. This is due to the increase in the effective buoyancy force as the component of

gravity in the flow direction increases. Furthermore, from the flow visualization, the

angles within this region are found to promote counter-current flow which can lead to an

increased flow rate. For all experiments it is found that the maximum ingress rate occurs

at an angle between 30° and 45°. For higher break angles, it is observed that the ingress

rate decreases with increasing break angle. This is due to the mixing structure occurring

within the break which becomes chaotic and three-dimensional as the counter-current

flow no longer occurs in two distinct layers. As can be seen from the figure, the ingress

rate in the initial region is slowest for the vertical case where the three dimensional

mixing is the most extreme.

55

Figure 3-10: Effect of Break Angle and Length on Ci

0

0.01

0.02

0.03

0.04

0.05

0.06

0 15 30 45 60 75 90

Slo

pe

in i

nit

ial

stag

e, C

i

Angle of Inclination (deg)

Ci, L/D=0.63

Ci, L/D=3

Ci, L/D=5

56

In investigating the transition time, ttr,i, where the density change becomes non-

linear, it is found that the time is not affected by inclination for break angles less than 45°

as shown in Fig. 3-11. However, at angles above 45°, the transition time increases with

break angle because the ingress rate is decreasing, as shown by the decreasing Ci values.

Figure 3-11: Effect of Break Angle on ttr,i.for L/D = 3.

0

100

200

300

0 15 30 45 60 75 90

Tra

nsi

tion t

ime

t tr,

i[s

ec]

Break Angle (deg)

57

Furthermore, it is found that the transition point can be related to a density

difference ratio given by:

(3-2)

where ρL is the density of the lower compartment and ρ*

tr,i is the density difference at the

point of transition normalized with respect to the initial density difference. The value of

this parameter is found to be within the same range for all 21 break angle and break

length combinations as shown in Fig. 3-12.

58

Figure 3-12: Effect of Break Angle and Length on ρ*

tr,i.

0.0

0.5

1.0

0 15 30 45 60 75 90

Den

sity

dif

fere

nce

rat

io a

t tr

ansi

tion ρ

*tr

,i

Break angle (deg)

L/D = 0.63

L/D = 3

L/D = 5

0.74 + 5%

0.74 - 5%

59

Using the derivative of Eq (3-1) to obtain the rate of change of density, the

volumetric flow rate in the initial region can be expressed as:

(3-3)

where is the initial density difference between the two fluids. Based on this

expression, it can be noted that the volumetric flow rate will increase as time progresses

as shown in Fig. 3-13. This occurs since the numerator is constant due to the constant

density change, while the denominator becomes smaller as time progresses. However,

additional experimental studies are necessary to validate this result.

60

Figure 3-13: Volumetric exchange rate in initial region for L/D = 3, 600.

0.E+00

1.E-06

2.E-06

3.E-06

4.E-06

5.E-06

0 100 200

Volu

met

ric

flow

rat

e Q

[m3/s

]

Time [sec]

61

3.3.2 Non-linear Region (ttr,i < t ≤ tf)

For the non-linear region of the exchange, the time rate of change for density is

found to decrease as the exchange progresses. This is physically consistent as the density

difference driven flow should cease when there is no density difference. The change in

density for this region is found to be well represented by a logarithmic decay relation of

the form:

(3-4)

where t, ttr,i and Ctr are the time into the exchange, the time at which the transition occurs

and a constant to fit the experimental data. The values of Ctr are varied to obtain the best

fit to the experimental data for each break length and angle. Additionally, an effort was

made to minimize the slope difference between the linear and logarithmic fits near the

transition in the selection of Ctr values to represent a continuous exchange. An example

of the data and correlation for L/D = 3, 600 is shown in Figs. 3-14 and 3-15 on regular

and logarithmic scale respectively.

62

Figure 3-14: Experimental data and correlation for L/D = 3, 600

on regular axes. (Error

bars are ±0.1%)

1010

1015

1020

1025

1030

0 150 300 450 600 750 900 1050 1200 1350 1500

ρH

[kg/m

3]

Time [sec]

Data

Time dye interface reaches Break Theoretical final

density

ρ = 1028 - 0.0350t

ρ = ρtr,i - Ctrln(t/ttr,i)

where Ctr = 3.6ρtr,i

63

Figure 3-15: Experimental data and correlation for L/D = 3, 600

on log axes. (Error bars

are ±0.1%)

1010

1 150

ρH

[kg/m

3]

Time [sec]

Data

Time dye interface reaches Break

ρ = 1028 - 0.0350t

ρ = ρtr,i - Ctrln(t/ttr,i)

where Ctr = 3.6

ρtr,i

64

It can be seen from Fig. 3-16 that the value of Ctr does not vary greatly in the

range of 3 to 4 for all break angles and lengths. This implies that in the transient region

the dependence of ingress rate on break angle and length may not be as significant as in

the initial region. The average value of Ctr for all experiments is found to be 3.43 ± 9%.

Figure 3-16: Effect of Break Angle and Length on Ctr.

0

1

2

3

4

5

0 15 30 45 60 75 90

Slo

pe

of

non l

inea

r re

gio

n,

Ctr

Angle of Inclination (deg)

L/D=0.63

L/D=3

L/D=5

3.4 - 9%

3.4 + 9%

65

Since the values of Ctr and ρ*

tr are found to be constant for all break angles and break

lengths, the density as well as the time scales can be normalized with respect to the

transition density and time respectively, in the non linear region so as to follow similar

trend as shown in Fig. 3-17.

Figure 3-17: Normalized density and time in non-linear region for L/D = 3.

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 0.2 0.4 0.6 0.8 1

Tra

nsi

tion R

egio

n N

orm

aliz

ed D

ensi

ty,

ρn

orm

Transition Region Normalized Time, tnorm

Correlation

0 deg

15 deg

30 deg

45 deg

60 deg

75 deg

90 deg

66

An example of the curve fit for density measurement in both regions is shown in

Figs. 3-18 (a) and (b) for the vertical case with L/D = 3 on a regular and logarithmic scale

respectively. It is clearly seen that the linear and logarithmic fits obtained from Eq (3-1)

and (3-4) respectively, describe the experimental data very well within ±0.1%. Similar

graphs showing the data and correlation for density measurement for all the conditions of

the test matrix are available in Appendices A3, A4 and A5.

67

(a) Data and correlation, L/D = 3, 900, regular scale. (b) Data and correlation, L/D = 3, 90

0, log scale.

Figure 3-18: Comparison of experimental data with correlation for L/D = 3, 90°. Error bars shown are ±0.1%.

1010

1015

1020

1025

1030

0 2000 4000 6000

Upper

com

par

tmen

t den

sity

ρH

[kg/m

3]

Ingress time [sec]

Theoretical end of

ingress (tf)

ρ = 1026 - 0.0111t

ρ = (ρH)tr,i - Ctrln(t/ttr,i)

where Ctr=3.4

(ρH)tr,i

1010

1 2000U

pper

com

par

tmen

t den

sity

ρH

[kg/m

3]

Ingress time [sec]

ρ = 1026 - 0.0111t

ρ = (ρH)tr,i - Ctrln(t/ttr,i)

where Ctr=3.4

(ρH)tr,i

68

Since in the prototypic condition, the ingress is continuous, it is necessary to

ensure that the current method of stopping the ingress to take measurement and starting it

again, has minimal effect on the ingress rate. In order to take into account, the effect of

starting and stopping of the ingress in small time intervals, additional experiments have

been performed with larger time intervals of continuous ingress. The results of these

experiments are shown in Fig. 3-19 for L/D = 5, vertical break. The data has been

collected in three separate experiments with the smallest time intervals of 30s, 120s and

300s respectively. In order to compare the results, the data has been normalized with

respect to the initial density for each experiment. As can be seen from the figure, the

results are found to be within ±0.1% of the value from the proposed correlation in all

cases. The data points match well with the correlation in the initial region but start

deviating towards the end of the experiment. In this region, however, the density

measurement becomes less reliable as the change in density may be lower than the 0.5

kg/m3 resolution of the hydrometer. Furthermore, the values of Ci, ρ

*tr and Ctr are found

to be within the limits described above, for all the three runs. Overall, the data is

repeatable and little effect of the static measurement procedure is observed.

69

Figure 3-19: Variation of density with time with multiple time steps for L/D = 5, vertical

break. Error bars are ±0.1%.

0.985

0.99

0.995

1

1.005

0 1000 2000 3000 4000 5000 6000 7000

Norm

aliz

ed d

ensi

ty

Time [sec]

30 s

120 s

300 s

Proposed Correlation

Normalized density =

Smallest time interval

70

The variation of normalized density with time for different break angles is shown

in Figs. 3-20 and 3-21 for L/D = 3. The density is normalized with respect to the

measured initial density of the upper compartment, ρH,i. As can be seen from Fig. 3-20,

the rate of change of density increases as the angle of inclination is increased for smaller

angles up to 300 due to the increased counter-current flow. However, the reverse trend is

observed in Fig. 3-21 for angles above 450

due to the mixing structure inside the break.

Figure 3-20: Experimental data and correlation for L/D = 3, θ ≤ 300. (Dotted lines

represent correlations).

0.980

0.985

0.990

0.995

1.000

0 200 400 600 800 1000 1200 1400

Norm

aliz

ed d

ensi

ty

Time [sec]

0 deg

15 deg

30 deg

Increasing break angle

Normalized density =

71

Figure 3-21: Experimental data and correlation for L/D = 3, θ ≥ 450. (Dotted lines

represent correlations).

0.99

0 500 1000 1500 2000

Norm

aliz

ed d

ensi

ty

Time [sec]

45 deg 60 deg

75 deg 90 deg

Increasing break angle

Normalized density =

72

Using the derivative of Eq (3-4) to evaluate the rate of change of density, the

volumetric flow rate in the transient region can be expressed as:

(3-5)

where is the lower compartment density at the transition time estimated through a

mass balance of the fluid in the apparatus. The mass balance of the fluid in the apparatus

can also provide the final theoretical density in the system when the compartments are

completely mixed. When this density is substituted into Eq (3-4), the total mixing time,

tf, can then be estimated. The derivation of the theoretical final density and time is

included in Appendix A1-2.

3.3.3 Mathematical treatment for final region (ttr,f < t ≤ tf)

A mathematical evaluation of Eq (3-5) shows that as the exchange progresses in

time, it reaches a point of inflexion where the derivative of Q becomes zero. Similarly,

the Froude number evaluated from Q also reaches a point of inflexion at time ttr,f such that

(3-6)

73

The solution of Eq (3-6) results in the value of the time ttr,f at which this inflexion

occurs. The density difference ratio at this point of inflexion is evaluated by the

expression

(3-7)

where is the density difference between the two fluids at the point of inflexion.

The values of obtained for all the conditions of the test matrix are shown in Fig. 3-

22. It is observed from Fig. 3-22 and Eq 3-7 that the density ratio at which inflexion

occurs is constant for all break angles and lengths.

74

Figure 3-22: Effect of Break Length and Angle on ρ* tr,f

0

0.5

1

0 15 30 45 60 75 90

Den

sity

dif

fere

nce

rat

io a

t in

flex

ion ρ

*tr

,f

Tilt angle (deg)

L/D = 0.63

L/D = 3

L/D = 5

75

Since this inflexion occurs while mixing is continuing, the application of Eq (3-5)

after this inflexion point is non-physical. Furthermore, as the exchange approaches

completion, i.e, as t approaches tf, both the numerator and denominator terms in Eq (2-4)

for Q approach zero. This denotes a point of singularity in the expression. Hence, a

suitable mathematical treatment needs to be applied to evaluate Q for the later stage.

Therefore, L’Hospital’s Rule is applied in the treatment of Eq. (3-5) to evaluate

the value of Q at the limiting condition. Furthermore, it is known that at tf the volumetric

exchange is zero since the density difference is zero. Utilizing this boundary condition

and the L’Hospital’s rule, the expression for Q for the final mixing stage is given by:

for ttr,f < t ≤ tf (3-8)

Hence, the Froude number can be obtained for the final mixing stage by:

for ttr,f < t ≤ tf (3-9)

The detailed derivation of this formulation is included in Appendix A1-3.

76

3.3.4 Volumetric Flow Rate and Froude Number Analysis

The volumetric flow rate and Froude number values obtained are shown in Figs.

3-23 and 3-24, respectively, for the vertical break with L/D=3. In each figure, there exist

three different time regions as shown by the vertical lines. These are, the initial region

with increasing exchange flow rate i.e. 0 < t ≤ ttr,i, the intermediate region with non

linearly decreasing exchange rate i.e. ttr,i < t ≤ ttr,f and the final region after the inflexion

occurs i.e. ttr,f < t ≤ tf. Similar graphs of the variation of exchange parameters with time

for all conditions of the test matrix are included in Appendices A6, A7 and A8.

77

Figure 3-23: Volumetric exchange rate for L/D = 3, 90°.

78

Figure 3-24: Froude number for L/D = 3, 90°.

Volumetric Flow Rate and Froude number for entire mixing period

The variation of volumetric flow rate and Froude number with time for the

horizontal, inclined and vertical pipes for L/D = 3 are shown in Figs. 3-25 and 3-26. As

can be seen from the figures, the exchange rate Q and Froude number are found to

increase very rapidly in the initial region for the inclined break (450) and also decrease

79

rapidly in the non linear region. The Q and Froude number for the vertical break are seen

to increase very slowly in the initial region and also decrease slowly in the non-linear

region. In case of the horizontal break, the rate of change of Q and Froude number are

intermediate between the vertical and inclined pipes. This is consistent with the ingress

mechanism for different break angles as discussed in Section 3.2.

Figure 3-25: Effect of break angle on volumetric exchange rate for L/D = 3.

0.E+00

1.E-06

2.E-06

3.E-06

4.E-06

5.E-06

6.E-06

0 1000 2000 3000 4000 5000

Q[m

3/s

]

Time [sec]

0 deg

45 deg

90 deg

80

Figure 3-26: Effect of break angle on Froude number for L/D = 3.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 1000 2000 3000 4000 5000

Fro

ude

Num

ber

Time [sec]

0 deg

45 deg

90 deg

81

The time averaged Froude number for the entire mixing period is obtained by

numeric integration. The average Froude number for all conditions of the test matrix is

shown in Fig. 3-27. The Froude number in general is found to increase as the break angle

is increased from the horizontal, until a peak value is reached. This peak Froude number

is found to occur at an angle between 30° to 45° depending on the break length. For

angles higher than the peak angle, the Froude number decreases until it reaches a

minimum value for the vertical break where the mixing is slowed down due to the three-

dimensional flow structure.

82

Figure 3-27: Effect of Break length and angle on Froude number of entire mixing period.

0.00

0.05

0.10

0.15

0.20

0.25

0 15 30 45 60 75 90

Fro

ude

Num

ber

Angle of Inclination (deg)

Avg Fr, L/D = 0.63

Avg Fr, L/D = 3

Avg Fr, L/D= 5

83

The average Froude number values from the present study are shown in Fig. 3-28

in comparison with the previous studies by Mercer & Thompson (1975) and Hishida et al

(1993). From the figure it can be seen that the general trend of the current results are

similar to those of the previous studies. The Froude number is found to gradually

increase as the break angle is increased from the horizontal until a peak value is reached

and thereafter decreases until reaching a minimum value for the vertical break. It can be

seen from the figure that the peak Froude number in case of previous experiments using

water and brine by Mercer & Thompson (1975) was found to occur at an angle smaller

than 15° for all break lengths, whereas for experiments using air and water by Hishida et

al. (1993), the peak Froude number for L/D = 10 is found to occur at an angle of 60°. For

the present study the peak Froude number occurs at angles between 30° to 45° for the

break lengths under consideration, which is closer to the air-water experiments by

Hishida et al. (1993). It is also observed that the Froude number predicted is lower in

general than the values previously obtained for the smaller angles, but compares well

with the data of Mercer & Thompson (1975) for angles above 30°. Furthermore, it can be

seen that the Froude number is almost constant for the horizontal break for all the three

L/D ratios under consideration, as observed in previous studies (Leach & Thompson,

1975).

84

Figure 3-28: Comparison of average Froude number with previous studies.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 15 30 45 60 75 90

Fro

ude

Nu

mber

Angle of Inclination (deg)

Avg Fr, L/D = 0.63

Avg Fr, L/D = 3

Avg Fr, L/D= 5

Mercer, L/D = 8

Hishida, helium-air, L/D = 0.05

Hishida, helium-air, L/D = 10

85

Froude number of initial region for all conditions

The Froude number calculated by considering only the initial linear region is

shown in Figure 3-29 for all conditions of the test matrix. The Froude number of the

initial region is found to compare well with previous studies as can be seen from the

figure.

Figure 3-29: Comparison of initial region Froude number.

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0 15 30 45 60 75 90

Fro

ude

Num

ber

Angle of Inclination (deg)

L/D=0.63, Initial region

L/D=3, Initial region

L/D=5, Initial region

Mercer, L/D = 5.5

Hishida, helium-air, L/D = 0.05

86

The effect of break length on Froude number for the vertical break is shown in

Fig. 3-30. The Froude number is found to decrease as the break length increases, similar

to what was observed in previous studies by Epstein (1988) and Hishida et al. (1993) for

the range of break lengths under consideration. However, the values of Froude number

obtained in the current study are lower than previously reported values. The time

averaged Froude number obtained by considering only the initial linear region is also

plotted in Fig. 3-30. As can be seen from the Figure, these values match very well with

the Froude number reported in previous studies.

Since the average value of Froude number for entire mixing period includes both

the linear and non linear regions, the Froude numbers calculated based on complete

exchange are much smaller than those calculated based on only the initial region. Similar

behavior is also seen from the calculated values of exchange rate. The time averaged

volumetric exchange rate calculated over the entire exchange period is found to be upto

70% lower than the value obtained by considering only the initial linear region.

87

Figure 3-30: Effect of Break Length on Froude Number

0.00

0.05

0.10

0.15

0 1 2 3 4 5 6

Fro

ude

Num

ber

Break L/D

"Current: Entire Stages"

"Current: Initial Stage"

Water-brine, Epstein

Air-He, Hishida et al.

88

CHAPTER 4

Conclusions and Recommendations for Future Work

89

4.1 Conclusions

The current study aims at performing a scoping study to investigate the effects of

the geometric parameters on the gravity driven exchange phenomena. The experiment is

performed using water and brine having density ratio, = 0.025 – 0.030, as the

mixing fluids. The geometric parameters under consideration are the break angle and the

break development length-to-diameter (L/D) ratio. This work is performed to establish

database to aid in the design of a heated helium-air facility to investigate the D-LOFC

event for the VHTR. Flow visualization and density measurement studies are performed

to estimate the ingress rate which is important for mitigative actions in the postulated

accident event. It is observed that the break angle has a significant effect on the ingress

mechanism. Furthermore, it is found that the finite mixing volume and the mixing

phenomena make significant impact in overall ingress characteristics. As such, the rate

of ingress is steady initially, but after the mixing phenomenon occurs and the brine

reaches the break height, the rate of ingress decreases. It is found that the transition

occurs at a critical density ratio of ρ* tr,i = 0.74 ± 5% for all conditions of the test matrix.

Furthermore, the slope of the change of density for the transient region was found to be a

constant value given by Ctr = 3.43 ± 9% for all the 21 test conditions under consideration.

The estimated Froude number, based on averaging of the entire ingress, for the various

conditions is compared to the previous studies. It is found that the average Froude

number in general follows similar trend as reported previously. However, the angle at

which the peak Froude number occurs is higher than previous data. The overall exchange

90

rate is found to be of the order of 70% lower than the exchange rate obtained by

considering only the initial linear region of the exchange. Furthermore, the Froude

number values of the initial region are found to match better with previous studies than

the average Froude number of the complete exchange. Thus, it is seen that the volume of

the mixing compartments as well as the break angle plays a major role in determining the

ingress mechanism and rate of ingress. However, it is possible that there may be some

distortion of the density data since the resolution of the hydrometer is limited to 0.5

kg/m3.

The flow visualization experiments performed using water and brine provides a

reliable estimation of the expected ingress mechanism in the proposed helium-air

experiments. However, there may be some differences in the flow phenomena for the

helium-air exchange flow due to the much higher density difference. The lock exchange

flow for the prototypic case is expected to follow the flow pattern described for non-

Boussinesq lock exchange flow as opposed to the present study which follows the

Boussinesq exchange flow pattern (Lowe et al., 2005). Furthermore, the molecular

diffusion may have a more significant effect than in the current study, since the

diffusivity of the helium-air system is about 104 times higher than the water brine system.

The dynamic viscosity of the helium air system is also about 102 times higher than the

current study and could have an effect on the ingress rate (Fumizawa, 1992). Although

the Froude number may be higher than what is predicted for water-brine exchange flow,

the overall exchange rate is expected to be lower than earlier predictions, since the

exchange rate is found to decrease after the initial rapid linear ingress stage.

91

4.2 Recommendations for Future Work

Future studies are recommended to perform experiments to obtain a continuous

measurement of the change in density during the exchange by using other instrumentation

in order to validate the results obtained in current study. Further experiments are

recommended to investigate other break geometries such as slit break, to simulate a crack

in a pipe and other break locations, such as primary pipe located close to the bottom of

the compartments. The scaling study needs to be performed to design the break in a

similar manner as in the current study, to maintain the exchange time close to that of a

prototypic VHTR. The conceptual design of such a slit break is shown in Figure 4-1.

The slit break should be designed in a manner that allows flow visualization without

optical distortion. The experiments for slit break need to be designed so as to obtain

continuous measurement of ingress rate during the exchange.

92

Figure 4-1: Conceptual design of slit break.

Slit: crack in

pipe

Pipe: rectangular

block

93

Future studies are recommended to utilize the results obtained from the current

study to design a heated helium-air test facility. The scaling analysis is performed for the

proposed test facility, similar to the scaling study for the water-brine test. Based on the

scaling study, the dimensions of the test apparatus are determined so as to maintain the

exchange time ratio between the test apparatus and the prototypic VHTR close to unity.

The test facility will consist of a cylindrical carbon steel vessel with the various breaks

installed on the top cover and on the vessel wall at various locations to simulate the

standpipe and primary pipe. The breaks will be installed with flanges that can be altered

to accommodate pipes at various inclined angles.

To monitor the rate of air ingress through the break, several instrumentations will

be utilized. Detailed temperature measurements can be taken at different radial and axial

locations within the vessel using thermocouples. Air concentration can be monitored by

an oxygen analyzer and local sampling probe at different axial levels. The flow velocity

in the break pipes can be acquired by a Laser Doppler Anemometer to relate local oxygen

concentration and velocity profiles to obtain the rate of air ingress.

Since the mixing is found to occur very rapidly until the mixture interface reaches

the height of the break, the proposed air-helium test facility needs to be designed in such

a manner that the measurement locations are more detailed up to the height of the break

and spread out thereafter. These experiments will provide reliable database for the rate of

air ingress and hence will aid in the future licensing of Very High Temperature Gas

Cooled Reactors.

94

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Birman, V., Martin, J. E. & Meiburg, E., 2005, The non-Boussinesq lock-exchange

problem. Part 2.High-resolution simulations. J. Fluid Mech. 537, 125–144.

Cholemari M. R., and Arakeri, J. H., 2005, Experiments and a model of turbulent

exchange flow in a vertical pipe, Int. J. Heat and Mass Transfer, 48, pp. 4467-4473.

Epstein, M., 1988, Buoyancy-driven exchange flow through small openings in horizontal

partitions, J. of Heat Transfer, 110, pp. 885-893.

Fumizawa, M., 1992, Experimental study of helium-air exchange flow through a small

opening, Kerntechnik, 57 (3), pp. 156-160.

Hishida, M., Fumizawa, M., Takeda, T., Ogawa, M., and Takenaka, S., 1993, Researches

on air ingress accidents of the HTTR, Nuc. Eng. Des., 144, pp. 317-325.

Karman, T. von, 1940, The engineer grapples with non-linear problems, Bull. Am. Math.

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Keller, J. J., and Chyou, Y.-P., 1991, On the hydraulic lock-exchange problem. J. Appl.

Math. Phys., 42, pp. 874-909.

95

Kim, S. and Talley, J.D., 2009, Quick Look Report Task 5: Air exchange and air ingress

experimental data, Scaling analysis and test apparatus design for water-brine scoping

experiments, prepared for the U.S Nuclear Regulatory Commission.

Kuhn, S.Z., Bernardis, R.D., Lee, C.H., and Peterson, P.F., 2001, Density stratification

from buoyancy-driven exchange flow through horizontal partitions in a liquid tank. Nuc.

Eng. Des., 204, pp. 337-345.

Leach, S.J., and Thompson, H., 1975, An investigation of some aspects of flow into gas

cooled nuclear reactors following an accidental depressurization, J. Br. Nucl. Energy

Soc., 14 (3), pp. 243-250.

Lowe, R. J., Rottman, J. W. and Linden, P. F., 2005, The non-Boussinesq lock-exchange

problem. Part 1. Theory and Experiments, J. Fluid Mech., 537, pp. 101-124.

Mac Donald P. E. et al., September 2003, NGNP Point Design – Results of the Initial

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Mercer, A., and Thompson, H., 1975, An experimental investigation of some further

aspects of the buoyancy-driven exchange flow between carbon dioxide and air following

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ducts, J. Br. Nucl. Soc., 14, pp. 327-334.

NO, H. C. et al., 2007. Multicomponent Diffusion Analysis and Assessment of GAMMA

Code and Improved RELAP 5 Code, Nuclear Engineeringand Design, 237, 997–1008.

Oh, C. H., Kim, E. S., Kang, H. S., No, H. C., and Cho, N. Z., December 2009,

Experimental validation of stratified flow phenomena, graphite oxidation and mitigation

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strategies of air ingress accidents, FY-09. Idaho National Laboratory INL/EXT-09-16465

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Oh, C. H., Kim, E. S., NO, H. C., and Cho, N. Z., December 2008, Experimental

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Turner, S., 1973, Buoyancy Effects in Fluids, Cambridge University Press, London.

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98

APPENDIX A1

Derivation of equations

99

Appendix A1-1: Volumetric exchange rate calculation (Epstein, 1988)

Mass balance of the two compartments gives:

(A1-1)

(A1-2)

Solving Eq. (A1-1) for Q gives:

(A1-3)

Adding Eq. (A1-1) and Eq. (A1-2) and integrating the result:

(A1-4)

(A1-5)

(A1-6)

Substituting Eq. (A1-6) in Eq. (A1-3) results in:

100

(A1-8)

Appendix A1-2: Theoretical final density

Initial condition at the start of experiment:

Density of upper compartment fluid = ρH,i

Density of lower compartment fluid = ρL,i

Final condition at the end of experiment:

Density of upper compartment fluid = ρH,f

Density of lower compartment fluid = ρL,f

Mass balance of the total volume of fluids between initial and final conditions gives:

(A1-9)

At the end of ingress, the densities of the two fluids are equal. Hence,

(A1-10)

Substituting Eq. (A1-10) in Eq. (A1-9) gives:

Since the compartment volumes are equal, this gives the expression for the theoretical

final density of the upper compartment as:

(A1-11)

101

Theoretical final time

The expression for change in density of the upper compartment in the non-linear region is

given by Eq. 3-4 in section 3.3.2 as:

(A1-12)

At t = tf, the theoretical final density is obtained from Eq. (A1-11) as discussed above.

Substituting this into Eq. (A1-12), we get:

(A1-13)

Hence, the theoretical final time for complete exchange to occur is obtained as:

(A1-14)

Appendix A1-3: Mathematical treatment for final region (ttr,f ≤ t ≤ tf)

The volumetric exchange rate is given by Eq. (A1-8) as:

Since the volumes of the mixing compartments are equal, this expression changes to:

(A1-15)

where = constant throughout the exchange

As t tf,

and

Hence, the limit of the expression for Q can be evaluated from Eq. (A1-15) as:

102

(A1-16)

Using L Hospital’s rule, the volumetric exchange rate is evaluated as:

(A1-17)

Taking the derivative of the correlation for density given by Eq. (A1-12):

(A1-18)

(A1-19)

Furthermore, it is also known that as the density difference between the two

compartments approaches zero, the exchange approaches completion and hence,

Q 0 as t tf (A1-20)

Hence, substituting Eqs (A1-18) and (A1-19) in Eq. (A1-17) and utilizing the boundary

condition given by Eq. (A1-20) the expression for volumetric exchange rate in the final

stage is obtained as:

, for t tr,f ≤ t ≤ tf (A1-21)

The Froude number in this final stage can thus be obtained from Eq. (A1-21) as:

, for t tr,f ≤ t ≤ tf (A1-22)

103

APPENDIX A2

Exchange Volume Calculations

104

Appendix A2-1: Exchange Volume calculation of prototypic GT-MHR (Quick

Look Report, 2009)

In order to calculate the global exchange time ratio for the current study as given

by Eq. (2-11), the prototypic GT-MHR (INEEL/EXT-03-00870, (2003)) is used as

reference to determine the total volume to be exchanged. A schematic diagram of the

reactor vessel is shown in Fig. A2-1 (a) and A2-1 (b). To simplify calculations, the

internal structures in the core and upper hemisphere and the shutdown cooling system in

the lower hemisphere are neglected.

In case of the rupture of the primary pipe, the ingressing air will replace the helium in the

vessel upto the highest point of connection between the primary pipe and the vessel, as

shown in Fig. A2-1 (b).

This volume is given by:

(A2-1)

where dvessel is the diameter of the vessel, and H is the diameter of the primary pipe.

105

(a) Reactor Vessel (dimensions are in inches) (b) Internal Structures

Figure A2-1: Schematic diagram of reference GT-MHR

Highest level

for primary

pipe

106

For the standpipe, the volume to be exchanged is the total free volume in the

vessel for any inclination of the break. This volume includes the core region for which

the cross-sectional layout is given in Fig. A2-2 and the fuel block dimensions are given in

Fig. A2-3.

Figure A2-2: Cross-sectional view of core layout

107

Figure A2-3: Fuel Block (all dimensions are in inches)

108

For calculating the available exchange volume of helium in the core region the

following assumptions are made:

Coolant channels are only present in the active core fuel blocks.

The maximum diameter of the core is 22.5 feet (270 inches) and the volume

between the core and vessel wall is filled with helium.

Control rods are neglected and fuel handling holes.

Fuel handling holes are neglected.

Thus, the volume of helium in the core is given by:

(A2-2)

where Nfuelblocks, dcoolant,l, dcoolant,s and H are the number of fuel blocks in the active core,

diameters of the larger and smaller coolant channels and the fuel block height

respectively. The helium filling the annular gap region (with a height of 10 fuel blocks)

between the vessel and core is given by:

(A2-3)

109

The volume of the cylindrical region above the core and the upper hemisphere is

calculated as:

(A2-4)

Hence, the total volume to be exchanged during a standpipe break is:

(A2-5)

Exchange volume calculation of water-brine test facility

To calculate the exchange time ratio, the exchange volume is calculated in the

water-brine experiment by considering the orientation of the tank with respect to gravity.

During the exchange in the tilted apparatus, two volumes will not be exchanged

completely as shown in Fig. A2-4. Here, a volume of heavy brine will remain in the

upper compartment, and a volume of lighter water will remain in the lower compartment

due to their position with respect to the break.

110

Figure A2-4: Exchange volume in inclined water-brine apparatus.

The volume that is not exchanged can be found using the geometry of the

apparatus. The area of the triangle formed by an unmixed volume is given by:

(A2-6)

where b is the base of the triangle and h is height. The base is found by:

Unmixed

water

Water-brine

mixture

Unmixed

brine

Water-brine

mixture

θ

h b

111

(A2-7)

where w is the width of the compartment and d is the circular break diameter, or the

length of the slit geometry. The height of the triangle is given by:

(A2-8)

where is the inclination or tilt angle. Thus the unmixed volume in one compartment is

obtained from:

(A2-9)

where s is the gap size. Eq. (A2-9) is applied for the tilting angles between 15 and 75

degrees. For the horizontal and vertical, the unmixed volume is assumed to be the entire

volume or half the total volume, respectively.

112

APPENDIX A3

Experimental data and correlation

for L/D = 0.63

(Note: For all figures in this appendix, the horizontal dotted line represents the theoretical

final density and the vertical dotted line represents the time at which the mixture interface

reaches the break as described in Fig. 3-14 in Section 3.3. All error bars are ±0.1%.)

113

Figure A3-1 (a): Experimental data and correlation for L/D = 0.63, 00

on regular axes.

Figure A3-1 (b): Experimental data and correlation for L/D = 0.63, 00 on log axes.

1010

1015

1020

1025

1030

0 200 400 600 800 1000 1200

ρH

[kg/m

3]

Time [sec]

ρ = 1026 - 0.0333t

ρ = 1023 - 2.85ln(t/ttr,i)

1010

1 200

ρH

[kg/m

3]

Time [sec]

ρ = 1026 - 0.0333t

ρ = 1023 - 2.85ln(t/ttr,i)

Mixture interface

reaches break Theoretical final

density

Mixture interface

reaches break

114

Figure A3-2 (a): Experimental data and correlation for L/D = 0.63, 150

on regular axes.

Figure A3-2 (b): Experimental data and correlation for L/D = 0.63, 150

on log axes.

1010

1015

1020

1025

1030

0 200 400 600 800 1000 1200 1400

ρH

[kg/m

3]

Time [sec]

ρ = 1028.5 - 0.0408t

ρ = 1025.5 - 3.2ln(t/ttr,i)

1010

1 200

ρH

[kg/m

3]

Time [sec]

ρ = 1028.5 - 0.0408t

ρ = 1025.5 - 3.2ln(t/ttr,i)

115

Figure A3-3 (a): Experimental data and correlation for L/D = 0.63, 300 on regular axes.

Figure A3-3 (b): Experimental data and correlation for L/D = 0.63, 300 on log axes.

1010

1015

1020

1025

1030

0 200 400 600 800 1000

ρH

[kg/m

3]

Time [sec]

ρ = 1025 - 0.0423t

ρ = 1022.5- 3.0ln(t/ttr,i)

1010

1 200

ρH

[kg/m

3]

Time [sec]

ρ = 1025 - 0.0423t

ρ = 1022.5- 3.0ln(t/ttr,i)

116

Figure A3-4 (a): Experimental data and correlation for L/D = 0.63, 450 on regular axes.

Figure A3-4 (b): Experimental data and correlation for L/D = 0.63, 450 on log axes.

1010

1015

1020

1025

1030

0 500 1000

ρH

[kg/m

3]

Time [sec]

ρ = 1025.5 - 0.0440t

ρ = 1021.5 - 3.4ln(t/ttr,i)

1010

1 400

ρH

[kg/m

3]

Time [sec]

ρ = 1025.5 - 0.0440t

ρ = 1021.5 - 3.4ln(t/ttr,i)

117

Figure A3-5 (a): Experimental data and correlation for L/D = 0.63, 600 on regular axes.

Figure A3-5 (b): Experimental data and correlation for L/D = 0.63, 600 on log axes.

1010

1015

1020

1025

1030

0 400 800 1200 1600

ρH

[kg/m

3]

Time [sec]

ρ = 1027 - 0.0408t

ρ = 1024 - 3.3ln(t/ttr,i)

1010

1 400

ρH

[kg/m

3]

Time [sec]

ρ = 1027 - 0.0408t

ρ = 1024 - 3.3ln(t/ttr,i)

118

Figure A3-6 (a): Density versus curve fit for L/D = 0.63, 750 on regular axes.

Figure A3-6 (b): Density versus curve fit for L/D = 0.63, 750 on log axes.

1010

1015

1020

1025

1030

0 300 600 900 1200 1500 1800 2100 2400 2700

ρH

[kg/m

3]

Time [sec]

ρ = 1024.5 - 0.029t

ρ = 1021- 3.2ln(t/ttr,i)

1010

1 300

ρH

[kg/m

3]

Time [sec]

ρ = 1024.5 - 0.029t

ρ = 1021- 3.2ln(t/ttr,i)

119

Figure A3-7 (a): Experimental data and correlation for L/D = 0.63, 900 on regular axes.

Figure A3-7 (b): Experimental data and correlation for L/D = 0.63, 900 on log axes.

1010

1015

1020

1025

1030

0 500 1000 1500 2000 2500 3000 3500

ρH

[kg/m

3]

Time [sec]

ρ = 1027 - 0.0191t

ρ = 1023- 3.2ln(t/ttr,i)

1010

1 500

ρH

[kg/m

3]

Time [sec]

ρ = 1027 - 0.0191t

ρ = 1023- 3.2ln(t/ttr,i)

120

APPENDIX A4

Experimental data and correlation

for L/D = 3.0

(Note: For all figures in this appendix, the horizontal dotted line represents the theoretical

final density and the vertical dotted line represents the time at which the mixture interface

reaches the break as described in Fig. 3-14 in Section 3.4. All error bars are ±0.1%.)

121

Figure A4-1 (a): Experimental data and correlation for L/D = 3, 00 on regular axes.

Figure A4-1 (b): Experimental data and correlation for L/D = 3, 00 on log axes.

1010

1015

1020

1025

1030

0 400 800 1200 1600

ρH

[kg/m

3]

Time [sec]

ρ = 1028.5 - 0.0333t

ρ = 1025.5 3.3ln(t/ttr,i)

1010

1 400

ρH

[kg/m

3]

Time [sec]

ρ = 1028.5 - 0.0333t

ρ = 1025.5 - 3.3ln(t/ttr,i)

122

Figure A4-2 (a): Experimental data and correlation for L/D = 3, 150 on regular axes.

Figure A4-2 (b): Experimental data and correlation for L/D = 3, 150 on log axes.

1010

1015

1020

1025

1030

0 150 300 450 600

ρH

[kg/m

3]

Time [sec]

ρ = 1027 -0.044t

ρ = 1023.5- 3.6ln(t/ttr,i)

1010

1 150

ρH

[kg/m

3]

Time [sec]

ρ = 1027 - 0.044t

ρ = 1023.5- 3.6ln(t/ttri)

123

Figure A4-3 (a): Experimental data and correlation for L/D = 3, 300 on regular axes.

Figure A4-3 (b): Experimental data and correlation for L/D = 3, 300 on log axes.

1010

1015

1020

1025

1030

0 500 1000 1500

ρH

[kg/m

3]

Time [sec]

ρ = 1027.5 - 0.05t

ρ = 1023.5 - 3.9ln(t/ttr,i)

1010

1 500

ρH

[kg/m

3]

Time [sec]

ρ = 1027.5 - 0.05t

ρ = 1023.5 - 3.9ln(t/ttr,i)

124

Figure A4-4 (a): Experimental data and correlation for L/D = 3, 450 on regular axes.

Figure A4-4 (b): Experimental data and correlation for L/D = 3, 450 on log axes.

1010

1015

1020

1025

1030

0 300 600 900 1200

ρH

[kg/m

3]

Time [sec]

ρ = 1026 - 0.0390t

ρ = 1022.5- 3.7ln(t/ttr,i)

1010

1 150

ρH

[kg/m

3]

Time [sec]

ρ = 1026 - 0.0390t

ρ = 1022.5- 3.7ln(t/ttr,i)

125

Figure A4-5 (a): Experimental data and correlation for L/D = 3, 600 on regular axes.

Figure A4-5 (b): Experimental data and correlation for L/D = 3, 600 on log axes.

1010

1015

1020

1025

1030

0 500 1000 1500

ρH

[kg/m

3]

Time [sec]

ρ = 1028 - 0.0350t

ρ = 1024 - 3.7ln(t/ttr,i)

1010

1 500

ρH

[kg/m

3]

Time [sec]

ρ = 1028 - 0.0350t

ρ = 1024- 3.7ln(t/ttr,i)

126

Figure A4-6 (a): Experimental data and correlation for L/D = 3, 750 on regular axes.

Figure A4-6 (b): Experimental data and correlation for L/D = 3, 750 on log axes.

1010

1015

1020

1025

1030

0 500 1000 1500

ρH

[kg/m

3]

Time [sec]

ρ = 1025 - 0.0202t

ρ = 1022- 3.7ln(t/ttr,i)

1010

1 200

ρH

[kg/m

3]

Time [sec]

ρ = 1025 - 0.0202t

ρ = 1022 - 3.7ln(t/ttr,i)

127

Figure A4-7 (a): Experimental data and correlation for L/D = 3, 900 on regular axes.

Figure A4-7 (b): Experimental data and correlation for L/D = 3, 900 on log axes.

1010

1015

1020

1025

1030

0 1000 2000 3000 4000 5000 6000

ρH

[kg/m

3]

Time [sec]

ρ = 1026 - 0.0111t

ρ = 1023- 3.4ln(t/ttr,i)

1010

1 1000

ρH

[kg/m

3]

Time [sec]

ρ = 1026 - 0.0111t

ρ = 1023- 3.4ln(t/ttr,i)

128

APPENDIX A5

Experimental data and correlation

for L/D = 5.0

(Note: For all figures in this appendix, the horizontal dotted line represents the theoretical

final density and the vertical dotted line represents the time at which the mixture interface

reaches the break, as described in Fig. 3-14 in Section 3.3. All error bars are ±0.1%.)

129

Figure A5-1 (a): Experimental data and correlation for L/D = 5, 00 on regular axes.

Figure A5-1 (b): Experimental data and correlation for L/D = 5, 00 on log axes.

1010

1015

1020

1025

1030

0 300 600 900 1200 1500 1800

ρH

[kg/m

3]

Time [sec]

ρ = 1027 - 0.027t

ρ = 1024 - 3ln(t/ttr,i)

1010

1 300

ρH

[kg/m

3]

Time [sec]

ρ = 1027- 0.027t

ρ = 1024 - 3ln(t/ttr,i)

130

Figure A5-2 (a): Experimental data and correlation for L/D = 5, 150 on regular axes.

Figure A5-2 (b): Experimental data and correlation for L/D = 5, 150 on log axes.

1010

1015

1020

1025

1030

0 400 800 1200 1600

ρH

[kg/m

3]

Time [sec]

ρ = 1027.5 - 0.043t

ρ = 1024- 3.1ln(t/ttr,i)

1010

1 400

ρH

[kg/m

3]

Time [sec]

ρ = 1027.5 - 0.043t

ρ = 1024- 3.1ln(t/ttri)

131

Figure A5-3 (a): Experimental data and correlation for L/D = 5, 300 on regular axes.

Figure A5-3 (b): Experimental data and correlation for L/D = 5, 300 on log axes.

1010

1015

1020

1025

1030

0 200 400 600 800 1000

ρH

[kg/m

3]

Time [sec]

ρ = 1025 - 0.0423t

ρ = 1022.5 - 3.0ln(t/ttr,i)

1010

1 200

ρH

[kg/m

3]

Time [sec]

ρ = 1025 - 0.0423t

ρ = 1022.5 - 3.0ln(t/ttr,i)

132

Figure A5-4 (a): Experimental data and correlation for L/D = 5, 450 on regular axes.

Figure A5-4 (b): Experimental data and correlation for L/D = 5, 450 on log axes.

1010

1015

1020

1025

1030

0 200 400 600 800 1000 1200 1400 1600

ρH

[kg/m

3]

Time [sec]

ρ = 1024.5 - 0355t

ρ = 1022 - 3.1ln(t/ttr,i)

1010

1 200

ρH

[kg/m

3]

Time [sec]

ρ = 1024.5 - 0355t

ρ = 1022 - 3.1ln(t/ttr,i)

133

Figure A5-5 (a): Experimental data and correlation for L/D = 5, 600 on regular axes.

.

Figure A5-5 (b): Experimental data and correlation for L/D = 5, 600 on log axes.

1010

1015

1020

1025

1030

0 300 600 900 1200 1500 1800 2100

ρH

[kg/m

3]

Time [sec]

ρ = 1025 - 0.0305t

ρ = 1022- 3.3ln(t/ttr,i)

1010

1 300

ρH

[kg/m

3]

Time [sec]

ρ = 1025 - 0.0305t

ρ = 1022- 3.3ln(t/ttr,i)

134

Figure A5-6 (a): Experimental data and correlation for L/D = 5, 750 on regular axes.

Figure A5-6 (b): Experimental data and correlation for L/D = 5, 750 on log axes.

1010

1015

1020

1025

1030

0 400 800 1200 1600 2000

ρH

[kg/m

3]

Time [sec]

ρ = 1025.5 - 0.021t

ρ = 1022.5 - 3.3ln(t/ttr,i)

1010

1 400

ρH

[kg/m

3]

Time [sec]

ρ = 1025.5 - 0.021t

ρ = 1022.5 - 3.3ln(t/ttr,i)

135

Figure A5-7 (a): Experimental data and correlation for L/D = 5, 900 on regular axes.

Figure A5-7 (b): Experimental data and correlation for L/D = 5, 900 on log axes.

1010

1015

1020

1025

1030

0 900 1800 2700 3600 4500 5400

ρH

[kg/m

3]

Time [sec]

ρ = -0.0063t + 1026.5

ρ = 1022.5 - 3.9ln(t/ttr,i)

1010

1 600

ρH

[kg/m

3]

Time [sec]

ρ = 1026.5 -0.0063t

ρ = 1022.5- 3.9ln(t/ttr,i)

136

APPENDIX A6

Volumetric Flow Rate (Q) and Froude Number (Fr)

for L/D = 0.63

137

Figure A6-1 (a): Volumetric flow rate for L/D = 0.63, 00.

Figure A6-1 (b): Froude number for L/D = 0.63, 00.

0.E+00

2.E-06

4.E-06

6.E-06

8.E-06

0 1000 2000 3000

Q[m

3/s

]

Time [sec]

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1000 2000 3000

Fro

ude

Num

ber

Time [sec]

t tr,i t tr,f tf

t tr,i t tr,f tf

138

Figure A6-2 (a): Volumetric flow rate for L/D = 0.63, 150.

Figure A6-2 (b): Froude number for L/D = 0.63, 150.

0.E+00

2.E-06

4.E-06

6.E-06

8.E-06

0 500 1000 1500 2000 2500 3000

Q[m

3/s

]

Time [sec]

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 500 1000 1500 2000 2500 3000

Fro

ude

Num

ber

Time [sec]

t tr,i t tr,f tf

t tr,i t tr,f tf

139

Figure A6-3 (a): Volumetric flow rate for L/D = 0.63, 300.

Figure A6-3 (b): Froude number for L/D = 0.63, 300.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 500 1000 1500 2000

Fro

ude

Num

ber

Time [sec]

0.E+00

2.E-06

4.E-06

6.E-06

8.E-06

0 500 1000 1500 2000

Q[m

3/s

]

Time [sec]

t tr,i t tr,f tf

t tr,i t tr,f tf

140

Figure A6-4 (a): Volumetric flow rate for L/D = 0.63, 450.

Figure A6-4 (b): Froude number for L/D = 0.63, 450.

0.E+00

2.E-06

4.E-06

6.E-06

8.E-06

0 500 1000 1500

Q[m

3/s

]

Time [sec]

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 500 1000 1500

Fro

ude

Num

ber

Time [sec]

t tr,i t tr,f tf

t tr,i t tr,f tf

141

Figure A6-5 (a): Volumetric flow rate for L/D = 0.63, 600.

Figure A6-5 (b): Froude number for L/D = 0.63, 600.

0.E+00

2.E-06

4.E-06

6.E-06

8.E-06

0 500 1000 1500 2000

Q[m

3/s

]

Time [sec]

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 500 1000 1500 2000

Fro

ude

Num

ber

Time [sec]

t tr,i t tr,f tf

t tr,i t tr,f tf

142

Figure A6-6 (a): Volumetric flow rate for L/D = 0.63, 750.

Figure A6-6 (b): Froude number for L/D = 0.63, 750.

0.E+00

2.E-06

4.E-06

6.E-06

8.E-06

0 1200 2400 3600 4800

Q[m

3/s

]

Time [sec]

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1000 2000 3000 4000 5000

Fro

ude

Num

ber

Time [sec]

t tr,i t tr,f tf

t tr,i t tr,f tf

143

Figure A6-7 (a): Volumetric flow rate for L/D = 0.63, 900.

Figure A6-7 (b): Froude number for L/D = 0.63, 900.

0.E+00

2.E-06

4.E-06

6.E-06

8.E-06

0 1000 2000 3000 4000 5000

Q[m

3/s

]

Time [sec]

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1000 2000 3000 4000 5000

Fro

ude

Num

ber

Time [sec]

t tr,i t tr,f tf

t tr,i t tr,f tf

144

APPENDIX A7

Volumetric Flow rate (Q) and Froude Number (Fr)

for L/D = 3.0

145

Figure A7-1 (a): Volumetric flow rate for L/D = 3.0, 00.

Figure A7-1 (b): Froude number for L/D = 3.0, 00.

0.E+00

2.E-06

4.E-06

6.E-06

8.E-06

0 500 1000 1500 2000 2500 3000

Q[m

3/s

]

Time [sec]

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 500 1000 1500 2000 2500 3000

Fro

ude

Num

ber

Time [sec]

t tr,i t tr,f tf

t tr,i t tr,f tf

146

Figure A7-2 (a): Volumetric flow rate for L/D = 3.0, 150.

Figure A7-2 (b): Froude number for L/D = 3.0, 150.

0.E+00

2.E-06

4.E-06

6.E-06

8.E-06

0 500 1000 1500

Q[m

3/s

]

Time [sec]

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 500 1000 1500

Fro

ude

Num

ber

Time [sec]

t tr,i t tr,f tf

t tr,i t tr,f tf

147

Figure A7-3 (a): Volumetric flow rate for L/D = 3.0, 300.

Figure A7-3 (b): Froude number for L/D = 3.0, 300.

0.E+00

2.E-06

4.E-06

6.E-06

8.E-06

0 400 800 1200

Q[m

3/s

]

Time [sec]

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 400 800 1200

Fro

ude

Num

ber

Time [sec]

t tr,i t tr,f tf

t tr,i t tr,f tf

148

Figure A7-4 (a): Volumetric flow rate for L/D = 3.0, 450.

Figure A7-4 (b): Froude number for L/D = 3.0, 450.

0.E+00

2.E-06

4.E-06

6.E-06

8.E-06

0 200 400 600 800 1000 1200

Q[m

3/s

]

Time [sec]

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 200 400 600 800 1000 1200

Fro

ude

Num

ber

Time [sec]

t tr,i t tr,f tf

t tr,i t tr,f tf

149

Figure A7-5 (a): Volumetric flow rate for L/D = 3.0, 600.

Figure A7-5 (b): Froude number for L/D = 3.0, 600.

0.E+00

2.E-06

4.E-06

6.E-06

8.E-06

0 500 1000 1500 2000

Q[m

3/s

]

Time [sec]

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 500 1000 1500 2000

Fro

ude

Num

ber

Time [sec]

t tr,i t tr,f tf

t tr,i t tr,f tf

150

Figure A7-6 (a): Volumetric flow rate for L/D = 3.0, 750.

Figure A7-6 (b): Froude number for L/D = 3.0, 750.

0.E+00

2.E-06

4.E-06

6.E-06

8.E-06

0 400 800 1200 1600 2000 2400

Q[m

3/s

]

Time [sec]

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 400 800 1200 1600 2000 2400

Fro

ude

Num

ber

Time [sec]

t tr,i t tr,f tf

t tr,i t tr,f tf

151

Figure A7-7 (a): Volumetric flow rate for L/D = 3.0, 900.

Figure A7-7 (b): Froude number for L/D = 3.0, 900.

0.E+00

2.E-06

4.E-06

6.E-06

8.E-06

0 1000 2000 3000 4000 5000 6000

Q[m

3/s

]

Time [sec]

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0 1000 2000 3000 4000 5000 6000

Fro

ude

Num

ber

Time [sec]

t tr,i t tr,f tf

t tr,i t tr,f tf

152

APPENDIX A8

Volumetric Flow rate (Q) and Froude Number (Fr)

for L/D = 5.0

153

Figure A8-1 (a): Volumetric flow rate for L/D = 5.0, 00.

Figure A8-1 (b): Froude number for L/D = 5.0, 00.

0.E+00

2.E-06

4.E-06

6.E-06

8.E-06

0 1000 2000 3000 4000

Q[m

3/s

]

Time [sec]

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1000 2000 3000 4000

Fro

ude

Num

ber

Time [sec]

t tr,i t tr,f tf

t tr,i t tr,f tf

154

Figure A8-2 (a): Volumetric flow rate for L/D = 5.0, 150.

Figure A8-2 (b): Froude number for L/D = 5.0, 150.

0.E+00

2.E-06

4.E-06

6.E-06

8.E-06

0 500 1000 1500 2000

Q[m

3/s

]

Time [sec]

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 500 1000 1500 2000

Fro

ude

Num

ber

Time [sec]

t tr,i t tr,f tf

t tr,i t tr,f tf

155

Figure A8-3 (a): Volumetric flow rate for L/D = 5.0, 300.

Figure A8-3 (b): Froude number for L/D = 5.0, 300.

0.E+00

2.E-06

4.E-06

6.E-06

8.E-06

0 200 400 600 800 1000 1200

Q[m

3/s

]

Time [sec]

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 200 400 600 800 1000 1200

Fro

ude

Num

ber

Time [sec]

t tr,i t tr,f tf

t tr,i t tr,f tf

156

Figure A8-4 (a): Volumetric flow rate for L/D = 5.0, 450.

Figure A8-4 (b): Froude number for L/D = 5.0, 450.

0.E+00

2.E-06

4.E-06

6.E-06

8.E-06

0 500 1000 1500

Q[m

3/s

]

Time [sec]

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 500 1000 1500

Fro

ude

Nu

mber

Time [sec]

t tr,i t tr,f tf

t tr,i t tr,f tf

157

Figure A8-5 (a): Volumetric flow rate for L/D = 5.0, 600.

Figure A8-5 (b): Froude number for L/D = 5.0, 600.

0.E+00

2.E-06

4.E-06

6.E-06

8.E-06

0 500 1000 1500 2000

Q[m

3/s

]

Time [sec]

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 500 1000 1500 2000

Fro

ude

Num

ber

Time [sec]

t tr,i t tr,f tf

t tr,i t tr,f tf

158

Figure A8-6 (a): Volumetric flow rate for L/D = 5.0, 750.

Figure A8-6 (b): Froude number for L/D = 5.0, 750.

0.E+00

2.E-06

4.E-06

6.E-06

8.E-06

0 1200 2400 3600

Q[m

3/s

]

Time [sec]

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 1200 2400 3600

Fro

ude

Num

ber

Time [sec]

t tr,i t tr,f tf

t tr,i t tr,f tf

159

Figure A8-7 (a): Volumetric flow rate for L/D = 5.0, 900.

Figure A8-7 (b): Froude number for L/D = 5.0, 900.

0.E+00

2.E-06

4.E-06

6.E-06

8.E-06

0 2000 4000 6000 8000

Q [

m3/s

]

Time [sec]

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 2000 4000 6000 8000

Fro

ude

Num

ber

Time [sec]

t tr,i t tr,f tf

t tr,i t tr,f tf

160