Chapter 8
Energy Transport
Background
• Chapter 5 dealt with how the neutron
population varies in time
• Chapter 6 and 7 dealt with the spatial
distribution of neutrons in the core
– It was determine that a critical reactor could
operate at any power level and that the
equilibrium would hold
•N
Nott entirely
ti l ttrue!!
• At high power levels, temperature changes will
create important transients
Objectives
• Find simple expressions to approximate
fuel temperature and coolant temperature
in a reactor
– Steady-state temperatures
–Transients
Transients
• Determine temperature effect on reactivity
Core Power Density
Density
P''' = P/V
• P’’’ is the power density
• P is the total core power
power
• V is the core volume
Power Peaking Factor
Factor
Fq = P'''
max/P'''
• Fq is the power peaking factor
• P’’’max is the maximum power density in
the core
P=
P'''
max
Fq
V
• We can combine the two previous eq
quations and
find the above relation
• Cores are designed to operate at a given power
P
– Maximizing the ratio P’’’max / Fq will allow for smaller
reactor design a
att given power P
P
• Reduces construction cost
– Main job of reactor physicist is to maximize this ratio •
•
•
•
Control rods
New designs
Varying enrichment
…
• P’’’ max depends primarily on material
material
properties
– Temperature and pressure that can be
tolerated by fuel, coolant, structure
• Fq can be lowered by playing with
•
enrichment loading, position of control
rods poisons,
rods,
poisons reflector
reflector, …
For a uniform bare reactor
– Fq = FrFz
f r (r)
f z (z)
From chapter 7, we’ve seen the
solution for such a reactor
– F r = 2.32
– F z = 1.57
More complicated geometries
will have higher peaking due to
local variations
f q (r,z) = f r (r)f z (z)
Image by MIT OpenCourseWare.
Simple heat transfer on fuel ellement
• Define q’
q has the linear heat rate (kW/m)
– Thermal power produced per unit length
– P’’’ = q
q’ / Acell where A
A
cell is the area of the fuel
element
q’ = PFqAcell / V = PFq / NH
N: Number of fuel pins
H: Height of the core
Steady
Steady-state
state temperatures
• Temperature drop from fuel to coolant is
proportional to linear heat rate
Tfe(r, z) - Tc(r,z) = Rf'e q'(r,z)
–R’fe is the fuel element thermal resistance,
details of which are found in Appendix D
• function
f
ti off the th
thermallcon d
ductivity
ti it off the fuel
f l and
d
the cladding as well as the heat transfer coefficient
• If we average over the volume
Tf - Tc = Rf P
wh
here
1
R'fe
Rf =
NH
It should be noted that when averaging q’
using the previous relation q’ = PFq / NH, we
must note that the average
g core peaking
p
g
factor is 1.
We know Rf and P, but we need more information to evalute Tf and Tc Outlet temperature = T0
Coolant heat balance
Wchcp[T0(r) - Ti] =
_
H/2
ò -H/2 q' (r, z')dz'
Replacing q' and solving for T0
Flow rate = Wch
H/2
1
P
fz(z)dz + Ti
f (r)
T0(r) =
Wcp NH r
-H/2
ò
_
Define total core flow rate
W = NWch
Inlet temperature = Ti
Image by MIT OpenCourseWare.
• We get
T0(r) =
1
Pf (r) + Ti
Wcp r
• If we average radially
T0=
1
P + Ti
Wcp
• Average coolant temperature
Tc=
1
(To + Ti )
2
• We can than get an expression for the coolant
temp
perature
1
Tc =
2Wcp
P + Ti
• And we can replace it in our previous expression
(
Tf
=
Rf +
(
1
P + Ti
2Wcp
Fuel Thermal transients
• If cooling is turned off
off, we can
approximate that the fuel will heat up
adiabatically
Mf cf
d
1
[T (t) - T (t)]
Tf (t) = P(t) c
dt
Rf f
Mf is the total fuel mass
cf is the fuel specific heat
• Bounding cases
– Steady-state d/dt =0 P = (Tf - Tc)/Rf
– No cooling (Rf tends to infinity)
Mf cf
d
T (t) = P(t)
dt f
• All the power stays in the fuel, fuel temperature increases and can eventually
y lead to melting
g
• More convenient form
d
1
P(t) - 1 [Tf (t) - Tc(t)]
Tf (t) =
dt
Mf cf
τ
IJ is
i the
th core thermal
h
l time
i
constant
Coolant thermal transient
• Consevation equation in the coolant
coolant
1 [T (t) - T (t)] - 2Wc [T (t) - Ti]
Mc cp d Tc(t) =
c
p c
Rf f
dt
Heating term
1 [T (t) - T (t)]
c
Rf f
Cooling term
2Wcp [Tc (t) - Ti]
• We
e cca
an re
ewrite
e ha
as
d
1 [T (t) - T (t)] - 1 [T (t) - T ]
Tc(t) =
c
i
dt
τ' f
τ'' c
where
τ' =
Mc cp
Mf cf
τ
and
τ'' =
Mc cp
2Wcp
:
• Coolant usually
y follows fuel surface
transient quite rapidly
–We
We can ignore the energy storage term of the
coolant equation
Tc(t) =
1
[2Rf Wcp Ti + Tf (t)]
1 + 2Rf Wcp
• Combining with the fuel transient
expression
d
1
1
P(t) - ~ [Tf (t) - Ti ]
Tf (t) =
dt
Mf cf
τ
Chapter 9
Reactivity Feedback
Background
• Temperature increase will create feedback
feedback
mechanisms in the reactor
–Doppler
pp broadening
g
– Thermal expansion
– Densityy changes which will induce sp
pectral
shifts
• These changes will impact the reactivity,
thus causing transients
Reactivity Coefficients
• Dynamic reactivity was defined by
by
k(t) - 1
ρ(t) =
k(t)
• We can relate a change in reactivity to a
chang
ge in k
dp = dk/k2 ~ dk/k = d(ln k)
• The advantage is that we change a
multiplication of terms into a sum of terms
Fuel Temperature Coefficient
• Doppler broadening of
the resonance capture
cross-section of U-238 is
the dominant effect in
LWR reactors
– Lots of U-238 (red)
present
– Similar effect with
Th-232 (green)
Image removed due to copyright restrictions.
Fuel Temperature Coefficient
• Effect is felt in resonance escape
p p
probability
y (p)
(p)
• No effect on ε,because …
• Minor effect on η and f
– Especially in the presence of Pu-239
• Doppler effect arises from the temperature
depend
dence off the
th cross-secti
tion on th
the rellative
ti
speed between neutron and nucleus
– Resonances are smeared in energy has temperature
increases.
– Thermal resonances are more important
Fuel Temperature Coefficient
• We can approximate it by
1 k ~1 p
αf =
k Tf p Tf
• You can evalluatte it usiing formullas from
Chapter 4 used to determine p (see book)
• Or, you can allso run siimullatitions att
different fuel temperatures and compare
the estimate of the eigenvalue
Moderator Temperature Coefficient
Coefficient
• We seek to evaluate
αm =
1 k
k Tm
• The biggest impact of the moderator
temperature comes from associated changes in
density
– As temperature increases, moderator (and coolant))
will see a decrease in their density
• Less water molecules, means less moderation, leading to a
spectral shift
Moderator Temperature Coefficient
• Decrease in slowing
g down efficiency
y will lead to
an increase in resonance absorption
– Value of p will decrease
• Lower coolant density will also have an impact
•
on the thermal utilization (f)
– Value of f will increase
• Fast fission will increase slightly, but eff
ffect is
negligible
• The combine effect is usually negative
negative, but in
some reactors with solid moderators (i.e.
graphite), the coefficient might be positive over
certain temperature ranges
ranges.
Coolant Void Reactivity Coefficient
• In LWRs and BWRs, this coefficient is alwayys neg
gative
– Coolant and moderator are the same, thus losing the coolant
also implies loosing all the moderation
• In CANDU and RBMK,, this coefficient is positive
– Loosing the coolant as very little impact on the moderation
• Causes slight increase in fast fission
• Causes slig
ght increase in resonance escap
pe probability
y
– Before Chernobyl, void reactivity coefficient of RBMK was 4.7
beta, after re-design it was lowered to 0.7 beta
– CANDU have a very small positive reactivity coefficient that can
be controlled easily
Fast Reactor Coefficients
• Leakage plays a more important role in fast
reactor transients
– Decreasing density will make the spectrum harder
• Larger value of η, thus increase in k
– Migration length would also increase
• More leakage,
leakage thus
th s decreasing k
– Overall effect is usually positive
• Doppler effect is smaller in magnitude
•
– Thermal resonances are more affected
Isothermal Temperature Coefficient
Coefficient
• In manyy reactors, the entire core is broug
ght veryy
slowly from room temperature to the operating
inlet coolant temperature
– Reactor at low power
– External heat source
– Decay heat
• Reasonabl
ble approximation
i ti is
i tto assume that
th t th
the
core behaves isothermally
Tff = Tc = Ti
• We can thus define the isothermal temperature
coefficient
αT =
dρfb
dT
=
1 k 1 k
+
k Tf k Tc
αT = αf + αc
Temperature Defect
Defect
• This coefficient allows us to estimate the
amount of reactivity needed to maintain
criticality at high temperature (hot zero
power)
• This reactivity is obtained by integrating
the isothermal temperature coefficient
from room temperature to hot temperature
DT =
Ti α (T)dT
T
Tr
Power coefficient
coefficient
• A far more useful coefficient
coefficient, it takes into
into
account impact of temperature changes
when reactor is operating at full power
power
αP =
dρfb
dP
=
1 k dTf 1 k dTc
+
k Tf dP k Tc dP
• If we assume that power changes are slow
compared to the time required for heat
removal, we can use the steady-state
temperature profiles from Chapter 8 and
derive them with respect to Power
Tc=
(
1
P + Ti
2Wcp
Tf= Rf +
)
1
P + Ti
2Wcp
dTc
1
=
dP 2Wcp
dTf
1
= Rf +
2Wcp
dP
Power coefficient
coefficient
• The power coefficient is thus expressed in
in
terms of both the fuel coefficient and the
moderator coefficient
(
αP = Rf +
1 1 k
1 k
1
+
2Wcp k Tf 2Wcp k Tc
)
αP = Rf αf + (2Wcp )-1 (αf + αc)
• Thus
Thus, as power is increased
increased, positive
reactivity is required to overcome negative
coefficients and maintain criticality
Power Defect
Defect
• As power increases to its operating level,
level,
additional negative reactivity is introduced
perature
byy an increase in temp
• We can evaluate the power defect by the
following
g
Dp =
Tf (p)
Ti
αf (T f )dTf +
Tc (p)
αe (Tc)dTc
Ti
where Tf(P) and Tc(P) are the fuel and coolant temperatures
at power P
Typical values
• Temperature Defect
• Power Defect
• Good exercise: Lewis 9
9.4
4
Excess Reactivity
• Defined as the value of rho if all control poisons
poisons
and rods were removed from the core
– Large excess reactivity are avoided because they
need lots of poison to compensate at BOC (beginning
of cycle) and require extra care
– Creates dangerous scenarios (e
(e.g.
g high worth control
rods become a problem if ejected)
– Strict limits are thus placed on excess reactivityy and
on the reactivity limits of control devices
• Large amount of small control rods
–
–
–
–
•
Temperature feedback causes excess
reactivity to decrease
– Need to pull out control rods
•
Cold shutdown (a)
Cold critical (b)
Hot zero power critical (c)
Full power (d)
If you shutdown, temperature
decreases and excess reactivity is
increased
– Need to insert control rods as you
reduce power
Shutdown m
argin
Temperature defect
ρex
• Negative temperature
coefficients are nice
from a stability and
safety point of view
view,
large negative values
can create excess
reactivity problems
• Plot depicts
b
a
Power defect
c
d
BOL
Time
EOL
Image by MIT OpenCourseWare.
Shutdown Margin
• A minimum shutdown margin is imposed by the NRC
– Reactitivity
it requiired
d to
t sh
hutd
tdown th
the reactor
t no matter
tt in whi
hich
h condi
dition
ti
(cold critical is the one with the most excess reactivity)
– The stuck rod criteria is usually applied
– Normally 1-5% of excess reactivity
• Going from curve a to b removes the excess margins to get to cold
critical
• As the core is heated, the excess reactivity curve goes from b to c,
with the difference being the temperature defect
– Slow temperature increase to reduce mechanical stresses on pipes and
pressure vessel
• As the reactor goes up in power, we approach curve d
– Remaiiniing excess reactitiviitty is wh
hatt allllows tth
he core tto operate
t for a
given cycle
– Typical LWR cycle 1-2 years
• Core designers try to predict excess reactivity curves
– Schedule outages
– Prepare reloading
• Cores are usually reloaded in 3-4 batches, thus in a PWR you
rep
place about 60 assemblies at each cy
ycle
• Typical assemblies will thus stay in the core for 3 cycles or 4.5
years
• Fuel is then sent to spent fuel pools for at least 5 years
• Pool
P l configuration
fi
ti is
i important
i
t t to
t avoid
id criticality
iti lit accident
id ts
• When pool is full, oldest spent fuel elements are put in dry casks
• If they fall short on reactivity, they can reduce
reduce
power to reduce temperature and increase
excess reactivity
• If they under predict the excess reactivity, it
indicates that they loaded more fresh fuel
bundles than they needed
– Reactor is still shutdown on schedule due to
mobilization of workforce
– $$$
• Outages usually last 3-4
•
3-4 weeks
Reactor Transients
• If rapid changes of power occur, steady-
steady
state temperatures cannot be used
–Rod ejjection
– Loss of coolant
–Loss of flow
• We can develop a simple reactor
ynamics model based on the kinetics
dy
relation, and the temperature transient
models
• Power
• Precursors
[ρ(t) - β]
~
d
P(t) + λi Ci (t)
P(t) =
dt
Λ
i
Σ
d C~ (t) = βi P(t) - ~
λi Ci (t)
Λ
dt i
• Fuel temperature
i = 1, 2, 3, 4, 5, 6
d T (t) = 1 P(t) - 1 [T (t) - T ]
i
~
Mf cf
τ f
dt f
• Coolant temperature
Tc(t) = Ti +
1
T (t)
2Rf Wcp f
Feedback effects
effects
• The reactivity will also have to include the
the
temperature feedback effects
ρ(t) = ρi (t) - |αf | [Tf (t) - Tf (0)] - |αc | [Tc (t) - Tc (0)]
• If the reactor is initially critical at power P0
we can evaluate the temperatures and
precursor concentrations using the steadystate relations
Demo – Step insertion
• Beta = 0
0.0065
0065
• Fullll power = 3000 MWth
– T inlet = 300 C
– T fuel = 1142 C
Step of 0.2$
0 2$ - FullPower
Power
Neutron population over all time
Neutron population over all time
5000
3800
4500
3600
4000
3400
3500
3200
3000
2.4
0
x
1
2
3
4
5
6
7
8
9
10
Precursor Population
107
2.02
0
x
1
2
3
4
5
6
7
8
9
10
8
9
10
Precursor Population
107
2
2.2
1.98
2
1.8
3000
1.96
0
1
2
3
4
5
6
No Feedback
7
8
9
10
1.94
0
1
2
3
4
5
6
7
Feedback
Image by MIT OpenCourseWare.
• T fuel at 10 seconds = 1173 C
• Prompt jump brings power to 3700 MWth
MWth
– Stabilizes to 3100 MWth with feedback
Step of 0
0.2$
2$ - Low Power (1 MWth)
Neutron population over all time
1.8
1.6
100
1.4
50
1.2
1
0
1
2
3
4
5
6
7
8
9
10
0
Precursor population
8000
8
7500
6
7000
4
6500
2
6000
Neutron population over all time
150
0
1
2
3
4
5
Feedback - 10s
6
7
8
9
10
0
0
x
0
100
200
300
105
100
400
500
600
700
800
900 1000
700
800
900
Precursor population
200
300
400
500
600
1000
Feedback - 1000s
Image by MIT OpenCourseWare.
• Fuel temperature eventually reaches 333 C
• Power eventually stabilizes to 120 MWth
Step of 1$ at Full power
x
10
Neutron population over all time
105
3
8
4
0
x
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
Precursor population
1017
2.2
2
2.1
1
2.0
0
Neutron population over all time
1
2
3
104
2
6
0
x
0
0.5
1
1.5
2
2.5
No Feedback
3
3.5
4
4.5
5
1.9
0
x
0
0.5
1
1.5
2.5
3
3.5
4
4.5
5
4
4.5
5
Precursor population
107
0.5
2
1
1.5
2
2.5
3
3.5
Feedback
Image by MIT OpenCourseWare.
• Power spikes to 27500 MWth
• Stabilizes to 3567 MWth
• Fuel temperature reaches 1293 C
Ramp insertion 1$/s with Feedback
Neutron population over all time
8000
6000
10000
4000
5000
2000
2.6
0
x
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0
0
Precursor population
107
6
2.4
x
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
3.5
4
4.5
5
Precursor population
106
4
2.2
2
2.0
1.8
Neutron population over all time
15000
0
0.5
1
1.5
2
2.5
Full Power
3
3.5
4
4.5
5
0
0
0.5
1
1.5
2
2.5
3
Low Power
Image by MIT OpenCourseWare.
• Fuel temperature increase is greater at high power
• At low power, negative reactivity feedback is too slow, thus reactor
reaches prompt critical, until temperature increases
• Both
th sit
ituati
tionconverge to the
th same power eventtually
ll
Shutdown (-5*Beta)
( 5 Beta)
Neutron population over all time
3000
2000
2000
1000
1000
0
2
0
x
1
2
3
4
5
6
7
8
9
10
Precursor population
107
0
2
1.5
1
Neutron population over all time
3000
0
x
1
2
3
4
5
6
7
8
9
10
7
8
9
10
Precursor population
107
1.5
0
1
2
3
4
5
6
No Feedback
7
8
9
10
1
0
1
2
3
4
5
6
Feedback
Image by MIT OpenCourseWare.
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http://ocw.mit.edu
22.05 Neutron Science and Reactor Physics
Fall 2009
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