Lectures on Applied Reactor Technology and Nuclear Power Safety
Lecture No 3
Title:
Neutron Poisons
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology
KTH
Spring 2005
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 1
Outline of the Lecture
•
•
•
•
•
•
•
•
•
•
Fixed Burnable Poisons
Soluble Poisons
Non-burnable Poisons
Fission Product Poisons
Production and Removal of Xenon-135
Xenon-135 Response to Reactor Power Changes and
Shutdown
Xenon-135 Oscillations
Production and Removal of Samarium-149
Samarium-149 Response to Reactor Shutdown
Other Neoutron Poisons
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 2
Fixed Burnable Poisons (1)
• During operation of a reactor the amount of fuel
contained in the core constantly decreases
• If the reactor is to operate during long periods, fuel in
excess of that needed for exact criticality must be added
• The positive reactivity due to the excess fuel must be
balanced with negative reactivity from neutron-absorbing
material
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 3
Fixed Burnable Poisons (2)
•
Moveable control rods containing neutron-absorbing materials are
one method used to offset the excess fuel
•
However, using control rods alone may be impractical
– E.g. there is physically insufficient room for the control rods and their
large mechanisms
•
To control large amounts of excess fuel burnable poisons are used
•
Burnable poisons are materials that have a high neutron
absorption cross section that are converted into materials of
relatively low absorption cross section as a result of neutron
absorption
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 4
Fixed Burnable Poisons (3)
• Due to the burnup of the poison material, the negative
reactivity of the poison decreases over core life
• Ideally, these poisons should decrease their negative
reactivity at the same rate the fuel’s excess positive
reactivity is depleted
• Fixed burnable poisons are usually used in the form of
compounds of boron or gadolinium that are shaped into
separate lattice pins or plates, or introduced as additives
to the fuel
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 5
Soluble Poisons (1)
• Soluble poisons, also called chemical shim, produce a
spatially uniform neutron absorption when dissolved in
the water coolant
• The most common soluble poison in PWRs is boric acid
(”soluble boron” or ”solbor”)
• The boric acid in the coolant decreases the thermal
utilization factor, causing the decrease in reactivity
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 6
Soluble Poisons (2)
• By varying the concentration of boric acid in the coolant
(a process referred to as boration and dilution) the
reactivity of the core can be easily varied
• If the boron concentration is increased (boration), the
coolant/ moderator absorbs more neutrons, adding
negative reactivity
• If the boron concentration is reduced (dilution), positive
reactivity is added
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 7
Non-Burnable Poisons (1)
• Non-burnable poison is one that maintains a constant
negative reactivity worth over the life of the core
• While no neutron poison is strictly non-burnable, certain
materials can be treated as non-burnable poisons under
certain conditions – for example hafnium
• The removal – by absorption of neutrons – of one
isotope of hafnium leads to the production of another
neutron absorber, and continues through a chain of 5
absorbers – resulting in a long-lived burnable poison
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 8
Non-Burnable Poisons (2)
•
It is possible to make the reactivity of a poison material that is
usually a burnable poison more uniform over core life through the
use of self-shielding
•
In self-shielding the poison material is thick enough that only the
outer layer of the poison is exposed to the neutron flux
•
The absorptions that take place in the outer layers reduce the
number of neutrons that penetrate to the inner material
•
As the outer layers of poison absorb neutrons and are converted to
non-poison materials, the inner layers begin absorbing more
neutrons, and the negative reactivity of the poison is fairly uniform
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 9
Fission Product Poisons (1)
• Fission fragments generated at the time of fission decay
to produce a variety of fission products
• Fission products are of concern because:
– they become parasitic absorbers of neutrons
– Result in long term source of heat
• Xenon-135 and samarium-149 have the most substantial
impact on reactor design and operation
• Both these poisons have impact on the thermal
utilization factor and thus keff and reactivity
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 10
Production and Removal of Xenon-135 (1)
• The neutron absorption cross section of xenon-135 is
equal to 2.6 x 106 barns
• It is produced directly by some fissions, but it is more
commonly a product of the tellurium-135 decay chain
β−
135
52
Te
→
19 s
β−
135
53
I
→
6.57 hr
β−
135
54
Xe
→
9.10 hr
β−
135
55
Cs
→
2.3 ⋅ 106 yr
135
56
Ba (stable)
• The half-life of Te-135 is so short that it can be assumed
that iodine-135 is produced directly from fission
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 11
Production and Removal of Xenon-135 (2)
• Iodine-135 is not a strong neutron absorber, but decays
to form the neutron poison xenon-135
• 95% of all the xenon-135 comes from the decay of
iodine-135
• Therefore, the half-life of iodine-135 plays an important
role in the amount of xenon-135 present
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 12
Production and Removal of Xenon-135 (3)
• The rate of change of iodine (dI/dt; I is the concentration
of iodine-135) is equal to the rate of production minus
the rate of removal
• The rate of production is just equal to yield from fission =
γI Σf φ, here γI = 0.061 is the fission yield
• The rate of removal is equal to the decay rate (λI I; λI is
the decay constant) plus the burnup rate (σI I φ)
dI
= γ I Σ f φ − λ I I − σ I Iφ
dt
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 13
Production and Removal of Xenon-135 (4)
• Since the microscopic absorption cross section σI is
quite small, the equation for the iodine-135 concentration
can be written as follows
dI
= γ I Σ f φ − λI I
dt
• When the rate of production of iodine equals the rate of
removal, equilibrium exists – the iodine concentration
remains then constant and equal to I0
γ IΣ fφ
0 = γ I Σ f φ − λI I 0 ⇒ I 0 =
λI
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 14
Production and Removal of Xenon-135 (5)
• Since the equilibrium iodine concentration is proportional
to the neutron flux, φ, it is also proportional to reactor
power level
• The rate of change of the xenon-135 concentration
(dX/dt) is equal to:
–
–
–
–
(+) Xenon-135 production from fission
(+) iodine-135 decay
(-) xenon-135 decay
(-) xenon-135 burnup
γ XΣ fφ
λI I
λX X
σ Xφ X
dX
= γ X Σ f φ + λI I − λ X X − σ X φ X
dt
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 15
Production and Removal of Xenon-135 (6)
• The xenon burnup term σ X φ X refers to neutron
absorption by xenon-135 by the following reaction
135
54
Xe+ 01n →
Xe + γ
136
54
• Xenon-136 is not a significant neutron absorber –
therefore neutron absorption by xenon-135 consitutes
removal of poison from the reactor
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 16
Production and Removal of Xenon-135 (7)
• At equilibrium:
γ X Σ f φ + λI I 0
0 = γ X Σ f φ + λI I 0 − λ X X 0 − σ X φ X 0 ⇒ X 0 =
λX + σ X φ
• Since
γ IΣ fφ
I0 =
λI
• Then the xenon-135 concentration at equilibrium is:
X0 =
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
(γ I + γ X )Σ f φ
λX + σ X φ
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 17
Production and Removal of Xenon-135 (8)
• Compare the equilibrium concentrations of iodine-135
and xenon-135:
γ IΣ fφ
I0 =
λI
X0 =
(γ I + γ X )Σ f φ
λX + σ X φ
• Iodine concentration at equilibrium is linearly proportional
to the neutron flux, and thus to the reactor power
• Xenon-135 concentration increases only with lower rate
than linear when reactor power increases
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 18
Production and Removal of Xenon-135 (9)
γ IΣ fφ
I0 =
λI
X0 =
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
(γ I + γ X )Σ f φ
λX + σ X φ
Slide No 19
Xenon-135 Response to Reactor Power
Changes and Shutdown (1)
• When a reactor is shutdown, the neutron flux is reduced
essentially to zero
• Therefore, after shutdown, xenon-135 is no longer
produced by fission and is no longer removed by burnup
•
The only remaining production mechanism is the decay
of the iodine-135 which was in the core at the time of
shutdown
• The only removal mechanism for xenon-135 is decay
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 20
Xenon-135 Response to Reactor Power
Changes and Shutdown (2)
•
•
•
•
•
Because the decay rate of iodine-135 is faster than the decay rate of
xenon-135, the xenon concentration builds to a peak
The peak is reached when the product of the terms λINI is equal to
λXeNXe (in about 10 to 11 hours)
Subsequently, the production from iodine decay is less than the
removal of xenon by decay, and the concentration of xenon-135
decreases
The greater the flux level prior to shutdown, the greater the
concentration of iodine-135 at shutdown; therefore, the greater the
peak in xenon-135 concentration after shutdown
This phenomenon can be seen in Figure on next slide, which
illustrates the negative reactivity value of xenon-135 following
shutdown from various neutron flux levels
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 21
Xenon-135 Response to Reactor Power
Changes and Shutdown (3)
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 22
Xenon-135 Response to Reactor Power
Changes and Shutdown (4)
• During periods of steady state operation, at a constant
neutron flux level, the xenon-135 concentration builds up
to its equilibrium value for that reactor power in about 40
to 50 hours
• Figure on next slide illustrates a typical xenon transient
that occurs as a result of a change in reactor power level
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 23
Xenon-135 Response to Reactor Power
Changes and Shutdown (5)
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 24
Xenon-135 Response to Reactor Power
Changes and Shutdown (6)
• At time zero, reactor power is raised from 50% power to
100% power
• When the reactor power is increased, xenon
concentration initially decreases because the burnup is
increased at the new higher power level
• Because 95% of the xenon production is from iodine-135
decay, which has a 6 to 7 hour half-life, the production of
xenon remains constant for several hours
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 25
Xenon-135 Response to Reactor Power
Changes and Shutdown (7)
• After a few hours (roughly 4 to 6 hours depending on
power levels) the rate of production of xenon from iodine
and fission equals the rate of removal of xenon by
burnup and decay
• At this point, the xenon concentration reaches a
minimum
• The xenon concentration then increases to the new
equilibrium level for the new power level in roughly 40 to
50 hours
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 26
Xenon-135 Response to Reactor Power
Changes and Shutdown (8)
• It should be noted that the magnitude and the rate of
change of xenon concentration during the initial 4 to 6
hours following the power change is dependent upon:
– the initial power level
– the amount of change in power level
• The xenon concentration change is greater for a larger
change in power level
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 27
Xenon-135 Response to Reactor Power
Changes and Shutdown (9)
• When reactor power is decreased from 100% to 50%
power (t = 55 hours), the process is reversed
• There is an immediate decrease in xenon burnup, which
results in an increase in xenon-135 concentration
• The iodine-135 concentration is still at the higher
equilibrium level for 100% power and is therefore still
producing xenon-135 at the higher rate
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 28
Xenon-135 Response to Reactor Power
Changes and Shutdown (10)
• The xenon-135 concentration continues to rise until the
rate of production of xenon-135 becomes equal to the
rate of removal (roughly 7 to 8 hours after the initial
reduction in power level)
• The xenon-135 concentration then gradually decreases
to the new equilibrium level in about 50 to 60 hours
• The magnitude of the xenon peak is greatest if the initial
power level is very high
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 29
Xenon-135 Response to Reactor Power
Changes and Shutdown (11)
• Maximum peak xenon occurs when a reactor that is
operating at 100% equilibrium xenon concentration is
suddenly shut down
• The most rapid possible burnout of xenon occurs when a
reactor is started up and operated at full power while this
maximum peak xenon condition exists
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 30
Xenon-135 Oscillations (1)
• Large thermal reactors with little flux coupling between
regions may experience spatial power oscillations
because of the non-uniform presence of xenon-135
• The mechanism is described in the following four steps:
– (1) An initial lack of symmetry in the core power distribution (for
example, individual control rod movement or misalignment)
causes an imbalance in fission rates within the reactor core, and
therefore, in the iodine-135 buildup and the xenon-135
absorption
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 31
Xenon-135 Oscillations (2)
– (2) In the high-flux region, xenon-135 burnout allows the flux to
increase further, while in the low-flux region, the increase in
xenon-135 causes a further reduction in flux. The iodine
concentration increases where the flux is high and decreases
where the flux is low
– (3) As soon as the iodine-135 levels build up sufficiently, decay
to xenon reverses the initial situation. Flux decreases in this
area, and the former low-flux region increases in power
– (4) Repetition of these patterns can lead to xenon oscillations
moving about the core with periods on the order of about 15
hours
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 32
Xenon-135 Oscillations (3)
• With little change in overall power level, these
oscillations can change the local power levels by a factor
of three or more
• In a reactor system with strongly negative temperature
coefficients, the xenon-135 oscillations are damped quite
readily
• This is one reason for designing reactors to have
negative moderator-temperature coefficients
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 33
Production and Removal of Samarium-149 (1)
• Samarium-149 is the second most important fissionproduct poison because of its high thermal neutron
absorption cross section of 4.1 x 104 barns
• Samarium-149 is produced from the decay of the
neodymium-149 fission fragment as shown in the decay
chain below
β−
149
60
Nd
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
→
1.72 hr
β−
149
61
Pm
→
53.1 hr
149
62
Sm (stable)
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 34
Production and Removal of Samarium-149 (2)
• For the purpose of examining the behavior of samarium149, the 1.73 hour half-life of neodymium-149 is
sufficiently shorter than the 53.1 hour value for
promethium-149 that the promethium-149 may be
considered as if it were formed directly from fission
• This assumption, and neglecting the small amount of
promethium burnup, allows the situation to be described
as follows
• Rate of change of 149Pm = yield from fission - decay
149Pm concentration
• therefore:
dP
= γ P Σ f φ − λP P
dt
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 35
Production and Removal of Samarium-149 (3)
• At equilibrium:
γ PΣ f φ
0 = γ P Σ f φ − λP P0 ⇒ P0 =
λP
• As can be seen, the equilibrium concentration of
promethium-149 is linearly increasing with the neutron
flux and thus with power
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 36
Production and Removal of Samarium-149 (4)
• The rate of samarium-149 formation is described as
follows:
dS
= γ S Σ f φ + λP P − σ S φ S
dt
• Since the fission yield of samarium-149 is nearly zero,
therefore the equation becomes:
dS
= λP P − σ Sφ S
dt
• And at equilibrium:
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
λP P0 γ P Σ f
0 = λP P0 − σ Sφ S0 ⇒
=
σ Sφ
σS
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 37
Samarium-149 Response to Reactor Shutdown (1)
• Since the neutron flux drops to essentially zero after
reactor shutdown, the rate of samarium-149 production
becomes the following:
dS
= λP P
dt
• Because samarium-149 is not radioactive and is not
removed by decay, it presents problems somewhat
different from those encountered with xenon-135, as
illustrated in Figure on the next slide
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 38
Samarium-149 Response to Reactor Shutdown (2)
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 39
Samarium-149 Response to Reactor Shutdown (3)
• The equilibrium concentration and the poisoning effect
build to an equilibrium value during reactor operation
• This equilibrium is reached in approximately 20 days
(500 hours), and since samarium-149 is stable, the
concentration remains essentially constant during
reactor operation
• When the reactor is shutdown, the samarium-149
concentration builds up as a result of the decay of the
accumulated promethium-149
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 40
Samarium-149 Response to Reactor Shutdown (4)
•
The buildup of samarium-149 after shutdown depends upon the
power level before shutdown. Samarium-149 does not peak as
xenon-135 does, but increases slowly to a maximum value
•
After shutdown, if the reactor is then operated at power, samarium149 is burned up and its concentration returns to the equilibrium
value
•
Samarium poisoning is minor when compared to xenon poisoning
•
Although samarium-149 has a constant poisoning effect during longterm sustained operation, its behavior during initial startup and
during post-shutdown and restart periods requires special
considerations in reactor design
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 41
Other Neutron Poisons (1)
• There are numerous other fission products that, as a
result of their concentration and thermal neutron
absorption cross section, have a poisoning effect on
reactor operation
• Individually, they are of little consequence, but "lumped"
together they have a significant impact
• These are often characterized as "lumped fission product
poisons" and accumulate at an average rate of 50 barns
per fission event in the reactor
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 42
Other Neutron Poisons (2)
• In addition to fission product poisons, other materials in
the reactor decay to materials that act as neutron
poisons
• An example of this is the decay of tritium to helium-3
• Since tritium has a half-life of 12.3 years, normally this
decay does not significantly affect reactor operations
because the rate of decay of tritium is so slow
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 43
Other Neutron Poisons (3)
• However, if tritium is produced in a reactor and then
allowed to remain in the reactor during a prolonged
shutdown of several months, a sufficient amount of
tritium may decay to helium-3 to add a significant
amount of negative reactivity
• Any helium-3 produced in the reactor during a shutdown
period will be removed during subsequent operation by a
neutron-proton reaction
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 44
Exercises (1)
• Exercise 7: Write the differential equation that describes
the xenon-135 concentration change after reactor shutdown
• Solution: The differential equation that describe the
xenon-135 concentration is as follows:
dX
= γ X Σ f φ + λI I − λ X X − σ X φ X
dt
Since after reactor shut-down the neutron flux is zero,
the equation becomes as:
dX
= λI I − λ X X
dt
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 45
Exercises (2)
The differential equation that describe the iodine-135
concentration is as follows:
dI
= γ I Σ f φ − λI I
dt
Since after reactor shut-down the neutron flux is zero,
the equation becomes as:
dI
= − λI I
dt
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 46
Exercises (3)
So in summary, after shutdown the xenon-135 and
iodine-125 concentrations are described with the
following equations:
dI
= − λI I
dt
dX
= λI I − λ X X
dt
− λI t
I
=
I
e
0
Concentration of I-135 can be readily found as:
which substituted to the equation for the Xe-135
concentration yields the answer:
dX
= λI I 0 e −λI t − λ X X
dt
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 47
Exercises (4)
• Exercise 8: Solve the differential equation that describes
the xenon-135 concentration change after reactor shutdown
• Solution: The differential equation that describe the
xenon-135 concentration after reactor shut-down has
been found in Exercise 7 and is as follows:
dX
= λI I 0 e −λI t − λ X X
dt
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 48
Exercises (5)
Multiplying both sides of this equation by so called
integrating factor eλ X t dt yields
dXeλ X t = λI I 0e − (λI −λ X )t dt − λ X Xeλ X t dt ⇒
dXeλ X t + λ X Xeλ X t dt = λI I 0e −(λI −λ X )t dt ⇒
(
)
(
)
d Xeλ X t = λI I 0e −(λI −λ X )t dt ⇒ ∫ d Xeλ X t = ∫ λI I 0e −(λI −λ X )t dt ⇒
Xe
λX t
λI I 0 − (λ − λ
=−
e
λI − λ X
I
X
)t
+C
here C is the integration constant, which is found using
the contition: X = X0 @ t = 0:
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 49
Exercises (6)
After substitution, the constant is found as:
λI I 0
C = X0 +
λI + λ X
and the expression for the xenon-135 concentration
becomes:
λI I 0 − λ t
λI I 0 −λ t
−λ t
+
=
X =−
e + X 0e
e
λI − λ X
λI − λ X
λI I 0
(
e −λ t − e λ t ) + X 0 e −λ t
λI − λ X
I
X
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
I
X
X
X
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 50
Exercises (7)
Answer: the xenon-135 concentration after reactor shutdown is described by the following equation:
λI I 0
(
X (t ) =
e −λ t − e λ t ) + X 0 e −λ
λI − λ X
X
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
I
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Xt
Slide No 51
Exercises (8)
• Exercise 9: Derive an expression for the time to attain
the maximum concentration of xenon after shutdown
• Hint: At maximum, the time derivative of the
concentration is equal to zero, that is: dX/dt = 0
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 52
Exercises (9)
• Exercise 10: A homogenized reactor as in Exercise 3
has been operating for a time at an average neutron flux
of 2 x 1018 neutrons/m2s; how long after shutdown will
the xenon concentration reach a maximum and what is
the concentration at that time? Given:
λI = 2.9 × 10−5 s −1 , λ X = 2.1 × 10−5 s −1 , γ I = 0.061, γ X = 0.003, σ f = 580 [b],
σ X = 2.6 × 106 [b]
Hint: Use the number of atoms of U-235 per unit reactor
volume found in exercise 3 to calculate the macroscopic
fission cross section. Next use the expression derived in
Exercise 9.
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 53
Home Assignment #2 due 05-02-02
• A homogenized reactor as in Home Assignment #1 has
been operating for a long time at an average neutron flux
of 2 x 1019 neutrons/m2s;
Given, in addition to HA1 data:
λI = 2.9 × 10−5 s −1 , iodine decay constant
λ X = 2.1 × 10−5 s −1 , xenon decay constant
γ I = 0.061, fission yield of iodine
γ X = 0.003, fission yield of xenon
σ aX = 2.6 ⋅ 106 [b], absorption cross section of xenon
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 54
Home Assignment #2 due 05-02-02
• Problem 1 (5 points):
– Plot the xenon-135 concentration as a function of time after
reactor shut-down in a time range from 0 to 50 hours
– How long after shut-down will the xenon concentration reach a
maximum and what is the concentration at that time?
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 55
Home Assignment #2 due 05-02-02
• Problem 2 (5 points):
– For a poisoning effect on reactivity in a homogenized reactor one
can use the following approximation:
Σ ap
Xσ aX
∆ρ ≈ −
≈−
Σa
Σa
where X is the xenon concentration and Σa is the total absorption
cross section in the reactor. What will be the maximum reactivity
defect ∆ρ after shut-down?
Applied Reactor Technology and
Nuclear Power Safety - Lecture 3
Henryk Anglart
Nuclear Reactor Technology Division
Department of Energy Technology, KTH
Slide No 56