COUPLED NEUTRONICS AND HEAT TRANSFER IN ...neams.rpi.edu/jiw2/papers/PHYSOR16 Composite.pdffollowing...

COUPLED NEUTRONICS AND HEAT TRANSFER IN ACCIDENT TOLERANT COMPOSITE MATRIX NUCLEAR FUELS

Christopher G. Morrison, Jaron Senecal, and Wei Ji*

Department of Mechanical, Aerospace, and Nuclear Engineering

Rensselaer Polytechnic Institute, Troy, New York, USA

[email protected]; [email protected]; [email protected]

ABSTRACT

Composite nuclear fuels containing a heterogeneous material distribution open the door to

engineering fuels that exhibit tailored characteristics. In the context of fault-tolerant safe-

ty-driven nuclear fuel design, composite fuels can provide functionality such as fission prod-

uct encapsulation where an inert matrix material surrounds fissile fuel kernels to provide a

barrier to fission product release.

This research proposes and explores another possible safety functionality for composite nu-

clear fuel forms. During a reactivity insertion accident it is theorized that heat can be isolated

in the fissile fuel kernels to enhance prompt Doppler reactivity feedback. The goal is to de-

velop a fuel that during steady state operations has a high thermal conductivity, but during

short reactivity transients the heat can be trapped within the fissile fuel kernel to establish

non-equilibrium prompt Doppler feedback that can counteract the reactivity insertion acci-

dent. The composite matrix fuel‟s thermal conductivity and geometry can be engineered to il-

licit different reactivity responses. Ideally a composite nuclear fuel may be able to be devel-

oped that can respond to prompt supercritical situations without fuel meltdown.

The purpose of this paper is to layout and discuss a time-dependent, multi-scale, coupled

neutronic and thermal heat transfer model that is required to analyze the non-equilibrium

Doppler feedback. Initial results with the model show the non-equilibrium Doppler feedback

following a prompt supercritical reactivity insertion for an infinite reactor. Later research will

focus on rigorous parameter analysis and realistic reactor designs.

Key Words: Multi-physics, time-dependent neutron transport, composite fuels

1. INTRODUCTION

Changes in temperature in nuclear fuel cause changes in the neutronic behavior of the fuel in a pro-

cess known as thermal feedback. Thermal feedback in nuclear systems, such as TRIGA research

reactors [1] and the TREAT reactor [2], play an important role in reactor operation. By engineering

the nuclear fuel in a certain way, designers can use thermal feedback to improve the response of the

nuclear system to accident scenarios.

In reactivity insertion accidents, a large amount of positive reactivity is inserted into the reactor

system causing the reactor to go prompt-supercritical. During a prompt-supercritical event, the

power of the reactor system is rapidly increasing typically doubling at a microsecond timescale.

* Corresponding author. Research group website: neams.rpi.edu.

1722PHYSOR 2016, Sun Valley, ID, May 1–5, 2016

mailto:[email protected]:[email protected]:[email protected]

This is happening at a time scale that is too fast for human action to control. If the nuclear fuel is

designed with prompt negative reactivity response to the thermal feedback then the result of the ac-

cident is not catastrophic and even benign. Otherwise, the resulting transient in a reactor will result

in a spike in power and likely cause core meltdown, such as what happened in the infamous Cher-

nobyl accident.

Particle dispersed fuels are a unique nuclear fuel form containing a fissile fuel kernel surrounded by

an inert matrix that can encapsulate fission products. These fuel forms offer reactor designers the

capability to maximize thermal feedback into a reactor system. Figure 1 below depicts the compo-

site fuel matrix fuel form. In the figure from left to right there is an increasing physical scale.

Figure 1: Graphic depiction of the physical composite matrix fuel form.

On the kernel scale the composite nature of the fuel can be easily emphasized with the yellow fissile

kernel encapsulated by the blue matrix. At the full core scale, the fuel appears to be almost homog-

enous, however the kernel scale will influence the full core attributes. The purpose of this research

is to explore how the kernel scale will impact the full core and how to use to engineer accident tol-

erant fuels by taking advantage of these kernel scale influences.

TRISO fuels are one notable example of composite matrix fuels and some research has been done

into heat transfer characteristics of the fuel form [3-4]. It is the hope of this research to go one step

further and analyze the reactivity feedback characteristics in TRISO fuels.

2. TEMPERATURE FEEDBACK MECHANISMS

Coupled neutronics and thermal hydraulic models have been heavily used in the design and control

of reactors such as the TRIGA reactor [1], the TREAT reactor[2], the EBRII reactor[5], and others.

These reactor designs have utilized thermal-neutronic feedback mechanisms such as:

Thermal Expansion: The thermal expansion of nuclear fuel and moderator can reduce reac-tivity. For example, in light water and other liquid moderated reactors, higher temperatures

decrease the density of the moderator reducing moderation.

Discrete Nuclear

Fuel Kernels

Continuous

Non-Fissile Matrix

Full Core Kernel Scale Fuel Block


Doppler Broadening: Higher temperatures cause Doppler broadening of fuel‟s resonance absorption cross sections and leads to the decrease in reactivity.

Neutron Temperature Shifting: As the temperature of nuclear fuels and moderators increase, the thermal equilibrium temperature of a neutron with the environment increases. This

causes an increase in the mean free path which increases leakage. The TREAT reactor in

Idaho uses neutron thermal temperature shifting to handle large reactivity excursions. TRI-

GA research are pulsed with a large amount of reactivity and use neutron temperature shift-

ing associated with the hydrogen in the U-ZrH fuel to increase leakage.

Each of the feedback effects described can be important under different circumstances. In the case

of particle dispersed fuels, the Doppler broadening is the key feedback mechanism that this research

will focus upon.

2.1. Doppler Feedback

Neutrons begin their life with a large amount of kinetic energy and slow down by colliding with

other nuclei. Some nuclei, especially heavy nuclei, have neutron energy resonances. If a neutron‟s

energy is within the energy resonance there is a large chance that the neutron will be absorbed in-

stead of scattered. If a material gets hot, the nuclei inside of it begin to vibrate. These vibrations

take the resonance (which was once within a very narrow energy range) and spread the resonance

out. When the resonance is spread out, neutrons over a wider energy range can be caught within the

resonance. For non-fissile materials this Doppler broadening has the effect of causing a decrease in

reactivity. For example, the presence of non-fissile U-238 atomically adjacent to the fissile U-235

causes more neutrons to be captured in the U-238. This reduces the reactivity almost instantaneous-

ly in the event of a rise in temperature and is known as a “prompt” feedback.

2.2. Composite Fuel Prompt Non-Equilibrium Feedback

Non-equilibrium conditions inside of a composite matrix nuclear fuel can be defined as a difference

in the temperature of the matrix and the fuel kernel. The goal of prompt non-equilibrium Doppler

feedback is to insulate the fuel kernel such that it cannot quickly release large amounts of power in

the event of a highly supercritical reaction. The insulation causes the fuel kernel to heat producing

prompt Doppler feedback. The insulation also must be design such that during steady state opera-

tions, the temperature peaking of the fuel kernel is still small. To use an analogy, the goal is to de-

sign the composite fuel such that it operates like a metallic fuel during steady state operation but

like a ceramic under transient situations. Table 1 below gives a notional response of a composite

matrix fuel to a prompt supercritical reactivity insertion.

Table 1: Notional transient response for a composite matrix fuel with a high thermal conductivity

matrix and a low thermal conductivity fuel kernel.

Stage 1 Stage 2 Stage 3

Timescale Microseconds Milliseconds Seconds


Thermal

Description

Majority of fission

heating in the fuel kernel

Kernel promptly heats, fis-

sion heat source

decreasing

Fission heat source

diminishes heat enters

the matrix

Temperature

Graphic

Neutronic

Description

Large positive reactivity

insertion

Larger negative Doppler

feedback

Reactivity is zero or

negative

Notional

Reactivity

Response

Graph

In Table 1 there are three stages of a transient. Beginning with a large reactivity insertion the fuel

begins to heat with several milliseconds. The heat is temporarily trapped within the fuel kernel for

several hundred milliseconds due to low thermal transport. During the time that the heat is trapped

in the particle there is a strong negative reactivity insertion reigning in the prompt supercritical re-

activity to a sub-prompt supercritical and to a negative reactivity where the power generated in the

kernel is decreasing. After a few seconds the heat trapped in the kernel transfers into the matrix but

now the power generated in the kernel has been decreased to manageable level. Materials and

geometry are independent variables that can be changed to modify the response of the composite

matrix to transient situations. Figure 2 below further explains the parameters that can be modified

and optimized to produce different responses.

Rea

ctiv

ity [

$]

-0.5

0 0.5

1

0.01 0.1 1 10 100 1000

Time [ms]


Figure 2: Composite matrix engineering design parameters.

In Figure 2, an additional layer has been defined as the fuel kernel particle coating and can be seen

as the black ring around the fissile fuel kernel. The different layers each have associated material

properties including thermal conductivity, density, and specific heat with a „k‟ for kernel, „m‟ for

matrix and „c‟ for coating.

2.3. Modeling Temperature Feedback

The ability to utilize the temperature imbalance between the fuel particles and the moderator matrix

can allow for a prompt, customized thermal feedback in the fuel particles. In order to analyze the

thermal feedback, a model must be able to couple time-dependent heat transfer and time-dependent

neutron transport over a period of time.

Figure 3:Multi-physics coupling flow chart

Coupling the thermal heat transfer and neutron transport is accomplished by solving the thermal

transfer physics separately from the neutron transport physics. Small time steps are taken and data is

fed back and forth into each physics solver for future time steps. Time-dependent multi-scale heat

transfer methods will be used in that they treat heat transfer in the fuel kernel scale separately from

the bulk macro scale heat transfer. Time dependent neutron transport methods are complex due to

the intricate physics involved. The next section will review and select a time dependent neutron

transport approach.

3. SELECTION OF TIME-DEPENDENT NEUTRON TRANSPORT METHODS

The neutron transport equation is the fundamental time-dependent neutron equation. The transport

can be solved in many different ways with various level of accuracy. Different methods are useful

under different circumstances. Table 2 lists several of the well-known methods.

Heat Transfer

Neutron Transport

Variable Symbol

Pitch 𝑝

Kernel Size 𝑅𝑘

Thermal Conductivity 𝑘𝑘, 𝑘𝑐, 𝑘𝑚

Thermal Inertia 𝜌𝑘 𝑐𝑝𝑘, 𝜌𝑐 𝑐𝑝𝑐, 𝜌𝑚 𝑐𝑝𝑚

Coating Thickness 𝑡𝑐

Fissile Fuel Kernel

Matrix Material

Kernel Coating

Fission Heating

Temperature


Table 2: Various neutron transport methods [6-8].

Name Description

Dynamic Monte

Carlo

Simulating individual neutron taking into account time progression.

Highly computational but represents the underlying physics with high

fidelity [9].

Flux Factorization

Methods

Splitting the neutron flux equation into a magnitude and shape function

and using the neutron kinetics equation. The improved quasi-static

method is a well-known flux factorization technique [10].

Point Reactor Model Easy to implement but ineffective at modeling feedback.

The flux factorization method was chosen. The main reason for the selection was the availability of

software packages with Doppler feedback capabilities (MCNP6) giving this research the ability to

begin collecting results quickly without needing to develop or request a new code base.

3.1. Flux Factorization Methods

The neutron flux 𝜙 can be described by a amplitude function 𝑝 and a flux shape function 𝜓 as shown in Equation (1).

𝜙(𝑟 𝑡) 𝑝(𝑡) 𝜓(𝑟 𝑡) (1)

Where the amplitude function is only a function of time and the shape function integral over space

and energy at a given time is unity as shown:

∫ ∫ 𝑡 (𝑡 ) ∫ 𝜓(𝑟 𝑡)

𝑝 𝑐𝑒

(2)

In transient analysis we are concerned with the time rate of change of the neutron flux. Taking the

time derivative of the flux function we obtain the following equation:

𝜙(𝑟 𝑡)

𝑡

𝑑𝑝(𝑡)

𝑑𝑡𝜓(𝑟 𝑡) + 𝑝(𝑡)

𝜓(𝑟 𝑡)

𝑡 (3)

The derivative of the magnitude function can now be expressed by a point (0-D) kinetics equation

where Λ is the neutron generation time, 𝜌 is the reactivity, β is the effective delayed neutron fraction, and 𝑆𝑑 is the delayed neutron source.

( )

( ) ( )

( ) ( ) +

( ) ( )

(4)

Equations (5,6) define the six delayed neutron groups that constitute the delayed neutron source.


The 𝜂 is the energy per fission, 𝜈 is the neutrons per fission, 𝐶𝑘(𝑡) is the number of delayed neu-tron sources in group k.

𝑆𝑑(𝑡) 𝜂

𝜈∑ 𝜆𝑘𝐶𝑘(𝑡)

6

𝑘=1

(5)

𝑑𝐶𝑘(𝑡)

𝑑𝑡 𝜆𝑘𝐶𝑘(𝑡) +

𝜈𝑑𝑘𝑝(𝑡)

𝜂 (6)

The neutron kinetics equation simplifies kinetics based methods by giving a relatively straight for-

ward differential equation to solve to determine the amplitude function. The shape function can be

solved for using diffusion or Monte Carlo methods.

3.1.1. Diffusion kinetics methods

Diffusion based kinetic methods use the diffusion equation to account for the time derivative of the

shape function. The diffusion equation contains the prompt fission source operator 𝐹𝑝, the migra-

tion and loss operator 𝑀, and the neutron velocity 𝑣.

𝑣( ) 𝜙(𝑟 𝑡)

𝑡 [ 𝐹𝑝(𝑟 𝑡) 𝑀(𝑟 𝑡) ]𝜙(𝑟 𝑡) + 𝑆𝑑( 𝜙(𝑟 𝑡𝑜 … 𝑡𝑡) ) (7)

Combining the diffusion and kinetics equations into the time rate of change yields the fundamental

equation for the improved quasi static kinetics based method as given in the following equation.

*𝐹𝑝(𝑟 𝑡) 𝑀(𝑟 𝑡) 𝛼(𝑡)

𝑣( )

𝑣( ) Δ𝑡 + 𝜓(𝑟 𝑡)

𝑆𝑑( 𝑟 𝑡 )

𝑝(𝑡)

𝜓(𝑟 𝑡 Δ𝑡)

𝑣( )Δ𝑡 (8)

Where:

𝛼(𝑡) 𝑑𝑝(𝑡)

𝑑𝑡

𝑝(𝑡) (9)

During a transient, the shape of the neutron flux distribution has a weak dependence upon time, and

the amplitude has a strong dependence upon time. The quasi-static flux factorization method can be

solved efficiently by calculating the kinetics equation on a short time scale and calculating the shape

function on a longer time scale. The improved quasi static method can be simplified, by ignoring

certain terms, into the quasi-static method, adiabatic method, pre-calculated adiabatic method, and

point reactor method. More information can be found in reference [7].

3.1.2. Monte Carlo flux factorization methods

Monte Carlo can be used in combination with the kinetics equation to give time-dependent neutron

transport. Monte Carlo can be used to obtain the neutron generation time, delayed neutron fraction,


reactivity, and the flux shape function.

Monte Carlo codes typically solve steady state neutron problems. The delayed sources may be in-

correct for some transient situations. For a higher fidelity simulation, delayed neutron sources could

be manually tallied and added to the simulation in a source problem. This level of complexity how-

ever may not be required to obtain reasonable results. Prompt supercritical transients are driven by

prompt neutrons and happen on a short time scale. The delayed precursors change on a larger time

scale. This would suggest that for short prompt supercritical transients starting from a steady state,

steady-state, delayed neutron sources should give a good approximation for the transient delayed

neutron sources.

For the purposes of this research a Monte Carlo flux factorization method was chosen because of

the simplicity of implementation, the availability of Doppler broadening for cross sections, and the

availability of codes such as MCNP6, KENO, Serpent, and OpenMC.

4. COUPLED MODEL DESCRIPTION

A flow chart of the coupled physics model used in the paper is shown in Figure 4. A key distinction

of this model is the premise that there is not a thermal equilibrium between the fuel particles and the

matrix material surrounding the fuel. It is not practical to model each fissile fuel kernel explicitly

because of the small size and large number of kernels. This model uses a multi-scale heat transfer

model wherein a small number of fuel particles are explicitly modeled. These microscopic cells are

then used to inform macro scale aggregate attributes. The multi-scale model differentiates the tem-

perature of the fissile fuel 𝑇𝑓 and the temperature of the matrix 𝑇𝑚. Figure 4 is a much more de-

tailed version of Figure 1.


Figure 4: Flow chart of the physics model used to model the time dependent response of the com-

posite matrix fuel to reactivity insertions.

4.1. Neutronics Model

The Monte Carlo kinetics based time dependent neutron transport method is depicted in the light

blue section of Figure 4. Each of the individual pieces of the neutronics model is described in the

following subsections.

4.1.1. K-eff calculation with fission distribution tally

Given the temperature and geometry of the problem the effective neutron multiplication factor, de-

layed neutron fraction, neutron generation time, and the shape distribution of the fission energy

deposition. For the delayed neutron sources steady state neutron sources were used (see the discus-

sion in Section 3.1.2).

MCNP6 is being used as the Monte Carlo program. In MCNP6 there is a Doppler broadening utility

that can properly modify the cross sections response to changes in temperature. The effective multi-

plication factor from MCNP6 𝑘𝑚𝑐 is directly given. The neutron generation time can be calculated

𝑞 𝑓(𝑟 𝑡) 𝑘𝑜

𝑇𝑓(𝑟 𝑡)

Kinet-

ics

Equa-

tions K-eff

Calculation

with Fis-

sion Tally

Distribu-

tion

Power

Distribution

𝑞 (𝑡)

𝜓𝑓(𝑟 𝑡)

𝑘𝑚𝑐(𝑡)

β(𝑡) Λ(𝑡)

𝑘𝑒𝑓𝑓(𝑡)

Particle

Scale

Conduc-

tion Heat

Transfer

Cell

Scale

Heat

Transfer

Take a

Time

Step

𝜕𝑇𝑓(𝑟 𝑡)

𝜕𝑡

𝑞 𝑚(𝑟 𝑡)

𝑇𝑚(𝑟 𝑡)

𝑇𝑚(𝑟 𝑡)

𝑇𝑓(𝑟 𝑡) 𝜕𝑇𝑝(𝓇 𝑡)

𝜕𝑡

𝜕𝑇𝑚(𝑟 𝑡)

𝜕𝑡

K-eff

Multiplier


by taking the prompt neutron lifetime in the MCNP6 output file and dividing by the 𝑘𝑒𝑓𝑓. In the

current phase of this research a constant 𝛽 of 0.00660 has been assumed based on the estimated beta of U-235 [7]. For the neutron kinetics equation, a term known as 𝛽𝑒𝑓𝑓 is used which accounts

for delayed neutrons not having the same probability of fission as prompt neutrons. The 𝛽𝑒𝑓𝑓 can

be as much as a ten or twenty percent different from 𝛽 in practice and is reactor dependent [11]. Research suggests that a good approximation for 𝛽𝑒𝑓𝑓 can be found from the fractional increase in

𝑘𝑒𝑓𝑓 in a Monte Carlo simulation with delayed neutrons verses without delayed neutrons [12].

4.1.2. K-eff multiplier

The effective multiplication factor from the Monte Carlo is modified before it is used in the kinetics

equation. The initial excess reactivity 𝜌𝑜 establishes the reactivity at the start of the simulation. This reactivity may be at odds with the reactivity that would be calculated by the 𝑘𝑚𝑐. When the simulation starts, a constant virtual multiplier Υ is established that is then used to modify the 𝑘𝑚𝑐 to account for the initial reactivity insertion.

Υ 𝑘𝑜

𝑘𝑚𝑐( 𝑡 ) (10)

𝑘𝑒𝑓𝑓(t) Υ 𝑘𝑚𝑐(𝑡) (11)

Using the constant virtual multiplier can be justified as a change in one of the parameter in the

six-factor formula. The virtual multiplier could be likened to a forced change in one of the six fac-

tors such as a leakage terms 𝑝𝑓𝑛 and 𝑝𝑡𝑛 or thermal utilization factor 𝑓.

𝑘𝑒𝑓𝑓 𝜖 𝑝 𝑓 𝜈 𝑃𝑓𝑛 𝑃𝑡𝑛 (12)

Using Υ as a virtual multiplier could have a real world representation as a control rod movement or loss of reflector liquid. For realistic reactor configurations Υ should be close to unity.

4.1.3. Kinetics equations

The kinetics equations block solves the six group kinetics equation (see Equations 4-6) over a given

time period. The kinetics equations can have large prompt jumps in power based on reactivity inser-

tion, so small time steps on the order of tens of nanoseconds are used. In contrast, the Monte Carlo

parameters fed into the kinetics equations are solved on a millisecond time scale because. The

Monte Carlo parameters change with thermal temperature swings which happen over a millisecond

time scale. This is beneficial from a computational standpoint as the kinetics equations are cheap to

integrate while the Monte Carlo parameters are expensive.

4.1.4. Power Distribution

The final step to determine the power density in the 𝑞 is to take the normalized fission deposition shape function 𝜓𝑓 and multiply by the kinetics amplitude function.


𝑞 𝜓𝑓(𝑟 𝑡) p(t)

(13)

Almost all of the heat is deposited in the fuel 𝑞 𝑓 though some gamma deposition and collision en-

ergy is deposited in the matrix 𝑞 𝑚.

4.2. Thermal Model

The thermal model is depicted in the orange section of Figure 4. The thermal model is broken into

two different scales to simplify the complexity of modeling many small fuel kernels. Figure 5 de-

picts the multi-scale regions for an example problem of a cylindrical fuel rodlet with a center cool-

ant channel.

Figure 5: Multi-Scale Example

The concentric macroscopic cells physically encapsulate many microcells, however a single micro-

cell is used to represent each macroscopic cell. Each of the individual pieces of the thermal model is

described in the following subsections.

4.2.1. Particle scale conduction heat transfer

Within each fuel kernel conduction is the dominant form of heat transfer. The conduction equation

is given in Equation (15).

(𝑇𝑝𝜌𝑝𝑐𝑝𝑝)

𝑡 �⃗⃗� ⋅ (𝑘𝑝 𝛻𝑇𝑝) + 𝑞𝑝

(14)

Since the fuel particles are small it is a reasonable assumption that the heat transfer is one dimen-

sional in the radial direction. This allows differentiating Equation (14) for radial spherical coordi-

nates and gives Equation (15).

2𝑅 𝑜

2𝑅 𝑖

𝐶𝑜𝑜𝑙𝑎𝑛𝑡 𝐶ℎ𝑎𝑛𝑛𝑒𝑙 𝑐𝑒𝑙𝑙𝑗=1 𝑐𝑒𝑙𝑙𝑗=2

Representative

Microcell for

Each Cell

𝑐𝑒𝑙𝑙𝑗=⋯

𝑚𝑖𝑐𝑟𝑜𝑐𝑒𝑙𝑙𝑖=7

𝑚𝑖𝑐𝑟𝑜𝑐𝑒𝑙𝑙𝑖=5


𝑇𝑝(𝓇 𝑡)

𝑡

* 𝑘𝑝(𝓇 𝑇𝑝)

𝑇𝑝( 𝑇𝑝(𝓇 𝑡)

𝑟*2

𝓇2 + 𝑘𝑝(𝓇 𝑇𝑝) (2𝓇 𝑇𝑝(𝓇 𝑡)

𝑟+ 𝑟2

2𝑇𝑝(𝓇 𝑡) 𝑟2

) + 𝓇2 𝑞 𝑝(𝓇 𝑡)+

𝓇2 ( 𝜌𝑝(𝓇 𝑇𝑝)𝑐𝑝𝑝(𝓇 𝑇𝑝) + 𝑇𝑝(𝑟 𝑡) ( 𝜌𝑝(𝓇 𝑇𝑝)

𝑇𝑝 𝑐𝑝𝑝(𝓇 𝑇𝑝) +

𝑐𝑝𝑝(𝓇 𝑇𝑝) 𝑇𝑝

𝜌𝑝(𝓇 𝑇𝑝)))

(15)

The cursive 𝓇 denotes the radial coordinate system with its origin in the center of the fuel kernel. A finite fourth order difference code was written to integrate Equation (16) and compute the temperature of a micro-

cell over a period of time. The boundary condition on the fuel particle is proportional to the difference be-

tween the temperature of the microcell at the boundary and the aggregate temperature of the matrix material

represented by the cell.

𝑇𝑝(𝑟 𝑅 𝑡)

𝑟

𝜇(𝑇𝑚 𝑇(𝑟 𝑅 𝑡))

𝜌𝑚𝑐𝑝𝑚

(16)

4.2.2. Cell scale heat transfer

The fundamental conduction equation is given in Equation (18) and comes from Vogl [13]. The

macroscopic cell scale has an additional heat term that represents the heat added to the matrix by the

fuel kernels.

𝑡(𝜌𝑚𝑐𝑝𝑚𝑇𝑚)

𝑆𝑝 𝑛𝑝 𝜇

𝜙𝑚(𝑇𝑝(𝑟 𝑅) 𝑇𝑚(𝑟)) + �⃗⃗� ⋅ 𝑘𝑚𝛻𝑇𝑚 + 𝑞 𝑚(𝑟 𝑡)

(17)

Where 𝑆𝑝 is the surface area of the fissile fuel kernel, 𝑛𝑝 is the number of particles per unit vol-

ume, 𝜙𝑚 is the matrix volume fraction, and 𝜇 is the contact conductance.

4.2.3. Take a time step

The fuel temperature time derivative is determined by the particle scale heat conduction transfer and

the matrix temperature time derivate by the cell scale heat transfer. The derivatives are integrated

giving a new temperature.

The kinetics equation is integrated on the order of nanoseconds and the results are fed into the heat

transfer model immediately. The heat transfer model is also integrated on a similar scale. The Monte

Carlo k-effective is recalculated on a millisecond time. The shape function (with the fission tally) is

calculated on a tens-of-millisecond time scale to save computational power.

4.3. Miscellaneous

The coupled model has a built in library of temperature dependent properties several different fuels

and matrix material which includes the materials data from previous research [14]. A C++ program

was written that implements the coupled methods described in this section.


5. RESULTS

An equivalent cell composite matrix reactor problem was created to test the coupled physics model

and begin to explore the nature of the prompt Doppler feedback. The problem consists of a 20 per-

cent enriched UO2 fuel kernel surrounded by a graphite matrix. The fissile kernel radius was 200

micrometers with a kernel pitch of 4 millimeters. Initial power was set to 200 MW/m3. A $1.45 re-

activity insertion was initiated at the start of the simulation. Figure 6 shows a graphic depiction of

the infinite composite matrix.

Figure 6: Equivalent Cell Matrix Problem

The outer boundary conditions for the equivalent cell are reflected for neutrons and set to a fixed

temperature derivative representing a power flux out of the cell equal to the initial power density.

The initial temperature conditions and the temperature derivative boundary condition was found by

iterating the heat transfer model for a steady power and fixed outer boundary temperature condition

of 800 degrees K. This process of determining initial condition by iterating to a stead state is show

in Figure 7. The kernel starts at a constant temperature of 800 K and iterates for approximately 300

milliseconds until the approximate equilibrium temperature is reached with a maximum temperature

of 1300 K.

Figure 7: Iteration to Steady State Initial Conditions.

Once the initial temperatures and temperature derivative boundary condition are known, the prob-

lem is ready to have a reactivity insertion. A $1.45 cent reactivity insertion is made which corre-

sponds to a k-effective of 1.01 as shown at time equals zero seconds in Figure 8.

�⃗⃗�

Kernel Matrix


Figure 8: Left: K-eff vs. Transient Time. Right: Power vs. Transient Time

The k-effective during the transient quickly drops to a well below prompt supercritical within 100

milliseconds of the reactivity insertion. Power seems to spike by approximately a factor of four but

drop down to 150 percent of starting power within one second. The prompt Doppler feedback seems

to be effective almost instantaneously, stopping the prompt supercritical excursion. A simulation

was also conducted to emphasize the effect of the multi-scale composite model against a homoge-

nized power model. Figure 9 demonstrates the reduction in the power excursion between the models

for a simulation with all other factors constant.

Figure 9: Comparison of Power Transient for Homogenized Power and Kernel Power

6. CONCLUSIONS

Within this document a method for coupled thermal heat transfer and neutronic transport is de-

scribed that has application in modeling non-equilibrium prompt Doppler feedback for composite

matrix nuclear fuels. This method was successfully utilized to analyze a reactivity transient for a

simple composite nuclear fuels equivalent cell. Future work ahead includes running many more test

cases to complete a parameter analysis to better understand the engineering design space for com-

posite matrix fuels. A larger fuel rod test case is planned. The methods described in paper this will

be used to attempt to design new reactor composite fuels with fewer or perhaps no feasible reactivi-

ty accident meltdown scenarios from supercritical reactivity insertion.


ACKNOWLEDGMENTS

Thank you to the Nuclear Energy University Program Integrated University Program (NEUP-IUP)

for funding this research.

REFERENCES

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[3] A. GARCIA-BERROCAL, “Temperature Transients in TRISO Type Fuel,” Annals of Nuclear Energy, 76, pp. 172-176 , 2015

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