European Nuclear Society
e-news Issue 20 Spring 2008
http://www.euronuclear.org/e-news/e-news-20/neutron-flux.htm

Calculation of the neutron flux, fuel and moderator temperature transients for Research Reactors

F. REISCH
Nuclear Power Safety, KTH, Royal Institute of Technology
Alba Nova, Roslagstullsbacken 21, S-106 91Stockholm – Sweden

ABSTRACT

When withdrawing or inserting control rods in the core of a research reactor generally only the end values of the resulting neutron flux are calculated. This code offers a possibility to - in advance - describe the whole course of the changes of the neutron flux, the fuel temperature and the moderator temperature. The reactor kinetics equations are used with six delayed neutron groups, the fuel and moderator thermal dynamics equations, first in the form of Laplace transform with simple time delays and than as first degree differential equations. This set of nine differential equations coupled together is solved numerically.

1. Introduction

The classical reactor kinetic equations with six groups of delayed neutrons (point kinetics) are not solved analytically. In the current program the fuel and the moderator thermal dynamic equations are coupled to the reactor kinetic equations. The equation system is solved numerically. This short program is suitable for use by nuclear engineering students when practicing at research reactors. The parameters to be used depend of course on the reactor design.

2. Simplified neutron kinetics equations

are

Here

N(0)=1

Table 1: Delayed neutron data for thermal fission in U²³⁵ 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	0.0124	0.0305	0.111	0.301	1.14	3.01

Table 2: The initial values of the delayed neutrons’ precursors are;

Using the MATLAB notations; x(1)=N x(2)=c1 ………… x(7)=c₆

3. Fuel

The fuel temperature change (T_FUEL) follows after the power with a time delay ()

The differential equation form is

;

At steady state (equilibrium) d/dt=0 N(0)=1
Suppose that at zero power the fuel temperature changes by 0.001 ⁰C when N=1 and thereby c_NF=0.001

Suppose =5 sec =0.2 =0.0002 ⁰C/sec

With the MATLAB notation x(8) = T_FUEL
and the neutron kinetics equations can be expanded to include the fuel dynamics
0.0002*x(1)-0.2*x(8)

3.1 The Doppler reactivity of the fuel is

Here

The reactivity of the Fuel’s Doppler effect is

with MATLAB notation; DeltaKfuel = – 3.1.10^-5 *x(8) + 0.0031.10^-5

4. Moderator

The differential equation for the moderator is similar to that of the fuel, when the moderator thermal time constant is much bigger then the fuel thermal time constant >>

With the MATLAB notation x(9) = T_Moderator; and the neutron kinetics equations can be expanded to include the moderator dynamics too; 0.000005*x(1)-0.01*x(9)

4.1 Moderator reactivity contribution from temperature change

Here

The reactivity contribution from the changing moderator temperature is

= -0.6.10^-5.(T_Moderator– 0.0005)

5. Control Rods

5.1 The reactivity balance with the control rods, the fuel’s Doppler effect andthe moderator’s temperature effect

The reactivity balance with MATLAB notation;
DeltaK = DeltaKcr + DeltaKfuel + DeltaKmoderator

6. Results of the Computation

In Figure 1 there is a diagramof the control rod reactivity used in the calculations
In Figure 2 the calculated relative neutron flux is displayed
In Figure 3are displayed the characteristics of the fuel and moderator temperature increase. The values are very small as here the calculations are performed for zero power operation when practically no power is generated in the fuel and transferred into the moderator. However, the curves clearly demonstrate that the fuel’s thermal time constant is much smaller than that of the moderator’s

Figure 1, Schematic of the control rod reactivity

Figure 2, Relative neutron flux

Figure 3, Characteristics of the fuel and moderator temperature increase

7. The Code

contains two parts

Part one

%Save as xprim9FM.m

function xprim = xprim9FM(t,x,i)

DeltaKcr=i*10^-5;
DeltaKfuel=-3.1*10^-5*x(8)+0.0031*10^-5;
if t>=0 & t<10
DeltaKcr=((i*10^-5)/10)*t;
end
if t>60 & t<70
DeltaKcr=(10^-5)*(i-8*(t-60));
end
if t>70
DeltaKcr=-30*(10^-5);
end
DeltaKmoderator=-0.6*10^-5*x(9)+0.0003*10^-5;
DeltaK=DeltaKcr+DeltaKfuel+DeltaKmoderator;
xprim=[(DeltaK/0.001-6.502)*x(1)+0.0124*x(2)+0.0305*x(3)+0.111*x(4)+0.301*x(5)+1.14*x(6)+3.01*x(7);
0.21500*x(1)-0.0124*x(2);
1.424000*x(1)-0.0305*x(3);
1.274000*x(1)-0.1110*x(4);
2.568000*x(1)-0.3010*x(5);
0.748000*x(1)-1.1400*x(6);
0.273000*x(1)-3.0100*x(7);
0.000200*x(1)-0.2000*x(8);
0.000005*x(1)-0.0100*x(9)];

Part two
%Save as ReaktorKinFM.m

a=50;
b=10;
c=60;

figure
hold on
for i=a:b:c %i is the max Control Rod reactivity i pcm
[t,x]=ode45(@xprim9FM,[0 80],[1; 17.3387; 46.6885; 11.4775; 8.5316; 0.6561; 0.0907;0.001; 0.0005],[] ,i);
plot(t,x(:,8))
end
hold off

figure
hold on
for i=a:b:c %i is the max Control Rod reactivity i pcm
[t,x]=ode45(@xprim9FM,[0 80],[1; 17.3387; 46.6885; 11.4775; 8.5316; 0.6561; 0.0907;0.001; 0.0005],[] ,i);
plot(t,x(:,9))
end
hold off

figure
hold on
for i=a:b:c %i is the max Control Rod reactivity i pcm
[t,x]=ode45(@xprim9FM,[0 80],[1; 17.3387; 46.6885; 11.4775; 8.5316; 0.6561; 0.0907;0.001; 0.0005],[] ,i);
plot(t,x(:,1))
end
hold off

figure
hold on
for i=a:b:c
x=[0,10,60,70,80];
y=[0,i,i,-30,-30];
plot(x,y)
end
hold off

8. References

University textbooks on nuclear engineering, thermal dynamics and control engineering contain the applied equations. Textbooks on information technology and numerical analyses contain the applied method used to solve the differential equations..

© European Nuclear Society, 2008