depressurized conduction coooodo a gldown event scaling– to reach this depressurized state, the...
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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
Develop ΠGroups andSimilarityCriteria
Significant ScalingDistortions inImportant Π
Groups?
Significant ScalingDistortions inImportant Π
Groups?
System DesignSpecifications
System DesignSpecifications
(6)
(5)
(4)
NoNo
YesYes
Develop ΠGroups andSimilarityCriteria
Significant ScalingDistortions inImportant Π
Groups?
System DesignSpecifications
No
Yes
Develop ΠGroups andSimilarityCriteria
Significant ScalingDistortions inImportant Π
Groups?
System DesignSpecifications
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
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
Depressurizaton Scaling Analysis
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
,&
&& =+
ostr
strstr q
,&
&& =+
,0
lossloss
loss
q+ =
&&
&
Depressurizaton Scaling Analysis
( )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& & &
Loop Momentum Equation
Loop Energy Equation
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
Characteristic Ratio
EquationPrototype
ValuePrototype to Model Ratio
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
q+ =
&&
&
0,loss
lossloss q
&
&& =+
Natural CirculationNon Dimensional Natural Circulation Equations
Loop Momentum Equation
( ) ( ) ( ) ( )( )
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 ⎢ ⎥⎨ ⎬⎜ ⎟ ⎜ ⎟ − −⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎪ ⎪ ⎣ ⎦⎣ ⎦⎩ ⎭∑&
& &
Loop Energy Equation
( )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
Characteristic Ratio
EquationPrototype
ValuePrototype to Model Ratio
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
Questions?Q