A Numerical Study of High Temperature and High Velocity

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

mailto:[email protected]


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].


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.


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,


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


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


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


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


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


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


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



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




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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Documents

A Numerical Study of High Temperature and High Velocity