Coupled first-order IVPs are frequently used in many parts of engineering and sciences. We present a “solver” including three computer programs which were joint with the MATLAB software to solve and plot solutions of the first-order coupled stiff or nonstiff IVPs. Some applications related to IVPs are given here using our MATLAB-linked solver. Muon catalyzed fusion in a
Coupled first-order IVPs are frequently used in many parts of engineering and sciences [
The main aim of the present research is to give a MATLAB-linked solver to solve first-order coupled differential equation which is used in many subjects of the nuclear engineering. Therefore, in the present study, some dynamical problems (which mathematically are coupled first-order IVPs) are studied as examples of the present solver ability. First we explain Muon Catalyzed Fusion (
Basically, consider a first-order coupled IVPs such as
Entering the number of differential equations (unknowns). Inserting initial values of Inserting start and end points of the computations, or in another words independent variables interval. The type of coupled differential equation should be specified. The answer includes “Stiff” or “Nonstiff” cases. The next question is the method in which user wants for executing. Answers includes “ode45 method", “ode23 method", “ode113 method" for the nonstiff case, and also “ode15s method", “ode23s method", “ode23t method", “ode23tb method" for stiff case so that for more information about these MATLAB commands, refer to the MATLAB help [
The basic process of the muon catalyzed fusion in a
Reaction cycle of the Muon Catalyzed Fusion,
Now, consider a homogeneous media in which the
Constant values for solving (
Process | Parameter | Value |
---|---|---|
Relative concentration of deuterium | 0.5 | |
Relative concentration of tritium | 0.5 | |
Muon decay constant | ||
Muonic-atom formation rate | ||
Media ion density | Liquid hydrogen density | |
Sticking coefficient | 0.008 | |
Fusion rate of | ||
Formation rate of | ||
Time of leakage | ||
Muon exchange rate between |
We have solved these coupled dynamics equations in time range of [0 to
At the end of running, neutron concentration is plotted in our calculated time interval and is given in Figure
:
Consider a PWR which has been operating in a suggested time interval. In a PWR reactors, nuclear fuel is Fission fragments production = Fission yields of
Therefore, according to our defined parameters, the coupled first-order differential equations which describes plutonium and uranium isotopes concentrations are given as:
Masses and atom densities of each fuel isotopes at the time of reactor startup.
Material | Mass, kg | Concentration, |
---|---|---|
93246 | 0.0217391 | |
3058.043 | 0.0007700 | |
84314.620 | 0.0209645 |
Effective properties of nuclide for thermal neutron in 1000 MWe PWR (3000 MW thermal power).
Nuclide | Subscript | |||
---|---|---|---|---|
25 | 556 | 1.96 | 0.2398 | |
28 | 2.2342 | 2.3432 | 0.1907 | |
49 | 1618.2 | 1.86 | 0.5430 | |
40 | 2616.8 | |||
41 | 1567.3 | 2.223 | 0.3765 | |
To solve the set of above IVP, ( Effective cross sections remain constant throughout the core and during fuel lifetime. Average neutron flux within the core is constant and is considered to be equal to The time duration at which reactor fuel has to be replaced with the fresh fuel, due to neutronic and/or thermal hydraulic reasons, is about
Using data given in Table
Fuel composition in a PWR after 1 year irradiation. All values are in terms of kg.
Material | BOC | EOC |
---|---|---|
3058 | 2632 | |
84314 | 83604 | |
0 | 210 | |
Fission fragments | 0 | 926 |
The fission fragments are highly radioactive which undergo
As shown in Figure
Nuclear properties for
Nuclide | Nuclide indication | Half-life | Absorption cross section (b) | Direct yield from |
---|---|---|---|---|
11.1 d | 0 | 0.0236 | ||
2.62 y | 845.72 to | 0 | ||
274.2 | 0 | |||
42 d (7 | 31964 | 0 | ||
5.4 d | 13858 | 0 | ||
53.1 h | 1105.6 | 0.0113 | ||
73635 | 0 | |||
2.7 h | 0 | 0 | ||
158.38 | 0 | |||
87 y | 9734.5 | 0.0044 | ||
813.01 | 0.00281 |
The fission-product chain leading to Sm-
According to Figure
The set of equations of (
(a) Uranium-
Another poison of our interest, as the greatest fission product in a nuclear reactor, is xenon
Nuclear properties of fission products of mass
Nuclide | Nuclide indication | Half life | Radioactive decay constant | Absorption cross-section (b) | Direct yield from |
---|---|---|---|---|---|
0 | 0 | ||||
0.0032 | |||||
17.2 | 0 | ||||
stable | 0 | 0 | 0 |
Xenon
We have presented a computer package to solve first-order IVPs with constant and variable coefficients using MATLAB software, in which the solution of a given stiff or nonstiff coupled differential equations with known initial values were found and plotted. In the present paper, some well-known nuclear engineering dynamical problems, related to the IVPs, were given. A major application of IVPs to a real problem is the fuel depletion in a suggested PWR, where it is computed by the present MATLAB-linked “solver”. We used matrices form such as
Our aim here is to bring out a MATLAB-linked solver for researchers to solve coupled IVPs numerically where it is appeared frequently in many cases of nuclear engineering problems. Reader may refer to the appendix to find our written MATLAB-linked solver program.
Three programs which are named
The second program is
The 3rd program is:
This work is supported under academic Grant no. 88-GR-ENG-6. The corresponding author wishes to acknowledge the Research Council of the Shiraz University for their financial support.