Page 308 - Dynamics and Control of Nuclear Reactors

310 APPENDIX F State variable models and transient analysis

(sometimes multiplying each other), the sensitivity analysis can be performed sys-
tematic perturbation of one or more parameters and making several simulation runs.
Because of the availability of fast personal computing, the latter approach is popular
for sensitivity analysis. Examples of typical parameters include, heat transfer coef-
ficients, thermal properties, feedback coefficients, areas and masses used in thermal-
hydraulic equations, and time constants.

F.6 Numerical solutions of ordinary differential equations

Numerous numerical techniques have been developed for solving ordinary differen-
tial Equations [6, 7]. There are three main concerns in numerical solutions:

1. Truncation: This is the error due to approximating an infinite series with a finite
number of terms.
2. Round-off: This is the error due to the necessary carrying of a finite number of
digits.
3. Stability: Stability is concerned with the tendency of errors to increase or
decrease as the solution proceeds.

Numerical techniques are classified as one-step methods or multi-step methods. In
one-step methods, the approximation depends only on information available at
time, t. In multi-step methods, the solution at time, (t+Δt), depends on information
at more than one previous time steps. Some of the more important techniques are the
following.
One-step methods:

1. Euler
2. Runge-Kutta order two
3. Runge-Kutta order four
4. Runge-Kutta Fehlberg

Multi-step methods:

1. Adams-Bashforth
2. Adams-Moulton
3. Adams fourth-order predictor-corrector
4. Milne
5. Simpson
Available solution software packages use one or more of these techniques. Users do
not need to program a solution method when using prepared computer software, but
they should know what goes on in a computer solution. Two of the methods are
described below to illustrate the general procedure. The basic idea is to estimate
the average slope of the solution between time step i and time step (i+1).