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**INTRODUCTION:**

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.

**CORE FUEL ROD ALLOY:**

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 mobile 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:

416 rods (X%) + 280 rods (Y%) = (416 + 280)rods (20%)

Equal amounts of Pu-239 in the upper and lower core zones gives:

416 rods(X%) = 280 rods (Y%)

Thus combining the above two equations gives:

2 X 416 rods (X%) = 696 rods (20%)

or

**X%** = (696 /832) (20%)

= **16.73%**

Similarly:

2 X 280 rods (Y%) = 696 rods (20%)

or

**Y%** = (696 / 560) 20%

= **24.86%**

Thus the movable active fuel bundles need core rods with 24.86% Pu-239 whereas the fixed active fuel bundles need core rods with 16.73% 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.

**FUEL ROD INITIAL ALLOY:**

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.

**Zr REDISTRIBUTION:**

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.

**FUEL TUBE INTERACTION:**

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**.

**HEAT FLUX AND TEMPERATURE PROFILE:**

Define:

T = fuel rod temperature at radius R from its center line;

Tc = temperature on the center line of the core fuel rod;

Ts = temperature on the surface of the core fuel rod;

R = a radius from the fuel rod center line;

Ro = core fuel rod radius = 4.5 mm;

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;

or

H = P / (Pi Ro^2 L)

Thermal balance gives:
Pi R^2 L H = 2 Pi R L (- dT / dR) Kc

or

R H = 2 (- dT / dR) Kc

or

R dR = 2 (- dT) Kc / H
or

Ro^2 / 2 = 2 (Tc - Ts) Kc / H

Substituting for H gives:

Ro^2 / 2 = 2 (Tc - Ts) Kc (Pi Ro^2 L)/ P

or

1 / 2 = 2 (Tc - Ts) Kc (Pi L) / P

Rearranging this equation gives:

**(Tc - Ts) = (P / L)[1 / 4 Kc Pi)]**

which is the temperature difference between the inside and outside 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

Hence:

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

= **13.75 W / m deg C**

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

On the web page titled: FNR Fuel Tubes

the operating temperature drop across the fuel tube wall is chosen to be 8 deg C which gives a fuel tube thermal flux of:

**230,037 W / m^2**

The corresponding value of (P / L) is:

**P / L** = 230,037 W / m^2 X Pi X 0.500 inch X .0254 m / inch

= **9178 W / m**

Hence the temperature drop between the fuel rod center line and the fuel rod outer surface is:

**(Tc - Ts)** = (P / L)[1 / (4 Kc Pi)]

= (9178 W / m) [1 / ( 4 X 13.75 W / m deg X Pi)]

= **53.11 deg C**

Since the maximum fuel center line temperature is 560 degrees C the maximum liquid sodium coolant working temperature is:

560 deg C - 53.11 deg C - 8 deg C

= 498.88 deg C

which is the design peak liquid sodium working temperature in a power FNR. To allow for the sodium coolant boundary layer and for component tolerances it is probably prudent to design **the FNR to operate at a primary liquid sodium top surface temperature at full rated power of about 490 degrees C.**

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.

**FIND TEMPERATURE AS A FUNCTION OF R:**

From above:

R dR = 2 (- dT) [Kc / H]

or

Integral from R = 0 to R = R of

R dR

= Integral from T|R = 0 to T |R = R of:

2 (- dT) Kc / H

or

R^2 / 2 = - 2 [Kc / H] [T|R = R - T|R = 0]

or

[T|R = 0 - T|R = R] = [H / Kc][R^2 / 4]

or

T = T|R = 0 -[H / Kc][R^2 / 4]

Average temperature Ta is given by:
Ta = Integral from R = 0 to R = Ro of

(T 2 Pi R dR) / Pi Ro^2

= T|R = 0

- Integral from R = 0 to R = Ro of

[H / Kc][R^2 / 4](2 Pi R dR) / Pi Ro^2

= Tc

- [H / Kc][1 / 4](2 Pi Ro^4 / 4) / Pi Ro^2

= Tc

- [H / Kc][1 / 8] (Ro^2 )

Recall that:

H = P / (Pi Ro^2 L)

Hence:

Ta = Tc - (P / L)(1 / Pi Ro^2)(1 / Kc)[1 / 8] Ro^2

= Tc - (P / L)(1 / 4 Kc Pi)[1 / 2]

Recall that:

(Tc - Ts) = (P / L)[1 / 4 Kc Pi)]

Hence:

**Ta = Tc - [(Tc - Ts) / 2]**

**PLUTONIUM DEPLETION:**

Let:

Fp = weight fraction of Pu-239 in fuel

Fu = weight fraction of U-238 in fuel

Then initially:

Fu / Fp = 7 / 2

= 3.5

and

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.

**FUEL ROD DIAMETER:**

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. The nominal core fuel smear density is:

[9.00 / 10.92]^2 = **0.6792**

The maximum 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.

**FUEL ROD TAPERING AND BEADING IN FIXED FUEL BUNDLES:**

The 9.0 mm OD core fuel rods used in fixed fuel bundles are fabricated to have:

short tapered ends closest to the center of the nuclear reaction to provide pockets where liquid sodium collects. In a prompt neutron critical condition this location is where the fuel is hotest, causing vaporization of the liquid sodium in these pockets. The resulting sodium vapor pressure tends to disassemble the core fuel. This effect is enhanced by forming a larger diameter bead on the cooler end of each core fuel rod. The

0.420 inch OD = 10.67 mm OD

beads on the fuel rod ends furthest from the nuclear reaction improve the fixed fuel bundle core fuel disassembly characteristics on sudden application of Cs and/or Na vapor pressure as will occur in a prompt neutron critical condition. The bead OD must be tightly controlled. The beads will cause high pressure sodium vapor produced in or near the nuclear reaction center to propel the fixed fuel bundle fuel rods upward toward the fuel tube plenum. The tapers amd beads are a critical safety features and it is important to check during fuel assembly that the fuel rods with tapers and beads are properly dimensioned and oriented. Note that the tapers and beads must be axially short so that the taper does not diminish thermalpower output and so that bead swelling due to fast neutron action does not cause excessive pressure on the inside of the fuel tube.

The tapering and beading is achieved by brief controlled melting of the ends of each core fuel rod.

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 fixed fuel bundle blanket fuel rods separate like blowgun projectiles. Projectile braking is achieved by compression of the gas trapped in the fuel tube plenum.

**CORE FUEL ROD DENSITY:**

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**

**FUEL ROD VACUUM CASTING:**

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 should be significantly longer than the core fuel rods. However, during operation the core fuel rods grow in length. Hence the blanket rods should each be made 35.0 cm, 35.5 cm, 36.0 cm, 36.5 cm and 37.0 cm long. The core rods can each be initially 30 cm long with the objective of realizing 2 core rods from each vacuum casting of core rod material.

**CORE FUEL ROD LENGTH:**

The core fuel rod stack 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 rod stack consists of two 0.30 m long core fuel rods. 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:

(490 C - 340 C) = **100 C**

over a chimney height of 4.0 m.

**FUEL ROD VERTICAL DISTRIBUTION:**

At the bottom of each active fuel tube is a bottom plug with an overall length of 0.05 m and a fuel tube overlap of 0.025 m. Above this plug are **5 X 0.35 m long X 10.0 mm diameter blanket rods** oriented beads down initially consisting of 90% uranium and 10% zirconium, **2 X 0.30 m long X 9.00 mm diameter core rods** oriented beads at opposite end from nuclear reaction, initially consisting of 70% uranium-20% plutonium-actinide-10% zirconium alloy and then another **5 X 0.35 m long X 10.0 mm diameter blanket rods** oriented beads up initially consisting of 90% uranium and 10% zirconium. Note that for future role identification purposes the blanket rods should actually be 0.35 m, 0.355 m, 0.36 m, 0.365 m and 0.37 m long. However, each stack of five such rods is 1.8 m long.

**FISSILE MATERIAL CONSERVATION:**

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

**SODIUM CONTENT:**

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.

**CORE FUEL RODS:**

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

or

** 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 0.30 m / rod X 16.006 g / cm^3 X 10^6 cm^3 / m^3 X 1 kg / 10^3 g

= **0.305477 kg / core rod**

Hence:

Average mass Mu of U-238 in each core fuel rod is:

**Mu** = .7 (0.305477 kg)

= **.213834 kg**

Average mass of Pu in each core fuel rod is:

**Mp** = 0.2 (0.305477 kg)

= **0.0610954 kg**

Mass Mz of Zr in each core fuel rod is:

**Mz** = 0.1 (0.305477 kg)

= **0.0305477 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.

**BLANKET FUEL RODS:**

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 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 each blanket fuel rod is given by:

Pi X (10.00 X 10^-3 m / 2)^2 X 0.36 m / rod X 15.884 g / cm^3 X 10^6 cm^3 / m^3 X 1 kg / 10^3 g

= **0.4491095 kg / blanket rod**

Hence:

Mass Mub of U-238 in each blanket fuel rod is:

**Mub** = .9 (0.4491095 kg)

= **.4041985 kg**

Mass of Zr in each blanket fuel rod is:

**Mzb** = 0.1 (0.4491095 kg)

= **0.04491095 kg**

This web page last updated April 5, 2021

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