3254957 Senior Design

Small Mobile Reactor Analysis for Underdeveloped Countries and Remote Locations

April 11, 2007Team 1:

Brittany SmithCarlos Juarez

Peter McKinnisJamie Anderson

Derek MannJames Henkel

NE 472 – Nuclear System DesignDr. M. Grossbeck

Abstract

The concept of a small modular reactor represents the next frontier in reactor

evolution. Currently, nuclear reactors are constructed on a high output industrial scale,

typically about 1000 MWe. Many smaller and less industrial developed countries are not

able to handle such massive electricity generators nor do they have the technical

knowledge to safely operate such a massive reactor. Small reactors should offer a safe

tool to train developing countries in reactor operation.

The goal of this project was to design a long-life core (at least 10 years without

refueling) that produced about 100 MWe and most importantly could be transported by

truck. Of the previously mention three design criteria, the ability to be transported by

truck was considered the most important. The entire reactor system did not have to be

transported by truck, but the major components (core, heat exchanger, turbines) needed to

fit on a truck.

A liquid metal cooled reactor was designed to meet these goals. Initially, this

reactor type was chosen because of its high linear power density and fuel longevity.

However, a liquid metal cooled reactor has its own unique set of problems. For example,

most liquid metal fuels are chemically active in air; the neutron spectrum is much faster,

leading to much quicker transients; and the coolants are generally not solid at room

temperature. Regardless, at the end of the design sessions, the reactor had a 20 year core

life (10 years proved to be easier transported by truck), produced 100 MWe, and the

major components could be fitted onto a truck. A detail list of the final core parameters

are provided in Appendix D of the attached paper.

Table of Contents

1 Introduction ....................................................................................................................... 1 1.1 Reactor Types ............................................................................................................ 1

2 Material Selection ............................................................................................................. 3 2.1 Fuel Selection ............................................................................................................. 3

2.1.1 Oxide Fuels ......................................................................................................... 3 2.1.2 Uranium Metal Fuel ............................................................................................ 4 2.1.3 Other Fuels .......................................................................................................... 4 2.1.4 Fuel Summary ..................................................................................................... 5

2.2 Coolant Selection ....................................................................................................... 6 2.2.1 Liquid Metal Coolants ........................................................................................ 6

2.3 Cladding Selection ..................................................................................................... 9 2.4 Control Rod Selection .............................................................................................. 10

3 Neutron Transport ........................................................................................................... 11 3.1 Fuel Assembly selection .......................................................................................... 11 3.2 Developing a Core Model ........................................................................................ 13 3.3 Results ...................................................................................................................... 17 3.4 Reactivity Coefficients ............................................................................................. 20

4 Heat Transfer .................................................................................................................. 21 4.1 Coolant Temperature Profile .................................................................................... 22 4.2 Cladding Temperature Profile .................................................................................. 23 4.3 Fuel Temperature Profile ......................................................................................... 25 4.4 Results ...................................................................................................................... 26 4.5 Pumping Power ........................................................................................................ 28

5 Shielding ......................................................................................................................... 30 6 Secondary System ........................................................................................................... 33

6.1 Steam Generators ..................................................................................................... 34 6.1.1 Steam Generator Design ................................................................................... 34 6.1.2 Steam Generator Sizing .................................................................................... 35 6.1.3 Steam Generator Weight ................................................................................... 39 6.1.4 Steam Generator Weight ................................................................................... 40

7 Safety Considerations ..................................................................................................... 40 8 General Plant Design ...................................................................................................... 41 9 Conclusions and Further Research .................................................................................. 43 References.........................................................................................................................45

Appendix A Sample SCALE input for core neutronics .............................. 1 Appendix B Matlab files for Heat Transfer Correlations ........................... 1 Appendix C Sample Scale input for shield calculations ............................. 1 Appendix D Summary of Core Parameters: ............................................... 1

List of Figures

Figure 1 - Boron Neutron Cross Section versus neutron energy. [10]...............................10Figure 2 - Diagram of a square pitch and a triangular pitch..............................................12Figure 3 - Cross section view of Core (not to scale)..........................................................14Figure 4 - Core size and End of Life k-eff.........................................................................17Figure 5 - k-eff versus core life..........................................................................................19Figure 6 - Temperature profiles for a linear power of 2500 W/m and a coolant velocity of 2.34 m/s..............................................................................................................................28Figure 7 - Dose rate as a function of distance from cap shield..........................................32Figure 8 - General Core Layout.........................................................................................43

List of Tables

Table 1 - Summary of Reactor Characteristics [1]..............................................................2Table 2 - Summary of Fuel Properties.................................................................................5Table 3 - Summary of Coolant Parameters..........................................................................9Table 4 - Core size for triangular and square pitches........................................................13Table 5 - Linear Power Ratings and k-eff..........................................................................16Table 6 - Core Parameters..................................................................................................18Table 7 - Bi-yearly k-eff....................................................................................................18Table 8 - Void coefficient of Reactivity and Doppler Coefficient....................................20Table 9 – Thermophysical Properties of Coolant at Inlet [3]............................................23Table 10 - Cladding Properties..........................................................................................24Table 11 - Summary of Fuel parameters............................................................................26Table 12 - Temperature Limits for Core............................................................................26Table 13 - Maximum Component temperatures for an average core linear power of 2500 W/m ...................................................................................................................................27Table 14 - Pumping Power Calculations...........................................................................30Table 15 - Thermophysical Properties Sodium 500 °C.....................................................35Table 16 - Thermopysical Properties Water 100 °C..........................................................35

List of Equations

Equation 1 - Pumping Power Parameter..............................................................................6Equation 2 - Temperature Range Ratio...............................................................................7Equation 3 - Specific Activity..............................................................................................8Equation 4 - Core diameter for a square pitch...................................................................12Equation 5 - Length of Fuel Pin equation [2]....................................................................14Equation 6 - Total fuel length equation.............................................................................14Equation 7 - Number of fuel pin equation.........................................................................15Equation 8 - hoop stress equation......................................................................................15Equation 9 - Conversion Ratio...........................................................................................18Equation 10 - Void Coefficient of Reactivity....................................................................21Equation 11 - Doppler Coefficient.....................................................................................21Equation 12 -Coolant Temperature Profile as Function of Vertical Position....................22Equation 13 - Outer Cladding Temperature.......................................................................23Equation 14 – Westinghouse Nusselt Number for 1.1≤P/D≤1.4 and 10≤Pe≤5000 [3].....24Equation 15 - One Dimensional Steady State Conduction................................................25Equation 16 - Fuel Centerline Temperature.......................................................................25Equation 17 - Pressure drop across a single channel [1]...................................................29Equation 18 - Friction factor formula [1]..........................................................................29Equation 19 - Pumping power formula [1]........................................................................29Equation 20 - Thermal Resistance Equation......................................................................35Equation 21 - Overall Heat Transfer Coefficient Equation...............................................36Equation 22 - Reynolds Number formula..........................................................................36Equation 23 - Nusselt Number formula.............................................................................36Equation 24 - Average heat transfer coefficient between water and pipe.........................36Equation 25 - Governing Energy Balance Equations........................................................37Equation 26 - Mass flow rate of sodium............................................................................38Equation 27 - Mass flow rate of water...............................................................................38Equation 28 - C parameter calculation...............................................................................38Equation 29 - Effectiveness of steam generator calculation..............................................38Equation 30 - NTU calculation..........................................................................................39Equation 31 - Heat transfer area calculation......................................................................39Equation 32 - Weight of Stainless Steel pipes in Steam Generator...................................39Equation 33 - Weight of sodium in steam generator.........................................................40Equation 34 - Weight of Steam Generator housing...........................................................40Equation 35 - Natural Circulation equation [4].................................................................41

1 Introduction

The purpose of this project was to examine the preliminary design steps

associated with reactor design. The goal was to design a small reactor that could be

transported by truck to a remote location. Initial requirements were that the reactor

should produce about 100 MWe, fit on a standard tractor trailer, have a core that would

last at least 10 years, and not be a major proliferation risk. In order to fulfill the last

requirement, the enrichment of the uranium fuel was held below 20%.

The major focus was developing a core with the previously listed characteristics.

Therefore, only limited attention was given to the secondary system. Without careful

attention to the secondary system, it is impossible to determine the true efficiency of the

plant. The efficiency of the plant was assumed to be approximately the same as standard

industrial power reactors of the same type. Furthermore, given the wide scope and

relatively limited amount, an in-depth economical evaluation was not feasible. When

possible, general assumptions about standard materials were compared in an effort to

minimize cost.

1.1 Reactor Types

To meet the previously listed requirements, a high power density and long fuel

cycle were the considered to be the most important characteristics. Also, a low system

operating pressure was desired to limit the thickness of a pressure vessel.

1

Table 1 - Summary of Reactor Characteristics [1]

Pressurized Water Reactor

Boiling Water Reactor

Gas Cooled Fast Reactor

Liquid Metal Fast Breeder Reactor

Candu Reactor

Thermal Output (MW)

3411 3579 2530 2410 1612

Efficiency (%) 33.7 33.5 39.5 39.0 31.0Fuel Enrichment (%)

2.9 2.7 10-15 10-15Natural

Uranium

Average Core Power Density (kW/l)

104 56.0 297 380 12.4

Operating Pressure (bar)

155 72 86 14 89

Some of the important characteristics for various reactor designs are summarized

in Table 1. From this information the reactor type was narrowed to a Liquid Metal Fast

Breeder Reactor (LMFBR). A Boiling Water Reactor (BWR) and Candu reactor were

eliminated because of their low power densities. A Pressurized Water Reactor (PWR)

had many favorable characteristics, but the high operating pressure would require an

extremely thick and heavy pressure vessel. Both a Gas Cooled Fast Reactor (GCFR) and

a LMFBR both have very high efficiencies and power densities. However, a GCFR’s

operating pressure would again require a thicker pressure vessel, therefore limiting the

amount a fuel weight available for a given core size. The LMFBR also tend to have

higher conversion ratios, favoring core longevity. The purpose of this reactor is power

production, not plutonium breeding. When designing a LMFBR an effort was made to

limit the breeding ratio and therefore the amount of plutonium available in the core.

2

2 Material Selection

When designing the core, four main components were identified: the fuel, the

coolant, the cladding, and the control rods. The materials selection process for these

components initially focused on thermal properties. Then efforts were made to maximize

compatibility and favorable nuclear properties.

2.1 Fuel Selection

Nuclear fuels come in a variety of compositions. Three basic types were

considered: oxide fuels, metal fuels, and other fuels. Each fuel type has various

advantages and disadvantages for a given design. For instance, while oxide fuels have an

extremely high melting point, the amount of fissile atoms in 1 gram atom of UO2 is

significantly less then in a similar amount of pure uranium metal, meaning that a core

consisting of UO2 would have to be much larger than a core utilizing pure uranium metal.

The following sections outline the different fuel types examined and the parameters used

in selecting the fuel composition

2.1.1 Oxide Fuels

Oxide fuels are commonly used in two basic forms: mixed oxide (MOX) fuels or

UO2. MOX fuels consist of a blend of plutonium with natural or depleted uranium.

These fuels have a high melting point but a low thermal conductivity. The thermal

conductivity is affected by the burn-up and porosity of the pellet. The low thermal

conductivity can result in overheating in the fuel centerline. Fuel swelling is limited at

high temperatures because fission gas products migrate to the center of the pellets,

3

forming central voids. At temperatures above about 1600 degrees Celsius, most volatile

fission gases will migrate to the previously described central voids.

MOX fuels have been discussed as an alternative to low enriched fuel in Light

Water Reactors (LWRs). MOX fuels provide a way to effectively utilize left over

plutonium and therefore serve as a means of proliferation control. UO2 is the standard

fuel used in most commercial power reactors and has been researched much more in

depth than MOX fuel.

2.1.2 Uranium Metal Fuel

Uranium metal fuel has the advantage of a much higher thermal conductivity. A

higher thermal conductivity will decrease the fuel centerline temperatures and therefore

allow for higher power densities. Also, uranium metal fuel has a higher fissile atom

density than any other fuel, again allowing for a higher power density. However uranium

metal undergoes a structural phase change at 668 degrees Celsius. Because the structural

properties of a metal are unstable during a structural phase change, this temperature limits

the acceptable operating temperature of uranium metal rather than the melting

temperature.

2.1.3 Other Fuels

Other fuel forms are uranium carbide (UC) and uranium nitride (UN). These

fuels have the advantage of higher melting points than uranium metal fuel and higher

thermal conductivities than oxide fuels. However, these fuels are more prone to swelling

and have not been as extensively studied as oxide and metal fuels.

4

UC fuels were originally studied for LMFBR reactors in the mid 1960’s, but

failed to gain widespread acceptance due to the success with UO2 fuel. The major

downfall for UC fuel stems from the lack of research done on this fuel type and the lack

of facilities equipped to produce this fuel.

UN fuels have found a niche in NASA reactor designs. They have a melting point

on par with UO2 and a thermal conductivity nearly an order of magnitude better than

UO2. Also, they have a higher fissile mass density than UO2. However, the nitrogen in

UN fuels can cause an excess of C-14 buildup as N-14 undergoes a proton-neutron

reaction. This reaction can be limited by using N-15 instead of the much more common

N-14. Due to the high cost of N-15 enrichment, it is unlikely that this fuel could

economically competitive with UO2.

2.1.4 Fuel Summary

While each fuel type has clear advantages and disadvantages, UO2 has emerged as

the industry standard. Table 2 provides a summary of various fuel materials

Table 2 - Summary of Fuel Properties

MaterialThermal Conductivity

(W/m*C)Melting Point

(C)U 0.27 1135

UO2 0.036 2800UC 0.23 2390UN 0.20 2800 (decom.)

UO2 was chosen as the fuel because of its favorable thermal properties and its continued

successful use in commercial power generation.

5

2.2 Coolant Selection

The coolant choices were limited to four categories: gases, organic liquids, water,

and liquid metals. As described in section 1, a liquid metal cooled reactor seemed most

in line with our guidelines. Therefore all subsequent analysis focuses on liquid metal

coolants.

2.2.1 Liquid Metal Coolants

Liquid metals are generally chosen as the coolants for breeder reactors. Liquid

metals offer high heat transfer coefficients with low neutron moderating power.

However, liquid metals have their own unique problems as coolants. Most liquid metals

used in reactors are alkali metals which react explosively with air and water, meaning

that an inert cover gas is needed in all reactor operations that expose the coolant. Also

most liquid metals are not liquid at room temperatures, therefore during reactor start-up

the coolant must be heated significantly before it can actually be pumped.

When choosing a liquid metal coolant three different parameters are generally

examined. The first is the Pumping Power Parameter and is defined by the following

equation.

Equation 1 - Pumping Power Parameter

HeatSpecificC

Density

ityVis

ParameterPowerPumpingP

CP

p

p

====

⋅=

ρµ

ρµ

cos

8.22

2.0

This equation examines the scales the amount of pump work necessary to remove a

certain amount of heat. The Pumping Parameter carries units; therefore the units of each

6

analyzed coolant must be the same when evaluating this parameter. Lower Pumping

Power Parameters are desired.

The second parameter is the Temperature Range Ratio and is defined by the

following equation.

Equation 2 - Temperature Range Ratio

eTemperaturOutletactorT

PoMeltingCoolantT

PoBoilingCoolantT

RatioRangeeTemperaturTR

T

TTTR

w

m

b

w

mb

Re

int

int

)(

====

−=

The Temperature Range Ratio relates the temperature span over which the coolant

remains liquid to the outlet temperature of the reactor. Higher Temperature Range Ratios

are desired.

Because the coolant passes through a high neutron flux, it is susceptible to

induced activity. This induced activity also contributes to the need for an intermediate

loop. Even without a fuel cladding rupture the coolant becomes contaminated due to the

neutron flux and must be shielded as it passes through piping. The Specific Activity

quantifies the amount of induced activity within the coolant. The following equation is

used to calculate the Specific Activity.

7

Equation 3 - Specific Activity

sC

disK

isotopethioflifehalf

timenirradiatio

coolantfortimecycle

cycleoneduringfluxneutronintimeresidencecoolant

coolantofweightatomicA

levelenergythjtheandisotopethithefortioncrosscmicroscopi

levelenergythjinfluxneutron

NumbersAvogadroN

coolantinisotopethioffractionatomicF

ActivitySpecificS

A

NF

KS

i

i

o

r

w

ij

j

o

i

irij

ow

joi

⋅⋅=

−===

==

=

==

==

⋅−−⋅⋅

⋅⋅= ∑

10107.3

'

''sec

'

'

'

693.0exp1

1

θθττ

σφ

θθτσ

τφ

Because breeding requires higher-energy neutrons, the coolant moderating power

is also an important parameter. While some of the atomically lighter elements have very

favorable characteristics, their moderating power renders them ineffective as a coolant

because lower energy neutrons do not breed as well as higher energy neutrons. Table 3

summarizes the before mentioned parameters for several liquid metal coolants. The

properties for the coolants were taken at about 550 degrees Celsius. Lithium has the most

favorable Pumping Power Parameter and the lowest Induced Activity of all the coolants,

however because its moderating power is relatively high it is ineffective as a breeding

reactor coolant. When accounting for moderating power, sodium has one of the lowest

pumping powers and highest temperature range ratio.

8

Table 3 - Summary of Coolant Parameters

Coolant Pumping PowerTemperature

RatioModerating

PowerSpecific

Activity (C/s)Sodium .0097 .97 .00568 .20

Potassium .050 .86 .00124 .11NaK .0316 .98 .00205 .11

Lithium .0011 1.40 .01729 .03Lead .039 1.74 .00179 .09

Mercury .0261 .94 .00178 .28

Also, most of the research associated with liquid metal coolants has focused on

sodium. For the previously described reasons and the amount of available technical data,

sodium was selected as the coolant. Lead may have also been a favorable choice, but its

high melting point (~330 degrees Celsius) would have required a more complex startup

and shutdown system.

2.3 Cladding Selection

Having selected the coolant and fuel materials, the available cladding materials

were examined. When choosing a cladding material, the two main considerations are the

temperature limits and the reactivity of the cladding with the coolant. The standard

cladding material in PWRs is Zircaloy. Zircaloy offers a very low neutron cross section,

however at it undergoes a structural phase change at higher operating temperatures.

Austenitic stainless steels offer superior structural properties at high temperatures and are

also highly compatible with sodium. The drawbacks of these steels are their low thermal

conductivity and higher neutron absorption cross sections. However the high heat

transfer coefficient of sodium more than compensates for the low thermal conductivity

and the higher fuel enrichment minimizes the effect of the higher neutron cross sections.

Furthermore the radiation damage of these steels is very well documented and studied.

9

Stainless Steel Type 316 was chosen as cladding material because of its favorable

thermophysical properties and because it is the standard cladding used in LMFBRs.

2.4 Control Rod Selection

The final structural material examined was for the control rods. The material

selection process for the control rods was not as dependent on thermophysical properties

as the previously described structures. For the control rods the only thermodynamic

requirement was that the material remains solid at the maximum examined coolant

temperature. Much more important than the thermophysical properties were the nuclear

properties. The effectiveness of a neutron poison for fast neutrons is generally much

lower than for thermal neutrons. Figure 1 displays the Boron neutron absorption cross

section as a function of incident neutron energy.

Figure 1 - Boron Neutron Cross Section versus neutron energy. [10]

10

The previous figure clearly shows that control rod materials lose their effectiveness as

neutron speed increases. However, this does not eliminate boron completely as a control

rod material but instead required that more boron be present to provide a large enough

negative reactivity insertion to ensure k-eff drops below 1.0. Boron carbide was chosen

as the control material because the industry has considerable experience with control rods

and because boron carbide has a melting point of approximately 2400 degrees Celsius,

well above any temperature expected in the core.

3 Neutron Transport

Once the materials had been selected, a working core model using the neutronics

code SCALE was developed. When determining the working core model, the changing

parameter was the linear power density. Fuel enrichment, thermal output, fuel element

size, and pitch to diameter ratio were all held constant. The fuel enrichment and thermal

output were held constant at 15% and 273 MW respectively. The thermal output was

calculated assuming a reasonable efficiency of 33%. The fuel element parameters were

based on the Clinch River Breeder Reactor (CRBR) design. These were used as starting

points and then adjusted as necessary based on the fluid dynamic calculations.

3.1 Fuel Assembly selection

To further reduce core size, the effect of a square pitch versus a hexagonal pitch

was also analyzed. Hexagonal, also known as triangular, pitches fit more fuel in a given

core area. Using a triangular pitch minimized the volume of the core while maximizing

the amount of fuel. The sodium blanket surrounding the core provided sufficient

11

reflection to minimize the neutron leakage associated with a more compact core. Figure

2 provides a visual of a square and triangular pitch.

Figure 2 - Diagram of a square pitch and a triangular pitch

The length of the pitch remains the same in both the square pitch and the triangle

pitch; however, the size of the core is smaller when using the triangle pitch. The fuel pin

length was calculated using Equation 5 in section 3.2. The diameter of the core for a

square pitch was then calculated using Equation 4. The diameter calculation for a

triangular pitch is described in section 3.2.

Equation 4 - Core diameter for a square pitch

cmpinfuelofLengthL

cmpitchP

cmMWdensitypowerLinearP

MWoutputthermalactorR

cmcoreofDiameterD

LP

PRcmpitchsquareD

fp

dot

thtot

core

fpdot

Totcore

,

,

/,

,Re

,

4),(

2

==

===

⋅⋅⋅⋅

=π

12

All the other parameters, such as the fuel pin length, number of fuel pins and the linear

power, remained constant for the comparison of the triangle pitch and the square pitch.

Table 4 summarizes the results for a triangular versus square pitch. Using a triangular

pitch reduced the volume of the core by approximately 16% in comparison to a similar

core with a square pitch.

Table 4 - Core size for triangular and square pitchesTriangle Pitch Square Pitch

Linear Power(W/m)

Active Core Radius (cm) Active Core Radius (cm)

2500 97.99 105.304000 83.78 90.035500 75.34 80.967000 69.52 74.71

3.2 Developing a Core Model

As previously stated, the initial core dimensions were either assumed constant

from the CRBR design or calculated based on an assumed linear power density. The

initial core layout consisted of an active core surrounded by a blanket of sodium, then

surrounded by the core vessel, and finally a depth of concrete. The purpose of the

sodium blanket was to limit the neutron leakage and hopefully create a negative void

coefficient during a loss of coolant accident. Figure 3 shows a cross sectional view of

the modeled core, excluding the gap and concrete shell surrounding the core vessel.

13

Figure 3 - Cross section view of Core (not to scale)

The size of the core, including the sodium blanket, core vessel, void and concrete,

the number of fuel pins, and the fuel pin length each were each affected by changing the

linear power density. Higher power densities decreased the core size, but often the core

would not last the specified amount of time, nor could it be adequately cooled.

The equations used to determine each of the parameters are as follows:

Equation 5 - Length of Fuel Pin equation [2]

( ) 3

22 )924.0(60sin4)(

dot

Totfp P

PRcmLPinFuelofLength

⋅⋅⋅⋅⋅

=π

Where:TotR = Reactor Power (MW) = 273 MWth

P = Pitch (cm) = 0.76cm

dotP = Linear Power Density (MW/cm)

The Number of Fuel Pins could then be calculated using Equation 6 and Equation 7.

Equation 6 - Total fuel length equation

dot

Tot

P

RcmLengthFuelTotal =)(

14

Equation 7 - Number of fuel pin equation

fpL

engthTotalFuelLPinsFuelofNumber =

The height to diameter ratio of the core was assumed to be 0.924. This ratio minimizes

the leakage for any cylindrical geometry. The height of the core was taken as the length

of the fuel pins.

The thickness of the sodium blanket was kept constant at 30.48 cm and the

stainless steel that surrounds the sodium would be 1.27 cm. A thickness of 1.27 was used

to ensure the maximum tensile strength of the steel (515 MPa) would not be exceeded.

The following equation was used to calculate the hoop stress:

Equation 8 - hoop stress equation

thicknessvesselt

radiusouterr

MPapressureP

MPastresshoopt

rP

====

⋅=

,

,σ

σ

There is a void that is in between the stainless steel reactor vessel and the concrete

containment, which was arbitrarily assumed to be 15.24 cm thick. The concrete

containment was also assumed 30.48 cm thick. These values could be arbitrarily chosen

because their purpose was to minimize leakage and overcome some of the assumptions in

the SCALE sequence. The SCALE sequence used assumed a perfect reflector boundary

when the outermost geometric shell was reached. The calculated k-eff would be

overestimated if there was not enough material between the outer radius of the active core

and the outer radius of the outermost geometric shell. The thickness of concrete and void

used in this analysis was not representative of the actual thickness determined from

15

shielding analysis, but rather was used simply to negate the effect of a perfect reflection

boundary.

Table 5 provides an outline of the different core models in SCALE showing the

different linear power densities and their effect on the End-of-Life (EOL) k-eff. The

criterion for acceptable EOL k-eff was determined to be an EOL k-eff over 1.01. The

EOL k-eff represents the k-eff at the end of 20 years. Originally, the core life was

designed for 10 years of operation. However, a 100 MWe, 10 year core could easily be

transported on a truck. Therefore, the core lifetime was increased to 20 years.

Table 5 - Linear Power Ratings and k-eff

Linear Power (W/m)

Core Size- Radius

(cm)Number ofFuel Pins

k-effBOL

k-effEOL

2500 97.99 60304 1.1037 1.01984000 83.78 44083 1.1036 0.92385500 75.34 35651 1.0750 0.87787000 69.52 30356 1.0508 0.8372

These results indicate the maximum allowable linear power would be 2500 W/m.

This criterion ensures enough excess reactivity for the reactor to remain critical even with

the depleted fuel. The chosen linear power was checked against the heat transfer codes

described in section 4 to ensure that none of the structural materials exceeded their

maximum allowable temperature. Figure 4 shows how decreasing the linear power, and

therefore increasing the core size, affected the EOL k-eff.

16

End-of-Life k-eff verses Core Size

0.8

0.85

0.9

0.95

1

1.05

60.00 65.00 70.00 75.00 80.00 85.00 90.00 95.00 100.00 105.00 110.00

Core Size (cm)

EO

L k

-eff

Figure 4 - Core size and End of Life k-eff

The total thermal power remained constant. Decreasing the linear power

increases the total amount of fuel needed, therefore increasing the total size of the core.

The core size influences leakage. A larger core decreases leakage, which increases k-eff,

as represented in Figure 4.

3.3 Results

As stated earlier, the linear power is the limiting parameter for determining a core

that will operate for 20 years with out refueling and using an enrichment of fuel at 15%

U-235. The fuel material used in the core is Uranium Oxide (UO2). The linear power

determined for this fast reactor core design is 2500W/m. The complete SCALE input file

for the normal operating core can be found in Appendix A. Table 6 provides a list of

core parameters compiled for the design of the reactor core.

17

Table 6 - Core ParametersLinear Power Density 2500 (W/m)

Core Volume 5.46 m3

Core Height 1.81 mCore Radius 0.9799 m

Core Power 273 MWth

Height/diameter 0.924Assumed Efficiency 33%

Net Power 100 MWe

Number of Fuel Pins 60304Coolant Sodium

Conversion Ratio .506Weight (including

vessel) 36 tons

The conversion ratio was calculated using the following equation:

Equation 9 - Conversion Ratio

nConsumptioAtomFissileofRateAverage

oductionAtomFissleofRateAverageCR

Pr=

Once all the parameters were set for the core a detailed model was created in

SCALE to track k-eff during the core life. The SCALE sequence used calculated the k-

eff at different times during the operation of the reactor. Fuel burn-up and production is

accounted for in these calculations. Table 7 contains the k-eff for every other year of

operation of the reactor.

Table 7 - Bi-yearly k-effYear k-eff

0 1.10372 1.09934 1.09056 1.08178 1.0729

10 1.06412 1.055214 1.046316 1.037418 1.028620 1.0198

18

Figure 5 graphs the decrease in k-eff as a function of core life.

k-eff verses Age of Core

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

1.09

1.1

1.11

0 2 4 6 8 10 12 14 16 18 20

Time (years)

k-ef

f

Figure 5 - k-eff versus core life

19

As mention in section 2.4, boron carbide was selected as the control rod material.

The control rods were assumed to be cylindrical sleeves that split the active core area into

two separate regions. Each cylindrical sleeve is radially divided into approximately 4 cm

section that can be manipulated individually. Figure 3 shows the geometry for the core

with the control rod fully inserted. Various cases were ran to ensure that the control rod

would provide a large enough negative reactivity insertion to shut down the reactor no

matter what the core age was. Two 1.5 cm thick control rods kept k-eff below 0.95

through the core life span.

3.4 Reactivity Coefficients

There are two inherent reactivity controls for nuclear reactor. One is the void

coefficient of reactivity, and the second is the Doppler coefficient. These inherent

reactivity controls are important for any reactor because they serve as passive safety

systems. The void coefficient relates the change in reactivity of a system to a decrease in

the effective density of the coolant, either due to temperature increase or coolant leakage.

The Doppler coefficient is also known as the fuel temperature coefficient and measures

the change of reactivity due to an increase in fuel temperature. Both of these inherent

control systems of reactivity need to be a negative value so that any increase in power

creates a negative feedback in the core.

Table 8 provides the values for the void coefficient and the Doppler coefficient

for the previously described reactor design.

Table 8 - Void coefficient of Reactivity and Doppler Coefficient

Parameter ResultsVoid Coefficient of Reactivity -18 pcm/%Na Reduction

Doppler Coefficient -0.8 pcm/K

20

The void coefficient and the Doppler coefficient were calculated using the

following equations:

Equation 10 - Void Coefficient of Reactivity

510Re%

12 ⋅−

=ductionNa

kkreactivityoftCoefficienVoid

effeff

Where:

1effk = k-eff at the Beginning-of-Life (BOL) for the normal operating core

2effk = k-eff at the BOL for the core with a % reduction of Na

Equation 11 - Doppler Coefficient

512 10)()(⋅

∆−

=T

TkTktCoefficienDoppler effeff

Where,

)( 1Tkeff = k-eff at BOL for normal operating core

)( 2Tkeff = k-eff at BOL for core with increased fuel temperature

T∆ = increase in temperature of the fuel

Both the void coefficient and the Doppler coefficient are negative; therefore both

provide negative feedback in the event of a power increase. When modeling the void

coefficient, the sodium density was decreased by 10%. When modeling the Doppler

coefficient the fuel temperature was increased by 100K and the coolant density was held

constant.

4 Heat Transfer

The heat transfer calculations are closely related to the core neutronics calculations.

The thermodynamic calculations ensure that the heat generated by the assumed linear

power can be safely carried away quickly enough to ensure that the limiting temperatures

21

of the fuel, cladding, and coolant are not exceeded. An adequately fast coolant velocity

will provide cooling at just about any power. However, increasing the coolant velocity

increases the required pumping power, resulting in a decrease of overall plant efficiency.

The goal is to strike a balance between thermodynamic demands and neutronic demands.

The majority of the design work involved testing and retesting neutron calculations

against thermodynamic calculations until a favorable combination was achieved.

4.1 Coolant Temperature Profile

The coolant temperature is only a function of the average linear power, the specific

coolant thermodynamic properties, and velocity of the coolant. When analyzing the

coolant temperature profile, the thermodynamic properties of sodium were assumed

constant over the entire heated length. Equation 12 from Todreas [3] was used to

calculate the coolant temperature as a function of axial position.

Equation 12 -Coolant Temperature Profile as Function of Vertical Position

( )

+

′+=

ee

e

p

oincoolant L

L

L

zL

cm

qTzT

2sinsin

πππ

Where:

q’o

= average linear power density

Le

= extrapolated fuel length (taken as actual fuel length)

Tin

= coolant inlet temperature

22

L = fuel rod axial position

Because the fuel pin is long, the extrapolated length (Le) was taken as the actual fuel pin

length. The following thermophysical properties were used:

Table 9 – Thermophysical Properties of Coolant at Inlet [3]

Tin

Inlet Temperature400 ˚C

€

ρDensity 832.96

€

kg

m 3

€

µViscosity

2.11E-4

€

N ⋅ s

m 2

€

cp

Specific Heat1276.12

€

J

kg ⋅ K

€

kc

Thermal Conductivity66.57

€

W

m ⋅ K

These properties were evaluated at the average moderator temperature, approximately

530 ˚C.

4.2 Cladding Temperature Profile

The outer cladding temperature is dependent on the coolant temperature, the coolant

heat transfer coefficient, and the average fuel rod heat flux. Again, the cladding profile

was calculated based on varying the coolant velocity and holding all other parameters

constant over the entire length of the fuel element. Equation 13 from Todreas was used

to model the outer cladding temperature.

Equation 13 - Outer Cladding Temperature

€

Tco z()=Tmz()+ ′ q o2πRcoh

cosπzLe

Where:

Tco = cladding outer temperature

23

Tm(z) = axial coolant temperature

Rco = Cladding outer radius

h = heat transfer coefficient of coolant

The other parameters were described in Equation 12. The heat transfer coefficient was

again calculated from the following correlation:

Equation 14 – Westinghouse Nusselt Number for 1.1≤P/D≤1.4 and 10≤Pe≤5000 [3]

€

Nu=4.0+0.33PD( )3.8Pe100( )0.86+0.16PD( )5.0

Where the Pe is the Peclet number, equal to the product of the Reynolds number and

Prandtl number, and P/D is the pitch to diameter ratio. For the previously described core,

the P/D ratio and Pe number fell well within the acceptable range of this correlation.

Table 10 summarizes the various parameters used to calculate the outer cladding

temperature.

Table 10 - Cladding Properties

Fuel Pin Length

Le

1.81 m

Pitch to Diameter Ratio

(P/D)

1.3

Cladding Inner Diameter

Ric

0.254 cm

Cladding Outer Diameter

Roc

0.2921 cm

Cladding Thermal

Conductivity

kc,clad

21.5

€

W

m⋅K

The inner cladding temperature was calculated assuming radial heat conduction.

24

Equation 15 - One Dimensional Steady State Conduction

dx

dTkq −=′′

Where:

q” = heat flux

k = thermal conductivity

4.3 Fuel Temperature Profile

The final temperature profile considered was that of the fuel centerline. The fuel

centerline temperature was calculated with the same assumptions described in the

previous two sections. Equation 16 [3] models the fuel centerline temperature.

Equation 16 - Fuel Centerline Temperature

( )

efggci

co

ccoo

eep

eoinCL

L

z

khRR

R

khRq

L

L

L

z

cm

LqTzT

⋅

⋅

+⋅⋅

+⋅

+⋅⋅

⋅′+

⋅+⋅⋅⋅

′+=

πππππ

πππ

cos4

1

2

1ln

2

1

2

1

2sin

2sin

Where:

Rg = radius of the gap

hg = heat transfer coefficient of the gap

Rco = cladding outer radius

Rci = cladding inner radius

kc = thermal conductivity of the cladding

kf = thermal conductivity of the fuel

Tcl = fuel centerline temperature

25

The other parameters have been previously described early in the section. Table 11

summarizes the input parameters used for the fuel centerline temperature profile.

Table 11 - Summary of Fuel parameters

Fuel Outer Diameter

Foc

0.254 in

Fuel Thermal Conductivity

kc,fuel2.163

€

W

m⋅K

Because there was gap modeled during the code implementation, the outer diameter of

the fuel pellet was taken as the inner diameter of the cladding. This assumption proved to

be conservative because the cladding temperature, not the fuel centerline temperature,

was the limiting parameter.

4.4 Results

With the MATLAB code provided in Appendix B, the temperature profiles for the

coolant, cladding and fuel centerline were determined. Table 12 summarizes the limiting

temperatures for each component. The coolant temperature is limited by its boiling point,

while the cladding and centerline temperatures are limited by structural phase changes

and strength associated with their respective materials. The temperature rise was

arbitrarily limited to lessen the thermal stress associated with a high temperature gradient.

Table 12 - Temperature Limits for Core

Temperature Rise

260 ˚C

Cladding 660 ˚C

26

Fuel Centerline 2200 ˚CCoolant 881 ˚C

The neutronics calculations described in section 3 indicate that a linear power of

2500 W/m will result in a core with a 20 year life span. By using the previously

mentioned linear power and varying the coolant flow rate, the maximum temperature of

each element and the temperature rise across the core can be calculated. The following

table outlines the maximum temperatures of each component as a function of coolant

velocity. Figure 6 shows a graph of the temperature profile for each component for a

coolant velocity of 2.34 m/s and a linear power of 2500 W/m. The outer and inner

cladding temperatures are nearly indistinguishable due to the small cladding thickness

(0.0381 cm). Higher power densities could still be adequately cooled, but the core

lifetime associated with higher power densities did not adequately meet the goals

previously outlined.

Table 13 - Maximum Component temperatures for an average core linear power of 2500 W/m

Coolant Velocity (m/s) 1.0 1.5 2.0 2.34 2.5

∆T(˚C) 468.4 312.2 234.2 200.2 187.2Tmax, coolant(˚C) 868.4 712.2 634.2 600.2 587.3Tmax, cladding(˚C) 868.4 712.3 634.2 600.2 587.4

Tmax, fuel centerline(˚C) 937.6 804.2 742.5 717.0 707.7Heat Transfer Coefficient

€

W

m2K

70421.9 74510.1 78408.1 80979.2 82170.6

27

Figure 6 - Temperature profiles for a linear power of 2500 W/m and a coolant velocity of 2.34 m/s

4.5 Pumping Power

Because the coolant medium is a liquid metal, an electromagnetic pump can be

used rather than standard centrifugal pumps. Electromagnetic pumps operate on the

principle that a force is exerted on a current-carrying conductor in a magnetic field. The

high electrical conductivity of liquid metals allows a pumping force to be developed

within the metals when they are confined in a duct or channel and subjected to a magnetic

field and to an electric current. These pumps were designed principally for use in liquid-

metal-cooled reactor plants where liquid lithium, sodium, potassium, or sodium-

potassium alloys are pumped. The absence of moving parts within the pumped liquid

eliminates the need for seals and bearings that are found in conventional mechanical

pumps, thus minimizing leaks, maintenance, and repairs, and improving reliability.

28

Electromagnetic pumps with a capacity of up to several thousand gallons per minute have

operated without maintenance for decades.

The pumping power required to cool the core is related to the pressure drop across

a single channel. The pressure drop across a channel can be calculated by the following

equation:

Equation 17 - Pressure drop across a single channel [1]

)5.1(5.0 2_

HD

LfvP +=∆ ρ

Where,

_

ρ = average coolant density

v = coolant velocity through the core

f = friction factor for a bare rod bundle

L = length of the single channel

DH = hydraulic diameter of the channel

The value 1.5 in the previous equation corresponds to the maximum form losses for the

entrance and exit of the core. The friction factor for a bare rod bundle is given by the

equation:

Equation 18 - Friction factor formula [1]

2168.0Re

243.0=f

With the pressure drop across the core evaluated, the required pumping power is given by

the following equation:

Equation 19 - Pumping power formula [1]

corexspump PAvW _∆=

This equation is the theoretical pump work for a given velocity. Given the uncertainties

in this analysis and the lack of data on electromagnetic pumps, the theoretical pump work

29

was increased by 20% to act as a margin of safety. Table 14 shows the pumping power

variables and calculations.

Table 14 - Pumping Power CalculationsAverage Coolant Density (kg/m^3) 832

Flow Velocity (m/s) 2.5Fiction Factor 0.023

Pressure Drop (kPa) 21.840Calculated Pump Work (MW) 0.171

120% Calculated Pump Work (MW) 0.206Ratio of Pump Work to Electrical Output 0.21%

Even with a high factor of safety, the required pump work is less than half a percent of

the total electrical output.

5 Shielding

The reactor will be placed in the ground in order to minimize shield requirements.

The hole will be concrete lined for mechanical purposes, but the surrounding soil will be

the primary radial shielding. Because the reactor is assumed to be in a remote location

the dose to the surrounding ground and the activation of the soil can be addressed at a

later time, but the possibility of ground water contamination must be addressed for each

specific reactor site.

The only shielding requirements come from the top of the reactor. A concrete lid

with a diameter of 185 cm will be cast on site and placed on top of the hole. The lid is

bigger than the hole housing the core to ensure that there will be no leakage under any

edges. The thickness of the lid was adjusted as necessary to create a suitably low dose to

a person standing directly on the center of the lid. The required thickness was calculated

using sas1x code in SCALE. A sample input deck is provided in Appendix C. This code

conservatively treats the reactor as a sphere. The code outputs the dose normalized to the

30

number of fission neutrons, therefore to get the actual dose the output must first be

multiplied by the number of fission neutrons. The sphere radius was adjusted to conserve

the volume of the actual core.

The cap shield, with a thickness of 114 cm resulted in a dose rate of 4 rems/year directly

on top of the cap shield. This is well below the yearly occupational dose limit of 5

rems/year. The thickness of the cap shield can be adjusted to further suppress the dose

rate. The thickness of the cap shield could be reduced if a no-walk zone was constructed

around the reactor. This no-walk zone could be as simple as a chain link fence deterring

people from approaching closer. However, from a safety stand point, using the full

concrete cap shield provides the surest way to reduce does. But if economics and

licensing allows it, constructing a no-walk zone would be a cheaper way of reducing the

dose workers are expected to receive. These doses reflect the dose from both gammas

and neutrons. The following figure shows expected dose rate as a function of distance

from the cap shield:

31

Dose Rate as a function of Distance

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 50 100 150 200 250 300 350 400 450

Distance from Cap Shield (cm)

Do

se R

ate

(rem

/yea

r)

Figure 7 - Dose rate as a function of distance from cap shield.

As shown by the graph, the dose rate exponentially as a person moves further away.

Therefore limiting the amount of time spent near the reactor or creating a no-walk zone

around the reactor could be substituted instead of increasing the amount of concrete. The

other shield concern results from the induced activity of the coolant. However, Table 3

has the specific activity of sodium, and this activity is much lower than radiation from

fission (e.g. prompt gammas). Since the coolant activity contributes little to the core

radioactivity, no additional shielding is required. Also, because of the negligible

radioactivity of the coolant, an intermediate loop was deemed unnecessary. However,

because the steam generator will be located above ground, there will be additional

shielding necessary to reduce the dose associated with being in close proximity to the

steam generator.

32

6 Secondary System

The overall efficiency and effectiveness of the reactor is directly proportional to

the secondary system’s ability to convert the thermal output of the plant into electrical

output. As previously stated, the main focus of this project was to design a core, not a

secondary system. Therefore, little in depth analysis was done to optimize the secondary

system.

There were two options for a secondary system: a gas cycle, and a steam cycle.

Gas cycles offer many advantages, the greatest being that helium gas is chemically inert

while water reacts vigorously with sodium. Also, gas cycles are simpler to model and

evaluate because there is no phase change across the heat exchanger. However, while gas

cycles offer the promise of efficiencies, actually engineering such a system is very

difficult due to the problems associated with trying to pump gases at a high mass flow

rate. Also, there is little research information available on the performance of closed

cycle gas turbines. For these reasons, a steam secondary system was chosen for this

reactor. Steam cycles have their own inherent difficulties, the most notable being that

water and sodium undergo a highly exothermic reaction when brought into contact.

Nevertheless, previous liquid metal plants have successfully used steam secondary

systems, and the performance of steam generators and turbines is well understood. The

main concern with a steam secondary system is the weight of the steam generator. The

following sections address this concern.

33

6.1 Steam Generators

LMFBRs generally employ a three loop system. In the three loop system there is

an intermediate loop between the primary loop and the steam loop. The purpose of the

intermediate loop is to isolate the radiation associated with the induced activity of the

coolant on the primary side. However, because this is a small reactor with a relatively

small amount of coolant, the dose from the induced activity of the coolant does not

provide a significant radiation risk. Also, because the reactor will be in a remote

location, there should be very limited access to the radiation field of the reactor.

Therefore the intermediate loop was deemed unnecessary and removed to reduce capital

cost and weight.

There are three main types of steam generators: once-through, u-tube, and helical.

Once-through steam generators are generally cheaper and lighter, but are not very space

efficient. U-tube and helical steam generators are more space efficient and have better

heat transfer performance, but are much more complex. To minimize maintenance and

cost, a once-through steam generator was chosen.

6.1.1 Steam Generator Design

One of the most important design constraints is weight. Therefore it is necessary

to estimate the size of the steam generator to characterize its weight. The overall

construction of the steam generator will be a counter current flow in which sodium will

flow down through small circular pipes, and water will flow up across the pipes.

The core is designed such that the sodium coolant will enter the core at 400 °C and exit at

600 °C. Therefore the steam generator must cool the sodium by 200 °C requiring a total

power of 290 MWt. The steam generator will be designed such that 10% of the water

34

will vaporize per pass. Because heat transfer to vapor is much smaller than that to the

water the exit quality is kept low to increase overall heat transfer characteristics and

decrease the steam generators size. The excess water will be recirculated through the

steam generator while the steam will pass to the turbine.

6.1.2 Steam Generator Sizing

Many of the thermo physical properties of water and sodium are used in the

following calculations. The properties are outlined in the following tables.

Table 15 - Thermophysical Properties Sodium 500 °CParameter Symbol Units ValueDensity ρ Kg/m^3 897Specific Heat Cp kJ/kgC 1.334Prandtl Pr dimensionless 6.8957E-06Thermal Conductivity K W/mC 80.09Kinimatic Viscosity ϖ m^2/s 2.945E-07Dynamic Viscosity µ Kg/ms 0.000414

Table 16 - Thermopysical Properties Water 100 °CParameter Symbol Units ValueDensity ρ Kg/m^3 985.1Specific Heat Cp kJ/KgC 4.215Prandtl Pr dimensionless 1.76Thermal Conductivity k W/mC 0.6775Dynamic Viscosity µ Kg/ms 0.2822

The overall heat transfer can be calculated using a thermal resistance model. In

this model one calculates the resistance to heat transfer as thermal energy travels from the

hot fluid to the cold fluid. The total heat transfer coefficient is then:

Equation 20 - Thermal Resistance Equation

2211

11

RRRRRRUA

fwf ++++=

Σ=

Where:

U = total heat transfer coefficient

35

A = total heat transfer area

R1 = thermal resistance from the sodium to the pipe

R2 = thermal resistance from the pipe to the water

Rw = thermal resistance across the pipe

Rf1 = thermal resistance due to fouling on the inside of the pipe

Rf2 = thermal resistance due to fouling on the outside of the pipe

Substituting various expressions for R (and dividing through by the total heat transfer

area on can derive the following expression for the total heat transfer coefficient in a

pipe:

Equation 21 - Overall Heat Transfer Coefficient Equation

Cm

kW

hd

dF

d

d

k

dddF

hU

od

id

od

id

w

idodid

°=

++++=

−

2

1

221

1

2562

)/ln(1

where h1 and h2 are the heat transfer coefficients from the fluid to the pipe, F1 and F2 are

the fouling factors, did and dod are the inner and outer diameters of the pipe, and kw is the

thermal conductivity of the pipe.

The heat transfer coefficient from the water to the pipe (h1) may be calculated

from the following equations as suggested by Schmidt [6]: The definition of parameters

is given in Table 16.

Equation 22 - Reynolds Number formula

==µpdV

Re 4990

Equation 23 - Nusselt Number formula

+−=

3

2

4.087.0 1Pr)280(Re012.L

dNu h =10.2

Equation 24 - Average heat transfer coefficient between water and pipe

Cm

kW

d

kNuh

°=⋅=

21 230

36

Where:

hd = hydraulic diameter of the flow channel

L = length of the flow channel

Since heat transfer to liquid metals is excellent, the heat transfer coefficient for sodium

brings a negligible thermal resistance. i.e.: 1/h2 ≈ 0.

The fouling is a major source of error in this calculation. Schmidt [6]

recommends a value of 0.0002 for F1 (flow of hot feed water fouling factor, m2 oC/W).

However the way the fouling factor builds over the lifetime of the plant is unknown.

Further work is planned to achieve a better estimate of the total buildup over 20 years of

operation. Additionally, there are no recommendations for F2. F2 was assumed to be the

same value as F1 in the following calculations.

The thermal resistance may then be used to calculate the total heat transfer area

necessary to achieve the desired power using the number of transfer units (NTU) model

[6]. This method uses empirical relationships to calculate the effectiveness of a steam

generator. The effectiveness (ε ) is a factor by which one can multiply the maximum

efficiency of the steam generator by to realize the actual efficiency. Efficiency is a

function of the number of transfer units, the heat capacities of the primary and secondary

streams, and the geometry of the steam generator.

Equation 25 - Governing Energy Balance Equations

)(

)(

min

minmax

cihic

cihi

TTCQ

TTCQ

−=

−=

ε

Where:

maxQ = maximum rate of heat transfer of the system

minC = minimum specific heat (between the two fluid mediums)

37

hiT = maximum temperature of the two fluid medium

ciT = minimum temperature of the two fluid medium

cQ = Rate of heat transfer from the hot fluid

An energy balance can be used to calculate the mass flow rates of both water and sodium.

The mass flow rate for sodium is calculated below.

Equation 26 - Mass flow rate of sodium

( ) s

Kg

CCCkg

kW

TTc

Qm

hohiphh 1068

400600kJ

1.334

000,273

)(=

°−°°

=−

=

In this analysis the energy from the thermal energy of the sodium was assumed to only

cause a phase change of the water, not increase the water’s temperature:

Equation 27 - Mass flow rate of water

s

kg

kg

kJkW

xh

Qm

fgc 1209

)1.0(2257

000,273 ===

Where hfg is the enthalpy of vaporization and x is the exit quality. The stream heat

capacities can then be calculated.

Equation 28 - C parameter calculation

pcmC =

max5098215.41209 CCs

kJ

CKg

kJ

s

kgcmC phhh =

°=

°==

min1364334.11068 CCs

kJ

CKg

kJ

s

kgcmC pccc =

°=

°==

Since Cmin is known as well at the total power delivered to the cold stream the

effectiveness of the steam generator can be calculated using Equation 29.

Equation 29 - Effectiveness of steam generator calculation

65.0)100600(1364

000,273

)(min

=°−°

°

=−

=CC

C

kWkW

TTC

Q

cihi

cε

38

The NTU is a function of ε and Cmin/Cmzx, however since there is a change of phase in

Cmax is considered to be infinite and Cmin is set equal to 0 [6]. The following equation

calculates NTU as a function of ε and Cmin/Cmzx.

Equation 30 - NTU calculation

NTUCCNTU

CCNTU

eeCC

e −−−

−−

−=−

−= 1)/(1

1)/1(

maxmin

)/1(

maxmin

maxmin

ε

0367.1)45ln(.)1ln( =−=−−= εNTU

By definition the number of transfer units is:

Equation 31 - Heat transfer area calculation

minC

UANTU =

2min 44.5 mU

CNTUA =

⋅=

The previous equation describes the total area required for heat transfer to remove the

thermal energy from the reactor into the secondary system, which is directly related to the

number of pipes required.

6.1.3 Steam Generator Weight

The total weight of the steam generator includes the weight of the water, sodium,

the copper piping, the housing material, and the steam dryer. Additionally there must be

piping to and from the turbine and core. This analysis ignores the weight of the steam

dryers and is simply shows that the total weight of the steam generator is much less than

the 50 ton limit.

The Weight of the Stainless Steel Type 316 to be used in the piping is:

Equation 32 - Weight of Stainless Steel pipes in Steam Generator

kgLrrtubesVm io 36383)(# 22 =⋅−⋅⋅== ρπρ

39

The weight of the Sodium is:

Equation 33 - Weight of sodium in steam generator

kgLrtubesVm i 1131# 2 =⋅⋅⋅⋅== ρπρ

The total weight of the housing material using 4 in steel casing is:

Equation 34 - Weight of Steam Generator housing

[ ]kg

thicknessLtubespitchtubespitchVpm

1639

#4)#(2 2

=⋅⋅⋅+⋅== ρ

A four-inch steel housing was chosen to keep the hoop stress of the steam generator

below the maximum tensile strength of Stainless Steel Type 316, using Equation 8. The

water may be added to the exchanger on site and therefore was not included in the

weight.

6.1.4 Steam Generator Weight

The total weight of the steam generator is on the order of 43 tons. The steam

generator is expected to be the heaviest part of the secondary system and based on this

analysis should be transportable on a truck. A steam generator and condenser will have

to be transported on a separate truck.

7 Safety Considerations

Because the reactor will be placed in a remote location, safety and reliability are

essential for successful operation. Basic safety strategies were implemented wherever

possible. For instance, the reactivity coefficients for the reactor were engineered to be

negative and there is a large factor of safety for temperature and pressure limitations.

One specific safety case examined was a loss of forced flow accident. The goal was to

create a system in which the natural circulation of the sodium would suffice in cooling

40

the reactor. It was assumed that during a loss of forced flow accident that the reactor

would immediately scram, and the only heat would be from the decay heat of the reactor.

The decay heat is greatest right after the reactor is first scrammed and amounts to about

6.0% of the original reactor thermal power. Using the same heat transfer codes described

in section 4, a conservative coolant velocity of 0.3 m/s would remove the decay heat. To

achieve natural circulation the heat sink and heat source must have some vertical offset.

The equation relating the height difference of the heat source and sink to the velocity of

the coolant is given by the following equation:

Equation 35 - Natural Circulation equation [4]

( ) )/*(***5.0** 2haveragehc dLfvhg ρρρ =∆−

A height difference of approximately one meter provided the necessary coolant velocity.

In fact, with a little more tweaking it seemed possible to run the reactor completely on

natural circulation and remove the pumps all together.

8 General Plant Design

Originally, the reactor was designed using the Clinch River Breeder Reactor (CRBR)

as a model. However, the purpose of the CRBR and the purpose of this reactor differed

in one main aspect: the CRBR was designed to breed fuel; this reactor was designed to

produce power. Therefore many of the features from the CRBR were deemed

unnecessary. For instance, the CRBR core consists of three concentric fuel zones: an

enriched uranium zone, meant to keep the required neutron multiplicity, an outer depleted

zone, optimized for breeding, and then a third outer depleted zone, designed for shielding

purposes. Aside from the fuel enrichment varying from zone to zone, other parameters

such as the pitch, fuel pin diameter, and cladding diameter also varied accordingly.

41

However, while the CRBR was unsuitable for direct scale down, many of its

characteristics were useable in this design. Most notably, the core materials (fuel,

cladding, and coolant), the fuel pin diameter, cladding thickness, and the temperature rise

across the core from the CRBR’s enriched zone were used as guidelines and/or starting

points in this design. In general, the core and subsequent systems were designed as

simply as possible, relying on proven technologies to provide a more robust estimate of

the reliability and feasibility of the reactor. Also, relying on simple, proven technologies

should help reduce the capital cost of the reactor.

As mentioned in the opening sections, weight was one of the most important issues.

Each component has been designed so that it could fit on a standard lowboy tractor trailer

(approximately 50 ton weight limit). The reactor and subsequent systems would

probably require about six trucks to transport all the required material. The core and heat

exchanger could be transported with the sodium already inside, however the connecting

piping would have to be pieced together, vacuum sealed, and then filled with liquid

sodium from an external tank. Also, external heaters will be required throughout the

connection pipes because sodium is not a liquid until about 100 degrees Celsius. The

following figure provides a general layout of the reactor system:

42

Figure 8 - General Core Layout

As mentioned previous, the core will reside below ground level, the hole will be

lined with concrete, and there will be a large concrete lid. The core hole should be

backfilled with an inert gas, such as helium. No inert gas is required in the steam

generator concrete structure because the steam generator was designed with water

flowing on the outside with the hot sodium flowing on the inside, which should minimize

the probability of the sodium coming in direct contact with the air. Also, a second

concrete hole can be dug onsite to provide a radioactively safe storage area for a spent

core. All major components will be crane lifted into place. The final heat sink will be

site specific. Ideally, a river or lake could serve as the final heat sink, but in the event

that a suitable large body of water is not present, small cooling towers will have to be

transported to the site. These cooling tours will require extra trucks to transport. A list of

the final core parameters are provided in Appendix D.

9 Conclusions and Further Research

While preliminary efforts seem promising, there is still much more work to be

done. A list of final core parameters is provided in Appendix D. The core neutronics

43

were modeled assuming a homogeneous core and the heat transfer correlations only

measure the single channel profile. Furthermore the shielding calculations only roughly

estimate the doses. The most amount of research still lies in the secondary system. For

the steam generator analysis only general heat exchanger correlations were used rather

than steam generator specific correlations. Also, no analysis was spent on the steam

turbine. Also, very little attention was given to the logistics of this reactor. For instance,

the specific plant layout and the construction time were only remotely considered.

Licensing and economic requirements were only briefly touched.

Some of the initial reasons for choosing a liquid metal cooled reactor did not turn

out to be as important as previously thought. Most notable, a liquid metal coolant was

chosen for breeding purpose, but in fact the breeding ration for this reactor was only

about 0.5, less than that of commercial LWRs. Also, contrary to initial thoughts, the high

power density seemed to be more a result of the fuel enrichment instead of the liquid

metal coolant. With these considerations, it seems feasible that a PWR type reactor could

also be built to meet these specifications. For example, waters high specific heat can help

offset its lower thermal conductivity than sodium. Also, because LWRs operate at lower

temperatures, it may be possible to substitute UO2 fuel for straight Uranium fuel.

Switching to straight uranium fuel, in addition to using 15% enriched fuel may even

increase the linear power density of an PWR to greater than that of an LMFBR and still

result in a similar life span. Before this reactor design is built, more analysis should go

into the advantages and disadvantages of different reactor types to verify that LMFBRs

have a distinct advantage over the other reactor types.

44

References

[1] “Tang, Y.S., Coffield, R.D., and Markley, R.A.” Thermal Analysis of Liquid Metal Fast Breeder Reactors. American Nuclear Society, 1978

[2] “Todreas, Neil E., Kazimi, Mujid S.” Nuclear Systems I: Thermal Hydraulic Fundamentals. Hemisphere Publishing Company, 1990

[3] “Todreas, Neil E., Kazimi, Mujid S.” Nuclear Systems II: Elements of Thermal Hydraulic Design. Hemisphere Publishing Company, 1990

[4] “Kutateladze, S.S., Borishanskii, V.M., Novikov, I.I., and Fedynskii, O.S.” Liquid-Metal Heat Transfer Media. Atomic Press, Moscow, 1958

[5] “Schmidt, Frank W., Henderson E. Robert, Wolgemuth, Carl H.” Introduction to Thermal Sciences. John Wiley & Sons, Inc, 1984

[6] “Yevick, John G., Amorsi A.” Faster Reactor Technology: Plant Design. Massachusetts Institute of Technology, 1966

[7] “Foust, O.J.” Sodium-NaK Engineering Handbook Volume II: Sodium Flow, Heat Transfer, Intermediate Heat Exchangers, and Steam Generators. Gordon and Breach, Science Publishers, Inc., 1976

[8] “Cochran, Thomas B.” The Liquid Metal Fast Breeder Reactor: An Environmental and Economic Critique. Resources for the Future, Inc., 1974

[9] “Division of Reactor Development and Technology, United States Atomic Energy Commission” Technical Problems of Fast Reactors. Atomic Energy Publishing House, 1969

[10] “National Nuclear Data Center.” Brookhaven National Laboratory, http://www.nndc.bnl.gov/

45

46

Appendix A Sample SCALE input for core neutronics

=sas2h parm='skipshipdata'*************************************************' LMFBR from Duderstadt & Hamilton App. H.' Case run using a square pitch' changed power to 273 MWth reactor' Pdot is currently 2500(W/m)'*************************************************LMFBR44groupndf5 Latticecelluo2 1 1.0 1316 92235 15 92238 85 endag-109 1 1.e-20 1316 endag-111 1 1.e-20 1316 endba-138 1 1.e-20 1316 endbr-81 1 1.e-20 1316 endce-140 1 1.e-20 1316 endce-141 1 1.e-20 1316 endce-142 1 1.e-20 1316 endce-143 1 1.e-20 1316 endce-144 1 1.e-20 1316 endcs-133 1 1.e-20 1316 endcs-135 1 1.e-20 1316 endcs-136 1 1.e-20 1316 endcs-137 1 1.e-20 1316 endeu-153 1 1.e-20 1316 endeu-155 1 1.e-20 1316 endeu-156 1 1.e-20 1316 endi-127 1 1.e-20 1316 endi-129 1 1.e-20 1316 endi-131 1 1.e-20 1316 endkr-83 1 1.e-20 1316 endkr-84 1 1.e-20 1316 endkr-85 1 1.e-20 1316 endkr-86 1 1.e-20 1316 endla-139 1 1.e-20 1316 endla-140 1 1.e-20 1316 endmo-95 1 1.e-20 1316 endmo-97 1 1.e-20 1316 endmo-98 1 1.e-20 1316 endmo-99 1 1.e-20 1316 endmo-100 1 1.e-20 1316 endnb-95 1 1.e-20 1316 endnd-143 1 1.e-20 1316 endnd-144 1 1.e-20 1316 end

1

nd-145 1 1.e-20 1316 endnd-146 1 1.e-20 1316 endnd-147 1 1.e-20 1316 endnd-148 1 1.e-20 1316 endnd-150 1 1.e-20 1316 endpd-105 1 1.e-20 1316 endpd-107 1 1.e-20 1316 endpd-108 1 1.e-20 1316 endpm-147 1 1.e-20 1316 endpm-148 1 1.e-20 1316 endpr-141 1 1.e-20 1316 endpr-143 1 1.e-20 1316 endrb-85 1 1.e-20 1316 endrh-103 1 1.e-20 1316 endru-101 1 1.e-20 1316 endru-102 1 1.e-20 1316 endru-103 1 1.e-20 1316 endru-104 1 1.e-20 1316 endru-106 1 1.e-20 1316 endsb-125 1 1.e-20 1316 endsb-126 1 1.e-20 1316 endse-80 1 1.e-20 1316 endse-82 1 1.e-20 1316 endsm-149 1 1.e-20 1316 endsm-151 1 1.e-20 1316 endsm-152 1 1.e-20 1316 endsm-154 1 1.e-20 1316 endsn-125 1 1.e-20 1316 endsn-126 1 1.e-20 1316 endsr-88 1 1.e-20 1316 endsr-89 1 1.e-20 1316 endsr-90 1 1.e-20 1316 endtc-99 1 1.e-20 1316 endte-128 1 1.e-20 1316 endte-130 1 1.e-20 1316 endte-132 1 1.e-20 1316 endxe-131 1 1.e-20 1316 endxe-132 1 1.e-20 1316 endxe-133 1 1.e-20 1316 endxe-134 1 1.e-20 1316 endxe-135 1 1.e-20 1316 endxe-136 1 1.e-20 1316 endy-89 1 1.e-20 1316 endy-91 1 1.e-20 1316 endzr-91 1 1.e-20 1316 endzr-92 1 1.e-20 1316 end

2

zr-93 1 1.e-20 1316 endzr-94 1 1.e-20 1316 endzr-95 1 1.e-20 1316 endzr-96 1 1.e-20 1316 endss316 2 1.0 865 endna 3 den=.832 1.0 769 endORCONCRETE 4 1.0 293 endend compsquarepitch .76 .560 1 3 .660 2 .584 0 endNPIN/ASSM=75978 FUELNGTH=143.73 NCYCLES=1 NLIB/CYC=7 PRINTLEVEL=4 INPLEVEL=3 NUMZTOTAL=6 end3 0.001 500 118.19 3 148.67 2 149.94 0 165.18 4 195.66BON endNIT endXSDWeighted cross sectionsI4= -1 3 0 9X5= .0001 .00001 1. 0. 0. 1.42 128.7 endPOWER=273.0 BURN=3650 DOWN=15 endend

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Appendix B Matlab files for Heat Transfer Correlations

clc;clear;

l_fp = 1.81; %mpd_ratio = 1.3;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%cladding propertiesclad_id = .2*.0254; %mclad_od = .23*.0254; %mkclad = 21.5; %W/m*K

pitch = pd_ratio * clad_od; %m%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%coolant properties

%taken at about 1000F or 540 C%assumed constant

v = 2.34; %m/s NEED TO VARY THIS PARAMETERtin = 400; %celsiusrow = 832.960095; % kg/m^3mu = 0.000210823225; %N*s/m^2cp = 1276.12; %J/(kg*K)kc = 66.56744; % W/(m*K)

z = -l_fp/2:.01:l_fp/2;%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% calculate q' - qz

qavg=2500;qmax = 2*qavg; %W/m %NEED TO CHANGE THIS PARAMETER

qz = qmax*cos(pi().*z./l_fp);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Fluid Temperature

1

%dh = 4*Af/Pw - Af = Flow Area, Pw = Wetted PerimeterAf = (sqrt(3)*pitch^2/4) - (pi()*clad_od^2/8);Pw = pi()*clad_od / 2;dh = 4 * Af / Pw;

%Reynolds Number calcRe = v * dh * row/mu;mdot = row * v * Af ;%Prandtl Number calcPr = cp * mu / kc;%Peclet numberPe = Re*Pr;

%Nu number from pg 451 of todreasNu = 4.0+0.33*pd_ratio^3.8 * (Pe/100)^0.86 + 0.16*pd_ratio ^ 5;

%h = Nu*k/dhh = Nu*kc/dh;

%coolant temperaturetmz = tin + (qmax*l_fp/ (mdot*cp* pi()) * (sin(pi().*z./l_fp) + sin(pi().*l_fp ./ (2.* l_fp))));

%cladding outer temperatureRco = clad_od / 2;tcoz = tmz + qmax / (2*pi()*Rco*h) * cos(pi().*z./l_fp);

%cladding inner temperatureRci = clad_id / 2;tciz = tcoz + qz/l_fp * (Rco - Rci)/ kclad;

%Centerline Temperaturekfuel = 2.163; %w/m*KRg = Rci;tclz = tmz + qz*(1/(4*pi()*kfuel) + 1/(2*pi()*kclad)*log(Rco/Rci) + 1/(2*pi()*Rco*h));

figure(1);plot([tmz;tcoz;tciz;tclz],z);legend('Coolant','Outer','Inner','Fuel')title(['Sodium Speed = ',num2str(v),' m/s'])xlabel('Temperature - Celsius');ylabel('Verical Position');grid;

[m1 n1] = max(tmz);fprintf('Max Coolant Temp is %f at position %f \n', m1, z(n1))fprintf('Temperature Rise Across Core is %f \n', m1 - tin)

2

[m2 n2] = max(tcoz);fprintf('Max Cladding Outer Temp is %f at position %f \n', m2, z(n2))

[m3 n3] = max(tciz);fprintf('Max Cladding Inner Temp is %f at position %f \n', m3, z(n3))

[m4 n4] = max(tclz);fprintf('Max Fuel Centerline Temp is %f at position %f \n', m4, z(n4))

fprintf('\nThis configuration is with a coolant velocity of %f (m/s)\nand a heat tranfer coefficient of %f (W/(m^2*K))\n',v,h);

if (m1-tin<263)&(m2<660)&(m3<660)&(m4<2200) fprintf('\nThis configuration meets the requirements.\n');else fprintf('\nThis configuration does not meet the requirements.\n');end

3

Appendix C Sample Scale input for shield calculations

#sas1x parm='size=900000'sphereical reactor with concrete shielding27n-18couple multiregion'' multiregion must be specified to run combined criticality/shielding problem.'uo2 1 0.76 1316 92235 15 92238 85 endna 1 den=0.8320 0.24 769 endss316 2 1.0 865 endorconcrete 3 1.0 293 endactivities 3 0 1.e-24 endend comp'' the criticality calculation inputspherical vacuum end1 282 end zone' isn=16 is specified to match the angular quadrature in the shielding calc.more data isn=16 end more dataendlastreactor shielding'' the shielding calculation input'spherical' first mixture must be void of 1 interval with outer dimension that matches' outer dimension of shielding calculation.' flags indicate boundary source will be input from xsdrnpm criticality calc.0 282 1 1 0 0 02 283.27 20 03 397.57 100 0end zoneread xsdoseend

1

Appendix D Summary of Core Parameters:

Core ParametersLinear Power Density 2500 (W/m)

Core Volume 5.46 m3

Core Height 1.81 mCore Radius 0.9799 mCore Power 273 MWth

Height/diameter 0.924Assumed Efficiency 33%

Net Power 100 MWe

Number of Fuel Pins 60304Conversion Ratio .506

Fuel Pin Length 1.81 m

Cladding Inner Diameter 0.254 cm

Cladding Outer Diameter 0.2921 cm

Fuel Pin Diameter 0.254 inPressure Drop 21.840 kPaPump Work .206 MW

Inlet Temperature 400 ˚CTemperature Rise 200.2 ˚C

Cladding Max Temperature 600.2 ˚CFuel Centerline Max Temperature 717.0 ˚C

Coolant Max Temperature 600.2 ˚CCore Weight (including vessel) 36 tons

Materials ChosenFuel (Enrichment) Uranium Dioxide (15%)

Cladding Stainless Steel Type 316Coolant SodiumVessel Stainless Steel Type 316

Secondary Water

Steam GeneratorMaterial Stainless Steel Type 316

Tube thickness 1.0 cmNumber of Passes 5000

Height 3.46 mThickness of Housing 10 cm

Water Inlet Temperature 290 ˚CSteam Exit Quality 10%

Weight 43 tons

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