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NESTet 2008
4 - 9 May 2008 in Budapest, Hungary


30.9. - 3.10. 2008 in Dubrovnic, Croatia
















































































































Taming the Chernobyl Avalanche

Frigyes Reisch
Nuclear Power Safety, KTH, Royal Institute of Technology
S-10691Stockholm – Sweden


Nneutron kinetics equations are one of the cornerstones of the theory of nuclear reactors. The awareness of nuclear engineering students of its importance is a precondition that the new generation will handle the currently operating and future nuclear power plants safely. They have to learn how to design control systems for reactors sensitive to an avalanche situation like power increase.

The classical neutron kinetic equations with six delayed groups are not solved analytically. Here they are solved both numerically and with a corresponding block diagram and applied to a Chernobyl type reactor. The results are displayed graphically.

The Chernobyl type reactors have positive void coefficients. When water is replaced with steam the power is increasing. A sudden increase of the steam content causes a rapid power surge.

The importance of choosing the magnitude of the void coefficient and the parameters for the automatic control system is demonstrated.

1. Introduction

Now, 22 years after the Chernobyl accident it is important for today’s and tomorrow’s generations of nuclear engineers to learn to design control systems for reactors with "runaway" characteristics.

The Chernobyl type of reactor (RBKM) core is a huge graphite cylinder (7 m high, 12 m diameter) and within some 1600 channels with water and steam cooled fuel rods 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 is reduced, causing a positive reactivity contribution. This is the positive void coefficient. After the accident the enrichment of the fuel was increased and the neutron spectrum became harder resulting in a lower positive void coefficient

the Chernobyl Reactor
Figure 1, the Chernobyl Reactor

2. The experiment

With 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.

3. Simplified neutron kinetics equations



time (sec)


neutron flux (proportional to the reactor power)


change of the effective neutron multiplication factor (keff)


sum of the delayed neutron fractions (here 0.006502)

ß i

the i:th delayed neutron fraction


neutron mean lifetime (here 0.001 sec)


i:th decay constant (sec-1)


concentration of the i:th fraction of the delayed neutrons’ precursors,


steady state, when time is zero t=0 all time derivatives are equal to zero,


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 U235 is used








Fraction ßi
0.000215 0.001424 0.001274 0.002568 0.000748 0.000273
Decay constant i


The initial values of the delayed neutrons’ precursors are;











4. Using the MATLAB notations

x(1)=N x(2)=c1………… x(7)=c6 the code is
%Save as xprim7A.m
function xprim =xprim7A(t,x,i)
DeltaK=i*0.010*0.50; %voidcoef=i*0.010pcm/percent void change, void increase 50percent

To study the importance of the magnitude of the void coefficient, it is enough to plot the first colon of the x matrix. The rows of the x matrix are the time steps.
%Save as ReaktorKinA.m
hold on
for i=0:1:3
[t,x]=ode45(@xprim7A,[0 0.2],[1; 17.3387; 46.6885; 11.4775; 8.5316; 0.6561; 0.0907],[] ,i);
hold off

5. The result

Is given in the following plot;

Figure2. Power Increase at the Insertion of 50% Void at Different Void Coefficients

6. Block diagram

Using the same parameters a block diagram is created here with SIMULINK

Delayed group 1
Gain1A = 0.2150 Gain1B = 0.0124 Gain1C = 0 0.0124 Integrator1 = 17.3387 (is the initial value of the first delayed group)

Delayed group 2
Gain 2A = …………………….

GainN = -1.502 [= 6.502 – 5 (= the void reactivity perturbation)]
IntegratorN = 1 (is the initial value of N)

The controller is represented with a zero pole block; (s – 1)/s(s – 1)
Here are the; zeros: [1], poles: [0 1], gain: [1]
Here the absorber rods are represented with an amplifier, the gain is 50

Block diagram of the neutron kinetics

Figure 3. Block diagram of the neutron kinetics (with six delayed groups) and the automatic control system with a PID (Proportional and Integrating “1/s” and Differentiating “s”) controller

In this case study, a 10 % pcm/% void coefficient is used and the perturbation is as earlier, a50% void increase. The system response without an automatic control system is like an avalanche

Figure 4. System response without an effective automatic control system.
Relative neutron flux (Power) vs. time (sec)

There is of course no construction which can take a 500 times power increase in 10 seconds. During the Chernobyl accident the result was a disaster with an elapsed time much less than 10 sec.

7. Automatic control

In theory one can specify a control system and a connected absorber rod actuator to team this transient. To achieve this, the control action must be extremely fast and must start effectively within a fraction of a second. However, to realize a mechanical absorber rod operating device with the required speed is very difficult. For this example a PID controller is chosen and the absorber rods are represented with an amplifier which follows the output of the controller. The result is quit reasonable.

Figure 5. System response with an automatic control system (PID).
Relative neutron flux (Power) vs. time (sec)

The Chernobyl disaster demands many analyses to really understand what happened there and how to avoid anything similar in the future. This article is one such analysis.

8. References

University textbooks on nuclear engineering contain the applied equations. Textbooks on information technology and numerical analyses contain the applied methods.

To be published, ENS 2008
Computing the Chernobyl Avalanche
Calculation of the neutron flux, fuel and moderator temperature transients for Research Reactors
Proceeding of the NESTet, Nuclear Engineering Science and Technology, energy technology Budapest, Hungary 4 -8 May 2008

ENS NEWS, Issue: 2006/13, Neutron Kinetics of the Chernobyl Accident

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