Page 313 - Dynamics and Control of Nuclear Reactors

APPENDIX F State variable models and transient analysis 315

input to the system. The solution vector T is then solved iteratively (Eq. (F.65)) for a
large sparse system of equations.
The technique described here can be generalized to a three-dimensional body
with appropriate nodalization and the knowledge of boundary conditions. See
Ref. [6, 8] for more details of solutions to partial differential equations.

F.7.3 Solution of partial difference equations using the finite
element method
Solution of partial differential equations of irregular geometries and boundary con-
ditions with derivatives of the spatial variables lead to increased complexity in prob-
lem formulation. Irregular shapes of boundaries and the need to approximate
derivative conditions by finite differences require establishing proper grid points
[6]. The finite element method (FEM) can overcome some of the numerical issues
faced by the FDM, and has been used for multi-dimensional neutron diffusion solu-
tions [9]. The FEM is used for solutions of problems in various engineering disci-
plines, including analysis of civil and aerospace structures, fluid mechanics, heat
transfer, electrostatics, electromagnetics, wave propagation, and others [10].

Exercises

F.1 Calculate the Laplace transform vector, X(s), for the state variable matrix
equation.
dx
¼ Ax + f
dt

12 1 3
A ¼ , f ¼ , X 0ðÞ ¼
3 4 2 4
F.2 A state variable matrix differential equation is defined as follows.

dx
¼ Ax + bf
dt

12 1
x 1
x ¼ , A ¼ , b ¼ ,
x 2 3 4 2
Determine the transfer function vector. Simplify your answer.
2 3
X 1 sðÞ
6 UsðÞ 7
6 7
6 7
X 2 sðÞ
4 5
UsðÞ