Copyright © 2014 by ASME 1
A Numerical Study of High Temperature and High Velocity Gaseous Hydrogen Flow in a Cooling Channel of a Nuclear Thermal Rocket Core
K. M. Akyuzlu
Department of Mechanical Engineering, University of New Orleans, New Orleans, LA 70148, USA Phone: (504) 280 6186
e-mail: [email protected]
ABSTRACT
Two mathematical models (a one-dimensional and a two-
dimensional) were adopted to study, numerically, the thermal
hydrodynamic characteristics of flow inside the cooling channels
of a Nuclear Thermal Rocket (NTR) engine. In the present
study, only a single one of the cooling channels of the reactor
core is simulated. The one-dimensional model adopted here
assumes the flow in this cooling channel to be steady,
compressible, turbulent, and subsonic. The physics based
mathematical model of the flow in the channel consists of
conservation of mass, momentum, and energy equations subject
to appropriate boundary conditions as defined by the physical
problem stated above. The working fluid (gaseous hydrogen) is
assumed to be compressible through a simple ideal gas relation.
The physical and transport properties of the hydrogen is assumed
be temperature dependent. The governing equations of the
compressible flow in cooling channels are discretized using the
second order accurate MacCormack finite difference scheme.
Convergence and grid independence studies were done to
determine the optimum computational cell mesh size and
computational time increment needed for the present
simulations. The steady state results of the proposed model were
compared to the predictions by a commercial CFD package
(Fluent.) The two-dimensional CFD solution was obtained in
two domains: the coolant (gaseous hydrogen) and the ZrC fuel
cladding. The wall heat flux which varied along the channel
length (as described by the nuclear variation in the nuclear
power generation) was given as an input.
Numerical experiments were carried out to simulate the
thermal and hydrodynamic characteristics of the flow inside a
single cooling channel of the reactor for a typical NERVA type
NTR engine where the inlet mass flow rate was given as an
input. The time dependent heat generation and its distribution
due to the nuclear reaction taking place in the fuel matrix
surrounding the cooling channel. Numerical simulations of flow
and heat transfer through the cooling channels were generated
for steady state gaseous hydrogen flow. The temperature,
pressure, density, and velocity distributions of the hydrogen gas
inside the coolant channel are then predicted by both one-
dimensional and two- dimensional model codes. The steady state
predictions of both models were compared to the existing results
and it is concluded that both models successfully predict the
steady state fluid temperature and pressure distributions
experienced in the NTR cooling channels. The two dimensional
model also predicts, successfully, the temperature distribution
inside the nuclear fuel cladding.
NOMENCLATURE
Symbols
A cross sectional area
As cladding surface area
a acoustic speed
cv specific heat at constant volume
D diameter of pipe
f Fanning friction coefficient
k thermal conductivity
L channel (axial) length
m mass flow rate
Ma Mach number
nf number of layers
P pressure of the gas
q heat flux
r radial distance
R gas constant, radius of cooling channel
Re Reynolds number
t time, thickness of cladding
T temperature
u axial velocity
x axial coordinate
Greek Symbols
convergence criterion
absolute viscosity
Proceedings of the ASME 2014 International Mechanical Engineering Congress and Exposition IMECE2014
November 14-20, 2014, Montreal, Quebec, Canada
IMECE2014-38438
Copyright © 2014 by ASME 2
ν kinematic viscosity
density
σ safety factor
τ shear stress
Subscripts
i inlet of channel
e exit of channel
w wall
INTRODUCTION
Nuclear Thermal Propulsion (NTP) has been identified as a
high NASA technology priority area by the National Research
Council (NRC). NTP could be a relatively near-term enabling
technology to reduce human transit time (and mission risk) to
Near-Earth Objects (NEOs) and Mars. Multi-mission nuclear
power and propulsion technologies are key enabling
technologies for future NASA exploration missions.
A review of the experiences gained from the Space Nuclear
Rocket Program (Rover) is given by Koenig [1]. Research on
Nuclear Thermal Rockets (NTR) started in 1959 and went until
1972. The research engines developed during this period were in
two categories, small like Kiwi type engines, and full size ones
like Phoebus which had a thermal power of 5320 Mw and a
thrust of 1123 KN. Also under NERVA, a program which
covered the period from 1964 to 1969, various types of nuclear
reactors and engines for NTR were built and tested (like NRX,
XE, and XEXF.) The description/specs of these engines and the
test results of the experiments on these engines can be found in a
final report by Koenig [1].
Figure 1 -Typical NERVA Derived NTR Engine [2]
A typical NTR engine (see Figure 1) is composed of
turbopumps to pressurize the cryogenic hydrogen, external
shield, nuclear reactor (reactor core, control drum, internal
shield), nozzle, and the nozzle extension. See references [2, 3,
and 4]. The nuclear core is composed of a matrix of fuel
elements with cooling channels through which the pressurized
cryogenic fluid (Hydrogen) flows (see Figure 2). Typical fuel
elements are hexagonal shape and made of composite fuel (UC2
coated with ZrC) graphite matrix shown in Figure 3. The fuel
matrix could also be in coated particle matrix form or as a
composite matrix form. The cooling channels are one tenth of an
inch (2.54 mm) in diameter and each fuel element has 16 of
them as shown in Figure 3. The fuel core also contains tie tubes
which extracts additional thermal energy from the nuclear core
to drive the turbopumps (TPA) [4].
Figure 2 - NTR Fission Reactor Cross-Section [4]
Figure 3 - NTR Fission Reactor Fuel Element and Tie Tube Cross-Sections [3]
The fluid attains high temperatures while passing through
the cooling channels of the core and then expands in the
converging–diverging nozzle. The temperatures in the core can
reach 2500 K. Maximum hydrogen temperature is very close to
this value. The operating pressure of a typical NTR core is
around 3 MPa. Some experimental NTRs have operated at
higher pressures.
Thermal power of the nuclear reactor which creates a thrust
of 337 kN is 1570 MW for the NERVA engine. Hydrogen flow
rate is 41.6 kg/sec and expansion ratio through the nozzle is
100:1. Chamber pressure and temperature for this engine is 3.1
MPa and 2360 K, respectively [5 and 6]. The operating
characteristics of a typical NTR engine (Rover/NERVA) are
given in Table I. The power, hydrogen, and wall temperature
distributions of a typical NTR engine are shown in Figure 4 [5].
Copyright © 2014 by ASME 3
Table I – Engine Design Characteristics for a Typical NTR (NERVA Engine Specs – from Koenig [1])
Characteristics
Units Value
Thrust kN 337
Specific Impulse secs 825
Thermal Power MW 1570
Turbopump Power MW 6.9
Turbopump Speed rpm 23920
Pump Discharge Pressure MPa 9.36
Engine Flow Rate kg/sec 41.9
Chamber Temperature K 2360
Chamber Pressure MPa 3.1
Expansion Ratio 100:1
Core Diameter m 0.57
Core Length m 0.89
Be Reflector OD m 0.95
Be Reflector Thickness mm 134
Pressure Vessel O.D. m 0.98
Pressure Vessel Length m 1.7
Pressure Vessel Thickness mm 25.4
Weight (w/o ext shield) kg 11,250
Total Operating time min 600
Number of Cycles 60
Reliability 0.995
Figure 4 - NERVA NTR engine Power and Temperature Distribution [5]
During the development of the NTR engines under the
Rover program, various fuel material problems were observed
that were not completely resolved. These material problems limit
the performance and the reliability of these engines. The results
of the NERVA engine tests indicated that these problems were
not due to the irradiation from fission process. Basically the
damage to the fuel elements was due to the high temperatures
attained at the fuel surface. It is now understood that many
interrelated and competing physical mechanisms do act in
concert to degrade the structural integrity of the fuel element and
accelerate the fuel mass loss. Among these processes are the (i)
melting of the fuel (formation of liquid), (ii) vaporization/
sublimation, (iii) creep of material cracks, (iv) corrosion,
and (v) structural degradation.
The challenge to design a high performance NTR engine
requires the understanding of these complex physical
phenomena and then develop core materials (fuel matrix and
coating) that can stand high temperatures (greater than 3000 deg
K) and high mass flow rates (greater than 50 kg/sec) of hydrogen
environment with minimum corrosion and avoid breakage from
vibration and thermally induced stresses under high pressures
(greater than 3 MPa) [7, 8, and 9].
Since the degradation of the fuel elements and the structural
failure of the nuclear reactor core is believed not to be caused by
irradiation from the fission process, the development and testing
of new fuel matrices, fuel coatings, compatibility of the fuel and
coating materials, and resistance of other reactor and engine
parts to high temperature gas flow can be studied in a non-
nuclear test environment provided that the hydrogen gas flow
has comparable (preferably higher) temperature and pressures to
those attained in a typical NTR engine.
To this end, NASA has invested in design and construction
of various NTR Environment simulators at various NASA
centers to understand the mechanisms that are involved in the
failure of fuel elements and also test new fuel materials for fuel
matrices that may ultimately improve the performance,
reliability, and durability of NTR engines. Such simulators
presently exist at various NASA centers [10, 11, and 12]. One of
such test facilities is located at the Idaho National Laboratory
[12]. The Hot Hydrogen Test Facility (HHTF) located in this
lab is suitable to test core materials in 2500 C hydrogen flowing
at 15000 liters per minute. This facility is intended to test non-
uranium containing materials and therefore is suitable to test
potential fuel cladding and coating materials. It can also be used
to understand the thermal-hydrodynamic behavior and stability
of the core, reflector, moderator, and the shielding materials.
The thermal and hydrodynamics characteristics of the
cooling channels of NTRs have been studied in the past. In 1992,
M.L. Hall et al. studied the thermohydraulics of the gaseous flow
in the nuclear core using KLAXON code [13]. They investigated
the hydrogen flow from the storage tanks through the reactor
core out of the NTR nozzle using an integral model. A shock-
capturing numerical methodology was used to model the gas
flow in the cooling channels. Their one-dimensional model was
able to predict the pressure distribution from the inlet of the
reactor core to the exit of the converging-diverging nozzle. They
also predicted the steady-state Mach number distribution for a
generic NTR.
E. Schmidt and et al. used KINETIC (which is a collection
of computer programs written for the purpose of analyzing start-
up transients in nuclear reactor) system code to analyze the
transients experienced in NTR engines [14]. This code consists
of a point reactor model and nodes to describe the fluid
dynamics and heat transfer mechanics in the cooling channel of
the NTR. With this code they were able to carry out a viable
transient analysis of a start-up and shutdown behavior of the
NTR engine.
J.E. Fittje of NASA Glenn Research Center used an
updated version of the Nuclear Engine System Simulation
(NESS) code to conduct integrated neutronic and thermal-fluid-
structural analysis of the NTR reactor core components [15].
This code uses the Monte Carlo N-Particle (MCNP) transport
code to determine the reactor inputs. Data obtained from the
MCNP is used to carry out the fuel elements heat transfer
analysis and propellant flow rate determination.
Copyright © 2014 by ASME 4
J.A. Webb and et al. used the MCNP code to determine
the volumetric heating rates within the nuclear core [16]. The
heating rates were then imported to STAR-CCM+ fluids code to
carry out the thermal hydraulic analysis of the cooling channels
of the nuclear core. Successful coupling of these two codes
enabled the authors to determine the spatial steady-state
temperature profile within the coolant channels. This
information was used to determine the optimum coolant channel
surface area to volume ratio to cool the rocket engine operating
at a high specific impulse.
The author of this paper has used a one-dimensional
mathematical model of a hybrid rocket motor to investigate the
instabilities due to coupling of acoustics and hydrodynamic
oscillations [17]. A modified version of this model was used to
simulate the thermal-hydrodynamic transients in the cooling
channels of the nuclear thermal propulsion engine [18]. Also, an
analytical study was carried out to explore the fluid-material
interactions in a nuclear thermal rocket [19].
Present study aims at simulating the thermal hydrodynamic
characteristics of a single cooling channel of the nuclear core of
a Nuclear Thermal Rocket engine where high mass flow rate
gaseous hydrogen is expanded to very high temperatures ( 2500
K) at high pressures around 3 MPa. For this purpose, the existing
one dimensional (in-house developed) computer program was
modified and adopted to the study of hot gaseous hydrogen flow
through the cooling channel. Also, a commercial CFD package
(Fluent) was employed to study the heating of the hydrogen gas
flow in similar settings. Simulations were carried out at
conditions similar to the ones experienced in nuclear thermal
reactors like NERVA by using both of these models to predict
the temperature in the hot gases and in the nuclear fuel cladding
material.
MATHEMATICAL MODEL
In the present study, thermal-hydrodynamic modeling of
very high temperature hydrogen gas flow at high pressures has
been sought. Also, we have included the thermal modeling of the
cooling channel wall (fuel cladding) and the heat transfer due to
conduction and radiation in our physics based comprehensive
model so that we can predict high temperatures the fuel cladding
is exposed to. No melting of the fuel coating (cladding) is
modeled in this phase of the project.
Two analytical models were developed for this study; one-
dimensional (1D) model which can model the steady and
unsteady flow of gaseous hydrogen (GH2) through the cooling
channel and a two dimensional (2D) two domain model which
can predict the GH2 (fluid domain) and the ZrC coating (solid
domain) temperatures in two dimensions at steady state.
One-Dimensional Mathematical Model
The GH2 flow in the cooling channel is modeled using a
one-dimensional multi-node computational domain. The
mathematical model for this computational domain is given
below.
The conservation equations for a one-dimensional,
unsteady, viscous, compressible, subsonic turbulent flow in a
channel with circular cross section can be written in terms of
primitive variables ρ, u, T, and P as follows:
Continuity:
0)(
u
xt
(1)
Momentum:
wx
Pu
xu
t
)(2 (2)
Energy:
A
Aq
x
Tk
xx
uPTuc
xTc
t
swvv
)( (3)
The equation of state
TRP (4)
is used for the closure of the one dimensional compressible
viscous flow model described by the conservation laws.
Wall shear stress per volume in Eq. 2 is given by
D
ww
4 (5)
where the wall shear stress in this equation is modeled by:
2
8
1ufw (6)
Darcy friction factor f in the above equation is determined from
Blasius correlation [20]
25.0Re
316.0f
(7)
for fully turbulent flows with high Reynolds numbers in conduits
with smooth walls.
The fluid thermodynamic and transport properties were
assumed to be function of temperature. For hydrogen, these
properties are given by NIST [21] for temperatures up to 1600
K. Linear extrapolation techniques were adopted to determine
the physical and transport properties of hydrogen at temperatures
higher than this value.
Numerical Solution Procedures 1D Model
For the compressible flow of gaseous hydrogen, a second-
order accurate (in time and space) numerical scheme (modified
MacCormack) is used to solve the conservation of mass,
Copyright © 2014 by ASME 5
momentum, and energy equations presented in the previous
section. This technique is well suited to solve unsteady
compressible flow equations for high velocities. It is a two-step
numerical scheme where the primitive variables of the problem
are first predicted using a forward in time scheme and then are
corrected using a backward in time scheme.
The numerical stability criterion for the MacCormack
scheme is given by
Re/21
CFLt
t
(12)
where is the safety factor and this factor is taken as 0.9.
The Courant-Frederics-Levy stability condition is given by :
ua
xt CFL
(13)
where “a” is the local speed of sound. “Re” is the minimum
mesh Reynolds number given by
Re whereReminRe
xuxx
(14)
The steady state solution of the above governing equations for
the 1D model was obtained using false transient technique. In
this technique, the gas in the channel is assumed to be stagnant,
initially, and then increased gradually until the steady state inlet
mass flow rate value is reached. Then the calculations are
continued until the primitive variables of the problem don’t
change with time.
Convergence and Mesh Independence Study for 1D Model A time convergence study was carried out for the 1D
numerical model which is second order accurate in time and
space. Once the grid size is set, the time computational time
convergence is determined by satisfying the convergence criteria
given in Eq. 12. For the present code validation study, a non-
dimensional computational time increment of 5 x 10-6 was
considered. It was found that the time increment less than this
value did not result in any significant changes in quantitative
results. (The quantitative comparison of the results indicated less
than 0.1 % difference.) In order to validate the accuracy and
convergence of the computer code, a grid independence study
was also conducted. The grid size chosen for the present study
has 41 nodes. To verify that the converged solutions were
independent of the grid chosen two more studies were carried
out with grid sizes with 31 nodes and 51 nodes. The comparison
of distribution of the mean fluid temperature along the length of
the channel indicated a maximum of 0.12 % difference between
the predictions of these three different mesh sizes. 2D Model A commercial CFD package (Fluent) was also used to
predict the temperature distribution inside the axisymmetric flow
of gaseous hydrogen in the coolant channels of the NTR. For this
purpose, two different computational models have been adopted: a
two-dimensional (2D) one- domain model where only the working
the fluid (gaseous hydrogen) flow is modeled and a 2D two-
domain model where both the ZrC cladding and the hydrogen are
separately modeled. Both of these computational models are based
on a two-dimensional (2D) mathematical model of the working
fluid because the flow in the coolant channel is assumed to be
axisymmetric; that is, no flow or property variations are assumed
to exist in the circumferential direction. Same is assumed for the
heat flux boundary condition. However, the computations are
carried out in three dimensions.
The mesh for the computational domain (fluid and the
solid) is generated using the Workbench mesh generator and has
around 250K cells (see Figure 5 and 6.) As boundary conditions,
the mass flow rate and temperature of the GH2 are specified at the
inlet, and the pressure is specified at the exit of the channel.
Conservation equations are solved using pressure-velocity
coupling. Second order upwind scheme is used in the
discretization of the conservation equations and the k-ε turbulence
model equations. (A summary of Fluent setting are given Table 2.)
Figure 5 - Cross-sectional view of the Three Dimensional Computational Domain for the 2D Two-Domain Model
Figure 6- Isometric View of the Three Dimensional Computational Domain for the 2D Two-Domain Model
Copyright © 2014 by ASME 6
Table 2- CFD Solver (Fluent) Setting
Description Settings
Problem Setup – Solver Pressure-Based
Turbulence Model k-ε
Viscous Standard
Viscous Heating on
Pressure-Velocity Coupling Coupled
Gradient Discretization Least Square Cell-Based
Pressure Discretization Standard
Density Discretization Second Order Upwind
Momentum Discretization Second Order Upwind
Turbulent k-ε Discretization Second Order Upwind
Energy Second Order Upwind
Residual: Criteria 1E-08
Mesh Independence Study for 2D CFD Model A mesh independence study was carried out using two
domain two-dimensional model, each with three different cell
sizes and radial direction layers. Table 3 compares the outlet
results between the two domain models with three different cell
sizes of 1x 10-3, 2x10-3 and 5x10-3 with five layers in radial
direction in the fluid domain and one layer in the solid domain.
Table 3- Mesh Independence (Cell Size) Study for 2D- Two-Domain
Model (nf = 5)
Axial
Cell
Size, Δx
[m]
No. of
Cells
Reynolds
Number,
Re
(x105)
Inlet
Pressure,
Pi
[MPa]
Outlet Centerline
Fluid
Temperature, Te
[K]
1x10-3 200,400 3.18 8.197 1926.3
8x10-4 217,000 3.18 8.198 1926.71
5x10-4 348,000 3.18 8.30 1931.81
The effect of number of layers in the radial direction and in the
solid was also studied. The result of these studies is given in
Tables 4 and 5.
Table 4 - Mesh Independence (Fluid Layer) Study for 2D-Two-
Domain Model (Δx = 1x10-3 m)
Table 5 - Mesh Independence (Solid Layer) Study for 2D Two-
Domain Model (Δx = 1x10-3 m, nf = 5)
Variable Property Study The effect of variable properties of gaseous hydrogen (transport
and thermodynamic) was also studied using the 2D two-domain
model. The values for constant physical properties at 300 K are
obtained from the default setting in Fluent and the values for
variable physical properties are calculated using the polynomial
relations as a function of temperature from the NIST [21]. The
result of this study is summarized in Table 6. The mean fluid
temperature is considerable higher for the variable property case
as shown in Figure 7.
Table 6 - Physical Parameter (cp, k, and µ) Study for 2D Model (Δx = 5x10-4 m, Mesh size = 348,000, nf = 5, turbulent flow)]
Parameter
cp , k,
and µ
Reynolds
Number,
Re
(x105)
Inlet
Pressure,
Pi
[MPa]
Outlet
Centerline Fluid
Temperature, Te
[K]
Outlet
Velocity,
ue
[m/s]
Constant at 300 K
3.18 7.909 1791.97 2708.76
Variable 3.18 8.30 1931.81 2921.21
Figure 7- Comparison of Mean Fluid Temperature Distribution Predictions Based
on Constant (at 300 K) and Variable Properties Boundary Conditions and Run Parameters
In the present study, thermal-hydrodynamic analysis of a
single cooling channel of the NTR core is considered. The
cooling channel is 1.2 meters long with an inner diameter of 2.54
mm. The thickness on the cooling channel wall (fuel cladding) is
0.125 mm. The fluid (gaseous hydrogen) comes in at a constant
temperature (300 K) and a constant mass flow rate (0.005
kg/sec) and exits at constant pressure, 3.1 MPa. The temperature
and the velocity of the fluid in the channel gradually increase as
it travels along the channel length due to the heat flux from the
NTR core. Considering the Mach number and Reynolds’s
number, the flow in the channel can be categorized as turbulent
and subsonic. No slip conditions are assumed on the wall of the
channel and the wall is assumed to be impermeable. Figure 8
Axial Length, x [m]
Flu
idT
em
pera
ture
,T
[K]
0 0.2 0.4 0.6 0.8 1 1.2
250
500
750
1000
1250
1500
1750
2000
2250
Variable
Constant
No. of
Layers,
nf
No. of
Cells
Reynolds
Number,
Re
(x105)
Inlet
Pressure,
Pi
[MPa]
Outlet
Centerline Fluid
Temperature, Te
[K]
5 212,577 3.18 8.608 2076.79
10 319,200 3.18 8.394 2074.79
13 386,400 3.18 8.372 2076.85
No. of
Layers
in solid,
ns
No. of
Cells
Reynolds
Number,
Re
(x105)
Inlet
Pressure,
Pi
[MPa]
Outlet
Centerline Fluid
Temperature, Te
[K]
5 340,800 3.18 8.503 2071.73
8 403,200 3.18 8.503 2071.74
10 459,600 3.18 8.503 2071.74
Copyright © 2014 by ASME 7
shows the wall heat flux distribution along the length of the
cooling channel of a NERVA type NTR nuclear core [5]. 1D and
2D CFD models are modified and adopted to study the flow of
hydrogen gas through the cooling channel of the NTR core
under the effect of similar non-uniform wall heat flux. The
predicted temperature distributions by these models are then
compared to the ones given in Figure 4 [5]. The geometrical and
operational parameters used in the case studies are summarized
in Table 7 and 8.
Figure 8 - Heat Flux Distribution along the Cooling
Channel of a NTR Core
Table 7- Geometrical Parameters for Case Studies Parameter Symbol Value
Length L [m] 1.2
Diameter D [m] 0.00254
Cladding Thickness t [m] 0.000125
Table 8- Operational Parameters for Case Studies Parameter Symbol Value
Inlet Mass Flow Rate m [kg/sec] 0.005
Inlet Temperature Ti [K] 300
Exit Pressure Pe [MPa] 3.1
Non-Uniform Heat Flux qmax [kW/m2] 22,000
RESULTS AND DISCUSSION
Results of NTR Channel Flow Simulations
1 Dimensional Model Study
The 1D model with 41 nodes and a time increment of 5 x
10-6 seconds was used to simulate (by false-transient techniques),
the steady state gaseous Hydrogen flow at the same geometrical
and operational parameters given in Tables 2 and 3. A heat flux
distribution similar to Figure 8 was imposed as the wall
boundary condition. The inlet pressure is predicted to be 7.079
MPa and decreases along the length of the channel to its set
value of 3.1 MPa at the exit. The pressure distribution along the
channel is shown in Figure 9.
Figure 9.- Pressure Distributions Along the Axial Length as Predicted by 1D and 2D
Models
Also, the density of the gaseous hydrogen decreases from its
initial value of 5.805 kg/m3 at the inlet to 0.345 kg/m3 at the
outlet of the channel. The distribution of density along the length
of the channel is shown in Figure 10.
Figure 10 - Density Distributions Along the Axial Length as Predicted by the 1D
and 2D Models
High heat flux also results in high axial velocity of the
fluid in the channel. The axial velocity increases from 169.9 m/s
Axial Length, x [m]
HeatF
lux,q
do
t[W
/m2
]
0 0.2 0.4 0.6 0.8 1 1.2
2.5E+06
5E+06
7.5E+06
1E+07
1.25E+07
1.5E+07
1.75E+07
2E+07
2.25E+07
Heat Flux
Axial Length, x [m]P
ressu
re,p
[MP
a]
0 0.2 0.4 0.6 0.8 1 1.20
1
2
3
4
5
6
7
8
9
10
2D one domain model
1D model
Axial Length, x [m]
Den
sity,R
o[k
g/m
3]
0 0.2 0.4 0.6 0.8 1 1.20
1
2
3
4
5
6
7
8
9
10
2D One Domain Model
1D Model
Copyright © 2014 by ASME 8
to 2846.3 m/s. The distribution of the axial velocity of the fluid
along the length of the channel is shown in Figure 11.
Figure 11- Axial Velocity Distributions Along the Axial Length as predicted by the
1D and 2D Models
The 1D model has an inlet fluid temperature of 300 K. With
the non-uniform heat flux applied at the surface of the channel,
the temperature of the fluid gradually increases along the length
of the channel as shown in Figure 12. The maximum mean
temperature of the fluid is 2253.54 K and the mean temperature
of the fluid at the outlet is 2216.63 K.
Figure 12- Mean Fluid Temperature Distributions Along the Axial Length
As Predicted by the 1D and 2D Models
In above figures, Figures 9 through 12, the pressure, and
mean values (averaged across the cross sectional area of the
cooling channel) of density, velocity, and temperature
distributions as predicted by the 2D one-domain model are also
presented. In summary, above results indicate that the 1D model
presented here fairs well when compared to a 2D one-domain
CFD model. Maximum deviations between the predictions of
each model occurs at the entry region of the flow as expected
since the 1D model assumes the flow to be fully developed from
the entrance to the exit of the channel.
Furthermore, the 1D model is computationally efficient.
(The CPU time for the steady state simulations for the 1D model
was found to be almost two orders of magnitudes smaller than
the one for the 2D one-domain CFD model.)
2D Two-Domain Model Study For the 2D 2 domain model (where gaseous hydrogen, the
cooling fluid, is assumed as one domain and the cooling channel
coating - that is, the metal ZrC cladding - is modeled as a
separate domain), the mesh depicted in Figures 6 and 7 was
used. This mesh had 5 layers in the radial direction in the fluid
domain, and had single layer in the solid, and resulted in
212,577 computation cells with Δx=1x10-3 m for the physical
domain describe in Table 7. Runs were carried out using the
Fluent settings given in Table 2 and operation parameters for
these runs are given in Table 8. The variable wall heat flux
imposed on the outside diameter of the cladding is is shown in
Figure 8. The results of this run are presented below.
The inlet pressure is calculated to be 8.762 MPa and it
decreases along the length of the pipe from its predicted value to
3.1 MPa at the outlet of the pipe as shown in Figure 13.
Figure 13- Pressure Distribution along the Axial Length as Predicted by the 2D
Two-Domain Model
Similarly, the density of the gaseous hydrogen also
decreases from its initial value of 7.179 kg/m3 at the inlet to
0.362 kg/m3 at the outlet of the channel as shown in Figure 14.
Axial Length, x [m]
Velo
city,u
[m/s
ec]
0 0.2 0.4 0.6 0.8 1 1.20
500
1000
1500
2000
2500
3000
2D One Domain Model
1D Model
Axial Length, x [m]
Tem
pera
ture
,T
[K]
0 0.2 0.4 0.6 0.8 1 1.2
250
500
750
1000
1250
1500
1750
2000
2250
2500
2750
2D One Domain Model
1D Model
Axial Length, x [m]
Pre
ssu
re,P
[Pa]
0 0.2 0.4 0.6 0.8 1 1.20
1E+06
2E+06
3E+06
4E+06
5E+06
6E+06
7E+06
8E+06
9E+06
Copyright © 2014 by ASME 9
Figure 14 - Density Distribution Along the Axial Length as Predicted by the 2D Two-
Domain Model
High heat flux also results in high axial velocity of the
fluid in the channel. The axial velocity increases from 139.64
m/s to 3239.25 m/s. The distribution of the axial velocity of the
fluid along the length of the channel is shown in Figure 15.
Figure 15 - Velocity Distribution Along the Axial Length as Predicted by the
2D Two-Domain Model
With the non-uniform heat flux applied at the surface of
the channel, the temperature of the fluid gradually increases
along the length of the pipe. The maximum mean fluid
temperature is 2313.49 K at 1.1 meters and the mean
temperature of the fluid at the outlet is 2213.71 K. The
distribution of the mean fluid temperature of the fluid along the
length of the channel is shown in Figure 16.
Figure 16 - Temperature Distribution Along the Axial Length as Predicted by the 2D
Two Domain Model
The 2-D Two-Domain model has different temperature
distribution between the inner wall and the outer wall of the
cooling channel due to the thickness. The wall temperatures are
much higher as compared to the temperature of the fluid flowing
inside. The outer wall and the inner wall have a maximum
temperature of 2653.06 K and 2525.01 K respectively. The outer
wall and the inner wall temperature converge at the outlet of the
cooling channel at around 2386 K. The comparison of wall
temperature distribution between the inner and the outer wall is
shown in Figure 17.
Figure 17- Inner and Outer wall Temperature Distributions Along the Axial Length
as Predicted by the 2D Two-Domain Model
The temperature profile at the outlet of the coolant
channel is shown in Figure 18. The outer-wall temperature of the
fuel cladding is 2424.90 K and the inner-wall temperature is
2386 K. The centerline temperature is 2075.96 K. Similarly, the
velocity profile at the outlet of the pipe is shown in Figure 19.
Since, the channel has no moving boundary and no slip
conditions the velocity at the interface is zero. The maximum
velocity of the fluid at the outlet is at the centerline of the
channel and has a magnitude of 3135.67 m/s.
Axial Length, x [m]
Den
sity,ro
h[k
g/m
3]
0 0.2 0.4 0.6 0.8 1 1.20
1
2
3
4
5
6
7
8
Axial Length, x [m]
Axia
lV
elo
city,u
[m/s
]
0 0.2 0.4 0.6 0.8 1 1.2
500
1000
1500
2000
2500
3000
3500
Axial Length, x [m]
Mean
Flu
idT
em
pera
ture
,T
m[K
]
0 0.2 0.4 0.6 0.8 1 1.2
250
500
750
1000
1250
1500
1750
2000
2250
2500
Axial Length, x [m]
Wall
Tem
pera
ture
,T
w[K
]
0 0.2 0.4 0.6 0.8 1 1.2
200
400
600
800
1000
1200
1400
1600
1800
2000
2200
2400
2600
2800
3000
Inner Wall Temperature
Outer Wall Temperature
Copyright © 2014 by ASME 10
Figure 18- Fluid Temperature Profile at the Outlet as Predicted by the
2D Two-Domain Model
Figure 19- Axial Velocity Profile at the Outlet as Predicted by the
2D Two-Domain Model
The temperature contours for the gaseous hydrogen gas
flowing in the cooling channel are shown in Figure 20.
Figure 20 - Fluid Temperature Contours inside the Cooling Channel Predicted by
2D Two-Domain Model
Comparison of the 1D Model and 2D Two Domain Model Results
A comparison of the predicted temperature, pressure,
velocity, and density distributions for steady state flow through
the channel for the same geometrical and operational parameters
is presented here. The temperature in all the models gradually
increases along the length of the channel. At 1.1 meters, the
experimental model has the highest mean fluid temperature of
2425 K. At the outlet, the experimental model has the highest
mean fluid temperature of 2400 K and the 2-D two-domain
(base) model and the one-dimensional model have quite
identical mean fluid temperature at the outlet with 2213.71 K
and 2216.63 K respectively. The quantitative comparison of
mean fluid temperature distribution based on the predictions
from the 1D and 2D Two-Domain, and the NERVA Study
results shown in Figure 4 [5] are illustrated quantitatively in
Table 9 and graphically in Figure 21.
Table 9 - Mean Fluid Temperature Distribution Along the Axial Length As Predicted by the Present Models and the Experimental Study [5]
Axial
Distance,
x
[m]
Temperature
Tm [K]
1D Model
Temperature
Tm [K]
2D Two-Domain
Model
Temperature
Tm [K]
NERVA
Figure 4 [5]
0 300 299.31 300
0.1 424.28 345.24 390
0.2 627.74 460.88 490
0.3 804.59 645.62 700
0.4 997.14 884.18 950
0.5 1267.99 1142.17 1250
0.6 1472.75 1396.17 1450
0.7 1670.51 1650.23 1750
0.8 1909.62 1891.22 2000
0.9 2059.94 2094.95 2175
1.0 2175.51 2242.47 2350
1.1 2253.41 2312.17 2425 1.2 2216.63 2213.71 2400
Figure 21- Comparison of the Mean Fluid Temperature Distributions as
Predicted by1D model, 2D Two Domain model, and the Experiment
Temperature, T [K]
Rad
ialL
en
gth
,r
[m]
2100 2200 2300 2400
-0.0015
-0.001
-0.0005
0
0.0005
0.001
0.0015 Outerwall
Innerwall
Axial Velocity
Rad
ialL
en
gth
,r
[m]
0 500 1000 1500 2000 2500 3000
-0.0015
-0.001
-0.0005
0
0.0005
0.001
0.0015 Outerwall
Innerwall
Axial Length, x [m]
Rad
ialL
eng
th,r
[m]
0 0.2 0.4 0.6 0.8 1 1.2-0.005
-0.004
-0.003
-0.002
-0.001
0
0.001
0.002
0.003
0.004
0.005
temperature: 446.356 740.655 1034.95 1329.25 1623.55 1917.85 2212.15 2506.45
Axial Length, x [m]
Mean
Flu
idT
em
pera
ture
,T
m[K
]
0 0.2 0.4 0.6 0.8 1 1.2
250
500
750
1000
1250
1500
1750
2000
2250
2500
2750
Mean Fluid Temperature - 3D Two-Domain
Mean Fluid Temperature - 1D
Mean Fluid Temperature - Experimental
Copyright © 2014 by ASME 11
Results of the Parametric Study
A parametric study was conducted using the 2D two-
domain CFD model to determine the effect of boundary
conditions (inlet mass flow rate and wall heat flux) on the
pressure, velocity, and temperature distributions along the
channel length.
Effects of Inlet Mass Flow Rate
The model adopted for this study had fluid domain with 5
layers in radial direction and a solid domain with one layer. A
mesh generated has an axial direction cell size of 1 x 10-3 meters
and has 212,577 cells in total. The inlet mass flow rate was
varied between 0.004 and 0.006 kg/sec to determine the highest
temperature in the fluid without choking the flow. The results of
this parametric study are presented in quantitative form in Table
10 and graphically in Figure 22. This data indicates a decrease in
the exit temperature of the gaseous hydrogen as the mass flow
rate is increased.
Table 10 – The Results of The Effect of Inlet Mass Flow Rate Study
Mass Flow
rate, mdot
[x10-3
kg/s]
Reynolds
Number,
Re (x105)
Inlet
Pressure,
Pi
[MPa]
Mean Outlet
Fluid
Temperature,
Te [K]
Outlet
Velocity,
ue
[m/s]
4.0 2.54 7.62 2597.69 3145.17
5.0 3.18 8.60 2076.79 3132.66
6.0 3.81 9.35 1724.06 3099.28
Figure 22 – Results of the Effect of Inlet Mass Flow Rate Study
Effects of Wall Heat Flux
In this study, a two-dimensional, two domain model
was used to study the effect of variable heat flux on temperature
and velocity distributions along the cooling channel. Predicted
pressure, axial velocity and fluid temperature distributions at
different wall heat flux values are presented quantitatively in
Table 11 and graphically in Figure 23. The reader should refer to
reference [22] for details of the parametric study.
Table 11- Results of the the Effect of Wall Heat Flux Study
Reynolds
Number,
Re
(x105)
Heat
Input, Q
[kJ/s-
m2]
Inlet
Pressure,
Pi
[MPa]
Outlet
Centerline
Fluid
Temperature,
Te [K]
Outlet
Velocity,
ue
[m/s]
3.18 20,00 8.25 1920 2900
3.18 22,000 8.60 2076 3132
3.18 30,000 9.67 2642 4126
Figure 23- Results of the Effect of Wall Heat Flux Study
CONCLUSIONS
A modified MacCormack scheme was successfully
employed to solve the governing differential equations of
turbulent flow inside a cooling channel of a NTR. The results of
the numerical study carried out using the proposed one-
dimensional mathematical model and the solution procedure fair
well when compared to results of the two dimensional CFD
model. The NERVA engine data (the published results) show a
higher mean temperature distribution along the length of the
cooling channel when compared to the predictions of the one-
dimensional and two dimensional models. In summary, it is
concluded that both models successfully predict the fluid
temperature distribution in the NTR cooling channel. The
parametric study carried out by using the two proposed models
indicate that increasing inlet mass flow rate may stop the
gaseous hydrogen attain the expected exit gas temperatures. The
parametric study using both mathematical models also indicate
that it is possible to increase the exit gaseous hydrogen
temperature up to 2750 K (without choking the flow in the
channel) by increasing the wall heat flux while keeping the the
mass flow rate around 0.005 kg/sec.
Axial Length, x [m]
Flu
idT
em
pera
ture
,T
[K]
0 0.2 0.4 0.6 0.8 1 1.2
250
500
750
1000
1250
1500
1750
2000
2250
2500
2750
3000
Mass Flow Rate = 0.005 kg/s
Mass Flow Rate = 0.004 kg/s
Mass Flow Rate = 0.006 kg/s
Axial Length, x [m]
Flu
idT
em
pera
ture
,T
[K]
0 0.2 0.4 0.6 0.8 1 1.2
250
500
750
1000
1250
1500
1750
2000
2250
2500
2750
3000
Heat Flux = 22000 kW/m2
Heat Flux = 20000 kW/m2
Heat Flux = 30000 kW/m2
Copyright © 2014 by ASME 12
ACKNOWLEDGEMENTS
This study is an extension of the work that was carried out
for NASA Stennis Space Center under Contract No.
NNS10AA92B, between June 15 and November 15, 2012. The
author would like to thank to David Coote of NASA SSC for his
continuous support of the Combustion and Cryogenics
Laboratories at the University of New Orleans. The author
would also like to thank graduate student Sajan Singh for
carrying out the 2D analysis using Fluent CFD simulations.
Graduate student Wesley Carleton’s assistance in 2D simulations
is also acknowledged.
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