XYLENE POWER LTD.

FNR REACTIVITY

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

FNR PHYSICS:
A FNR core zone, where the fixed and movable fuel bundles overlap, produces an excess of neutrons which diffuse out of the core zone and into the adjacent blanket zones. The blanket zones absorb these excess neutrons and produce more fissile fuel atoms. When the core fuel is new about half of the fission neutrons generated in the core zone diffuse into the blanket zones. As the core fuel ages the thickness of the core zone is gradually increased to maintain the FNR setpoint temperature and the fraction of fission neutrons that diffuse from the core zone into the blanket zones gradually decreases. When this fraction approaches zero it is no longer possible to breed new fissile atoms or maintain the desired reactor temperature setpoint To, so the core fuel must be replaced.

Typically core fuel replacement is required after about 15% of core fuel mass has transmuted to become fission products.
 

CORE ZONE THICKNESS:
On average each U-235 fission produces Gu = 2.6 neutrons and each Pu-239 fission produces Gp = 3.1 neutrons. In order for criticality to be maintined on average one of these fission neutrons must cause fission of a fissile atom. Since most of the fissile atoms are in the core zone generally that fission must happen before the remaining neutrons leave the core zone.

Once a neutron leaves the core zone the probability of it being captured by U-238 within the adjacent blanket zone must be very high. There is some probability of a neutron being captured by a Pu-239 atom in the blanket zone, but that probability is barely sufficient to sustain the chain reaction in that zone.

In a FNR started with U-235 fuel (Gu = 2.6) there is very little margin for neutron loss. Plutonium (Gp = 3.1) is a much more practical FNR fissile fuel.

Roughly speaking, if La is the average total neutron travel distance from neutron emission to fission in travelling distance La the neutron goes through about 50 scattering events so its linear travel distance is much smaller than La.
 

REACTIVITY:
A FNR must simultaneously meet two important reactivity goals. Firstly, subject to fuel geometry modification, the FNR must potentially provide a reactivity:
R = 0
at temperature:
T = To
throughout the FNR operating life in spite of fuel aging. Secondly, at all times the inequality:
(dR / dT) < 0
must be met so that the fisssion chain reaction shuts down as the FNR temperature T rises above its temperature setpoint To.

These criteria lead to the fuel tube grid center to center spacing of (9 /16) inch which is a critical dimension in FNR design.

At any instant in time t and at temperature T the number of free neutrons N in a FNR can be expressed as:
dN = N R dt
or
dN / N = R dt
or
Ln(N / No) = R (t - to)
or
[N / No] = Exp[R (t - to)]
or
N = No Exp[R (t - to)]
where:
No = number of free neutrons N at time:
t = to
and
R = FNR net reactivity which is a function of FNR temperature T.

dN / dt = R No Exp[R (t - to)]

Note that when:
R = 0
then:
dN / dt = 0
which implies constant FNR thermal power output.

The net reactivity R of a FNR is a mathematical indication of the rate of growth or rate of decline of the FNR's free neutron population N. If the net reactivity R is positive the free neutron population N will grow exponentially over time. If the net reactivity R is negative the number of free neutrons N will decay exponentially over time. If the net reactivity R is zero the rate of free neutron production exactly equals the rate of free neutron loss so that the free neutron population remains constant over time. Hence at steady state the FNR thermal power output is constant and the rate of heat removal from the FNR by the liquid sodium coolant exactly equals the FNR thermal power output.
 

WEBPAGE DIRECTION:
This web page sets out the conditions for FNR thermal stability and safety. It is shown that Pu-239 is a much better FNR fissile fuel than U-235.

Another approach is to use a different fuel fabrication methodology in order to minimize the positive void coefficient effect of sodium both inside and outside the active fuel tube. This issue is discussed at Breed and Burn Reactor theory and at Cross Comparison of Fast Reactor Concepts with Various Coolants.
 

BASIC FNR OPERATION:
A FNR should spontaneously adopt an average fuel temperature:
T = To
where To is the average fuel temperature at which:
R = 0.

Provided that:
[(dR / dT)|T = To] < 0,
when the FNR average fuel temperature T rises above the setpoint temperature To the free neutron concentration in the reactor core zone decays which shuts down the fission reaction and hence the thermal power output. If the average fuel temperature T falls below the setpoint temperature To the free neutron concentration in the reactor core zone grows which increases the fission power thermal power output. The system should quickly converge to a steady state condition where the fission thermal power exactly equals the rate of heat extraction by the coolant and
T = To.
 

FNR THERMAL CONTROL LOOP:
In normal steady state operation the reactivity R of a FNR should be zero and:
dR / dT < 0
so that the FNR maintains a constant temperature T = To independent of the thermal power load.

If there is a small increase in thermal load there will be a brief drop in FNR fuel temperature T making:
T - To < 0
which should increase the FNR reactivity R. The consequent increase in free neutron population will increase the FNR thermal power output sufficiently to restore the FNR fuel temperature T to its setpont To, which will cause cause the reactivity R to return to zero.

Similarly, if there is a small decrease in thermal load then there will be a brief increase in FNR fuel temperature T causing:
T - To > 0
which should reduce the FNR reactivity R. The consequent decrease in free neutron population will decrease the FNR thermal power output sufficiently to restore the FNR fuel temperature T to its setpont To which will cause cause the reactivity R to return to zero.

Note that the quantity [- dR / dT] sets the response rate of the FNR control loop.

An issue to check is that the fission power quickly converges to a steady value rather than oscillating. In this respect delayed fission neutrons may play an important role.

A complicating issue in a FNR is that while the thermal response of the core fuel is almost instantaneous, the thermal response of the coolant liquid sodium lags the core fuel by a few seconds, and the thermal response time of the steel and blanket fuel lags the thermal response of the sodium by yet more seconds.

Thermal expansion of the fissile fuel injects -ve reactivity into R whereas thermal expansion of the sodium and steel injects positive reactivity onto R. For FNR stability we must ensure that the -ve reactivity injection caused by thermal expansion of the core fuel is larger than the total positive reactivity injection caused by thermal expansion of the other FNR components. Particularly problematic is sodium that due to its relatively large thermal coefficient of expansion injects significant positive reactivity on a rise in temperature. Sodium is potentially much more of a problem if it is permitted to approach its boiling point.

In order to guarantee a sustained negative value of [dR / dT] the concentration of fissile atoms, the fissile atom fast neutron fission cross section and the fissile fuel thermal coefficient of expansion must all be sufficient. As the average fissile fuel atom concentration in the core zone decreases, FNR control stability potentially becomes more of a problem. This issue can potentially limit the fuel cycle time (time between successive fuel reprocessings).

An important issue in FNR design is ensuring sufficient ongoing sodium circulation and making allowance for potentially obstructed liquid sodium cooling channels.
 

FNR PANCAKE MODEL:
A FNR's core zone can be mathematically modelled as a flat pancake of core fuel, sodium, iron and chromium atoms. At steady state and hence at constant thermal power output the net reactivity R = 0. There is a density of free neutrons that is highest at the pancake thickness center and declines along the vertical axis toward the pancake faces. At a constant thermal power the density of free neutrons at any point along the vertical axis is less than the free neutron density at the pancake thickness center.

In steady state operation this pancake emits a fraction of its fission neutrons out its flat faces. Assuming that the pancake has constant uniform atomic concentrations, the thicker the pancake the smaller the fraction of fission neutrons that are emitted via the pancake flat faces. Hence a key parmeter to control is the pancake thickness which is equal to the axial overlap between the FNR's movable and fixed fuel rods.

Adjusting the pancake thickness (the movable and fixed fuel rod overlap) at a particular temperature adjusts the emitted fission neutron flux fraction. At steady state, where net reactivity R = 0, a certain fraction of the fission neutrons are transmitted from the core to the blanket via the pancake surfaces. The FNR is designed so that decreasing its temperature causes thermal contraction which increases R and hence increases the thermal power output. The corresponding increase in FNR temperature causes thermal expansion which passively restores the condition R = 0.
 

FNR BASIC CRITICALITY REQUIREMENTS:
1) When the movable fuel bundles are fully inserted into the matrix of fixed fuel bundles and the core fuel is nearly fully depleted the reactor will still be critical. This is the depleted fuel condition. However, the reactor must still be thermally stable in the depleted fuel condition.

2) When the movable fuel bundles are 1.1 m withdrawn and the core fuel is new the reactor must be dependably sub-critical. This is the new fuel cold shutdown condition. In this condition the movable active fuel bundles form a lower core fuel layer and the fixed active fuel bundles form an upper core fuel layer. The two layers are separated by about 0.5 m of blanket material. Both the upper core fuel layer and the lower core fuel layer and their combination must be subcritical.

This requirement for subcriticality in the upper and lower core fuel zones limits the maximum Pu-239 concentrations and hence the fissile material weight fractions in the core fuel rods.
 

CORE ZONE REACTIVITY OVERVIEW:
The function of the FNR core is to maintain a nuclear chain reaction via fissioning of Pu-239, Pu-240 and other transuranium actinides while emitting surplus neutrons to the FNR blanket. A fundamental question from a practical reactor engineering perspective is: "What is the proper range of core zone thickness?"

When the core fuel is new there is a surplus of fission neutrons and about half of the fission neutrons should diffuse out of the core zone and into the blanket zone. When the core fuel is old the core zone will be thicker and most of the fission neutrons remain in the core zone.

An important technical issue that must be addressed to answer the aforementioned core zone thickness question is: "What is the ratio of neutron random walk path length to core zone thickness?" This path length will vary as the core fuel rods age causing the average Pu-239 concentration in the core zone to change.

Neutrons diffuse through the core zone by scattering. At each scatter a neutron loses a small fraction of its kinetic energy. Between successive scatters the number of neutrons slightly reduces due to neutron absorption. Our first concern is that at about (1 / 3) of the neutrons that are released in the core zone must be absorbed by Pu in the core zone causing fission reactions to maintain the overall chain reaction. Hence to sustain reactor criticality the neutron random walk path length in the core zone must be long enough to cause 33% absorption by fissile Pu atoms.

When the core fuel is new about (1 / 6) of the fission neutrons are absorbed by U-238 in the core zone. As the core fuel ages this fraction gradually rises to about (1 / 2).

Neutrons that are not absorbed in the core should be almost totally absorbed in the blanket zone.

The required blanket thickness is relatively independent of reactor power.

The average concentrations of Pu-239 and U-238 atoms in the core are functions of the core fuel design. These concentrations determine the rate of absorption of neutrons along a neutron random walk path.
 

BULK REACTIVITY:
Bulk reactivity Rb is the theoretical reactivity that would pertain if the FNR core material extended infinitely far in each direction. In a real operating FNR the bulk reactivity Rb must always be positive.

For thermal power stability with prompt fast neutrons it is esssential that:
[dRb / dT] < 0.
This condition imposes important constraints on the FNR's core fuel design.
 

NET REACTIVITY:
At normal reactor operating conditions R = 0, and neutron leakage out the core zone pancake flat faces effectively reduces the fission neutron multiplication ratio G. An important safety issue is ensuring that:
[dR / dT] < 0
 

REACTIVITY DESIGN TARGETS:
Important facets of FNR design are realizing a FNR net reactivity R that:
1) Is mechanically adjustable to compensate for long term fuel aging;
2) In the normal FNR operating temperature range the net reactivity R must instantly decrease with increasing temperature;
3) At onset of a sudden prompt neutron critical condition the fuel temperature rapidly rises causing thermal expansion/vaporization of adjacent in-fuel tube liquid sodium. The resulting liquid sodium pressure wave causes instantaneous axial fuel disassembly which injects negative reactivity that prevents prompt critical equipment damage. Note that the fixed fuel rods have top beads which enhance the fuel stack disassembly in a prompt critical condition.
 

CORE ZONE THICKNESS ADJUSTMENT:
Major changes in reactivity are realized by mechanical axial repositioning of movable fuel rod bundles with respect to fixed fuel rod bundles, which changes the movable and fixed fuel bundle overlap. That repositioning is done by changing the insertion of the movable fuel bundles in the matrix of fixed fuel bundles. The change in fuel rod overlap changes the number of atoms in the pancake shaped core zone.

An issue of great importance is the instantaneous decrease in FNR net reactivity with increasing fuel temperature. This decrease may rely on use of plutonium as a fissile material which has a large Thermal Coefficient of Expansion (TCE).
 

ASSEMBLY DETAIL:
Note that the bottom end of each active fixed bundle core fuel rod is positioned at the top of the lower blanket fuel rod stack. Each such core fuel rod has a top bead.

Note that the top end of each movable bundle core fuel rod is positioned at the bottom end of the upper blanket fuel rod stack. These fuel rods have no top bead.
 

NUMBER OF FREE NEUTRONS N:
Note that for a pancake shaped FNR with thick top and bottom blanket zones and a vertical axis Z which is zero at the middle of the reactor core zone then to a good approximation:
N = Integral from Z = - infinity to Z = + infinity of:
n(Z) A dZ
where:
n(Z) = free neutron concentration as a function of vertical position Z,
and
A = cross sectional area of FNR in the X-Y plane.

The FNR described herein consists of a pancake shaped reactor core zone with a geometry such that with new core fuel about half of the fission neutrons produced diffuse out of the core zone and into the surrounding reactor blanket. The temperature independence of this neutron diffusion flux affects the FNR reactivity temperature dependence. Significant issues are the effect of the high thermal coefficient of expansion of sodium and the sodium volume fraction required to provide sufficient cooling with natural circulation of liquid sodium.
 

FISSION MATHEMATICS:
Each neutron absorbed by a fissionable Pu-239 atom causes a fission which produces on average G neutrons where:
G ~ 3.1

Consider an initial burst of neutrons N propagating in the core zone at velocity Vn.

For an absorbing species a in the core zone:
dN = N (- Na Sigmaa dX)
where:
dX = distance along a neutron propagation path
Na = atomic concentration of absorbing species a
and
Sigmaa = absorption cross section of absorbing species a

For a fissioning species f in the core zone:
dN = N (Nf Sigmaf (G - 1) dX)
where:
Nf = atomic concentration of fissioning species f
and
Sigmaf = cross section of fissioning species f

Neutrons which exit the core zone in effect reduce G.

Hence:
dN = N [(Nf Sigmaf (G - 1) dX) - (Na Sigmaa dX) - (Nb Sigmab dX)]
= N [(Nf Sigmaf (G - 1)) -(Na Sigmaa) - (Nb Sigmab)] dX

dX = Vn dt
where Vn is the neutron velocity

Hence:
Ln[N / No] = [(Nf Sigmaf (G - 1))) -(Na Sigmaa) - (Nb Sogmab)] Vn (t - to)
or
N = No Exp{[(Nf Sigmaf (G - 1)) -(Na Sigmaa) - (Nb Sigmab)] Vn (t - to)}
= No Exp{R (t - to)}
which gives:
R = [(Nf Sigmaf (G - 1)) - (Na Sigmaa) - (Nb Sigmab)] Vn

Note that in a fast neutron reactor Vn is large which causes N to grow or shrink very quickly.

At the FNR operating point:
R = 0
or
(Nf Sigmaf (G - 1)) -(Na Sigmaa) - (Nb Sigmab) = 0
or
(Nf Sigmaf (G - 1)) = (Na Sigmaa) + (Nb Sigmab)
 

MINIMUM BULK REACTIVITY:
Note that if:
(Nf Sigmaf (Gm - 1)) - (Na Sigmaa) - (Nb Sigmab) > 0
is not satisfied then there can be no nuclear reaction because any level of external neutron leakage prevents maintenance of fission criticality. For Pu-239 Sigmaf and (Gm - 1) are both larger than for U-235. Hence for Pu-239 the required Nf value is much smaller than for U-235. This issue has important implications in terms of reducing the fissile start fuel requirement for FNRs.

In normal reactor operation: R = 0 = (Nf Sigmaf (G - 1)) Vn - [(Na Sigmaa) + (Nb Sigmab)] Vn
or
1 = (Na Sigmaa) + (Nb Sigmab) / (Nf Sigmaf (G - 1))
where due to neutron leakage G < Gm.

In general form this normal reactor operating equation is:
1 = [Np Sigmaap + Nu SigmaaU + Nz Sigmaaz + Ns Sigmaas + Ni Sigmaai + Nc Sigmaac] / [ Npf Sigmafp (G - 1)]
 

CORE ZONE REACTIVITY OVERVIEW:
The function of the FNR core is to maintain a nuclear chain reaction via fissioning of Pu-239, Pu-240 and other transuranium actinides while emitting surplus neutrons to the FNR blanket. A fundamental question from a practical reactor engineering perspective is: "What is the proper range of core zone thickness?"

When the core fuel is new there is a surplus of fission neutrons and about half of the fission neutrons should diffuse out of the core zone and into the blanket. When the core fuel is old the core zone is thicker and most of the fission neutrons remain in the core zone.

An important technical issue that must be addressed to answer the aforementioned core zone thickness question is: "What is the ratio of neutron random walk path length to core zone thickness?" This path length will vary as the core fuel rods age causing the average Pu-239 concentration in the core zone to change.

Neutrons diffuse through the core zone by scattering. At each scatter a neutron loses a small fraction of its kinetic energy. Between successive scatters the number of neutrons slightly reduces due to neutron absorption. Our first concern is that at about (1 / 3) of the neutrons that are released in the core zone must be absorbed by Pu in the core zone causing fission reactions to maintain the overall chain reaction. Hence to sustain reactor criticality the neutron random walk path length in the core zone must be long enough to cause 33% absorption by fissile Pu atoms.

When the core fuel is new about (1 / 6) of the fission neutrons are absorbed by U-238 in the core zone. As the core fuel ages this fraction gradually rises to about (1 / 2).

Neutrons that are not absorbed in the core should be almost totally absorbed in the blanket zones.

The required blanket thickness is relatively independent of reactor power.

The average concentrations of Pu-239 and U-238 atoms in the core are functions of the core fuel design. These concentrations determine the rate of absorption of neutrons along a neutron random walk path.
 

FNR BASIC CRITICALITY REQUIREMENTS:
1) When the movable fuel bundles are fully inserted into the matrix of fixed fuel bundles and the core fuel is nearly fully depleted the reactor will still be critical. This is the depleted fuel condition. However, the reactor must still be thermally stable in the depleted fuel condition, especially if it is fueled with U-235 instead of Pu-239.

2) When the movable fuel bundles are 1.1 m withdrawn and the core fuel is new the reactor must be dependably sub-critical. This is the new fuel cold shutdown condition. In this condition the movable active fuel bundles form a lower core fuel layer below the core zone and the fixed active fuel bundles form an upper core fuel layer above the core zone. The two layers are separated by 0.5 m of blanket rod material. Both the upper core fuel layer and the lower core fuel layer and their combination must be subcritical.

This requirement for subcriticality in the upper and lower core fuel zones limits the maximum Pu-239 concentrations and hence the fissile material weight fractions in the core fuel rods.
 

CRITICALITY MAINTENANCE:
It is necessary to maintain criticality over a range of Nf values that during the fuel life drop by almost a factor of two. Recall that for steady state criticality:
(Nf Sigmaf (G - 1)) = (Na Sigmaa) + (Nb Sigmab)

As the fuel ages Nf falls from an initial value of Nfo to a final value of:
~ Nfo / 2,
while G effectively rises from about 1.5 to about 2.5

During that same period:
(Ns Sigmas) drops to zero. Hence:
(Nfo Sigmaf (G - 1)) = 2 [(Na Sigmaa) + (Nb Sigmab)]
or
Nfo = 2 [(Na Sigmaa) + (Nb Sigmab)] / [Sigmaf (G - 1)]

The generalized form of this equation is:
Npo = 2 [Nu Sigmaau + Np Sigmaap + Nz Sigmaaz + Ns Sigmaas + Ni Sigmaai + Nc Sigmaac] / [Sigmafp (Gp - 1)]
= 2 {[(16.070 X 10^26 Pu-239 atoms / m^3 X 40 X 10^-3 b)
+ (56.4813 X 10^26 U-238 atoms / m^3 X 250 X 10^-3 b)
+ (21.052 X 10^26 Zr atoms / m^3 X 6.6 X 10^-3 b)
+ (142.07 X 10^26 Na atoms / m^3 X 1.4 X 10^-3 b)
+ (156.657 X 10^26 Fe atoms / m^3 X 8.6 X 10^-3 b)
+ (25.885 X 10^26 Cr atoms / m^3 X 14 X 10^-3 b)]
/ [ 1700 X 10^-3 b X (3.1 - 1)]}
 
 
= {2 X 10^26 / m^3} {[(16.070 X 40)
+ (56.4813 X 250)
+ (21.052 X 6.6)
+ (142.07 X 1.4)
+ (156.657 X 8.6)
+ (25.885 X 14)]
/ [ 1700 X (3.1 - 1)]}
 
 
= {2 X 10^26 / m^3} {[(642.8)
+ (14120.3)
+ (138.9)
+ (198.9)
+ (1347.25)
+ (362.39)]
/ [3570]}
 
 
= {2 X 10^26 / m^3} {[(16810.54)
/ [3570]}
 
 
= {9.417 X 10^26 / m^3}
 

Recall that a 20% fissile fraction resulted in:
Np = 16.070 X 10^26 Pu-239 atoms / m^3

Thus the optimal Pu fraction in the core fuel is about:
[9.417 / 16.070] X 20% = 11.72%

The initial thickness of the core zone should be consistent with this Pu concentration. As the Pu concentration decreases the core zone thickness required to maintain criticality increases.
 

REACTOR CRITICALITY MAINTENANCE:
In order for an FNR to operate there must be a self-sustaining chain reaction in the core zone. This requirement imposes constraints on the average atomic densities of Pu-239, U-238, Na-23, Fe, Cr in the core zone and the core zone thickness.

In normal reactor operation the neutron concentration is highest at the center of the core zone, gradually diminishes near the top and bottom surfaces of the core zone and then rapidly diminishes in the adjacent blanket zones. The subcritical regions immediately adjacent to the core zone assist in distributing the reactor heat output over the core fuel rod stack length.

When the core zone is too thin neutron losses by diffusion into the adjacent blanket zones will prevent core zone criticality. Maintenance of criticality in the core zone requires a minimum core zone thickness together with a minimum plutonium density within the core zone. From each fission fissile atoms in the core zone must capture at least one neutron to sustain the chain reaction and will likely lose an additional 0.5 neutrons to U-238 absorption in the core zone. As core zone thickness increases there is proportionately less neutron diffusion out of the core zone and the amount of reactor power control provided by change in fuel temperature decreases. To prevent reactor power runaway the core zone must not be permitted to get too thick.

At criticality when the neutron gain, neutron absorption and neutron diffusion loss mechanisms are in balance then:
dM /dt = 0
or
Mo Vn [(Gk - 1) Nk Sigmafk - Ni Sigmaai - Nj Sigmaaj] - Mo Vn Kn = 0
or
[(Gk - 1) Nk Sigmafk - Ni Sigmaai - Nj Sigmaaj - Kn] = 0

This equation for the FNR criticality condition gives insight into FNR design.

In a real reactor the core zone thickness is mechanically adjusted so that Kn equals the sum of the other terms at the desired operating temperature. Due to the long-term gradual change in Nk with time during each fuel cycle it is necessary to periodically mechanically adjust Kn over a total of about a 2:1 range. Note that Kn is proportional to the rate of neutron diffusion out of the reactor core zone which increases with increasing temperature. To explicitly solve for Kn we need to solve the neutron diffusion equations in both the core and blanket zones.

Note that in order to have temperature control of the criticality condition via the temperature dependence of Nk, Ni and Nj the first term should be at least twice the sum of the second two terms. Hence:
Kn ~ [(Gk - 1) Nk Sigmafk / 2]

If Kn is too small there is reduced temperature control which makes the FNR dangerously unstable. If Kn is too large core zone criticality cannot be attained.

This is an enormously important result in terms of practical FNR design. Contemplatiion of an initial FNR core zone that is significantly thicker than the EBR-2 core zone is treading on dangerous ground. This issue together with fuel tube material properties significantly limits FNR power per unit of core zone area.

During the working life of a fuel bundle operated to 15% burnup the Pu fraction drops from 20% to about 12.7%. Thus Nk (average concentration of Pu-239) gradually decreases from its initial value of Nko to its final value of 0.635 Nko. Simultaneously Ni (average concentration of U-238) changes. To maintain reactor reactivity it is necessary to compensate for these changes by reducing Kn by further insertion of the movable core fuel bundles into the matrix of fixed fuel bundles.

A related issue is that Nko can vary from fuel bundle to fuel bundle due to variations in the fuel bundle life and in the initial Pu atom concentration.

Note that Nk, Ni and Nj are average atomic concentrations which are weak functions of fuel temperature. The core zone thickness is mechanically set so that the threshold of criticality occurs at a fuel temperature of slightly over 500 degrees C. Then circulation of 490 degree C coolant sodium past the fuel draws heat out of the fuel which increases Nk, Ni and Nj which causes the number of neutrons in the core zone to increase which causes the reactor to dissipate sufficient thermal power in the fuel to maintain the fuel temperature.

Thus the reactor thermal power is proportional to the difference between the fuel temperature and the coolant temperature. This issue potentially leads to a very high thermal flux if the coolant temperature is too low with respect to the reactor fuel temperature setpoint. In a real reactor the local thermal flux on the lower side of the core zone where the coolant is at a relatively low temperature is mitigated by diffusion of the consequent excess neutrons through the thickness of the core zone.

Recall that in normal reactor operation:
[(Gk - 1) Nk Sigmafk - Ni Sigmaai - Nj Sigmaaj - Kn] = 0

The moment to moment reactor power control via FNR temperature relies on:
the decrease in the term:
[(Gk - 1) Nk Sigmafk - Ni Sigmaai - Nj Sigmaaj]
with increasing temperature.

Note that if the liquid sodium coolant channels are enlarged to improve flow characteristics Nj is increased. Hence the fuel rod diameter must also be increased so that Nk is increased in order to keep the first term at least twice the sum of the second two terms.

The values of Sigmaai, Sigmaaj, Gk are atomic constants. The values which the reactor designer can affect are Ni, Nj, Nk and Kn. Thus it is necessary to calculate these parameters under various practical fuel geometries. Computation of the corresponding atomic concentrations is the subject of the web page FNR CORE.

The requirement that:
(Gk - 1) Nk Sigmafk - Ni Sigmaai - Nj Sigmaaj = Kn
determines the thickness of the core zone. As the fuel ages Nk decreases and to compensate the core zone is made thicker causing Kn to decrease. The decrease in Nk as the fuel ages reduces the change in criticality with interatomic spacing and hence with temperature. This effect reduces the strength of reactor temperature control at low values of Nk when the fuel is old and the core zone thickness has approximately doubled.

Fine adjustment of reactor reactivity by fuel thermal expansion is the subject of the web page FNR Reactivity.
 

REACTOR CRITICALITY MAINTENANCE:
In order for an FNR to operate there must be a self-sustaining chain reaction in the core zone. This requirement imposes constraints on the average atomic densities of Pu-239, U-238, Na-23, Fe, Cr in the core zone and the core zone thickness.

In normal reactor operation the neutron concentration is highest at the center of the core zone, gradually diminishes near the top and bottom surfaces of the core zone and then rapidly diminishes in the adjacent blanket zones. The subcritical regions immediately adjacent to the core zone assist in distributing the reactor heat output over the core fuel rod stack length.

When the core zone is too thin neutron losses by diffusion into the adjacent blanket zones will prevent core zone criticality. Maintenance of criticality in the core zone requires a minimum core zone thickness together with a minimum plutonium density within the core zone. From each fission fissile atoms in the core zone must capture at least one neutron to sustain the chain reaction and will likely lose an additional 0.5 neutrons to U-238 absorption in the core zone. As core zone thickness increases there is proportionately less neutron diffusion out of the core zone and the amount of reactor power control provided by change in fuel temperature decreases. To prevent reactor power runaway the core zone must not be permitted to get too thick.

At criticality when the neutron gain, neutron absorption and neutron diffusion loss mechanisms are in balance then:
dM /dt = 0
or
Mo Vn [(Gk - 1) Nk Sigmafk - Ni Sigmaai - Nj Sigmaaj] - Mo Vn Kn = 0
or
[(Gk - 1) Nk Sigmafk - Ni Sigmaai - Nj Sigmaaj - Kn] = 0

This equation for the FNR criticality condition gives insight into FNR design.

In a real reactor the core zone thickness is mechanically adjusted so that Kn equals the sum of the other terms at the desired operating temperature. Due to the long-term gradual change in Nk with time during each fuel cycle it is necessary to periodically mechanically adjust Kn over a total of about a 2:1 range. Note that Kn is proportional to the rate of neutron diffusion out of the reactor core zone which increases with increasing temperature. To explicitly solve for Kn we need to solve the neutron diffusion equations in both the core and blanket zones.

Note that in order to have temperature control of the criticality condition via the temperature dependence of Nk, Ni and Nj the first term should be at least twice the sum of the second two terms. Hence:
Kn ~ [(Gk - 1) Nk Sigmafk / 2]

If Kn is too small there is reduced temperature control which makes the FNR dangerously unstable. If Kn is too large core zone criticality cannot be attained.

This is an enormously important result in terms of practical FNR design. Contemplatiion of an initial FNR core zone that is significantly thicker than the EBR-2 core zone is treading on dangerous ground. This issue together with fuel tube material properties significantly limits FNR power per unit of core zone area.

During the working life of a fuel bundle operated to 15% burnup the Pu fraction drops from 20% to about 12.7%. Thus Nk (average concentration of Pu-239) gradually decreases from its initial value of Nko to its final value of 0.635 Nko. Simultaneously Ni (average concentration of U-238) changes. To maintain reactor reactivity it is necessary to compensate for these changes by reducing Kn by further insertion of the movable core fuel bundles into the matrix of fixed fuel bundles.

A related issue is that Nko can vary from fuel bundle to fuel bundle due to variations in the fuel bundle life and in the initial Pu atom concentration.

Note that Nk, Ni and Nj are average atomic concentrations which are weak functions of fuel temperature. The core zone thickness is mechanically set so that the threshold of criticality occurs at a fuel temperature of slightly over 500 degrees C. Then circulation of 490 degree C coolant sodium past the fuel draws heat out of the fuel which increases Nk, Ni and Nj which causes the number of neutrons in the core zone to increase which causes the reactor to dissipate sufficient thermal power in the fuel to maintain the fuel temperature.

Thus the reactor thermal power is proportional to the difference between the fuel temperature and the coolant temperature. This issue potentially leads to a very high thermal flux if the coolant temperature is too low with respect to the reactor fuel temperature setpoint. In a real reactor the local thermal flux on the lower side of the core zone where the coolant is at a relatively low temperature is mitigated by diffusion of the consequent excess neutrons through the thickness of the core zone.

Recall that in normal reactor operation:
[(Gk - 1) Nk Sigmafk - Ni Sigmaai - Nj Sigmaaj - Kn] = 0

The moment to moment reactor power control via FNR temperature relies on:
the decrease in the term:
[(Gk - 1) Nk Sigmafk - Ni Sigmaai - Nj Sigmaaj]
with increasing temperature.

Note that if the liquid sodium coolant channels are enlarged to improve flow characteristics Nj is increased. Hence the fuel rod diameter must also be increased so that Nk is increased in order to keep the first term at least twice the sum of the second two terms.

The values of Sigmaai, Sigmaaj, Gk are atomic constants. The values which the reactor designer can affect are Ni, Nj, Nk and Kn. Thus it is necessary to calculate these parameters under various practical fuel geometries. Computation of the corresponding atomic concentrations is the subject of the web page FNR CORE.

The requirement that:
(Gk - 1) Nk Sigmafk - Ni Sigmaai - Nj Sigmaaj = Kn
determines the thickness of the core zone. As the fuel ages Nk decreases and to compensate the core zone is made thicker causing Kn to decrease. The decrease in Nk as the fuel ages reduces the change in criticality with interatomic spacing and hence with temperature. This effect reduces the strength of reactor temperature control at low values of Nk when the fuel is old and the core zone thickness has approximately doubled.

Fine adjustment of reactor reactivity by fuel thermal expansion is the subject of the web page FNR Reactivity.
 

CORE ZONE THICKNESS:
It is desirable to minimize the Pu-239 concentration consistent with long fuel cycle time to maximize the core zone thickness. A thicker core zone reduces the thermal flux through the fuel tubes which allows longer fuel tube life and potentially higher reactor thermal power.
 

MOVABLE FUEL BUNDLE OVER INSERTION:
The main risk in power FNR deployment is rapid over insertion of movable fuel bundles into the matrix of fixed fuel bundles which could potentially force To up to the boiling point of liquid sodium at which point the increase in FNR reactivity due to sodium void instability could potentially blow the fuel assembly apart.

It may be possible to suppress such sodium boiling by longitudinal fuel disassembly, but it is much better to simply avoid such a condition in the first place by properly controlling To.
 

FUEL AGING:
The FNR core zone, where the fixed and movable fuel bundles overlap, produces an excess of neutrons which diffuse out of the core zone and into the adjacent blanket zones. The blanket zones absorb neutrons. When the core fuel is new about half of the fission neutrons generated in the core zone diffuse into the blanket zones. As the core fuel ages its fissile fuel concentration Nf decreases. To compensate the thickness of the core zone is gradually increased but the fraction of fission neutrons that diffuse from the core zone into the blanket zones gradually decreases. When this fraction approaches zero it is no longer possible to maintain reactor thermal stability at the desired reactor temperature setpoint To so the core fuel must be replaced.

Typically with Pu-239 based core fuel replacement is required after 15% of the core fuel mass has become fission products.
 

FNR SAFETY:
For FNR safety the FNR must be designed so that:
[dR / dT] < 0
so that at any available fuel assembly geometry as the FNR average fuel temperature T increases the FNR reactivity decreases. Then the FNR should naturally seek an above ambient operating temperature:
T = To
where the reactivity R is:
R = 0.
If the reactivity R is negative at room temperature the FNR thermal power output will always be zero.

The operating temperature setpoint To is a function of the FNR fuel assembly geometry. The FNR should be physically designed such that at any accessible fuel geometry To remains within a range which is safe for the FNR materials. For sodium cooled FNRs the setpoint temperature To should be adjustable from below room temperature up to about 500 degrees C. The upper operating temperature limit is chosen to be 500 degrees C to ensure that the peak temperature in the fuel does not exceed 602 degrees C at which point a Pu phase transition occurs prior to Pu melting at about 640 degrees C. Hopefully the phase transition will lower the Pu density which should inject some negative reactivity. Note that once fuel centerline melting occurs pure Pu will tend to concentrate near the fuel center line. ______
 

TEMPERATURE DEPENDENCE OF REACTOR REACTIVITY:
The core zone thickness is the length by which core fuel in the movable fuel bundles overlaps core fuel in the fixed fuel bundles. The temperature dependence of the reactor reactivity depends on:
a) The geometry and concentrations of the atomic species;
b) The temperature coefficients of expansion of the materials the core zone.
 

QUALITATIVE EXPLANATION FOR THE TEMPERATURE DEPENDENCE OF FNR REACTIVITY:
Assumptions:
1) The pancake thickness is the active length of the core fuel rods;
2) The pancake thickness is comparable to a neutron diffusion length;
3) The pancake diameter is much much larger than a neutron diffusion length;
4) The neutron interaction cross sections are independent of temperature;
5) The FNR is designed so that thermal expansion causes the net reactivity to decrease;
6) A component of the FNR reactivity increases with increasing temperature due to thermal expansion of the the sodium, steel and U-238;
7) A component of the FNR reactivity decreases with increasing temperature due to the thermal expansion of the fissile plutonium.

In order to ensure a negative slope FNR reactivity versus temperature curve it is essential that the negative reactivity injection of the fissile fuel exceeds the positive reactivity injection of the other components.

In order to meet these criteria:
a) The fissile fuel needs to have a relatively large TCE;
b) The fissile fuel cross sectional area must be sufficiently large compared to the cross sectional area of the other components, particularly the sodium.
 

RESPONSE TIME:
An issue of concern is the thermal response time of the FNR to a change in reactivity. 1) Major changes in reactivity involve changing the fuel rod overlap by changng the insertion of movable fuel bundles into the matrix of fixed fuel bundles. This process is driven by a mechanical actuator, not by temperature.

2) The fast response change in bulk reactivity with temperature is used to keep the FNR at a constant operating temperature during normal thermal load variations. The negative slope of the net reactivity versus temperature curve needs to be as high as possible to over ride aging effects. An important issue in FNR design is that the effect of sodium is to increase reactivity with increasing temperature.

3) Axial fuel disassembly is used to prevent equipment damage during minor prompt critical events such as fuel overheating during sodium pool warmup. If the core fuel suddenly goes prompt critical it will vaporize part of the sodium internal to the fuel tube. In the fixed fuel bundles this sudden localized rise in fuel tube pressure will cause core fuel axial disassembly toward the fuel tube plenum which will temporarily reduce the reactivity and hence stop the prompt critical condition.

This issue points to preference to use a bead material such as brass which has a high thermal coefficient of expansion. Hence at low temperatures the beaded material can easily slide inside the fuel tube whereas at high temperatures the bead makes a sodium fluid seal to the fuel tube inner wall.
 

FNR THERMAL STABILITY:
The temperature dependence of R originates in the temperature dependence of the atomic concentrations via thermal expansion / contraction.

A fundamental FNR design requirement is choice of FNR physical parameters such that at FNR's desired steady state operating temperature To the reactivity R as a function of temperature T satisfies the equations:
R(To) = 0
and
{[dR(T) / dT]|(T = To)} < 0

From a reactor safety and thermal stability perspective it is essential that the reactivity R always have a negative temperature coefficient.

FNRs rely on a significant flux of neutrons exiting from the core zone to provide thermal stability. That requirement for a high exit neutron flux is further increased by the relatively high temperature coefficient of expansion of liquid sodium coolant. This web page sets out the reason for the high exit neutron flux from the FNR core zone, which significantly influences the fuel geometry of a FNR.
 

TEMPERATURE DEPENDENCE OF BULK REACTIVITY:

dR / dT = d{[(Nf Sigmaf (G - 1)) - (Na Sigmaa) - (Nb Sigmab)] Vn} / dT
= [(dNf / dT)Sigmaf (G - 1) - (dNa / dT)Sigmaa -(dNb / dT) Sigmab] Vn

For the FNR to be temperature stable:
dR / dT < 0

Note that due to thermal expansion:
(dNf / dT) < 0
and
(dNa / dT) < 0
and
(dNb / dT) < 0

Assume that the neutrons are so fast that Vn is not a function of temperature.

Thus the criteria for bulk thermal stability is:
[(dNf / dT)Sigmaf (G - 1) - (dNa / dT)Sigmaa -(dNb / dT) Sigmab] < 0
or
[- (dNa / dT) Sigmaa -(dNb / dT) Sigmab] < [- (dNf / dT) Sigmaf (G - 1)]
or
- (dNf / dT) > [- (dNa / dT) Sigmaa - (dNb / dT) Sigmab] / [Sigmaf (G - 1)]

This is a key equation that must be satisfied by the FNR geometry.

Hence the stability inequality can be expressed as:
Nf TCEf > [Na TCEa Sigmaa + Nb TCEb Sigmab] / [Sigmaf (G - 1)]
or
1 > [Na TCEa Sigmaa + Nb TCEb Sigmab] / [TCEf Nf Sigmaf (G - 1)]
 

Note that:
Due to fuel aging initially:
G ~ (Gm / 2);
where Gm is the maximum value of G
and
due to Pu fissioning Nf gradually decreases to (Nf / 2).
Hence it is important that the above inequality be satisfied with G ~ Gm / 2.

If the fissile isotope is Pu-239 the generalized form of the thermal stability inequality is:
1 > [Np Sigmaap TCEp + Nu SigmaaU TCEu + Nz Sigmaaz TCEz + Ns Sigmaas TCEs + Ni Sigmaai TCEi + Nc Sigmaac TCEc] / [TCEp Npf Sigmafp (G - 1)]

If the fissile isotope is U-235 the generalized form of the thermal stability inequality takes the form:
1 > [Np Sigmaap TCEp + Nu Sigmaau TCEu + Nz Sigmaaz TCEz + Ns Sigmaas TCEs + Ni Sigmaai TCEi + Nc Sigmaac TCEc] / [TCEu Nuf Sigmafu (G - 1)]

At the normal operating condition with Pu-239 fissile:
Npf Sigmafp (G - 1)] = [Np Sigmaap + Nu SigmaaU + Nz Sigmaaz + Ns Sigmaas + Ni Sigmaai + Nc Sigmaac]

Hence the reactivity requirement is:
(G - 1)] > [Np Sigmaap + Nu SigmaaU + Nz Sigmaaz + Ns Sigmaas + Ni Sigmaai + Nc Sigmaac] / [Npf Sigmafp]

The thermal stability requirement for plutonium as:
1 > [Np Sigmaap TCEp + Nu SigmaaU TCEu + Nz Sigmaaz TCEz + Ns Sigmaas TCEs + Ni Sigmaai TCEi + Nc Sigmaac TCEc]
/ {[TCEp][Np Sigmaap + Nu SigmaaU + Nz Sigmaaz + Ns Sigmaas + Ni Sigmaai + Nc Sigmaac]}
or
1 > [Np Sigmaap + Nu SigmaaU (TCEu / TCEp) + Nz Sigmaaz (TCEz / TCEp) + Ns Sigmaas (TCEs / TCEp) + Ni Sigmaai (TCEi / TCEp) + Nc Sigmaac (TCEc / TCEp)]
/ {[[Np Sigmaap + Nu SigmaaU + Nz Sigmaaz + Ns Sigmaas + Ni Sigmaai + Nc Sigmaac]}

A potential problem is that TCEs > TCEp so for this inequality to be valid there is a hard upper limit on Ns which means that there is an upper limit on X.

These equations are of great importance in FNR design because they impose an upper limit on Ns which will limit the maximum grid spacing X between fuel tubes. As the grid spacing X increases Ns increases but Np, Npf, Nu, Nz, Ni, Nc and Nuf all decrease. It is necessary to itterate through various grid spacings from (3 / 8) inch up to (9 / 16) inch in (1 / 32) inch steps to find the optimum grid spacing for Pu fissile fuel. Potential grid spacings are: (12 / 32) inch, (13 / 32) inch, (14 / 32) inch, (15 / 32) inch, (16 / 32) inch, (17 / 32) inch, and (18 / 32) inch. The grid spacing affects the coolant sodium circulation.

Note that the fuel geometry is affected by both thermal expansion of the fuel and by thermal expansion of the fuel bundle steel and the sodium coolant.

Various mechanisms affect the average concentration of fissionable atoms in the reactor core region but only the fuel temperature can respond to changes in fission power at the rate necessary to suppress a prompt critical condition.

NUMERICAL EVALUATION FOR X = 0.500 INCH:

Evaluate the thermal stability inequality for (3 / 8) inch OD fuel tubes, Pu = 20% and X = 0.500

On the web page named FNR Core
it is shown that for (3 / 8) inch OD fuel tubes, X = 0.500 and Pu = 20%:
THE AVERAGE ATOMIC CONCENTRATION DATA SUMMARY:
Ns = 0.14637 X 10^29 Na atoms / m^3
Ni = 0.183236 X 10^29 Fe atoms / m^3 Nc = 0.02683636 X 10^29 Cr atoms / m^3
Nu = 0.04234 X 10^29 U atoms / m^3
Np = .012046 X 10^29 Pu atoms / m^3
Nz = 0.01578 X 10^29 Zr atoms / m^3
 

Sigmafp = Pu-239 fast neutron fission cross section = 1700 X 10^-3 b
Sigmafu = U-238 fast neutron fission cross section = 41 X 10^-3 b
Sigmafu = U-235 fast neutron fission cross section = 960 X 10-3 b

From Kaye & Laby the cross sections for fast neutron absorption in a FNR are:
Sigmaap = Pu-239 fast neutron absorption cross section = 40 X 10^-3 b
Sigmafu = U-238 fast neutron fission cross section = 41 X 10^-3 b
Sigmaau = U-238 fast neutron absorption cross section = 250 X 10^-3 b
= U-235 fast neutron absorption cross section
Sigmaas = Na fast neutron absorption cross section = 1.4 X 10^-3 b
Sigmaai = Fe fast neutron absorption cross section = 8.6 X 10^-3 b
Sigmaac = Cr fast neutron absorption cross section = 14 X 10^-3 b
Sigmaaz = Zr fast neutron absorption cros section = 6.6 X 10^-3 b

NUMERICAL EVALUATION FOR (3 / 8) inch fuel tubes, X = 0.5 inch, 20% Pu.70% U, 10% Zr:

Np Sigmaap = 0.012046 X 40 X 10^26 atoms b / m^3 = 0.48184
Nu Sigmaau = .04232 X 250 X 10^26 atoms b / m^3 = 10.58
Nz Sigmaaz = 0.01578 X 6.6 X 10^26 atoms b / m^3 = 0.104148
Ns Sigmaas = 0.14637 X 1.4 X 10^26 atoms b / m^3 = 0.204918
Ni Sigmaai = 0.183236 X 8.6 X 10^26 atoms b / m^3 = 1.5758296
Nc Sigmaac = 0.02683636 X 14 X 10^26 atoms b / m^3 = 0.375709

Volumetric TCE values:
Pu - TCEp = 141 X 10^-6 / deg C
U - TCEu = 40.2 X 10^-6 / deg C
Zr - TCEz = 17.1 X 10^-6 / deg C
Na - TCEs = 210 X 10^-6 / deg C
Fe - TCEi = 36.0 X 10^-6 / deg C
Cr - TCEc = 19.5 X 10^-6 / deg C
 

(TCEu / TCEp) = 40.2 / 141 = 0.2851
(TCEz / TCEp) = 17.1 / 141 = 0.1212766
(TCEs / TCEp) = 210 / 141 = 1.4893
(TCEi / TCEp) = 36 / 141 = 0.25532
(TCEc / TCEp) = 19.5 / 141 = 0.1383

The stability requirement for plutonium fissile is:
1 > [Np Sigmaap TCEp + Nu SigmaaU TCEu + Nz Sigmaaz TCEz + Ns Sigmaas TCEs + Ni Sigmaai TCEi + Nc Sigmaac TCEc]
/ {[TCEp][Np Sigmaap + Nu SigmaaU + Nz Sigmaaz + Ns Sigmaas + Ni Sigmaai + Nc Sigmaac]}
or
1 > [Np Sigmaap + Nu SigmaaU (TCEu / TCEp) + Nz Sigmaaz (TCEz / TCEp) + Ns Sigmaas (TCEs / TCEp) + Ni Sigmaai (TCEi / TCEp) + Nc Sigmaac (TCEc / TCEp)]
/ {[[Np Sigmaap + Nu SigmaaU + Nz Sigmaaz + Ns Sigmaas + Ni Sigmaai + Nc Sigmaac]}

NOW CHECK FOR STABILITY AT X = (9 / 16) INCHES

We need to evaluate the stability inequality for Pu = 20% and X = 0.5625 inch

We must ensure that this inequality is still valid at Pu = 12%

From FNR Core

SUMMARY FOR X = 0.5625 INCH:
Ns = 0.162328 X 10^29 atoms Na / m^3
Ni = 0.15784 x 10^29 atoms Fe / m^3
Nc = 0.0231169 X 10^29 atoms Cr /m^3
Nu = 0.033453 X 10^29 atoms U-238 / m^3
Np = 0.00951795 x 10^29 atoms Pu / m^3
Nz = 0.0124656 X 10^29 atoms Zr / m^3
 

Sigmafp = Pu-239 fast neutron fission cross section = 1700 X 10^-3 b
Sigmafu = U-238 fast neutron fission cross section = 41 X 10^-3 b
Sigmafu = U-235 fast neutron fission cross section = 960 X 10-3 b

From Kaye & Laby the cross sections for fast neutron absorption in a FNR are:
Sigmaap = Pu-239 fast neutron absorption cross section = 40 X 10^-3 b
Sigmafu = U-238 fast neutron fission cross section = 41 X 10^-3 b
Sigmaau = U-238 fast neutron absorption cross section = 250 X 10^-3 b
= U-235 fast neutron absorption cross section
Sigmaas = Na fast neutron absorption cross section = 1.4 X 10^-3 b
Sigmaai = Fe fast neutron absorption cross section = 8.6 X 10^-3 b
Sigmaac = Cr fast neutron absorption cross section = 14 X 10^-3 b
Sigmaaz = Zr fast neutron absorption cros section = 6.6 X 10^-3 b

NUMERICAL EVALUATION FOR (3 / 8) inch fuel tubes, X = 0.5625 inch, 20% Pu.70% U, 10% Zr:

Np Sigmaap = 0.00951795 X 40 X 10^26 atoms b / m^3 = 0.380718
Nu Sigmaau = 0.033453 X 250 X 10^26 atoms b / m^3 = 8.36325
Nz Sigmaaz = 0.0124656 X 6.6 X 10^26 atoms b / m^3 = 0.08227296
Ns Sigmaas = 0.162328 X 1.4 X 10^26 atoms b / m^3 = 0.2272592
Ni Sigmaai = 0.15784 X 8.6 X 10^26 atoms b / m^3 = 1.357424
Nc Sigmaac = 0.0231169 X 14 X 10^26 atoms b / m^3 = 0.3236366

(TCEu / TCEp) = 40.2 / 141 = 0.2851
(TCEz / TCEp) = 17.1 / 141 = 0.1212766
(TCEs / TCEp) = 210 / 141 = 1.4893
(TCEi / TCEp) = 36 / 141 = 0.25532
(TCEc / TCEp) = 19.5 / 141 = 0.1383

1 > [Np Sigmaap + Nu SigmaaU (TCEu / TCEp) + Nz Sigmaaz (TCEz / TCEp) + Ns Sigmaas (TCEs / TCEp) + Ni Sigmaai (TCEi / TCEp) + Nc Sigmaac (TCEc / TCEp)]
/ {[[Np Sigmaap + Nu SigmaaU + Nz Sigmaaz + Ns Sigmaas + Ni Sigmaai + Nc Sigmaac]}

This indicates thermal stability.

Now check for reactivity at X = (9 / 16) inch.

If the grid spacing is too large the reactivity is insufficient at Pu = 20%. If the fissile fuel is not plutonium the required negative slope reactivity versus temperature relationship is not realized.

The reactivity requirement is:
(G - 1)] > [Np Sigmaap + Nu SigmaaU + Nz Sigmaaz + Ns Sigmaas + Ni Sigmaai + Nc Sigmaac] / [Npf Sigmafp]

this inequality is only barely satisfied as G falls from 3.1 to (12 / 20) 3.1 = 1.86

Thus the reactivity requirement limits the maximum value of X to (9 / 16) inches. At that dimension the FNR is thermally stable. This (9 / 16) inch dimension is carried back to the fuel bundle design.

SUMMARY
The requirement of thermal stability keeps the various atomic components in a fixed ratio with respect to one another. If the initial fuel alloy is 20% Pu and the fuel tube is (3 / 8) inch OD then there is an optimum fuel tube center to center spacing X. Increasing X increases the sodium coolant flow channel cross section which is desirable but may also increase the required Pu inventory per unit of reactor thermal power which is undesirable.

Thus there is a fundamental tradeoff between Pu inventory and sodium cooling channel size. Successful application of sodium cooled FNRs requires focus on the sodium filter issue and maintenance of a minimum sodium temperature of 340 degrees C to prevent precipitation of NaOH in the sodium cooling channels. However, there will still be a problem with KOH (melting point = 360 C) precipitation inside the intermediate heat exchange tubes which must be addreswsed by filtering.

A better solution is to operate the FNR at a minimum sodium temperature of 340 C, a minimum NaK temperature of 330 C and to focus on excluding water from the NaK and filtering of the NaK. With this temperature choice filtering of the sodium becomes less important and NaK gaskets are easier to deal with. From time to time the FNR must be operated at low power so that the minimum sodium temperature rises above 370 degrees C. That will cause solid KOH that precipitates in the NaK loop to melt allowing internal cleaning of the intermediate heat exchange tubes. That NaK then needs to be transferred to a dump tank where its temperature should be reduced to 350 C to precipitate the KOH, which then needs to be removed via a filter cartridge.
 

At the normal operating condition with U-235 fissile:
Nuf Sigmafu (G - 1)] = [Np Sigmaap + Nu SigmaaU + Nz Sigmaaz + Ns Sigmaas + Ni Sigmaai + Nc Sigmaac]
which gives the thermal stability requirement for U-235 as:
1 > [Np Sigmaap TCEp + Nu SigmaaU TCEu + Nz Sigmaaz TCEz + Ns Sigmaas TCEs + Ni Sigmaai TCEi + Nc Sigmaac TCEc]
/ {[TCEu][Np Sigmaap + Nu SigmaaU + Nz Sigmaaz + Ns Sigmaas + Ni Sigmaai + Nc Sigmaac]}

The problem in the case of U-235 is that TCEu is much smaller than TCEp which severely restricts the size of Ns and hence the grid spacing X between the fuel tubes.

Hence a naturally circulated FNR is only practical with Pu-239 fissile core fuel. With Pu-239 the grid spacing is also limited by the requirement to maintain reactivity as the Pu-239 concentration drops from 20% to 12%.
 

DATA
LINEAR TCE VALUES:
TCEU = 13.4 X 10^-6 / deg C
TCEPu = (47 - 54) X 10^-6 / deg C
TCEZr = 5.7 X 10^-6 / deg C
TCENa = 70 X 10^-6 / deg C
TCEFe = 12.0 X 10^-6 / deg C
TCECr = (6 - 7) X 10^-6 / deg C

Volumetric TCE values:
Pu - TCEp = 141 X 10^-6 / deg C
U - TCEu = 40.2 X 10^-6 / deg C
Zr - TCEz = 17.1 X 10^-6 / deg C
Na - TCEs = 210 X 10^-6 / deg C
Fe - TCEi = 36.0 X 10^-6 / deg C
Cr - TCEc = 19.5 X 10^-6 / deg C
 

NUCLEAR PARAMETER DEFINITIONS:
Define:
Gu = number of neutrons emitted per average U-235 atomic fission
Gp = number of neutrons emitted per average Pu-239 atomic fission
Ns = average concentration of Na atoms in core zone
Ni = average concentration of Fe atoms in core zone
Nc = average concentration of Cr atoms in core zone
Np = average concentration of Pu atoms in core zone
Nz = average concentration of Zr atoms in core zone
Nu = average concentration of U-238 atoms in core zone
Nf = average number of fission product atoms in core zone

On the web page named FNR Core
it is shown that for (3 / 8) inch OD fuel tues, X = 0.500 and Pu = 20%:

NEW FUEL AVERAGE ATOMIC CONCENTRATION DATA SUMMARY:
Ns = 0.14637 X 10^29 Na atoms / m^3
Ni = 0.183236 X 10^29 Fe atoms / m^3 Nc = 0.02683636 X 10^29 Cr atoms / m^3
Nu = 0.04234 X 10^29 U atoms / m^3
Np = .012046 X 10^29 Pu atoms / m^3
Nz = 0.01578 X 10^29 Zr atoms / m^3
 

Sigmafp = Pu-239 fast neutron fission cross section = 1700 X 10^-3 b
Sigmafu = U-238 fast neutron fission cross section = 41 X 10^-3 b
Sigmafu = U-235 fast neutron fission cross section = 960 X 10-3 b

From Kaye & Laby the cross sections for fast neutron absorption in a FNR are:
Sigmaap = Pu-239 fast neutron absorption cross section = 40 X 10^-3 b
Sigmafu = U-238 fast neutron fission cross section = 41 X 10^-3 b
Sigmaau = U-238 fast neutron absorption cross section = 250 X 10^-3 b
= U-235 fast neutron absorption cross section
Sigmaas = Na fast neutron absorption cross section = 1.4 X 10^-3 b
Sigmaai = Fe fast neutron absorption cross section = 8.6 X 10^-3 b
Sigmaac = Cr fast neutron absorption cross section = 14 X 10^-3 b
Sigmaaz = Zr fast neutron absorption cros section = 6.6 X 10^-3 b

CONSIDER A Pu-239 BASED FNR:
Recall that the FNR thermal stability inequality for fissile Pu is:
1 > {[Np Sigmaap TCEp + Nu SigmaaU TCEu + Nz Sigmaaz TCEz + Ns Sigmaas TCEs + Ni Sigmaai TCEi + Nc Sigmaac TCEc] / [TCEp Np Sigmafp (G - 1)]}

Numeric evaluation of this thermal stability inequality for _________ gives:
1 > {[(16.070 X 10^26 Pu-239 atoms / m^3 X 40 X 10^-3 b X 150 X 10^-6 / deg C)
+ (56.4813 X 10^26 U-238 atoms / m^3 X 250 X 10^-3 b X 40.2 X 10^-6 / deg C)
+ (21.052 X 10^26 Zr atoms / m^3 X 6.6 X 10^-3 b X 17.1 X 10^-6 / deg C)
+ (142.07 X 10^26 Na atoms / m^3 X 1.4 X 10^-3 b X 210 X 10^-6 / deg C)
+ (156.657 X 10^26 Fe atoms / m^3 X 8.6 X 10^-3 b X 36.0 X 10^-6 / deg C)
+ (25.885 X 10^26 Cr atoms / m^3 X 14 X 10^-3 b X 19.5 X 10^-6 / deg C)]
/ [141 X 10^-6 / deg C X 16.070 X 10^26 Pu-239 atoms / m^3 X 1700 X 10^-3 b X (3.1 - 1)]}
 
 
= {[(16.070 X 40 X 150 X 10^-6 / deg C)
+ (56.4813 X 250 X 40.2 X 10^-6 / deg C)
+ (21.052 X 6.6 X 17.1 X 10^-6 / deg C)
+ (142.07 X 1.4 X 210 X 10^-6 / deg C)
+ (156.657 X 8.6 X 36.0 X 10^-6 / deg C)
+ (25.885 X 14 X 19.5 X 10^-6 / deg C)]
/ [141 X 10^-6 X 16.070 X 1700 X (3.1 - 1)]}
 
 
= {[(96420 X 10^-6 / deg C)
+ (567,637 X 10^-6 / deg C)
+ (2376 X 10^-6 / deg C)
+ (41,769 X 10^-6 / deg C)
+ (48,501 X 10^-6 / deg C)
+ (7067 X 10^-6 / deg C)]
/ [141 X 10^-6 X 57,370]}
 
 
= {[(763,770 X 10^-6 / deg C)
/ [141 X 10^-6 X 57,370]}
 
= {(13.313 X 10^-6 / deg C)/ 141 X 10^-6 / deg C}
 
Thus, based on the above calculation a plutonium based FNR is thermally stable down to a very low plutonium concentration. This stability is enabled by the large TCE of plutonium.
 

CONSIDER A U-235 BASED FNR:
In a U-235 based FNR the Pu-239 is replaced by U-235. Recall that for a U-235 based FNR the thermal stability inequality is:

If the fissile isotope is U-235 the generalized form of the thermal stability inequality takes the form:
TCEu > {[Nu Sigmaau TCEu + Nz Sigmaaz TCEz + Ns Sigmaas TCEs + Ni Sigmaai TCEi + Nc Sigmaac TCEc] / [Nuf Sigmafu (Gu - 1)]
 

Numeric evaluation of this thermal stability inequality gives:
40.2 X 10^-6 / deg C > {[(16.070 X 10^26 U-235 atoms / m^3 X 250 X 10^-3 b X 40.2 X 10^-6 / deg C)
+ (56.4813 X 10^26 U-238 atoms / m^3 X 250 X 10^-3 b X 40.2 X 10^-6 / deg C)
+ (21.052 X 10^26 Zr atoms / m^3 X 6.6 X 10^-3 b X 17.1 X 10^-6 / deg C)
+ (142.07 X 10^26 Na atoms / m^3 X 1.4 X 10^-3 b X 210 X 10^-6 / deg C)
+ (156.657 X 10^26 Fe atoms / m^3 X 8.6 X 10^-3 b X 36.0 X 10^-6 / deg C)
+ (25.885 X 10^26 Cr atoms / m^3 X 14 X 10^-3 b X 19.5 X 10^-6 / deg C)]
/ [16.070 X 10^26 U-235 atoms / m^3 X 960 X 10^-3 b X (2.8 - 1)]}
 
 
{[(16.070 X 250 X 40.2 X 10^-6 / deg C)
+ (56.4813 X 250 X 40.2 X 10^-6 / deg C)
+ (21.052 X 6.6 X 17.1 X 10^-6 / deg C)
+ (142.07 X 1.4 X 210 X 10^-6 / deg C)
+ (156.657 X 8.6 X 36.0 X 10^-6 / deg C)
+ (25.885 X 14 X 19.5 X 10^-6 / deg C)]
/ [16.070 X 960 X (2.8 - 1)]}
 
 
{[(161,504 X 10^-6 / deg C)
+ (567,637 X 10^-6 / deg C)
+ (2376 X 10^-6 / deg C)
+ (41,769 X 10^-6 / deg C)
+ (48,501 X 10^-6 / deg C)
+ (7067 X 10^-6 / deg C)]
/ [27,769]}
 
 
{[(161,504 X 10^-6 / deg C)
+ (567,637 X 10^-6 / deg C)
+ (2376 X 10^-6 / deg C)
+ (41,769 X 10^-6 / deg C)
+ (48,501 X 10^-6 / deg C)
+ (7067 X 10^-6 / deg C)]
/ [27,769]}
 
 
= {[(828,854 X 10^-6 / deg C)]
/ [27,769]}
 
= 29.848 X 10^-6 / deg C

Based on the above calculation a U-235 based FNR is thermally stable but there is not a big safety margin for U-235 concentration degradation during the fuel life.

This issue will likely force a decrease in the fuel cycle time (time between successive fuel reprocessings).

Since one of the absorbing species (sodium) has an abnormally high TCE (thermal coefficient of expansion) the FNR relies on Nf being sufficiently large for compliance with the thermal stability inequality. As the U-235 fuel ages there is a distinct limit on how small Nf can be permitted to fall. With U-235 fuel a liquid lead coolant may be better in terms of allowing a longer fuel cycle time.

Note that for U-235 based FNRs there may be a dangerous range of fuel concentration where the requirement for FNR criticality is satisfied but the requirement for FNR thermal stability is not. This issue needs further investigation. The best solution is to use fissionable fuel containing at least some plutonium. If the fuel is initially 20% U-235 in 80% U-238 there may be enough breeding of plutonium to solve the problem.
 

FIX FROM HERE ONWARD

No = number of free neutrons at time t = to;
Ka = reactivity for prompt neutrons
Kb = reactivity for delayed neutrons.
G = Gp + Gd
Gp = average number of prompt neutrons per fission
Gd = average number of delayed neutrons per fission
 

DELAYED NEUTRONS:
In real nuclear fissions a fraction of 1% of the fission neutrons actually come from fission products rather than from the original fission. On average the delayed neutrons are not emitted until about:
to = 3 seconds
after the original fission. Hence the differential equation takes the form:
dM = {M [(Gkp - 1) Nk Sigmafk - Ni Sigmaai - Nj Sigmaaj - Kn]
+ M|(t - to)[Gkd Nk Sigmafk]} dX

Typically:
(Gkd / Gkp) < 0.01

In stable reactor operation:
dM = 0
M|t = M|(t - to)
[(Gkp - 1) Nk Sigmafk - Ni Sigmaai - Nj Sigmaaj - Kn] < 0
and
[(Gdk + Gkp - 1) Nk Sigmafk - Ni Sigmaai - Nj Sigmaaj - Kn] = 0

As long as:
[(Gkp - 1) Nk Sigmafk - Ni Sigmaai - Nj Sigmaaj - Kn] < 0
the rate of neutron population growth is limited by the rate of formation of successive generations of delayed neutrons, each of which generations are separated by about 3 seconds. Since each generation of neutron population growth is then separated by about three seconds it becomes practical to control a nuclear reactor using mechanical means. In a FNR Kn is mechanically fixed. However, care must still be taken to ensure that the inequality:
[(Gkp - 1) Nk Sigmafk - Ni Sigmaai - Nj Sigmaaj - Kn] < 0
is always valid. The reactor power will rise very rapidly out of control if:
[(Gkp - 1) Nk Sigmafk - Ni Sigmaai - Nj Sigmaaj - Kn] > 0
as might be caused by a fuel geometry instability that causes Kn to rapidly decrease. The delayed neutrons provide only about a 0.3% adjustment range in the term:
[(Gk - 1) Nk Sigmafk]
and the temperature dependence of Nk, Ni and Nj provides about a 0.3% adjustment range in the term:
[(Gkp - 1) Nk Sigmafk - Ni Sigmaai - Nj Sigmaaj].

Note that:
Nk = Nko / [1 + Ak (T - To)]^3
and
Ni = Nio / [1 + Ai (T - To)]^3
and
Nj = Njo / [1 + Aj (T - To)]^3
where:
Ak, Ai, Aj are the material linear coefficients of expansion of materials k, i, j.

The significance of this issue is that the mechanical means of adjusting Kn must be slow, smooth and free of hysterisis. In normal reactor operation Kn is varied by slowly changing the insertion depth of the mobile fuel bundles over about a 0.35 m range. Hence the mechanical fuel bundle insertion control should be stable to less than 1 mm of insertion.
 

PROMPT NEUTRON CRITICALITY SUPPRESSION:
Note that in normal operation the reacting end of the core fuel rod is hotter than the opposite end. If the hot end of a core fuel rod rapidly gets too hot it will vaporize the fission product Cs and then vaporize the adjacent liquid Na contained inside the fuel tube. In the movable fuel bundles the sodium vapor pressure tends to cause internal liquid sodium to blow the upper blanket rods into the plenum while holding the core fuel rods in place. In the fixed fuel bundles the sodium vapor pressure tends to blow both the core fuel rods and the upper blanket rods into the plenum. This action, which happens in a fraction of a ms time frame, separates the fixed fuel bundle core fuel rods from the movable fuel bundle core fuel rods, which reduces the reactor reactivity and will suppress a prompt critical condition.

Note that absent core fuel rod cool end beading, due to vapor leaking past the core fuel rods the aforementioned prompt criticality suppression mechanism might not operate reliably until the core fuel rods in the fixed fuel bundles have swelled enough to fill the fuel tube. That swelling relies on formation of inert gas bubbles in the fuel which may take weeks to fully form. Until the core fuel rods in the fixed fuel bundles have swelled enough to fully fill the fuel tube core fuel rod end beading must be relied upon to suppress a prompt critical condition.
 

15% CORE FUEL BURNUP:
An issue is maintaining criticality in the core zone at 12.7% Pu while being certainly subcritical when half the movable fuel bundles are withdrawn at 20% Pu. We rely on net neutron diffusion out of the core zone for reactor operating temperature setpoint control. However, the subcritical regions immediately above and below the core zone assist in distributing the reactor heat output over the core fuel rod length.
 

ATOMIC CONCENTRATION SUMMARY FOR FNR CORE AT 15% FUEL BURNUP:
Ns = 1.4207 X 10^28 sodium atoms / m^3
Ni = 1.56657 X 10^28 iron atoms / m^3
Nc = 0.258851207 X 10^28 chromium atoms / m^3
Nu = 0.502683676 X 10^28 uranium atoms / m^3
Np = 0.102044479 X 10^28 plutonium atoms / m^3
Nf = 0.24104995 X 10^28 fission product atoms / m^3
Nz = 0.209966455 X 10^28 zirconium atoms / m^3
 

Re-evaluate terms:
Np [(Gp - 1) Sigmafp - Sigmaap]
= [10.2044 X 10^26 Pu atoms / m^3] [2.1 (1.70) - 0.040] (b / Pu atom) [10^-28 m^2 / b]
= 36.0215 X 10^-2 / m
= 0.360215 / m

Thus with old fuel:
Kn = Np [(Gp - 1) Sigmafp - Sigmaap] + Nu [(Gu - 1) Sigmafu - Sigmaau]
- [Ns Sigmaas + Ni Sigmaai + Nc Sigmaas + Nz Sigmaaz]
 
= 0.360215 / m - 0.0926948 / m - 0.0204747989 / m
= 0.24704 / m

Thus Kn should be adjustable over about a 2 : 1 range.
 

During the working life of a fuel bundle operated to 15% burnup the Pu fraction drops from 20% to about 12.7%. Thus Nf varies from its initial value of Nfo to its final value of 0.635 Nfo. To maintain a reactivity of ~ 1.0 it is necessary to compensate for the change in Nf by increasing Fc which is adjusted by further insertion of movable fuel bundles. Typically when the fuel is new Fc ~ 0.5 and when the fuel is ready for reprocessing Fc ~ 0.9. Most of the nuclear heat is injected into the middle core zone. A complication with this strategy is that the aging of each fixed fuel bundle is determined by the aging of the various adjacent movable fuel bundles. Thus keeping track of the aging of the various fuel bundles is a complicated process.

Determine the fraction of fission neutrons absorbed by sodium both in the reactor core and blanket and via neutron leakage into the guard band.
 

REFERENCE:
Reactivity Coefficients in BN-600 Core with Minor Actinides
 

This web page last updated August 7, 2023.