# XYLENE POWER LTD.

## FNR REACTIVITY

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

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 start fuel than U-235, but over sufficient time a FNR started with U-235 will form its own Pu-239.

Another approach is to use a different fuel fabrication methodology in order to minimize the positive void coefficient effect of sodium inside the active fuel tube. This issue is discussed at Breed and Burn Reactor theory.

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

The 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 reactivity R is positive the free neutron population N will exponentially increase over time. If the reactivity R is negative the free neutron population N will exponentially decay over time. If the 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. Hence at steady state the FNR thermal power output is constant and the rate of heat removal from the FNR by the primary liquid sodium coolant exactly equals the FNR thermal power output.

TEMPERATURE DEPENDENCE OF REACTOR REACTIVITY:
The temperature dependence of the reactor reactivity depends on two parameters:
a) The concentrations of the atomic species, which vary due o their tempeature coefficients of expansion;
b) The temperature coefficient of the width of the core zone. This width is the length by which core fuel rods in the movable fuel bundles overlap core fuel rods in the fixed fuel bundles. An important issue in fuel tube design is that the effect of fuel rod length expansion with increasing temperature is to increase the reactor reactivity. To compensate for this issue the decrease in fuel atomic concentration with increasing temperature must be larger than simple calculations indicate. That issue points to use of Pu as the fissile fuel due to its high thermal coefficient of expansion.

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 equals the rate of heat extraction by the coolant and
T = To.

FNR 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 thermal power output, which for a constant thermal load results in a constant fuel temperature:
T = To

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 increases 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 decreases 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. In a practical reactor we do not want [-dR / dT] to be either too large or too small.

An issue to check is that with a constant thermal load the fission power quickly converges to a steady value rather than oscillating. This isssue may be primary sodium circulation dependent. 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 liquid sodium lags the core fuel by a few seconds, and the thermal response time of the steel and blanket fuel lags the sodium by yet a few 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 eventual total positive reactivity injection caused by thermal expansion of the other FNR components. Particularly problematic is sodium due to its relatively large thermal coefficient of expansion which injects 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 be sufficient. As the average fissile fuel atom concentration in the core zone decreases, FNR control stability potentially becomes more of a problem, particularly with U-235 fissile fuel. This issue can potentially limit the cycle time (time between successive fuel reprocessings) of U-235 based FNR fuel.

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

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.

COMPUTATION OF N:
Note that for a pancake shaped FNR with thick top and bottom blanket zones and a vertical dimension 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 zone. 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 an absorbing species b in the core zone:
dN = N (- Nb Sigmab dX)
where:
Nb = atomic concentration of absorbing species b
and
Sigmab = absorption cross section of absorbing species b

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 = fission cross section of species f

For a species which scatters neutrons out of the core zone:
dN = N (- Ns Sigmas dX)
where:
Ns = atomic concentration of scatter loss species s
and
Sigmas = cross section of scatter loss species s

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

dX = Vn dt
where Vn is the neutron velocity

dN = N [(Nf Sigmaf (G - 1)) -(Na Sigmaa) - (Nb Sigmab) - (Ns Sigmas)] Vn dt
or
dN / N = [(Nf Sigmaf (G - 1)) -(Na Sigmaa) - (Nb Sigmab) - (Ns Sigmas)] Vn dt

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

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

Note that if:
(Nf Sigmaf (G - 1)) > (Na Sigmaa) + (Nb Sigmab)
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 (G - 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 huge implications in terms of reducing the fissile start fuel requirement for FNRs. A typical U-235 fueled FNR uses uranium enriched to almost 20% U-235. To achieve the same performance a FNR needs only about 8% Pu-239.

A further implication is that if a FNR is started with U-235 it will self breed Pu-239 in the core zone for so long that the fuel cycling time will probably be set by inert gas accumulation and by fuel tube deterioration rather than by fuel deterioration.

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

Assume that the neutrons are fast so the probability of a neutron being scattered out of the core zone is not a function of temperature. Then:
d(Ns Sigmas) / dT = 0

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

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

The condition for the FNR to be thermally stable is:
(dR / dT ) < 0
or
{[(Sigmaf (G - 1))(dNf / dT) - (Sigmaa)(dNa / dT) - (Sigmab)(dNb / dT)] Vn} < 0
or
(Sigmaf (G - 1))(- dNf / dT) > (Sigmaa)(- dNa / dT) + (Sigmab)(- dNb / dT)
or
(- dNf / dT) > [(Sigmaa)(- dNa / dT) + (Sigmab)(- dNb / dT)] / [Sigmaf (G - 1)]

Recall that:
At the FNR steady state operating point:
R = 0
or
(Nf Sigmaf (G - 1)) = (Na Sigmaa) + (Nb Sigmab) + (Ns Sigmas)
or
[Sigmaf (G - 1)] = [(Na Sigmaa) + (Nb Sigmab) + (Ns Sigmas)] / Nf

Hence the condition for FNR thermal stability becomes:
[(1 / Nf)(- dNf / dT)] > [(Sigmaa)(- dNa / dT) + (Sigmab)(- dNb / dT)]
/ [(Na Sigmaa) + (Nb Sigmab) + (Ns Sigmas)]
or
[(1 / Nf)(- dNf / dT)] > [(Na Sigmaa){(1 / Na)(- dNa / dT)} + (Nb Sigmab){(1 / Nb)(- dNb / dT)}]
/ [(Na Sigmaa) + (Nb Sigmab) + (Ns Sigmas)]

(Nf Sigmaf (G - 1)) = (Na Sigmaa) + (Nb Sigmab) + (Ns Sigmas)

Hence the condition for FNR stability becomes:
[(1 / Nf)(- dNf / dT)]
> [(Na Sigmaa){(1 / Na)(- dNa / dT)} + (Nb Sigmab){(1 / Nb)(- dNb / dT)}]
/ [Nf Sigmaf (G - 1)]

[(1 / Nf)(- dNf / dT)] = volumetric TCE of species f;
{(1 / Na)(- dNa / dT)} = volumetric TCE of species a;
{(1 / Nb)(- dNb / dT)} = volumetric TCE of species b;

Hence the condition for FNR thermal stability becomes:
TCEf > {[Na Sigmaa TCEa + Nb Sigmab TCEb] / [Nf Sigmaf (G - 1)]}

For calculation simplicity ignore fussion of U-238.

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

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

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 fuel bundle coolant.

Both 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.

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 = 150 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
Sigmafp = Pu-239 fast neutron fission cross section
Sigmaap = Pu-239 fast neutron absorption cross section
Sigmafu = U-238 fast neutron fission cross
Sigmaau = U-238 fast neuton absorption cross section
Sigmaas = Na fast neutron absorption cross section
Sigmaai = Fe fast neutron absorption cross section
Sigmaac = Cr fast neutron absorption cross section
Gu = number of neutrons emitted per average U-235 atomic fission
Gp = number of neutrons emitted per average Pu-239 atomic fission

From Kaye & Laby the cross sections for fast neutron absorption in a FNR are:
Pu-239 - Sigmaap = 40 X 10^-3 b
U-238 = Sigmaau = 250 X 10^-3 b
u-235 = Sigmaau = 250 X10^-3 b___
Zr - Sigmaaz = 6.6 X 10^-3 b
Na - Sigmas = 1.4 X 10^-3 b
Fe - Sigmaai = 8.6 X 10^-3 b
Cr - Sigmaac = 14 X 10^-3 b

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

Neutrons per fission:
Gp = 3.1
Gu = 2.8___

AVERAGE ATOMIC CONCENTRATIONS IN THE FNR CORE ZONE:
The values of Ns, Ni, Nc, Nu, Np and Nz in the core zone with new fuel are calculated on the web page titled:
FNR CORE

NEW FUEL CORE ZONE DATA SUMMARY:
Ns = 142.07 X 10^26 Na atoms / m^3
Ni = 156.657 X 10^26 Fe atoms / m^3
Nc = 25.885 X 10^26 Cr atoms / m^3
Nu = 56.4813 X 10^26 U-238 atoms / m^3
Np = 16.070 X 10^26 Pu-239 atoms / m^3
Nz = 21.052 X 10^26 Zr atoms / m^3
Nuf = 16.070 X 10^26 U-235 atoms / m^3

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

Numeric evaluation of this thermal stability inequality gives:
150 X 10^-6 / deg C > {[(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)]
/ [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)]
/ [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)]
/ [57,370]}

= {[(763,770 X 10^-6 / deg C)
/ [57,370]}

= {(13.313 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.

CRITICALITY MAINTENANCE:
In a Pu-239 fueled FNR 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) + (Ns Sigmas)

As the fuel ages Nf falls from an initial value of Nfo to a fianal value of:
~ Nfo / 2

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)]
/ }

= {2 X 10^26 / m^3} {[(16810.54)
/ }

= {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 gradually decreases the core zone thickness required to maintain criticality increases.

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 therough 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 liquid sodium vapor pressure could blow the fuel assembly apart.

This risk is in part mitigated by use of a thick core zone.

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.

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 600 degrees C at which point fuel melting processes commence.

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.

NUCLEAR DATA:
From Kaye & Laby for a FNR core:
Sigmaas = fast neutron absorption cross section of sodium = 0.0014 b
Sigmass = fast neutron scatter cross section of sodium = 2.62 b___3.7 b
Sigmaai = fast neutron absorption cross section of iron = .0086 b
Sigmasi = fast neutron scatter cross section of iron = 4.6 b ____3.8 b
Sigmaac = fast neutron absorption cross section of chromium = 0.014 b
Sigmasc = fast neutron scatter cross section of chromium = _____4.2 b
Sigmaaf = fast neutron absorption cross section of fission products = ____ b
Sigmaau = fast neutron cross section of U-238 = 0.25 b (absorption) + 0.041 b (fissioning)
Sigmasu = fast neutron scatter cross section of U-238 = 9.4 b
Sigmaap = fast neutron absorption cross section of plutonium-239 = 0.040 b
Sigmaaz = fast neutron absorption cross section of zirconium = 0.0066 b
Sigmafp = fast neutron fission cross section of plutonium-239 = 1.70 b
Sigmafu = fast neutron fission cross section of uranium-238 = 0.041 b

FNR TEMPERATURE SETPOINT:
A FNR differs from water cooled reactors in that in a FNR the reactivity R is a well defined function of temperature T. The setpoint temperature To is the value of T at which R = 0.

At R = 0 the reactor design ensures that:
(dR / dT) < 0
which causes the nuclear process to passively maintain the average fuel temperature:
T = To.

The temperature setpoint To is primarily a function of the reactor fuel, coolant and fuel bundle structural geometry.

A practical concern in liquid sodium cooled reactors is that near the boiling point of liquid sodium the thermal coefficient of expansion of liquid sodium increases. This increase might be sufficient to change the sign of (dR / dT) which could cause a prompt critical accident. The liquid sodium would boil which would blow the fuel assembly apart. To prevent that happening it is essential to:
a) Keep the setpoint temperature To far below the boiling point of liquid sodium;
b) To ensure that there is no local loss of liquid sodium coolant flow by ensuring that there are no hot spots in the sodium coolant which might allow the local temperature to rise to the boiling point of liquid sodium;
c) To operate with a sufficiently negative value of [dR / dT] such that even if the sodium locally boils the local value of [dR / dT] remains negative. This object is most easily achieved by using Pu-239 instead of U-235 as the fission fuel.

One method of raising the boiling point of liquid sodium in the core zone is to keep a head of almost 10 m of liquid sodium above the core zone, which raises the sodium boiling point in the reactor core zone.

Another method is to monitor the temperture everywhere in the core zone with sufficient resolution to detect any hot spots. This method involves scanning the core zone to find T as a function of position. At steady state :
T ~ To, so in effect this method finds To as a function of position in the core zone.

The object is to adjust the movable fuel bundle insertions so that in each reactor movable fuel bundle:
To ~ 460 degrees C.

A key issue in FNR safety is to design the fuel bundles so that each fuel tube is cooled by multiple adjacent fuel channels. Blocked fuel channels will create a local hot spot and a local reduction in gamma output. Thus the combination of a higher discharge temperature and a lower gamma output than other fuel bundles is indicative of a problem.

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.

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.

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.

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]

Expressed simply, the neutron flux lost by diffusion should be about half the net neutron generation flux. The corresponding FNR fuel assembly is more like a pancake than an upright cylinder.

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:
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 one of these fission neutrons must be captured by a plutonium atom. That capture must happen before the neutron leaves 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 not sufficient to sustain a chain reaction in that zone.

In a FNR started with U-235 fuel there is very little margin for neutron loss from the core zone. Plutonium is a much more practical FNR start 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.

P>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 net 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 the thickness of the core zone is gradually increased 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 maintain reactor power control at the desired reactor temperature setpoint To so the core fuel must be replaced.

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

FNR BASIC CRITICALITY REQUIREMENTS:
1) When the movable active fuel bundles are fully inserted in the matrix of fixed active fuel bundles and the core fuel is nearly fully depleted (average 12.7% Pu) the reactor must still be critical. This is the depleted fuel condition.

2) When the movable fuel bundles are 1.1 m withdrawn and the core fuel is new (average 20% Pu) the reactor must be reliably sub-critical. This is the new fuel cool____ shutdown condition. In this condition the mobile 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 0.5 m of blanket rod material. Both the upper core fuel layer and the lower core fuel layer must each be individually subcritical.

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

PROMPT NEUTRON CRITICALITY SUPPRESSION:
Note that in normal operation the reacting end of the core fuel rod stack is hotter than the opposite end. If the hot end of a core fuel rod stack 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 vapor pressure tends to blow the upper blanket rods into the plenum while holding the core fuel rods in place. In the fixed fuel bundles the 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 fuel rod sodium trapping and end beading are relied upon to suppress a prompt critical condition.

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 required fissile atom concentration in new fuel?"

When the core fuel is new there is a surplus of fission neutrons and about half of the fission neutrons 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 fissile atom concentrtion 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 decrease.

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 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 to maintain the 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 is a function of the core fuel design. These concentrations determine the rate of absorption of neutrons along a neutron random walk path.

REQUIRED DATA:
From Kaye & Laby the cross sections for high energy neutron scattering in a FNR core are:
Sigmass = 3.7 b
Sigmasi = 3.8 b
Sigmasc = 4.2 b
Sigmasu = 9.4 b
Sigmasz = ____

Data from Kaye & Laby for a FNR core:
Sigmaas = fast neutron absorption cross section of sodium = 0.0014 b
Sigmass = fast neutron scatter cross section of sodium = 2.62 b___3.7 b
Sigmaai = fast neutron absorption cross section of iron = .0086 b
Sigmasi = fast neutron scatter cross section of iron = 4.6 b ____3.8 b
Sigmaac = fast neutron absorption cross section of chromium = 0.014 b
Sigmasc = fast neutron scatter cross section of chromium = _____4.2 b
Sigmaaf = fast neutron absorption cross section of fission products = ____ b
Sigmaau = fast neutron absorption cross section of U-238 = 0.25 b + 0.041 b (fissioning)
Sigmasu = fast neutron scatter cross section of U-238 = 9.4 b
Sigmaap = fast neutron absorption cross section of plutonium-239 = 0.040 b
Sigmaaz = fast neutron absorption cross section of zirconium = 0.0066 b
Sigmafp = fast neutron fission cross section of plutonium-239 = 1.70 b
Sigmafu = fast neutron fission cross section of uranium-238 = 0.041 b

Gu = number of neutrons emitted per average U-235 atomic fission = 2.6
Gp = number of neutrons emitted per average Pu-239 atomic fission = 3.1

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

+ Nu [(Gu - 1) Sigmafu - Sigmaau]
= [50.2683676 X 10^26 U atoms / m^3] [1.6 (41 X 10^-3 b) - 250 X 10^-3 b] (b / U atom) [10^-28 m^2 / b]
= - 9269.48 X 10^-5 / m
= -0.0926948 / 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.

SUMMARY OF INITIAL CROSS SECTIONAL AREA FRACTIONS:
From the web page titled FNR Fuel Bundles the core zone cross sectional area fractions are:
Sodium = 0.6013977
Core Fuel = 0.180882
Steel = 0.21772

This web page last updated June 28, 2022.