Depressurized Conduction Coooodo a gldown Event Scaling– To reach this depressurized state, the system must go through a variety of mass and energy states as the pressure decreases

Depressurized Conduction Cooldown Event Scalingoo do a g

VHTR Cooperative AgreementProgram Review Meeting

February 25, 2009

DCC Phases

• DCC can be divided into three distinct stages: – Depressurization, – Air-ingress, and – Natural circulation. Ai ing ess stage can be f the di ided into• Air ingress stage can be further divided into…– Air-ingress by lock exchange, and – Air-ingress by molecular diffusion. – The extent to which the prototypical plant

experiences air-ingress by lock exchange depends on the location and the size of the break.

DCC ScalingSpecify Experiment

Objectives

Depressurized ConductionCooldown PIRT

Perform Scaling Analysis(3)

(2)

(1)

VesselDepressurizationScaling Analysis

Air IngressScaling Analysis

NaturalCirculation

ScalingAnalysis

RCCS ScalingAnalysis

Evaluate Key PIRTProcesses to Prioritize System

Design Specification

Gas Reactor Test Section (GRTS)Design Specifications

(8)

(7)

Develop ΠGroups andSimilarityCriteria


Significant ScalingDistortions inImportant Π

Groups?


Groups?

System DesignSpecifications


(6)

(5)

(4)

NoNo

YesYes



Groups?


No

Yes



Groups?


No

Yes

Depressurizaton Scaling Analysis• HTTF designed as a reduced pressure facility.

– Initial state of full system pressure cannot be duplicated in the test facility.

– Many important phenomena of interest in DCC occur at low pressures.

– To reach this depressurized state, the system must go through a variety of mass and energy states as the pressure decreases.

– Determine a set of initial and boundary conditions so that the trajectory of the system states can be duplicated on a scaled basis.

• Double-ended break of the annular hot gas duct assumed.– Results in a very fast primary side depressurization.

Depressurizaton Scaling Analysis


gdMm

dt= −∑ &

( )dU dV∑

Governing Equations for VHTR Vessel Depressurization

Conservation of Mass

Conservation of Energy ( )g core str loss

dU dVmh q q q P

dt dt= − + + + −∑ & & & &

( )g g core str lossv P

e dP eM m h e v q q q

P dt v

⎛ ⎞∂ ∂⎛ ⎞ ⎛ ⎞= − − + + + +⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠∑ & & & &

Conservation of Energy

Depressurization Rate Equation


gg

o

MM

M+ =

mm

∑∑∑ =+

&

&&

gP

gP

gP o

eh e v

veh e v

v eh e v

v

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

Normalized Variables

om∑∑ &

oP

PP =+

,

v

v

v o

ee P

ePP

+∂⎛ ⎞

⎜ ⎟∂ ∂⎛ ⎞ ⎝ ⎠=⎜ ⎟ ∂∂ ⎛ ⎞⎝ ⎠⎜ ⎟∂⎝ ⎠

ocore

corecore q

qq

,&

&& =+

ostr

strstr q

qq

,&

&& =+

,0

lossloss

loss

qq

q+ =

&&

&


( )dep g g core core str str loss lossv P

e dP eM m h e v q q q

P dt vετ++ + ++ + + +⎛ ⎞∂ ∂⎛ ⎞ ⎛ ⎞Π = − − + + Π + Π + Π⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠

∑ & & & &

Non Dimensional Depressurization Equation

Assuming Helium acts as perfect gas

odep

o

M

mτ =

∑ & γε1=Π

( ) ( )0

1core ocore

o

q

RT m

γγ

−Π =

∑&

&

( ) ( )1str ostr

o o

q

RT m

γγ

−Π =

∑&

&

( ) ( )1loss oloss

o o

q

RT m

γγ

−Π =

∑&

&

g p g

Break Flow Area• Break size flow area is a key parameter of the test facility. • Choked flow.

– For helium, when the ratio of the absolute upstream pressure to the absolute downstream pressure is equal to or greater than 2.049.

• HTTF capable of exceeding this ratio.

• Non-choked flow.– Transitions from choked to non-choked flow up upon sufficient decrease in

vessel pressurevessel pressure.– Break flow modeled by normal accidental release source term equations

that are a function of both upstream and downstream pressures. • Fraction of HTTF blowdown time with choked flow less than that for

the VHTR. – Not considered important for double-ended break as it is expected that

the entire depressurization process occurs in under one second. – Transition from choked to non-choked flow for smaller break flows may be

important because of their reduced depressurization rates.

Break Flow Area

( )iBrkBrkDio aGCm =,&

( ) [ ]G P=

Choked Flow

•Complete fluid property similitude will not be assumed. S ifi h t ti i i il( ) [ ],0 0Brk RR

G P=

( )Brk RR

Va

t⎛ ⎞= ⎜ ⎟⎝ ⎠

•Specific heat ratio is primarily a function of temperature

Assuming…•Identical break geometry (CD),•an ideal gas.

Depressurizaton Scaling RatiosTime Scale Ratio

( ) 1: 4BD RBrk R

Vt

a

⎛ ⎞= =⎜ ⎟⎝ ⎠

Ch t i ti P t t P t t t

Characteristic Time Ratios

Characteristic Ratio

EquationPrototype

ValuePrototype to Model Ratio

Distortion

~1.6 1 0

<<1 NA NA

<<1 NA NA

<<1 NA NA

During the prototypical blowdown, the residence time will be so short that the power scale ratios will be much less than unity and thus will be negligible contributors during the blowdown transient.

Exchange Flow Scaling• Significant ingress of air

into the VHTR vessel may occur. – Upon completion of the

depressurization phase.

Hot Helium Out ofBreak

Cold Air Ingressthrough Break

Plume IntrusionOnset of Counter

Current Flow

Duct Exchange Flow

– Depending on the break size and location.

• Mechanisms of air-ingress.– Duct exchange flow.– Molecular diffusion.

Thermally Stratified HeliumInitial Condition

for MolecularDiffusion Portion

of Transient

Mixing Period

VHTR LOWERPLENUM

Exchange Flow Scaling

uLPd

huH

( ) 2mu

t

ρ ρ κ ρ∂ + ∇ = ∇∂

( ) ( )uuu P g S

t

ρρ ρ μ

∂+ ∇⋅ = −∇ + + ∇⋅

∂

Conservation of Mass

Conservation of Momentum

Governing equations for buoyant jets

Exchange Flow ScalingNormalized Variables

C

ρρρ

+ =

x

( )0 C H

P PP

P gd ρ ρ+ = =

−

xx

d+ =

( )1

2bC H

C

u uu

ugd ρ ρ

ρ

+ = =⎛ ⎞−⎜ ⎟⎝ ⎠

b

SdS

u+ =

btutt

dτ+ = =

Exchange Flow ScalingNon Dimensional Buoyant Jet Equations

( )21

Pet

ρ ρ ρ+

+ + + ++

∂ + ∇ = ∇∂ Π

( ) ( )R

1 1

1

uu u P S

t

ρρ ρ

+ +

+ + + + + + + + ++

∂ ⎛ ⎞+ ∇ ⋅ = −∇ + + ∇ ⋅⎜ ⎟⎜ ⎟∂ − Π Π⎝ ⎠

Characteristic Time Ratios

Re1t ρ∂ Π Π⎝ ⎠

H

Cρ

ρρ

Π = bPe

m

u d

κΠ = Re

b

C

u d

υΠ =

Exchange Flow Scaling• Fluid property similarity can be assumed.

– Density ratio equal to unity.• For exchange flow velocity high compared to mass diffusivity…

– Peclet number would tend to be large.• Diffusive mixing of the gas streams could be ignored.

• Assume inertial force much greater than viscous force.Reynolds Number would tend to be very large– Reynolds Number would tend to be very large.

• For this idealized, inviscid scenario, the exchange flow velocity would be determined primarily by the density difference between the two gases.

( )C

HCLP

gdu

ρρρ −= 44.0

Exchange Flow ScalingInviscid Flow

( )2

1

RCRLP

du ⎟⎟

⎠

⎞⎜⎜⎝

⎛ Δ=ρ

ρ

( ) ( ) 21

RRLP du =

General velocity ratio

Assuming fluid property similarity( ) ( )RRLP

( ) ( ) ( ) ( ) ( )512 2

LP Brk BrkR R R R RQ u a d a d= = =

52

R

R R

V Vt

Q d

⎛ ⎞⎛ ⎞= = ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

Exchange flow time scale ratio

Volumetric flow ratio

Exchange Flow Scaling• Prototypical Reynolds number is large.

– Approximates inviscid flow.• Model Reynolds number is much

smaller. – Inviscid flow assumption may or may

not be applicable. – Difficult to match Reynolds numbers in

( ) ( ) ( )3 2Re bR R R

u d dΠ = =

Difficult to match Reynolds numbers in a reduced scale facility.

• For proper simulation of the air ingress phenomena the flow in the duct during exchange flow should be in the same regime.

• Turbulent flow.– Mixing at the interface will most likely

not be driven by molecular diffusion.

bPe

t

u d

κΠ =

( ) ( ) ( )t j bR RRv bu du= =

Exchange Flow Scaling Ratios• It is important to determine if

the prototype and the model are operating in similar flow regimes during the exchange flow process.– This will drive both the

turbulent mixing and the t b l

( ) ( )32

Re 1: 20.7R

R

dΠ = =

Re Ratio

momentum balance. • Prototypical Re = 200,000.

– 1000°C Helium in the lower plenum.

– 100°C air in the reactor cavity.• Model Re = 10,000.

( ) 52

1:1.46EF R

R

Vt

d

⎛ ⎞= =⎜ ⎟⎝ ⎠

Time Scale Ratio

Molecular Diffusion• End state of the exchange flow mixing process is the establishment of

an air-helium interface in the lower plenum.– Note that for an upper vessel break the end state of the exchange flow

process may result in the air-helium interface at elevations greater than the top of the lower plenum.

• Assumptions: – All fluid flows are one-dimensional along the loop axis, therefore fluid

properties are uniform at every cross-section.properties are uniform at every cross section.– The Boussinesq approximation is applicable.– The fluid is incompressible. (Mach number < 0.3)– The cold side and hot side temperatures are constant. – The diffusion coefficients are independent of gas concentration.– The molar average velocity, w, can be used in the momentum equation.

• No effort has been made in this analysis to capture local convection effects such as those that might be caused by the graphite oxidation process.

Molecular Diffusion

HOTSIDE

HELIUM

AIR

Lo

DC

DH

SIDE

COLDSIDE

Molecular DiffusionGoverning Equations for Air Ingress by Molecular Diffusion

Gas Mixture Continuity imm && =

Continuity Equation for Air 2

2

z

XD

z

Xw

t

X AAA

∂∂

=∂

∂+

∂∂

2∂∂∂

Loop Momentum Equation

Continuity Equation for He 2

2

z

XD

z

Xw

t

X HeHeHe

∂∂

=∂

∂+

∂∂

( ) ∑∑== ⎥

⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+−−=⋅⎟⎟

⎠

⎞⎜⎜⎝

⎛ N

i i

c

ihcavgDHC

N

i i

i

a

aK

d

fl

a

mLg

dt

md

a

l

1

2

2

2

1 2

1

ρρρ &&

( )1A A A HeM

PX M X M

RTρ ⎡ ⎤= + −⎣ ⎦

( )( )M C

vg sys pg H C loss str

d T TC M mC T T q q

dt

− = − + +& & &

Gas Density Equation

Loop Energy Equation

Molecular DiffusionNormalized Variables

00

0

Da

Lm

m

mm

co ρ&

&

&& ==+

D

wLww o==+

0

avgavg

ρρ

ρ+ =

DD

LL

L+ =

0

DD

D+ =

sys syssys

M MM

M L+ = =

0Dwo

oL

zz =+

o

N

i i

c

ih

N

i i

c

ihN

i i

c

ih

a

aK

d

fl

a

aK

d

fl

a

aK

d

fl

∑

∑∑

=

=+

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

1

2

1

2

1

2

2

1

2

1

2

1

( ) ( )( )

0

C HC H

C H

ρ ρρ ρ

ρ ρ+ −

− =−

0L ,sys

sys o o c oM a Lρ

0

( )( )

( )M C

M CH C

T TT T

T T+ −− =

−

0

( )( )

( )H C

H CH C

T TT T

T T+ −− =

−

Molecular DiffusionNon Dimensional Molecular Diffusion Equations

2

2

+

++

+

++

+

+

∂

∂=

∂∂

+∂∂

z

XD

z

Xw

t

X AAA

2

2

+

++

+

++

+

+

∂

∂=

∂∂

+∂

∂

z

XD

z

Xw

t

X HeHeHe

Continuity Equation for Air

Continuity Equation for He

∂∂∂ zzt

( )22

1

1

2

NcF

G Gr C H Diavg h ii

amdm flL K

dt d aρ ρ

ρ

+++

+ ++ +

=

⎧ ⎫⎡ ⎤⎛ ⎞ ⎛ ⎞Π ⎪ ⎪⎢ ⎥Π = Π − − +⎨ ⎬⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎪ ⎪⎣ ⎦⎩ ⎭

∑&&

, ,

( )1( )M CH C loss diff loss str diff str

d T Tm T T q q

dtγ

++ + + +

+

− = − + Π + Π& & &



Molecular DiffusionCharacteristic Time Ratios for Molecular Diffusion

∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛=Π

N

i io

ciG aL

al

1

pg

vg

C

Cγ =

( ) 3

20 0

C H ooRi

g L

D

ρ ρρ−

Π =

o

N

i i

c

ihF a

aK

d

fl∑= ⎥

⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+=Π

1

2

2

1

,0 0,

0 0 0( )loss

loss diffc p H C

q L

a D C T TρΠ =

−&

,0 0,

0 0 0( )str

str diffc p H C

q L

a D C T TρΠ =

−&

Molecular Diffusion Scaling Ratios• Primary heat removal mechanism.

– Prismatic block core.• Conduction.

– Pebble bed core.• Radiation.

– Reflectors.• Conduction for both core designs.g

• Thermal resistance scales inversely to the core decay power during natural circulation.

( ),

11:113.7core diff R

th R

qR

⎛ ⎞= =⎜ ⎟⎝ ⎠

&

Molecular Diffusion Scaling Ratios• Richardson number.

– Accounts for the relative effect of buoyancy versus mass diffusion.– Assuming fluid property similitude.

( ) ( )30 1: 64Ri R RLΠ = =

• There will be some distortion due to the reduced scale of facility.• Important that the phenomena represented by the Richardson number should be of

the same qualitative character for both model and prototype. • Prototypical Richardson number is approximately 3.5x1010.

• 1000°C helium as the hot fluid.• 100°C air as the cold fluid.• Diffusion coefficient for air and helium.

• Model Richardson number will be closer to 5.5x108. • Both represent a highly stratified system where there is resistance to diffusive

mixing due to the density differences of the fluid.

Molecular Diffusion Scaling RatiosLength Scale of Individual Components

1i c i

o i oR R

l a l

L a L

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠∑ ∑

( ) 1: 4il =( )i

Area Scale of Individual Components

( ) 85.56:1=Ria

Molecular Diffusion Scaling Ratios


EquationPrototype


Distortion

NA 1 0

Time Scale Ratio

Characteristic Scale Ratios

( ) ( )2 1:16D oR Rt L= =

NA 1 0

3.5x1010 1:64 0.98

NA 1 0

~1.6 1 0

NA 1 0

NA 1 0

Natural Circulation• Single phase natural circulation heat transfer in the core.

– Natural circulation phase begins when air ingress is sufficient to start natural circulation flow upward through the core.

– Following analysis applicable for both DCC and PCC events. • Assumptions:

– Flow is one-dimensional along the loop axis.Fluid properties are uniform at each cross-section– Fluid properties are uniform at each cross-section.

– Fluid densities assumed equal to an average fluid density except for those that comprise the buoyancy term and acceleration due to expansion across the core.

– Fluid is incompressible. (Mach number < 0.3)– Pressure losses in the core dominate the loop resistance.– Viscous dissipation is negligible.

Natural Circulation

Natural CirculationGoverning Equations for Natural Circulation

Conservation of Momentum

( ) ( )22 21

1

1N NH Ci c

H C h

T Tl adm m fl mg T T L K

ββ ρ

⎡ ⎤ −⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥= − − + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟∑ ∑& & &

Conservation of Energy

( ) ( )1 2 21 1 12 1c H C th

i ii avg c h i c c H Ci

g T T L Ka dt a d a a T T

β ρρ ρ β= =

⎢ ⎥+⎜ ⎟ ⎜ ⎟ ⎜ ⎟ − −⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦∑ ∑

( ) ( )1M C

vg sys pg H C loss str

d T TC M mC T T q q

dt

−= − + +& & &

Natural CirculationNormalized Variables

1 1 0

N Nsys sysi

loop ii ii c co c

M Ml

u m u aτ τ

ρ= =

= = = =∑ ∑ &

0m

mm

&

&& =+

( ) ( )( )0CM

CMCM TT

TTTT

−−=− +

( ) ( )( )01

11

CH

CHCH TT

TTTT

−−=− +

o

N

i i

c

ih

N

i i

c

ihN

i i

c

ih

a

aK

d

fl

a

aK

d

fl

a

aK

d

fl

∑

∑∑

=

=+

=

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

=⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+

1

2

1

2

1

2

2

1

2

1

2

1

( )01 CH

( )( )

( )( )

( )( )

1

11

1 1

1 0

1

1

1

H C

H CH C

H C H C

H C

T T

T TT T

T T T T

T T

βββ

β ββ

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

,0

strstr

str

qq

q+ =

&&

&

0,loss

lossloss q

qq

&

&& =+

Natural CirculationNon Dimensional Natural Circulation Equations


( ) ( ) ( ) ( )( )

22 2 1

11 1

1

2 1

NH Cc

L Ri H C F Ei h i H Ci

T Tadm flT T m K m

dt d a T T

ββ

+ ++

+ + ++

=

⎧ ⎫⎡ ⎤ ⎡ ⎤−⎛ ⎞ ⎛ ⎞⎪ ⎪⎢ ⎥Π = Π − −Π + +Π ⎢ ⎥⎨ ⎬⎜ ⎟ ⎜ ⎟ − −⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎪ ⎪ ⎣ ⎦⎣ ⎦⎩ ⎭∑&

& &


( )1 1i h i H Ciβ⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎪ ⎪ ⎣ ⎦⎣ ⎦⎩ ⎭

( )1

1 MH C str str loss loss

dTm T T q q

dtγ

+++ + +

+ = − − Π − Π& & &

Natural CirculationCharacteristic Time Ratios for Natural Circulation

∑=

=ΠN

i iref

ciL al

al

1

sysref

c c

Ml

aρ=

( )2

01 thCHRi u

LTTg −=Π

β ,03

core thRi

gq L

a C u

βρ

Π =&

Loop Reference Length Number where

Loop Richardson Number orcou c c pg coa C uρ

2

1

1

2

Nc c

Fiavg h ii o

aflK

d a

ρρ =

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

∑

FRi Π=Π3

1

⎟⎟⎠

⎞⎜⎜⎝

⎛

Π=

Fpc

thcoco Ca

gLqu

ρβ &

Loop Resistance Number

Steady State Solution

Natural CirculationCharacteristic Time Ratios for Natural Circulation

Ratio of Specific Heats

Thermal Expansion Number

pg

vg

C

Cγ =

( )( )

1

11H C

EH C

T T

T T

ββ

⎡ ⎤−Π = ⎢ ⎥− −⎢ ⎥⎣ ⎦

Stored Energy Transport Number

Loop Heat Loss Number

( )11 H C oT Tβ⎢ ⎥⎣ ⎦

( ),0 ,0

0 ,00

str strstr

P M C core

q q

m C T T qΠ = =

−& &

& &

( ),0 ,0

0 ,00

loss lossloss

P M C core

q q

m C T T qΠ = =

−& &

& &

Natural Circulation Scaling Ratios

Velocity Scale Ratio

( ) 1: 2R

NC R

Lu

t

⎛ ⎞= =⎜ ⎟⎝ ⎠

Time Scale Ratio

( )1

32

,0

1: 2

R

c FNC R

core

a lt

q

⎛ ⎞Π= =⎜ ⎟⎜ ⎟⎝ ⎠&

Natural Circulation Scaling Ratios

Characteristic Scale Ratios


EquationPrototype


Distortion

NA 1 0

>>1 1 0>>1 1 0

NA 1 0

~1.6 1 0

~2.0 1 0

NA 1 0

NA 1 0

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Depressurized Conduction Coooodo a gldown Event Scaling– To reach this depressurized state, the system must go through a variety of mass and energy states as the pressure decreases