Neutron Kinetics of the Chernobyl Accident
The Chernobyl type of reactor has a positive
void coefficient, which means that when a part of the water is
replaced by steam the power will increase. At the Chernobyl experiment
the steam content in the coolant channels increased suddenly causing
a catastrophic power excursion. The presented analyses give details
about the importance of the magnitude of the void coefficient.
Also the delayed neutrons behaviour is described.
The classical reactor kinetic equations with
six groups of delayed neutrons are not solved analytically. Here
they are solved numerically with MATLAB and applied to the Chernobyl
accident, the results are presented graphically.
Now, 20 years after the accident it is important
for today’s and tomorrow’s generations of nuclear
engineers to learn not to design reactors with runaway characteristics
which can cause an avalanche like power excursion
The Chernobyl type of reactor core is a huge
graphite cylinder (7 m high, 12 m diameter) and within some 1600
channels with water and steam cooled fuel inside. The fission
neutrons are slowed down (thermalised) mainly in the graphite
and a portion of them is absorbed in the water. When a part of
the water is replaced by steam (void) the absorption becomes less,
causing a positive reactivity contribution. This is the positive
void coefficient. After the accident the enrichment of the fuel
was increased the neutron spectrum became harder resulting in
a lower positive void coefficient
At the Chernobyl experiment due to the abrupt
decrease of the speed of the main circulation pumps and the sudden
drop of the reactor pressure at low reactor power and heavy Xenon
poisoning the steam (void) content in the coolant channels increased
suddenly from a few percent to about 50%. Thus the positive void
coefficient  about 30 pcm/%  caused a large reactivity insertion.
The neutron flux and thereby the reactor power
increased very fast. Due to the thermal inertia of the fuel and
the small value of the fuel temperature coefficient the Doppler
effect could not break the power excursion. Therefore, to characterize
the process at the initial phase, to use only the reactor kinetics
equations is sufficient.
The simplified neutron kinetics equations

and 

Here
t time (sec)
N neutron flux (proportional to the reactor power)
k change
of the effective neutron multiplication factor (k_{eff})
ß sum of the delayed neutron fractions (here 0.006502)
ß_{i} the i:th delayed neutron fraction
l neutron mean lifetime (here 0.001 sec)
i i:th
decay constant (sec^{1})
ci concentration of the i:th fraction of the delayed neutrons’
precursors,
At steady state, when time is zero t=0 all time derivatives are
equal to zero,
all d/dt=0 and the initial value of the relative power equals
unity N(0)=1, and also
no reactivity perturbation is present k=0
Delayed neutron data for thermal fission in U^{235}
is used
Group 
1 
2 
3 
4 
5 
6 
Fraction ßi 
0.000215 
0.001424 
0.001274 
0.002568 
0.000748 
0.000273 
Decay constant i 
0.0124 
0.0305 
0.111 
0.301 
1.14 
3.01 
The initial values of the delayed neutrons’ precursors
are;
i 
1 
2 
3 
4 
5 
6 
ci(0) 
17.3387 
46.6885 
11.4775 
8.5316 
0.6561 
0.0907 
If you are interested in the MATLAB code used to generates the
figures below, please click
on the following link
The result
Is given in the following plot
It is obvious that, the more positive the void
coefficient is, the faster the power is increasing. When the void
coefficient is zero the power is not increasing at all. The Western
type boiling water reactors have strong negative void coefficient
meaning that a similar transient would quickly lead to the shut
down of the reactor by itself
Delayed neutrons
To study the course in time of the concentration
of the delayed neutrons’ precursors (c_{1} to c_{6})
is also interesting. The following plot is for +30 pcm/% void
coefficient showing the colons 2 to 7 of the x matrix
It is clear that the precursor concentration
of the fastest growing group of the delayed neutrons is the 6th
where the decay constant is the largest [_{6}=3.01]
and the largest concentration of the delayed neutron’s precursors
at time zero is the 2nd group [c_{2}(0)=46.6885]
To compare the evaluation of the behaviour in
time of the fission neutrons and the six groups of delayed neutrons’
precursors at +30 pcm/% void coefficient all seven colons of the
x matrix are plotted here
Remarkably at the beginning of the transient
[t=0] some of the delayed neutrons’ precursors [c_{1}(0),
c_{2}(0), c_{3}(0), c_{4}(0)] are larger
than the fission neutrons’ flux [N(0)=1], while later, as
the transient evolves the fission neutrons overwhelm grossly the
delayed neutrons’ precursors as expected.
Neutron Kinetics of the Chernobyl
Accident
Frigyes Reisch
Nuclear Power Safety
KTH, Royal Institute of Technology
Stockholm, Sweden
