Simulation of Uranium-235 Nuclear Fission Reactor Core
using quasi-Monte Carlo Techniques
with a special appendix
Chaos in the Computational Laboratory:
New Fractal Discovered while Simulating Uranium-235 Fission
Daniar Hussain
CSAP 415
Department of Mathematics
New Trier High School
Winnetka, Illinois
January 1999
Abstract
A computational model of uranium-235 fission was developed using certain simple physical assumptions. Experiments were performed on this model to determine how certain initial conditions (such as temperature) affect the reaction rate, final equilibrium position, and other properties in an effort to understand conditions that lead to a runaway meltdown. In addition, a new fractal was discovered while working with this model.
Table of Contents
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Introduction
Objective
The Reaction
Physical Assumptions
The Reactor
Reactor Simplifications
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Normal Operation
Catastrophic Conditions
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10
Appendix A: Source Code . . . . . . . . . . . . . . . . . . . . . .11
Appendix B: New Fractal . . . . . . . . . . . . . . . . . . . . . . 17
Introduction
Source Code
Initial Conditions
Sample Images
Appendix C: Data . . . . . . . . . . . . . . . . . . . . . . . . . . . .22
Normal Opeartion
Catastrophic Conditions
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24
Background
Introduction
In recent years, nuclear reactions have become an integral part of our lives, whether we realize it or not. The Cold War (1945-1990) between the United States and the former USSR was largely driven by the competition for nuclear weapons. And presently, 8% of American primary energy and as much as 19% of electrical energy comes from nuclear sources. There is still much basic science that needs to be understood before nuclear energy can replace conventional sources, and before nuclear reactors become safe and small enough for daily use.
Objective
To simulate the state of a uranium-235 nuclear fission reactor core under normal operation. To investigate the condition of a runaway reactor moments before meltdown, as occurred in Chernobyl, USSR in 1986.
The Reaction
Uranium undergoes fission if it collides with a neutron with sufficient kinetic energy. This energy is called the activation energy, and is required to attain the activated complex, or the state of highest energy. Energy must be supplied to overcome the large strong nuclear force that exists between the nucleotides (protons and neutrons). Once enough kinetic energy overcomes this strong force, the electrostatic repulsion between positive protons tears the nucleus apart. The precise reaction mechanism does not concern us, just the energy released and final products. The resulting products weight less than the initial reactants, and this mass is converted into energy in the manner described by Einstein’s reactions E = m∙c2. The products are also radioactive, and emit more energy through beta and gamma decay. The following is just one of the many ways in which uranium can undergo fission.
1 235 141 92 1
n + U à Ba + Kr + 3 n ∆H = -3.5 ∙ 10-11 J
0 92 56 36 0
Notice that three neutrons are released in this reaction; these neutrons continue the reaction by colliding with other uranium nuclei. In another variant of uranium fission, only two neutrons may be released. These neutrons make it possible to have a self-sustained fission process, or a chain reaction. In order for the fission process to be self-sustained, at least one neutron from each fission event must go on to split another nucleus. If, on average, less than one neutron causes another fission event, the reaction dies out, and the reaction is said to be subcritical. If exactly one neutron from each fission event causes another fission event, the process sustains itself at the same level and is said to be critical. If more than one neutron from each fission event causes another fission event, the process accelerates rapidly and the heat buildup causes a violent explosion. This happened in the Chernobyl nuclear disaster, and is described as supercritical. This number will be one of the variables that we will investigate.
Note also the large amount of energy released in this reaction (-3.5∙10-11 J/event, which translates to 2.1∙1013 J/mol U-235), compared to the combustion of methane, which releases only 8.0∙105 J/mol CH4.
Physical Assumptions
KE = ½ * m * v2
KEprod – KEreact = -∆H
KEave = NA ∙ (1/2 ∙ m ∙ uave2)
where NA = Avagadro’s number = 6.022∙1023 particles/mole
uave = average velocity of all particles
m = mass of particle
KEave = 3/2 ∙ R ∙ T
T = 2/3 ∙ KEave ∙ 1/R
where R = universal gas constant = 8.31451 J/(mol∙K)
T = temperature of core in degrees Kelvin
The Reactor
To harbor the tremendous energies involved, reactors were designed in which controlled fission could occur. The captured energy is used to heat water to produce steam to run turbine generators, in much the same way that a coal-burning power plant, to generate electricity. In the reactor core, either uranium that has been enriched to approx. 3% U-235 (natural uranium occurs mostly as U-238) or pure uranium-235, is housed in cylinders. A moderator surrounds the cylinders to slow down the neutrons so that the uranium fuel can capture them more efficiently. Control rods, composed of substances that absorb neutrons, are used to regulate the power level of the reactor. The reactor is designed so that should a malfunction occur, the control rods are automatically inserted into the core to stop the reaction. A liquid (usually water) is circulated throughout the core to extract the heat generated by the energy of fission; the energy can then be passed via a heat exchanger to water in the turbine system. A layer of material called the reflector surrounds the core, which reflects escaping neutrons back into the core. In turn the reflector is surrounded by a thermal and biological shield. This extra layer absorbs stray neutrons and radiation to protect the personnel working around the reactor; it also 
functions to contain any radiation during an accidental core meltdown.
Reactor Simplifications
Results
Normal Operation
Since we are measuring the affect of a change, rather than absolute quantities, all units have been scaled up to allow easier manipulations. The following conditions were used as reasonable starting values:
|
Volume of reactor (Vr) |
100 x 100 x 100 |
|
Volume of uranium (Vu) |
2.35 x 2.35 x 2.35 |
|
Volume of neutron (Vn) |
0.01 x 0.01 x 0.01 |
|
Mass of uranium (Mu) |
0.235 |
|
Mass of neutron (Mn) |
0.001 |
|
Total number of particles (Np) |
10000 |
|
Ratio of uranium to neutrons (Ru:n) |
99::1 |
|
Number of neutrons released per fission event (Nn) |
2 |
|
Activation energy (Eact) |
0 |
This simulation was started on Saturday, January 02, 1999 at approx. 10:44 pm, and was run overnight for 10 time units with a time step of 0.01. On Sunday, January 03, 1999 at approx. 10:46 am the simulation was prematurely halted at time unit 8.00. The graphed results can be seen below (for the full data set, see Appendix C).
Catastrophic Conditions
The reactor has reached normal operating conditions under the estimated working values, and should maintain at equilibrium indefinitely. To investigate the condition of a runaway reactor, we can model the removal of all control rods by modifying the Nn (number of neutrons released per fission event) variable. By increasing this value from two to four, we see that the reactor temperature increases out of control.
Note that only after 1.2 time units, the temperature of the core has reached a celestial 250,000 temperature units, and no end is in sight. At that high a temperature, it is clear that a meltdown would occur. This is exactly what happened at Chernobyl. After fixing a problem with the coolant distribution system, the workers at Chernobyl decided to test the whole reactor during a short downtime period. The operators thought that the reactor temperature was not increasing fast enough, so they raised most of the rods, in an effort to increase energy output. When they lowered the last rod, the safety system activated, releasing the emergency rods, and the reaction halted. The operators decided to get the manager to shutdown the failsafe system, which he did. The operators proceeded to raise the rods once again. As soon as the last rod was raised, they realized that the temperature was increasing out of control, but it was too late… seconds later, the reactor exploded, sending the ten ton reactor lid five miles into the air, and spewing millions of tons of radioactive material from the core. This was a disaster that could have been avoided; this simulation gives us an insight into the kinds of core temperatures just before meltdown.
Conclusion
There is still much to be learned from simulations of this kind. An interesting direction is to investigate the affect of initial temperature, concentrations of reactants, ratio of neutrons to uranium nuclei, and activation energy on the properties of the reactor throughout time. Another possibility is to investigate a more general reactor – such as simulating more general chemical equations. This would involve assigning to each particle a representative Schrodinger equation, and following its eigenstate through time. Simulation of some large-scale structure in the universe, such as our Solar System, is another feasible direction with minor adjustments. Both of these extensions would involve the addition of inter-particle forces to account for the affect of gravity or electrostatic attraction and repulsion.
Simulation of all physical parameters would be difficult, however, and most likely impossible using current technology. Simulating only 10,000 particles took an entire day, and real reactions contain millions of billions of particles. Even using a Cray Supercomputer, a useful simulation would take years. For that reason, simulation as a means of predicting experimental values precisely remains a dream for the future.
The great velocity (pun intended) of computational improvement in recent years, as well as the development of the quantum computer (which can simulate 2N classical bits in just N quantum bits), may open new doors in physical simulation. I dream of a day when computer scientists (having become physicists!) will be able to predict the future states of our Universe. But until that day comes, chemists will have to continue to play with their test tubes, and physicists will not be able to abandon their particle accelerators just yet. But the day will come when humanity will ask of the greatest computer ever conceived the Ultimate Question of Life, the Universe, and Everything, and I can almost imagine what the answer several million years hence will be… 42.
Appendix A: Source Code
/* Fission.H */
#ifndef fission_h
#define fission_h
#include <math.h> // for sqrt()
#include <stdlib.h> // for rand(), srand()
#include <time.h> // for time()
#include <iostream.h>
#include <fstream.h>
template<class field>
struct vector3d
{
field x;
field y;
field z;
void operator=( const vector3d& v )
{
x = v.x;
y = v.y;
z = v.z;
};
vector3d operator*( double s )
{
x *= s;
y *= s;
z *= s;
return *this;
};
};
struct particle
{
char name; // identifier
double mass; // atomic mass
vector3d<double> size; // dimensions
vector3d<double> loc; // 3D coordinate location
vector3d<double> vel; // 3D velocity vector
double KE() const // kinetic energy
{
double norm = sqrt(vel.x*vel.x+vel.y*vel.y+vel.z*vel.z);
// |(x,y,z)| = sqrt(x^2+y^2+z^2)
return 0.5*mass*norm*norm;
// KE = 1/2*m*u^2
};
void update(double dt)
// update coordinates based on velocity and elapsed time
{
loc.x += vel.x*dt;
loc.y += vel.y*dt;
loc.z += vel.z*dt;
};
void operator=( const particle& p )
{
name = p.name;
mass = p.mass;
size = p.size;
loc = p.loc;
vel = p.vel;
};
};
class reactor
{
public:
reactor( double w, double l, double h );
void configCore( particle* c, int length );
void configInjector( double x, double y, double z );
void configNuclearParam( double E_act, double num_nuetron );
void run( double length, double step );
// precondition: reactor has been configured
private:
particle* core; // reactor core
int num_part; // number of particles in core
vector3d<double> size; // dimensions of core
vector3d<double> injector; // location of nuetron injector
double temp; // core temperature
double press; // core pressure
double E_act; // activation energy (in joules)
double num_nuetron; // number of nuetrons released per fission event
bool test( double ax, double ay, double az, double w, double l, double h, double bx, double by, double bz );
bool collideParticle( particle& a, particle& b );
void reboundParticle( particle& a, particle& b);
bool collideContainer( particle& a );
void updateDisplay( double time );
};
inline double random( double lower, double upper )
// returns random decimal between lower and upper
{
return double(rand())/pow(2,15)*(upper-lower)+lower;
};
#endif
/* Fission.CPP */
#include "fission.h"
particle U, n; // global definitions of Uranium-235 and nuetron
ofstream fout("data.txt"); // output file
reactor::reactor( double w, double l, double h)
{
size.x = w;
size.y = l;
size.z = h;
num_part = 0;
srand( (unsigned) time(NULL) ); // seed random generator
}
void reactor::configCore( particle* c, int length )
{
num_part = length;
core = new particle[length];
for( int i=0; i<num_part; i++ )
core[i] = c[i];
}
void reactor::configInjector( double x, double y, double z )
{
injector.x = x;
injector.y = y;
injector.z = z;
}
void reactor::configNuclearParam( double Act_e, double N_n )
{
E_act = Act_e;
num_nuetron = N_n;
}
void reactor::run( double length, double step )
{
int i,j;
double time=0;
while( time<length )
{
time += step;
temp = 0;
press = 0;
for( i=0; i<num_part; i++ )
{
temp += core[i].KE();
core[i].update(step);
collideContainer(core[i]);
for( j=i+1; j<num_part; j++ )
{
if( collideParticle(core[i],core[j]) )
reboundParticle(core[i],core[j]);
}
}
temp = temp/num_part;
updateDisplay( time );
}
}
bool reactor::collideParticle( particle& a, particle& b )
// returns true if a collides with b
{
return(
test(a.loc.x,a.loc.y,a.loc.z,a.size.x,a.size.y,a.size.z,b.loc.x,b.loc.y,b.loc.z) ||
test(a.loc.x,a.loc.y,a.loc.z,a.size.x,a.size.y,a.size.z,b.loc.x+a.size.x,b.loc.y,b.loc.z) ||
test(a.loc.x,a.loc.y,a.loc.z,a.size.x,a.size.y,a.size.z,b.loc.x,b.loc.y+a.size.y,b.loc.z) ||
test(a.loc.x,a.loc.y,a.loc.z,a.size.x,a.size.y,a.size.z,b.loc.x,b.loc.y,b.loc.z+a.size.z) ||
test(a.loc.x,a.loc.y,a.loc.z,a.size.x,a.size.y,a.size.z,b.loc.x+a.size.x,b.loc.y+a.size.y,b.loc.z) ||
test(a.loc.x,a.loc.y,a.loc.z,a.size.x,a.size.y,a.size.z,b.loc.x+a.size.x,b.loc.y,b.loc.z+a.size.z) ||
test(a.loc.x,a.loc.y,a.loc.z,a.size.x,a.size.y,a.size.z,b.loc.x,b.loc.y+a.size.y,b.loc.z+a.size.z) ||
test(a.loc.x,a.loc.y,a.loc.z,a.size.x,a.size.y,a.size.z,b.loc.x+a.size.x,b.loc.y+a.size.y,b.loc.z+a.size.z)
);
}
void reactor::reboundParticle( particle& a, particle& b)
{
if( (a.name=='n' && b.name=='U') ||
(a.name=='U' && b.name=='n') &&
a.KE()+b.KE() > E_act )
{
/* if collision of Uranium-235 and nuetron
and energy of collision greater than activation energy
then products created and energy released */
a = U;
a.name = 'X'; // Ba-141
a.mass = 0.141;
a.vel.x = random(a.vel.x*1.05,a.vel.x*1.1);
a.vel.y = random(a.vel.y*1.05,a.vel.y*1.1);
a.vel.z = random(a.vel.z*1.05,a.vel.z*1.1);
b = U;
b.name = 'Y'; // Kr-92
b.mass = 0.092;
b.vel.x = random(b.vel.x*1.05,b.vel.x*1.1);
b.vel.y = random(b.vel.y*1.05,b.vel.y*1.1);
b.vel.z = random(b.vel.z*1.05,b.vel.z*1.1);
while( num_nuetron>0 )
{
n.loc = a.loc;
n.vel.x = random(a.vel.x*1.05,a.vel.x*1.1);
n.vel.y = random(a.vel.y*1.05,a.vel.y*1.1);
n.vel.z = random(a.vel.z*1.05,a.vel.z*1.1);
core[num_part] = n;
num_part++;
num_nuetron--;
}
}
else
{
/* if collision ineffective, particles rebound elastically */
vector3d<double> temp;
temp = a.vel;
a.vel = b.vel;
b.vel = temp;
}
}
bool reactor::test( double ax, double ay, double az, double w, double l, double h, double bx, double by, double bz )
{
return((ax<=bx)&&(bx<=ax+w) && (ay<=by)&&(by<=ay+l) && (az<=bz)&&(bz<=az+h));
}
bool reactor::collideContainer( particle& a )
{
if( a.loc.x <= 0 || a.loc.x >= size.x )
a.vel.x = -a.vel.x;
else if( a.loc.y <= 0 || a.loc.y >= size.y )
a.vel.y = -a.vel.y;
else if( a.loc.z <= 0 || a.loc.z >= size.z )
a.vel.z = -a.vel.z;
else
return false;
press++;
return true;
}
void reactor::updateDisplay( double time )
{
cout << time << '\t' << temp << "\t\t" << press << endl;
fout << time << '\t' << temp << "\t\t" << press << endl;
}
void main( )
{
particle* C;
C = new particle[10000];
U.name = 'U';
U.mass = 0.235;
U.size.x = 2.35;
U.size.y = 2.35;
U.size.z = 2.35;
for( i=0; i<9900; i++ ) {
U.loc = random(0,100);
U.vel = random(90,110);
C[i] = U;
}
n.name = 'n';
n.mass = 0.001;
n.size.x = 0.01;
n.size.y = 0.01;
n.size.z = 0.01;
for( i=9900; i<10000; i++ ) {
n.loc = random(0,100);
n.vel = random(90,110);
C[i] = n;
}
reactor R(100,100,100);
R.configCore(C,10000);
R.configNuclearParam(0,2);
R.run(10,0.01);
}
Appendix B: New Fractal
Introduction
While simulating uranium-235 fission, I noticed strange behavior of the particles when I installed a force-field generator. I decided to plot the motion of just a single particle under the influence of this force. I coded this routine in Visual Basic because I thought I was only debugging my C++ program. I plotted just the two-dimentional cross-section where z=0, and I got something I did not quite expect!
I was thinking… "bug in program while simulating particle in force-field!" That reminded me of the fractal that the mathematician Mandelbrot discovered when he thought he found a bug in his program. After checking his calculations by hand, he realized it was not a bug, but a fractal! That also brought to mind the work of the physicist Lorenz who discovered another fractal while working with thermodynamics data. It was too much of a coincidence that the image that I saw reminded me of a fractal, that I thought it was only a bug, and that I was working with physical data! I decided to investigate further…
Even though Mandelbrot and Lorenz’ fractals were in full color, the one I saw was only in black and white. I thought about how Newton’s fractal generates its color – by mapping the number of iterations to reach a root to a color. This gave me an idea – I decided to plot the course of the particle, and to color any single location with the kinetic energy of the particle at that moment in time. (Kinetic energy is the energy of motion and is defined as ½*mass*velocity2.)
The results I got really surprised me – I was looking at a full color fractal! The image was highly sensitive to initial conditions, a property all fractals share. It was also self-contained, in that any section exactly replicated the whole, another property of fractals. I think some of the images make wonderful wallpaper, a third property many fractals share.
Source Code
Sub Main
' Randomize
' random "data.txt", num_par, picMain
load "data.txt", txtNum.Text
run picMain
End Sub
Public Const num_par = 1
Public par(1 To num_par) As Particle
Type Particle
pos_x As Double
pos_y As Double
vel_x As Double
vel_y As Double
force_x As Double
force_y As Double
radius As Double
mass As Double
End Type
Sub draw(par As Particle, pic As PictureBox)
Dim color As Double
color = 1 / 2 * par.mass * (par.vel_x ^ 2 + par.vel_y ^ 2) * &HFFFFFF
pic.FillColor = color
pic.Circle (par.pos_x, par.pos_y), par.radius, color
End Sub
Sub load(filename As String, num_loc As Integer)
Dim i As Integer, j As Integer
Dim blank As Double
Open filename For Input As #1
For j = 1 To num_loc
For i = 1 To num_par
Input #1, par(i).pos_x
Input #1, par(i).pos_y
Input #1, par(i).vel_x
Input #1, par(i).vel_y
Input #1, par(i).force_x
Input #1, par(i).force_y
Input #1, par(i).radius
Input #1, par(i).mass
Next i
Input #1, blank
Next j
Close #1
End Sub
Sub random(filename As String, num_par As Integer, pic As PictureBox)
Dim i As Integer
Open filename For Output As #1
For i = 1 To num_par
Print #1, Rnd * pic.Width 'pos_x
Print #1, Rnd * pic.Height 'pos_y
Print #1, Rnd * 100 'vel_x
Print #1, Rnd * 100 'vel_y
Print #1, Rnd * 0.000001 'force_x
Print #1, Rnd * 0.000001 'force_y
Print #1, 10 'radius
Print #1, 0.0000005 'mass
Next i
Close #1
End Sub
Sub run(pic As PictureBox)
Dim color As Double
color = &HFFFFFF
Do
For i = 1 To num_par
draw par(i), pic
update par(i), pic
Next i
Loop
End Sub
Sub update(ByRef par As Particle, pic As PictureBox)
par.pos_x = par.pos_x + par.vel_x
par.pos_y = par.pos_y + par.vel_y
If par.pos_x - par.radius < 0 Or par.pos_x + par.radius > pic.Width Then
par.vel_x = -par.vel_x
End If
If par.pos_y - par.radius < 0 Or par.pos_y + par.radius > pic.Height Then
par.vel_y = -par.vel_y
End If
par.vel_x = par.vel_x + par.force_x
par.vel_y = par.vel_y + par.force_y
End Sub
Initial Conditions
Initial conditions can be generated randomly by a call to random() or entered manually into ‘data.txt’. Here are some interesting initial conditions that produce wonderful fractals.
98.12349
5325.065
40.7245
35.4519
4.53527569770813E-06
4.14032697677612E-05
10
0.00005
6646.896
2548.131
26.24342
38.35558
5.35045266151428E-06
5.92458248138428E-05
10
0.00005
4938.833
3733.968
28.97593
14.47812
3.01948010921478E-05
7.74740099906921E-05
10
0.00005
2913.152
5764.135
22.19111
24.67633
2.99430310726166E-05
9.54168200492859E-05
10
0.00005
6720.758
3285.285
2.402815
46.5107
6.16199314594269E-05
9.9836802482605E-05
10
0.00005
1532.918
2752.736
27.48037
24.03073
1.74836337566376E-05
5.67940354347229E-05
10
0.00005
5100
3991.31
26.33623
7.52297
9.86713230609894E-05
7.93076276779175E-05
10
0.00005
2913.152
5764.135
22.19111
24.67633
2.99430310726166E-05
9.54168200492859E-05
10
0.00005
6981.496
3293.477
16.54002
17.41086
1.88483655452728E-05
9.27633166313171E-07
10
0.00000005
373.9109
3403.953
5.328688
21.32683
1.11606419086456E-06
2.79510855674744E-05
10
0.0000005
154.3172
1243.764
43.27494
38.76859
8.13270032405853E-06
1.67262017726898E-05
10
0.0000005
6601.022
5285.118
73.17905
36.00884
9.18186008930206E-05
3.83914709091187E-06
10
0.0000005
6404.5
5680.715
3.967282
5.251133
3.86817395687103E-05
4.63190972805023E-05
10
0.0000005
1861.048
1020.668
35.60353
35.30312
2.57404148578644E-05
8.42517614364624E-06
10
0.0000005
Sample Images
Appendix C: Data
Normal Operation
(time, temperature)
0.01, 3508.92
0.02, 6283.89
0.03, 6195.85
0.04, 6183.65
0.05, 6395.21
0.06, 6542.14
0.07, 6653.35
0.08, 7050.34
0.09, 7086.44
0.1, 6938.86
0.11, 7328.89
0.12, 7484.74
0.13, 7728.36
0.14, 7631.29
0.15, 7675.44
0.16, 7952.15
0.17, 8345.08
0.18, 8034.85
0.19, 8077.46
0.2, 8497.84
0.21, 8386.79
0.22, 8669.32
0.23, 8402.69
0.24, 8606.12
0.25, 8711.55
0.26, 9021.29
0.27, 8829.74
0.28, 8851.72
0.29, 8806.3
0.3, 8925.08
0.31, 8798.5
0.32, 8984.9
0.33, 9080.31
0.34, 9150.35
0.35, 8947.18
0.36, 9252.83
0.37, 9002.4
0.38, 9135.5
0.39, 9134.15
0.4, 9180.44
0.41, 9110.67
0.42, 9289.74
0.43, 9242.41
0.44, 9133.39
0.45, 9262.61
0.46, 9000.99
0.47, 9200.97
0.48, 9063.71
0.49, 9273.15
0.5, 9235.52
0.51, 9309.81
0.52, 9310.72
0.53, 9267.76
0.54, 9232.68
0.55, 9281.8
0.56, 9485.15
0.57, 9298.51
0.58, 9256.91
0.59, 9433.94
0.6, 9188.18
0.61, 9630.65
0.62, 9277.19
0.63, 9270.53
0.64, 9390.12
0.65, 9268.53
0.66, 9563.56
0.67, 9444.23
0.68, 9218.11
0.69, 9332.75
0.7, 9381.9
0.71, 9338.15
0.72, 9232.68
0.73, 9381.73
0.74, 9223.74
0.75, 9367.51
0.76, 9000.39
0.77, 9206.23
0.78, 9398.03
0.79, 9319.96
0.8, 9334.31
0.81, 9273.6
0.82, 9097.87
0.83, 9241.14
0.84, 9236.6
0.85, 9272.18
0.86, 9180.06
0.87, 9382.34
0.88, 9178.52
0.89, 9398.3
0.9, 9163.89
0.91, 9164.42
0.92, 9077.98
0.93, 9192.16
0.94, 9532.71
0.95, 9549.43
0.96, 9192.06
0.97, 9545.76
0.98, 9488.36
0.99, 9267.78
1, 9192.68
1.01, 9476.69
1.02, 9257.37
1.03, 9256.65
1.04, 9512.48
1.05, 9317.6
1.06, 9352.45
1.07, 9510.28
1.08, 9334.53
1.09, 9220.17
1.1, 9428.79
1.11, 9385.46
1.12, 9073.44
1.13, 9564.1
1.14, 9366.04
1.15, 9321.2
1.16, 9449.63
1.17, 9538.38
1.18, 9488.96
1.19, 9344.95
1.2, 9365.96
1.21, 9301.63
1.22, 9261.71
1.23, 9232.84
1.24, 9523.2
1.25, 9311.69
1.26, 9430.34
1.27, 9264.38
1.28, 9363.13
1.29, 9369.3
1.3, 9349.58
1.31, 9489.07
1.32, 9406.83
1.33, 9454.54
1.34, 9302.6
1.35, 9516.42
1.36, 9350.76
1.37, 9197.83
1.38, 9515.38
1.39, 9627.62
1.4, 9403.47
1.41, 9311.98
1.42, 9303.44
1.43, 9424.92
1.44, 9419.46
1.45, 9340.45
1.46, 9341.93
1.47, 9501.34
1.48, 9431.3
1.49, 9376.04
1.5, 9448.08
1.51, 9249.12
1.52, 9179.99
1.53, 9379.22
1.54, 9211.52
1.55, 9521.79
1.56, 9311.94
1.57, 9340.79
1.58, 9290.74
1.59, 9608.71
1.6, 9286.05
1.61, 9528.73
1.62, 9492.6
1.63, 9456.25
1.64, 9388.25
1.65, 9340.43
1.66, 9501.64
1.67, 9359.51
1.68, 9379.92
1.69, 9428.98
1.7, 9382.5
1.71, 9429.75
1.72, 9613.52
1.73, 9274.84
1.74, 9411.09
1.75, 9420.43
1.76, 9124.85
1.77, 9525.1
1.78, 9317.19
1.79, 9392.07
1.8, 9413.63
1.81, 9316.01
1.82, 9361.48
1.83, 9459
1.84, 9449.89
1.85, 9453.5
1.86, 9336.55
1.87, 9355.44
1.88, 9367.03
1.89, 9463.35
1.9, 9457.09
1.91, 9381.05
1.92, 9449.51
1.93, 9317.03
1.94, 9279.21
1.95, 9332.31
1.96, 9423.03
1.97, 9423.16
1.98, 9399.12
1.99, 9388
2, 9457.96
2.01, 9470.19
2.02, 9305.61
2.03, 9415.35
2.04, 9332.25
2.05, 9520.23
2.06, 9363.22
2.07, 9380.19
2.08, 9331.67
2.09, 9449.25
2.1, 9343.84
2.11, 9252.56
2.12, 9355.06
2.13, 9612.21
2.14, 9321.37
2.15, 9376.91
2.16, 9392.55
2.17, 9273.38
2.18, 9416.45
2.19, 9446.44
2.2, 9436.77
2.21, 9304.87
2.22, 9315.47
2.23, 9557.6
2.24, 9364.54
2.25, 9478.66
2.26, 9359.37
2.27, 9438.36
2.28, 9316.5
2.29, 9449.9
2.3, 9312.42
2.31, 9360.41
2.32, 9347.35
2.33, 9639.16
2.34, 9321.01
2.35, 9490.66
2.36, 9351.37
2.37, 9358.05
2.38, 9416.89
2.39, 9322.4
2.4, 9376.91
2.41, 9576.83
2.42, 9318.43
2.43, 9294.94
2.44, 9478.13
2.45, 9332
2.46, 9352.61
2.47, 9445.21
2.48, 9367.03
2.49, 9382.93
2.5, 9449.78
2.51, 9342.73
2.52, 9342.27
2.53, 9466.2
2.54, 9418.14
2.55, 9352.03
2.56, 9336.67
2.57, 9432.43
2.58, 9415.18
2.59, 9345.06
2.6, 9298.15
2.61, 9472.01
2.62, 9380.06
2.63, 9486.79
2.64, 9484.58
2.65, 9395.22
2.66, 9414.92
2.67, 9475.95
2.68, 9490.97
2.69, 9365.55
2.7, 9349.04
2.71, 9451.92
2.72, 9430.1
2.73, 9286.76
2.74, 9406.25
2.75, 9398.07
2.76, 9389.99
2.77, 9360.06
2.78, 9353.18
2.79, 9450.17
2.8, 9465.57
2.81, 9368.44
2.82, 9420.98
2.83, 9405.08
2.84, 9350.29
2.85, 9347.79
2.86, 9399.11
2.87, 9600.18
2.88, 9309.6
2.89, 9350.24
2.9, 9389.24
2.91, 9440.65
2.92, 9483.18
2.93, 9415.02
2.94, 9348.89
2.95, 9407.1
2.96, 9479.69
2.97, 9410.83
2.98, 9435.17
2.99, 9461.15
3, 9460.95
3.01, 9367.06
3.02, 9367.72
3.03, 9335.13
3.04, 9424.97
3.05, 9360.19
3.06, 9477
3.07, 9487.77
3.08, 9337.84
3.09, 9344.98
3.1, 9295.89
3.11, 9539.6
3.12, 9473.33
3.13, 9313.33
3.14, 9284.98
3.15, 9364.12
3.16, 9367.57
3.17, 9408.33
3.18, 9279.58
3.19, 9329.17
3.2, 9202.95
3.21, 9419.9
3.22, 9473.54
3.23, 9406.11
3.24, 9494.96
3.25, 9381.61
3.26, 9419.93
3.27, 9263.34
3.28, 9434.96
3.29, 9363.25
3.3, 9396.25
3.31, 9525.6
3.32, 9414.49
3.33, 9223.5
3.34, 9471.17
3.35, 9361.5
3.36, 9380.67
3.37, 9353.45
3.38, 9368.47
3.39, 9417.47
3.4, 9405.78
3.41, 9377.7
3.42, 9364.24
3.43, 9375.1
3.44, 9363.97
3.45, 9359.58
3.46, 9380.52
3.47, 9280.45
3.48, 9386.79
3.49, 9377.34
3.5, 9398.28
3.51, 9347.39
3.52, 9279.7
3.53, 9427.76
3.54, 9409.99
3.55, 9431.59
3.56, 9435.06
3.57, 9355.43
3.58, 9403.61
3.59, 9435.78
3.6, 9341.63
3.61, 9433.67
3.62, 9426.94
3.63, 9479.59
3.64, 9422.97
3.65, 9349.48
3.66, 9466.46
3.67, 9348
3.68, 9383.86
3.69, 9492.16
3.7, 9387.33
3.71, 9348.96
3.72, 9382.12
3.73, 9327.01
3.74, 9297.18
3.75, 9520.77
3.76, 9382.39
3.77, 9350.35
3.78, 9454.42
3.79, 9398.29
3.8, 9363.53
3.81, 9372.67
3.82, 9469.05
3.83, 9362.23
3.84, 9382.79
3.85, 9391.67
3.86, 9466.33
3.87, 9393.87
3.88, 9312.56
3.89, 9436.22
3.9, 9382.28
3.91, 9350.39
3.92, 9353.1
3.93, 9435.91
3.94, 9383.38
3.95, 9435.56
3.96, 9396.23
3.97, 9344.13
3.98, 9382.36
3.99, 9309.08
4, 9337.99
4.01, 9443.8
4.02, 9399.12
4.03, 9377.92
4.04, 9273.81
4.05, 9332.23
4.06, 9363.81
4.07, 9444.13
4.08, 9365.61
4.09, 9409.06
4.1, 9380.39
4.11, 9362.44
4.12, 9384.97
4.13, 9434.18
4.14, 9434.47
4.15, 9391.16
4.16, 9423.24
4.17, 9465.15
4.18, 9424.7
4.19, 9431.83
4.2, 9311.21
4.21, 9301.87
4.22, 9382.71
4.23, 9332.93
4.24, 9431.74
4.25, 9389.42
4.26, 9370.85
4.27, 9379.55
4.28, 9345.6
4.29, 9515.23
4.3, 9342.09
4.31, 9390.46
4.32, 9467.96
4.33, 9419.95
4.34, 9359.63
4.35, 9364.59
4.36, 9408.7
4.37, 9393.04
4.38, 9433.81
4.39, 9419.47
4.4, 9381.89
4.41, 9363.2
4.42, 9379.65
4.43, 9379.36
4.44, 9366.83
4.45, 9349.82
4.46, 9380.75
4.47, 9392.9
4.48, 9308.17
4.49, 9346.52
4.5, 9409.26
4.51, 9392.79
4.52, 9397.56
4.53, 9395.4
4.54, 9427.85
4.55, 9392.4
4.56, 9423.2
4.57, 9359.29
4.58, 9381.36
4.59, 9395.54
4.6, 9422.24
4.61, 9352.18
4.62, 9421.86
4.63, 9352.77
4.64, 9467
4.65, 9348.72
4.66, 9422.96
4.67, 9408.74
4.68, 9378.8
4.69, 9347.39
4.7, 9377.38
4.71, 9393.46
4.72, 9379.9
4.73, 9387.3
4.74, 9383
4.75, 9391.07
4.76, 9342.12
4.77, 9420.99
4.78, 9458.38
4.79, 9362.75
4.8, 9380.95
4.81, 9389.88
4.82, 9384.16
4.83, 9366.02
4.84, 9380.93
4.85, 9398.09
4.86, 9407.88
4.87, 9316.52
4.88, 9426.88
4.89, 9406.94
4.9, 9411.78
4.91, 9390.67
4.92, 9441.63
4.93, 9446.66
4.94, 9396.65
4.95, 9503.84
4.96, 9355.46
4.97, 9404.45
4.98, 9397.72
4.99, 9402.36
5, 9312.65
5.01, 9405.89
5.02, 9395.53
5.03, 9499.05
5.04, 9436.3
5.05, 9405.25
5.06, 9460.8
5.07, 9388.52
5.08, 9350.4
5.09, 9389.28
5.1, 9378.75
5.11, 9390.94
5.12, 9369.33
5.13, 9495.19
5.14, 9388.28
5.15, 9406.72
5.16, 9426.29
5.17, 9404.18
5.18, 9482.02
5.19, 9365.5
5.2, 9397.04
5.21, 9362.17
5.22, 9397.55
5.23, 9347.99
5.24, 9425.98
5.25, 9403.29
5.26, 9435.99
5.27, 9437.11
5.28, 9397.27
5.29, 9406.67
5.3, 9369.72
5.31, 9408.16
5.32, 9357.17
5.33, 9404.34
5.34, 9398.59
5.35, 9373.88
5.36, 9398.38
5.37, 9404.33
5.38, 9386.49
5.39, 9404.46
5.4, 9499.21
5.41, 9397.8
5.42, 9355.6
5.43, 9405.84
5.44, 9397.83
5.45, 9473.34
5.46, 9396.92
5.47, 9346.48
5.48, 9417.44
5.49, 9404.98
5.5, 9357.39
5.51, 9404.32
5.52, 9395.46
5.53, 9407.71
5.54, 9371.73
5.55, 9381.83
5.56, 9372.17
5.57, 9365.8
5.58, 9395.11
5.59, 9405.26
5.6, 9381.08
5.61, 9352.17
5.62, 9353.06
5.63, 9404.73
5.64, 9481.81
5.65, 9437.88
5.66, 9397.65
5.67, 9404.56
5.68, 9398.83
5.69, 9405.79
5.7, 9395.76
5.71, 9403.55
5.72, 9397.31
5.73, 9405.37
5.74, 9397.39
5.75, 9406.56
5.76, 9396.66
5.77, 9448.54
5.78, 9366.27
5.79, 9406.63
5.8, 9393.97
5.81, 9435.13
5.82, 9394.92
5.83, 9376.8
5.84, 9396.8
5.85, 9405.22
5.86, 9396.84
5.87, 9436.35
5.88, 9397.26
5.89, 9406.07
5.9, 9397.77
5.91, 9406.3
5.92, 9395.48
5.93, 9380.39
5.94, 9395.58
5.95, 9408.43
5.96, 9451.51
5.97, 9384.44
5.98, 9343.66
5.99, 9384.6
6, 9301.61
6.01, 9382.42
6.02, 9374.45
6.03, 9382.08
6.04, 9373.06
6.05, 9383.65
6.06, 9372.99
6.07, 9381.95
6.08, 9376.65
6.09, 9383.36
6.1, 9373.87
6.11, 9383.06
6.12, 9370.91
6.13, 9413.41
6.14, 9373.95
6.15, 9385.33
6.16, 9342.52
6.17, 9336.98
6.18, 9372.19
6.19, 9384.98
6.2, 9400.74
6.21, 9353.9
6.22, 9341.58
6.23, 9384.74
6.24, 9372.2
6.25, 9384.73
6.26, 9341.63
6.27, 9384.31
6.28, 9372.28
6.29, 9382.52
6.3, 9402.34
6.31, 9383.93
6.32, 9372.1
6.33, 9413.65
6.34, 9372.62
6.35, 9382.68
6.36, 9343.6
6.37, 9380.5
6.38, 9374.4
6.39, 9384.16
6.4, 9372.44
6.41, 9383.32
6.42, 9372.97
6.43, 9381.21
6.44, 9373.59
6.45, 9385.54
6.46, 9372.11
6.47, 9382.09
6.48, 9373.21
6.49, 9384.76
6.5, 9373.32
6.51, 9408.72
6.52, 9373.34
6.53, 9384.2
6.54, 9326.61
6.55, 9386.42
6.56, 9371.51
6.57, 9385.14
6.58, 9372.42
6.59, 9382.93
6.6, 9371.77
6.61, 9348.46
6.62, 9342.94
6.63, 9382.8
6.64, 9375.09
6.65, 9382.25
6.66, 9372.19
6.67, 9385.98
6.68, 9402.22
6.69, 9352.38
6.7, 9373.06
6.71, 9384.93
6.72, 9371.44
6.73, 9385.49
6.74, 9371.5
6.75, 9382.73
6.76, 9412.76
6.77, 9341.27
6.78, 9461.98
6.79, 9384.77
6.8, 9412.55
6.81, 9384.9
6.82, 9331.95
6.83, 9386.01
6.84, 9331.12
6.85, 9385.99
6.86, 9371.65
6.87, 9385.11
6.88, 9372.52
6.89, 9382.67
6.9, 9372.01
6.91, 9383.33
6.92, 9373.52
6.93, 9422.7
6.94, 9370.82
6.95, 9384.96
6.96, 9372.04
6.97, 9383.16
6.98, 9372.09
6.99, 9382.8
7, 9375.49
7.01, 9428.89
7.02, 9334.94
7.03, 9383.06
7.04, 9411.64
7.05, 9384.72
7.06, 9374.04
7.07, 9344.65
7.08, 9412.61
7.09, 9346.04
7.1, 9331.62
7.11, 9346.18
7.12, 9372.54
7.13, 9384.79
7.14, 9370.74
7.15, 9346.68
7.16, 9413.71
7.17, 9387.7
7.18, 9370.63
7.19, 9384.31
7.2, 9410.6
7.21, 9382.63
7.22, 9372.37
7.23, 9382.76
7.24, 9374.53
7.25, 9383.4
7.26, 9371.23
7.27, 9385.4
7.28, 9374.4
7.29, 9384.84
7.3, 9372.51
7.31, 9386.73
7.32, 9330.46
7.33, 9430.79
7.34, 9371.42
7.35, 9381.95
7.36, 9371.41
7.37, 9345.17
7.38, 9410.75
7.39, 9384.27
7.4, 9330.73
7.41, 9384.45
7.42, 9372.37
7.43, 9383.44
7.44, 9333.73
7.45, 9423.24
7.46, 9334.31
7.47, 9422.16
7.48, 9373.09
7.49, 9422.73
7.5, 9373.37
7.51, 9382.93
7.52, 9372.54
7.53, 9383.95
7.54, 9372.25
7.55, 9383.69
7.56, 9372.45
7.57, 9383.05
7.58, 9373.95
7.59, 9336.44
7.6, 9411.59
7.61, 9414.24
7.62, 9334.84
7.63, 9383.34
7.64, 9412.92
7.65, 9383.7
7.66, 9372.62
7.67, 9383.91
7.68, 9341.68
7.69, 9384.18
7.7, 9373.98
7.71, 9384.61
7.72, 9372.07
7.73, 9383.99
7.74, 9372.2
7.75, 9384.55
7.76, 9373.03
7.77, 9343.96
7.78, 9413.46
7.79, 9383.63
7.8, 9372.65
7.81, 9383.67
7.82, 9372.27
7.83, 9383.5
7.84, 9372.26
7.85, 9384.65
7.86, 9371.45
7.87, 9385.05
7.88, 9372.84
7.89, 9384.76
7.9, 9370.98
7.91, 9385.34
7.92, 9371.17
7.93, 9385.94
7.94, 9373.38
7.95, 9354.51
7.96, 9347.24
7.97, 9383.2
7.98, 9371.76
7.99, 9382.22
8, 9373.58
Catastrophic Conditions
(time, temperature)
0.01, 3508.95
0.02, 143484
0.03, 143649
0.04, 143699
0.05, 143880
0.06, 144055
0.07, 144334
0.08, 144436
0.09, 264942
0.1, 128980
0.11, 129286
0.12, 129932
0.13, 130043
0.14, 165339
0.15, 165843
0.16, 257079
0.17, 137927
0.18, 137866
0.19, 138187
0.2, 139222
0.21, 173568
0.22, 139848
0.23, 229937
0.24, 140155
0.25, 140334
0.26, 140623
0.27, 140945
0.28, 163513
0.29, 129274
0.3, 219526
0.31, 129685
0.32, 188211
0.33, 189886
0.34, 189915
0.35, 212206
0.36, 189910
0.37, 266239
0.38, 176069
0.39, 210850
0.4, 212170
0.41, 211355
0.42, 211573
0.43, 188978
0.44, 246744
0.45, 216223
0.46, 216475
0.47, 215757
0.48, 216771
0.49, 216886
0.5, 216927
0.51, 274060
0.52, 292810
0.53, 247333
0.54, 282784
0.55, 247387
0.56, 188924
0.57, 248186
0.58, 247432
0.59, 323268
0.6, 247332
0.61, 283281
0.62, 247945
0.63, 248201
0.64, 306452
0.65, 247570
0.66, 248033
0.67, 171634
0.68, 282568
0.69, 247125
0.7, 247958
0.71, 306365
0.72, 247925
0.73, 248162
0.74, 247202
0.75, 190459
0.76, 155689
0.77, 190586
0.78, 306222
0.79, 247879
0.8, 247789
0.81, 248187
0.82, 248197
0.83, 248788
0.84, 247971
0.85, 270278
0.86, 189583
0.87, 247807
0.88, 248223
0.89, 248034
0.9, 248071
0.91, 247946
0.92, 247745
0.93, 190504
0.94, 270942
0.95, 248084
0.96, 247653
0.97, 248104
0.98, 248217
0.99, 324580
1, 247587
1.01, 305750
1.02, 249207
1.03, 248451
1.04, 248778
1.05, 248503
1.06, 248228
1.07, 248120
1.08, 248127
1.09, 248043
1.1, 247319
1.11, 248405
1.12, 247887
1.13, 248094
1.14, 248137
1.15, 248427
1.16, 248352
1.17, 248030
1.18, 248272
1.19, 248168
1.2, 248149
1.21, 247607
1.22, 248159
1.23, 248558
1.24, 248314
1.25, 248484
References
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Gale, Dr. Robert Peter, 1988. Final Warning: The Legacy of Chernobyl, Warner Books, Inc.
Kauzmann, Walter, 1957. Quantum Chemistry: An Introduction, Academic Press Inc.
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Rickard, Graham, 1989. The Chernobyl Catastrophe, The Bookwright Press
Schichtel, Bret A., Husar, Rudolf B., 1995. Regional simulation of atmospheric pollutants with the capita monte carlo model. Center for air pollution and trend analysis, Washington University, St. Louis, Montana.
Zumdahl, Steven S., 1993. Chemistry, 3rd ed. D.C. Heath and Co., Massachusetts. pp. 995-996, 1016-1023.
Compton’s Interactive Encyclopedia, 1995. Compton’s NewMedia, Inc.