Subchannel Analysis Method at a Glance

1

Subchannel Analysis Method at A Glance

Lecture Note on Nuclear Reactor Thermal Hydraulics

Syeilendra Pramuditya

Department of Nuclear EngineeringTokyo Institute of Technology

December 2012

2

Subchannel Analysis Method

Estimation of:Fuel temperature (conduction eq.)Clad temperature (heat transfer & conduction eq.)Coolant temperature (transport eqs.)Flow field (transport eqs.)

Fuel Subassembly

3

Subchannel Analysis MethodGoverning equations: time-averaged transport equations

c c v J c vtρ ρ ρφ ρ∂ ′ ′+∇ ⋅ = ∇ ⋅ + −∇ ⋅∂

unsteady term + convection term = diffusion term + source term + fluctuation term

c J φ

Mass 1 0 0

Linear momentum v pI τ− + g

Internal energy e q′′− ( )1 q p v φρ

′′′ − ∇ ⋅ +

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Three types of subchannel

axial planesubchannel number under consideration

adjacent subchannel number ( 1, 2,3)gap number between subchannel and

limk kk i mk

≡≡≡ =

≡

Three types of control volume (CV)

The basic equations are discretized based on 2D non-orthogonal subchannel geometryBasically it is a finite volume coarse grid CFD

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Subchannel Analysis MethodDiscretized equations

{ } { } { }

1 1

, , , , 1 1, , , , ,

31 1 1 1

, ,, , 1/ 2 , 1/ 2 ,1

Internal energy

n n n n

i l i l i l i l n ni l E c i l w i l

n n n n nk t i li l i l i l k l

k

e eV Q Q

t

p w A w A u A Q

ρ ρψ

σ

+ +

+ +

+ + + +

+ −=

−+ = +

Δ⎡ ⎤

− − + −⎢ ⎥⎣ ⎦∑

{ }( ) { }( )1

, 1/ 2 , 1/ 2 1 1, 1/ 2 , 1 ,

1/ 2

, 1/ 2 , 1/ 2, 1/ 2

Axial momentum

n n

i l i l n ni l Mz i l i l

n nz i l i li l

w wV p A p A

t

R V g

ρ ρψ

ρ

+

+ + + ++ +

+

+ ++

− ⎡ ⎤+ = − −⎢ ⎥⎣ ⎦Δ

+ −

{ }( ) { }( )1

, , 1 1, 1/ 2 , , ,

1/ 2

,

Transverse momentum

n nn k l k l n n

i l Mx kk l mk l i l

n

x k l

u uV p A p A

t

R

ρ ψ σ+

+ ++

+

− ⎡ ⎤+ = − −⎢ ⎥⎣ ⎦Δ

+

Need modeling

Need modeling

Need modeling

6


Four equations and twelve unknowns (unsolvable)

7


8


( )2

Axial flow resistance

2fs

zz z z

eA

A wR pn n dA p fV D

ρτ= − + ⋅ = −Δ =∫

( )2

Transverse flow resistance

2fs

xx x x

VA

A uR pn n dA p fV D

ρτ= − + ⋅ = −Δ =∫

Pipe modelsEngel/Novendstern modelsRehme modelCheng-Todreas model

Gunter-Shaw

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*,

Turbulent term of the internal energy equation

ff

t ij H p i jA

Q e v n dA W C T Tρ ′ ′ ⎡ ⎤= ⋅ = −⎣ ⎦∫

COBRA modelCheng-Todreas model

* *, ,

,0.125 ,

Re

ij H ij M i ij k

mix V ki

ij k

W W S G

C D

β

βλ

= =

=

Inter-channel mixing of energy

10

Subchannel Analysis MethodSubstituting the two momentum equations into mass and energy equations:

( )( )

( )

1 1 1, 1/ 2 0 1 , , 1

1 1 1, 1/ 2 0 1 , 1 ,

1 1 1, 0 1 , ,

n n ni l i l i l

n n ni l i l i l

n n nk l mk l i l

w a a p p

w b b p p

u c c p p

+ + ++ +

+ + +− −

+ + +

= + −

= + −

= + −

Two equations and two unknowns (solvable)

We obtain:

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To solve this algebraic equation system, the mass and energy equations can be considered as “zero-valued” non-linear functions:

( )( ), 0

, 0mass mass

energy energy

F F p T

F F p T

≡ =

≡ =

This system is then solved by using the multivariable Newton-Raphsonmethod:

, , , 1 , 1 1, 2, 3,

, , , 1 , 1 1, 2,

r r

r r r r r r rmass mass mass mass mass mass mass

i l i l i l i l m l m l m lrr r r r r r

energy energy energy energy energy energy energ

i l i l i l i l m l m l

J X F

F F F F F F Fp T p p p p p

JF F F F F F Fp T p p p p

δ

− +

− +

× = −

∂ ∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂

=∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂ 3,

, , , 1 , 1 1, 2, 3,

ry

m l

T

i l i l i l i l m l m l m l

rmassrr

energy

p

X p T p p p p p

FF

F

δ δ δ δ δ δ δ δ− +

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

∂⎢ ⎥⎣ ⎦

⎡ ⎤= ⎣ ⎦

⎡ ⎤− = − ⎢ ⎥

⎢ ⎥⎣ ⎦

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With some matrix manipulation:

, 1

, 1, 11 12 13 14 15 1

1,, 21 22 23 24 25 2

2,

3,

, 11 , 1 12 , 1 13 1, 14 2, 15 3, 1

, 2 21 , 1 22 ,

i l

i li l

m li l

m l

m l

i l i l i l m l m l m l

i l i l i l

pp

p c c c c c gp

T c c c c c gpp

p c p c p c p c p c p g

T g c p c p

δδ

δδ

δδδ

δ δ δ δ δ δ

δ δ δ

−

+

− +

−

⎡ ⎤⎢ ⎥⎢ ⎥⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥+ =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦⎣ ⎦⎢ ⎥⎢ ⎥⎣ ⎦

+ + + + + =

= − +( )1 23 1, 24 2, 25 3,m l m l m lc p c p c pδ δ δ+ + + +

Discrete Poisson equation for pressure correction:

( )2 pδ ψ∇ =

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1,1 1,2 1 1

2,1 2,2 2,3 2 2

3,2 3,3 3,4 3 3

, 1 , , 1

1, 2 1, 1 1, 1 1

, 1 ,

l l l l l l l l

L L L L L L L L

L L L L L L

A A P B

A A A P B

A A A P B

A A A P B

A A A P B

A A P B

− +

− − − − − − −

−

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜⎜ ⎟⎜ ⎟ ⎜=⎜ ⎟⎜ ⎟ ⎜⎜ ⎟⎜ ⎟ ⎜⎜ ⎟⎜ ⎟ ⎜⎜ ⎟⎜ ⎟ ⎜⎜ ⎟⎜ ⎟ ⎜⎜ ⎟⎜ ⎟ ⎜⎜ ⎟ ⎜⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

⎟⎟⎟⎟⎟⎟⎟⎟⎟

1, 2, , ,

T

l l l i l I lP p p p pδ δ δ δ⎡ ⎤= ⎣ ⎦… …

1, 2, , ,

T

l l l i l I lB b b b b⎡ ⎤= ⎣ ⎦… …

11

,

,

l l

I I

aA

a

⎛ ⎞⎜ ⎟

= ⎜ ⎟⎜ ⎟⎝ ⎠

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The Block Tridiagonal Matrix system is then solved either by direct or iterative methods.

Solution procedures:- Gaussian elimination (direct)- Gauss-Seidel (iterative)

15


START

Steady state

Time counter

Output

FINISH

Newton-Rhapson

Poisson solver

time = stop time

time = print time

SOLVER

PRE-PROCESSORTranslating the problem underconsideration into appropriateinput files (e.g. tabular data of SC, rod, gap numbering, power density at each cell, form friction factor at arbitrary locations, etc).

POST-PROCESSORDisplaying the calculation result into something meaningful to the user (e.g. nice fancy colorful images and graphs).

Initialize

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Subchannel Analysis MethodPre-processor

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Axial Power Profile (EOC)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

-500 0 500 1000 1500 2000 2500 3000z [mm]

Rel

ativ

e po

wer

Bundle axial nodalization

Rod radial nodalization

Example of problem specification

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Subchannel Analysis MethodVelocity vector

x [cm]210-1-2

z [c

m]

500

480

460

440

420

400

380

360

340

320

300

280

260

240

220

200

180

160

140

120

100

80

60

40

20

100% power & flowRe 16200Radial skew 3.0Wall heat loss 0%

Heatloss

Power Skew

x

19


1% power & flowRe 162Radial skew 3.0Wall heat loss 10%

Heatloss

Power Skew

x

Velocity vector

x [cm]210-1-2-3

z [c

m]

500

480

460

440

420

400

380

360

340

320

300

280

260

240

220

200

180

160

140

120

100

80

60

40

20

0

Flow recirculation

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Comparison with Experimental Data

Bundle Characteristics

UniformPower profile

34.1Flat-to-flat distance [mm]

52.2H/D

304.8Wire axial pitch (H) [mm]

1.4224Wire diameter [mm]

1.265W/D

7.3914Edge pitch (W) [mm]

1.243P/D

7.2644Pitch (P) [mm]

5.842Rod diameter (D) [mm]

533.4Heated length [mm]

19Number of rods

SodiumFluid

Parameters

Experiment Parameters

*M.H. Fontana, R.E. MacPherson, P.A. Gnadt and L.F. Parsly, “Temperature distribution in the duct wall and at the exit of a 19-rod simulated LMFBR fuel assembly (FFM Bundle 2A)”, Nuclear Technology, Vol. 24, 1974, pp. 176–200.

1586009.755022472-1143

976004.842012172-1338

1306004.832012172-1447

1906000.945020372-1054

1906000.894.2020372-1443

1756000.542.6020472-0949

1926000.462.1020472-1118

1936000.150.74020472-1459

dT [F]Tin [F]Power

kW/ft.rodFlow rate gal/minRun

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Subchannel and rod numbering Gap numbering

Calculation domain:19 pins42 subchannels60 gaps38/74 axial planes

Boundary conditions:Inlet coolant temperatureTotal mass flow rateOutlet pressureConstant wall heat flux

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Max Error: 10.2%RMS error: 6.9%


,

* ,i in

out avg in

T TTT T

−=

−

1,

1

,

N

i i i ii

out avg N

i i ii

T w AT

w A

ρ

ρ

=

=

=∑

∑

Mass flow rate 2.86 kg/s

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

-20 -15 -10 -5 0 5 10 15 20Corner-to-corner distance [mm]

Nor

mal

ized

tem

pera

ture

Cold dimensionsHot dimensionsExp. data


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Bundle Characteristics

1.21Axial power peaking

Chopped cosinePower profile

50.4Flat-to-flat distance [mm]

47.2H/D

307Wire axial pitch (H) [mm]

1.32Wire diameter [mm]

1.23W/D

8.003Edge pitch (W) [mm]

1.21P/D

7.87Pitch (P) [mm]

6.50Rod diameter (D) [mm]

930Heated length [mm]

37Number of rods

SodiumFluid

Parameters

1.341300001370L37P43

1.34810002810G37P25

1.34207007520G37P22

1.17886001530F37P27

1.17800002810F37P20

1.17201007610E37P17

1.0061100739E37P13

1.00528002810C37P06

1.001360011200B37P02

Power skew [max/avrg]GrReRun No.

Experiment Parameters

*F. Namekawa, A. Ito, K. Mawatari, Buoyancy Effects on Wire-Wrapped Rod Bundle Heat Transfer in an LMFBR Fuel Assembly, National Heat Transfer Conference, 1984.

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HEA

TED

LE

NG

TH93

0 m

m

L2

915

mm

Subchannel and rod numbering Gap numbering

Calculation domain:37 pins78 subchannels114 gaps38 axial planes (dz = 2.583 cm)

Boundary conditions:Inlet coolant temperatureInlet total mass flow rateOutlet pressureConstant wall heat flux

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Comparison with Experimental DataRe 7610, Gr 20100, Skew 1.17

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

-25 -15 -5 5 15 25Flat-to-Flat Distance [mm]

Nor

mal

ized

Tem

pera

ture

Exp. dataSimulation

Re 7520, Gr 20700, Skew 1.34

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


Nor

mal

ized

Tem

pera

ture

Exp. dataSimulation 5.6 (Hot)9.3 (Hot)Skew 1.34

3.4 (Hot)5.4 (Hot)Skew 1.17

8.5 (Cold)5.5 (Hot)

9.8 (Cold)8.1 (Hot)

Skew 1.00

ErrorRMS [%]ErrorMAX [%]

Re 11200, Gr 13600, Skew 1.0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


Nor

mal

ized

Tem

pera

ture

Exp. dataCase C04 (Hot)Case C04 (Cold)

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Subchannel Analysis Method at a Glance

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