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

This web page deals with metallic FNR fuel rods. Important issues with these fuel rods are that:
a) The required Pu concentration is readily achieved via electrolytic fuel reprocessing;
b) The fuel rods are compatible with sodium bonding;
c) The fuel rod lengths are compatible with vacuum casting;
d) The fuel rods can be cast using silica tube forms.

The FNR core fuel rods are initially nominally an alloy consisting of 70% metallic U-238, 20% Pu-239 and 10% zirconium by weight. The purpose of the Pu-239 is to fuel the nuclear reaction. The purposes of the U-238 are to absorb surplus neutrons to breed more Pu-239 and to dilute the Pu-239. The purpose of the 10% zirconium is to prevent formation of a low melting point eutectic (410 deg C at an iron atom fraction of 0.1) between plutonium and iron. (Til & Yoon P.105 - 106) and Hofman et al 1996. The fraction of Zr cannot be further increased due to an upper limit on the fuel rod material melting point imposed by use of silica tubing molds for casting the fuel rods.

In reality there are two core rod alloys. For core rods used in the movable fuel bundles the Pu-239 / U-238 ratio needs to be larger than for core rods used in fixed octagonal active fuel bundles. This requirement is necessary to realize equal reactivities above and below the core zone. Otherwise the region above the core zone will be dangerously reactive when the fuel is new while the region below the core zone has unnecessarily low reactivity. The object is for the core zone to be able to achieve criticality while the average Pu-239 core rod fraction varies from 20%___ down to about 12.7%___.


At 15% fuel burnup:
20%(initial average Pu fraction) - 15%(burnup) + 7.7% (U-238 to Pu-239 conversion) = 12.7%

Note that a key reactor design requirement is that when the movable fuel bundles are fully inserted the core zone achieves criticality with an average Pu-239 fraction in the core fuel of 12.7%. This core zone criticality requirement is checked on the web page titled: FNR CRITICALITY

An average core rod Pu-239 fraction of 20% gives:
384 rods (X%) + 248 rods (Y%) = (384 + 248)rods (20%)

Equal amounts of Pu-239 in the upper and lower core zones gives:
384 rods(X%) = 248 rods (Y%)

Thus combining the above two equations gives:
2 X 384 rods (X%) = 632 rods (20%)
X% = (632 / 768) (20%)
= 16.4583 %

2 X 248 rods (Y%) = 632 rods (20%)
Y% = (632 / 496) 20%
= 25.484%

Thus the movable active fuel bundles need core rods with 25.48% Pu-239 whereas the fixed active fuel bundles need core rods with 16.45% Pu-239

The melting points of the fuel rod elemental components are:
Pu = 639. 4 deg C
U = 1134 deg C
Zr = 1855 deg C.

The fuel rod casting process involves relatively rapid cooling of the entire fuel rod, so that during the casting process the U, Pu and Zr remain uniformly distributed.

Hoffman has shown that during fuel rod use in a FNR the Zr in the central portion of the fuel rod moved radially toward the fuel rod center displacing U and leaving a Zr depleted U enriched region in a ring around the fuel rod center. This behavior is likely a result of melting of the central portion of the fuel rod. When the melt cooled, which due to the large thermal mass of the surrounding liquid sodium coolant likely occured over a long period of time, the outer part of the melt was cooler than the center and the fusing U-Pu alloy excluded Zr, which caused Zr to concentrate along the fuel rod center line. This action raised the melting point on the fuel rod center line but lowered the melting point in the Zr depleted ring. The subsequent fuel rod thermal expansion was then dominated by the character of the ring around the fuel rod center which contains no Zr and contains U-Pu enriched in U.

However, we cannot operate the fuel close to the melting point of this ring due to the large thermal coefficient of expansion due to phase changes below its melting point. The resulting tensile hoop stress on the outer portion of the fuel rod will likely crack the fuel rod material and might cause the fuel rod diameter to expand so much that the fuel rod can no longer slide inside the fuel tube.

The aformentioned rapid expansion of the U-Pu alloy will commence at the temperature at which the alloy becomes phase unstable. This phase transition point is 560 degrees C.

In summary, when fuel rod center line melting occurs the fuel forms a ring around its center line that is depleted in Zr. From that time forward the radial thermal expansion of the fuel rod is dominated by the thermal expansion of that ring when its temperature exceeds 560 degrees C. We do not know the radius of the Zr depleted ring, so for certainty in FNR design we need to assume that its radius is small. Hence we need to design the FNR to have a maximum fuel center line temperature of 560 degrees C.

The stable U-Pu phase of interest is below 560 degrees C for uranium atom fractions in the range 0.74 to 0.85.

The importance of phase stability is that when the phase changes the alloy becomes less dense. A decrease in fuel rod density near its center leads to large tensile hoop stress near the rod outer surface. We do not want the fuel rod to crack in a manner that jams it in the fuel tube. Hence 560 degrees C is the maximum allowable temperature along the fuel rod center line.

The issue of a potential interaction between the plutonium and the iron in the fuel tube is mitigated by:
a) Keeping the U-Pu fuel in a stable phase at all times;
b) Ensuring that the steel fuel tube wall is sufficiently thick to safely accommodate any low temperature Pu-Fe eutectic that does form;
c) By adding up to 10% Zr to raise the U-Pu-Zr eutectic melting point as high as the silica fuel rod casting forms will permit.

A problem with Zr addition is that it likely will not be uniformly incorporated into the fuel crystal structure. However, uranium-zirconium alloys are phase stable below 607 degrees C.

The melting point of binary couple Pu-Zr increases with increasing Zr fraction. The lowest melting point of this couple is 639.4 degrees C.

Of all the aforementioned temperature constraints the dominant one is the 560 degree C constriant for a phase stable U-Pu alloy. Hence for core fuel stability we need to keep the temperature on the fuel center line less than 560 degrees C.

An important issue in FNR design is the full load temperature difference between the fuel rod center line and the adjacent cooling sodium. A FNR with fuel rods on a square grid has four sodium filled cooling channels associated with each fuel tube. In order to make the FNR durable it must be able to safely function with two of those coolong channels blocked. Under that circumstance the temperature difference between the fuel rod center line and the sodium will be twice its normal value. To prevent fuel damage the maximum allowable fuel center line temperature is 560 degrees C. Hence in order to realize a primary sodium discharge temperature of 460 degrees C at full load the normal temperature difference between the fuel rod center line and the sodium is limited to 50 degrees C. That temperature difference in combination with the fuel dimensions results in a power per unit fuel rod length. From that power we can calculate the required core zone thickness. From that thickness we can calculate the required Pu-239 concentration in the fuel rods.

R = radius from fuel rod center line;
Ro = core fuel rod radius = 4.5 mm;
Tf = fuel rod temperature at radius R from its center line;
Tfc = temperature on the center line of the core fuel rod;
Tfs = temperature on the surface of the core fuel rod;
Kc = thermal conductivity of the core fuel rod material;
L = length of core fuel rod;
H = thermal power per unit volume generated within the fuel rod;
P = total thermal power generated in the core fuel rod;

P = Pi Ro^2 L H
H = P / (Pi Ro^2 L)

Thermal balance at radius R gives:
Pi R^2 L H = 2 Pi R L (- dTf / dR) Kc
R H = 2 (- dTf / dR) Kc
R dR = 2 (-dTf Kc / H or
Ro^2 / 2 = 2 (Tfc - Tfs) Kc / H

Substituting for H gives:
Ro^2 / 2 = 2 (Tfc - Tfs) Kc (Pi Ro^2 L)/ P
1 / 2 = 2 (Tfc - Tfs) Kc (Pi L) / P

Rearranging this equation gives:
(Tfc - Tfs) = (P / L)[1 / 4 Kc Pi)]
which is the temperature difference between the fuel rod center line and the outside surface of the core fuel rod.

Due to formation of fission product gas bubbles in the core fuel:
Kc ~ (Kcb / 2)
where Kcb is the thermal conductivity of fuel material not containing fission product gas bubbles.

For uranium:
Kcb = 27.5 W / m deg C

Kc = (27.5 W / m deg C) / 2
= 13.75 W / m deg C

(Tfc - Tfs) = (P / L)[1 / (4 Kc Pi)]
= (P / L)[1 / (2 Kcb Pi)]

Note that the temperature drop across the fuel tube wall is also proportional to (P / L).

For the fuel tubes:
Let Kt = Thermal Conductivity of fuel tube material
= 26.2 W / m deg C

Let Ts = adjacent sodium temperature
P = [(Tfs - Ts)(surface area) Kt] / (wall thickness)
= [(Tfs - Ts) (2 Pi Rt L) Kt] / (wall thickness)
[P / L] = [(Tfs - Ts) (2 Pi Rt) Kt] / (wall thickness)
(Tfs - Ts) = [P / L][(wall thickness) / (2 Pi Rt Kt)]

Thus the temperature drop from the fuel center line to the liquid sodium is:
(Tfc - Tfs) + (Tfs - Ts)
= {P / L}{[1 / (2 Kcb Pi)] + [(wall thickness) / (2 Pi Rt Kt)]}
{P / L}
= (Tfc - Ts) / {[1 / (2 Kcb Pi)] + [(wall thickness) / (2 Pi Rt Kt)]}
= [2 Pi (Tfc - Ts)] / {[1 / Kcb] + [(wall thickness) / (Rt Kt)]}

This formula allows calculation of the important parameter [P / L].

{P / L} = [2 Pi (Tfc - Ts)] / {[1 / Kcb] + [(wall thickness) / (Rt Kt)]}
= [2 Pi(50 deg C)] /{[1 / (27.5 W / m-deg C)] +[0.035 in / (0.25 in 26.2 W / m deg C)]}
= {[314.159] / [0.036363 + .005343]} W / m
= {[314.159] / [0.041706]} W / m
= 7532.70 W / m

As shown on the web page titled FNR Geometry the total number of active fuel tubes is 299,776.

Hence to achieve 1000 X 10^6 Wt each active fuel tube must output:
1000 X 10^6 W / 299,776 active fuel tubes = 3335.8 W / active fuel tube.

Recall that the maximum heat output from a fuel tube is:
7532.70 W / m

Hence the minimum active fuel tube length is:
[3335.8 W / active fuel tube] / [7532.70 W / m] = 0.4428 m
which is the minimum core zone width for full power operation.

If there is a beyond specification high temperature transient there will be fuel center line phase change but the fuel center line temperature must increase by a further
605 C - 560 C = 45 degrees C
to enable plutomium melting anywhere in the uranium and
607 C - 560 C = 47 C
to enable zirconium mobility anywhere in the uranium.

From above:
R dR = 2 (- dT) [Kc / H]
Integral from R = 0 to R = R of
R dR
= Integral from Tf|R = 0 to Tf |R = R of:
2 (- dT) Kc / H
R^2 / 2 = - 2 [Kc / H] [Tf|R = R - Tf|R = 0]
[Tf|R = 0 - Tf|R = R] = [H / Kc][R^2 / 4]
Tf = Tf|R = 0 -[H / Kc][R^2 / 4]

= Tfc -[H / Kc][R^2 / 4]

Average temperature Tfa is given by: Tfa = Integral from R = 0 to R = Ro of
(Tf 2 Pi R dR) / Pi Ro^2
= Tf|R = 0
- Integral from R = 0 to R = Ro of
[H / Kc][R^2 / 4](2 Pi R dR) / Pi Ro^2
= Tfc
- [H / Kc][1 / 4](2 Pi Ro^4 / 4) / Pi Ro^2
= Tfc
- [H / Kc][1 / 8] (Ro^2 )

Recall that:
H = P / (Pi Ro^2 L)

Tfa = Tfc - (P / L)(1 / Pi Ro^2)(1 / Kc)[1 / 8] Ro^2
= Tfc - (P / L)(1 / 4 Kc Pi)[1 / 2]

Recall that:
(Tfc - Tfs) = (P / L)[1 / 4 Kc Pi)]

Tfa = Tfc - [(Tfc - Tfs) / 2]

Assume 20% Pu, 70% U:
Fp = weight fraction of Pu-239 in fuel
Fu = weight fraction of U-238 in fuel
Then initially:
Fu / Fp = 7 / 2
= 3.5
Fu + Fp = 0.90

Assume that during the fuel cycle there is 15% core rod burnup. Then due to Pu-239 fissioning the core rod Pu-239 initial fraction drops from 20% to 5% but due to new Pu-239 formation the core rod Pu-239 concentration increases by:
[(number of Pu-239 atoms formed) / (number of Pu-239 atoms fissioned)] X 15%
= [(number of U-238 atoms)(U-238 fast neutron capture cross section) / (no. of Pu-239 atoms)(Pu-239 fast neutron fission cross section)] X 15%
= [(70)(0.25 b) / (20)(1.7 b)] X 15%
= 7.72%

Hence the final Pu-239 concentration is about:
(5% + 7.72%) = 12.72%

We must check that the reactor core can remain critical at this Pu-239 fraction. There will be some neutron absorption by fission products, fuel bundle steel and liquid sodium.

The nominal fuel tube ID is:
0.500 inch - 2 (.035 inch) = 0.430 inches
= 10.92 mm.
Note that this is a standard steel tube ID. The reactor core fuel rods are 9.00 mm nominal diameter to 9.4 mm maximum diameter metallic rods that loosely slide into the steel fuel tubes. The 9.00 mm is a standard silica tube ID. Note that the core rods are thinner than the blanket rods which permits a longer core rod than a blanket rod. For a core rod to slide into the fuel tube the maximum deviation of a fuel tube from being dead straight is:
10.92 mm - 9.4 mm = 1.5 mm over a length of 0.60 m.

The blanket fuel rods are nominally 10 mm in diameter. For a blanket rod to slide into a fuel tube the maximum deviation of a fuel tube from being dead straight is:
10.92 mm - 10.0 mm = 0.92 mm over a fuel tube length of 0.30 m

The nominal core fuel smear density is:
[9.00 / 10.92]^2 = 0.6792

The maximum core rod smear density is:
[9.40 / 10.92]^2 = 0.740

This smear density allows for experimentally observed fuel swelling. Inside the steel fuel tubes, along with the fuel rods is liquid sodium, which provides a good thermal contact between the fuel rods and the steel fuel tubes independent of the core fuel rod diameter and which chemically absorbs the otherwise gaseous and corrosive fission products iodine and bromine.

The 9.0 mm OD core fuel rods are fabricated with smoothly rounded hot ends and flat cool ends to provide a recess adjacent to a ceramic ball where liquid sodium collects. In a prompt neutron critical condition this location becomes very hot causing rapid vaporization of the liquid sodium in this recess. The resulting sodium vapor pressure tends to disassemble the core fuel. Ceramic balls
0.420 inch OD = 10.67 mm OD
at the core fuel rod hot ends further improve the fuel disassembly characteristics. A prompt neutron critical will cause a sudden heat pulse which will vaporize the Na and any Cs. The ceramic ball OD must closely match the tube ID. The vapor pressure will tend to propel fuel rods above the ceramic ball upward toward the fuel tube plenum. The same effect tends to push fuel rods below the ball downward. Since the ceramic balls are at the bottom of the core rods for fixed fuel bundles and at thetop of the core rods for movable fuel bundles, the net effedt of this sodium vaporization is to further separate the core rods of the fixed and movable fuel bundles. The ceramic ball position is an important safety feature, so it is necessary to check during fuel assembly that the ceramic balls are properly dimensioned and positioned.

The blanket fuel rods are initially 90% uranium plus 10% zirconium. Zirconium is used to prevent formation of a Pu-Fe eutectic. Since fissioning in the blanket rods is minimal their nominal initial outside diameter is 10.00 mm, which is about 91.57% of the initial fuel tube ID. Note that 10.00 mm is a standard commercial silica tube ID. There must be sufficient clearance between the blanket fuel rod OD and the fuel tube ID to allow for fabrication tolerances and for a loose sliding fit. The objective is to make the fuel rods located above the ceramic balls move toward the plenum like blowgun projectiles. Projectile braking is achieved by compression of the gas trapped in the fuel tube plenum.

In order for the sodium vapor to lift the weight of the stack of fuel rods above it The vapor pressure must reach about:
(1.8 m + 0.6 m) / (0.646 m Atmosphere) = 4 atmospheres.
The sodium temperature must reach about 1000 degrees C.

Thus the practical effect of the sodium vapor expansion due to prompt neutron criticality is to blow the core rods in the fixed fuel bundles toward the plenum spaces. Note that for this mechanism to work reliably the core rod should be a single piece. Otherwise some of the core fuel rods in a moveable fuel bundle could be blown in the wrong direction.

On average the core fuel rods are initially by weight 10% zirconium, 20% plutonium and 70% uranium by weight. The density of zirconium is:
6.52 gm / cm^3
The density of plutonium is about:
19.8 gm / cm^3
The density of uranium is about 18.9 gm / cm^3

Vz = volume of zirconium in a core rod
Vp = volume of plutonium in a core rod
Vu = volume of uranium in a core rod.
Mz = mass of zirconium in a core rod
Mp = mass of plutonium in a core rod
Mu = mass of uranium in a core rod

Total volume V is given by:
V = Vz + Vp + Vu

Core fuel rod mass M is given by:
M = Mz + Mp + Mu

Mz = 0.1 M
Mp = 0.2 M
Mu = 0.7 M

Mz / Vz = 6.52 gm / cm^3
Mp / Vp = 19.8 gm / cm^3
Mu / Vu = 18.9 gm / cm^3

The average initial core rod density is:
M / V = (Mz + Mp + Mu) / (Vz + Vp + Vu)
= (Mz + Mp + Mu) / ((Mz / 6.52) + (Mp / 19.8) + (Mu / 18.9))
= M / [(0.1 M / 6.52) + (0.2 M / 19.8) + (0.7 M / 18.9)]
= 1 / ((0.1 / 6.52) + (0.2 / 19.8) + (0.7 / 18.9))
= 1 / (.015337 + .010101 + .037037)
= 1 / .062475
= 16.006 gm / cm^3

The density of mercury at room temperature is:
13.6 gm / cm^3

Hence the maximum length of a core rod made by vacuum casting is:
(13.6 / 16.006) X 760 mm = 646 mm
= 0.646 m

Thus it is practical to design a FNR with core rods about 0.60 m long.

The preferred method of fuel rod fabrication is vacuum casting from a liquid alloy melt. A silica tube is used as the mold. The lower end of the tube is dipped in the liquid alloy and a vacuum is applied to the upper end of the tube. If the furnace pressure is one atmosphere the alloy will rise in the tube. The maximum height of that rise is:
76 cm X (density of mercury) / (density of alloy)
= 76 cm X (13.6 g / cm^3) / (16.006 gm / cm^3) = 64.57 cm

This fuel rod fabrication method limits the maximum posssible length of each fuel rod.

However, this length is further reduced by the vapor pressure at the alloy melt temperature of impurities in the alloy melt. There is a further issue that it is desirable to be able to easly sort used fuel rods based on their length. The blanket rods are much shorter than the core fuel rods. For economy in blanket rod formation try to get two blanket rods out of each cast. Hence the blanket rods should each be made 29.0 cm, 31.0 cm, 29.5 cm, 30.5 cm, 28.5 cm, 31.5 cm long. Note that the average blanket rod length is 30 cm. The core rods can each be initially 60 cm long.

The core fuel rod height (0.60 m) must be enough that the reactor can go critical when the Pu weight fraction in the fuel is as low as 12.7%. The core fuel rods must provide for sufficient heat transfer to the liquid sodium at partial fuel bundle overlap when the average Pu weight fraction in the fuel is as high as 20%. The amount of heat that can be removed from the fuel rod stack is limited by the chimney effect operating through the differential temperature of:
(460 C -410 C) = 50 C
over a chimney height of 3.65 m.

At the bottom of each active fuel tube is a bottom plug with an overall length of 0.10 m and a fuel tube overlap of 0.05 m. Then for all active fuel tubes there are 6 X 0.30 m long X 10.0 mm diameter blanket rods initially consisting of 90% uranium and 10% zirconium. Then for the fuel tubes used in fixed fuel bundles there is a close fitting ceramic ball. Then 1 X 0.60 m long X 9.00 mm diameter core rod, initially consisting of 70% uranium-20% plutonium-actinide-10% zirconium alloy. Then for the fuel tubes in movable fuel bundles there is a close fitting ceramic ball. Then for all the active fuel tubes another 6 X 0.30 m long X 10.0 mm diameter blanket rods initially consisting of 90% uranium and 10% zirconium.

The core rods should not be too long to avoid tying up the available inventory of fissile material.

Each fuel tube contains sufficient liquid sodium to cover the core and blanket rods up to at least a height of 4.3 m to ensure good thermal contact between the fuel rods and the enclosing fuel tube.

The steel fuel tube initial ID is 0.430 inches. The initial core fuel rod diameter is 9.00 mm. Hence the smear density (Till & Yang P. 123) is:
[9.00 mm / (.430 inch X 25.4 mm / inch)]^2 = 0.679017

In terms of allowance for core fuel rod swelling:
1 / [(0.679017)^0.5] = 1.21355
21.355% linear core rod swelling before the core fuel causes significant hoop stress on fuel tube walls.

The mass of each core fuel rod is given by:
Pi X (9.00 X 10^-3 m / 2)^2 X 0.60 m / rod X 16.006 g / cm^3 X 10^6 cm^3 / m^3 X 1 kg / 10^3 g
= 0.610954 kg / core rod

Average mass Mu of U-238 in each core fuel rod is:
Mu = .7 (0.610954 kg)
= .4276678 kg

Average mass of Pu in each core fuel rod is:
Mp = 0.2 (0.610954 kg)
= 0.1221908 kg

Mass Mz of Zr in each core fuel rod is:
Mz = 0.1 (0.610954 kg)
= 0.0610954 kg

This plutonium can be obtained by reprocessing of spent CANDU fuel.

The amount of plutonium readily available from spent CANDU fuel is about:
0.0038 X 60,000 tonnes = 228 tonnes. Hence there is presently enough plutonium available to start about:
228 tonnes / (45 tonnes / reactor) = 5 FNRs.
It is clear that in FNR planning a very important objective is breeding additional plutonium for starting future FNRs.

The blanket rods must slide easily into the fuel tubes but are subject to much less swelling because their only fissionable content comes from breeding. Hence the initial blanket rod diameter is:
10.000 mm

The blanket rods rely on:
0.42 inch OD = 10.668 mm OD
cool end beads for high pressure sodium vapor to propel the fixed fuel bundle blanket rods into the fuel tube plenums on the occurance of prompt neutron criticality.

The blanket fuel rods should have rounded hot ends to create small recesses for liquid sodium to collect. A the point where the hot end of a core fuel rod touches the hot end of a blanket furel rod the sodium in this pocket should vaporize to cause fuel disassembly.

The blanket fuel rods are nominally 10% zirconium, 90% uranium by weight.
The density of zirconium is:
6.52 gm / cm^3
The density of uranium is about 18.9 gm / cm^3

Vzb = volume of zirconium in a blanket rod
Vub = volume of uranium in a blanket rod.
Mzb = mass of zirconium in a blanket rod
Mub = mass of uranium in a core rod

Total volume V is given by:
Vb = Vzb + Vub

Blanket fuel rod mass Mb is given by:
Mb = Mzb + Mub

Mzb = 0.1 Mb
Mu = 0.9 Mb

Mzb / Vzb = 6.52 gm / cm^3
Mub / Vub = 18.9 gm / cm^3

The average blanket rod density is:
Mb / Vb = (Mzb + Mub) / (Vzb + Vub)
= (Mzb + Mub) / ((Mzb / 6.52) + (Mu / 18.9))
= Mb / ((0.1 Mb / 6.52) + (0.9 Mb / 18.9))
= 1 / ((0.1 / 6.52) + (0.9 / 18.9))
= 1 / (.015337 + .047619)
= 1 / .062956
= 15.884 gm / cm^3

The mass of an average blanket fuel rod is given by:
Pi X (10.00 X 10^-3 m / 2)^2 X 0.30 m / rod X 15.884 g / cm^3 X 10^6 cm^3 / m^3 X 1 kg / 10^3 g
= 0.3742579 kg / blanket rod

Mass Mub of U-238 in each average blanket fuel rod is:
Mub = .9 (0.3742579 kg)
= .336832 kg

Mass of Zr in each average blanket fuel rod is:
Mzb = 0.1 (0.3742579 kg)
= 0.03742579 kg

This web page last updated June 28, 2022

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