Download - Reactor physics
Reactor physics
Reactor training course Institut für Kernchemie
www.kernchemie.uni-mainz.de
Binding energy
• Binding energy
EB = E - E = - E
E Energy of the free nucleus; E = 0
E Energy of the compound nucleus
E > E
• Mass defect
m = EB / c²
• Mass of the nuclei
m = Z mp + N mn – EB / c²
Fission
Fusion
Fusion: 1 to3.5 MeV / nucleus ca. 20 MeV / fusion
Fission: about 1 MeV / nucleon ca. 200 MeV / fission
Neutron induced fission
Abstand
Po
ten
tiel
le E
ner
gie
Ef = limit for fission
Ef 5 MeV for Z > 90
Neutron induced fission
• Capture of a free neutron
excited compound nucleus (ZK) with excitation energy
EA = EB + Ekin,n Ef
• EB(gg-ZK) > EB(ug-,gu-ZK)
• ug-nuclei: uneven number of n and even number of p or opposite
• gg–nuclei: even number of n and p
Fission of heavy nucleons with neutrons
Nucleus
Compound nucleus
Necessaryneutron energy for fission in MeV
slow neutrons
fast neutrons
Neutron energies
• Slow (thermal) neutrons
Etherm ≤ 0.4 e V
typical thermal energy Etherm = 0.025 eV
• Epithermal neutrons
0.4 eV < Eepi < 10 keV
• Fast neutrons
Efast ≥ 10 keV
Application of research reactors
Uranium
1 g 238U • 20 spontaneous fissions per hour (tunnel effect)• 106 times more -decays
Natural Uranium• 0.72 % 235U• 99.28 % 238U
Enrichment of 235U• Power reactors: 3-4%• Research reactors: 20 % (LEU)
> 20 % (HEU) fuel development
Fission products
U235
Number of fission products nf for 235U
Fission product nf [%]131I (8.05 d) 3.1132Te (77 h) 4.7133Sb (4.1 min) 4.0133Te (63 min) 4.9133I (21 h) 6.9133Xe (5.27 d) 6.6134Te (44 min) 6.9135I (6.7 h) 6.1137Cs (29 a) 6.15140Ba (12.8 d) 6.44143Ce (33 h) 5.7144Ce (285 d) 6.0
a) thermal fission of 233U and 239Pu
b) thermal und 14 MeV-fission of 235U
c) fission by prompt neutrons of 232Th and 238U
Activation
• Structural components of the reactorAluminumStainless steelConcrete
• Air, water
Fission of 235U – operation of reactors
Prompt fission neutrons
= average number of prompt neutrons produced by the fission
For U-235:(E)= 2.43 + 0.106 x EwithE = excitation energy for the neutrons
Fission spectrum
Energy spectrum of the prompt neutrons by thermal fission of 235U
Maxwellverteilung mit
EW = 0,7 MeV
Ē = 2 MeV
Delayed neutrons
After the emission of the 2 n the excitation energy of the nucleus is too small to emit an other neutron. The stability for the decay products is reached by - decay.
Properties of delayed neutrons
Group
Half decay time
s
Mean
energy
keV
Fractional yield for thermal fission of 233U 235U 239Pu
% % %
1 55 250 0.022 0.021 0.007
2 23 560 0.077 0.140 0.063
3 6,2 430 0.065 0.126 0.044
4 2,3 620 0.072 0.253 0.068
5 0,61 420 0.013 0.074 0.018
6 0,23 ----- 0.009 0.027 0.009
0.258 0.641 0.209
Cross section
Probabilities for the neutron
Total cross section =
Probabilities for the interaction Neutron - nucleus
Unit: Barn (b), 1 b = 10-24 cm²
Different microscopic cross sections for the different processes
Microscopic cross sections
For small neutron energies in thermal reactors is , p and 2n 0
scattering
absorption
fission
capture
-radiation
elastic
inelastic
2n-emission
p-emission
Cross section
Cross sections for 235U
fission
fission
total
total
capture
scattering
Cross section for 238U
totalcapture
fission
Cross section for Cadmium
Measurements with and without Cd:Separation of thermal and epithermal neutrons
Characteristics for fissionable materials
For fissionable materials in reactors three important characteristics are
= average number of neutrons produced per fission = ratio of the number of neutrons captured by the fuel
= average number of neutrons produced per neutron capture by the fuel
= / (1 + )
Neutron regeneration for thermal neutrons
Fuel
U( natural ) 2.47 0.837 1.33
U (5% U-235) 2.47 0.272 1.94
U (20% U-235) 2.47 0.202 2.05
U-235 2.47 0.183 2.09
U-233 2.55 0.132 2.29
Pu-239 2.91 0.416 2.02
Multiplication factor k
number of fissions in one generation number of fissions in previous generation
k =
k < 1 under critical power level
k = 1 constant power level
k > 1 over critical power level
Reactivity ρ
• Definition of the reactivity ρ: ρ = (k - 1) / k = k/k
• Unit Percentage [% k/k] or number [k/k] or Dollar [$] and Cent [¢]
• Calculation: 1 $ = 100 ¢ = 0,0073 k/k
1 k/k = 137 $ = 13699 ¢
Reaktivität ρ
< 0 under critical power level
= 0 constant power level
> 0 over critical power level
Four – factor formula
For large reactors an infinite multiplication factor is defined
k = . . p . f
= number of neutrons produced per neutron absorbed in the fuel
= fast fission factor, a correction factor to take into account the fact that some fissions will be produced by fast neutrons
p = resonant escape probability, the probability that a neutron will escape from capture while it is being slowed through the resonance energy range (approximately 1 to 100 eV).
f = the thermal utilization, the fraction of thermal neutrons which are absorbed in in the fuel
Four – factor formula - examples
Homogenous mixture of natural uranium and graphite, both being powder:Number of neutrons produced per absorbed neutron = 1.33Fast fission factor 1.0, Product of resonant escape probability p and thermal utilization f: p.f 0.6
k = . . p . f = 1.33 . 1 . 0.6 = 0.8
Impossible to use such a combination in a reactor
If the uranium is used in rods with diameters of 1 to 2 inch in a matrix of solid graphite (heterogeneous system) p is increased and the product p.f 0.8 Fast fission factor 1.03
k = 1.33 . 1.03 . 0.8 = 1.09
Possible to construct such a reactor
Effektive multiplication factor
Real reactor: Escape of neutrons through the surface (neutron leakage), Absorption of neutrons in 238U or (n,) reactions
keff = k . Ps . Pth
Ps and Pth number of neutrons, which do not escape (probabilities for slow and fast neutrons)
With the reactivity: ρ = (keff - 1) / keff in the unit $
Reactor period T
• Reactor operation at constant power level: keff = 1, = 0
• Suddenly multiplication factor changed by keff
Increase dn of the number of neutrons n per unit volume dn in the time dt :
dn/dt = keff . n / l
with l = mean lifetime of the neutrons (time between generations)
Solution of the differential equation:
n = n0 . exp( keff . t / l)
with l / keff = T = Reactor period or e-folding time
Neutron flux: n = n0 . exp(t/T)
without delayed neutrons
Reactor period T
Example :
Fission neutrons in natural Uranium: l = 0.001 s
Increase of power level of keff = ½ % = 0.005
Reactor period:T = l / keff = 0.001 s / 0.005 = 0.2 s
Neutron flux without delayed neutrons n = n0 . exp(t/T)
Increase of the power level per second of exp(5)
Control of the reactor is not possible
Reactor period T
Delayed neutrons caused an increase of the mean life time l of 0.1 s
Increase of the power level of keff = ½ % = 0. 005
Reactor period: T = l / keff = 0.1 s / 0.005 = 20 s
With delayed neutrons
Increase of the power per second of exp(0.05)
Control of the reactor is possible
Reactor period T
n = n0 . exp(t/T)
Reactor period T or e-folding time: Time, in which the neutron flux changed of the factor e = 2.72
Relative changes of the flux = (1 / T) . 100
Reactor period [s] -50 -100 100 50 20
Rel. flux changes [% s-1] -2 -1 0 1 2 5
Inhour equation
The „inhour equation“ gives the
relationship between the reactivity and
the reactor period in terms of the
delayed neutrons (regarding 6 groups)
and the prompt neutron lifetime.
Inhour equation
Using a time dependent diffusion equation and taking into account the delayed neutrons, it is possible to derive the following equation
ρ = (keff - 1) / keff = l / (T keff) + i / (1 + T/i ) i
withi = number of delayed neutrons of the group i
i = life time of the delayed neutrons of the group i
System of 7 linear in-homogenous differential equations of first order, the differential equations of the reactor kinetics.
Inhour equation
Reactivity: ρ = l / (T keff) + / (1 + T/ )
= mean lifetime of the delayed neutrons = i
i
Case I: Large positive or negative TT >> , 1 << T, ρ <<
ρ = / T ρ 1 / T
Inhour equation
Beispiel: TRIGA ρ = / T
= 0.0073
= 12.3 s
T = 1 h
= 0.0073 . 12.3 / 3600 2.5 . 10-5 (inverse hour)
Inhour equation
Reactivity: ρ = l / (T keff) + / (1 + T/ )
Case II: Small positive T, large reactivities ρ
0 < T << , ρ >
ρ = l /( keff . T) + T = l / ( keff ( - ))
If k is not too far from unit, then T = l / ( - )
If k exceed 1+ß, then the reactor will be critical on prompt neutrons alone. The reactor is prompt critical.
Inhour equation
Für TRIGA
1 dollar [$] = = 0.0073 , 1 cent = 0.01 dollar
2 Dollar - Pulsρ = 2 (2 $) = 0.0146, T = 13.5 ms
Inhour equation for the TRIGA
Rod calibration
Fission of 235U with Thermal Neutrons
1. Neutron absorber rods (k=1, steady state)
2. Neutron moderator (1 MeV 0.025 eV)
1 MeV
0.025 eV
TRIGA Fuel Moderator Elements
TRIGA Fuel:
91 % Zr
1 % H
8 % U (20% U-235)
TRIGA Fuel:
91 % Zr
1 % H
8 % U (20% U-235)
Atomic Ratio:
Zr/H 1/1
Protons in U-Zr-Matrix act as Moderator
Atomic Ratio:
Zr/H 1/1
Protons in U-Zr-Matrix act as Moderator
Decrease of reactivity:
per °C
= -1,2 x 10-4 keff/keff
.
at T = 100°C
= -1.2 x 10-2 keff/keff = - 1.64 $
For comparision: 1 ¢ per kW increase of the power level
TRIGA Fuel Moderator Elements
Prompt negative temperature coefficient
Xenon poisoning
Reason for the Xenon poisoning:
Decay of I-135 into Xe-135 (large capture cross section for thermal neutrons)
• Operation at constant power level: Production of I-135 and Xe
• Increase of the power level: increase of I-135 and Xe
• Decrease of the power level: Xe concentration increases due to the decay of the I-135 and neutrons will be captured by Xe. With a delay time of the half lifetime of I-135 the capture of neutrons decreases and also the power decreases slowly.
• Operation of the reactor after the shut down is not possible when the absorption by Xe is too large.
• Operation possible, when the Xe-135 production is negligible (about 20 h)
Xenon poisoning
a start of reactor operation with not poisoned core
b after fast shut down of the reactor
c after reduction of the power level
d after increase of the power level to the previous level
Time (h)
Xen
on
co
nce
ntr
atio
n
Core excess reactivity
At a power level of 100 kW following contributions a necessary for the core excess reactivity:
0.40 $ for operation
0.50 $ for compensation of the temperature effect
0.60 $ for compensation of the poison (Xe-135)
0.20 $ for compensation of the burn-up
Up to 1.00 $ for compensation of the neutron absorption in samples for irradiation positions
__________
2.70 $
Summary
1. Nuclear fission
2. Reactivity
3. Ideal (infinite) and real (finite) reactor
4. Inhour equation
5. TRIGA fuel (moderation, prompt negative temperature coefficient, inherent safe reactor)
6. Xenon poisoning
More information to the operation of the TRIGA Mainz
1. Structure
2. Instrumentation
3. Cooling- and purification circuits
4. Radiation protection
5. Safety
6. Checks (internal – external)
7. Special incidents
8. Documentation
9. Organization