A Study of Intermittent Buoyancy Induced Flow

Phenomena in CANDU Fuel Channels

by

Zheko Petrov Karchev

A thesis submitted in conformity with the requirements

for the degree of Master of Applied Science

Department of Chemical Engineering and Applied Chemistry

University of Toronto

© Copyright by Zheko Petrov Karchev 2009

ii

A Study of Intermittent Buoyancy Induced Flow Phenomena

in CANDU Fuel Channels

Zheko Petrov Karchev

Master of Applied Science

Department of Chemical Engineering and Applied Chemistry

University of Toronto

2009

Abstract

The present work focuses on the study of two-phase flow behavior called “Intermittent

Buoyancy Induced Flow” (IBIF) resulting from the loss of coolant circulation in a

CANDU nuclear reactor core. The main objectives are to study steam bubble formation

and migration through the pressure tube and into the feeder tubes and headers, and to

study the effect of pressure tube sagging on the two-phase flow behavior during IBIF.

Experiments are conducted using air and water flow at atmospheric pressure to

qualitatively examine the IBIF phenomena. The test showed oscillating periodic behavior

in the void fraction as the air vents.

In addition to this a mathematical model based on a simplified momentum balance for the

liquid and gas phases was formulated. The model was further solved and compared to the

experimental data. The model predictions showed a reasonable agreement within the

investigated range of void fractions.

iii

Acknowledgments

I wish to express my appreciation to Professor Masahiro Kawaji for his support,

professional guidance and excellent supervision throughout the course of this work.

I would like to acknowledge the financial support of Ontario Power Generation,

Bruce Power Inc. and Atomic Energy of Canada.

I would also like to express my gratitude to Mr. Laurence Leung, Mr. Muhammad

Ali and Mr. Marc Kwee for their valuable input on this work

iv

Table of Contents

Abstract ii

Acknowledgements iii

Table of Contents iv

List of Figures vi

List of Tables viii

Nomenclature viii

List of Appendices viii

1. Introduction 1

1.1 Nuclear Power Plant - Overall View 1

1.2 CANDU Nuclear Reactor Core 2

1.3 Project Objectives and Scope 3

2. Literature Review 7

2.1 Overview of Two – Phase Flow Models 7

2.2 Bubble Formation and Propagation 9

2.3 Void Fraction Measurements 10

2.4 CANDU Reactor behaviour in case of circulation outage 11

3. Experimental Design 12

3.1 Facility Design 12

3.2 Experimental Set-Up 15

3.2.1 Overall View 15

3.2.2 Pressure Tube and Fuel Bundles Design 17

4. Experimental Results and Discussion 19

4.1. Preliminary Observations 19

4.2. Effect of Air Injection Nozzle Location 21

4.3. Combined Effect of Feeder Water Level and Air-Injection Rate on the

Venting Time 25

4.4. Effect of Multiple Air Injection on the Venting Time 27

4.4.1. Air Injections through Two Air-Injection Nozzles 28

4.4.2. Air Injections through Three Air-Injection Nozzles 29

v

4.5. Effect of Pressure Tube Sagging on the Venting Time 30

4.6. Air-Lift Effect 32

4.7. Oscillatory Behavior 35

4.8 Summary of Experimental Results 39

5. Mathematical Model 40

5.1. Mathematical Description 40

5.2. Numerical solution of the mathematical model 44

5.2.1. Calculating the basic geometric parameters 44

5.2.2. Numerical Solution of the model 51

5.3. Calculation Results 53

5.4 Summary of Model Development 57

6. Conclusions 58

References 60

Appendices 63

vi

List of Figures

Fig. 1-1 Nuclear power plant – overall diagram

Fig. 1-2 CANDU nuclear reactor core

Fig. 1-3 Decay power 6 hours to 29 days after reactor shutdown

Fig. 3-1 Experimental set-up – block diagram

Fig. 3-2 Experimental set-up – overall view

Fig. 3-3 CANDU nuclear reactor steam supply system

Fig. 3-4 Photograph of a CANDU pressure tube with a fuel bundle placed inside

Fig. 3-5 Photograph of the simulated fuel bundle used in the current design

Fig. 4-1 Schematics of the experimental set-up

Fig. 4-2 Photograph of the system behavior upon air injection

Fig. 4-3 Photograph of the slug rising in the vertical feeder

Fig. 4-4 Schematics of the experimental set-up – effect of the air injection location

Fig. 4-5 Consecutive photographs of the bubble propagation front (65 ms time interval

between the frames)

Fig. 4-6 Effect of air injection location at different simulated power level (SPL)

Fig. 4-7 Schematic of the experimental set-up to investigate the combined effect of feeder

water level and air-injection rate

Fig. 4-8 Venting time as a function of water level in the feeder line at different simulated

power levels

Fig. 4-9 A typical CANDU reactor pressure tube axial heat flux distribution

Fig. 4-10 Schematic of the experimental set-up for air injection through two air- nozzles

Fig. 4-11 Schematic of the experimental set-up – air injections through three air- nozzles

Fig. 4-12 Schematic of the experimental set-up to study the effect of pressure tube

sagging

Fig. 4-13 Photograph of the inclined pressure tube (sagging of 5.08 cm (2”) in the mid

section

Fig. 4-14 Effect of pressure tube sagging on the venting time

Fig. 4-15 Schematic of the experimental set-up for studying the air-lift effect

Fig. 4-16 Air-lift effect on flow velocity

Fig. 4-17 Effect of water level on the frequency of oscillations (SPL 1.1 kW)

vii

Fig. 4-18 Effect of the air injection rate (SPL) on the frequency of oscillations (Feeder

Line Water Level – 230 cm)

Fig. 5-1 Diagram of the modeled two-phase system

Fig. 5-2 Diagram of the gas liquid interface

Frig. 5-3 Schematics of the simulated pressure tube with 37 acrylic rods

Fig. 5-4 Schematics of the gas-liquid interface

Fig. 5-5 Schematics of the gas-liquid interface with the rods placed inside

Fig. 5-6 Wall wetted perimeter schematics

Fig. 5-7 Schematics of the wetted perimeter calculation with rods placed inside

Fig. 5-8 Newton-Raphson method - graphical representation

Fig. 5-9 Variations of the liquid-solid and gas-solid interface lengths as a function of void

fraction

Fig. 5-10 Variation in the gas-liquid interface length as a function of the void fraction

Fig. 5-11 Variations of the liquid and gas-solid cross sectional areas as a function of the

void fraction

Fig. 5-12 Schematic of the experimental set-up for model validation

Fig. 5-13 Variation of the liquid phase velocity with the void fraction in the pressure tube

Fig. 5-14 Comparison between the predicted and the calculated liquid phase velocity

viii

List of Tables

Table 3-1 Comparison of test section and CANDU reactor component dimensions

Table 4-1 Effect of the air injection location on the venting time

Table 4-2 Venting time data for air injections through two air- nozzles

Table 4-3 Venting time for air injections through three air- nozzles

Table 4-4 Effect of the pressure tube sagging

Table 4-5 Air-lift effect on flow velocity (m/s)

Table 4-6 Air-lift effect on header water level

Table 4-7 Effect of water level on the frequency of oscillations (SPL 1.1 kW)

Table 4-8 Effect of air injection rate (SPL) on the frequency of oscillations

Table 4-9 Effect of air injection rate (SPL) on the frequency of oscillations (Ten fold

decrease in the header tank volume)

Nomenclature

IBIF Intermittent Buoyancy Induced Flow

SPL Simulated Power Level

List of Appendices

Appendix 1: Numerical Code for Interfacial Area Calculation

Appendix 2: Numerical Code for Liquid Velocity Calculation

Appendix 3: User Input Function

1

1. Introduction 1.1. Nuclear Power Plant - Overall View

CANDU – PHWR is an essential part of Ontario’s Power system. Twenty reactors of this

type generate about 15,000 MW of electricity. The CANDU nuclear technology

combines cost efficiency, low capital costs and a design which has proven its reliability

and safety for more than 30 years.

A nuclear power plant usually includes three basic compartments illustrated in Fig 1-1:

Vacuum building, reactor building and a building housing the steam turbine and the

electrical generator. The ‘heart’ of the plant is the nuclear reactor in which the heat

generated as a result of the fission reaction is used to produce high pressure steam which

is transferred to the steam turbine. The turbine transforms the energy carried by the steam

into mechanical energy which is subsequently transferred to the electrical generator and

converted into electrical energy.

As a safety measure the reactor building is connected through a large diameter pipe to the

Vacuum Building. The purpose of this structure is to ensure fast seam condensation in

case of Loss of Coolant Accidents (LOCA) in the reactor building. In order to prevent

even small releases of radioactive materials from the nuclear power plant, all the air

released into the atmosphere is filtered through a filter system.

Fig 1-1. Nuclear power plant – overall diagram

Vacuum Building 

Reactor Building

Filter System  Reactor

Steam Turbine

Generator

Secondary Loop 

Primary Loop

Boiler 

Cooling Water from Lake 

2

1.2 CANDU Nuclear Reactor Core

The CANDU Reactor uses two independent water loops for removing the heat from the

nuclear core. The primary loop is indicated in Fig. 1-2 with yellow color and uses heavy

water as a heat transfer fluid. The heavy water flows through the pressure tubes (#10) in

which the fuel bundles (#1) are placed. The heat released as a result of nuclear fission

reactions is transferred to the water. The water in the primary loop is kept under high

pressure which allows it to be heated to higher temperature and more heat to be removed

from the core. The water temperature in the primary loop is lower than the boiling

temperature which means that there is a single-phase flow through the core. The heavy

water flows through the pressure tubes, gets heated and reaches the steam generator (#5).

In the steam generator the hot heavy water is used to heat up light water in the secondary

loop (#12) and generate steam (#11) which is fed to the steam turbine.

1. Fuel Bundles; 2. Calandria; 3. Control Rods; 4. Pressurizer; 5. Steam Generator

6. Light Water Pump; 7. Heavy Water Pump; 8. Fuel Loading Machine; 9. Moderator

10. Pressure Tube; 11.High Pressure Steam (to Steam Turbine); 12. Water

Condensate (from Condenser); 13.Reactor Containment Building; 14 Primary Loop

Fig 1-2. CANDU nuclear reactor core (source: http://en.wikipedia.org/wiki/Candu)

14

3

All of the pressure tubes are placed inside the Calandria (#2) containing heavy water

which is used as a moderator. The control rods (#3) are 28 cadmium rods which serve as

an emergency shutdown system. In case of an accident they are submerged into to the

heavy water moderator in the Calandria by a gravitational drop. This is possible since the

moderator is kept at low pressure. As a backup to the primary shutdown system the

CANDU reactor uses a second system, which injects gadolinium through 6 nozzles into

the Calandria. The gadolinium with a large neutron absorption cross section acts as a

neutron poison and can rapidly terminate the fission reaction.

One of the main design characteristics of the CANDU reactor is the use of a horizontal

core containing many small diameter pressure tubes (about 4” ID) in which the uranium

fuel bundles are placed. This allows on-line re-fueling of the reactor at full power. A re-

fueling machine (#8) attached to both ends of the pressure tube push in a new fuel bundle

at one end and removes an old bundle at the other end. In contrast, light water reactors

which are more popular in other countries, must be shut down for re-fueling purposes.

The horizontal pressure tubes are typically 9 m long and can sag in the middle after many

years of service. The horizontal orientation of the pressure tubes, fuel bundles and

sagging phenomenon are closely related to the objectives and scope of the present project

as discussed below.

1.3 Project Objectives and Scope

The complete understanding of the thermal-hydraulic phenomena taking place in the

nuclear reactor core is a requirement for the safe operation of all nuclear reactors. The

current investigation is focused on the thermal-hydraulics of a CANDU reactor under loss

of coolant circulation conditions. In such an event, the reactor is shut down and the

pumps circulating the heavy water coolant through the primary loop will cease their

operation. Consequently, the coolant will become stagnant inside the pressure tubes for

an extended period of time, while the fuel rods continue generating a varying amount of

heat due to decay heat.

4

The reactor has several safety systems which are designed to terminate the fission

reaction shortly after any accident occurs. This leads to a drastic decay in the power level

inside the core. For example, the decay power 6 hours after the CANDU reactor with 37-

element fuel assemblies is shut down is just 0.794% of the full power. The heat

generation continues inside the fuel long after the shut-down although the power level is

significantly low (0.105% of full power 29 days after the shut-down) as shown in Fig. 1-3.

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 

Fig. 1-3 Decay power from 6 hours to 29 days after reactor shutdown (Ref. Report # N-

REP-03500.2-10002, page 61, Darlington NGS)

In case of a loss of coolant circulation event, the pressure in the primary loop is rapidly

reduced so that an over-pressurization can be avoided. Under these conditions it is

expected that the stagnant water boils in the pressure tube. The main safety concern in

this case is the uncovery of fuel rods due to the formation of large vapor bubbles, and the

time interval between the vapor bubbles formation and their venting through the feeder

tubes located at both ends of the pressure tube. Venting is important since the bubbles

rising in feeder tubes induce two phase flow inside the feeder tubes and consequently the

Pow

er L

evel

per

Cha

nel,

kW

Days after shutdown

5

pressure tube which facilitates the heat removal from the fuel bundles. This two phase

flow resulting from a loss of coolant circulation accident is referred to as “Intermittent

Buoyancy Induced Flow” (IBIF).

The formation of steam bubbles inside the pressure tubes under normal operation is not

desirable since it can lead to an increase in the nuclear fission rate. This phenomenon is

referred to as positive void reactivity and it results in an increase in the reactor power

level. This property of the CANDU reactor is significantly different from the Light Water

Reactors which have a negative void reactivity and the fission rate decreases with the

void volume. This positive void coefficient of CANDU reactors is not important as long

as the emergency shutdown systems (Cadmium control rods and Gadolinium Injection)

are activated.

After long term operation, the horizontal pressure tubes inside the CANDU reactor core

tend to sag slightly due to the thermal and mechanical stresses to which they are

subjected. The degree of sagging is at maximum about 5.0 cm (2 inches) measured at the

centre of the pressure tube. This phenomenon is expected to enhance the venting of the

steam bubbles formed in the pressure tubes as a result of boiling induced by the loss of

coolant circulation event. This assumption is based on experimental studies of two phase

flow behavior according to which the bubbles remain stagnant in horizontal tubes and

also the bubble rise velocity in inclined pipes generally exceeds the value of this

parameter in vertical pipes [10]. Despite all this, the effect of pressure tube sagging has

not been experimentally studied.

The present objectives and scope of the current project can be summarized as follows:

• Design and construct an experimental facility which nearly exactly duplicates the

geometry of a CANDU reactor pressure tube together with the thirteen replicas of 37-

element fuel bundles placed inside.

• Study of bubble formation and migration throughout the system by injecting air

bubbles into the rod bundles.

• Study of the effect of pressure tube sagging on the IBIF phenomena and two-phase

flow behavior in feeder tubes.

6

• Study on the possible existence of buoyancy-induced two phase flow circulation inside

the system.

7

2. Literature Review The two phase flow behavior has been an object of extensive theoretical and

experimental studies in energy systems. The reason behind this is the fact that this type of

flow occurs in wide industrial applications including nuclear reactors, boilers, oil wells

and pipelines, etc. Despite the large amount of work done on studying the behavior of the

two- and multi – phase flow systems there is still some uncertainty associated with them.

2.1 Overview of Two-Phase Flow Models There are different mathematical descriptions proposed for two-phase flow systems.

Some of them are based on theoretical considerations and others are derived from

experimental investigations. Due to a high level of complexity arising from the presence

of the two phases the practical models apply different simplifications which result in

certain models applicable only to specific flow patterns.

The simplest model used for describing a two-phase flow is the so called homogeneous

flow model. It treats the two-phase system as a homogeneous mixture with averaged

properties [1]. This model leads to relatively simple equations but it is applicable to a

limited number of systems in which the homogeneous equilibrium flow assumption can

be justified. The model predictions tend to strongly deviate from the experimental data as

the void fraction increases [2].

A little more sophisticated two-phase flow model is the drift flux model based on the

concept of a drift flux which uses a reference plane moving with a given velocity. The

flow of each phase is referred to this plane. This significantly simplifies the analysis and

allows a much wider range of flow patterns to be covered. The drift flux model shows a

good agreement with the experimental data for co-current and counter-current flow as

well as for stratified flow [2].

Taitel and Duler [3] combined the momentum balance equations for the liquid and the

gas phases into a single expression. The model accounts for the interfacial and fluid to

8

wall momentum interactions as well as for the effect of flow channel inclination. It is

especially well suited for the stratified flow pattern.

As a further extension of the stratified flow model, Sadatomi et al. [4] included the

interfacial level gradient into the momentum equations for both phases. The model

showed good agreement with the experimental data for some specific applications

involving stratified flow in large diameter horizontal pipes.

In addition to the more fundamental mathematical descriptions discussed above, some

simplified one-dimensional two-phase flow models applicable to a specific flow pattern

have also been formulated. One such model applicable to the bubbly flow was proposed

by Wallis [1]. The model can be applied to a wide range of void fractions. Another model

of this type is the one proposed by Nicklin et al. [5] and applicable to the slug flow.

The major limitation of the models discussed above is the fact that their application is

restricted to a given flow pattern. In order to cover a wider range of two-phase flow

conditions, Lockhart and Martinelli [6] proposed an empirical correlation relating the

void fraction to a parameter called the Lockhart-Martinelli parameter. This parameter is

defined as the square root of the ratio of the friction pressure drop of liquid to that of the

gas under the assumption that each phase flows alone in the channel. The Lockhart-

Martinelli approach has been widely applied to a large spectrum of two-phase systems

such as air-lift pumps [7, 8], two-phase heat exchangers, etc.

9

2.2 Bubble Formation and Propagation Bubble formation and propagation problems have been investigated extensively. A

pioneering study in this field was performed by Bretherthon [9] who investigated the

propagation of a long axisymmetric isothermal bubble of inviscid gas through a viscous

liquid. A solution was obtained under the assumption of strong surface tension effects

and it showed that the bubble dynamics is dominated by two ‘capillary-statics’ regions

located at both ends of the bubble. The analysis subsequently translates into a transitional

region in which the surface tension forces are balanced by the viscous ones. Finally a

third region is defined in which the viscous forces are dominant and this results in a thin

liquid film surrounding the gas bubble [9]. Wilson et al. [10] expanded Bretherton’s

analysis focusing on the unsteady mass-transfer expansion and contraction of a two-

dimensional vapor bubble located in-between subcooled or superheated plates. In a

subsequent study Kenning et al. [11] investigated the growth of a bubble in a capillary

tube under superheated conditions and proposed a mathematical model describing the

bubble generation and propagation under these conditions.

A quite comprehensive study on the growth and departure of bubbles from a submerged

needle was performed by Hassan et al. [12]. The study employed a simplified model for

the bubble growth based on the Rayleigh-Plesset equation. The results showed the

existence of two separate bubble growth regimes which depend on the rate at which the

gas flow is injected into the bubble. In the case of high velocity gas injection the bubbles

formed at the injection nozzle tend to elongate in the direction of the gas flow due to the

axial momentum of the gas. Under these conditions the bubble growth can be modeled

well enough by an ellipsoid expanding upwards [13].

Yuan et al. [14] defined a model for the vapor bubble growth and collapse in a small

channel connecting two reservoirs. The model was based on the potential flow theory and

did not account for the initial variations in the internal bubble pressure assuming that the

bubble dynamics is inertia controlled. The bubble growth in a confined space causes a

displacement of the surrounding liquid phase. This behavior can result in unidirectional

liquid transport upon careful timing of the bubble generation and collapse. Yuan at al.

10

[15] further investigated the possibility of existence of a net pumping action under these

conditions. The study showed that under certain system parameters it is possible for the

liquid transport to be initiated. Ory et al. [16] performed a similar study defining an

analytical model for the bubble growth and collapse in a narrow tube which showed a

good agreement with the experimental result. As a further investigation Alira et al. [17]

proposed a one-dimensional model incorporating the coupled heat transfer and phase

change phenomena in a narrow channel. The mathematical description was divided into

several time periods, covering the bubble generation, growth, collapse and channel refill

and provided an insight into the effect of the fluid viscosity on the system behavior.

2.3 Void Fraction Measurements The complex nature of the two phase flows introduces significant difficulties in the

experimental studies. The main parameters under investigation in these systems are

pressure drop, heat transfer coefficient, mass-transfer coefficient and void fraction. The

accurate measurement of the void fraction is especially important. The fraction of the

flow channel occupied by the gas or vapor phase is related to the flow regime of the two-

phase system and enters directly into the gravitational and acceleration terms of the

pressure drop calculations. In nuclear reactor engineering calculations the void fraction

is a major design parameter due to the fact that it affects the neutron absorption rate.

The most widely used method for void fraction measurements is based on the attenuation

of high energy electromagnetic waves (gamma or X-rays) passing through the flow

channel. The main difficulty associated with this method arises from the need to handle

radiation. In addition to this there is a significant error resulting from the gas-liquid

interface orientation, the effect of the tube wall, the effect of the temporal fluctuations,

etc. [18].

Another class of techniques employed in the void fraction measurements is based on the

fact that the electrical conductance and capacitance of the fluid depend on the

concentration of the phases. These methods are known as impedance methods. In order

for the readings to be accurate it is required for the electrical field between the electrodes

11

to be homogeneous, the electrodes to be located close enough and there should be no

discontinuity in the channel cross section [18].

The void fraction can also be measured by mounting fast acting valves at the entrance

and the exit of the flow channel of interest. By simultaneously closing both valves one

can easily measure the fraction of the volume occupied by each phase. The method is

precise but it is not applicable to online measurements [18].

In addition to the above mentioned techniques there are other methods such as acoustic

techniques, electro-magnetic measurements, optical methods, etc. [17].

2.4 CANDU Reactor Behaviour in Case of Circulation

Outage There is not much information available in the literature regarding CANDU reactors

under a loss of the forced circulation of the coolant through the core. Due to the

complexities of the system involved and the phenomena to be studied as well as the fact

that they are specific to the CANDU design only a limited number of papers have been

focused on this problem.

Previous experimental studies and analytical investigations showed that under a loss of

coolant circulation there are three modes of core cooling that can be induced – single-

phase thermo-siphoning, two-phase thermo-siphoning and intermittent buoyancy induced

flow (IBIF) [19, 20]. Feyginberg et al. proposed a lumped parameter model based on a

transient energy balance for a single channel in a CANDU reactor core. The model

divided the heat transfer in each channel into five time periods including preheating, local

saturation, venting and channel refill. Subsequently each stage was described in terms of

a transient heat balance. In order to validate the model the researchers used an

experimental setup in which a 6 m long channel was heated up and IBIF was induced.

The model showed good agreement with the basic trend observed during the

12

experimental studies but the predicted results were generally higher than the values

obtained from the experiment [20].

3. Experimental Design 3.1 Facility Design

The phenomenon under investigation is referred to as the “Intermittent Buoyancy

Induced Flow” (IBIF). In order to investigate this phenomenon resulting from a loss of

coolant circulation in a CANDU reactor core, an experimental set up was designed and

constructed. The experimental set-up simulates a CANDU reactor pressure tube as

schematically shown in Fig. 3-1. The boiling phenomenon was simulated by injecting air

bubbles into the stagnant water inside the pressure tube. The air was distributed into three

parallel lines and injected into the pressure tube through 12 nozzles mounted throughout

the length of the pressure tube.

In order to measure the air flow rate, one air velocity meter was mounted on each air line

(#4). The velocity meters used in the design were high precision ones with integrated

temperature and pressure correction. This meter allows the air flow rate to be measured

precisely. In addition, there were 12 measuring needle valves (#8) with a Vernier handle

mounted on each air injection nozzle. The power density in the actual CANDU pressure

tube is not uniform throughout the whole length. By using the needle valves the air

injection at each nozzle location could be precisely adjusted to closely simulate the real

power distribution.

A differential pressure transducer (DP) with a range of 6.87 kPa (1.0-psid) was installed

on each of the two vertical feeder pipes to measure the collapsed water level in each pipe.

Based on that, the instantaneous void fraction in the feeder pipes could be calculated. In

addition to this, two pressure transducers were added to both feeders, allowing the

changes in the water level inside the water tanks at the top of the feeder pipes to be

detected. This would allow the amount of water dragged by the air flow during its venting

to be measured.

13

Other instrumentation included three 0 - 25.4 cm (0 - 10 inch) water level differential

pressure (DP) transducers mounted at different locations throughout the length of the

pressure tube. They allowed the horizontal liquid level distribution and respectively the

void fraction inside the pressure tube to be measured. In addition, by comparing the

signals of two adjacent differential pressure transducers the propagation velocity of the

waves formed inside the pressure tube could be measured. There were 8 pairs of ball

valves mounted on the pressure tube which allowed the differential pressure transducers

to be used at different locations.

The two water tanks were connected through a pipe connection (#5). This allowed the

possible existence of a continuous IBIF to be studied. Through the valve (#6), the

connection between the two tanks could be interrupted which permitted the investigation

of different system configurations.

14

1. Water Tank 2. Feeder Tube 3. Pressure Tube 4. Air Distribution Line 5. Water Tank Pipe Connection 6. Valve

DP – Differential Pressure Transducer PT – Pressure Transducer F – Air Flow-Meter T - Thermocouple

Fig. 3-1 Experimental set-up – block diagram

1

Fe

Air Flow

1

D D

P

2

3

T

2

7

5 6

4

F F

44

8

F

D DD

9

15

Header Tanks

Ball Valve

Pipe Connection

Feeder

Air Injection

Feeder

3.2 Experimental Set-Up

3.2.1 Overall View

The test section was a close simulation of a real CANDU pressure tube. It was 9.0 meters

long made of 10.0 cm (4-inch) ID acrylic resin tubing. It consisted of six sections

connected to each other by flanges. Two 2.2-m long and 5.08 cm (2.0-inch) ID vertical

pipes were attached at the end sections of the pressure tube to simulate the feeder pipes.

Each feeder pipe was connected to a rectangular water tank with a 0.3 m3 volume,

simulating the header tank in the CANDU reactor.

Fig. 3-2 Experimental set-up – overall view

In the CANDU reactor two pressure tubes work in parallel forming a closed loop for the

heavy water coolant circulating between the reactor core and the steam generators. Under

these conditions there is a possibility for the existence of a continuous flow throughout

the loop in case of boiling inside the reactor core resulting from the loss of coolant

16

circulation. In order to investigate this both tanks were connected with a pipe with a valve

in-between. This connection represented a simplified model of the second pressure tube

working in parallel with the one being studied. This would allow us to study the possible

existence of the above mentioned phenomenon.

Fig. 3-3 CANDU nuclear reactor steam supply system [21]

Light Water Steam

Steam Generators

Primary Pumps

Calandria

Fuel Channel Assembly

Heavy Water Moderator

Heavy Water Coolant

Light Water Condensate

17

3.2.2 Pressure Tube and Fuel Bundles Design

The dimensions of the simulated pressure tube and the fuel bundles were selected as close

as possible to those of the real pressure tube. An acrylic resin was selected as the main

material of construction since its transparency would allow us to directly view and record

the two-phase flow phenomena taking place inside the system.

A comparison between the actual pressure tube used in a typical CANDU reactor and the

one used in the current design is presented in Figure 3-4.

a) Real pressure tube b) Simulated pressure tube

Fig. 3-4 Photograph of a CANDU pressure tube with a fuel bundle placed inside (Ref.

Report # N-REP-03500.2-10002, page 61, Darlington NGS)

Thirteen simulated fuel bundles were fabricated from acrylic rods and placed inside the

pressure tube. Each bundle was composed of 37 acrylic rods with 12.7-mm OD

simulating a 37-Fuel Rod Assembly. The acrylic rods were attached to actual zircaloy

end plates provided by Bruce Power Inc. A photograph of the fabricated fuel bundle is

shown in Fig. 3-5.

Zircaloy end plate Fuel Rod 

18

Fig. 3-5 Photographs of the simulated fuel bundle used in the current design

The dimensions of the simulated pressure tube and fuel rods are summarized in Table 3-1

and compared to those of the CANDU reactor. As it can be seen from this Table the

dimensions of the simulated pressure tube and fuel bundles were chosen as close as

possible to the real ones in order to more accurately simulate the IBIF phenomena in the

real system.

Table 3-1. Comparison of test section and CANDU reactor component dimensions

Simulated Actual Pressure Tube ID, mm 101.6 102.4 Fuel Bundle OD, mm 87.9 99.7 Fuel Rod OD, mm 12.7 13.08

19

4. Experimental Results and Discussion 4.1. Preliminary Observations A schematic of the flow phenomena studied is shown in Fig. 4-1. The approach employed

in the current experimental study was to investigate the effect of different process

variables on the two-phase flow behavior. The main parameters that were varied included

the air-injection rate which corresponded to the simulated reactor power level, the water

level inside the feeder pipes and the depth of sagging of the pressure tube in the middle

from a horizontal position. A system of pressure transducers and video cameras were used

to collect the data. This information was subsequently processed and analyzed to present

the results.

Fig. 4-1 Schematics of the experimental set-up

Preliminary studies showed that the air bubbles injected into the pressure tube rapidly

rose towards the top, merged together and formed a continuous layer rather than

remaining as discrete bubbles. The observed behavior is illustrated in Fig. 4-1 and shown

in Fig. 4-2. Air then flowed along the pressure tube towards the end section and vented

into the feeder pipes forming a slug flow in the feeder pipes (Fig. 4-3).

Water Tank

Pressure Tube

Water Tank 

Air Injection

20

a) bubble merging b) air-layer formation

Fig. 4-2 Photographs of the system behavior upon air injection

Fig. 4-3 Photograph of a gas slug rising in the vertical feeder

21

4.2. Effect of Air Injection Nozzle Location The goal of this series of experiments was to investigate the effect of air injection

location on the gas venting time. Air was injected at different locations throughout the

length of the pressure tube (Fig. 4-4) and the time interval between the start of the air

injection and air venting was measured. In Figure 4-5 are shown two photographs of the

bubble propagation front taken at a 65 ms time interval between the frames. In order for

the results to be consistent the air injection rate as well as the water levels inside the

water tanks were kept constant for each run. The only parameter which was varied in

different runs was the location of the air injection. Each measurement presented is an

average of five separate runs which allows the error to be minimized. The experiments

were performed at three different simulated power levels and the results are summarized

in Table 4-1 and Fig. 4-6.

Fig. 4-4 Schematics of the experimental set-up – effect of the air injection location

Air Injection

Venting DistanceWater Tank

Feeder Tube

Pressure Tube

22

Fig. 4-5 Consecutive photographs of the bubble propagation front

(65 ms time interval between the frames)

Table 4-1. Effect of the air injection location on the venting time

Simulated Power

Level(SPL), kW

Average Venting Time, s

Venting Distance, cm

Bubble Expansion

Velocity, cm/s

Average Bubble

Expansion Velocity, cm/s

5.55 345 62.16 4.06 258 63.5 1.1

2.46 150 60.97 62.2

5.37 345 64.25 4.03 258 64.02 1.5 2.38 150 63.03

63.8

5.11 345 67.51 3.92 258 65.82

2.0 2.24 150 66.96 66.8

Bubble Front Bubble Front

23

40

50

60

70

80

100 150 200 250 300 350 400

Average VelocitySeries1Series3Series4

40

50

60

70

80

100 150 200 250 300 350 400

AverageSeries1Series3Series4

a) SPL – 1.1 kW

b) SPL – 1.5 kW

Bub

ble

Exp

ansi

on V

eloc

ity, c

m/s

Venting Distance, cm

Venting Distance, cm

Bub

ble

Exp

ansi

on V

eloc

ity, c

m/s

Average Velocity

Measured Velocity

Average Velocity

Measured Velocity

24

40

50

60

70

80

100 150 200 250 300 350 400

AverageVelocitySeries1Series3Series4

c) SPL – 2.0 kW

Fig. 4-6 Effect of air injection location on bubble expansion velocity for different

Simulated Power Levels (SPL)

The observed system behavior showed that the air bubbles introduced into the pressure

tube rapidly rose towards the top and formed a continuous layer rather than remaining as

discrete bubbles. Air then flowed along the pressure tube towards the end section and

vented into the feeder pipes forming a slug flow in the feeder pipes. The current results

showed only slight variations of the bubble expansion velocity as a function of the air-

injection location. The experiments were performed at three different injection rates and

the results were consistent. Therefore, we can consider the bubble front to move with a

constant velocity.

Venting Distance, cm

Bub

ble

Exp

ansi

on V

eloc

ity, c

m/s

Average Velocity

Measured Velocity

25

345 cmWater Tank Feeder Tube

Pressure Tube

4.3. Combined Effect of Feeder Water Level and Air-

Injection Rate on the Venting Time

The goal of this series of experiments was to investigate how the water level inside the

feeder pipes and the rate of air injection would affect the venting time. The experiments

were performed at three different simulated power levels and at nine different water

levels in the feeder pipes. Following the results from the previous experiments which

showed that the bubble would not accelerate along the pressure tube, all the tests were

performed by injecting air through a single injection point. The venting distance was kept

constant and equal to 345 cm (Fig. 4-7). The total number of runs was 135 and each data

point is an average of five tests performed under the same conditions. The results are

presented graphically in Fig. 4-8.

Fig. 4-7 Schematic of the experimental set-up to investigate the combined effect of feeder

water level and air-injection rate

Water Level

Air Injection

26

3.5

4

4.5

5

5.5

6

6.5

45 95 145 195 245

SPL - 1.1 kWSPL - 2 kWSPL - 2.85 kW

Fig. 4-8 Venting Time as a function of water level in the feeder line at different simulated

power levels

As expected, the experimental results showed an increase in the venting time with an

increase in the water level inside the feeder pipes and with the decreasing simulated

power level, i.e. the air injection rate. The experimental data were subjected to a

multivariable regression analysis in order to derive a correlation between the parameters

under investigation.

T = 3.54 - 0.177*SPL + 0.011*WL (4-1)

where T = venting time [s]

SPL = simulated power level at the point of injection [kW]

WL = water level in the feeder line [cm]

The maximum absolute deviation of the data from the correlation was MaxErr = 0.68 [s].

By taking into account the venting distance we can express the bubble expansion velocity

(BEV) as a function of the SPL and WL as follows.

BEV = 0.10 -1.95*SPL + 30.4*WL (4-2)

where BEV = bubble expansion velocity [cm/s]

Feeder Line Water Level, cm

Ave

rage

Ven

ting

Tim

e, s

27

4.4. Effect of Multiple Air Injection on the Venting Time

As it was noted earlier, the power density in the real pressure tube is not uniformly

distributed throughout its length. The heat generation rate is higher in the central sections

of the pressure tube and lower at the ends. A typical axial heat flux distribution in a

CANDU Reactor pressure tube is presented in Fig. 4-9.

Fig. 4-9 A typical CANDU reactor pressure tube axial heat flux distribution [21]

Under these conditions it is expected that in the event of a loss of coolant circulation, the

steam generation would initially occur at the centre of the pressure tube and subsequently

in the other sections. The goal of the current experiment was to investigate how the

venting time would be influenced if the steam generation occurs simultaneously at

several different locations.

The experiments were conducted with two or three simultaneous air injections. In each

case the total air injection rate was kept the same and equal to SPL = 1.1 kW and the

water level in the feeder pipes equal to WL = 234 cm. The results could be compared

with those of the single injection tests.

For each case twenty runs were conducted at three different locations of the air injection

points. The data were analyzed to determine how the relative distance between the

injection points would affect the venting time.

Rel

ativ

e H

eat F

lux,

loca

l/ave

rage

Relative Location, x/L

28

4.4.1. Air Injection through Two Air-Injection Nozzles The two air-injection locations used are shown in Fig. 4-10. The first air-injection

location was fixed at the middle of the pressure tube and the second air-injection was

varied.

Fig. 4-10 Schematic of the experimental set-up for air injection through two air- nozzles

The results shown in Table 4-2 indicate that the simultaneous air injection leads to a

decrease in the venting time. The decrease is more significant when air is injected

through nozzles which are located close to each other. One explanation for the observed

behavior can be the fact that the bubbles formed above the two injection points interacted

with each other. This interaction restricted the bubble expansion towards each other and

the bubbles predominantly expanded towards the feeder tubes. This behavior is expected

to enhance the pressure tube venting since in the real reactor the steam bubbles will be

first formed predominantly in the centre of the pressure tube.

345cm Water Tank Feeder Tube

Pressure Tube Variable Distance

WL= 234cm

Air Injection 1Air Injection 2

29

Table 4-2 Venting time data for air injections through two air- nozzles

Venting Distance, cm

Venting Time (dual injection), s

Venting Time (single injection), s

Change in the Venting Time, %

307 4.6 5.41 -14.9 258 3.95 4.55 -13.2 200 3.06 3.52 -13.1 150 2.39 2.64 -9.5

4.4.2. Air injections through Three Air-Injection Nozzles An experimental schematic is shown in Fig. 4-11 and the results are summarized in Table

4-3. The same venting distance from the feeder pipes was used for the second and third

air-injection locations.

Fig. 4-11 Schematic of the experimental set-up – air injections through three air- nozzles

Table 4-3 Venting time for air injections through three air- nozzles

Venting Distance, cm

Venting Time (triple injection), s

Venting Time (single injection), s

Decrease in the Venting Time, %

307 4.37 5.41 -19.2 258 3.86 4.55 -15.2 200 3.01 3.52 -14.5 150 2.31 2.64 -12.5

345cm Water Tank Feeder Tube

Pressure Tube

Variable Distance

WL= 234cm

Air Injection 1

Air Injection 2Variable Distance

Air Injection 3

30

The experimental results show that the simultaneous injection through three nozzles

further reduces the venting time. Similar to the two air-injection location experiments,

with an increase in the relative distance between the injection points the effect of the

multiple air injection gets weaker. It can be expected that the addition of more injection

points will lead to a further decrease in the venting time which is expected to enhance the

pressure tube venting in the CANDU reactor core due to the significant number of

nucleation sites existing on the fuel rods inside the pressure tube during the loss of

coolant circulation event.

4.5. Effect of Pressure Tube Sagging on the Venting Time The goal of this experiment series was to investigate how the pressure tube sagging

would affect the venting time. In order to study this phenomenon we gradually lowered

the supports of the pressure tube at the centre as shown in Figs. 4-12 and 4-13. As a result,

it was possible to achieve smooth sagging of the tube in the middle between the two ends.

This configuration is considered to be very close to the one in the CANDU reactor.

The experiments were conducted by injecting air through a single nozzle keeping the

water level in both feeder pipes constant and equal to 234 cm. The air injection rate was

also kept constant and equal to an equivalent of 1.1 kW of Simulated Power Level. The

experimental results are summarized in Table 4-4. Each data point is an average of five

runs performed under the same conditions.

Fig. 4-12 Schematic of the experimental set-up to study the effect of pressure tube

sagging

345cmWater Tank Feeder Tube

Pressure Tube

Air Injection

WL= 234cm Sagging Distance

Inclination Angle

31

Fig. 4-13 Photograph of the inclined pressure tube (sagging of 5.08 cm (2”) in the mid

section)

The experimental results are summarized in Table 4-4 and Fig. 4-14. The average venting

time data showed that even small sagging of the pressure tube in the middle by 12.52 mm

(0.5 inch) could cause a significant decrease in the venting time by 8% as compared to

the horizontal pressure tube. Upon further increases in the pressure tube sagging the

venting time was further reduced, however, the effect became less significant as the depth

of sagging was increased to 1.0 and 2.0-inches (25.4 and 50.8 mm). Although it was not

possible to perform additional experiments at greater depths of sagging due to the risk of

fracturing the pressure tube at the midpoint, the venting time is expected to continue to be

reduced as shown by the trend seen in Fig. 4-14.

Table 4-4. Effect of the pressure tube sagging

Sagging Distance,

Inch

Inclination Angle,

deg

Venting Time (sagged tube),

s

Venting Time (horizontal tube),

s

Decrease in the Venting Time,

% 0.5 0.16 5.57 8 1.0 0.32 5.40 11 1.5 0.43 5.28 13 2.0 0.65 5.23

6.08

14

Horizontal Line Inclined Pressure Tube

32

5.25.35.45.55.65.75.85.9

66.1

0 0.5 1 1.5 2

Fig. 4-14 Effect of pressure tube sagging on the venting time

4.6 Air-Lift Effect The goal of this set of experiments was to investigate the possibility of inducing a

continuous IBIF when the two header tanks are connected as shown in Fig. 4-15. In order

to perform the experiments, air was injected asymmetrically through 5 nozzles so that the

injection was dominant in the left half of the pressure tube. This way the venting would

occur preferentially through the left feeder pipe.

Fig. 4-15 Schematic of the experimental set-up for studying the air-lift effect

Sagging Distance, inch

Ave

rage

Ven

ting

Tim

e, s

L1

Feeder Tube

L1

Asymmetrical Air Injection

Water Tank

Pressure Tube

UF

33

Under these conditions the air venting caused an increase in the water level inside the left

header tank due to the air lift effect caused by the venting air on the stagnant water inside

the system. It was noticed that this behavior was similar to the one of an air-lift pump [7,

8]. As the total amount of water remained constant, an increase in the water level in one

of the headers resulted in a decrease in the water level in the other header. The difference

in the water levels between the two headers caused a continuous flow from the left to the

right header (or vice versa) through the pipe connecting the two header tanks. This

system configuration resulted in a continuous flow of water circulating throughout the

system, so an ultrasonic flow meter (UF) was used to measure the flow rate of water in

the connecting tube.

It was observed that when the initial water level inside the header tanks was low, the

results of the experiments were altered. When the water level is low the water in the non-

venting tank gets completely depleted which does not allow for the continuous flow of

water to be sustained since the venting starts occurring simultaneously through both

feeders. The experiments were performed at 6 different initial water levels and 5 different

air-injection rates, i.e. simulated power levels. The results are presented in Table 4-5 and

Fig. 4-16, where each data point represents an average of 3 repeated experiments

performed under exactly the same conditions. A total of 90 experiments were conducted

in this experiment.

Table 4-5 Air-Lift Effect on Flow Velocity (m/s)

0.7 1.6 1.8 2.5 4.5

34 0.53 0.61 0.67 0.73 0.95 30 0.56 0.64 0.68 0.69 0.87 26 0.51 0.59 0.66 0.71 0.88 22 0.52 0.62 0.64 0.75 0.92 18 0.59 0.61 n.a. n.a. n.a. 16 n.a. n.a. n.a. n.a. n.a.

Simulated Power Level

kW

  Header Tank

Initial Water Level, cm

 

34

The results showed that when the initial level of water inside the headers is high enough

the induced flow velocity depends linearly on the air injection rate.

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

0.5 1 1.5 2 2.5 3 3.5 4 4.5

Initial Water Level - 34 cmInitial Water Level - 30 cmInitial Water Level - 26 cmInitial Water Level - 22 cmInitial Water Level - 18 cm

Fig. 4-16 Air-lift effect on flow velocity

With an increase in the air injection rate, more water was transported to one of the header

tanks resulting in a higher water level (L1) inside that tank and respectively a lower level

(L2) in the other tank. This caused a higher static pressure difference between the two

header tanks which resulted in a higher flow velocity.

Table 4-6 Air-lift effect on header water level

Simulated Power Level,

kW

Flow Velocity (cm/s)

L1 (cm)

L2 (cm)

∆L (cm)

0.7 0.53 32 22 10 1.6 0.61 35 23 12 1.8 0.67 36 22 14 2.5 0.73 37 20 17 4.5 0.95 40 13 27

Simulated Power Level, kW

Flow

Vel

ocity

, m/s

35

When the initial amount of water inside the water tanks was reduced to lower than 18 cm

from the tank bottom the venting started occurring simultaneously through both feeders.

Under these conditions the continuous flow could not be sustained between the two

header tanks any longer.

4.7 Oscillatory behavior Upon symmetrical air-injection throughout the pressure tube, the air layer that had

formed on top of the stagnant water in the pressure tube propagated simultaneously to

both feeders. Once the air front reached the feeder pipes the venting of air occurred. Due

to the fact that complete symmetry could not be achieved the venting actually occurred

through one of the feeders slightly earlier than through the opposite feeder pipe. Once the

air injection was initiated the venting would occur through one of the feeders on a

random basis. The air venting caused water accumulation in the header tank due to the

air-lift effect. As a result, the water level in this header tank would increase and the level

in the opposite tank would decrease. This means that a net amount of water is being

transported between the header tanks. As the water level in the header tank increases the

hydrostatic pressure in the corresponding part of the pressure tube increases, and the

pressure in the opposite header tank decreases. This process continues until a critical

water level is reached in the header tank, and the venting process switches directions and

starts occurring through the opposite feeder. Afterwards the switching process repeats

itself. This oscillatory behavior can be characterized by a certain frequency of oscillations.

The goal of this series of experiments was to investigate how the air injection rate

and the initial water level inside the header tanks would affect the oscillation frequency.

The experiments were performed by injecting air at five different air injection rates for

five different initial water levels inside the feeder line. The system behavior was observed

and the frequencies of oscillations were recorded by measuring the time interval between

switching. The experimental results are presented in Table 4-7 and Fig. 4-17. A total of 75

experiments were performed and each data point represents an average of five runs

performed under the same conditions.

36

0.0059

0.006

0.0061

0.0062

0.0063

0.0064

0.0065

0.0066

220 222 224 226 228 230 232 234 236

Table 4-7 Effect of water level on the frequency of oscillations (SPL 1.1 kW)

Feeder Line Water Level,

Cm

Period of Oscillations,

s

Frequency of Oscillations,

Hz 234 165.9 0.0060 230 159.4 0.0062 226 155.4 0.0064 222 152.5 0.0065 210 n.a. n.a.

Fig. 4-17 Effect of water level on the frequency of oscillations (SPL 1.1 kW)

The experimental results show that with a decreasing water level in the feeder line the

frequency of oscillations increases following almost a linear trend. This flow behavior is

kept until the water level inside the header tanks is significantly low. At a water level of

210 cm inside the feeders the system started to vent preferentially through one of the

feeders and the oscillating pattern was not observed anymore. Under these conditions the

water level inside the feeders would change fast enough to balance the hydrostatic

pressure difference between both ends of the pressure tube. As a result, once the venting

occurs through one of the feeders it does not tend to switch directions.

The second part of this experimental study was to investigate the effect of the air

injection rate (simulated power level) on the observed frequency of oscillations. The

Feeder Line Water Level ,cm

Freq

uenc

y of

osc

illat

ions

, Hz

37

results are summarized in Table 4-8 and plotted in Fig. 4-18.

Table 4-8 Effect of air injection rate (SPL) on the frequency of oscillations

Simulated Power Level,

kW

Period of Oscillations

(s)

Frequency of Oscillations

(Hz) 0.5 118.7 0.0084 0.7 153.7 0.0065 1.1 159.4 0.0062 1.3 178.8 0.0056

00.0010.0020.0030.0040.0050.0060.0070.0080.009

0.4 0.6 0.8 1 1.2 1.4

Fig. 4-18 Effect of the air injection rate (SPL) on the frequency of oscillations (Feeder

Line Water Level – 230 cm)

The experimental results show that with an increase in the SPD or air injection rate, the

observed frequency of oscillations decreases. This behavior results from the fact that at

higher air injection rates the liquid is being accumulated faster inside the header tank.

This reduces the time interval between the subsequent switches in the venting direction.

The observed oscillatory behavior results from the air lift effect exerted by the air flowing

upward in the feeder pipes on the stagnant water inside the pressure tube. The air drag

causes water accumulation inside the header tank through which the venting occurs. This

results in an increase in the hydrostatic pressure in the corresponding part of the pressure

Simulated Power Level

Freq

uenc

y of

osc

illat

ions

, Hz

38

tube. Under these conditions it is expected that the size of the header tank will affect the

system behavior. A smaller tank should result in faster water accumulation which will

lead to a higher frequency of oscillations.

In order to validate this hypothesis, the above experiment was performed again with a

tank which had an internal volume ten times smaller. The results presented in Table 4-9

show significant increases in the frequency of oscillations for the smaller header tanks.

Table 4-9 Effect of air injection rate (SPL) on the frequency of oscillations (Ten fold

decrease in the header tank volume)

Simulated Power Level,

kW

Period of Oscillations

(s)

Frequency of Oscillations

(Hz) 0.5 28.6 0.0350 1.1 34.2 0.0292

The results of the experiment showed that the volume of the header tank has a significant

effect on the frequency of oscillations. This leads to the conclusion that in the actual

reactor the frequency of oscillations which would result from the air lift exerted by the

venting steam through the pressure tube would be relatively low due to the extremely

large volume of the header tanks.

39

4.8 Summary of Experimental Results The experiments conducted were aimed to cover the whole spectrum of IBIF phenomena

occurring upon a loss of coolant circulation accident. The first experiment was focused

on investigating the effect of air injection location on the bubble front propagation

velocity. The experimental results showed minor differences in the bubble velocity upon

changing the air injection location which allowed for the bubble velocity to be assumed

constant. As a further step, the effects of the air injection rate and the initial water level

inside the feeder lines were investigated. The collected data showed an increase in the

bubble propagation velocity with an increase in the air injection rate and decrease in the

feeder line water level. Since in the actual reactor, the steam generation occurs

throughout the whole length of the fuel bundles, an experimental study focusing on the

effect of simultaneous air injection through multiple points was also performed. The

results showed a decrease in the venting time due to the interaction between the adjacent

expanding bubbles. The effect was stronger when the air injection was through nozzles

located close to each other. Other findings are summarized as follows.

• The pressure tube sagging was found to significantly decrease the venting time.

This behavior is expected to enhance the venting since the pressure tubes in the in

actual reactor tend to sag with the year of service.

• The possibility of inducing continuous flow throughout the system was

investigated by connecting the two header tanks. Such a flow could be induced

and sustained within certain limits of the operating parameters. Considering the

fact that in the actual reactor a large number of pressure tubes are connected

together this phenomenon is expected to have a positive effect on the heat

removal in the event of a loss of coolant circulation.

• The final experiments studied the phenomena resulting from the air lift effect

exerted by the venting air onto the stagnant water inside the pressure tube. The

system showed oscillatory behavior characterized by alternating directions of air

venting. The frequency of oscillations was observed to depend on the air injection

rate, the initial amount of water inside the system and on the volume of the

venting tanks.

40

5. Mathematical Model In this Chapter, a mathematical model of the flow of liquid in the pressure tube

containing a fuel bundle as illustrated in Fig. 5-1 is developed and the model predictions

will be compared with the measurements.

Fig. 5-1 Diagram of the modeled two-phase system

5.1. Mathematical Description The mathematical description of the system is based on the model proposed by Taitel and

Dukler (1975). In their work they formulated a simplified momentum balance for each

phase as follows:

Liquid phase:

(5-1)

Gas phase:

(5-2)

Gas Phase

Liquid Phase Liquid-Solid

Interface

Gas-Solid Interface

Inclination Angle

Pressure Tube

0)( =+−− iiLWLL SSdxdPA ττ

0)( =−−− iiGWGG SSdxdPA ττ

Gas-Liquid Interface

Fuel Rod

41

0g.sinASS)dxdP(A LLiiLWLL =θρ−τ+τ−−

where:

The equations (5-1) and (5-2) can be equated assuming that the pressure gradient and the

magnitudes of the interfacial drag for both phases are equal. This assumption has been

shown to be reasonable for most two-phase flow regimes [22, 23].

In order for the effect of the pressure tube sagging to be included in the above

expressions an additional term accounting for the gravitational effect must be added to

equations (5-1) and (5-2). With this modification the momentum balance for each phase

can be written as follows:

Liquid phase:

(5-3)

Gas phase:

(5-4)

By equating equations (5-3) and (5-4) the overall momentum balance for the two-phase

flow can be written in the following form

(5-5)

where:

L

G

L

G

W

W

i

AASS

L

G

τ

ττ

, gas-surface shear, kg/m.s2

, gas-liquid shear, kg/m.s2

, length of gas-solid interface, m

, length of liquid-solid interface, m

, gas cross-section, m2 , liquid cross-section, m2

, interfacial shear, kg/m.s2

0.sin)( =−−−− gASSdxdPA GGiiGWG G

θρττ

( ) 0g.sin.A1

A1.S.

AS

.AS

. GLLG

iiL

LW

G

GW LG

=θρ−ρ−⎟⎟⎠

⎞⎜⎜⎝

⎛−τ+τ−τ

g

G

L

θρρ , liquid density, kg/m3

, gas density, kg/m3 , inclination angle, deg

, gravitational acceleration, m/s2

Si, length of gas-liquid interface, m

42

The shear stresses in the above expressions are evaluated in a conventional manner as

follows:

2

2LL

LWuf

L

ρτ = (5-6)

2

2GG

GWuf

G

ρτ = (5-7)

2)( 2

LGGii

uuf −=

ρτ (5-8)

where:

Further information is required for the friction factors GL ff , and if . These

parameters are evaluated using the Blasius-type equations [22]:

n

L

LLLL

uDCf −= )(υ

(5-9)

m

G

GGGG

uDCf −= )(υ

(5-10)

where LD and GD are the hydraulic diameters. The liquid is presented as if it flows in an

open channel and with this assumption LD can be evaluated as follows:

LLL SAD /4= (5-11)

The gas phase is assumed to flow in a closed duct and under these conditions GD is

evaluated as follows:

)/(4 iGGL SSAD += (5-12)

i

L

G

G

L

fffuu , liquid velocity, m/s

, gas velocity, m/s , gas friction factor

, interfacial friction factor , liquid friction factor

43

The main difficulty in applying the mathematical model described above comes from the

need to evaluate the interfacial friction factor, if . There were a number of studies in

which different correlations for evaluating if have been proposed [21]. Andritos et al.

suggested that the ratio between the interfacial friction factor and the friction factor for

the gas is almost unity if there are no waves in the system as illustrated in Fig. 5-2.

a) b)

Fig. 5-2 Diagram of the gas liquid interface: a) no waves; b) wavy interface

In the case of roll waves existing on the gas-liquid interface, the ratio between both

friction factors is no longer close to unity. Previous experimental studies showed that

these waves appear above a critical superficial gas velocity of approximately 5 m/s at

atmospheric pressure [22]. They proposed the following correlation (5-13) for the

g

if

f ratio which has been employed in the current model,

)1()(151,

5.00 −+=tG

G

G

i

UU

Dh

ff (5-13)

where:

tGUDh

,

0

The superficial gas velocity is defined as the ratio between the volumetric flow rate of the

gas and the cross sectional area of the tube:

, liquid phase height, m , tube diameter, m , critical superficial gas velocity, m/s

0h

Gas Phase

Liquid Phase

Gas-Liquid

Interface (flat) 0h

Gas Phase

Liquid Phase

Gas-Liquid

Interface (wavy)

44

tube

gG A

VU

.

= (5-14)

where:

tube

g

A

V.

The critical superficial gas velocity is specified as tGU , = 5 m/s.

From equation (5-12) it follows that if the superficial gas velocity is lower than the

critical velocity, the ratio g

if

f is very close to unity. When the superficial gas velocity

starts to approach the critical velocity, roll waves are expected to appear. Under these

conditions the gas-liquid interface is no longer flat and this is accounted for by the second

term in equation (5-13) which becomes significant.

5.2. Numerical solution of the mathematical model

All the above mentioned parameters should be introduced into equation (5-5). By using

the experimental data obtained for the void fraction the model can be solved for the liquid

phase velocity.

5.2.1. Calculating the basic geometric parameters

The main difficulty with the solution of equation (5-5) comes from the complexity of the

geometry under study as illustrated in Fig. 5-3. The change in the water level inside the

pressure tube results in a change in the length of the gas-liquid interface (Si), gas-solid

wall (SG/S) and liquid-solid wall (SL/S) interfaces as well as the gas (AG) and liquid (AL)

cross sectional areas.

, volumetric gas flow rate, m3/s , tube cross sectional area, m2

45

Frig. 5-3 Schematic of the simulated pressure tube with 37 acrylic fuel rods

The complexity of the geometry requires a numerical solution for the change in the above

mentioned parameters: Si, SG/S, SL/S, AG, AL to be calculated as a function of the water

level inside the system. Due to the symmetry of the system the solution was obtained

only for the right half of the pressure tube cross section. The equations for the eighteen

and a half rods placed inside were solved simultaneously with the equation of the line of

the water level inside the tube.

Y

Pressure Tube

Acrylic Rod

12

3

4

5

6

78

9

1011

12

13

14 15

16

17

18

19

X

46

5.2.1.1 Calculating the gas-liquid interface length, Si

a) Length of the gas-liquid interface (assuming there are no rods inside)

Fig. 5-4 Schematics of the gas-liquid interface

Depending on the height of the water level h0 inside the pressure tube (Fig. 5-4) the

length of the interface BD was calculated as follows:

For Rh 0 ≤ :

20

2 )hR(R.2BD −−= (5-15)

For Rh 0 > :

20

2 )Rh(R.2BD −−= (5-16)

•A

R

h0

•  • • B

C

D γ

47

b) Length of the gas-liquid interface (accounting for the presence of the rods)

Fig. 5-5 Schematic of the gas-liquid interface with the rods placed inside

The point of intersection between the rod and the line corresponding to the water level

inside the pressure tube as shown in Fig. 5-5 was obtained by solving simultaneously the

equation of the circle describing the given rod with the equation of the line of the water

level. If the centre point of the fuel rod circle is G (x0, y0) then the solution for a single

rod placed inside the pressure tube will require a simultaneous solution of the following

system of two equations:

Equation of the circle positioned at G(x0, y0):

20

20 r)yy()xx( =−+− (5-17)

Equation of the line representing the average water level inside the tube:

0hy = (5-18)

The simultaneous solution of equations (5-17) and (5-18) will give us the points of

intersection between the line 0hy = and the circle – pt. E and F. In this case the length of

the interface will be determined as follows:

EFBDSi −= (5-19)

The same approach was applied to all the rods placed inside the pressure tube. The

geometry was solved for the right half of the pressure tube cross section. This required

•A

R

h0

•  • • B

C

D γ

•  • E F • 

r

• G

48

for the nineteen equations of the circles representing the fuel rods on the right side of the

tube (Fig. 5-3) to be solved simultaneously with the equation of the line representing the

water level inside the pressure tube. As a result of this, real solutions can be obtained only

for the rods for which the water level line intersects the circles.

5.2.1.2. Calculating the liquid-solid interface length, SL/S

The liquid/solid interface includes the wetted perimeter, WP , of the pressure tube inner

wall plus the sum of all the wetted perimeters of the rods inside the pressure tube, ∑ iRP .

5.2.1.2.1 Calculating the wetted perimeter, WP , of the

pressure tube inner wall

The tube wall wetted perimeter, WP is equal to the length of the arc BED :

γ== .RBEDPW (5-20)

where R is the tube radius, γ [rad] is half the angle determined by the points of

intersection between the water level line, the pressure tube wall and the centre of the

pressure tube.

Fig. 5-6 Wall wetted perimeter schematics

•A

R

h0

•  • • B

C D

γ

• E

49

The length of the wetted perimeter depends on the angle γ which is determined by the

height of the water level inside the pressure tube. Depending on the height of the water

level h0 with respect to the middle of the tube the angle γ was calculated as follows:

For Rh 0 ≤ :

( )⎥⎦⎤

⎢⎣⎡ −=γ R

hRarctan.2 0 (5-21)

For Rh 0 > :

( )⎥⎦⎤

⎢⎣⎡ −−π=γ R

Rharctan.2.2 0 (5-22)

5.2.1.2.2 Calculating the wetted perimeters of the rods inside

the pressure tube, ∑ iRP

The calculation of ∑ iRP requires initially the number of fuel rods in the bundle which

are entirely submerged in the water. This calculation involves counting the number of

circles for which the following condition is met:

rjh0 +≥ (5-23)

where j is the ‘y’ coordinate of the centre points )j,i(O of a given circle corresponding to

the fuel rod placed inside the pressure tube.

After determining the total number of rods which are completely submerged, the

liquid/solid interface needs to be calculated for the rods which are partially submerged.

This calculation uses the coordinates of the points obtained from the simultaneous

solution of the equation of the line representing the water level with the equations of the

circles representing the rods inside the pressure performed in section 5.2.1.1.

For each circle for which the simultaneous solution generates real results, there are points

of intersection between the water line and the circumference of the circle. In this case the

length of the interface is determined as the product of the radius of the rod and the

corresponding central angle. For example, if we have a single rod which is partially

50

submerged as illustrated in Fig. 5-7, the length of the arc HKF is calculated as follows:

β== .rHFPiR (5-24)

where r is the radius of the rod, and β is the central angle, [rad].

Fig. 5-7 Schematic of the wetted perimeter calculation with rods placed inside

Analogous to 5.2.1.2.1 the value of the central angle is determined based on the

relative position of the water level with respect to the center point G(i,j) of the circle

representing the fuel rod.

For jh0 ≤ :

( )⎥⎦⎤

⎢⎣⎡ −=β r

hrarctan.2 0 (5-25)

For Rh 0 > :

( )⎥⎦⎤

⎢⎣⎡ −−π=β r

rharctan.2.2 0 (5-26)

• A

R

h0

•  • • B

C

D γ

•  • E F • 

r

• G

h0

G • 

•  • 

β

H F

r

K • 

51

5.2.2. Numerical Solution of the model For the purpose of the current study, the mathematical model was solved by employing

the Newton-Raphson method. This technique is widely used for locating roots of

nonlinear functions which is the case with our model. Similar to the majority of

numerical techniques, the Newton-Raphson method uses a Taylor series expansion of the

function under investigation f(x) around a given point xi:

( ) ( ) ( )( ) ( )( ) ( )( ) ...!3

xxxf!2

xxxfxxxfxfxf3

ii'''2

ii''

ii'

i +−

+−

+−+= (5-27)

We can linearize the function f(x) by taking into account only the first two terms from the

Taylor series expansion. With this approximation the above equation can be rearranged

and solved for the value of x which leads to the following result:

( )i'

ii xf

)x(fxx −= (5-28)

Equation (5-27) can be rewritten in an iterative form as follows:

( )i'

ii1i xf

)x(fxx −=+ (5-29)

In this form the new value of xi+1 can be calculated from the old value xi by correcting it

with the expression ( )( )i

'i

xfxf . This iterative procedure is applied until the difference

between the values of x in two consecutive iterations converges to some initially

specified value ε, which is the convergence criterion:

ε≤−+ i1i xx (5-30)

The Newton-Raphson technique is graphically represented in Fig. 5-8.

Fig. 5-8 Newton-Raphson method - graphical representation

x2 x3

x•

x1

f(x1)

f(x2)

f(x3) f(x•)

52

( )

( )

( ) ( )L

L)n1(

LLn

L

nLL

LGiGiGL

'

GL

LGi

2LGGi

L

L2LLL

G

G2GGG

AS

.2u.

).n2.(D.C

A1

A1.S..f.uuxF

g.sin.A1

A1S.

2)uu.(.f

AS

.2

u..fAS

.2

u..fxF

−

−

− ρ−

υ−⎟⎟

⎠

⎞⎜⎜⎝

⎛−ρ−=

θρ−ρ−

−⎟⎟⎠

⎞⎜⎜⎝

⎛−

−ρ+

ρ−

ρ=

As it can be seen from the above figure every subsequent value of x is determined by the

tangent of the function at the previous value of x. For the current model the function

which is supposed to be solved and its derivative are as follows:

(5-31)

(5-32)

53

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

40.00

0 0.2 0.4 0.6 0.8 1

5.3 Calculation Results The above model was solved in the range of void fractions from 0 to 1 and the results are

presented in the figures below.

0.00200.00400.00600.00800.00

1000.001200.001400.001600.001800.002000.00

0 0.2 0.4 0.6 0.8 1

Liquid-Surface InterfaceGas-Surface Interface

Fig. 5-9 Variations of the liquid-solid and gas-solid interface lengths as a function of

void fraction

Fig. 5-10 Variation in the gas-liquid interface length as a function of void fraction

Void Fraction

Inte

rfac

e, m

*103

Void Fraction

Liquid-solid interface Gas-solid interface

Gas

-Liq

uid

Inte

rfac

e, m

*103

54

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

0 0.2 0.4 0.6 0.8 1

Liquid Interfacial AreaGas Interfacial Area

As it can be seen from Fig. 5-10 the gas-liquid interface does not change smoothly. There

are peaks which result from the rapid changes in the number of rods submerged in the

water as a result of the change in the void fraction.

The variations in the cross sectional area for the liquid and gas phase as a function of the

void fraction are presented in Fig. 5-11.

Fig. 5-11 Variations of the liquid and gas cross sectional areas as a function of the

void fraction

Finally, the present model was solved for the liquid phase velocity at different void

fractions and zero degree of pressure tube inclination. In order for the model to be

validated a number of tests were performed allowing for the liquid phase velocity to

be measured at different void fractions. A schematic of the experimental set-up is

presented in Fig. 5-12.

Cro

ss S

ectio

nal

Are

a, m

2 *103

Void Fraction

Cross Sectional Area (liquid) Cross Sectional Area (gas)

55

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1

Fig. 5-12 Schematic of the experimental set-up for model validation

The liquid phase velocity was measured using an Ultrasonic Flowmeter (UF)

mounted on the bottom of the pressure tube and correlated with the void fraction

measured inside the pressure tube trough the differential pressure transducer (DP).

The results are presented on Fig 5-13.

Fig. 5-13 Variation of the liquid phase velocity with the void fraction in the pressure

tube

Void Fraction

Liq

uid

Vel

ocity

, m/s

L1

Feeder Tube

L1

Asymmetrical Air Injection

Water Tank

Pressure Tube

DP

UF F

56

For the current range of operating parameters (void fractions and measured liquid

velocities) the Reynolds number for the liquid was calculated to range between 432

and 6665 for void fractions between 10 % and 80 %. As a result of this, it is expected

for the flow regime to change from laminar to turbulent with increase in the void

fraction.

The analytical predictions were further compared with the experimental results in

Fig.5-14.

.

Fig. 5-14 Comparison between the predicted and the calculated liquid phase velocity

As it can be seen from the figure above the agreement between the predicted and

measured liquid velocity values is reasonably good.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1 1.2

Calculated liquid veloctiyExperimental data

Liq

uid

Vel

ocity

, m/s

Void Fraction

57

5.4 Summary of Model Development A steady two-phase flow model based on a simplified momentum balance for each phase

was developed in this chapter. In addition to this, a numerical procedure for calculating

the major geometrical parameters as a function of the void fraction was employed. The

model was further solved numerically and the results were compared with the collected

experimental data. The model showed a reasonable agreement with the experimental

results for the low values of the void fraction and significant deviation for the void

fractions closer to 1.

58

6. Conclusions An experimental investigation of Intermittent Buoyancy Induced Flow (IBIF) has been

conducted to better understand the two-phase flow phenomena occurring during a loss of

coolant circulation event in CANDU reactors. An experimental apparatus was designed

and constructed to closely simulate a CANDU pressure tube containing a 37-element fuel

bundle and connected to vertical feeder pipes and header tanks in the end sections.

Acrylic pipes were used as the pressure tube and two feeder pipes to enable direct

observation and video recording of the IBIF phenomena.

In the experiments, air was injected into a water-filled pressure tube through small

nozzles inserted into the rod bundle to simulate steam generation in stagnant water during

the loss-of-coolant circulation event. The number and locations of air injection points, air

flow rate, water levels in the feeder pipes were varied to study their effects on the venting

of air and two-phase flow behavior in the entire system. An analytical model was also

developed to predict the liquid flow velocity induced in the pressure tube ny air injection

into the fuel bundle. The following conclusions can be drawn from the experimental

analytical results.

A. For a horizontal pressure tube the bubble expansion velocity is constant

throughout the entire length, i.e. no net acceleration has been observed throughout the

pressure tube.

B. The main parameters affecting the air venting time were found to be the air

injection rate and the water level in the feeder pipes.

C. Based on the experimental data collected, an empirical correlation relating the

above-mentioned parameters to the venting time has been derived.

D. The simultaneous air injection through multiple nozzles caused a decrease in the

venting time due to the interactions between the air-bubbles formed inside the tube. This

behavior is expected to benefit the steam venting of the CANDU pressure tube

59

considering the large number of nucleation sites from which steam would be generated on

the surface of the fuel rods.

E. The pressure tube sagging caused a decrease in the venting time ranging from 8%

to 14% for sagging depths of 12.5 mm to 50.4 mm below horizontal at the mid point of

the pressure tube. The effect was stronger for minor inclinations and became weaker with

a further increase in the sagging depth. Since the aging of CANDU reactors could lead to

greater sagging of pressure tubes by as much as 50 mm (2 inches), the steam venting time

would be expected to be reduced and safety margins would be increased in loss-of-

coolant circulation events.

F. An analytical model was also developed to predict the liquid velocity in the

pressure tube induced by air injection into the pressure tube. The presence of a 37-rod

bundle was also taken into consideration in the analysis. The model predictions were

shown to be in good agreement with the measured values.

60

References 1. G.B. Wallis, One-dimensional Two-phase Flow, McGraw Hill, New York

USA(1969)

2. G.F. Hewitt, “Pressure Drop” and “Void Fraction”, in Handbook of Multiphase

Systems, ed. G. Hetsroni, Hemisphere Publishing Corporation, New York, 1982

3. Y. Taitel, and A.E. Dunkler, “A Theoretical Approach to the Lockhart-Martinelli

Correlation for Stratified Flow”, International J. Multiphase Flow, Vol. 2, pp. 591-

595 (1976)

4. M. Sadatomi, M. Kawaji, C.M. Lorencez, and T. Chang, “Prediction of liquid level

distribution in horizontal gas-liquid stratified flows with interfacial gradient”,

International J. Multiphase Flow, Vol. 19, No.6, pp. 978 – 997 (1993)

5. D.J. Nicklin, J.O. Wilkes, and J.F. Davidson, “Two-phase flow in vertical

tubes”, Trans. Inst. Chem. Eng., Vol. 40, pp. 61–68 (1962)

6. R.W. Lockhart and R.C. Martinelli, “Proposed correlation of data for isothermal

two-phase, two-component flow in pipes”, Chemical Engineering Progress

Symposium Series 45 , pp. 39–48 (1949)

7. N.N. Clark and R.J. Dabolt, “A General Design Operating in Slug Equation for Air

Lift Pumps Flow”, AIChE Journal, vol. 32, pp. 56-64 (1986)

8. F. de Cachard, and J. M. Delhaye, “A slug-churn flow model for small-diameter

airlift pumps”, International Journal of Multiphase Flow, vol. 22, pp. 627-

649(1996)

9. F.P. Bretherton, “The motion of long bubbles in tubes”, Trinity College, Cambridge

(1960)

10. S. K. Wilson, H. Davis, and G. Bankoff, “The unsteady expansion and contraction

of a long two-dimensional vapor bubble between superheated or sub-cooled parallel

plates”, J. Fluid Mech., vol. 391, pp. 1-27 (1999)

11. D.B.R. Kenning, D.S. Wen, K.S. Das, and S.K. Wilson, “Confined growth of a

vapor bubble in a capillary tube at initially uniform superheat: Experiments and

modeling”, International Journal of Heat and Mass Transfer, vol. 49, pp. 4653–

4671 (2006)

61

12. H.N. Oguz and A. Prosperetti, “Dynamics of bubble growth and detachment from a

needle”, J. Fluid Mech., vol. 251, pp. 11 1-145 (1993)

13. E. Anagbo, J. K. Brimacombe, and A. E. Wraith, “Formation of Ellipsoidal

Bubbles at a Free-Standing Nozzle”, Chemical Engineering Science, vol. 46, No. 3,

pp. 781-788 (1991)

14. H. Yuan, H.N. Oguz, and A. Prosperetti, “Growth and collapse of a vapor bubble in

a small tube”, International Journal of Heat and Mass Transfer, Vol. 42, pp. 3643-

3657 (1999)

15. H. Yuan, and A. Prosperetti, “The pumping effect of growing and collapsing

bubbles in a tube”, J. Micromech. Microeng., vol. 9, pp. 402–413 (1999)

16. E. Ory, H. Yuan, and A. Prosperetti, “Growth and Collapse of a vapor bubble in a

narrow tube”, Physics of Fluids, Vol. 12, pp. 1268-1277 (2000)

17. A. Asai, T. Hara and I. Endo, “One-Dimensional Model of Bubble Growth and

Liquid Flow in Bubble Jet Printers”, Japanese Journal of Applied Physics, vol. 26,

pp. 1794 – 1801 (1987)

18. V. P. Carey, Liquid Vapor Phase Change Phenomena: An Introduction to the

Thermophysics of Vaporization and Condensation Processes in Heat Transfer

Equipment, 2nd edition, 2007

19. Y. Feyginberg , P. Sergejewich, W. I. Midvidy, A Method For Assessing Reactor

Core Cooling Without Forced Circulation, NSS File HIST/RE/2198, Nuclear

Studies and Safety Department, Ontario Hydro

20. Y. Jiyang, W. Songtao, and J. Baoshan, “Development of sub-channel analysis

code for CANDU-SCWR”, Progress in Nuclear Energy, vol. 49, pp. 334 – 350

(2007)

21. K.K. Fung and J.C. Mackinnon, OPG’s approach of crediting natural circulation in

outageheat sinks, Twenty Second Annual Conference of The Canadian Nuclear

Society,Toronto, Ontario, Canada, June 10-13, 2001

22. N.A. Vlachos, S.V. Paras, and A.J. Karabelas, Prediction of holdup, axial pressure

gradient and wall shear stress in wavy stratified and stratified/atomization

gas/liquid flow, Brief Communication, Department of Chemical Engineering and

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Chemical Process Engineering Research Institute, Aristotle University of

Thessaloniki (1998)

23. N. Andritsos, and T.J. Hanratty, Influence of interfacial waves in stratified gas-

liquid flows, AIChE J., Vol. 33, pp. 444-454 (1987)

63

Appendices

64

Appendix 1

Numerical Code for Interfacial Area Calculation

function [S_l S_g S_i A_l A_g] = interfacial_area(h)

% The function computes the interface between gas and liquid(S_i),

% liquid interface(S_l), gas interface(S_g), liquid cross section(A_l),

% gas cross setion(A_g)as a function of the water level h

% Initiation of the constants.All dimentions in milimeters.

rod_radius = 6.35;

tube_ir = 50.80;

single_rod_circumference = 2*pi*rod_radius;

total_circumference_bundle = 37*single_rod_circumference;

total_circumference_tube = 2*pi*tube_ir;

single_rod_cross_section = pi*rod_radius^2;

bundle_total_cross_section = 37* single_rod_cross_section;

tube_cross_section = pi*tube_ir^2;

% Calculating the points of intersection between the fuel rods and the

% water level line at y=h. The solution requires simultaneous solution of

65

% the equations of the circles representing half of the fuel rods placed

% inside the left half of the pressure tube and the equation of the line

% at y = h representing the water level inside the pressure tube.

% Generating a zero matrix to store the solutions.

b= zeros(19,2);

% Specifing the system of equations which determine the location of the

% circles inside the pressure tube

x(1,1) = 7.52 + sqrt(rod_radius^2-(h - 8.15)^2);

x(1,2) = 7.52 - sqrt(rod_radius^2-(h - 8.15)^2);

x(2,1) = 21.65 + sqrt(rod_radius^2-(h - 13.30)^2);

x(2,2) = 21.65 - sqrt(rod_radius^2-(h - 13.30)^2);

x(3,1) = 33.18 + sqrt(rod_radius^2-(h - 22.96)^2);

x(3,2) = 33.18 - sqrt(rod_radius^2-(h - 22.96)^2);

x(4,1) = 40.70 + sqrt(rod_radius^2-(h - 35.99)^2);

x(4,2) = 40.70 - sqrt(rod_radius^2-(h - 35.99)^2);

x(5,1) = 43.31 + sqrt(rod_radius^2-(h - 50.80)^2);

x(5,2) = 43.31 - sqrt(rod_radius^2-(h - 50.80)^2);

66

x(6,1) = 40.70 + sqrt(rod_radius^2-(h - 65.61)^2);

x(6,2) = 40.70 - sqrt(rod_radius^2-(h - 65.61)^2);

x(7,1) = 33.18 + sqrt(rod_radius^2-(h - 78.64)^2);

x(7,2) = 33.18 - sqrt(rod_radius^2-(h - 78.64)^2);

x(8,1) = 21.65 + sqrt(rod_radius^2-(h - 88.30)^2);

x(8,2) = 21.65 - sqrt(rod_radius^2-(h - 88.30)^2);

x(9,1) = 7.52 + sqrt(rod_radius^2-(h - 93.45)^2);

x(9,2) = 7.52 - sqrt(rod_radius^2-(h - 93.45)^2);

x(10,1) = 7.44 + sqrt(rod_radius^2-(h - 23.03)^2);

x(10,2) = 7.44 - sqrt(rod_radius^2-(h - 23.03)^2);

x(11,1) = 20.33 + sqrt(rod_radius^2-(h - 30.47)^2);

x(11,2) = 20.33 - sqrt(rod_radius^2-(h - 30.47)^2);

x(12,1) = 27.77 + sqrt(rod_radius^2-(h - 43.36)^2);

x(12,2) = 27.77 - sqrt(rod_radius^2-(h - 43.36)^2);

x(13,1) = 27.71 + sqrt(rod_radius^2-(h - 58.24)^2);

x(13,2) = 27.71 - sqrt(rod_radius^2-(h - 58.24)^2);

x(14,1) = 20.33 + sqrt(rod_radius^2-(h - 71.13)^2);

x(14,2) = 20.33 - sqrt(rod_radius^2-(h - 71.13)^2);

x(15,1) = 7.44 + sqrt(rod_radius^2-(h - 78.57)^2);

x(15,2) = 7.44 - sqrt(rod_radius^2-(h - 78.57)^2);

x(16,1) = 7.44 + sqrt(rod_radius^2-(h - 37.91)^2);

67

x(16,2) = 7.44 - sqrt(rod_radius^2-(h - 37.91)^2);

x(17,1) = 14.88 + sqrt(rod_radius^2-(h - 50.80)^2);

x(17,2) = 14.88 - sqrt(rod_radius^2-(h - 50.80)^2);

x(18,1) = 7.44 + sqrt(rod_radius^2-(h - 63.69)^2);

x(18,2) = 7.44 - sqrt(rod_radius^2-(h - 63.69)^2);

x(19,1) = 0 + sqrt(rod_radius^2-(h - 50.80)^2);

x(19,2) = 0;

% Calculating the total length of the inerface "L" assuming there are no

% bundles placed inside the pressure tube.The solution separates the

% pressure tub in three regions: bellow the center line, above the centre

% line and at the center line.

if (h < 50.80)

L = sqrt(50.80^2 - (50.80 - h)^2);

elseif (h > 50.80)

L = sqrt (50.80^2 - (h - 50.80)^2);

68

else

L = 50.80;

end

% Calculating the length of the lines that should be excluded from the

% total length of the interface due to the presence of the fuel bundles.

% The code traces the sollutions of the above system of equations and

% excludes the ones which have imaginary components.

for i = 1:19

for j = 1:2

if (imag(x(i,j))== 0)

b(i,j)= x(i,j);

end

end

end

% The lengths of the lines that should be excluded are calculated based

69

% on the coordinates of the points calculated from the system above.

for i = 1:19

l(i)= b(i,1)-b(i,2);

l = l';

end

% Calculating the sum of the sections that should be excluded

for i=1:18

a(1)=l(1);

a(i+1)= l(i+1)+a(i);

end

excluded_sections_length =a(19);

% The gas liquid interface 'S_i'is the difference between the total length

% of the interface 'L' and the sum of the sections that should be

% excluded 'a'.

S_i = (L-excluded_sections_length)/1000;

70

% Input of the 'y' coordinates of the bundles placed inside the pressure

% tube.

y(1,1) = 8.15;

y(2,1) = 13.30;

y(3,1) = 22.96;

y(4,1) = 35.99;

y(5,1) = 50.80;

y(6,1) = 65.61;

y(7,1) = 78.64;

y(8,1) = 88.30;

y(9,1) = 93.45;

y(10,1) = 23.03;

y(11,1) = 30.47;

y(12,1) = 43.36;

y(13,1) = 58.24;

y(14,1) = 71.13;

y(15,1) = 78.57;

y(16,1) = 37.91;

y(17,1) = 50.80;

71

y(18,1) = 63.69;

y(19,1) = 50.80;

% Calculating the curcumference for the liquid phase interphase. This

% parameter is determined by calculating the lenght of the arcs which are

% in contact with the liquid.

for i=1:18

if (b(i,1)~= 0)

if (y(i,1) > h)

w_arc_length(i,1) = 2*(acos((y(i,1)-h)/6.35))* 6.35;

elseif (y(i,1) < h)

w_arc_length(i,1) = single_rod_circumference - 2*(acos((h -

y(i,1))/6.35))*6.35;

else

w_arc_length(i,1) = pi*6.35;

72

end

else

w_arc_length(i,1) = 0;

end

end

if (b(19,1)~= 0)

if (y(19,1) > h)

w_arc_length(19,1) =(acos((y(19,1)-h)/6.35))* 6.35;

elseif (y(19,1) < h)

w_arc_length(19,1) = single_rod_circumference - (acos((h -

y(19,1))/6.35))*6.35;

73

else

w_arc_length(19,1) = pi*6.35/2;

end

else

w_arc_length(19,1) = 0;

end

% Summing all the arc lengths

for i=1:18

c_sum(1)= w_arc_length(1,1);

c_sum(i+1)= w_arc_length(i+1)+c_sum(i);

end

arcs_length_liquid = c_sum(19);

% Counting the number of bundles completely submerged in the liquid

74

if (57.5 <= h)

counter = - 0.5;

for i=1:19

if (y(i,1)+6.35 <= h)

counter = counter + 1;

else

end

end

else

counter = 0;

for i = 1:19;

if (y(i,1)+ 6.35 <= h)

counter = counter + 1;

else

end

end

end

% Calculating the liquid interface for the bundles.Since all calculations

% are performed for half of the tube a factor of 2 is applied to the final

% answer.

75

circumference_bunde_liquid = 2*(counter*single_rod_circumference +

arcs_length_liquid);

% Calculating the circumference of the tube contacting the liquid

% phase - 'wall_c_liquid'

if (h < tube_ir)

theta_wcl = acos((tube_ir - h )/tube_ir);

wall_c_liquid = 2*theta_wcl* theta_wcl;

elseif (h > tube_ir)

theta_wcl = acos((h - tube_ir )/tube_ir);

wall_c_liquid = total_circumference_tube - 2*theta_wcl* tube_ir;

else

76

wall_c_liquid = pi*tube_ir;

end

% Calculating the liquid circumference 'S_l'. This parameter is determined

% by the sum of the circumference of the tube wetted by the liquid, the

% circumference of the bundles completely submerged into the liquid as well

% as the circumference of the arcs which are in contact with the liquid

% phase.

S_l = (circumference_bunde_liquid + wall_c_liquid)/1000;

% The circumference of the tube wall "wall_c_gas" which is in contact with

% the gas phase is the difference between the total circumference and the

% circumference of the tube which is in contact with the liquid phase

wall_c_gas = total_circumference_tube - wall_c_liquid;

% Calculating the cross sectional area of the bundles submerged in

% the gas phase. This parameter is the difference between the total

% circumference of the bundle rods and the circumference which is in

77

% contact with the liquid phase

circumference_bundle_gas = total_circumference_bundle - circumference_bunde_liquid;

% Calculating the liquid circumference 'S_g'

S_g = (circumference_bundle_gas + wall_c_gas)/1000;

% Calculating the bundle fragments 'bundle_fragments_sub_liquid' submerged in

% the liquid

for i=1:18

if (b(i,1)~= 0)

if (y(i,1) > h)

theta(i)= 2*acos((y(i,1)-h)/rod_radius);

bundle_fragment_sub_liquid(i,1) = 1/2*rod_radius^2*(theta(i)- sin(theta(i)));

elseif (y(i,1) < h)

78

theta(i)= 2*acos((h - y(i,1))/rod_radius);

bundle_fragment_sub_liquid(i,1) = single_rod_cross_section -

1/2*rod_radius^2*(theta(i)- sin(theta(i)));

else

bundle_fragment_sub_liquid(i,1) = 1/2*(pi*rod_radius^2);

end

else

bundle_fragment_sub_liquid(i,1) = 0;

end

end

% Calculating the fragment of rod 19 submerged in the liquid

if (b(19,1)~= 0)

79

if (y(19,1) > h)

theta(19)= 2*acos((y(19,1)-h)/rod_radius);

bundle_fragment_sub_liquid(19,1) =1/4*rod_radius^2*(theta(19)-

sin(theta(19)));

elseif (y(19,1) < h)

theta(19)= 2*acos((h - y(19,1))/rod_radius);

bundle_fragment_sub_liquid(19,1) = 1/2*single_rod_cross_section -

1/4*rod_radius^2*(theta(19)- sin(theta(19)));

else

bundle_fragment_sub_liquid(19,1) = 1/4*(pi*rod_radius^2);

end

else

bundle_fragment_sub_liquid(19,1) = 0;

80

end

% Calculating the total cross section of the rods submerged into the

% liquid. This is the sum fo the rod cross section wich are completely

% submerged plus the cross sections of the segments of the rods wich are

% submerged.

% Calculating the sum of the bundle fragments submerged in water

'sum_bundle_fragments'

for i=1:18

sum_bundle_fragments_sub_liquid(1)= bundle_fragment_sub_liquid(1,1);

sum_bundle_fragments_sub_liquid(i+1)=

bundle_fragment_sub_liquid(i+1)+bundle_fragment_sub_liquid(i);

end

% Calculating the bundle cross section submerged into the liquid

% 'bundle_cross_section_sub_liquid'.

81

bundle_cross_section_sub_liquid = 2*counter* single_rod_cross_section +

2*sum_bundle_fragments_sub_liquid(19);

%Calculating the total cross section of the rods submerged into the gas.

%This area is the difference between the total bundle cross section and

%the cross section of the bundle which is submerged

bundle_cross_section_sub_gas = bundle_total_cross_section -

bundle_cross_section_sub_liquid;

% Calculating the liquid cross section. The area is determined as the

% difference between the water level fragment and the cross section of the submerged

bundles.

% Calculating the water level fragment

if (h < tube_ir)

theta_wl = 2*(acos((tube_ir - h )/tube_ir));

segment_wl = 1/2*tube_ir^2*(theta_wl - sin(theta_wl));

82

elseif (h > tube_ir)

theta_wl = 2*(acos((h - tube_ir )/tube_ir));

segment_wl = tube_cross_section -1/2*tube_ir^2*(theta_wl - sin(theta_wl));

else

segment_wl = 1/2*pi*tube_ir^2;

end

% Calculating the liquid cross section 'A_l'. This area is the

% difference between the segment cross section determined by the liquid

% level and the bundle cross section submerged in the liquid

A_l = (segment_wl - bundle_cross_section_sub_liquid)/10^6;

% Calculating the gas cross section 'A_g'. This area is the difference

% between the segment occupied by the gas(which is the difference betweent

83

%the tube cross section and the area of the segment determined by the water

%level) phase and the bundle cross section submerged in the gas

A_g = ((tube_cross_section - segment_wl) - bundle_cross_section_sub_gas)/10^6;

84

Appendix 2 Numerical Code for Liquid Velocity Calculation

function [value]= liquid_velocity

% Importing the user input function

[theta_incl wl spd C_l C_g m n] = user_input;

% Loading the void fraction data

load void_fraction.dat;

% Calculating the water level inside the pressure tube

tube_d = 101.6;

h = (1-void_fraction)*tube_d;

% Importing the function calculating the inferfacial area

%l = size(h);

%l = l(1);

fprintf('\n spd Inclination Angle Void Fraction Liquid Velocity\n');

fprintf('\n kW deg m/s \n\n');

85

%for i= 1:200

[S_l S_g S_i A_l A_g] = interfacial_area(h(1));

itermax = 100; % max # of iterations

iter = 0;

errmax = 0.001; % convergence tolerance

error = 1;

%Initiating the loop

% Density of water at 25 deg C

rho_l = 997.0479;

% The density of air can be calculated based on the ideal gas law and

% depending on the water level inside the feeder line

g = 9.81;

p = rho_l*g*wl + 101325;

R = 286.9;

T = 273.15 + 25;

rho_g = p/(R*T);

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% Viscosity of air(Ref. Viscosity Tables)(@ 25 deg C):

ni_g = 1.8616e-5;

% Viscosity of liquid(Ref. Viscosity Tables)(@ 25 deg C):

ni_l = 8.98e-4;

% Calculating the gas velocity 'u_g' is determined by dividing the

% volumetric air flow rate to the cross section for the gas phase

h_fg = 2484.5e3;

itermax = 100; % max # of iterations

iter = 0;

errmax = 0.001; % convergence tolerance

error = 1;

u_l =(1000*spd/(h_fg*rho_g))/A_g;

while error > errmax && iter < itermax

87

for i = 1:10

iter = iter + 1;

[f fprime] = fcn_nr(theta_incl, spd, C_l, C_g, m, n , S_l , S_g , S_i , A_l , A_g ,rho_l ,

rho_g ,ni_g ,ni_l , h_fg ,u_l(i),10);

if fprime == 0

fprintf('ERROR: deriv(x) = 0; can''t divide by zero\n')

break;

end

value(i) = f

end

u_l_new = u_l - f / fprime; % here is new root estimate

plot(u_l(2:10),value(2:10));

error = abs((u_l_new - u_l)/u_l_new) * 100; % find change from previous

u_l = u_l_new; % set up for next iteration

end

fprintf('\n %10.5f %10.6f %5.4f %10.6f \n',

spd,theta_incl,void_fraction(i),u_l);

%end

88

Appendix 3 User Input Function

function [theta_incl wl spd C_l C_g m n] = user_input

% Initializing :

% In order for the calculation to start it's needed the simulated power

% density data

% The user is required to enter the value for the water level inside the

% feeder line

theta_incl = input('Input the pressure tube inclination angle, [Deg] = ');

wl = input('Input the water level inside the feeder line, [m] = ');

spd = input('Input the simulated power level, [kW] = ');

C_l = input('Input the value for C_l /C_l = 0.046(turbulent), C_l = 16(laminar)/ = ');

C_g = input('Input the value for C_g /C_g = 0.046(turbulent), C_g = 16(laminar)/ = ');

m = input('Input the value for m /m = 0.2(turbulent), m = 1.0(laminar)/ = ');

n = input('Input the value for n /n = 0.2(turbulent), n = 1(laminar)/ = ');