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

i

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 Valves

3.6 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

ii

Nomenclature

A area (m2)

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

CD discharge coefficient (-)

C0 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 de/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/s2 K)

Re Reynolds number (=ude/)

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)

iii

W mass flow rate (kg/s)

x steam quality (-)

Xtt 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)

- 1 -

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 the

core 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 (m2) 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 horizontal flows

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 boilingSubcooled boiling

Liquid flow

Churn flow

Fuel pin

- 2 -

A

Qj

gg (1.1.1)

A

Qj 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.

PhaseLiquid

PhaseGas

0

1 (1.2.1)

V

dVV

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 l0 (

m/s

)

jg0 (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 StratifiedWavy

Annular

Bubbly

l

g

- 3 -

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

WW

Wx

(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

iix

(1.2.4)

We can define even negative quality as follows.

lg

sat

lg

sublsub i

ii

i

TCpX

(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

kk A

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

AA kk (1.3.2)

A

Qj 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

- 4 -

kkk uj (m/s) (1.3.4)

kkkkkk juG [kg/m2s] (1.3.5)

AGW kk [kg/s] (1.3.6)

j

jkk (1.3.7)

is called as the volumetric flow fraction.

Therefore, the steam quality is rewritten as follows.

lglggg

ggg

lllggg

ggg

lg

g

lg

g

uu

u

uu

u

GG

G

WW

Wx

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

11

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

11

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.

- 5 -

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

Measured

Correlation

Void fraction, 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 fraction of PWR fuel bundle

Bankoff

Thom

Armand

- 6 -

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

ii

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 ( 2100Re ) (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.

DR

: sand roughness (m)

In a computer code, the friction factor is approximated by the following equation. cReba (Re>4000) (1.4.3)

Fig. 1.4.1 Moody’s chart

Reynolds number, Re

Hydraulicallysmooth

TurbulentTransition

λ＝64/Re

Equ

ival

ent

rela

tive

roug

hne

ss, ε

/D

Laminar

Fric

tion

fact

or,

λ

Reynolds number, Re

Hydraulicallysmooth

TurbulentTransition

λ＝64/Re

Equ

ival

ent

rela

tive

roug

hne

ss, ε

/D

Laminar

Fric

tion

fact

or,

λ

- 7 -

RRa 53.0094.0 223.0 44.00.88 Rb 1340621 .R.c

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

103<Re<107.

3161020000100550

/

ReD.

(1.4.4)

When these correlations are applied to the reactor core, the equivalent diameter De should be used

instead of the pipe diameter D. This hydraulic equivalent diameter can be calculated using the

following definition.

he L

AD

4 (1.4.5)

A: flow area (m2)

Lh: perimeter (m)

In case of the two-phase flow, the pressure loss becomes larger than that in the single-phase flow with

the same mass flow rate. Usually, this increase can be evaluated using the two-phase multiplier 2

defined as follows.

SPF

TPF

z

P

z

P

2 (1.4.6)

The two-phase multiplier is defined by a ratio of the pressure loss in a two-phase condition to the

pressure loss in a single-phase flow when the same amount of flow rate is achieved in a flow path.

Lockhart-Martinelli conducted many studies about the two-phase pressure loss, and they proposed the

Lockhart-Martinelli parameter given by the following equation.

901050 .

g

l

.

g

l

.

l

gtt W

WX

(1.4.7)

The relationship between the parameter X and two-phase multiplier were studied by many researchers.

The example of the study is shown in Fig. 1.4.2. The following correlations can be used not only for

friction loss but also for the local losses such as bends and valves.

22 1

1XX

Cl (1.4.8)

22 1 XCXg (1.4.9)

The constant C is given by the following table.

- 8 -

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)

Rel<1000

Turbulent(t)

Reg>2000

501090

40

59

1.

l

g

.

g

l

.

g

l.g G

GRe

12

Turbulent(t)

Rel>2000

Laminar(v)

Reg<1000

505050

40

159

.

l

g

.

g

l

.

g

l.

gG

G

Re

10

Laminar(v)

Rel<1000

Laminar(v)

Reg<1000

505050 .

l

g

.

g

l

.

g

l

G

G

5

Vo

id f

ran

ctio

nα

Fig. 1.4.2 Relationship between parameter X and two-phase multiplier or void fraction

This parameter is used to express not only the pressure loss but also the heat transfer coefficient. The

two-phase multiplier is very much dependent on flow configurations such as pipes with various

roughnesses, elbows, and obstacles such as spacers of a fuel assembly. Therefore, the two-phase

multipliers should be measured experimentally for these configurations. Examples are shown in Figs.

1.4.3 and 1.4.4.

When we assume the homogenous flow, the following equation is derived by the theory.

- 9 -

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

gjgj

gjgg

ujC

ujj

juj

ujuu

0

(1.5.3).

j

jC

0

Ｐｒｅｓｓｕｒｅ；6.9 MPaＭａｓｓ　Ｖｅｌｏｃｉｔｙ

3000 ㎏/㎡ｓ22001500850

Thermal equilibrium steam qulity：x（－）

Two-phasemultiplier;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-phase multiplier 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

Two-phase

mul

tiplier

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

- 10 -

The above relations are re-written simply as follows.

gjg

g ujCj

u 0 (1.5.4)

gl jjj A

Qj g

g A

Qj l

l

ugj: vapor drift velocity to average superficial velocity (m/s), C0: 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 C0

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..

l

g

for annular tube

181350351

exp..

l

g

for rectangular tube

Churn-turbulent

flow 4

1

22

l

gl g

Slug flow 21

350

l

gl gD.

Annular flow 2

1

0150

1

4

1

l

gl

l

g .

gD

l

g

4

11

Droplet flow 41

22

g

gl g

1

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

- 11 -

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 gas7 MPa Nitrogen gas7 MPa Vapor

j (m/s)

Vg (

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

- 12 -

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

tm (2.1.1)

llgg uuG 1

dlA

singz

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)

fssc

sfcc

f

c TThA

FTTh

A

Fq (W/m3) (2.1.5)

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

1lgm (2.1.6)

lgm

xx

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

dPiiGPG

(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

Reflectoror thermal insulator

Gap

Fuel

Reflectoror thermal insulator

Cladding

Spring

Pellet

Reflectoror thermal insulator

Gap

Fuel

Reflectoror thermal insulator

- 13 -

fPcPP

PPPP qTTh

A

F

dt

dTCp (2.1.8)

cfcc

ccPP

c

Pccc TTh

A

FTTh

A

F

dt

dTCp (2.1.9)

sees

esfs

s

ssss TTh

A

FTTh

A

F

dt

dTCp (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

kkk

m uzt

(2.2.1)

i

iiwllll

llllll

L

A

AA

Persing

z

P

z

u

t

u

2

(2.2.2)

i

iiwgggg

gggggg

L

A

AA

Persing

z

P

z

u

t

u

2

(2.2.3)

The above two equations yields the following equation:

A

Persing

z

Pu

zu

t wmkkkkkk

2 (2.2.4)

A

PerqPeu

ze

t kkkkkm

(2.2.5)

Where e is total energy expressed by the following equation:

sinzguie kkk 2

2

1 (2.2.6)

kkgl ixixixi 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.

- 14 -

kkkkkk uzt

(2.3. 1)

ikkk

ikkkkkkkkkkkk uz

PgFPz

uz

ut

2 (2.3.2)

ikwkikikkk

ikkkkkkkkk

kkkkkkkkk

qquht

PguuFuPz

uuez

uet

2

22

2

1

2

1

2

1

(2.3.3)

where, Γk, Fk 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.

Thq (2.4.1)

T=Tf – Tc

q: heat flux(W/m2)

h: heat transfer coefficient (W/m2K)

Tf: fuel surface temperature（K）

Tc: 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.

- 15 -

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 numberk

dhNu 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

103104105106107

Nu Prg

-0.4=0.023(-Re)0.8

Nu Prg

-0.4=0.926(-Re)0.33

+20%

-20%

Minimumheattransfer

- - - - -

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

Nu

Pr g-0

.4

50

Nu Prg

-0.4=0.023Re0.8

Power dist.uniformchopped cosine

-30%+50%

P(MPa)0.3-0.9 5-7 Condition

Two-phaseVapor flow

- 16 -

Table 2.4.1 Heat transfer correlations

Regime Heat transfer coefficient Nomenclature and others

Subcooled Dittus-Boelter

40800230 .l

.l

e

l PrRe.d

kh

g

geg

l

gggx

l

lelx

l

lel

udRe

xxReRe

xudRe

udRe

1

1

Nucleate

boiling

Jens-Lottes

634

1

82.0p

x eqT

（Engineering unit）

ｐ(ata), q (kcal/m2 h)

Rohsenow

71

3

1

0130

.l

gl

l

lfgfg

xl

Pr

gH

q.

H

TCp

Thom

68821

02430 .p

x eq.T

（Engineering unit）

Schrock-Grossman

4080

105090

750

0230

1

152

.l

.lx

e

llx

.

g

l

.

l

g.

tt

lx

.

ttN

PrRe.d

kh

x

xX

hX

.h

Transition

boiling

FN h)(hh 1

)/texp( nucleate boiling to film boiling,

)/texp( 1 film boiling to nucleate boiling

Film boiling Dougall-Rohsenow

40800230 .g

.gx

e

gF PrRe.

d

kh

Super

heated

Dittus-Boelter

40800230 .g

.g

e

g PrRe.d

kh

Cp : specific heat [J/kg K]

x : quality [-]

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

de : equivalent diameter [m]

Re : Reynolds number [-]

Pr : Plandtl number [-]

Hfg : 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)

- 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.

constp

(2.5.2)

Eq. (2.5.1) yields the following.

RTp

c

(2.5.3)

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

RT

u

c

uM

(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 T0 is defined by the following equation.

2

2

0 2

11

2MT

C

uTT

p

(2.5.5)

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

relationships,

RTp (2.5.6),

RCC vp (2.5.7),

we can obtain the following equation.

p

TC p 1 (2.5.8)

p0, 0, c0pe, e, ue

Fig. 2.5.1 Flwo in a nozzle

02

2

1iui

Ideal gas i=CpT

- 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

112

1

p

p

ppu

(2.5.9)

12100

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

00 1

1

2

p

pcu e

e (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

000

1

00

1

2

11

2

p

p

p

pAc

p

pAcAum

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

- 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.

12

1

000 1

2

1

2

AcAcAum cccc (2.5.17)

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

2

1

1

1

00 1

2

p

A

mc (kg/m2 s) (2.5.18)

In another method, the continuity equation is

.constuA (2.5.19)

0A

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

dp

udu (2.5.21).

Therefore,

sdp

d

u

dp

A

dA 2

1 (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.

- 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:

mmmaxc d

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

dpuxxu

dz

dG 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 uxxup

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 SuuGxG (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

Sxx

SG

1

(2.5.32)

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

xxSS

Sxx

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 CD.

eD ppCG 02 (2.6.1)

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

- 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)

Dis

char

ge c

oeff

icie

nt, C

D(-

)

CD=G/(2 P)0.5

G: kg/m2sP: Pa

0.59

- 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

Heat flux, q

Twall

-TB

Nucleate boilingNon-boiling

Dryout or DNB point

Film boiling

Minimum heat flux point

Transition boiling

- 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.

qc = F(x, fp, fL, fsp, fe, fa, Fsub) (3.1.1)

qc: critical heat flux

x: average thermal equilibrium steam quality

fp: factor of pressure

fL: factor of local peaking

fsp: factor of spacer

fe: factor of eccentricity of fuel assembly

fa: factor of axial power distribution

fsub: 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.

2321 xaxaaqst (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

HTLSEL

Mass velocity (kg/m2s)

Tot

al p

ower

(M

W)

36-rod bundleP = 7 MPaT

in = 548 K

0

0.5

1

1.5

2

2.5

3

-1000 -600 -200 200 600 1000

Tot

al p

ower

(M

W)

Fig. 3.1.2 Critical power of a fuel assembly

- 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.

oxxo

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

Crit

ical

hea

t flu

x,O

pera

tiona

l hea

t flu

x

qo

qc

Thermal equilibrium steam quality

xo

xc

Operational condition

xi

Operational conditionPower increase

CHF correlation

CHF correlation

i) CHFR

ii) CPR

Crit

ical

hea

t flu

x,O

pera

tiona

l hea

t flu

x

Thermal equilibrium steam quality

Crit

ical

hea

t flu

x,O

pera

tiona

l hea

t flu

x

qo

qc

Thermal equilibrium steam quality

xo

xc

Operational condition

xi

Operational conditionPower increase

CHF correlation

CHF correlation

i) CHFR

ii) CPR

Crit

ical

hea

t flu

x,O

pera

tiona

l hea

t flu

x

Fig. 3.1.3 Safety index

- 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 22

604T

N

gHQT

GDdt

dNm

(3.2.1)

N: rotational speed（1/s）

GD2: pump and motor inertia (kg･m2)

g: gravitational acceleration (m/s2)

Tm: 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 (-)

T0: torque of friction (N･m)＝k･Tm (k: constant, eg.0.04)

The pump head is approximated by the following equations.

2

3212

n

qh

n

qhh

n

h

10

n

q (3.2.2)

2

3212'''

q

nh

q

nhh

q

h

10

q

n (3.2.3)

- 26 -

0N

Nn ,

0Q

Qq ,

0H

Hh

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

NnF

PT N

m (3.2.5)

P0: rated pump shaft power (kW)

ω: angular velocity (＝2N0) (rad/sec)

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

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

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.

4

32

1 gHNQns (3.2.6)

N: rotational speed (1/s), Q: volumetric flow rate (m3/s), g: gravitational acceleration (m/s2), H: head (m).

When NPSH is defined by Hsv, 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)

HnH ssv 3

4

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

- 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

dco W

WxatioCarryoverR (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 ATR Fig. 3.3.2 Carryover analysis by particle tracking

- 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)

- 29 -

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 the

Electric Hydraulic Controller (EHC).

G

Mainsteam

MSIV Main controlvalve

Drain separator

Highpressureturbine

Low pressureturbine

Generator

Condenser

Condensatepump

Feed water heatersFeed waterpump

Containmentvessel

Bypass valve

Feed waterControl 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

- 30 -

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

SA SC

TA A

B C

D

TB

Heat QH

Cooling QC

Work L2

Work L1Work 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

Pressure (MPa)

Step

1 2 3 4 5 6 7 8

Inlet

to low pressureturbine

Extract Extract

- 31 -

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

LLLL

LLLLQQQ CH

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

H

CH

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

f1f2

T

Ｓ0

AB

B’C’

C

D

SA SC

E

e1e2

f1f2

T

Ｓ0

AB

B’C’

C

D

SA SC

E

T

Ｓ0

AB

B’C’

C

D

SA S’C

D’

SC

E

Saturation curve

Liquid

Super- heatedvapor

Two-phase

Critical point647.3 K, 22.1MPa

T

Ｓ0

AB

B’C’

C

D

SA S’C

D’

SC

E

Saturation curve

Liquid

Super- heatedvapor

Two-phase

Critical point647.3 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

- 32 -

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 )BSCS'C'BB(AreaiiQ ACBC 1 (3.4.1)

Discharged heat )(2 ASADSAreaiiQ ACAD (3.4.2)

Power output DC iiL 1 (3.4.3)

Work by pump AB iiL 2 (3.4.4)

The effective work is expressed by the following equation.

)''(2121 CDACABBAreaQQiiiiLL ABDC (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

BSCSCBBArea

CDACABBArea

Q

QQ

Q

LL

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 ACB'C 1 (3.4.7),

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

Therefore, the efficiency is expressed as follows:

)'''(

)'''(

1

21

BSSCBBArea

ADCABBArea

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 f1 by heating. The other one represents extraction

- 33 -

at e2 and becomes condensate f2 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

the following equation.

1

1

fC

Dej

n

j jDC

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 Cv 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 Cv value is equal to W. The relationship

between the local loss coefficient ζ and the Cv value is expressed using the

following correlation.

2

481038.21

VC

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

- 34 -

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

12

ii+1

Tpi

Tpi+1

Tsi

Tsi+1Tti ⊿Z

N

Coolant: Shell side

Coolant: Tube side

12

ii+1

Tpi

Tpi+1

Tsi

Tsi+1Tti ⊿Z

N

- 35 -

equations.

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

flow direction;

2

2

z

Tk

A

qTT

A

K

z

TGC

t

TC

pl

ppt

p

pppp

ppp

(3.7.1)

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

2

2

z

TkTT

A

K

z

TGC

t

TC s

lsts

ssss

sss

(3.7.2)

For heat transfer tubes;

tst

stp

t

pttt TT

A

KTT

A

K

t

TC

(3.7.3)

where,

p,fpsp

p

tpp

p

hddd

dln

khd

NK

12

2

11

(3.7.4)

s,fss

sp

tss

s

hdd

ddln

khd

NK

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 (m2)

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

d: diameter of heat transfer tube (m) ｛(dp-ds)/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’=UHXPA(TP-TA) : (W/m)

PA: perimeter of shell side (m) Ｔ: temperature (K)

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

ρ: density (kg/m3)

Subscripts

- 36 -

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

ssf

o

i

o

t

o

pfp dh

d

dh

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)

PrRePe

/udRe e , k/Cpa/Pr , k/hdNu 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

- 37 -

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

Nu

Pe

Inlet vanes (controlled)Blower

Inlet damper (open-close)

Finned heat transfer tubes

Exit damper(controlled together with inlet vanes)

Approx.30m

~4.5m ~5.3m

~6.5m

Rated: 15MW

Fig. 3.7.4 Schematic of air cooler

- 38 -

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.

8.03 025.00.5 PeNu 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 /. PrRe.Nu Re<3000 (3.7.10) 3167020

2 13700 /. PrRe.Nu Re≧3000 (3.7.11)

Nu = min (Nu1, Nu2) (3.7.12)

Mochizuki (2007) obtained the above

correlation as shown in Fig. 3.7.5. Since the

effect of fin is removed from the above

correlations, the fin efficiency should be taken

into account to evaluate the heat transfer with

various fin configurations. The heat transfer

coefficient h obtained from the above

correlations has the following relationship

with the fin efficiency .

T

fbT A

AAhh

(3.7.13)

Where, AT, Ab, Af stand for total surface area

of finned heat transfer tube, surface area on the

heat transfer tube between fins, and surface area of fins. φ was studied by Gardner (1945). In the case

of the disk type fin, the fin efficiency is expressed by the following Bessel functions.

bb

bb

b

fb

f uKuI

uKuI

u

uu

00

11

2

1

2

(3.7.14)

f

f

uK

uI

1

1 ,

1

/

o

fb

d

dkyhH

u ,

o

fbf d

duu (3.7.15)

As you can see from Eq. (3.7.15), the fin efficiency is a function of the heat transfer coefficient.

Therefore, iterative calculations are necessary to obtain the final heat transfer coefficient from the

Fig. 3.7.5 Heat transfer of finned AC

0.1

1

10

100

1000

10 100 1000 104 105

50 MWSGMonjuJoyo

Nu/

Pr1/

3

Re

Nu/Pr1/3=0.1370Re0.6702

+20%

-20%

Data by Jameson

+10%

-10%

Nu/Pr1/3=9.796 10-3Re0.9881×

- 39 -

correlations.

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

the following equation.

ALTTgF aB (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 and water flows inside the heat transfer tubes. The previous correlations

can be applied to evaluate heat transfer for the straight-tube-type steam generators. However, in the case

of ‘Monju’, the heat transfer tube configuration is helical-coil-type. In this type, centrifugal force effects

on the heat transfer. Therefore, the heat transfer coefficient is different from that of straight tube.

There are only two empirical correlations for the helical-type heat transfer tube. Mori-Nakayama

(1967) studied the heat transfer coefficient for single-phase flow of liquid and gas. These correlations are

used coupling with Schrock-Grossman (1959) that can be used for subcooled boiling and nucleate boiling.

61

52

121

65

40 06101

41 .

coil

il

coil

il

.l

l

ill

D

dRe

.

D

dRe

Pr

k

dhNu (3.7.17)

(Mori & Nakayama),

,1

5.275.0

ltt

hX

h

(Schrock & Grossman) (3.7.18)

Where,

1.05.09.01

g

l

l

gtt x

xX

(3.7.19)

hl is given by the Mori-Nakayama correlation.

Heat transfer coefficient for vapor is given by the Mori & Nakayama (1967) correlation.

5

12

101

54

32

09801

0740226

coil

ig

coil

ig

g

g

g

igg

D

dRe

.

D

dRe

.Pr.

Pr

k

dhNu (3.7.20)

For film boiling heat transfer coefficient, Eqs. (3.7.17) and (3.7.20) are interpolated according to the

value of steam quality x. Heat transfer coefficient between sodium flow and the heat transfer tube is

- 40 -

evaluated by the Hoe et al. (1957) correlation.

3

2228.003.4 N

O

NN Pe

d

kh (3.7.21)

Using these correlations, temperature distributions

measured at the 50 MW SG are evaluated. The Results

are shown in Fig. 3.7.6 and Fig. 3.7.7.

200

250

300

350

400

450

500

-1 0 1 2 3 4 5 6 7

Exp. No. R18-0010

Water in downcomer (Exp.)Water in heater tube (Exp.)Sodium (Exp.)Temperature difference (Exp.)Water in dwoncomer (NETFLOW)Water in coil (NETFLOW)Sodium (NETFLOW)

Te

mp

era

ture

Position (m)Bottom Top of SG

Axial Temperature Profile of EV: 100% Load

（℃）

Effective helical coil region

A

BC D

A: Non-boiling heat transfer, B: Subcooled boiling, C: Nucleate boiling, D: Transition boiling to

Fig. 3.7.6 Temperature distribution in evaporator of 50MW SG

Photo 3.7.2 50MW SG Facility

450

460

470

480

490

500

-0.5 0 0.5 1 1.5 2

Exp. No. R18-0010

Sodium (Exp.)Temperature difference (Exp.)Sodium (NETFLOW)

Tem

pera

ture

Position (m)Bottom Top of SG

Axial Temperature Profile of SH: 100% Load

（℃

）

Effective helical coil region

Photo 3.7.1 Evaporator of 50MW SG

E

Fig. 3.7.7 Temperature distribution in super heater of 50MW SG

- 41 -

dryout, E: Gas flow.

3.8 Control rod

Configuration of control rod is different form

each plant type. In the BWR core, the control

rods are driven by water pressure from the bottom

of the core. In the PWR plant, they are driven by

motors from the top. While, in FBR, ATR,

CANDU, they are driven by motors from the top.

Since the environment pressure is atmospheric,

gravitation can be used to drop the rods. The

mechanism driving the rods is called as control rod

drive (CRD). A wire hanging the rod is

connected to CRD via an electro-magnetic clutch.

If this clutch could not be cut during an abnormal

transient, the reactor would encounter the very

crucial situation. This situation is called as

ATWS (anticipated transient without scram), and

countermeasures should be taken in the design.

The control rods must be inserted into the core within the

designed time even in the case of the earthquake.

Therefore, special tests are needed to confirm the capability

using a mock-up shaken by simulated acceleration. In the

case of LWR, the safety rods are inserted into the core

within 2 seconds.

Negative reactivity is inserted into the core as the control

rod insertion. Because of the axial distribution of the

neutron flux, a relationship between the reactivity and

position is S shape. Figure 3.8.1 shows negative

reactivity of the safety rod vs control rod position at the

experimental fast reactor ‘Joyo’. When relative value is

used, almost all the curves become similar.

Since the safety rod is a very important component to stop the neutron reaction, the different types of

rods must be provided in order to escape the common failure. In heavy water reactors, liquid poison is

injected into the heavy water moderator. In fast breeder reactor, a new type shutdown system is under

development. This system is called as self-actuated shutdown system (SASS) that uses a very special

magnetic clutch as shown in Fig. 3.8.2. The magnetic clutch is consists of specific magnet that loses

magnetic forth at the specific temperature. Therefore, if the exit temperature of the reactor increased, the

Coil

Temperature sensible Alloy(30Ni-32Co-Fe )

Connecting surface

Armature

Electric magnet

Iron

Top of CR

Self Actuated Shutdown System (SASS)

Coil

Temperature sensible Alloy(30Ni-32Co-Fe )

Connecting surface

Armature

Electric magnet

Iron

Top of CR

Self Actuated Shutdown System (SASS)

Fig. 3.8.2 SASS of FBR

Fig. 3.8.1 Characteristic of control rod

- 42 -

control rods were declutched automatically. The system is tested using the ‘Joyo’ reactor and shows the

good result.

3.9 Example of the plant

Figure 3.9.1 shows the reactor vessel and

internals. The coolant is re-circulated by

internal pumps housed in the reactor vessel,

and becomes two-phase mixture during

passing through the core. The two-phase

mixture is separated into steam and water using steam separators provided above the core. The separated

water returns to the pump through the downcomer between the reactor vessel and the shroud. The steam

pressure at approximately 7.2MPa (saturation temperature: 287℃) is taken out of the reactor vessel and is

supplied to the turbine. The vapor is condensed in the condenser and feed water at approximately 215℃

supplied to the reactor via the nozzle provided on the reactor vessel. A sparger is provided at the top of

the feed water pipe, and supplies subcooled water. Carry-under voids are condensed at this level, and

re-circulation flow becomes subcooled water at approximately 276℃ . The increase of coolant

temperature in the BWR core is only about 10℃ because energy generated in the core is used to evaporate

coolant. The coolant temperature increase in the PWR core is larger than that of BWR because the

generated energy is absorbed by coolant in the form of heat capacity.

Dryer

Separators

Fuelassemｂｌies

Internal pump

Control rod drive

Reactor vessel and internals

Dryer

Separators

Fuelassemｂｌies

Internal pump

Control rod drive

Reactor vessel and internals

Fig. 3.9.1 Reactor core of ABWR

Fuel rod

Spring

Cladding (Zircalloy)

Pellet

Pellet

Handle

Upper tie plateOuter spring

Fuel rod

Support grid

Channel box

Lower tie plate

Fuel rodWater rod

Control rod

Channel box

14cm

10 mm

10 mm

Fuel rod

Spring

Cladding (Zircalloy)

Pellet

Pellet

Handle

Upper tie plateOuter spring

Fuel rod

Support grid

Channel box

Lower tie plate

Fuel rodWater rod

Control rod

Channel box

14cm

10 mm

10 mm

Fig. 3.9.2 Fuel bundle and CR

- 43 -

In ABWR, 10 internal pumps are provided in the reactor vessel and each pump cools fuel assemblies in

one sector. The fuel assembly is housed in a channel box as shown in Fig. 3.9.2. The fuel assembly

consists of fuel rods and other components. Many pellets of 10mm in diameter and 10mm in height are

contained inside the fuel rod made of zircalloy. There is a space upper part of the fuel rod, i.e., cladding

in order to contain gases as fission products.

The control rods are driven by water from the bottom of the core. The control rod is inserted

in-between the four fuel assemblies. In side the + shaped control rod, B4C spheres are packed in the

claddings. Due to this mechanism, the ejection of the control rod is taking into account in the safety

design.

An example to calculate plant dynamics is shown in Fig. 3.9.3. Since the void reactivity of BWR is

large in negative value, the positive reactivity is inserted into the core when voids are collapsed. As for

anticipated transients and accidents, items listed in Table 3.9.1 are evaluated. In any abnormal transients,

it is not allowed to cause dryout of cladding surface. In accidents, the dryout is allowed, however,

maximum cladding temperature shall be kept less than 1200℃ by providing properly engineered safety

features. That temperature is the maximum temperature not to occur exothermal reaction between

zircalloy claddings and water.

Fig. 3.9.3 Example of analytical model of plant dynamics

Table 3.9.1 Transients and accidents in the safety analysis of BWR

Abnormal transients Accidents

Control rod withdrawal in start-up

Control rod withdrawal during power operation

Loss of onsite power

Loss of feed water heating

Malfunction of re-circulation flow control system

All re-circulation pump trip accident

Loss of coolant accident (LOCA)

Main steam pipe break accident

Control rod ejection accident

- 44 -

Re-circulation pump trip

Load rejection

Closure of MSIV

Malfunction of feed water control system

Malfunction of pressure control system

Loss of total feed water

Break of radioactive waste gas process unit

Fuel handling Accident

References

Gardner, K.A., 1945. Efficiency of Extended Surface, Trans. ASME, 67, 621-631.

Gellerstedt, J.S., Correlation of Critical Heat Flux in a Bundle Cooled by Pressurized Water, BAW-1000A

Topical Report, 1976.

Jameson, S.L., 1945. Tubing Spacing in Finned Tube Banks, Trans. ASME, 67, 633-642.

Hench, J.E. et al., Design Basis for Critical Heat Flux Condition in Boiling Water Reactors, APED-5266,

1966.

Hoe, R.J. et al., 1957. Heat-Transfer Rates to Crossflowing Mercury in a Staggered Tube Bank, Trans.

ASME, 79, 899-907.

Huges, E.D., A Correlation of Rod Budle Critical Heat Flux for Water in the Pressure Range 150 to 725

psia, IN-1412, 1970.

Lubarsky, B. and Kaufman S.J., 1955. Review of Experimental Investigations of Liquid-Metal Heat

Transfer, NACA TN 3336.

Lyon, R.N., 1949. Forced Convection Heat Transfer Theory and Experiments with Liquid Metals,

ORNL-361.

Lyon, R.N., 1951. Liquid metal heat-tranfer coefficients, Chemical Engineering Progress, 47, 2, 75-79.

Mochizuki, H. and Takano, M., 2009. Heat Transfer in Heat Exchangers of Sodium Cooled Fast Reactor

Systems, Nuclear Engineering and Design, 239, pp. (In press).

Mori, Y. and Nakayama, W., 1967. Study on Forced Convective Heat Transfer in Curved Pipes (2nd Report),

Int. J. Heat Mass Transfer, 10, 37-59.

Mori, Y. and Nakayama, W., 1967. Study on Forced Convective Heat Transfer in Curved Pipes (3rd Report),

Int. J. Heat Mass Transfer, 10, 681-695.

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.

Seban, R.A. and Shimazaki, T.T., 1951. Heat Transfer to a Fluid Flowing Turbulently in a Smooth Pipe

with Walls at Constant Temperature, Trans. ASME, 73, 803-809.

Subbotin, V.L., et al., 1962., A Study of Heat Transfer to Molten Sodium in Tubes, Translated from

Atomnaya Energiya, 13, 4, 380-382.

- 45 -

4. Plant stability

When a nuclear power plant is designed, the following items relating plant dynamics shall be evaluated

in the nuclear regulation law, and the results are reported in the commissioning document.

1) channel hydraulic stability

2) core stability

3) plant stability

4) xenon oscillation stability

Regarding these items, the attenuation ratios are regulated. The attenuation ratio is defined as the ratio

of second overshoot to the first overshoot when a stepwise input is added to the system. When this ratio is

less than 1, the oscillation is attenuated. However, this value is regulated less than 0.5 or 0.25 depending

upon the situation.

Regarding the channel hydrodynamic stability, since power might cause oscillation if a density wave

oscillation occurred, it must be shown that the operation conditions are within the stability region. It is

usual that this stability is evaluated by a frequency domain code. Regarding the core stability, it must

be shown that the oscillation coupling with nuclear characteristics does not occur. In the plant stability,

it must be evaluated that the oscillation coupled with control systems does not occur. To evaluate these

sort of oscillations, stepwise change of the reactor power, system pressure, water level and others are

added to the systems during the commissioning tests other than the analysis beforehand. Regarding

xenon oscillation, it must be shown by analysis that the oscillation does not occur. Furthermore, it must

be shown that the dryout of the fuel surface and cladding maximum temperature are within the regulation,

and suitable margin is kept.

4.1 Channel hydraulic stability and core stability

In general, the equation of oscillation is expressed by the following equation.

02 22

2

xndt

dx

dt

xdn (4.1.1)

n: angular frequency

ζn: attenuation constant

When ζn is larger than 0, the solution of the above

equation is attenuated. Whenζ n is 0, the solution

shows the limit cycle.

The hydraulic, core and plant stabilities are evaluated

by two parameters. One is the attenuation ratio shown

in Fig. 4.1.1. This ration is expressed as X2/X0. The

other one isζn. These two parameters mean the same

X0

X1

X2

Time

Value

X0

X1

X2

Time

Value

Fig. 4.1.1 Attenuated oscillation

- 46 -

thing.

(a) Regulated value

The plant shall be designed in order to satisfy the following conditions for all operating conditions.

Channel hydraulic stability 102 X/X , ζn＞0

Core stability 102 X/X , ζn＞0

Plant stability 102 X/X , ζn＞0

(b) Operational regulation

The safety of the plant is guaranteed by the above regulation. However, the following conditions are

targets for the plant operation.

Channel hydraulic stability 5002 .X/X , 110.n

Core stability 25002 .X/X , 22.0n

Plant stability 25002 .X/X , 220.n

In a sense, these are margins for the plant design. Therefore, there are some cases that these values are

not satisfied with real operation.

4.2 Ledinegg instability

In same cases, the pressure loss along the flow

passage shows not the quadratic characteristic but the

characteristic having a negative gradient as shown in

Fig. 4.2.1. If the pump characteristic is case-1 and

the pressure-loss characteristic crosses at three points

like case-1, the operation condition of the pump is

unstable. For example, if the pump is operated at the

point A, the operating condition has an excursion to B

or C when a small perturbation is added to the pump.

This flow instability is called the Ledinegg instability. In order to escape from this instability, the both

characteristics should be changed not to have plural solutions. The following methods are taken in

general; the pressure loss characteristic shown as case-2 by decreasing inlet subcooling. The orifice is

installed at the inlet of the core, the pump characteristic is changed like case-3. Since this curve does not

have plural solutions, the system is stabilized.

4.3 Density wave oscillation

In the some conditions of the two-phase flow, the density wave oscillation may occur. When a

perturbation at the inlet is added to the total flow rate W, flow rate changes to W+W. Due to this

perturbation, the boiling boundary moves upward. The pressure loss in the single-phase region increases

by P1 and decreases by P2 in the two-phase region. Naturally the pressure loss change in the

Fig. 4.2.1 Pressure loss of boiling two-phase flow

ΔP

Flow rate

Pump characteristic: case-2

Pressure loss: case-1

Pressure loss: case-2

Case-1B

C

E

DA

Pump characteristic:

Diff

eren

tial p

ress

ure

Pump characteristic: case-3

- 47 -

two-phase region is larger than in the single-phase region. Since the

pressure difference between the inlet and exit is normally kept at constant,

the flow changes to keep the following relationship.

021 PP (4.3.1)

Sound velocity of the two-phase flow is slower than that of single-phase

flow. Therefore, the flow in the single-phase region changes to take

balance, i.e., decreases the flow rate. Shortly, pressure disturbance of the

density change to increase the flow rate travels to the inlet. The phases of

both requirements are different each other. Due to this phase shift, the

flow is oscillated in some conditions. This is called the density wave

oscillation. During this oscillation, we can observe that the density change

of the two-phase flow propagates like a wave in the flow passage. In order

to stabilize the flow, an orifice at the entrance of the flow passage is

effective. In this case, the

relative pressure difference

in the two-phase region to

the single-phase region

becomes smaller than the

case where there is no

orifice. This type

oscillation in the natural

circulation condition was

investigated using a

full-scale heat transfer loop

(HTL) and a safety

experiment loop (SEL) of

the pressure-tube type

reactor Fugen as shown in Fig. 4.3.2, because this oscillation accompanies dryout in the core. It was

found that the density wave oscillation occurred in high quality regions. Q, W and Γ stand for power

(kW), flow rate (kg/s) and latent heat (J/kg), respectively. The abscissa stands for non-dimensional

subcooling defined as follows.

lg

sat

lg

sublsub i

ii

i

TCpX

(4.3.2)

The region inside the solid curve corresponding to individual pressure is stable. Measured data were

threshold to start oscillation.

Exit

Inlet

Tw

o-ph

ase

P２

Sin

gle-

phas

e

P１

Boundary of boiling

Con

stan

t pre

ssur

e di

ffer

ence

Fig. 4.3.1 Density wave oscillation

0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15 0.2

75317 (Present data)7531

Q/(W )

Xsub (-)

Xe = 1

Xe = 0

Low power oscillation

HTL

SEL

Pressure (MPa)

Stable for 1MPa

Stable

Stablefor 7MPa

Fig. 4.3.2 Threshold of density wave oscillation

- 48 -

4.4 Flow pattern induced oscillation

The flow pattern induced oscillation can

be seen in a flow passage having long

horizontal piping at the exit of the heating

section. This type oscillation was found

in the pressure-tube type reactor Fugen

when the plant was tripped as shown in

Fig. 4.4.1. By the tests using a full-scale

heat transfer loop of Fugen, it was

clarified that this oscillation was classified

into the low power oscillation. To

investigate the mechanism of the

oscillation, the test using water and air was conducted using HTL. Voids generated in the vertical section

flew into the horizontal section and coalesced to make a large plug. This plug escaped from the horizontal

section to the vertical section intermittently. This behavior caused the flow oscillation. In the case of

piping with 2-degree inclination, voids did not make a large plug but a small scale slug flew smoothly in

the line and the flow oscillation was not excited.

4.5 Geysering

In case of a long vertical piping with closed

end at the bottom or flow rate is very small,

sadden boiling would occur when subcooled

liquid is heated and reached to the super-heated

condition. Steam generated by the boiling

0

1

2

3

4

5

6

7

200

220

240

260

280

300

0 50 100 150 200 250 300 350 400Time (sec)

Ch.1175

Flo

w R

ate

(kg/

s), P

ress

ure

(MP

a)

Tem

pera

ture

(o C)

Measured Flow Rate

Header Pressure

Header Temperature

Saturation Temperature

Ch.1575

Fig. 4.4.1 Flow oscillation in Fugen reactor

Photo 4.4.1 Flow regime in horizontal piping

Pressure: atmospheric

Average water flow rate: 35 l/min

Average air flow rate: 15 l/min

Photo 4.4.2 Flow regime in piping with

2-degree inclination

Pressure: atmospheric

Average water flow rate: 35 l/min

Average air flow rate: 15 l/min

liquid

vaporheating

Condensation

Fig. 4.5.1 Illustration of geysering

- 49 -

flows out of the pipe and subcooled liquid flows into the vertical pipe. Due to the subcooled liquid,

boiling stops for a while. This type oscillation is called as geysering. The oscillation may be excited

under the conditions of the lower system pressure, the longer the downstream piping and the lower the inlet

flow rate.

4.6 Chugging

In the flow system where steam is supplied into the

subcooled water through a long piping as shown in Fig. 4.6.1, a

large flow oscillation may be excited under a certain condition.

This is the case in a steam relief pool of the nuclear reactor. In

this system, condensation induces flow fluctuation always.

However, steam can be supplied rather constantly. In the case

of the chugging, subcooled water reversed to the piping due to

violent condensation. The flow oscillation criteria are affected by steam velocity in the pipe, subcooling

of water adjacent to the exit of the pipe, ratio of non-condensable gas, compressibility of steam, etc.

Fig. 4.6.1 Illustration of chugging

- 50 -

5. Application of component modeling to the nuclear power plant

5.1 Plant transient in Liquid-metal-cooled fast reactors

5.1.1 Heat transfer between subassemblies

In case of a liquid-metal-cooled reactor, there are several

specific heat transfers to be considered. One of them is inter

subassembly heat transfer (ISHT). Since liquid-metal has

high thermal conductivity compared with water, there is a heat

flow across the core. Therefore, temperatures of all the fuel

assemblies and control rods should be evaluated taking into

account of ISHT. Otherwise, exit sodium temperature in the

center region is calculated higher than the reality and

temperatures in peripheral channels are calculated lower than

the reality. This phenomenon is outstanding during a low

primary flow condition in particular. The heat transfer

mechanism is shown in Fig. 5.1.1, i.e., the heat transfer based on heat conductions of both liquid metal and

wrapper tube and heat transfer by inter-wrapper flow in-between subassemblies.

When gap between the subassemblies isδg, and thickness of wrapper tube is t, the overall heat transfer

coefficient is expressed as follows for the concerned axial mesh j.

jk,sg

jk,liq

g

jk k

t

hkU

1 (5.1.1)

kliq: thermal conductivity of liquid metal

ks: thermal conductivity of structure (wrapper tube)

h: heat transfer coefficient by inter-wrapper flow

8002505 .

liq

e Pe.k

hdNu

l

ld

gge

2

l: facing peripheral length

The heat rate is evaluated by the following relation.

jk

ji,m

jki

jmi,k

jk TTUzlNQ

6

1

Nmk,i: number of subassemblies of channel group m facing to the face i of channel k

Δzj: length of axial mesh j

Tk

Tk＋1

t

k+1th layer

Tk-1

Center

k-1th layer

δg

Sodium flow

Tk

kth layer

Fig. 5.1.1 Inter-subassembly heat transfer

- 51 -

5.1.2 Turbine trip test of ‘Monju’

A turbine trip test at 45% thermal power of

Monju shown in Fig. 5.1.2 was conducted in

1995 to investigate the capability of the air

cooling system (ACS) in an actual situation.

In this test, an abnormal situation of the

turbine was assumed. The reactor was

scrammed by the signal of turbine trip, and

then pumps in primary and secondary loops

were tripped. Pony motors took over

operation when flow rates in primary and

secondary loops were approximately 10%

and 8%, respectively. The operation of the

blower of ACS was initiated to start cooling

of the secondary loop. The stop valve of

the steam generator (SG) was closed completely 30 seconds after the trip of the reactor, and at the same

Primary Heat Transport System (PHTS)Turbine system

Secondary sodium

Primarysodium

IntermediateHeat Exchanger(IHX)

Primary circulating

pump

Core

Air cooler（AC)

Evaporator(EV)

TurbineGenerator

Feed water pump

Sea water cooler

Condenser

Secondary Heat Transport System (SHTS)

Secondary circulating

pump

Super heater(SH)

PLUS CodeNETFLOW Code

NETFLOW++ code

Fig. 5.1.2 Schematic diagram of ‘Monju’

Fig. 5.1.3 Example of analytical model of ‘Monju’

7

Evaporator Super-heater

[13]

[1]～[7]

[47]

Monju 3-Loop Calculation Model

２10

-2

8 9

12

1

5

6

-1

[8] IHX

Air cooler

[28]

[29]

[30]

Highpressureplenum

Pump

Upper plenum

[31]

[32]

[33]

[15]

[16]

[34][35]

[17]

[14]

to turbine

11[10]

[11]

[12]

[9]

22.80

32.50

29.00

32.7533.05

23.8024.00

26.613

23.0021.62

21.35

28.45 26.85

31.90

34.20

29.20

26.20

38.66 38.42 38.42

49.30

31.41

31.45

42.20

43.30

37.11

28.20

28.84

34.8537.11

47.00

39.1735.27

50.9650.9647.68

47.68Pump

326.906

Link 1-Link 6: 1st to 6th layer (Inner driver)Link 7: 7th & 8th layer (Outer driver)Link 8: 9th to 11th layer (Blanket)Link 9: Center CRLink 10: CRsLink 11: Bypass

[24]

[18]

[19]

1

[46]

[23]

19

[49]

522

-4

2021

24

[42]

[43]

[44] [45]

23

[25]

[26]

[27]

13

[48]

416

-3

14

18

[37]

[38]

[39] [40]

17

[20]

[21]

[22]

1

[36]

[41]

to Loop-B

to Loop-C

Loop-ALoop-B

Loop-C from Loop-B, C

24.00

1 3

1

1

3

No. 2 feed water heater

No. 1 feed water heater

Feed water pump

Deaerator

Drain Drain

Extraction

Low-pressure turbine

Condenser(Pressureboundary)

3

４

15

2

(Flow boundary)

- 52 -

time turbine bypass valves were opened for a while to prevent the pressure increase and then closed

completely.

Figure 5.1.3 shows the calculation model of this even. One third of the core is represented by three

kinds of channels corresponding to inner core fuels, outer core fuels and blanket fuels etc. Shielding

channels are neglected in the calculation model. Total numbers of sub-assemblies considered in the

present calculation are 130, i.e., 36, 30 64 sub-assemblies for the above three kinds of regions. An upper

plenum model is not used in this calculation because this part is not cooled by dipped cooling heat

exchangers. In order to consider the inventory of sodium in the plenum that mixes with the main flow, a

junction to the free surface and to a flow passage for IHX is provided near the liquid surface. Mixed

sodium flows downward once, and flows out of reactor vessel. In the reactor vessel of Monju, there is

such a flow passage with holes at different heights. Flow coefficients of values, Cv, for the stop valve and

the bypass valve are given to the code by tables as a function of throttling. The vane is controlled by a

proportional-integral-derivative

(PID) controller model in the code.

A comparison between the test

and a calculation is shown in Fig.

5.1.4. Open symbols stand for the

test and closed symbols stand for the

calculation. P and S in parentheses

stand for the primary loop and the

secondary loop. Flow rate changes

in primary and secondary loops from

main motors to pony motors are

simulated. It is complicated to give

two characteristics for each pump to

calculate rated operation and low speed

operation. Due to this reason, flow

rate in the secondary loop is fluctuated

slightly. Flow rate of the primary loop

is simulated without any fluctuation.

Temperature behaviors in both loops are

also simulated with a satisfactory

accuracy. Temperature fluctuation

near 6000 seconds is due to the air

cooler operation mode change from

forced to natural circulation.

Measured temperatures and

calculated temperatures with the ISHT

300

350

400

450

500

0 2000 4000 6000 8000 1 10 4

1A11C12F12F23D33F14B34F14F25B1

5B36B16B66D1Cal. Layer 1Cal. Layer 2Cal. Layer 3Cal. Layer 4Cal. Layer 5Cal. Layer 6

Time (sec)

Temperature (℃）

Fig. 5.1.5 Temperatures at the exit of SAs

Fig. 5.1.4 Turbine trip at 45% thermal power

200

250

300

350

400

450

500

0

100

200

300

400

500

600

0 2000 4000 6000 8000 1 104

R/V exitIHX exit (P)IHX inlet (S)IHX exit (S)NETFLOWNETFLOWNETFLOWNETFLOWR/V Exit (SC)IHX exit (P) (SC)IHX inlet (S) (SC)

Flowrate (P) kg/sFlowrate (S) kg/sNETFLOWNETFLOW

Flo

w r

ate

(kg/

s)

Tem

pera

ure

(℃)

Time (sec)

- 53 -

model at the exit of subassemblies are shown in Fig. 5.1.5. They are corresponding to exit subassembly

temperatures for 1st to 6th layer, which are inner core driver fuel assemblies (SAs). Legends indicated with

the combination of an alphabet and figures are names of subassemblies: first figure indicates layer number.

All measured temperature trends from the center to the outer of the core are almost of the same magnitude,

and are predicted by the NETFLOW code with the same behavior. Although temperature curves without

the ISHT model are not illustrated in these figures, they are almost of the same magnitude as those with the

ISHT model due to the forced circulation using a pony motor. These trends are very much dependent on

coolant flow distributions by orifices at the entrance nozzle. Only a small temperature peak appears at

around 1100 seconds.

5.1.3 Natural circulation of sodium cooled reactors

In the case of a

liquid-metal-cooed fast

reactor, it is not difficult

to calculate a plant

transient under forced

circulation conditions.

However, it is generally

difficult to trace the plant

behavior under natural

circulation conditions.

The calculation should be

long. When the heat

transport system of the

fast reactor is operated

with natural circulation,

the system to remove decay heat of the core is operated with forced or natural circulation. These systems

should be correctly coupled in order to evaluate temperatures in the core and various portions of the plant.

In the present safety logic, the natural circulation has no credit. However, to show the capability of the

reactor cooling using the natural circulation is very important for the future reactor.

A natural circulation behavior is shown in Fig. 5.1.6. This shows the test in order to confirm the air

cooler (AC) capability when the secondary heat transport system is operated under the natural circulation

condition. The decay heat is simulated by heat produced in the primary pump. Therefore, the primary

heat transport system is operated under the forced circulation condition. All important plant parameters

are predicted with satisfactory accuracy. But this is not the actual natural circulation.

In fast reactors, it was estimated that temperature at the exit of the core might increase before the

establishment of the natural circulation because it took long period to establish buoyancy force depending

upon various temperatures in the plant. Therefore, a natural circulation test was conducted at the

Fig. 5.1.6 Natural circulation test-2 at ‘Monju’

200

250

300

350

400

0

50

100

150

200

250

300

350

400

0 2 4 6 8 10

R/V exit Temp. (Test)IHX second. inlet temp. (Test)IHX second. exit temp. (Test)ACS(B) Na exit temp. (Test)R/V exit temp. (Calc.)IHX second. inlet temp. (Calc.)IHX second. exit temp. (Calc.)ACS(B) Na exit temp. (Calc.)

ACS(B) Na Flow rate (Test)ACS(B) Na flow rate (Calc.)

Temperature (℃

)

Flowrate (kg/s)

Time (hours)

- 54 -

experimental

fast reactor

‘Joyo’ with

the 100MW

irradiation

core under

the

assumption

of total

blackout of

the plant.

In case of

turbine trip

or load

rejection

event,

emergency diesel system can be used in order to obtain alternative power source. However, no electricity

except DC power used for instrumentation is available in the case of the total blackout event. In light

water reactors, since this event is classified into the accident, no test is conducted in the real plant.

The important plant parameters and subassembly exit temperatures during the natural circulation are

calculated using the analytical model shown in Fig. 5.1.7. Eleven channel groups representing various

subassemblies in the core. The comparison of the plant parameters between the test and the analysis is

shown in Fig. 5.1.8 for the center and the 3rd subassemblies. As shown in the figure, subassembly exit

temperatures decrease due to forced flow by pump coast-down. However, the temperatures increase just

before the establishment of the natural circulation. Then, the temperatures decrease due to the natural

circulation and decreased decay heat.

Measured

SSC-L

400

440

480

520

560

0 50 100 150 200 250 300 350 400

NETFLOW++ with ISHT modelNETFLOW++ without ISHT model

Time (sec)

Te

mp

era

ture

( ℃

)

Fig. 5.1.8 Temperature behaviors at the exit of the center and the 3rd subassemblies

Link [1]: Center-subassemblyLink [2]: First layerLink [3]: Second layerLink [4]: Third layerLink [5]: 4th layerLink [6]: 5th layerLink [7]: Irradiation rigsLink [8]: Control rodsLink [9]: Inner reflectorsLink [10]: Outer reflectorsLink [11]: Bypass channel

JointSub-joint

2

6

3

UCSUpper plenum

-1

[1]-[7][8]-[10]

[13]

Low-pressure plenum

[20]

[11]

[17]

[18]

[19][24]

1High-pressure plenum

GL‐8.50

GL‐6.20

GL‐6.10

GL‐9.53

GL‐12.50

GL‐12.80

GL‐7.45

-3

Pump

Pump

IHX

[21] [22]

[23]

[26]

Air cooler 4Air cooler 3

7

８

-2

Pump

Pump

IHX

[15][14]

[16]

[25]

GL+2.50

GL+8.00

GL‐11.70

GL‐4.60

GL+8.62

Air cooler 1 Air cooler 2

7

８

45

[12]

Check valve

Fig. 5.1.7 Calculation model of ‘Joyo’

SSC-L

Measured

400

420

440

460

480

500

520

540

560

0 50 100 150 200 250 300 350 400

NETFLOW ++ with ISHT modelNETFLOW ++ without ISHT model

Time (sec)

Te

mp

era

ture

(

℃ )

- 55 -

References

[1] Mochizuki, H., 2007. Verification of NETFLOW code using plant data of sodium cooled reactor and

facility, Nuclear Engineering and Design, 237, pp.87-93.

[2] Mochizuki, H., 2007. Development of a Versatile Plant Simulation Code with PC, Proc. ICAPP2007.

[3] Mochizuki, H., 2007. Inter-subassembly Heat Transfer of Sodium Cooled Fast Reactors: Validation of

the NETFLOW Code, Nuclear Engineering and Design, 237, (2007), pp.2040-2053.

[4] Mochizuki, H., 2008, Expansion of a Versatile Plant Dynamics Analysis Code And Validation using

FBR Plant Data, NTHAS6, Okinawa Japan, N6P1001.

[5] Mochizuki, H. and M. Takano, 2009. Heat Transfer in Heat Exchangers of Sodium Cooled Fast Reactor

Systems, Nuclear Engineering and Design, 239, pp.295-307.

[6] H. Mochizuki, 2009. Code Validation for Decay Heat Removal using PRACS or DRACS in a Sodium

Loop, Proceedings of GLOBAL 2009, Paris, France, Paper 8137, pp.1674-1681.

[7] H. Mochizuki, 2009. Natural Circulation Characteristics in a Loop-Type LMFBR, Proceedings of

NURETH-13, Kanazawa, Japan, N13P1001.

- 56 -

5.2 Chernobyl accident

Figure 5.2.1

shows the

schematic of the

Chernobyl type

reactor that is

graphite-moderated

light-water-cooled

pressure-tube-type

reactor. At the

unit 4 of the

Chernobyl reactor,

the accident had

occurred on 26

April 1986. It has

passed two decades

since the Chernobyl

accident. However, the root cause of the Chernobyl accident was not clear enough according to the papers

published after the first official meeting held by IAEA (1986). So far, the accident was mainly

investigated from the following three scenarios:

1) the positive reactivity insertion by the flaw of the scram rods,

2) loss of pumping power by cavitation,

3) the positive reactivity insertion only by pump coast-down.

The mechanism of the positive reactivity insertion by the scram rods has been experimented and

calculated, and many researchers think that this is one of the causes of the accident. However, some said

that only the voiding only by the coast-down of re-circulation pumps was the root cause of the accident.

If we know the pump characteristics in

high-pressure condition, pump cavitation never

cause the Chernobyl accident. Flow rate

hardly decrease to zero during the cavitation

under the high pressure condition compared to

the characteristic under the atmospheric

condition. Furthermore, the accident was not

caused by only the scenario 3) when the

appropriate calculation is conducted.

Eventually it is shown by the analysis that the

root cause of the accident is a combination of

1. Core2. Fuel channels3. Outlet pipes4. Drum separator5. Steam header6. Downcomers7. MCP8. Distribution group headers

9. Inlet pipes10. Fuel failure detection equipment11. Top shield12. Side shield13. Bottom shield14. Spent fuel storage15. Fuel reload machine16. Crane

Electrical power 1,000 MWThermal power 3,200 MWCoolant flow rate 37,500 t/hSteam flow rate 5,400 t/h (Turbine)Steam flow rate 400 t/h (Reheater) Pressure in DS 7 MPaInlet coolant temp. 270 0COutlet coolant temp. 284 0CFuel 1.8%UO2Number of fuel channels 1,693

Fig. 5.2.1 Schematic of the Chernobyl reactor

0

500

1000

1500

2000

2500

3000

3500

25:0

0:00

:00

25:0

1:00

:00

25:1

3:05

:00

25:2

3:10

:00

26:0

0:28

:00

26:0

1:00

:00

26:0

1:23

:04

26:0

1:23

:40

The

rmal

Po

we

r (M

W)

Scheduled power level for experiment

30MW

Power excursion

secminhourday

Fig. 5.2.2 Power trend before the accident

- 57 -

the scenarios 1) and 3).

5

6

7

8

9

10

-600

-400

-200

0

200

400

0 60 120 180 240 300

Flowrate (m3/s)P (MPa)Flowrate (cal.)Pressure (cal.)

DS water level (mm)Feed water flowrate (kg/s)Reactor power (cal.)DS water level (cal.)

Flo

wra

te (

m3 /s

), P

ress

ure

(M

Pa

)

DS

wat

er le

vel (

mm

), F

eedw

ate

r flo

wra

te(k

g/s

)

Time (sec) Push AZ-5 button

Close stop valve:Turbine trip

Trend of parameters for one loop from 1:19:00 on 26 April 1986

1:19:00

Fig. 5.2.3 Plant behavior before the accident and power excursion at Chernobyl

It is very important to know the

plant conditions before the accident.

Because all data relating the

neutronics are made on the basis of

the operating condition before the

accident. It is also important to

trace the plant behavior before the

accident. Without this evaluation,

the simulation of the Chernobyl

accident seems to be impossible and

improper. We have to apply all

characteristics relating two-phase

flow and components to simulate the

plant transient.

Operators intended to conduct experiment to generate electricity to supply the important components like

pumps after the plant was scrammed. If the diesel system was reliable enough, this sort of experiment

was not necessary. They decreased the reactor power to the scheduled level as illustrated in Fig. 5.2.2.

Unfortunately, they encountered some troubles and they could not keep the power at the scheduled level.

The reactor power decreased to 30MW that was only 1/100 of the rated thermal power. They withdrew

the control rods from the core to increase the reactor power. However, the reactor power was only

200MW just before the accident due to the Xe build-up. According to the operation manual, they had to

8.0

1.０

5.0

1.5

Negative reactivity

Positive reactivity

Graphite displacer

Scram rod

Fuel 2×3.5m

8.0

1.０

5.0

1.5

Negative reactivity

Positive reactivity

Graphite displacer

Scram rod

Fuel 2×3.5m

Fig. 5.2.4 Control rods of the Chernobyl Reactor

- 58 -

remain certain number of control rods in the core. They might not know this regulation.

Figure 5.2.3 shows the plant trend approximately 5 minutes before the accident. Closed symbols in the

figure mean measured results by the SKALA system in the power station. In the accident behavior

analysis using a computer code, it is very important

to trace the operation from far before the accident in

order to obtain the similar transient as the accident.

The result analyzed by the NETFLOW code has good

agreement with the measured trend.

The safety rods of the RBMK reactor was inserted

into the core having more than 7m height with very

slow speed at 0.35m/s. The safety rod hanged a

displacer made of graphite to improve the neutron

economy when it was withdrawn. There was a

water column of 1m between the control rod and

displacer, and 1.5 m water column beneath the

displacer as shown in Fig. 5.2.4. Furthermore, there

was a non-heat generation region at the center of the

fuel because two short fuel bundles were connected

as shown in Fig. 5.2.5. Axial

power distribution showed a

double humped shape.

When the safety rod was

inserted into the core, the lower

water column was displaced from

the core and positive reactivity

was generated there. Without

this characteristic, the accident

never occurred. This positive

scram characteristic never

happens in the other reactors

except RBMK.

The power excursion occurred

at the lower part of the core at first, then expanded to the whole core. However, it was clarified by the

analysis that the accident was not such a crucial situation if the reactor power were in the scheduled level.

The calculated trends of thermal equilibrium steam qualities in the core are shown in Fig. 5.2.6, there

was no voids in the core just before the accident due to the system pressure increase by a steam control

valve or MSIV closure. As shown in chapter 1, the increment of the void fraction is larger when voids are

generated from the low steam quality. This means that the large power was generated in the core due to

the void reactivity.

Fig. 5.2.5 Fuel assembly of RBMK reactor

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

0 50 100 150 200 250 300

TopCenter

Tim e (sec)

Th

erm

al

eq

uili

bri

um

ste

am

qu

alit

y,

x (

-)

Fig. 5.2.6 Behavior thermal equilibrium steam quality

- 59 -

The cause of the accident is not very complex. But many researchers have reported that the root cause

of the accident is not clear enough even though they assume the positive scram. Because, they had to

assume a large positive reactivity to cause the same scale accident. It seems that they started calculation

just before the accident. In general, we have to take a steady state to start the calculation. However, if

take the steady state several seconds before the accident, such action may cause the difference of important

parameters from the reality.

References

[1[ IAEA, 1986. The accident at the Chernobyl nuclear power plant and its consequences. In: Information

compiled for the Exparts’ meeting, Vienna, Austria (INIS-mf-10523).

[2] Verikov, E.P., et al., 1991. The Chernobyl accident: current vision of its causes and development. In.:

Proceedings of the International Conference of Nuclear Accident and the Future of Energy, Paris,

France.

[3] Mochizuki, H., 2007. Analysis of the Chernobyl accident from 1:19:00 to the first power excursion,

Nuclear Engineering and Design, 237, pp.300-307.

- 60 -

Photos: Chernobyl Unit-4 after the

accident