Robert M. Edwards of Penn State presents exploratory modeling of nuclear electric propulsion (NEP) systems. The modeling includes reactor kinetics equations, fuel pin thermal dynamics, and integration with a Brayton power conversion system. Dynamic simulations are developed in Simulink to examine responses of reactor power and temperature to reactivity inputs, as well as pressure responses in the gas circulation loops. The modeling aims to integrate detailed component models for reactor cores, heat exchangers and power conversion to enable system-level simulations and control design studies of NEP concepts.

1. Exploratory NEP modeling
Robert M. Edwards
Penn State
814.865.0037
rmenuc@engr.psu.edu

2. Motivation: System Integration
Steam
Generator &
Electrical:
Pressure
Vessel &
Piping
Core Design:
neutronics
thermal
hydraulics
Mechanical
Pumps, valves
turbines
System
Integration
(Control
Engineering)
DETAILED MODELSDETAILED MODELS
DETAILED MODELS DETAILED MODELS

3. References
 Scoping Calculations of Power Sources for Nuclear Electric Propulsion,
ORNL CR-191133, 1994
 50 MW 4-year reactor example data
 Brayton Power Conversion System Parametric Design Modeling for NEP,
NASA contractor report CR-191135, 1993
 500 kWe Brayton PCU
 Modular Modeling System (MMS): A Code for the Dynamic Simulation
of Fossil and Nuclear Power Plants: Overview and General Theory, EPRI
CS/NP-2989, 1983
 Preliminary Results of a Dynamic System Model for a Closed-Loop
Brayton Cycle Coupled to a Nuclear Reactor, Steven Wright, Sandia
National Lab.
 “Dynamic Analysis and Control System Design for an Advanced Nuclear
Gas Turbine Power Plant”, a dissertation in Mechanical Engineering,
MIT 1990.

4. Reactor Kinetics Equations
 output nr, relative reactor power
 input r(t), reactivity
 from control devices
 feedback from temperature, etc
 b is fraction of neutrons that are “delayed”
 
  1,...6icn
dt
dc
cn
t
dt
dn
irri
ir
ir
6
1i
i
r
r


b


br
 


5. reactor power response to 10 cents
without temperature feedback
0 1 2 3 4 5 6 7 8 9 10
1
1.05
1.1
1.15
1.2
1.25
1.3
1.35
1.4
power response of reactor without feedback to 10 cents
seconds
relativereactorpower
  br 1.0t

6. Scoping Calculations of Power
Sources for Nuclear Electric
Propulsion, ORNL CR-191133, 1994
 50 MW, four year life
 82.24 cm diameter, 75.37 cm height
 82.92% enriched Uranium
 5871 fuel pins, 6.4 mm diameter
 Tantalum-181 clad, 0.6355 mm
 Tungsten liner, 0.127 mm
 Uranium-Nitride fuel, 4.826 mm
 Lithium coolant, 16.139 kg/s
 2.75 g/s per fuel pin
 500 oK inlet temperature, 1200 oK outlet
temperature

7. Reactor Fuel Pin Equations:
for 30 axial nodes, k
  ccinkp
k
ckcc
k
k
ck
f
kfk
f
f
kff
C/T)k(Tcm
R
)k(T)k(T
dt
)k(Td
dt
)k(dT
C/
R
)k(T)k(T
R
)k(T)k(T
dt
)k(dT
C/
R
)k(T)k(T
Q
dt
)k(dT


















 










 


Fuel (Tf)Clad (Tk)Coolant (Tc)
Tc(k)
Tc(k-1)
Rk Rf
Cc Ck Cf

8. Fuel temperature response to
10% step change in power
0 1 2 3 4 5 6 7 8 9 10
900
905
910
915
920
925
930
935
940
945
Fuel temperature response to 10% step in power
seconds
temperature(K)

9. SIMULINK Reactor Model

10. reactor power response to 10 cents
WITH temperature feedback
0 2 4 6 8 10
0.98
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
power response of reactor with feedback to 10 cents
seconds
relativereactorpower

11. reactor temperature response to 10
cents WITH temperature feedback
0 2 4 6 8 10
901.9
902
902.1
902.2
902.3
902.4
902.5
902.6
902.7
seconds
temperature(K)
temperature response of reactor with feedback to 10 cents

12. Brayton Power Conversion System
Parametric Design Modeling for NEP,
NASA contractor report CR-191135, 1993
 500 kWe unit
 Helium-Xenon with cp=0.5 cal/g-K
 Compressor inlet
 375 oK, 1339.18 kPa
 Turbine inlet
 1144.69 oK, 2355.46 kPa
 Lithium Intermediate Heat Exchanger
 1166.7 inlet, 1111.1 oK outlet temperatures
 (considerably smaller DT than 50 MW reactor)

13. Brayton Code Model
Duct 6
Duct 5
Duct 4
Duct 2
Duct 3
Duct 1
Gas
Cooling
Aux
Cooling
IHX
TurbComp Generator
Recuperator
16
1
5
6 7
1314
15
10
8
12
9

14. Brayton Code data
 Temperatures, K 375.00 500.67 500.67
500.67 500.67 500.67 871.19 869.82 1144.44
1141.69 1141.69 939.33 936.58 566.05
566.05 375.00 375.00
 Pressures, KPA 1339.18 1874.86 1874.86
2410.53 2410.53 2398.48 2379.17 2367.28
2362.54 2355.46 1867.82 1380.19 1376.05
1359.47 1352.68 1345.91 1339.18

15. Brayton Code data
 duct dimensions
 Duct 1 diameter, cm 12.91509
 Length, cm 193.7263
 Duct 2 diameter, cm 14.96846
 Length, cm 224.52680
 Duct 3 diameter, cm 18.38337
 Length, cm 275.75050
 Duct 4 diameter, cm 22.70933
 Length, cm 340.64000
 Duct 5 diameter, cm 17.61942
 Length, cm 264.29120
 Duct 6 diameter, cm 15.94148
 Length, cm 239.12220

16. MMS component equation set:
mass, momentum, energy
 
 
 
r






r

r

r
r





 r



r
r









r







r
P/huuses
dt
dP
V
dt
d
VhWqhmhm
V
1
dt
dh
hP,fuses
dt
d
VhWqhmhm
Vdt
d
dt
dP
K
m
PP
L
A
dt
md
mm
V
1
dt
d
oo
sooii
o
Ph
o
sooii
ho
o
2
i
oi
i
oi
o





17. Ideal Gas Assumption
RT
1
P
TRc
P
h
RT
P
Tch
h
P
2
pP
h
p


r



r

r


18. SIMULINK “duct” model

19. duct model equation implementation
 function sys=mdlDerivatives(t,x,u,V,L,Af,cp,R,K,mo)
 delp=u(1)-x(1);
 rhoo=x(1)/(R*x(2));rhoi=u(1)/(R*u(2));rhobar=(rhoo+rhoi)/2;
 if mo==0
 if delp<=0
 mdi=0;
 else
 mdi=K*sqrt(delp*rhobar);
 end
 else
 xd(3)=Af/L*(u(1)-x(1)-(u(3)/K)^2/rhobar);
 %Vi=x(3)/rhoi/Af;Vo=u(3)/rhoo/Af; this term creates numerical
 %problems and is neglected.
 %xd(3)=xd(3)+(x(3)*Vi-u(3)*Vo)/L
 mdi=x(3);
 end
 drholdt=(mdi-u(3))/V;
 Tbar=(u(2)+x(2))/2;
 Pbar=(u(1)+x(1))/2;
 dp=1/(R*Tbar);
 dh=-Pbar/(R*cp*Tbar^2);
 term1=(mdi*cp*u(2)-u(3)*cp*x(2)+u(4)-cp*Tbar*V*drholdt);
 xd(1)=(rhobar*drholdt-dh/V*term1)/(dh+dp*rhobar);
 xd(2)=(term1/(rhobar*V) + xd(1)/rhobar)/cp;
 sys=xd'

20. Duct model response to set
change in inlet pressure
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
2.35
2.4
2.45
2.5
2.55
2.6
x 10
6
outlet pressure response to inlet pressure step, duct 1
time in seconds
pressureinpascal

21. Response without the dynamic
form of the momentum equation
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
2.38
2.4
2.42
2.44
2.46
2.48
2.5
x 10
6
outlet pressure response to inlet pressure step, duct 1
time in seconds
pressureinpascal

22. Interconnecting the duct
components to form a loop

23. Interconnecting ducts to form
a heat exchanger

24. Compressor and Turbine models
from performance data
 A representation of Wright’s data
Mass flow
Speed 1
Speed 2
Pressure ratio
compressor
turbine
Speed 1

25. A linear generalization of
Wright’s data
1.2 1.4 1.6 1.8 2
2
4
6
8
10
12
14
16massflowkg/s
pressure ratio
compressor/turbine performance characteristics
compressor
turbine

26. Maps that execute
1.2 1.4 1.6 1.8 2
2
4
6
8
10
12
14
16
compressor
turbine
50000 RPM
50000 RPM

27. Compressor/Turbine Thermal
model
T4s
T3
T2
T2s
T
s
T1
T4

28. Compressor Thermal
equations
 Compressor
 Turbine
   

























1s2
2
12
1s2
k1k
1
2
1s2
1
s2
k1k
1
2
T)1(T
T
TT
TT
P
P
TT
T
T
P
P
 
 
 s4334
s43
43
k1k
4
3
3
s4
s4
3
k1k
4
3
TTTT
TT
TT
P
P
T
T
T
T
P
P



























29. SIMULINK compressor block
poscope
mdo mdi
Pi
Ti
mdo
N
Po
To
mdi
J
compressor2
ToTi
Pi
N J

30. SIMULINK Compressor model
4
J
3
mdi
2
To
1
Po
1/s
po
(u(1)-u(2))*u(3)*cp
mdot*cp*deltat
mp*u(1)+u(2)
mdot
(u(1)-u(2))*2500000
flow equalizer
mn*u(1)+b
Yint
u(3)*(u(1)/u(2))^((k-1)/k)
T2s
(u(1)-(1-eta)*u(2))/eta
T2
u(2)/u(1)
Pr
4
N
3
mdo
2
Ti
1
Pi

31. SIMULINK model of CBC

32. Compressor-recuperator
subsystem

33. turbine – IHX subsystem
constant heat input

34. Flow response to speed step
transient 28000->30000 RPM
10 10.2 10.4 10.6 10.8 11 11.2 11.4 11.6 11.8
6.5
6.6
6.7
6.8
6.9
7
7.1
7.2
7.3
flow response to speed step 28000->30000
time (sec)
flow(kg/s)

35. Turbine inlet temperature to speed
step transient 28000->30000 RPM
0 2 4 6 8 10 12 14 16 18 20
1100
1105
1110
1115
1120
1125
1130
1135
1140
1145
turbine inlet temperature to speed step 28000->30000
time (sec)
temperature(K)

36. Net power output response to speed
step transient 28000->30000 RPM
0 2 4 6 8 10 12 14 16 18 20
1
1.02
1.04
1.06
1.08
1.1
1.12
1.14
1.16
x 10
6
net power output to step in speed 28000 30000
time (sec)
power(watts)

37. CBC with reactor model
turbine
exhaust
recuperator
exhaust
high
pressurePi
Ti
md0
N
mdRx
rho
Po
To
mdi
J
nr
turbine_ihx
rho
mdot
Pi
Ti
mdo
N
Po
To
mdi
J
compressor-recuperator
Scope9
Scope8
Scope7
Scope6Scope5Scope4
Scope3
Scope2
Scope1
Scope
Output Point1
Output Point
N
Input Point2
Input Point1
Input Point

38. turbine-ihx with reactor model
5
nr
4
J
3
mdi
2
To
1
Po
Pi
Ti
mdo
N
Po
To
mdi
J
turbine2 -3.797e04
q2
-1.892e04
q
mdot
Ti
peak
Tout
Tf max
Tf av g
Tkav g
Tcav g
rhof b
neppin
Tg
Ti
mdot
Q
TiRx
ihxts
Pi
Ti
mdo
q
Po
To
mdi
ihx_shell
Pi
Ti
mdo
q
Po
To
mdi
duct3
Pi
Ti
mdo
q
Po
To
mdi
duct2
Scope7
Scope6
Scope5
Scope4
Scope3
Scope2
Scope1
Scope
1/0.646
Gain2
109.61
Gain1
.5
Gain
rho (dk) nr
6-delayed
groups
6
rho
5
mdRx
4
N
3
md0
2
Ti
1
Pi

39. mass flow Response to speed
step 28000 to 30000 RPM
10 10.2 10.4 10.6 10.8 11 11.2 11.4 11.6 11.8
6.5
6.6
6.7
6.8
6.9
7
7.1
7.2
7.3
flow response to step in speed to 30000 RPM
seconds
flow(kg/s)

40. turbine inlet temperature Response
to speed step 28000 to 30000 RPM
0 5 10 15 20 25 30 35 40 45 50
1130
1132
1134
1136
1138
1140
1142
turbine inlet temperature to step in speed to 30000 RPM
seconds
temperature(K)

41. Reactor power Response to
speed step 28000 to 30000 RPM
0 5 10 15 20 25 30 35 40 45 50
1
1.005
1.01
1.015
1.02
1.025
1.03
1.035
power response of reactor to step in speed to 30000 RPM
seconds
relativereactorpower

42. net power output Response to
speed step 28000 to 30000 RPM
0 5 10 15 20 25 30 35 40 45 50
1.06
1.07
1.08
1.09
1.1
1.11
1.12
1.13
1.14
1.15
x 10
6
seconds
netpoweroutput(watts)
net power output to step in speed to 30000 RPM

43. Simplified generator model
added:  
I
PP
2
dt
d outin
2 



44. Speed Response to step -15%
step decrease in load
0 50 100 150 200 250 300 350 400 450 500
2.8
2.9
3
3.1
3.2
3.3
3.4
3.5
3.6
x 10
4
speed response to -15% step in load
time (sec)
speed(RPM)

45. Turbine Inlet Temperature
response to -15% in load
0 50 100 150 200 250 300 350 400 450 500
1100
1105
1110
1115
1120
1125
1130
1135
1140
1145
turbine inlet temperature to -15% step in load
time (sec)
temperature(K)

46. Reactor Power Response to
-15% step in load
0 50 100 150 200 250 300 350 400 450 500
1
1.05
1.1
reactor power response to -15% step in load
time (sec)
relativereactorpower

47. Net Power Output response to
-15% step in load
0 50 100 150 200 250 300 350 400 450 500
9
9.2
9.4
9.6
9.8
10
10.2
10.4
10.6
x 10
5
Net power output to -15% step in load
time (sec)
power(watts)

48. Summary, exploratory NEP
modeling approach
 representative fuel pin from a 50 MW four
year core
 500 kWe CBC
 MMS equation set
 not suitable for low pressure drops and flows
 simplified compressor/turbine performance
curves
 results consistent with Wright
 more study needed

49. Possible Improvements
 T=f(h,P), r=f(h,P), h=f(T,P)
 compressor/turbine performance maps
 mass flow as a function of speed and Pr
 efficiency as a function of speed and Pr
 add component metal heat capacities
 data for another Brayton unit
 full power steady state temperatures and pressures around
the unit
 component dimensions, masses, heat capacities
 more detail on the generator, electrical power
distribution system, and ion propulsion