Core Scaling Analysis

VHTR Cooperative AgreementProgram Review Meeting

February 25, 2009

Prismatic Core Scaling• The dominant contributor to flow

resistance in the VHTR reactor vessel during the natural circulation phase is its core.

• The pressure drop across the core consists of frictional losses, form losses and acceleration losses due to gas expansion.

Frictional pressure is largest

( )( )

21

1

22 1

H Cc c ccore

c H C

T Tu fLP K

d T T

βρβ

⎛ ⎞−Δ = + +⎜ ⎟⎜ ⎟− −⎝ ⎠

– Frictional pressure is largest.

• If the losses due to gas expansion are neglected…

• In a reduced height facility, the designer can satisfy this criterion through the use of orifices to obtain the desired pressure drop around the loop.

[ ] ( )1core c H C thR RP g T T Lβ ρ⎡ ⎤Δ = −⎣ ⎦

[ ] [ ]core thR RP LΔ =

Applicable to Pebble Bed Core as well!

Prismatic Core Scaling

( ) ( ) ( )1

2 32

core c pg cR RRF R

lq C a T

βρ⎛ ⎞

= Δ⎜ ⎟Π⎝ ⎠&

Core Power Scaling

For Natural Circulation

( )1

2

,core c RRF R

lq a

⎛ ⎞= ⎜ ⎟Π⎝ ⎠

&

( ) ( )1

2

core cR RF R

lq Pa

β⎛ ⎞= ⎜ ⎟Π⎝ ⎠

&

Assuming fluid property similarity (DCC NC)

For PCC Natural Circulation

The predicted decay power for the 600 MWt prismatic block VHTR can be determined using the “American National Standard for Decay Heat Power in Light

Water Reactors”.

Prismatic Core Scaling• Radial heat conduction

and radiation heat transfer.

• Assumptions:– Axial heat transfer into the

upper and lower graphite reflectors via conduction

l bl

1 0d dT qr

r dr dr k

′′′⎛ ⎞ + =⎜ ⎟⎝ ⎠

&

negligible. – Radiation heat transfer from

the cylindrical portion of the vessel is dominant mode of system heat removal.

– Constant material thermal conductivity in each region.

Prismatic Core Scaling

14

4 coreS

qT T

A ε σ∞

⎡ ⎤= +⎢ ⎥⎣ ⎦

&1.0core

RAD S

q

A ε⎡ ⎤

=⎢ ⎥⎣ ⎦

&

To preserve the maximum fuel zone temperature and the vessel surface temperatures in the HTTF…

Vessel Surface Temperature for Prototypical Temperatures

RAD SA ε σ⎣ ⎦ RAD S RA ε⎣ ⎦

( ) ( )0( )Score

th

T Tq

R

− =∑

& 1.0core th Rq R⎡ ⎤ =⎣ ⎦∑&

Thermal Resistance for Prototypical Temperatures

Because of its high thermal conductivity, graphite cannot be used as the sole material for the HTTF core!

Prismatic Core Scaling

( ) ( )core RRS pS R

R

q tC

Vρ =

&

Stored Energy

Prismatic Core Scaling Assessment

VHTR Decay Power (MW) with G-Factor

45

HTTF Core Decay Power

350

• Comparison of the predicted decay heat profiles for the VHTR and HTTF. – HTTF profile is based on a one-half time scale and a 1:113.7 scaled power. – 300 kW corresponds to a decay power of approximately 6%. – The decay power curve can be programmed into the HTTF power control system.

0

5

10

15

20

25

30

35

40

1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06

Time (s)

Co

re P

ow

er (

MW

)

0

50

100

150

200

250

300

1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06

Time (s)

Co

re P

ow

er

(kW

)

Prismatic Core Scaling Assessment

• Comparison.– VHTR radial temperature profile developed analytically.– HTTF radial temperature profile predicted using FLUENT.

• Only radial heat transfer is considered.• Half a 1/8 axial cross section of the core was modeled.• Symmetry boundary condition at the centerline.

Prismatic Core Scaling Assessment

VHTR Comparison with Graphite Core GRTS

700

800

900

1000

VHTR and GRTS Radial Temperature Profiles

700

800

900

1000

Comparison between Graphite and Ceramic HTTF Reflectors and Core.

0

100

200

300

400

500

600

0 0.2 0.4 0.6 0.8 1

Normalized Radial Position

Tem

p (C

el)

GRTS

VHTR

HTTFVHTR

0

100

200

300

400

500

600

0 0.2 0.4 0.6 0.8 1

Normalized Radial PositionTem

p (C

el)

VHTR

GRTSHTTFVHTR

Graphite HTTF Reflector and Core. Ceramic HTTF Reflector and Core.

Pebble Bed Scaling• Porosity and packing factor.

– Packed beds of uniform sized spheres for tight randomly packed arrangements.

• Porosity: 0.38 to 0.41.• Packing factors: 0.62 to 0.59. • Local porosity can change significantly due to the effects of stacking spheres next to

walls. – Pebble bed has double wall effect in the radial direction.

P i ill ll h di f b f di f d l• Porosity oscillates near walls to the distance of about four diameters for moderately sized pebbles.

• Prototypical pebble bed core will experience the wall effects at the inner and outer reflectors and will have a region in the center of the annulus which is relatively free of pebble porosity wall effects.

– Capturing this effect in the HTTF will push the pebble diameter to scale as the facility diameter scale.

– Practical consideration of manufacture will tend to push in the opposite direction because the larger the pebble, the easier and cheaper they will be to construct.

Pebble Bed Scaling

0.6

0.8

1

acti

on

Pebble Bed Wall Effect

0

0.2

0.4

0 5 10 15 20

Diameters from center wall

Vo

id F

ra

Pebble Bed Scaling

Diameter Comparison for Pebble Diameter Selection

Prototypical Scaled Model

Pebble Diameter (in) 2.36 1/2" 3/8" 5/16"

Pebble Diameter (cm) 6.000 1.270 0.953 0.794

Number 507713.07 235690.88 558674.68 965389.85

Comparison of Diameter Choices

Porosity 0.36 0.36 0.36 0.36

Number of diameters, radial 15.50 9.72 12.96 15.55

Number of diameters, axial 132.17 156.10 208.14 249.76

A 3/8” pebble size would be more economical than the 5/16” while maintaining a significant region between the walls where the oscillatory behavior of porosity is non-

existent.

1: 6.30p Rd⎡ ⎤ =⎣ ⎦

Pebble Bed ScalingGoverning Equations for Pebble Bed Radial Heat Transfer

Steady-State Radial Heat Equation

( )22

30 (1 ) ( )p

eff p pb p g radp

dk T h T T q q

d

ε πε

⋅′′′= − ∇ − − − ∇ ⋅ +

r&

Pebble Bed Effective Thermal Conductivity

(1 )eff g sk k kε ε= + −

Thermal Radiation Transport Equation

( ( ) ( ))4 ( , ) ( )4 4 ( ) 4 ( )( ) ( ) ( )

sb brad

s s s

r rdI r s rq I I r I r

r ds r r

κ σπ κκ π π πσ σ σ

⎛ ⎞⎛ ⎞+∇ ⋅ = − + −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

r

Pebble Bed ScalingNormalized Variables

o

qq

q+ ′′′′′′ =

′′′&

&&

4,

ob o

I II

TI σπ

+ = =

pp

o

TT

T+ =

44

4p

po

TT

T+ =

p

ss

d+ =

Pebble Bed ScalingNon Dimensional Pebble Bed Heat Equation

( )

( )

3 22 4

3 2 3 2

( ) 4 1(1 ) (1 )

4 4 0(1 ) (1 ) (1 )

p o pp p g p

eff s eff

o p o p o ps

ff ff ff

d h T dT T T T

k k

T d T d q ddII q

k k ds k T

ε π σκκε σ ε

σ σκ κ σ κσ ε σ ε ε

+ + + +

++ +

+

⎡ ⎤⋅ ⎡ ⎤⎛ ⎞⎢ ⎥∇ − − − +⎢ ⎥⎜ ⎟− −⎢ ⎥ ⎢ ⎥⎝ ⎠⎣ ⎦⎣ ⎦

⎡ ⎤ ⎡ ⎤ ⎡ ⎤′′′+ ⎛ ⎞′′′+ + + =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎜ ⎟− − −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎝ ⎠⎣ ⎦ ⎣ ⎦ ⎣ ⎦

&&

(1 ) (1 ) (1 )s eff s eff eff ok k ds k Tσ ε σ ε ε⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎝ ⎠⎣ ⎦ ⎣ ⎦ ⎣ ⎦

3

, (1 )o p

Rad dIs eff

T d

k

κσσ ε⎡ ⎤

Π = ⎢ ⎥−⎢ ⎥⎣ ⎦

( )(1 )s

p

Nueff

h d

k

ε πε

⎡ ⎤⋅⎢ ⎥Π =

−⎢ ⎥⎣ ⎦

2

(1 )p o

qeff o

d q

k Tε′′′

Π =−&( ) psIRad dσκ +=Π ,

Core Scaling Ratios

( ) ( ) 12 1:113.7core RR R

q a L= =&

[ ] ( ) 1: 4core thR RP LΔ = =

Core Power For DCC Natural Circulation

Core Pressure Drop

( ) ( )1 1:113.7th Rcore R

Rq

= =∑&

( ) ( )1core RR

S pS RR

q tC

Vρ = =

&

Reflector and Prismatic Block Core Thermal Resistance

Energy Storage

( ) 1: 3.76core coreS R

RRAD R

q q

A Ldε

⎡ ⎤ ⎡ ⎤= = =⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

& &Vessel Emissivity Ratio

Pebble Bed Core Scaling Ratios

Pebble Diameter Ratio

Effective Thermal Conductivity Ratio

1: 6.30p Rd⎡ ⎤ =⎣ ⎦

( ) ( )2 1: 79.4eff p oR Rk d q′′′= =

Heat Transfer Coefficient Ratio( ) 1:12.6eff

Rp R

kh

d

⎡ ⎤= =⎢ ⎥⎢ ⎥⎣ ⎦

Assuming temperature similarity

1:12.6pR

eff R

d

kκ

⎡ ⎤= =⎢ ⎥⎢ ⎥⎣ ⎦

Absorption Coefficient Ratio

Questions?Q