Two Phase Flow 2009

8/20/2019 Two Phase Flow 2009

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13:20-14:50

Thermal-Hydraulics in Nuclear Reactors

International graduate course

Tokyo Institute of Technology

October 2009

Professor

Research Institute of Nuclear Engineering

University of Fukui

Hiroyasu MOCHIZUKI


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Contents

1. Two-phase flow

1.1 Flow regime

1.2 Void fraction and steam quality

1.3 Relationship between void fraction and steam quality

1.4 Pressure loss

1.5 Drift flux model

2. Thermal-hydraulics in the reactor

2.1 Homogeneous flow model

2.2 Separated flow model

2.3 Two-fluid model

2.4 Heat transfer correlations

2.5 Critical flow

2.6 Single-phase discharge

3. Thermal-hydraulic issues in components

3.1 Safety parameter of the fuel assembly

3.2 Pump

3.3 Steam separators

3.4 Turbine system

3.5 Valves3.6 Piping

3.7 Heat exchangers

3.8 Control rod

3.9 Example of the plant

4. Plant stability

4.1 Channel hydraulic stability and core stability

4.2 Ledinegg instability

4.3 Density wave oscillation

4.4 Geysering

4.5 Chugging

5. Application of component modeling to the nuclear power plant

5.1Plant transient in Liquid-metal-cooled fast reactors

5.1.1 Heat transfer between subassemblies

5.1.2 Turbine trip test of ‘Monju’

5.1.3 Natural circulation of sodium cooled reactors

5.2. Chernobyl accident


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Nomenclature

A area (m2)

C or Cp specific heat capacity (J/kg K)

C D discharge coefficient (-)

C 0 distribution parameter (-)

c sound velocity (m/s)

D diameter (m)

De equivalent diameter (m)

e total energy (J/kg) or parameter (-)

F heat transfer area per unit length (m2/m) or Force (N)

G mass velocity (kg/m2s)

GD2

pump inertia (kg m2)

g gravitational acceleration (m/s2)

H pump head (m)

h heat transfer coefficient (W/m2 K)

i enthalpy (J/kg)

j superficial velocity (m/s)

k thermal conductivity (W/m K)

L length (m) or perimeter (m)

M Mach number (=

u/c)

m mass flow rate (kg/s)

N rotational speed (rpm)

Nu Nusselt number (=h d e /k )

ns specific pump speed (-)

P or p pressure (Pa)

Pe Péclet number (= Re･Pr )

Pr Prandtl number (=Cp /k )

Q volumetric flow rate (m3/s) or heat rate (W)

q heat flux (W/m2)

R Gas constant (m2/s

2 K)

Re Reynolds number (=ud e / )

S slip ration (=ug /ul)

s entropy (J/K)

T temperature (K or ℃)

t time (s)

U overall heat transfer coefficient (W/m2 K)

u velocity (m/s)

V velocity (m/s)


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W mass flow rate (kg/s)

x steam quality (-)

X tt Martinelli parameter (-)

z coordinate (m)

void fraction (-)

volumetric flow fraction (-)

sand roughness (m)

local loss coefficient (-)

efficiency (-)

angle (rad)

ratio of specific heat capacity (Cp/Cv)

friction factor (-)

viscosity (N s/m2

)

density (kg/m3)

surface tension (N/s)

2 two-phase multiplier (-)

angular velocity (rad/s)


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1. Two-phase flow

A phase means one of states of matters. We can see the phases of

liquid, gas and solid in our daily life. When two out of three or all phases

flow simultaneously, we call the flow as a multi-phase flow. Therefore, a

two-phase flow is a simplest form of the multi-phase flow. There are

several kinds of two-phase flows in general, e.g., two-phase flow

consisting of liquid and gas, liquid and solid, and gas and solid. A flow

consisting of water and oil is a kind of the two-phase-flow as well.

Among them, we can see the two-phase flow consisting of liquid and

vapor in case of a nuclear reactor. In a boiling water reactor (BWR) and

a pressurized water reactor (PWR), flow regimes shown in Fig. 1.1 appear

in the core or piping during the normal operating and an accidental

condition, respectively.

1.1 Flow regime

We can distinguish the two-phase flow through flow regimes. The flow regimes appearing in a vertical

flow are shown in Fig. 1.1.1 and those appearing in a horizontal flow are shown in Fig. 1.1.2. In BWR, a

bubbly flow appears at the lower part of the core, and a churn-annular flow can be seen at the exit of thecore depending on the channel power. The void fraction is dependent on steam quality and pressure

(temperature).

These regimes can be classified by superficial velocities of gas and liquid. They are simply defined

using the ratio of volumetric flow rate Q (m3/s) to the total flow area A (m

2) as follows.

気泡流Bubbly flow

スラグ流Slug flow

チャーン流（フロス流）Churn flow

環状噴霧流 Annular -mist flow

噴霧流Mist flow

gg

Fig. 1.1.1 Flow regimes in vertical flows

Fig. 1.1.2 Flow regimes in horizontalflows

気泡流Bubbly flow

スラグ流Slug flow

波状流Wavy flow

環状流 Annular flow

プラグ流Plug flow

層状流Stratified flow

g

Fig. 1.1 Flow regimes in a heated channel

Gas flow

Dispersed flow

Dryout

Annular flow

Slug flow

Bubbly flow

Nucleate boiling

Subcooled boiling

Liquid flow

Churn flow

Fuel pin


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A

Q j

g

g (1.1.1)

A

Q j ll (1.1.2)

Using the above velocities, typical flow maps for liquid and gas are illustrated for the vertical and the

horizontal flows as shown in Fig. 1.1.3 and Fig. 1.1.4. Since these are maps in order to use the computer

calculation, a border is shown using a curve. However, the border has a band because it is decided by

several experiments and the judgment of the border is very much dependent on persons. Furthermore,

these flow maps depend on ratios of gas to liquid volumes, i.e., void fraction, velocities, physical properties

of liquid, and a configuration of flow passage, and there is no universal map. Most famous one is the

Baker’s chart for the horizontal flow. In order to evaluate pressure loss in a flow system, the flow regime

should be clarified at first and a proper pressure-loss evaluation method should be applied.

Fig. 1.1.3 Flow map for vertical flow Fig. 1.1.4 Flow map for horizontal flow

1.2 Void fraction and steam quality

The void fraction is a very important parameter to express the two-phase flow, and has a relationship

with steam quality in case of the two-phase flow in the reactor. The void fraction means a ratio of vapor

or gas to the total volume of the flow. When is defined as the following step function, the void fraction

is expressed as an average value of in the control volume.

Phase Liquid

PhaseGas

0

1 (1.2.1)

V dV V 1

(1.2.2)

When we observe the cross-section of the flow, the area of vapor to the total area means void fraction,

i.e., average value in flow area. The void fraction is defined as a time averaged fraction as well.There are several methods to measure the void fraction.

0.01

0.1

1

0.1 1 10

BS

SF

FA

Air/Water

7 MPa

j l 0 ( m / s )

j g0

(m/s)

Slug

Annular

Bubbly

Flow regime map for vertical flow

Froth

0.001

0.01

0.1

1

10

0.1 1 10 100 1000

Present data

j (m/s)

j (m/s)

SlugPlug

Stratified Smooth Stratified

Wavy

Annu lar

Bubbly

l

g


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1) One of old methods is to isolate a pipe using two quick shut valves provided upstream and downstream

of a flow passage.

2) The method of CT scan is sometimes used to measure average gas volume.

3) A neutron or -rays are used to measure existence of gas phase in the beam line.

4) Electric probes or optical fibers are sometimes used to measure the time average void fraction at the

specific positions.

The steam quality is defined as a ratio of the vapor mass to the total mass in a control volume. Usually

the quality is important parameter for the flow of one-component two-phase flow in nuclear reactors. We

call it the steam quality in case of the light water reactors.

gl

g

W W

W x

(1.2.3),

where W stands for mass flow rate (kg/s). The above quality is the flow quality that expresses the real

vapor ratio to the total flow rate. On the other hand, the steam quality under the assumption of the thermal

equilibrium is often used. This is called as the thermal equilibrium steam quality. This quality is

calculated if we can know the enthalpy of the fluid as follows.

lg

sat

i

ii x

(1.2.4)

We can define even negative quality as follows.

lg

sat

lg

sublsub

i

ii

i

T Cp X

(1.2.5)

Cpl：specific heat capacity（J/kg K ）

ilg：latent heat (J/kg)

1.3 Relationship between void fraction and steam quality

There are two definitions of velocity, i.e., real velocity and superficial velocity that was introduced

before. They are velocities for both phases, i.e., uk and jk .

k

k k

A

Qu (m/s) k=l, g (1.3.1)

A A k k (1.3.2)

A

Q j k k (m/s) (1.3.3)

Q stands for the volumetric flow rate (m3/s). From the above definitions, the following relations can be

derived.

P(MPa) Cp⊿ T (kJ/kg) ilg(kJ/kg)

0.1 4.1868 2265

7 5.373 1511

15 8.194 1024

22.1(Pc) 18.35 0


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k k k u j (m/s) (1.3.4)

k k k k k k juG [kg/m2s] (1.3.5)

AGW k k [kg/s] (1.3.6)

j

jk

k

(1.3.7)

is called as the volumetric flow fraction.

Therefore, the steam quality is rewritten as follows.

lglgggggg

lllggg

ggg

lg

g

lg

g

uu

u

uu

u

GG

G

W W

W x

1

(1.3.8)

The above equation is rewritten again in the case where the quality is the dependent variable.

x

x

u

u

ll

ggg

1

1

1

(1.3.9)

Since the ratio of the gas-phase velocity to the liquid-phase velocity is defined as the slip ratio S , the

above equation is written as follows.

S x x

l

gg

1

1

1

(1.3.10)

The slip ratio S is written as follows from the above equation.

g

l

l

g

x

x

u

uS

1

1 (1.3.11)

g

The relative velocity between ug and ul is called as the slip velocity.

lgr uuu (1.3.12)

In the case of the two-phase flow, there is a problem how to evaluate the void fraction from the steam

quality. If the slip ratio is assumed as 1, the flow is called as the homogenous flow, and the void fraction

is defined as a function of the steam quality. However, it is usual that the two-phase flow has a slip ratio

between the vapor and liquid phases. Smith studied the slip ratio based on his experiment, and proposed

the following correlation based on the theoretical discussion.


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50

11

1

1

.

g

l

x

xe

x

xe

eeS

(1.3.13)

In the above correlation, a parameter e was explained as follows.

e= (mass of water flowing into homogeneous mixture)/(total liquid mass)

He decided e=0.4 according to his experimental

observation. When we assume e=1, the slip ratio

becomes unity, i.e., homogenous flow, and when we

assume e=0, the slip ratio becomes as gl / , i.e., the

slip ratio proposed by Fauske. If we read Smith’s paper

carefully, the parameter e was not constant but a function

of mass velocity. In a sense, e=0.4 was an average value.

In another study carried out at JAEA, this value was

correlated as a function of steam quality as follows.

05005950 .) x.tanh(.e (1.3.13)

When the above correlation was applied to Eq. (1.3.10), the void fraction measured in a flow channel

containing a simulated fuel bundle was fitted as shown in Fig. 1.3.1. However, the correlation based on

the homogeneous assumption cannot express void fraction in the subcooled region. According to the

measurement by Hori (1995), void fraction at the thermal equilibrium quality was in the range 0.1-0.2 in

the case of PWR operating conditions. This characteristic is not always very important in the case of

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

MeasuredCorrelation

V o i d f r a c t i o

n , α

( - )

Thermal equilibrium steam quality, x (-)

Pressure 7MPa

Fig. 1.3.1 Example of void fraction above the core

z=0

Exit

Inlet α=1

Void fraction

Heat flux

Steam quality

Steam quality

in case of uniform heat flux

Fig. 1.3.3 Profile of void fraction and

steam quality in the core

Fig. 1.3.2 Example of void fractionof PWR fuel bundle

Bankoff

Thom

Armand


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BWR. Figure 1.3.2 shows an example of void fraction measurement using a CT scanner. It is obvious

that many voids exist when the thermal equilibrium quality is zero. Figure 1.3.3 illustrates examples of

the quality and void distributions in the core of BWR.

There is another approach to calculate the void fraction. That is the drift-flux model.

1.4 Pressure loss

The evaluation of pressure loss of the two-phase flow in a vertical pipe is very important to characterize

the flow. In case of the single-phase flow, a pressure loss can be evaluated for a pipe of diameter D and

length z by the following equation.

ii

iii

i

i

i

G

D

zu

D

zP

22

22

(1.4.1)

w: mass velocity (kg/m2s)

For a laminar single-phase flow, the pipe friction factor is given by the following correlation;

Re

64 ( 2100 Re ) (1.4.2)

Du Re

As for a turbulent flow, the Moody’s chart shown in Fig. 1.4.1 can be usually used in order to evaluate

is functions of the Reynolds number and the equivalent relative roughness R.

D R

: sand roughness (m)

In a computer code, the friction factor is approximated by the following equation.

c Reba ( Re>4000) (1.4.3)

Fig. 1.4.1 Moody’s chart

Reynolds number, Re

Hydraulicallysmooth

TurbulentTransition

λ＝64/Re

E q u i v a l e n t r e l a t i v e r o u g h n e s s , ε / D

Laminar

F r i c t i o n f a c t o r , λ

Reynolds number, Re

Hydraulicallysmooth

TurbulentTransition

λ＝64/Re

E q u i v a l e n t r e l a t i v e r o u g h n e s s , ε / D

Laminar

F r i c t i o n f a c t o r , λ


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- 7 -

R Ra 53.0094.0223.0

44.00.88 Rb 1340621 . R.c

In 1947, Moody also proposed an approximation that could be used in the Reynolds number ranging

103


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Table 1.4.1 Parameter X and constant C

Liquid Vapor X C

Turbulent(t)

Rel>2000

Turbulent(t)

Reg>2000

501090 .

l

g

.

g

l

.

g

l

G

G

20

Laminar(v)

Rel2000

501090

40

59

1.

l

g

.

g

l

.

g

l.

gG

G Re

12

Turbulent(t)

Rel>2000

Laminar(v)

Reg


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xg

l

112

(1.4.10)

1.5 Drift flux model

In all two-phase flows, the local velocity and the local void fraction vary across the channel dimension, perpendicular to the direction of flow. To help us consider the case of a velocity and void fraction

distribution (possibly different) it is convenient to define an average and void fraction weighted mean value

of local velocity. Let F be parameters, such as any one of these local parameters, and an area average

value of F across a channel cross-section would be given as:

AFdA AF 1

(1.5.1)

When a void fraction weighted mean value of F for drift flux parameters is defined as follows:

F

F (1.5.2).

A void fraction weighted gas velocity ug is expressed as follows;

gj

gj

gj

gjg

g

u jC

u j j

ju j

u juu

0

(1.5.3).

j

jC

0

Ｐｒｅｓｓｕｒｅ；6.9 MPaＭａｓｓＶｅｌｏｃｉｔｙ

3000 ㎏/㎡ｓ2200

1500850

Thermal equilibrium steam qulity：x（－）

T w o - p

h a s e m u

l t

i p

l i

e r

; φ

2 （

－）

Homogeneous model

18

16

14

12

10

8

6

4

2

00 0.2 0.4 0.6

Fig. 1.4.3 Example of two-phasemultiplier for spacer

Fig. 1.4.4 Twho-phase multiplier for pipes

0

2

4

6

8

10

12

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

1500220030005000

T w o - p

h a s e m u l t i p l i e r

Steam quality, x

Mass velocity (kg/m2 s)

D=0.05 mroughness: 20 mP: 7MPa

Homogeneous model

Diverging flow ζ≃1.0

Converging flow ζ=0.2～0.5

Drift velocity


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The above relations are re-written simply as follows.

gj

g

g u jC j

u 0

(1.5.4)

gl j j j A

Q

j

g

g A

Q

j

l

l ugj: vapor drift velocity to average superficial velocity (m/s), C 0: distribution parameter (-)

The distribution parameter is a parameter to express the effects of distributions of the void fraction and

velocity in the cross section of a pipe. The value of the distribution parameter is larger than unity when

the void fraction is large at the center region. On the other hand, the value is smaller than unity when the

void fraction is larger near the pipe wall. When the value of the distribution parameter is unity, it means

the uniform void distribution in the cross section of the flow passage. In general, the drift velocity and the

distribution parameter are dependent on the flow regime as shown in Table 1.5.1.

Table 1.5.1 Coefficients in the drift flux model

Flow regime Drift velocity ugj Distribution parameter C 0

Cap bubbly flow 2

1

540

l

gl gD.

1

D(m): pipe diameter

Bubbly flow

n

l

gl g

12

41

2

One example of n is 1.75.

1812021

exp.. lg

for annular tube

181350351

exp..

l

g

for rectangular tube

Churn-turbulent

flow 4

1

22

l

gl g

Slug flow 2

1

350

l

gl gD.

Annular flow

21

0150

1

4

1

l

gl

l

g .

gD

l

g

4

11

Droplet flow 4

1

22

g

gl g

1

D(m): pipe diameter, (N/m): surface tension


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Figure 1.5.1 shows the results of measurements under

the two-phase flow consisting of high-pressure nitrogen

gas /water and vapor/water in the fuel bundle region of a

test facility. Both results are on the line predicted by

the drift flux model. As shown in these results, the

drift flux model is very practical to express the

two-phase flow.

References

Henry, R.E. and Fauske, H.K., 1971. The Two-Phase Critical Flow of One-component Mixtures in Nozzles,

Orifices, and Short Tubes, J. of Heat Transfer, Trans. ASME, 179.

Hori, K., et al., 1995. Void Fraction in a Single Channel Simulating One Subchannel of a PWR Fuel

Assembly, Proceedings of the Two-Phase Flow Modelling and Experimentation 1995, pp.1013-1027.Moody, L.F., 1944. Friction Factors for Pipe Flow, Trans. ASME, 66, p.671-684.

Moody, L.F., 1947. An Approximate Formula for Pipe Friction Factors, Mechanical Engineering, 69,

pp.1005-1006.

Smith, S.L., 1969-70. Void Fraction in Two-Phase Flow: A Correlation Based upon An Equal

Velocity Head Model, Proc. Instr. Mech. Engrs

Wallis, G.B., 1969. One-Dimensional Two-Phase Flow, McGraw-Hill Book Company.

Zuber, N. and Findlay, J., 1965. Average Volumetric Concentration in Two-Phase Flow Systems, J. Heat

Transfer, 87, 453.

0

0.5

1

1.5

0 0.5 1

5 MPa Nitrogen gas

7 MPa Nitrogen gas

7 MPa Vapor

j (m/s)

V g

( m /

s )

Relationship between velocity of gas phase

and total flux of gas and liquid

Vg = 0.23 + 1.0 j

Fig. 1.5.1 Confirmation of drift flux model


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2. Thermal-hydraulics in the reactor

In general, thermal-hydraulics in the core for the light water reactors is treated as a one-dimensional

two-phase flow. Even in the case of the pressurized water reactor (PWR), the two-phase flow appears at

the exit of the core. The flow is sometimes assumed as a piston flow that has a flat velocity distribution in

the flow cross section. We can usually evaluate thermal-hydraulics in the core based on two methods, i.e.,

the homogeneous flow or separated flow model, and the two-fluid model.

2.1 Homogeneous flow model

In the reactor core, we have to consider not only the conservation

equations of continuity, momentum and energy of the fluid, but also

energy equations for pellet, cladding and structures like vessel and pipe.

Three conservation equations regarding coolant are derived as follows.

0

z

G

t

m (2.1.1)

llgg uuG 1

dl A

sing z

PG

zt

Gm

m

12 (2.1.2)

Dt DPqiG

zi

t m (2.1.3)

m

PG

zt

P

Dt

DP

(2.1.4)

f ssc

s f cc

f

c T T h A

F T T h

A

F q (W/m3) (2.1.5)

Average density of the fluid is expressed by the following equations.

1lgm (2.1.6)

lgm

x x

11 (2.1.6’)

Since the flow is in thermal equilibrium, the following equation of state is needed to close the equations.

dGG

dPP

dii

d PiiGPG

(2.1.7)

In the above equation, F stands for heat transfer area per unit length, and subscripts stand for as follows;

p: pellet, c: cladding, f : coolant (fluid), and s: structure. The energy equations for pellet, cladding and

structure is expressed by the following equations.

Fig. 2.1.1 Model of fuel pin

Cladding

Spring

Pellet

Reflector

or thermal insulator

Gap

Fuel

Reflector


Cladding

Spring

Pellet

Reflector


Gap

Fuel

Reflector



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f PcPP

PPPP qT T h

A

F

dt

dT Cp (2.1.8)

c f ccc

cPPc

Pc

cc

T T h A

F T T h

A

F

dt

dT Cp (2.1.9)

sees

es f s

s

ssss T T h

A

F T T h

A

F

dt

dT Cp (2.1.10)

In the above equation, subscript e stands for environment around the structure. The last term in Eq.

(2.1.10) expresses heat loss from the structure to the environment.

2.2 Separated flow model

In the separated flow model, the conservation equations of continuity, momentum and energy are given

as follows:

0

k k k m u

zt

(2.2.1)

i

iiwllll

llllll

L

A

A A

Per sing

z

P

z

u

t

u

2 (2.2.2)

i

iiwgggg

gggggg

L

A

A A

Per sing

z

P

z

u

t

u

2 (2.2.3)

The above two equations yields the following equation:

A

Per sing

z

Pu

zu

t wmk k k k k k

2 (2.2.4)

A

Per qPeu

ze

t k k k k k m

(2.2.5)

Where e is total energy expressed by the following equation:

sin zguie k k k 2

2

1 (2.2.6)

k k gl i xi xi xi 1 (2.2.7)

2.3 Two-fluid model

In the two-fluid model, each phase is assumed as an independent fluid, and conservation equations are

derived for each phase. In this model, hydraulic non-equilibrium such as slip between phases and thermal

non-equilibrium are evaluated through basic equations. The model is much precise compared to

equilibrium model or drift flux model. However, it needs many constitutive equations.


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k k k k k k u zt

(2.3. 1)

ik k k ik k k k k k k k k k k k u z

PgF P z

u z

ut

2

(2.3.2)

ik wk ik ik k k

ik k k k k k k k k

k k k k k k k k k

qquht

PguuF uP z

uue z

uet

2

22

2

1

2

1

2

1

(2.3.3)

where, Γ k , F k stand for mass transfer rate per unit volume due to phase change and interaction force

between phases and divergence of viscosity, respectively. Since pressure in the cross section is assumed

being equal, one pressure model is usually used.

PPPPP iliglg (2.3.4)

In order to close the equations, several constitutive equations are necessary. These equations effect on

the flow conditions. They are criteria of droplet generation and droplet diameter, equation to estimate

amount of phase change, frictions between phases and at the wall, heat transfer coefficient, and others.

2.4 Heat transfer correlations

As the coolant is passing through along the fuel bundle that has high temperature, temperature of the

coolant increases. The heat transfer coefficient is defined by the heat flux and the temperature

difference between the fuel surface and the bulk of the coolant.

T hq (2.4.1)

T =T f – T c

q: heat flux(W/m2)

h: heat transfer coefficient (W/m2K)

T f : fuel surface temperature（K ）

T c: coolant temperature（K ）

Many researchers have conducted experiments and proposed practical empirical correlations.

In the evaluation of the heat transfer, an appropriate correlation should be chosen according to the

boiling conditions. Correlations usually used in the light water reactors are listed in Table 2.4.1.

Historical correlations are contained in this table, that we have to use using engineering unit. Among

them, the transition boiling heat transfer coefficient is a little bit different from others. In order to

evaluate heater surface temperature accurately, it has been clarified by a blow-down test using a mock-up

with an electrically heated heater bundle that both nucleate boiling and film boiling seems to be mixed with

a certain ratio. The ratio is a function of time, and time constant is approximately one second.


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1) Heart transfer in subcooled flow

The heat transfer coefficient in single-phase flow is studied by Dittus-Boelter (1930). They proposed

the non-dimensional heat transfer number, i.e., the Nusselt number k

d h Nu e

, as shown in Table 2.4.1.

Physical properties of liquid shall be used.

2) Heat transfer in nucleate boiling

The heat transfer coefficient in the nucleate boiling is very large. During this boiling regime, the bulk

temperature is decided by the system pressure because of thermal equilibrium. Therefore, the coolant

temperature along the core is almost the same. There are several correlations to evaluate the nucleate

boiling as shown in Table 2.4.1. Among them, the correlation by Jens-Lottes is the most famous one.

3) Heat transfer in film boiling

When flow direction is upward, the heat transfer correlations listed in the table can be used. In case of

downward flow, the heat transfer coefficient is degraded by the effect of voids. Figure 2.4.1 shows the

Nusselt number in the film boiling heat transfer. For upward flow, the Dougall-Rohsenow correlation has

good agreement with measured data. However, for downward flow, the heat transfer coefficient is

degraded when the flow rate in the negative direction is small but returns to the Dougall-Rohsenow

correlation when the Reynolds number in negative direction increases.

10

100

1000

103

104

105

106

107

Nu Prg

-0.4=0.023(-Re)

0.8

Nu Prg

-0.4=0.926(-Re)

0.33

+20%

-20%

Minimum

heat

transfer

- - - - -

Fig. 2.4.1 Film boiling heat transfer coefficient for both flow directions

Nucleate boiling heat transfer is very large compared with other ones. Since the heat transfer

coefficient in film boiling is lower three order of magnitude than that in nucleate boiling, the proper

correlation should be chosen in temperature evaluation. In safety evaluation of the nuclear reactor, fuel

and cladding temperatures are estimated very high unless the proper correlation is chosen. That results in

too much conservatism in the fuel design, the design of emergency cooling systems and so on.

4) Heat transfer in super-heated flow

The super-heated flow is a kind of gas flow. Therefore, the Dittus-Boelter correlation can be used.

The physical properties of super-heated vapor shall be used.

10

100

103

104

105

106

500

Re

N u P r g

- 0 . 4

50

Nu Prg

-0.4=0.023Re

0.8

Power dist.uniform

chopped cosine

-30%+50%

P(MPa)

0.3-0.9 5-7 Condition

Two-phase

Vapor flow


20/64

- 16 -

Table 2.4.1 Heat transfer correlations

Regime Heat transfer coefficient Nomenclature and others

Subcooled Dittus-Boelter

4080

0230

.

l

.

le

l

Pr Re.d

k

h

g

ge

g

l

g

ggx

l

le

lx

l

le

l

ud Re

x x Re Re

xud Re

ud Re

1

1

Nucleate

boiling

Jens-Lottes

634

1

82.0

p

x eqT

（Engineering unit）

ｐ(ata), q (kcal/m2 h)

Rohsenow

71

3

1

0130

.

l

gl

l

l fg fg

xl

Pr g H

q.

H

T Cp

Thom

6882

1

02430 . p

x eq.T

（Engineering unit）

Schrock-Grossman

4080

105090

750

0230

1

152

.

l

.

lx

e

l

lx

.

g

l

.

l

g

.

tt

lx

.

tt

N

Pr Re.d

k h

x

x X

h X

.h

Transition

boiling

F N h)(hh 1

) / t exp( nucleate boiling to film boiling,

) / t exp( 1 film boiling to nucleate boiling

Film boiling Dougall-Rohsenow

40800230 .g

.

gx

e

g

F Pr Re.d

k h

Super

heated

Dittus-Boelter

40800230 .g

.

g

e

gPr Re.

d

k h

C p : specific heat [ J /kg K ]

x : quality [-]

h : heat transfer coef. [ Jl/m2s K ]

d e : equivalent diameter [m]

Re : Reynolds number [-]

Pr : Plandtl number [-]

H fg : latent heat [ J /kg]

P : pressure [Pa]

q : heat flux [W /m2]

T : Temperature [℃]

t : Time after dryout [sec] k : thermal conductivity [ J /m

K ]

: viscosity [kg/s m]

: density [kg/m3]

: kinematic viscosity [m2/s]

u : velocity [m/s]

l : surface tension [ N/m]

: transition time s]

( l: liquid g: gas)


21/64

- 17 -

2.5 Critical flow

When a pipe break accident occurs, the amount of coolant, i.e., inventory, should be evaluated accurately.

Otherwise, we cannot evaluate accurately plant parameters such as reactor water level, pressure, cladding

temperature and others. In the blowdown process, the discharged coolant evaporates and becomes the

two-phase flow due to depressurization. Therefore, we have to derive an equation for the critical flow in

the two-phase flow. However, the derivation of the equation is not simple.

1) Ideal gas

At first, the critical flow of the ideal gas is discussed. The

sound velocity c (m/s) is a pressure disturbance in a gas and is

expressed using pressure p and density as follows.

s

pc

(2.5.1)

For the isentropic change of the ideal gas obeys the following law.

const p

(2.5.2)

Eq. (2.5.1) yields the following.

RT p

c

(2.5.3)

When the velocity of the specific point is u, the Mach number is defined as follows.

RT

u

c

u M

(2.5.4)

When M is less than 1, this flow is called as the sub-sonic flow. While, the flow of M >1 is called as

super-sonic flow. The total temperature or stagnation temperature T 0 is defined by the following equation.

22

02

11

2 M T

C

uT T

p

(2.5.5)

In the above equation, T is called static temperature. Using the following famous thermo-dynamic

relationships,

RT p (2.5.6),

RC C v p (2.5.7),

we can obtain the following equation.

pT C

p 1 (2.5.8)

p0, 0, c0pe, e, ue

Fig. 2.5.1 Flwo in a nozzle

0

2

2

1iui

Ideal gas i=C p T


22/64

- 18 -

v

P

C

C

Therefore, Eq. (2.5.5) is changed using Eq. (2.5.2) to the following equation.

0

01

00

02

1121

p p p pu

(2.5.9)

121

00

2

11

M

T

T

p

p (2.5.10)

This pressure p0 is obtained when the flow is stopped by the isentropic change.

When we assume a flow from a tank at pressure p0 to the environment at pressure pe through a nozzle,

the velocity ue from the nozzle is calculated using the following equation.

1

0

0 11

2

p

pcu ee (2.5.11)

The mass flow rate from the nozzle is expressed by the following equation when the flow area is A,

1

0

2

0

00

1

0

0

1

2

11

2

p

p

p

p Ac

p

p Ac Aum

ee

eeee

(2.5.12)

Where, the non-dimensional mass flow rate is defined as follows.

Ac

m

001

2

(2.5.13)

Substituting Eq. (2.5.2) into Eq. (2.5.13),

1

0

2

0

p

p

p

p ee (2.5.14)

The above equation is 0 when pe is equal to p0, and has maximum when pe= pc. (This is derived by

0edp / d )

1

0 1

2

p

pc (2.5.15)

This condition is called critical, and the flow is the critical flow. When this condition is substituted into


23/64

- 19 -

Eq. (2.5.11), we have the following relationships.

ce ucu

1

20

(2.5.16)

Therefore, the critical mass flow rate is given by the following equation.

121

0001

2

1

2

Ac Ac Aum cccc (2.5.17)

When Eq. (2.5.3) is substituted into the above equation:

2

1

1

1

001

2

p

A

mc (kg/m2 s) (2.5.18)

In another method, the continuity equation is.const uA (2.5.19)

0 A

dA

u

dud

(2.5.19’)

The one-dimensional momentum equation is written as follows when the viscosity term is neglected.

z

p

Dt

Du

(2.5.20)

When we consider the steady state,

dz

dp

dz

duu

dpudu (2.5.21).

Therefore,

sdp

d

u

dp

A

dA

21

(2.5.22).

When the above equation is expressed in terms of the mass velocity G,

sdp

d

Gdp

G

dG

22

11 (2.5.23)

From the above equation, it is obvious that the maximum mass velocity occurs when dG/dp=0 or when

d

dpGmax (2.5.24)

This maximum flow rate occurs at the throat of a nozzle where dA/A=0.


24/64

- 20 -

2) Two-phase homogenous equilibrium model (HEM)

For the homogeneous flow, the critical flow rate occurs in the same manner as the single-phase flow:

m

mmaxcd

dpGG

(2.5.25)

If the slip ration of the two-phase flow is not unity, the momentum equation Eq. (2.5.21) should be

changed to the following equation.

dz

dpu x xu

dz

d G lg 1 (2.5.26)

Since the criterion of the critical flow is given by the following correlation:

0dp

dG (2.5.27),

11

lg u x xu

p

G (2.5.28)

The liquid velocity and vapor velocity are related by the slip ratio S :

lg Suu (2.5.29)

The mass velocity is expressed from the definition in the chapter 1 as

lggggg SuuG xG (2.5.30)

S

x

x

l

g

1

1

1

(2.5.31)

Combining Eq. (2.5.30) and (2.5.31) gives G as

lgl

glu

S x x

S G

1

(2.5.32)

Eliminating from Eq. (2.5.28) using the above equation and using the dG/dp=0 condition gives

x xS S

S x x

p

G

gl

gl

c

c

11

12

(2.5.33)

The above equation is general form of the critical flow. When we assume that the slip ratio is unity in the

above equation, we can obtain the same equation as Eq. (2.5.25).

2.6 Single-phase discharge

When a break diameter is small and a flow pass is rather short, the slightly subcooled coolant is

discharged from the system to the environment in the form of the single-phase flow. Boiling will happen

outside the heat transport system. In this case, the discharged flow rate can be estimated using the

Bernoulli equation with the discharge coefficient C D.

e D p pC G

02 (2.6.1)

The value of C D is measured by the experiment under high-pressure and high-temperature conditions as


25/64

- 21 -

shown in Fig. 2.6.1. In this experiment, the break holes were provided on the pipe with 60.5 mm in outer

diameter. Since thickness of the pipe was 5.5 mm, the ratio of flow passage length L to break diameter D

is approximately 0.25. The experimental result indicates that the discharge coefficient of approximately

0.6 can be used when L/D is less than 0.25. As the diameter becomes large, the boiling due to the

depressurization affected the discharge coefficient.

References

Bird, R. B., Stewart W.E. and Lightfoot E.N., Transport Phenomena, John Wiley & Sons, Inc., (1960).

Dittus, F.W. and Boelter, L.M.K., 1930. Heat Transfer in Automobile Radiators of the Tubular Tube, Univ.

Calif. Publs. Eng. 2, 13, p.443.

Dougall, R.S. and Rohsenow, W.M., 1963. Film Boiling on the Inside of Vertical Tube with Upward Flow

of Fluid at Low Qualities, MIT Report #9079-26.

Hsu, Yih-Yun, Graham R.W., 1976. Transport Processes in Boiling and Two-phase Systems, McGraw-Hill

Book Company.

Jens, W.H. and Lottes, P.A., 1951. Analysis of Heat Transfer, Burnout, Pressure Drop and

Density Data for High Pressure Water, ANL-4627.

Rohsenow, W.M., 1952. A Method of Correlating Heat Transfer Data for Surface Boiling Liquid, Trans.

ASME, 74, 969-975.

Schrock, V.E. and Grossman, L.M., 1959. USAEC report, TID-14639.

Thom, J.R.S., et al., 1966. Proc. of Inst. Mech. Engrs, 180, Pt 3C, p.226.

Fig. 2.6.1 Relationship between measured

discharge coefficient and equivalent diameter

0

0.5

1

0 10 20 30 40 50 60

Circular 2MPa

Circular 3MPa

Circular 4MPa

Circular 5MPa

Circular 6MPa

Circular 7MPa

Slit 7MPa

Ogasawara 7MPa

Equivalent diameter (mm)

D i s c h a r g e c o e f f i c i e n t , C

D ( - )

CD=G/(2 P)

0.5

G: kg/m2s

P: Pa

0.59


26/64

- 22 -

3. Thermal-hydraulic issues in components

3.1 Safety parameter of the fuel assembly

The design of the reactor core consists of various designs like neutronics, thermal-hydraulics, fuel,

structure and so forth. The heat balance of the plant is calculated based on required heat generation rate.

Then, number of fuel assemblies and pins par assembly are decided, and local heat generation distribution

of the fuel assembly is designed by the neutronic calculation. In this process, an axial power distribution,

a radial power distribution, and a peaking factor are decided. Using these data, the thermal hydraulic

calculation in the steady state is conducted. Then temperature distributions, void fraction distributions

and others are calculated. To keep the consistence in design between the thermal hydraulics and

neutronics, iterative calculations should be done between the two fields.

On the other hand, several plant transient calculations such as turbine trip, feed water trip and others

must be done using the immature data to clarify the most crucial event in terms of heat removal. The

difference of the minimum critical power ratio, MCPR, is calculated and OLMCPR (operational limit

minimum critical power ratio) is evaluated. This result is fed back to the above design. The method of

MCPR is one of safety indexes regarding the fuel assembly of BWR. In 1970’s, MCHFR (minimum

critical heat flux ratio) was used as the safety index. However, this method is taken over by the MCPR

method.

Figure 3.1.1 shows the schematic relationship between

heat flux of a wire in water and surface super-heat. As

the heat flux increases, wire cooled by the nucleate boiling

reaches the maximum. This point is called as the boiling

crisis. If the heat flux is increased much more, the

boiling regime changes from the nucleate boiling to the

film boiling. According to the transition, the surface

temperature increases drastically. That is the reason why

this point is called as burn-out point as well. The flux

corresponds to critical heat flux.

In the case of a flow system, this characteristic moves

upward. Since the critical heat flux of a fuel assembly

under the forced circulation is dependent on the system pressure, mass velocity, steam quality, spacer pitch,

local peaking, and others, we usually measure the critical heat flux using a mock-up. In case of the

two-phase flow, the sizes of voids are decided by the system pressure and temperature. Therefore, the

experiment using the mock-up is very important. Figure 3.1.2 shows an example of measured result

using 14 MW Heat Transfer Loop and 6 MW Safety Experimental Loop in O-arai. The critical heat fluxes

were measured both for upward and downward flows.

Fig. 3.1.1 Boiling curve

1000

104

105

106

0.01 0.1 1 10 100 1000 104

H e a t f l u x , q

Twall

-TB

Nucleate boilingNon-boiling

Dryout or DNB point

Film boiling

Minimum heat flux point

Transition boiling


27/64

- 23 -

The critical heat flux should be clarified in these methods. General Electric provides GEXL correlation,

and W-3 and W-3 correlations can be applied to PWR. Constants in the equations are confidential. We

can use the Hench-Levy’s correlation as well. In the case of Advanced Thermal Reactor (ATR) developed

in Japan, the following correlation form is provided based on the full-scale experiment using the 14MW

heat transfer loop.

q c = F ( x, f p , f L , f sp , f e , f a , F sub) (3.1.1)

qc: critical heat flux

x: average thermal equilibrium steam quality

f p: factor of pressure

f L: factor of local peaking

f sp: factor of spacer

f e: factor of eccentricity of fuel assembly

f a: factor of axial power distribution

f sub: factor of inlet subcooling

In the case where the fuel specifications are decided, the critical heat flux qst is fitted by the following

quadratic equation.

2

321 xa xaaqst (3.1.2)

Experimental conditions to make the correlation contain flow conditions predicted in the abnormal

transients. But it does not contain flow conditions in the accident such as loss of coolant accidents

(LOCAs). Therefore, we have to be careful in the application of the correlation.

0

5

10

-3000 -2000 -1000 0 1000 2000 3000

HTL

SEL

Mass velocity (kg/m2s)

T o t a l p o w e r ( M W )

36-rod bundle

P = 7 MPa

Tin = 548 K

0

0.5

1

1.5

2

2.5

3

-1000 -600 -200 200 600 1000

T o t a l p o w e r ( M W )

Fig. 3.1.2 Critical power of a fuel assembly


28/64

- 24 -

Critical power can be known through the specific

experiment. But it is difficult to conduct

experiment for all the conditions expected in the

operation. Therefore, the critical power is

evaluated as follows using the CHF correlation. At

first, the power distribution and flow conditions

such as pressure, flow rate and inlet enthalpy are

fixed. Then, the relationships between the cross

sectional average steam qualities and the heat fluxes

are calculated for the various power levels. And

the power which contacts with the CHF correlation

is called as the critical power. Figure 3.1.1 shows

the comparison between the methods of evaluation

using the MCHFR (minimum critical heat flux

ratio) and MCPR. The CHF correlation is a

function of steam quality, and has a characteristic

that decreases monotonously. The power

distribution of the fuel assembly is cross to the

cosine distribution as shown in the figure. In the

case of CHFR, the minimum ratio of the heat flux in this power to the heat flux by the CHF correlation is

defined as MCHFR.

o x xo

c

q

qCHFR

(3.1.3)

In the case of PWR, the DNB (departure from

nucleate boiling) correlation is prepared, and DNBR

(departure from nucleate boiling) is used as the

index instead of CHFR. The minimum value is

defined as MDNBR.

On the other hand, the critical power ratio is

defined as the ratio of the power that the power

distribution contacts the CHF correlation to the

operating power.

o

c

Q

QCPR (3.1.4)

As for parameters relating to the MCPR evaluation, they are accuracy of the CHF correlation, and

indeterminacies of pressure of the core, inlet enthalpy, axial and radial power distributions, channel flow

rate distribution, heat generation rate and others.

H

Q

Pressure loss

Q0

H0

H

Q

Pressure loss

Q0

H0

Fig. 3.2.1 Pump Q-H curve and pressure loss

Thermal equilibrium steam quality

C r i t i c a

l h e a t f l u x ,

O p e r a

t i o n a l h e a t f l u x

qo

qc


xo

xc

Operational condition

xi

Operational conditionPower increase

CHF correlation

CHF correlation

i) CHFR

ii) CPR

C r i t i c a l h e a t

f l u x ,

O p e r a t i o n a l

h e a t f l u x


C r i t i c a

l h e a t f l u x ,

O p e r a

t i o n a l h e a t f l u x

qo

qc


xo

xc

Operational condition

xi

Operational conditionPower increase

CHF correlation

CHF correlation

i) CHFR

ii) CPR

C r i t i c a l h e a t

f l u x ,

O p e r a t i o n a l

h e a t f l u x

Fig. 3.1.3 Safety index


29/64

- 25 -

3.2 Pump

The circulation pump is one of very important components of the reactor. Unless the pump and the

motor are properly designed, the required flow rate cannot be obtained in the neutronic and

thermal-hydraulic designs. In the case where the plant is tripped by an abnormal transient, drayout of the

fuel may occur if the inertia of the pump is small because of fast flow rate decrease. While, it is

disadvantageous when the flow coast-down is too slow because of extra coolant discharge during the

coast-down. Therefore, it is usual there are upper and lower limits for the specification.

The pressure loss of the reactor system is approximately proportional the square of flow rate. This

characteristic is dependent on head loss and two-phase multiplier as a function of flow rate. While, the

pump characteristic is expressed as a Q-H curve, and the pressure head is approximated as a function of

quadratic volumetric flow rate as shown in Fig. 3.2.1. The pressure loss characteristic is evaluated by the

designed flow rate and the distribution of the void fraction. The intersection of both curves is the

operating condition. Therefore, the proper pump should be chosen after the evaluation of the pressure loss

for the necessary flow rate. In general, the flow rate of the pump exceeds the design value taking into

account the aging of the pump.

In the pump characteristics evaluation for steady state and transients such as pump start-up and

coast-down, the kinetic equation with pump efficiency is solved.

02 2

2

604T

N

gHQT

GDdt

dN m

(3.2.1)

N : rotational speed（1/s）

GD2: pump and motor inertia (kg･m

2)

g: gravitational acceleration (m/s2)

T m: Torque of motor (N･m)

H : pump head (m)

ρ: density of fluid (kg/m3)

Q: volumetric flow rate (m3/s)

η: pump efficiency without friction (-)

T 0: torque of friction (N･m)＝k ･T m (k : constant, eg.0.04)

The pump head is approximated by the following equations.

2

3212

n

qh

n

qhh

n

h

10n

q (3.2.2)

2

3212'''

q

nh

q

nhh

q

h

10

q

n (3.2.3)


30/64

- 26 -

0 N

N n ,

0Q

Qq ,

0 H

H h

N : rotational speed (1/s)

Subscript 0stands for the rated condition.

The pump efficiency is approximated by a quadratic equation.

2

321

n

qa

n

qaa (3.2.4)

Constants should be decided by referring handbooks. The torque of the motor can be calculated by the

following equation.

0

0

' N

N

nF

P

T N

m (3.2.5)

P0: rated pump shaft power (kW)

ω: angular velocity (＝2 N 0) (rad/sec)

N N : rated rotational speed of motor (1/s)

n’ : ratio of pump/motor rotational speed （＝ N/N N ）

Another important item is NPSH (Net Positive Suction Head). The suction head of the pump should be

positive. Otherwise, cavitations may occur, and this results in flow rate decrease and corrosion of the

impellers and the casing. The value of NPSH requirement is decided for each pump. The specific speed

of pump ns is expressed as follows.

43

2

1 gH NQns (3.2.6)

N : rotational speed (1/s), Q: volumetric flow rate (m3/s), g: gravitational acceleration (m/s

2), H : head (m).

When NPSH is defined by H sv, cavitation coefficient is defined by the following equation.

H

H sv (3.2.7)

Since NPSH means the differential pressure between suction and saturation, the following relationship can

be established.

sinsv PPgH (3.2.8)

In experiences, the cavitations never occur if the cavitaion coefficient satisfies the following equation.

3

4

78.2 sn (3.2.9)

H n H ssv 34

78.2 (3.2.10)

Since the value evaluated by Eq. (3.2.10) gives the minimum required NPSH, the NPSH in all operating

conditions should exceed this value. In the recirculation system of BWR, the separation of steam from


31/64

- 27 -

liquid is carried out just above the core, and carryunder phenomenon may occur. The carryunder voids are

collapsed by feed water. However, saturation pressure tends to be increased.

When the pump is operated, coolant is heated due to the rotational energy. The heat input by the pump

is sometimes used other than the nuclear power in order to heat-up the system. It is shown by the

dimensional analysis that the amount of heat input is proportional to the product of density, cubic rotational

speed and 5th

power of the impeller diameter.

3.3 steam separators

Steam obtained by LWRs is saturated vapor. Therefore,

vapor should be separated from liquid using many steam

separators. The separator shown in Fig. 3.3.1 was

developed for ATR. The separator for BWR is designed

with the same principle as ATR, but the part of corrugated

separator is separated from the body.

The two-phase flow entering into the separator is rotated by vanes provided at the bottom of the separator.

Liquid pressed on the wall of the turbo-separator flow out of holes provided at upper region. The

collector is provided in order to catch the liquid film and not to pass upward. The two-phase flow

containing droplets are enter into the corrugated part, and the droplets are separated by inertia. However,

small amount of droplets is contained in the main steam. Therefore, the main steam has to pass through

several layers of meshed screens. The droplets that are not separated by these separators are called

carryover.

The carryover ratio is defined by the ratio of droplet flow rate to vapor flow rate as follows.

g

d

co W

W xatioCarryoverR (3.3.1)

In general, the carry over ratio is very small. Since the carryover causes transfer of radioactive

0.95

OD 0.33

Corrugated separator

Guide vaneCollector

Swirl vane

Turbo-separator

Perforation

Two-phase flow

Fig. 3.3.1 Steam separator of ATRFig. 3.3.2 Carryover analysis by particle tracking


32/64

- 28 -

materials to the turbine and corrosion of the turbine blades when it is large, the steam separator should be

designed properly not to generate too much carryover. In case of BWR, many human-power and time

schedule are needed for inspection if the carry over is large.

Figure 3.3.2 shows the analysis that traced many droplets generated by the Monte-Carlo method in the

vapor flow field. One trajectory can be calculated taking into account the drag force working on each

droplet. Calculated result was compared with test result using a mockup of the separator and its

surrounding space.

In the case of the separator for BWR, analyses

have been done using the two-fluid model as

shown in Fig. 3.3.3.

Fig. 3.3.3 Steam separator of BWR and analysis of two-phase flow

3.4 Turbine system

In almost all power stations in Japan, the turbine is used to generate electricity. Figure 3.4.1 illustrates

inside the turbine of the FBR Monju. A high-pressure turbine is shown on the left side, next low-pressure

turbines. Steam is expanded adiabatically, and energy is transmitted to the turbine blades. The blades of

the low-pressure turbine are long in order to catch effectively low-pressure vapor. The longest one is 52

inches (1.3m) for the

blades of the advance

nuclear power stations.

These blades rotate

with 1500 rpm

keeping some ten

microns with the

casing. Photo 3.4.1

shows low pressure

turbine used for a

nuclear power station

Fig. 3.4.1 Inside the turbine casing (Monju)


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in JAPC.

Figure 3.4.2 shows general arrangement of the

turbine system in LWR. The steam generated in

the reactor or the steam generator is introduced

into the high-pressure turbine via the main steam

isolation valves (MSIVs), and flows into the

low-pressure turbine after the elimination of

droplets generated in the high-pressure turbine.

In this case, a re-heater is provided in some

reactors in order to improve the quality of the

steam. Inside the turbine, the isentropic

expansion of the steam is taken place in order to

rotate the turbine blades. Beneath the turbine, a

condenser is provided to condense the steam by sea water. Therefore, pressure inside the condenser is

very low. In general, inside temperature is approximately 40℃. The condensate water is pumped and

fed to the bank of feed water heaters. Near the last stage of the heaters, feed water pumps are provided in

order to pressurize the feed water to high pressure, e.g., approximately 7 MPa for BWR. The number of

feed water heaters is usually 5 or 6. The feed water returns to the reactor via the feed water control valve

and the check valve. A turbine bypass valve drawn in the figure has a role to release the main steam to the

condenser in order to prevent overpressure after the turbine trip event and so forth. The upstream valve is

called as turbine control valve that control the rotational speed. These valves are controlled by theElectric Hydraulic Controller (EHC).

G

Main

steam

MSIVMain control

valve

Drain separator

High

pressure

turbine

Low pressure

turbine

Generator Condenser

Condensate

pump

Feed water heatersFeed water

pump

Containment

vessel

Bypass valve

Feed water

Control valve

Check valveSea water

Intercept valve

C’ or C

D

A

B

NSSS

BOP

Fig. 3.4.2 Outline of turbine system

Photo 3.4.1 Low pressure turbine blades used


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The inlet steam conditions are different

between LWRs and FBRs. Table 3.4.1 shows

the typical inlet conditions for PWR, BWR,

FBR and fire plant. As shown in the table, the

high-pressure turbine in FBR is really high

pressure compared with those of LWRs.

Figure 3.4.3 shows a pressure distribution in

the high-pressure turbine for FBR. After

steam passes second blades, the internal

pressure is equivalent to that of LWRs.

Exhausted steam pressure is only 0.8 MPa in

the rated condition.

Table 4.3.1 Steam inlet conditions

Reactor type Inlet pressure (MPa) Inlet temperature (℃) Steam condition

PWR Approx. 6.0 274 Saturated

BWR Approx. 6.6 282 Saturated

FBR Approx. 12.5 483 Super-heated

Fire Plant Approx. 12.5 538 Super-heated

These thermodynamic states in the turbine system are discussed using a chart drawn on the plain of T-S.

T stands for temperature and S stands for entropy. The basics theory of the engine was studied by Carnot.

The ideal cycle of the engine is called as the Carnot cycle as shown in Fig. 3.4.4. T and S stand for

temperature and entropy. This engine works under reversible cycle.

S

T

S A SC

T A A

B C

D

TB

Heat QH

Cooling QC

Work L2

Work L1 Work L3

Work L4

Fig. 3.4.4 Carnot cycle

QH

QC

Fig. 3.4.3 Pressure distribution in high-pressure

turbine of FBR

0

5

10

15

0 5 10

P r e s s u r e (

M P a )

Step

1 2 3 4 5 6 7 8

Inlet

to low pressureturbine

ExtractExtract


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1) Adiabatic compression process by work L1 from the outside.

2) Isotropic expansion process receiving heat QH from outside, and doing work L2

3) Adiabatic expansion process doing work L3

4) Isotropic compression process discharging heat QC, and receiving work L4

In the above cycle, heat remaining in the system can be expressed as follows.

4132

3412

L L L L

L L L LQQQ C H

Therefore, the efficiency of the cycle is calculated with the following equation.

H

C H

H Q

QQ

Q

Q

This cycle shows the maximum efficiency among engines.

In the case of actual turbine system, the chart of the cycle draws as illustrated in Fig. 3.4.5. This chart

is called as the Rankine cycle. The curve AB’ shows the condition of saturated water and the curve C’E

shows the condition of saturated vapor. The line AB shows the process of pressurization by the feed water

pump, BB’ the process of temperature increase by heating, B’C’ boiling under saturated condition, C’C

super heated process. These corresponding positions are illustrated in Fig. 3.4.5 in case of a fossil plant.

e1e2

f 1f 2

T

Ｓ0

A

B

B’ C’

C

D

S A SC

E

e1e2

f 1f 2

T

Ｓ0

A

B

B’ C’

C

D

S A SC

E

T

Ｓ0

A

B

B’ C’

C

D

S A S’C

D’

SC

E

Saturation curve

Liquid

Super- heated

vapor

Two-phase

Critical point

647.3 K, 22.1MPa

T

Ｓ0

A

B

B’ C’

C

D

S A S’C

D’

SC

E

Saturation curve

Liquid

Super- heated

vapor

Two-phase

Critical point

647.3 K, 22.1MPa

Fig. 3.4.5 Rankine cycle

In the case of LWR, it

cannot produce super-heated

steam. However, fast reactors,

can produce super-heated

steam as well as the fossil

plants, because of

high-temperature liquid metal

T G

Sea water

Feed water pump Condensate pump

AB

C’

B’ C

D

Fig. 3.4.6 Rankine cycle of fossil plant


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coolant. The steam expands with isentropic change and rotates turbine blades in the process of CD. The

process DA means the condensation in the condenser. In the case of LWR, the process C’D’ corresponds

to the isentropic expansion in the turbine.

When enthalpy is expressed by i, and these points are used as subscripts, the individual process is

expressed as follows.

Received heat ) BS CS ' C ' BB( AreaiiQ AC BC 1 (3.4.1)

Discharged heat )(2 AS ADS AreaiiQ AC A D (3.4.2)

Power output DC ii L 1 (3.4.3)

Work by pump A B ii L 2 (3.4.4)

The effective work is expressed by the following equation.

)''(2121 CDAC ABB AreaQQiiii L L A B DC (3.4.5)

The inside of the cycle corresponds to this area. Therefore, the efficiency of the cycle is evaluated by

the following equation.

)''(

)''(

1

21

1

21

BS CS C BB Area

CDAC ABB Area

Q

QQ

Q

L L

AC

(3.4.6)

Therefore, the area of the cycle should be enlarged in order to have a good efficiency. Re-heating of

steam and heating of feed water by extracted steam have good effect on the efficiency. However, these

countermeasures should be chosen based on cost-and-benefit. Since the thermal efficiency of LWR is

approximately 30%, 70% heat generated in the core is discharged into the environment.

In the case of LWR, the processes of receiving heat and discharging heat are different from the fossil plant

and FBR as follows:

Received heat ) BS ' S ' C ' BB( Areaii' Q AC B' C 1 (3.4.7),

Discharged heat ) AS ' S ' AD( Areaii' Q AC A' D 2 (3.4.8).

Therefore, the efficiency is expressed as follows:

)'''(

)'''(

1

21

BS S C BB Area

A DC ABB Area

Q

QQ

AC

(3.4.9).

The efficiency becomes lower than the cycle that can produce super-heated steam.

The right hand side figure in Fig. 3.4.4 shows an example of two-step extraction. There are two lines.

One represents extraction at e1 and becomes condensate f 1 by heating. The other one represents extraction


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at e2 and becomes condensate f 2 by heating. Remaining steam expands to the state-D and cooed to the

state-A. In this cycle, since heat discharged from the condenser decreases by the amount of extracted heat,

the efficiency increases. In general, in the case of n-step extraction, the thermal efficiency is expressed by


1

1

f C

Dej

n

j j DC

ii

iimii

(3.4.10)

In ABWR, there are four low-pressure and two high-pressure feed water heaters. And the re-heater is

provided at the drain separator to increase the steam quality.

3.5 Valves

Many types of valves are used in the plant. The C v value is used very

often in the design. This value is defined using psi and gallon units. When

water at 60F (15.6℃) flows W gallon/min through the valve and pressure

difference is 1 psi (6.89kPa), the C v value is equal to W . The relationship

between the local loss coefficient ζ and the C v value is expressed using the

following correlation.

2

48

1038.21V C

d (3.5.1)

d : diameter of valve (m)

The most common one is called the globe valve, and the shape is

shown in Fig. 3.5.1. The fluid should be flown into the valve

from the left, then flown between the seat and the body. If the

setting direction is reversed, it may cause problems. Because,

the high-pressure may cause the coolant leak from the

ground part of the valve through packings.

The pressure loss of the valve is calculated by the

following equation when the local loss coefficient is

ζ.

2

2

1uP (3.5.2)

u: velocity at the inlet (m/s)（ velocity in the

connected pipe）

Fig. 3.5.2 Check valve

Flow direction

Seat Disk

Arm

Fig. 3.5.1 Globe valve

0.1

1

10

100

1000

104

0.001 0.01 0.1 1 10

Velocity (m/s)

Fig. 3.5.3 Loss coefficient of a check valve


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The check valve or non-return valve is used in order to prevent reversal flow in the primary heat

transport system and in the feed water system. The check valve for the feed water system is efficient not

to lose coolant from the system in case of a pipe break accident. Figure 3.5.2 shows a simple swing-type

check valve that has a disk. When flow is regular, the loss coefficient of the valve is very small, but the

coefficient becomes very large during reversal flow and finally infinite as the disk is closed. Figure 3.5.3

shows an example of the measured result. Since this type valve is closed very rapidly, one has to take into

account the intactness of the valve. Because, the seat hits the body with an extraordinary speed.

One of important valves is the main steam isolation valve (MSIV). This valve is closed rapidly when

an abnormal situation happens in the reactor, and is required reliability. When ‘Fugen’ reactor was

constructed in 1970’s, there was no technology to produce MSIV. Therefore, one MSIV was installed in

the experimental blow-down facility at O-arai Engineering Center of PNC in order to develop. The

capability of the valve was checked through hundreds

steam line rupture experiments, and finally installed

at the ‘Fugen’ reactor. That one shows in Photo

3.5.1. Two MSIVs are provided in one steam line

inside and outside of the containment vessel. The

main steam flows from the left to the right direction.

Since the type of the valve is so called Y-type valve,

pressure loss of the valve in operation is very small

compared with the friction loss of piping.

3.6 Piping

In the plant design, one pipe is called using A or B. For example, in the case of approximately 50 mm

pipe, we have to find the pipe at 50A or 2B. The pipe size is based on the outside diameter, and inside

diameter is different depending upon pressure. The outside diameter of the pipe is close to the unit A in

mm, and the unit B in inches. Appropriate pipes should be chosen

according to the system pressure. This choice is done by Sch

(schedule) coded in U.S.A. The thickness of the pipe with Sch80 is

thicker than that of Sch30. Sch80 piping should be used in most

piping of BWR operated around 7MPa.

3.7 Heat exchangers

1) General theory

Many shell and tube type heat exchangers (HXs) with counter

flow are used in the nuclear power plant. The heat transfer

between the shell and the tube is evaluated using the follwoing

Photo 3.5.1 MSIV of Fugen

Fig.3.7.1 HX model

Coolant: Shell side

Coolant: Tube side

1

2

i

i+1

TpiTpi+1

Tsi

Tsi+1Tti ⊿Z

N

Coolant: Shell side

Coolant: Tube side

1

2

i

i+1

TpiTpi+1

Tsi

Tsi+1Tti ⊿Z

N


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equations.

For the primary flow, the energy equation is expressed taking into account the thermal conductivity in

flow direction;

2

2

z

T k

A

qT T

A

K

z

T GC

t

T C

p

l

p

pt

p

p p

p p

p

p p

(3.7.1)

For the secondary flow (flow inside heat transfer tubes);

2

2

z

T k T T

A

K

z

T GC

t

T C slst

s

ssss

sss

(3.7.2)

For heat transfer tubes;

t st

st p

t

pt t t T T

A

K T T

A

K

t

T C

(3.7.3)

where,

p , f ps p

p

t p p

p

hd d d

d

lnk hd

N K

12

2

11

(3.7.4)

s , f ss

s p

t ss

s

hd d

d d ln

k hd

N K

1

22

11

(3.7.5)

Nomenclatures used in the above equations are

ＡＰ: flow area of shell side (m2) ＡS : flow area of tube side (m

2)

Ａｔ: cross sectional area of tubes (m2) Ｃ: specific heat capacity (J/kg K)

d : diameter of heat transfer tube (m) ｛(d p-d s)/2: thickness｝

Ｇ: mass velocity (kg/m2 s) h: heat transfer coefficient (W/m2 K)

K : (W/m K)＝(overall heat transfer coefficient)×(heat transfer area per unit length)

k : thermal conductance (W/m K) N : number of heat transfer tubes (-)

q’: linear heat loss (W/m) q’=U HX P A(T P-T A) : (W/m)

P A: perimeter of shell side (m) Ｔ : temperature (K)

U HX : overall heat transfer coefficient of shell side (fluid to environment)(W/m2 K)

ρ: density (kg/m3)

Subscripts


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p: shell side s: tube side

t : heat transfer tube A: environment

f : fouling

Other than the above evaluation, the overall heat transfer coefficient is usually given by the following

equation.

ss

o

ss f

o

i

o

t

o

p f p d h

d

d h

d

d

d

k

d

hhU

,,

ln2

111 (3.7.6)

2) Liquid metal

In the case of liquid metal coolant, the heat transfer correlation is different from water due to the small

Plandtl number. Seban-Shimazaki (1951) proposed the following correlation.

8002505 .Pe. Nu (3.7.7)

Pr RePe

/ ud Re e , k / Cpa / Pr , k / hd Nu e

His correlation seems to give us the most proper value according

the many handbooks and studies. The similar correlation that has

constant 7 in the correlation was proposed by Lyon (1949). The

above correlation was proposed by Subbotin (1962) too, and

sometimes it is called the Subbotin’s correaltion. Furthermore,

the heat transfer coefficients for heavy metals are degraded

compared with sodium and other liquid metals. The cause of this

characteristic is not clear yet. Lubarsky & Kaufman (1955)

proposed the following correlation taking into account this fact.

406250 .Pe. Nu (3.7.8)

There are several components like heat exchangers and steam

generators to which we have to apply the heat transfer correlations

other than the reactor core. Since the flow system is complex, we

have to apply the heat transfer coefficient to the component and

confirm its applicability in advance. The almost of all the heat

transfer coefficients were measured using small-scale apparatuses

and the range of applicability is narrow in general. Therefore, it is

difficult to apply the correlations to the real-scale components even

though they are the non-dimensional forms. Figure3.7.2 shows a

Primary sodium

Secondary

~6m

~12m

Fig. 3.7.2 Schematic of IHX


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schematic of an intermediate heat exchanger (IHX) of the ‘Monju’ reactor. The Nusselt number based on

measured heat transfer coefficient at ‘Monju’ is shown in Fig. 3.7.3. It was clarified that the Nusselt

number is expressed by the correlation proposed by Seban-Shimazaki when the Péclet number is larger

than 30. Since the Péclet number is a product of the Reynolds number and the Plandtl number, the large

Péclet number means the large Reynolds number. On the other hand, the Nusselt number is lowered from

the Seban-Shimazaki’s correlation when the Péclet number is less than 30.

Fig. 3.7.3 Comparison of measured heat transfer coefficient and data in handbooks

3) Air coolers

In liquid metal cooled fast reactor, air

coolers (ACs) are used as one of

passive heat removal systems of decay

heat. Evaluation of the heat transfer

coefficient is generally difficult because

of a complex geometry. Figure 3.7.4

shows a schematic of the air cooler

provided at the second heat transport

system (HTS) of the ‘Monju’ reactor.

[1] Seban & Shimazaki

[2] Martinelli-Lyon

[3] Lubarsky & Kaufman

( Nu=0.625Pe0.4)

[3]

0.1

1

10

100

1 10 100 1000 104

105

PrimarySecondary

N u

Pe

Inlet vanes (controlled)Blower

Inlet damper (open-close)

Finned heat transfer tubes

Exit damper (controlled togetherwith inlet vanes)

Approx.30m

~4.5m ~5.3m

~6.5m

Rated: 15MW

Fig. 3.7.4 Schematic of air cooler


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Sodium in the secondary HTS flows inside the finned heat transfer tubes shown in the figure, and cooled by

air flow. Some studies have been done for the forced circulation heat transfer realized by a blower.

However, an appropriate heat transfer correlation is required to calculate the accurate temperature in the

case of the natural circulation. The heat transfer inside the heat transfer tubes are evaluated using the

following empirical correlation.

8.0

3 025.00.5 Pe Nu Seban & Shimazaki (1951) (3.7.9)

Heat transfer from finned heat transfer tube to air can be evaluated by the following empirical correlation

derived from the air-cooling experiment conducted at 50 MW steam generator facility and ‘Monju’.

31988103

1 107969 / .

Pr Re. Nu Re


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correlations.

In the case where AC is operated by natural circulation, the buoyancy force should be calculated using


ALT T gF a B (3.7.16)

A: flow area (m2), L: flow path length (m)

β: volume expansion rate of air (1/℃)

4) Steam generators

In this section, steam generators (SGs) for the fast breeder reactor is explained. In SG, high temperature

coolant flows outside the pipe a

Two Phase Flow 2009

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