Thermofluids Data Book

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MEC6422 Nuclear Thermal Hydrauli cs and Heat Transfer DATA BOOK S. He (2012)

1

MEC6422

Nuclear Thermal Hydraulics and Heat Transfer

(2012-2013)

DATA BOOK

Chapters 1, 2 & 3 Nuclear Power Generation Fundamentals




2




3

Power density

√

Specific power

Energy generation ∑

∑

where

Decay heat




4

Chapter 4 Thermal analysis of fuel elements




5




6

Thermal conductivity of fuel:

k ,

Loeb equation:

Modified Loeb equation: , where is between 2 and 5.

Biancharia model: , where α2=1.5.

Temperature distribution in solid and annular fuel:

where ()

Fuel restructure- Three zone distribution:

Zone 3:

Zone 2:

Zone 1




7

where

Two zone modelZone 2:

Zone 1:

Gapping effect

Overall temperature distribution in fuel

Chapter 5 Heat transfer to coolant – single phase

Governing equations

Laminar flow

Turbulent flow




8

Blasius relation (Re<30,000):

McAdams relation (30,000<Re<1,000,000): The Karman-Nikuradse:

( ) The empirical Colebrook equation:

Bare rod bundles

where

Pressure drop at spacers

DeStordeur model:

where Cs can be obtained from Figure 9-26.

Rehme model:

where Cv is the modified drag coefficient, which is again a function of Re. Rehme’ data show Cv = 9.5 for Re=104 and Cv=6.5 at Re=105.

Expansion and contraction:




9




10




11

Laminar heat transfer

∫ ∫

for q’’=const. for Tw=const.

The length of the thermal developing region:

An alternative expression due to Bhatti and Savery:

Turbulent heat transfer




12

Non-metallic fluids:

(1) For circular pipe the following expressions are often been used:

Dittus-Boelter equation:

where n = 0.4 when fluid is heated and n=0.3 when fluid is cooled; 0.7<Pr<100 and Re > 10,000 and L/D>60. Seider and Tate equation:

0.7<Pr<120 and Re > 10,000 and L/D>60.

Colburn equation:

0.7<Pr<100 and Re > 10,000 and L/D>60

(2) Rod bundles

General expression: where Nuc.t. is correlation for a circular tube, say, Dittus-Boelter.

Presser for infinite arrays:

for triangular array and 1.05≤P/D≤2.2, and

for square array and 1.05≤P/D≤1.9.

Weisman [51] for water in infinite arrays:

and

for triangular array and and

for square array and .

Markocz for a finite array (Fig 10-14): where Re is based on the hydraulic diameter defined as:

∑ ∑

and B=De/D and D is the diameter of the rod




13

(3) Entrance effectEntrance length:

Mean Nusselt number (McAdams):

for Re >10,000, 0.7<Pr<120 and square-edge entries,

Local Nusselt number:For uniform velocity and temperature at the inlet:

For abrupt entrance:

Metallic fluids:

(1) Circular tube with constant heat flux along and around the tube:

(2) Circular tube with constant axial wall temperature and uniform radial heat flux:

(3) Parallel plate with constant heat flux through one wall and adiabatic in the other:

(4) Concentric annuli with fully developed flow and the boundary condition of uniform heat flux in the inner

wall, D2/D1>1.4:

(5) Rod bundlesWestinghouse [25]




14

For 1.1≤P/D≤1.4, 10≤Pe≤5000, where P is pitch and D is diameter of the rod.

Schad-modified [25]

For 1.1≤P/D≤1.5, 150≤Pe≤1000 and

For Pe<150.

Chapter 6. Two-phase flow

6.1 Fundamentals of two phase flows

f g W W W .

AW G / , AW G f f / , AWg G

g / .

G

G

W

W x

g g ,

G

G

W

W x

f f 1

A

A g ,

A

A f f 1 .

f

g

u

u K ,

1

1 g

f

x

x K

f g R uuu

H g D uuu

g g g g u AW

f f f f u AW

f g

g

v x K xv

xv

)1(

f g

f

v x K xvv x K )1(

)1(1

A

Q j

g

g , A

Q j

f

f

g g g u AQ , f f f u AQ

f g

g

QQ

Q

,

f g

g

v x xv

xv

)1(

Av M u g g

g

, ])1([ f g g v x K xvGu




15

A

v M u

f f

f )1(

, ])1([ f g f v x K xv K

Gu .

g

g

ju ,

1

f

f

ju .

])1([ f g H v x xvGu

f g m )1(

f g m

x K x

x K x

/)1(/

)1(

f g H

x x

/)1(/

1

)1(

)1(

x K x

v x K xv

v

f g

m

f g H v x xvvv )1(

222

0

g

Fg

f

Ff

fo

Ff F dz

dp

dz

dp

dz

dp

dz

dp

1

0

2

0

Ff F

f dz

dp

dz

dp

, where

D

vG f

dz

dp f fo

Ff

2

0

2

Ff F

f dz dp

dz dp

2 , where

Dv xG f

dz dp f f

Ff

2

)]1([2

Fg F

g dz

dp

dz

dp

2 , where

D

vGx f

dz

dp g g

Fg

2][2

4/1Re079.0

fo fo f ,

4/1Re079.0

f f f ,

4/1Re079.0

g g f

f fo GD /Re , f f f DG /Re , g g g DG /Re ,

Fg

Ff

f

g

dz dp

dz dp

X /

/2

2

2

.

1 Note that in the lecture notes, ‘sf’ and ‘f’ were both used to refer to ‘liquid’, e.g,

Ff sf F dz dpdz dp //, and

similarly, ‘sg’=’g’ referring to gas, e.g., Fg sg F dz dpdz dp // ,




16

6.2 Flow Patterns

(1) Verti cal flow - Hewitt and Roberts (1969)

f

f f xG j

2

2 )1(

g

g g

xG j

22

Figure 1. Hewitt and Roberts (1969) map for

vertical flow

(2) Hori zontal f low - Baker (1954)2/1

w

f

A

g

,

3/12

f

w

w

f w

where ρw= 1000 kg/m3; ρA,=1.23 kg/m

3; μw = 0.001 Ns/m

2 and σw = 0.072 N/m.

Figure 2. Two-phase flow pattern map of Baker (1954) for horizontal tubes




17

(3) Taitel and Dukler (1976) uni fi ed flow pattern for hori zontal fl ows

Fsg

Fsf

dz dp

dz dp X

/

/2 ,

cos Dg

j Fr

g

g f

g

,

2/1

cos)(

)/(

g

dz dpT

g f

sf ,

2/1

2/1

Re sf

f

f Fr v

j D Fr K

Figure 3 Taitel and Dukler (1976) flow pattern for horizontal flows




18

6.3 The homogeneous and separated flow model

(1) The homogeneous model

Governing equations:

u AW

dz ud

AW g

dz F d

Adz dp sin1

dz g u

d diwq sin2

2

where dpvdE di .

Relationships for homogeneous fluid mixture:

f g uuu ,

jvGu ,

v

xv g

1][])1([

G

j xvvv x xv

W

Qv fg f f g

Static pressure drop:

dp

dv xG

vv xv

g

dz

dx

v

vvG

v

v x

D

vG f

dz

dp

g

f fg f f

fg

f

f

fg f TP

2

2

2

1

)]/(1[

sin1

2

Frictional pressure gradient

2

0

fo

Ff F dz

dp

dz

dp

where D

vG f

dz

dp f f

Ff

2

0

0

2

and

f

fg fo

v

v x1

2 (method 1) or

4/1

211

g

gf

f

fg

fo xv

v x

(method 2)




19

(2) The separated flow model

(i ) Static pressure gradient

2

2

2

222

2

2

2

22

2

2

)1(

)1(1

)1(

)1(

)1(

)1(22

sin)]1([2

g f

x

g

g f

p

f g

f g fo

f fo

v xv x

pdp

dv x

dz

dpG

v xv x

x

v x xv

dz

dxG

g D

vG f

dz

dp

(ii ) The Lockhart-Martinelli (1949) correlation

1) Graphic form

Figure 4. Lockhart-Martinelli (1949) correlation




20

2) Analytical form

2

2 11

X X

C f ,

221 X CX g

where the value of C is given below for the various flow regimes:

liquid gas Cturbulent turbulent (tt) 20

viscous turbulent (vt) 12

turbulent viscous (tv) 10

viscous viscous (vv) 5

Table 1 Lockhart-Martinelli (1949) correlation constants

(ii i) The Fri edel correlation (1979)

035.0045.0

321

2 24.3We Fr

A A A fo

where

fo g

go f

f

f x x A

22

1 )1(

224.078.0

2 )1( x x A

7.019.091.0

31

f

g

f

g

g

f A

2

2

gD

G Fr

DGWe

2

f g x x

/)1(/

1




21

(iv) The Mar tinell i-Nelson (1948) corr elation

(a) value of 2

fo as function of pressure and mass quality

(b) Void fraction as a function of quality α and absolute pressure for steam water

Figure 5. Martinelli-Nelson 1948 Correlation




22

(v) The Thom corr elation (1964)




23

(iv) The Baroczy correlation (1965)

)1356(

22

2G fo

f fo

F D

vG f

dz

dp

Baroczy (1965) model –

Graph (a)




24

Graph (b)

Figure 6 Baroczy (1965) model




25

(v) Two-phase flow in inclined pipes - Beggs and Brill (1973) model

4/1/079.0

GD f where ])1([ f f g g v x xv

Figure 7 Beggs and Brill (1973) model for two-phase flow in inclined pipes




26

Chapter 7 Heat transfer with phase change

Vapour bubbles

( )




27




28




29

Chapter 8 Core thermal-hydraulics analysis

1. One – dimensional analysis

||

:

||

Single phase:




Pressure

||

Temperature (for power rating )

Two-phase:

Location of onset of boiling:

( )

Mixture properties:

Pressure drop:

where

[ ] ||

||

2. Sub-channel analysis

(1) Continuity equation

(2) Single phase energy equation:

[ ]

(3) Axial momentum equation

( )

{}

(4) Transverse momentum equation

() () () { }

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