Hydrodynamic Effects of S-CO2 Property Variations in Nuclear Energy Systems Michael Z. Podowski and Tara Gallaway Steven P. Antal - contributor Center for Multiphase Research Rensselaer Polytechnic Institute Supercritical CO2 Power Cycle Symposium Boulder, Colorado, May 24–25, 2011 Center for Multiphase Research Rensselaer Polytechnic Institute
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Presentation Outline Application of supercritical fluids in Gen. IV nuclear energy systems Overview of generic issues associated with supercritical fluid systems Analysis of flow induced oscillations in supercritical fluid systems Effect of property variations on local flow and heat transfer at near-critical pressures Center for Multiphase Research Rensselaer Polytechnic Institute
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Gen. IV Supercritical Fluid Reactors Supercritical-CO2-Cooled Reactor
SCO2
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Sodium Fast Reactor (SFR) with S-CO2 Brayton Cycle SCO2
Compressor
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SCO2 Brayton Cycle
SCO 2
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Critical Issues for Application of Supercritical Fluid in Nuclear Energy Conversion Systems ¾ ¾
¾
¾
Highly ‘reactor power’-to-flow ratio Effect of system operating conditions (high temperature, neutronics) on material properties Dynamic response (oscillations and instabilities) of supercritical fluid systems, driven by fluid property variations Effect of property variations on local flow and heat transfer at near-critical pressures Center for Multiphase Research Rensselaer Polytechnic Institute
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Overall Goal of Present Study ¾
¾
Improve our understanding of fundamentals of fluid mechanics, heat transfer and system dynamics Provide mechanistic framework for a variety of applications, such as: Help to properly interpret experimental results Guide future experiments Develop scaling criteria
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Nature of flow-induced instabilities Predictions depend on ¾
¾
Good understanding of underlying mechanisms (instability modes) Combination of proper formulation of governing equations and the associated BCs
Numerical solutions using large (system) codes make it difficult to separate physics from numerics, and identify proper means to mitigate instabilities Instabilities in BWRs are still not well understood Rigorous solutions of well-defined problems are key to effective measures to avoid instabilities Center for Multiphase Research Rensselaer Polytechnic Institute
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Stability concept System response win
External perturbation Δp
t win t
t Center for Multiphase Research Rensselaer Polytechnic Institute
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Modeling of instabilities Linear models ¾ ¾
¾
Applicable up to marginal stability conditions Both time-domain and frequency-domain (FD) methods are available FD approach is very accurate
Nonlinear models ¾
¾
Required to understand range of stable response and properties of unstable response (limit cycle, period bifurcation, stability islands, etc.) Direct time-domain integration is a standard approach (accuracy is an issue) Center for Multiphase Research Rensselaer Polytechnic Institute
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Unstable Response of Loop with Parallel-Channel Heater Example of two parallel channels
Case (1): Unstable channels, stable external loop Center for Multiphase Research Rensselaer Polytechnic Institute
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Unstable Response of Loop with Parallel-Channel Heater
Case (2): Unstable channels, weakly unstable external loop
Case (3): Unstable channels, unstable external loop
Case (4): Stable channels, unstable external loop
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Multiple Channel Systems
Parallel-channel mode
Channel-to-channel mode Center for Multiphase Research Rensselaer Polytechnic Institute
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Systems with Combined Heated/Unheated Sections
Heated channel with riser
Heated channel with downcomer Center for Multiphase Research Rensselaer Polytechnic Institute
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1-D Model of SCW Channel Dynamics Continuity equation ∂ρ ∂G + =0 ∂t ∂z
Conservation of energy equation ∂ ( ρh ) ∂ ( Gh ) q′′Pw + = ∂t
∂z
Ac
Momentum equation ⎛ G2 ⎞ ∂⎜ ⎟ 2 2 2 ρ p f G G G ∂G ∂ + ⎝ ⎠ =− − − ρg − Kin in δ ( z ) − Kout out δ ( z − L ) ∂t ∂z ∂z 2 DH ρ 2ρin 2ρout
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Reason for Potential Instabilities: Property Variations 900
-4
12
Tin
1.2x10
800 10
700
μ [kg/m s]
500 400
CO 2
Tout
300 Water
200
Tin
CO2
2
0.75
0.85
0.95
1.05
1.15
1.25
1.35
0 0.65
T/Tpc
-5
8.0x10
-5
6.0x10
CO2
Tout
100 0 0.65
6 4
Tin Water
8
Pr
ρ [kg/m 3 ]
600
-4
1.0x10
-5
4.0x10
Water
Tout
-5
2.0x10 0.75
0.85
0.95
1.05
1.15
1.25
1.35
0.65
0.75
0.85
0.95
T/Tpc
1.05
1.15
1.25
1.35
T/Tpc
Analytical model of S-H2O and S-CO2 properties ζ(T , p ) = aζ ,i ( p ) + bζ ,i ( p )T + cζ ,i ( p )T 2 + d ζ ,i ( p )T 3 for
Ti ≤ T ≤ Ti +1 (i = 1, 2,..., K ) Center for Multiphase Research Rensselaer Polytechnic Institute
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Effect of pressure on S-H2O properties
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Linear Stability Analysis Model linearization: G ( z , t ) = Gss + δG ( z , t )
h ( z , t ) = hss ( z ) + δh ( z , t )
δρ ( z , t ) =
d ρ ss dh
δh ( z , t ) hss
0
4.1359x10-25 -5.000 x10-7
-10
d μ /dT
d ρ /dT
-1.000 x10-6
-20
-0.0000015 -0.000002 -0.0000025
-30
-0.000003
25 MPa -40
25 MPa
-0.0000035
300
350
400
Temperature [C]
450
500
300
350
400
450
500
Temperatue [C]
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Standard Approach: Channel Discretization 1
Continuity and energy equations are discretized An
2
ρ ss ,n 3
4
M N
dyn ( xn − xn−1 ) + =0 dt Δz
( yn − yn −1 ) = 0 dyn ⎛ x + xn −1 ⎞ + b⎜ n + G ss ⎟ dt Δz 2 ⎝ ⎠
⎧ xn = δG (t , zn ) where ⎨ ⎩ yn = δh(t , zn )
After algebraic manipulations and including the momentum equation dyn = λ n x0 + γ n ,1 y1 + γ n ,2 y2 + ... + γ n ,n yn dt dx0 = ψδΔP + βx0 + α1 y1 + α2 y2 + ... + α N y N dt or x0(
N)
+ a N −1 x0(
N −1)
where
xo = δGin (t )
% P + K + a1 x0 + a0 = ψδΔ Center for Multiphase Research Rensselaer Polytechnic Institute
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Stability Analysis based on Discretized Models Method 1: direct integration in time domain 1
2
dyn = λ n x0 + γ n ,1 y1 + γ n ,2 y2 + ... + γ n ,n yn dt dx0 = ψδΔP + βx0 + α1 y1 + α2 y2 + ... + α N y N dt
3
4
Method 2: frequency domain transfer function x0(
M N
N)
+ a N −1 x0(
N −1)
% P + K + a1 x0 + a0 = ψδΔ M
H dis ( s ) =
} = ∑b s
{ L {δΔPˆ ( s )} ∑a s L δGˆ in ( s)
m =0 N
n =0
m
m
n
n
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Analytic Approach: Direct Integration in Frequency-Domain Method 3 Taking s = jω, the real and imaginary components of individual variables are separated, Xˆ ( jω) = Xˆ R (ω) + jXˆ I (ω) d 2 Xˆ R dXˆ R dXˆ I −β − α( z ) ω + γ ( z )ωXˆ I = Ψ Re(δΔpˆ ) 2 dz dz dz
1 dXˆ I ˆ ωYR = − A dz
d 2 Xˆ I dXˆ R dXˆ I + α( z ) ω −β − γ ( z )ωXˆ R = Ψ Im(δΔpˆ ) 2 dz dz dz
1 dXˆ R ˆ ωYI = A dz
L L L ⎛G ⎡ δΔpˆ ⎤ Gss2 AL ˆ 2Gss Gss Gss2 AL ˆ ⎞ 2Gss ˆ ss ˆ ˆ ˆ ˆ ˆ ⎜ ⎡ ⎤ XR − Y − + ∫ ⎣C1 X R + C2YR ⎦dz + g ∫ AYR dz + K in + K out XR − Y ⎟ ⎢ ˆ ⎥ = −ω∫ X I dz + 2 R 2 R ⎜ ⎟ ρ ρ ρ ρ X 2 (ρ ss , L ) ss , L ss ,0 ss ,0 ⎣ 0 ⎦R (ρ ss ,L ) 0 0 0 ⎝ ss ,L ⎠
L L L ⎛G ⎡ δΔpˆ ⎤ 2Gss ˆ Gss2 AL ˆ Gss2 AL ˆ ⎞ ss ˆ ˆ ˆ ˆ ˆ ⎜ ⎡ ⎤ = ω + − + + + + − X dz X Y C X C Y dz g AY dz K X Y ⎟ 2 I⎦ R I I out I ⎢ ˆ ⎥ 2 I 2 I ∫ ∫ ∫ ⎣ 1 I ⎜ ⎟ ρ ρ X 2 (ρ ss ,L ) ss , L ⎣ 0 ⎦I (ρ ss ,L ) 0 0 0 ⎝ ss ,L ⎠
Characteristic function of the system becomes δΔpˆ ( jω) = Re G ( ω) + j Im G ( ω) G ( jω) = ˆ X 0 ( jω) Center for Multiphase Research Rensselaer Polytechnic Institute
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Comparison of Models and Methods 3 2
455 kW/m2 470 kW/m2 485 kW/m2
0.8
G = 746.04 kg/m2 s
0.6
Direct Integration Fully Nodalized
0.4
Imag(G)
1
δ Gin/Gss
G = 746.04 kg/m2 s
0 -1
0.2 ω = 1.9 0 485 kW/m2 470 kW/m2 455 kW/m2
-0.2 T = 3.3 s -2 -3 0
-0.4 DR455 kW/m2 = 0.82 10
20
t [s]
30
40
50
-0.6 -0.2
0
0.2
0.4
0.6
0.8
1
Real(G)
Frequency domain models are in agreement with each other and the time domain Natural frequency in the frequency domain, ω = 1.9 [rad/s], agrees with the oscillation period in the time domain, T = 3.3 [s] Center for Multiphase Research Rensselaer Polytechnic Institute
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Stability Maps 140 120
1.8
1600 kg/s 1700 kg/s 1800 kg/s 1900 kg/s
1.6 1.4
NSUB,SC
(Tpc - Tin) [K]
100
1600 kg/s 1700 kg/s 1800 kg/s 1900 kg/s
80 60 40
1.2 1 0.8
Kin = 10, Kout = 1 20 350
390
430
470 2
510
q" [kW/m ]
Dimensional Map
550
0.6 2.7
2.8
2.9
3
3.1
3.2
3.3
3.4
NPCH, SC
Nondimensional Map N SUB , SC =
h pc − hin Δhref
N PCH , SC =
q wΔhref
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Model Testing and Verification Power [MWt] 2000
2500
3000
30 MPa
4000
Pandey et al. Current Model
w = 1000 kg/s w = 1100 kg/s w = 1200 kg/s
150
(Tpc - Tin) [K]
3500
100
50
0 200
300
400
500
2
600
700
q" [kW/m ]
Comparison against simplified model of Pandey and Kumar
5
Pseudo-Subcooling-Number
1500 200
800
4
Ortega Gomez (Kin = 0, Kout = 0) Current Model (Kin=0, Kout=0) Current Model (Kin=0, Kout=0.6)
3
2
1 5
6
7
8
9
10
Pseudo-Phase-Change-Number
Comparison against artificial two-phase flow model of Ortega Gomez
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Multidimensional Modeling of Flow and Heat Transfer in Fluids at Supercritical Pressures Averaging Approach to Flow Turbulence Effect of Local Property Variations on Heat Transfer
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Turbulence Modeling at Supercritical Pressures: Reynolds Averaging Definitions:
φ = φ + φ′
1 φ= Δt
t0 +Δt
∫
φdt
t0
Momentum equation ∂ ( ρu ) ∂ ( ρu u ) ∂ (ρ′u ′ ) ∂ (ρ′u ′u ) ∂ (ρ′u′ u ) ∂ (ρ′u′u ′ ) + + + + + = i
i
i
j
∂t ∂x j 14442444 3
∂t
i
∂x j
I
j
j i
i
∂x j
(
j
∂x j
)
⎛ ∂uk ⎞ ∂ ρ ui′u ′j ⎤ ∂ ∂Pi ∂ ⎡ ⎛ ∂ui ∂u j ⎞ 2 ⎥+ − + + ⎢μ ⎜ ⎟ − δi , j ⎜ μ ⎟− ∂x ∂x j ⎣⎢ ⎜⎝ ∂x j ∂xi ⎠ 3 ∂ ∂ x x ⎥ ∂x j k ⎠ j ⎝ 14444444444 4244444444444 3⎦
⎡ ⎛ ∂u ′ ∂u′ ⎢μ ′ ⎜ i + j ⎢ ⎜⎝ ∂x j ∂xi ⎣
⎛ ∂uk′ ⎞ 2 ⎟ − δi , j ⎜ μ ′ ⎠ 3 ⎝ ∂xk
⎞⎤ ⎟⎥ ⎠ ⎥⎦
I
-0.5
Conclusion: variable fluid properties introduce several new terms
-1
3
∂ρ h′ui′ ∂h h
2
ρ′ui′ =
∂ ρ/∂ h [kg /kJm ]
where
0
-1.5 -2 -2.5 -3 -3.5 -4 -4.5 200
250
300
350
400
450
500
h [kJ/kg]
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Turbulence Modeling at Supercritical Pressures: Favre Averaging Definitions:
ρφ ρ′φ′ φ% = = φ+ ρ ρ
φ′′ = φ −
ρφ ρ′φ′ = φ′ − ρ ρ
Momentum equation ∂ ( ρu% ) ∂ ( ρu% u% ) ∂P + =− + i
i
j
∂t ∂x j 14442444 3 I
(
)
⎛ ∂u%k ⎞ ∂ ρui′′u ′′j ⎤ ∂ ⎡ ⎛ ∂u%i ∂u% j ⎞ 2 ⎥ + ⎢μ ⎜ ⎟ − δi , j ⎜ μ ⎟− x x ∂x ∂x j ⎣⎢ ⎝⎜ ∂x j ∂xi ⎠ 3 ∂ ∂ ⎥ k ⎠ j ⎝ 14444444444 4244444444444 3⎦ i
I
⎛ ∂ui′′ ∂u ′′j ⎞ 2 ⎛ ∂uk′′ ∂ ⎡ ⎛ ∂ui′′ ∂u ′′j ⎞ ∂uk′′ ⎞ ⎤ ⎢μ ⎜ ′ ′′ + + + μ + − δ μ + μ ⎟ ⎜⎜ ⎟⎥ ⎟ i, j ⎜ ∂x j ⎢ ⎝⎜ ∂x j ∂xi ⎠⎟ ∂ ∂ x x 3 x x ∂ ∂ k k j i ⎠ ⎝ ⎠ ⎥⎦ ⎝ ⎣
Conclusion: if lower-magnitude terms are ignored, momentum equation becomes similar to that for constant-property fluids
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Turbulence Modeling: Lessons Learned Application of Reynolds-averaging to supercritical fluids concept introduces several new terms which are practically impossible to express it terms of state variables Favre averaging yields conservation equations which can be approximated by s standard form used by CFD codes Unresolved issue: application of existing turbulence models (such as k-ε or similar) using Favre-averaged variables Next step: use DNS approach to verify/modify averaged turbulence models Center for Multiphase Research Rensselaer Polytechnic Institute
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Thermal Models of Turbulence (A) High-Reynolds k-ε Standard approach to near-wall heat transfer q′′ Tw = Tp + * u ρcp
where
⎡ Prt ⎛ y + ⎞ ⎤ + ln ⎜ + ⎟ ⎥ ⎢ Pr yo + κ ⎝ yo ⎠ ⎦ ⎣
yo+ ≈ 11.2
New thermal wall function (for constant property fluids) q′′ Tw = Tp + * u ρcp
where
⎡ 2/3 + Prt ⎛ y + 1/3 ⎞ ⎤ ln ⎜ + Pr ⎟ ⎥ ⎢ Pr yo + κ ⎝ yo ⎠⎦ ⎣
yo+ = yh+ Pr1 3
New thermal wall function (for variable property fluids) q′′ Tw = Tp + * u ρc p ,m
where
⎛ Prt ⎛ y + ⎞ ⎞ + ln ⎜ + ⎟ ⎟ ⎜ Prm yh + κ ⎝ yh ⎠ ⎠ ⎝
φm = (1 − ζ)φ p + ζφw Center for Multiphase Research Rensselaer Polytechnic Institute
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Thermal Models of Turbulence (B) Low-Reynolds k-ε Model Near-wall turbulence decay modeled inside laminar sublayer (for y + ≥ 1)
(
⎛ k2 ⎞ − c3 y + νt = cμ ⎜ ⎟ 1 − e ⎝ ε ⎠
)
Near-wall heat transfer is mainly by conduction Tw
qw′′ y +p = ∫ k (T )dT Tp
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Low Reynolds Radial Property Distributions in Developing Flow using Low Reynolds k-ε Model 1.4
46 x = 35 cm (297 kJ/kg) x = 85 cm (331 kJ/kg) x = 135 cm (366 kJ/kg)
44
x = 35 cm (297 kJ/kg) x = 85 cm (331 kJ/kg) x = 135 cm 366 kJ/kg)
1.2
42
1
u [m/s]
o T [ C]
40 38
0.8 0.6
36 0.4
34
0.2
32 30
0 0
0.0005
0.001
0.0015
0.002
0
0.0025
r [m]
800
0.001
0.002
0.0025
x = 35 cm (297 kJ/kg) x = 85 cm (331 kJ/kg) x = 135 cm (366 kJ/kg)
30
600
0.0015
r [m]
35
700
25
500
cp [kJ/kgK]
3 ° [kg/m ]
0.0005
400 300
20 15 10
200 x = 35 cm (297 kJ/kg) x = 85 cm (3313 kJ/kg) x = 135 cm (366 kJ/kg)
100
5
0
0 0
0.0005
0.001
0.0015
r [m]
0.002
0.0025
0
0.0005
0.001
0.0015
0.002
0.0025
r [m]
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Model Validation against Experiments
Heat transfer enhancement
Heat transfer deterioration
High-Reynolds k-ε Model predicts heat transfer enhancement but fails to predict heat transfer deterioration (yet still captures heat transfer recovery regime) Low-Reynolds k-ε Model predicts both heat transfer enhancement and deterioration, although heat transfer coefficient for the latter is underestimated Center for Multiphase Research Rensselaer Polytechnic Institute
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Conclusions - 1 Two different methods of stability analysis have been introduced for systems cooled using fluids at supercritical pressures: a time-domain method and a frequency-domain method, with two different methods of solution for the latter: a rigorous (exact) integration method and a multi-node approximation It has been demonstrated that all methods of solution give similar results for the stability characteristics of heated channels Center for Multiphase Research Rensselaer Polytechnic Institute
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Conclusions - 2 A consistent mechanistic approach to model local multidimensional phenomena for fluids at supercritical pressures has been discussed, including both kinematic and thermal aspects of turbulence It has been demonstrated that the proposed Favreaveraged models are capable of capturing both heat transfer enhancement and heat transfer deterioration phenomena observed in experiments Effect of variable properties of supercritical fluids on turbulence is not fully understood and more work is still needed to develop physically-base models Center for Multiphase Research Rensselaer Polytechnic Institute
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