Page 310 - Dynamics and Control of Nuclear Reactors

312 APPENDIX F State variable models and transient analysis

F.7.1 Examples of partial differential equations
A few examples of partial differential equations encountered in engineering appli-
cations are stated.
Neutron diffusion equation: The one-speed neutron diffusion equation, as a
function of space (r) and time (t) is given by
1 ∂φ X

r:D rðÞrφ + r ðÞφ r, tÞ ¼ S r, tÞ (F.50)
ð
ð
v ∂t a
The parameters in this equation are defined as in Appendix C. The important variable
in Eq. (F.50) is the space and time-dependent neutron flux φ (r, t).
One-dimensional heat conduction: One-dimensional heat conduction through a
thin plate as a function of distance from a surface (x) and time (t) has the form.
2
∂ T 1 ∂T
¼ (F.51)
∂x 2 α ∂t
T (x, t) is the temperature variation across the plate and α is the thermal conductivity
of the plate material.
Two-dimensional heat conduction: The partial differential equation for heat
conduction of a flat plate, for the temperature distribution is T(x, y, t), has the form.
2
2
∂ T ∂ T 1 ∂T
+ ¼ (F.52)
∂x 2 ∂y 2 α ∂t
Three-dimensional heat conduction: The temperature distribution {T(x, y, z, t)}in
an isotropic body (thermal conductivity at any point in the body is independent of the
direction of heat flow) is given by [6, 8].
2
2
2
∂ T ∂ T ∂ T 1 ∂T
+ + ¼ (F.53)
∂x 2 ∂y 2 ∂z 2 α ∂t
One dimensional wave equation: If a string is fixed between two points and
vibrates in a vertical plane, its vertical displacement x at time t is given by the dif-
ferential equation.
2 2
ð
∂ ux, tÞ ∂ ux, tÞ
ð
A ¼ (F.54)
∂x 2 ∂t 2

F.7.2 Solution of partial difference equations using the finite
difference method
Advances in numerical computing and techniques facilitate accurate and fast solu-
tion of complex boundary value problems. These methods are used to solve problems
encountered in diffusion theory, heat transfer, fluid mechanics, structural analysis,
electrostatics, magnetism, and other engineering fields. This section provides a
brief overview of the finite difference method (FDM) which is a popular and