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XYLENE POWER LTD.

FNR FUEL TUBE WEAR

By Charles Rhodes, P.Eng., Ph.D.

INTRODUCTION:
FNR fuel tubes are a key part of FNR design. The fuel tubes must be replaced every fuel cycle so the cost of the fuel tubes, their working life and their disposal are all significant issues. The fuel tube working life should be sufficiently long that it does not dictate the minimum frequency of FNR fuel reprocessing. One of the factors determining fuel tube working liife is fuel tube alloy swelling.

Under sustained fast neutron bombardment FNR fuel tubes will eventually swell due to phase changes in the fuel tube alloy resulting from fast neutron induced formation of He-4 and Cr-52. Inelastic scattering of fast neutrons excites fuel tube alloy nuclei to high energy states. In circumstances where these excited nuclei cannot readily emit sufficient energy via electron or positron emission these excited states relax via alpha particle (He-4 nuclei) formation.

A FCC crystal structure is too dense for storing much He-4. When the amount of He-4 produced exceeds the BCC fuel tube He-4 lattice storage capacity swelling will occur. The He-4 induced transition from an FCC to a BCC crystal structure causes its linear dimensions to increase by a factor of about 1.26 in addition to normal thermal expansion. Later transition from a BCC to an HCP crystal structure also causes unacceptable fuel tube swelling. Formation of Fe2Ti in a fuel tube alloy will also reduce the alloy's He-4 storage capacity. Hence the fuel tube alloy must be chosen so that it remains BCC throughout its working life.

This web page is concerned with choice of a fuel tube alloy that gives a long fuel tube working life while also meeting other necessary performance constraints.

The fuel tube alloy optimization strategy is to prevent fuel tube swelling by keeping the fuel tube alloy crystal structure in its same BCC (alpha plus alpha prime) phase at all times and at all operating temperatures and to choose fuel tube constituant elements that have low fast neutron inelastic scattering cross sections to minimize the He-4 production rate.

Zirconium is not a suitable fuel tube component because it is expensive and because Zr-93 with a half life of 1.5 X 10^6 years will eventually become a waste disposal problem.

Nickel is not a suitable fuel tube component because it is expensive, its fast neutron scattering cross section is much higher than iron, it forms the long lived isotope Ni-59 with a half life of 10^5 years which is a nuclear waste disposal problem and when the ratio of Ni to Fe exceeds about 5% there is a BCC to FCC phase change in the fuel tube alloy.

The cobalt concentration in the fuel tube must be kept under 70% to maintain a BCC lattice through the fuel tube operating temperature range.

The manganese (Mn) concentration in a fuel tube must also be kept under 3.0% to maintain a BCC crystal structure through the fuel tube's operating temperature range.

The Fe - Cr system has two BCC phases designated alpha and (alpha + alpha prime). The initial chromium (Cr) concentration in a fuel tube must be set at 12% to keep the fuel tubes in the (alpha + alpha prime) phase through there operating temperature range and throughout their operating life. If the fuel tubes were intended to operate only in the alpha phase their life would be limited by crossing of the phase boundary as the material ages.

Successful FNR fuel tubes have been made with an initial component ratio of 85% Fe, 12%Cr alloy known as HT-9.

There is merit in choosing the HT-9 alloy that has a BCC crystal structure throughout its operating temperature and component ratio range and hence has sufficient internal lattice space for storing He-4.

It is shown herein that for Fe-Cr fuel tubes the end of useful fuel tube working life occurs when the elemental fractions in the fuel tube alloy become about 60% Fe, 26% Cr, 11% Ti. At larger Ti fractions formation of Fe2Ti causes the He-4 storage capacity to rapidly diminish. At this point the He-4 storage capacity is:
1.5 [0.60 Ni - 2(0.11 Ni) + 0.26 Ni] = 1.5 [0.64 Ni]
= 0.96 Ni

During its working life the fuel tube alloy operating temperature must be kept less than the BCC to FCC transition temperature which varies from 720 C for a pure Fe down to about 460 C for a mixed Fe-Cr alloy with a Cr fraction ~ 0.2 . Hence with HT-9 fuel tubes the maximum FNR primary liquid sodium operating temperature should be about 445 C to allow for a 15 degree C temperature drop across the fuel tube wall.

Let Nt = number of fuel tube atoms.

The fuel tube rating places an upper limit on the fuel burn-up fraction per FNR fuel cycle.

Accumulation of interstial He-4 causes fuel tube embrittlement at lower operating temperatures (330 degrees K to 425 degrees K).

An important consideration in the choice of fuel tube alloy is that neutron interactions with the fuel tube material should not produce long lived low atomic weight isotopes that create long term waste disposal problems. This issue leads to rejection of Be, C and Ni as initial fuel tube components, in spite of the favorable physical characteristics that these elements can potentially provide to the fuel tube alloy.

In principle pure Ti fuel tubes might be better than Fe - Cr alloy fuel tubes due to the low density Ti HCP lattice and the relatively low He-4 production rate. However, there are major practical problems in fabricating FNR fuel tubes from pure Ti.
 

EFFECTS OF FAST NEUTRON IMPACTS ON FUEL TUBE ATOMS:
The effects of fast neutron impacts on fuel tube atoms are:
1) Addition of energy to the fuel tube material lattice which reduces the depth of its crystal lattice binding energy wells and hence reduces the material strength. This material strength reduction relates to stress from the radial heat flux and the internal gas pressure;
2) Reduction in the atomic numbers of the impacted fuel tube atoms because removal of of He-4 nuclei changes the overall fuel tube component mix and leaves He-4 imbedded in the crystal matrix;
3) Note that some elements are more likely to suffer He-4 removal than other elements due to different fast neutron scattering cross sections;
4) Multiple inelastic fast neutron scattering impacts change Fe-56 to Cr-52 and change Cr-52 to Ti-48.
5) Severe fuel tube alloy swelling occurs after He-4 atoms fill all the available BCC lattice spaces;
6) Severe fuel tube alloy swelling occurs on a FCC to BCC phase transition;
7) Severe fuel tube alloy swelling occurs on a BCC to HCP phase transition;
8)Unacceptable fuel tube swelling occurs on a BCC alpha to BCC (alpha + alpha prime) phase transition. 8) Note that at temperatures below 720 degrees C pure Fe has a BCC crystal structure. Pure Cr, pure Mn and pure V also have a BCC crystal structures but pure Ti has a HCP crystal lattice. Ti is held in a BCC lattice via the compound Fe2Ti. Hence at the end of the fuel tube working life the number of Fe atoms present must be more than twice the number of Ti atoms.

It is shown herein that a fuel tube material with a FCC crystal lattice can only absorb 1 He-4 atom for every 4 fuel tube atoms without significant fuel tube material swelling whereas a fuel tube material with a BCC crystal lattice can absorb up to 6 He-4 atoms for every 4 fuel tube atoms (1.5 He atoms per unbound (Fe + Cr) atoms without significant fuel tube material swelling.

Fast Neutron Reactor (FNR) tube wear can be expressed in terms of number of fast neutron scattering impacts per fuel tube atom (fnipa). The number Nk indicates the number of fast neutron impacts required to form one He-4 nucleus.
 

He-4 PRODUCTION:
He-4 production is the result of alpha particle emission. Alpha particles are only emitted in circumstances where an atomic nucleus decays via alpha particle emission instead of via electron or positron emission. There are 4 routes to alpha partile emission from the most common iron isotope Fe-56. In the following presentation KE on the left hand side of a reaction equation refers to kinetic energy acquired by a nucleus as a result of inelastic fast neutron scattering. KE on the right hand side refers to combined gamma and kinetic energy liberated during natural radio isotope decay. The isotope relative energy data comes from the Table of Isotopes, Sixth Edition by Lederer, Hollander and Perlman, 1967

Route #1:
a) Fe-56(-60.665 MeV) + KE(4.262 MeV) = Mn-56(-56.904 MeV) + positron(0.501 MeV)

b) Mn-56(-56.904 MeV) + KE(2.105 MeV) = Cr-56(-55.3 MeV) + positron(0.501 MeV)

c) Cr-56(-55.3 MeV) + KE(6.7858 MeV) = V-52(-51.44 MeV) + He-4(2.4248 MeV) + positron(0.501 MeV)

d) V-52(-51.44 MeV) = Cr-52(-55.411 MeV) + e(0.501 MeV) + KE(4.47 MeV) (natural decay, half life = 3.75 minutes)

In Route #1 the total KE requirement to produce one He-4 nucleus is:

4.262 MeV + 2.105 MeV + 6.7858 MeV = 13.1528 MeV
 

Route #2:
e) Fe-56(-60.665 MeV) + KE(5.136 MeV) = Co-56(-56.03 MeV) + e(0.501 MeV)

f) Co-56(-56.03 MeV) + KE(2.611 MeV) = Ni-56(-53.92 MeV) + e(0.501 MeV)

g) Ni-56(-53.92 MeV) + KE(8.00 MeV) = Fe-52(-48.33 MeV) + He-4(2.4248 MeV)

h) Fe-52(-48.33 MeV) = Mn-52(-50.70 MeV) + positron(0.501 MeV) + KE(2.871) (natural decay, half life = 8.2 h )

i) Mn-52(-50.70) = Cr-52(-55.411) + positron(0.501 MeV) + KE(4.21 MeV) (natural decay, half life = 5.6 days)

In Route #2 the total KE requirement to produce one He-4 nucleus is:
5.136 MeV + 2.6111 MeV + 8.00 MeV = 15.747 MeV
 

Route #3:
j) Fe-56(-60.665 MeV) + KE(4.262 MeV) = Mn-56(-56.904 MeV) + positron(0.501 MeV)

k) Mn-56(-56.904 MeV) + KE(4.41 MeV) = Cr-52(-55.411 MeV) + He-4(2.4248 MeV) + e(0.501 MeV)

In Route #3 the total KE requirement to produce one He-4 nucleus is:
4.262 MeV + 4.41 MeV = 8.672 MeV
 

Route #4:
l) Fe-56(-60.665 MeV) + KE(5.136 MeV) = Co-56(-56.03 MeV) + e(0.501 MeV)

m) Co-56(-56.03 MeV) + KE(3.545 MeV) = Cr-52(-55.411 MeV) + He-4(2.4248 MeV) + positron(0.501 MeV)

In Route #4 the total KE requirement to produce one He-4 nucleus is:
5.136 MeV + 3.545 MeV = 8.681 MeV
 

Reactions (c), (g), (k) and (m) are theoretical. On the basis of requiring less kinetic energy from inelastic fast neutron scattering Routes #3 and #4 appear to be dominant and each requires about 4 X 2.0 MeV inelastic fast fission neutron impacts per He-4 nucleus generated.

When the FNR is not operating:
In reactions (a) and (j) Mn-56 decays to Fe-56 by electron emission with a half life of 2.57 hours
In reaction (b) Cr-56 decays to Mn-56 by electron emission with a half life of 5.9 minutes
In reactions (e) and (l) Co-56 decays to Fe-56 by positron emission with a half life of 77.3 days
In reaction (f) Ni-56 decays to Co-56 by positron emission with a half life of of 6.1 days

In reaction (k) Mn-56 decays to Co-56 by electron emission with a half life of 2.57 hours

Note that Fe-56 has an excited state at 0.8469 MeV from which it can, via absorption of more KE, move to higher semi-stable energy states in Co-56 by electron emission or in Mn-56 by positron emission.

On all four reaction routes the steady state net reaction is:
Fe-56 + KE = Cr-52 + He-4

In Route #1 reaction (c) the reaction rate is enhanced because Cr-56 has excited state(s) which enable acquision of KE = 6.7858 MeV in multiple steps.

In Route #2 due to reaction (e) the time required to approach steady state conditions is about 3 months. In Route #2 the rate of reaction (g) is enhanced becasuse Ni-56 has excited state(s) which allow acquisition of KE = 8.00 MeV in multiple steps.

In Route #3 reaction (k) is theoretical but it is likely a main He-4 production path.

In Route #4 reaction (m) is theoretical but it is likely a main He-4 production path.

One of the most suitable FNR initial fuel tube components is iron. As fast neutron scattering removes He-4 atoms from iron the dominant stable isotope Fe-56 changes to stable Cr-52. Similarly stable Fe-54 changes to stable Cr-50. Similarly stable Fe-57 changes to stable Cr-53. Similarly stable Fe-58 changes to stable Cr-54. At temperatures below 460 degrees C the Fe-Cr system has a stable BCC crystal lattice configuration at all Fe/Cr ratios.

During the next He-4 removal step stable Cr-52 changes to stable Ti-48; stable Cr-50 changes to stable Ti-46; stable Cr-53 changes to stable Ti-49 and stable Cr-54 changes to stable Ti-50. To the extent of Fe availability the Ti forms Fe2Ti. When there is insufficient Fe to form Fe2Ti the Ti will form a HCP lattice causing fuel tube material swelling.

When there is insufficient Cr and unbound Fe to store the total amount of He-4 produced fuel tube alloy swelling will also occur.
 

FUEL TUBE MATERIAL NEUTRON ACTIVATION:
Stable Ni-58 + n becomes unstable Ni-59 which is an undesirable long lived isotope;

Stable Fe-54 + n becomes unstable Fe-55 which positron decays to stable Mn-55;
Stable Fe-56 + n becomes stable Fe-57;
Stable Fe-57 + n becomes stable Fe-58;
Stable Fe-58 + n becomes unstable Fe-59 which decays by electron emission into stable Co-59.

Stable Mn-55 + n becomes unstable Mn-56 which decays by electron emission into stable Fe-56 .

Stable Cr-52 +n becomes stable Cr-53;
Stable Cr-50 + n becomes unstable Cr-51 which decays by electron emission into Mn-51 which then decays by positron emission into stable V-51;
Stable Cr-53 + n becomes stable Cr-54;
Stable Cr-54 + n becomes unstable Cr-55 which decaysby electron emission into stable Mn-55.

Thus a fuel tube initially composed of a Fe + Mn alloy appears from a nuclear perspective to be very good.In large measure the working life of a fuel tube is terminated by formation of Ti. The fuel tube starts out as a Fe-Mn alloy but with sufficient neutron exposure becomes a Fe-Cr-Ti-Mn-V-Co alloy with interstial He-4.

Note that the amounts of carbon, oxygen, beryillium and nickel initially present in the fuel tube alloy should be minimized to avoid formation of long lived C-14, Be-10 and Ni-59.
 

ELEMENT PROPERTIES:
The room temperature crystal structures, densities and melting points of pure Ni, Fe, Mn, Cr, V and Ti are as follows:
Ni: FCC, 8.90 g / cm^3, 1455 C
Co: BCC for Co < 70% in Fe, T < 910 C; 8.90 gm / cm^3, 1495 C
Fe: BCC for T < 720 deg C, 7.874 g / cm^3, 1538 C
Mn: BCC for Mn < 1.5% in Fe, 7.21 g / cm^3, 1246 C
Cr: BCC at 0 to 20% in Fe for T < 512 C (problems for T > 460 C, 7.19 g /cm^3, 1907 C
V: BCC, 6.0 g / cm^3, 1910 C
Ti: HCP, 4.506 g / cm^3, 1668 C
Note the very low density of pure Ti as compared to pure Cr, which potentially leads to rapid fuel tube material swelling if the Ti fraction in Fe becomes too large.
 

FUEL TUBE LIFE:

Let Nt = initial number of fuel tube atoms.

The optimum fuel tube life maximization strategy is to start with nearly pure Fe and add enough Cr to stabilize the material in the (alpha + alpha prime) zone. Hence at the start of fast neutron exposure the fuel tubes consist of:
85% Fe + 12% Cr + 3% other.
Ensure that the fuel tube alloy is free of nickel, which has a relatively large fast neutron scattering cross section and which forms the problem waste Ni-59. The amounts of Be, C, and O present must also be minimized because they lead to formation of long lived isotopes Be-10 and C-14.

Assume 30% conversions. Then at the end of fuel tube working life exposure to a fast neutron flux has reduced the number of Fe atoms by a factor of 0.7 giving:
0.7 (.85 Ni) = 0.60 Ni

The number of new Cr atoms is 0.25 Ni.

The total number of Cr atoms remaining at the end of fuel tube life is:
0.7 [(0.25 Ni + 0.12 Ni) = 0.26 Ni

The number of Ti atoms formed is:
0.3 (0.37 Ni) = .11 Ni

Hence at the end of fuel tube working life the fuel tube component ratios are:
60% Ni, 26% Cr, 11% Ti + 3% other

Thus the total amount of He-4 produced is:
0.3 (0.85 Ni) + 0.3 (0.25 Ni + 0.12 Ni)
= 0.3 (1.22 Ni)
= 0.37 Ni

The He-4 storage capacity constraint is:
1.5 [0.60 Ni + 0.26 Ni - 2 (0.11 Ni)] = 1.5 (0.64 Ni)
= 0.96 Ni

Hence the amount of He-4 produced is less than one half the storage capacity constraint.

The Ti is bound by the compound Fe2Ti. Hence: 2 (0.11 Ni) Fe-56 + (0.11 Ni) Ti-50 provide no He-4 storage.

Hence only (0.38) of the original Fe-56 + (0.26) in the form of Cr-52 are available to store He-4 atoms.

Let Ni = initial number of Fe atoms.

The number of fast neutron impacts on fuel tube atoms is:
0.37 Ni Nk

The number of fast neutron impacts per fuel tube atom is:
(0.37 Ni Nk) / Ni = 0.37 Nk

Hence HT-9 fuel tube material is rated for a maximum of 0.37 Nk fnipa, where Nk is the average number of fast neutron scatters per fuel tube nucleus required to form one He-4 nucleus.

Brookhaven fast neutron elestic and inelastic scattering data for Fe-56 indicates that Nk ~ 7.

Hence the fuel tubes can withstand about:
7 (0.37) fnipa = 2.59 fnipa
before fuel tube swelling becomes a concern.

Judging by the storage margin the fuel tubes are probably good to past 3.0 fnipa.

The fuel tube rating places an upper limit on the fuel burn-up fraction per FNR fuel cycle.

Thus the fast neutron damage that fuel tubes initially consisting of (85% Fe + 12% Cr) can absorb without significant swelling is Nk (0.37) fast neutron impacts per fuel tube atom (fnipa).

After exposure to Nk (0.37) fnipa the fuel tube material will consist of atomic ratios of about:
60% Fe-56, 26% Cr-52, 11% Ti-48 and 3% other.

Formation of Ti within the FNR fuel tube material plays a role in FNR fuel tube swelling. If the Ti fraction exceeds twice the Fe fraction the Ti will form a lower density lattice which will cause fuel tube alloy swelling. To minimize the fuel tube swelling due to He-4 and Ti formation the fuel tube alloy lattice must always be BCC because a BCC lattice is less dense than a FCC lattice and hence provides more interstial space for He-4 atom storage. To both minimize the rate of He-4 production and maximize the He-4 storage capacity the nickel (Ni) concentration of the fuel tube lattice must be minimized. Minimizing nickel concentration also avoids formation of long half life Ni-59 which is a disposal problem.

If the fuel tube crystal lattice is initially FCC on addition of He-4 there will soon be a transition to BCC which reduces the fuel tube material density by a factor of two and which causes a linear swelling dimension increase factor of 1.26. This material swelling will obstruct the primary liquid sodium circulation. The solution to this problem is to fabricate the fuel tubes from components such as (Fe + Cr) that have not lattice phase change in the temperature range of interest.

A result of filling the fuel tube lattice with He-4 at lower temperatures is radiation embrittlement of fuel tube material.
 

Thus the fuel tube will not accommodate wear above Nk (0.37) fnipa. Hence it appears that fast neutron conversion of the fuel tube Fe to Cr and Ti and He-4 accumulation in the fuel tube lattice can potentially limit the fuel tube life and hence the burnup fraction per fuel cycle.
 

LATTICE MATHEMATICS:
Common crystal lattice configurations are:
HCP = hexagonal close packed
FCC = face centered cubic
BCC = body centered cubic

The normal low temperature crystal structure of pure iron is BCC with one side of a cube unit cell = 286.65 pm.
However, from 1185 deg K to 1667 K the crystal structure of iron becomes FCC with one side of a cube = 286.65 m

In FCC each of 8 cube Fe corner atoms is shared among 8 cubes for a net contribution of 1 Fe atom / cube
In FCC each of 6 cube face atoms is shared between 2 cubes for a net contribution of 3 Fe atoms/ cube
Thus in an FCC lattice there are 4 Fe atoms per cube.
A FCC lattice provides one He-4 atom storage space per FCC cube.
Hence a FCC lattice provides 1 He-4 atom storage space / 4 Fe atoms

In a BCC lattice each of 8 cube corner Fe atoms is shared among 8 cubes for a net contribution of 1 Fe atom/ cube.
In BCC each cube center Fe atom is associated only with that cube for a net contribution of 1 Fe atom/Cube;
Thus in a BCC lattice there are 2 Fe atoms per cube.

A BCC lattice can store 3 He-4 atoms / BCC cube.
Hence a BCC lattice provides 3 He-4 atom storage space / 2 Fe atoms
or
6 He-4 atom storage spaces / 4 Fe atoms

.

Thus the maximum He-4 storage capacity of the lattice is 1.5 X [number of Cr and free Fe fuel tube atoms].

For the same inter-atomic spacing a FCC lattice has 2 X the mass density than a BCC lattice. Thus if the Fe - Fe atom interatomic spacing remains nearly constant a FCC to BCC lattice conversion triggers a linear dimension expansion of (2)^0.333 = 1.26

The number of fast neutron impacts per fuel tube atom (fnipa) can be calculated by starting from the reactor power level, finding the rate of neutron impacts on fissionable atoms, and using relative fast neutron scattering cross sections of fuel tube atoms to find the rate of fast neutron impacts on FNR fuel tube atoms. However, Nk fast neutron impacts are required to form a He-4 nucleus.

Brookhaven elastic and inelastic neutron scattering data for Fe-56 and average fission neutron energy data indicates that Nk > 7. In doing this calculation we can assume that within a particular fuel tube bundle the various atomic types are uniformly distributed within the critical portion of the fuel bundle volume.
 

DEFINITIONS:
Nk = average number of fission neutron impacts with fuel tube atoms required to form one He-4 nucleus.
N = number of fast neutrons randomly moving through the fuel bundle
Np = number of Pu atoms in the fuel bundle;
Vol = volume of fuel bundle;
Sigmaf = 1.7 b = plutonium fission core neutron cross section;
Ef = energy release per Pu atom fission
P = thermal power of fuel bundle;
Ni = number of iron atoms in the fuel bundle;
Nn = nimber of nickel atoms in the fuel bundle;
Nc = number of chromium atoms in the fuel bundle;
Nt = number of titanium atoms in the fuel bundle;
Sigmais = 3.8 b = iron atom core neutron scattering cross section;
Sigmaia = 0.0086 b = iron atom core neutron absorption cross section;
Sigmacs = 4.2 b = chromium atom fast neutron scattering cross section;
Sigmaca = 0.014 b = chromium atom fast neutron absorption cross section;
Sigmats = 0 b = titanium atom fast neutron scattering cross section;
Sigmata = ___ b = titanium atom fast neutron absorption cross section;
Vn = neutron velocity;
 

Consider a single fast neutron passing through the fuel bundle. The number of fissions per unit time due to this single neutron is:
[(Sigmaf Np Vn) / Vol]

The number of fast neutron scatters per unit time off iron atoms due to a single neutron is:
[(Sigmais Ni Vn) / Vol]

The number of fast neutron scatters per unit time off chromium atoms due to a single neutron is:
[(Sigmacs Nc Vn) / Vol]

Hence the number of fast neutron scatters off fuel tube atoms per fission is:
[(Sigmais Ni) + (Sigmacs Nc)] / (Sigmaf Np)

Hence the number if He-4 atoms formed in fuel tubes per fission is:
[(Sigmais Ni) + (Sigmacs Nc)] / (Nk Sigmaf Np)

The maximum He-4 storage capacity is:
1.5 (Ni + Nc)

Hence formation of He-4 limits the maximum number of fissions to:
[1.5 (Ni + Nc) Nk] / {[(Sigmais Ni) + (Sigmacs Nc)] / (Sigmaf Np)}
= [1.5 (Ni + Nc)(Nk Sigmaf Np)] / [(Sigmais Ni) + (Sigmacs Nc)]

Hence formation of He-4 limits the reactor energy output per fuel tube working life to:
[1.5 Ef (Ni + Nc)(Nk Sigmaf Np)] / [(Sigmais Ni) + (Sigmacs Nc)]
= [1.5 Ef Nk (1 + (Nc / Ni))(Sigmaf Np)] / {[(Sigmais) + (Sigmacs (Nc / Ni))]}
= [Ef Np] Nk [1.5 (1 + (Nc / Ni))(Sigmaf)] / [(Sigmais) + (Sigmacs (Nc / Ni))]
= [Ef Np] Nk [1.5 (1 + (12 / 85))(1.7 b)] / [(3.8 b) + (4.2 b)(12 / 85)]
= [Ef Np] Nk [2.91 / 4.393]
= 0.6624 [Ef Np] Nk

If 100% of the fissionable atoms fissioned the total energy output would be:
(Np + Nu) Ef

Hence the burnup fraction if it is limited by fuel tube He-4 storage capacity is:
{0.6624 [Ef Np] Nk} / (Np + Nu) Ef
= {0.6624 Nk} / (1 + (Nu / Np))
= {0.6624 Nk} / (1 + (70 / 20))
= 0.1472 Nk
 

VALUE OF Nk:
Recall that Nk = average number of fast neutron scatters from 1 fuel tube nucleus required to produce 1 He-4 nucleus.
Note that Brookhaven data shows that almost half the fast neutron scatters are elastic and provide no energy transfer. The inelastic scatters can provide kinetic energy amounts of about 2 MeV per scatter. It takes at least 8 MeV of acquired kinetic energy to produce one He-4 nucleus. Hence, elastic and inelastic fast neutron scattering data from Brookhaven suggests that for Pu fission neutrons and Fe fuel tubes:
Nk ~ 7
 

For Nk = 7 gives:
(burnup fraction) = 0.1472 Nk
= 1.03

Hence for HT-9 fuel tubes fuel tube swelling is not a life limiting factor. The fuel tube working life will instead be limited by fission product accumulation.
 

SWELLING PROVISION:
Fuel tube swelling triggers potential issues with respect to the amount of plenum stored liquid sodium required to fill the internal clearance between the fuel rod OD and the fuel tube ID once swelling occurs. Due to reactor asymetries the fuel tube mounting arrangements should accommodate substantial fuel tube swelling late in the working life of a fuel bundle.
 

This web page last updated January 7, 2017.

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