Page 311 - Dynamics and Control of Nuclear Reactors

APPENDIX F State variable models and transient analysis 313

well-developed technique for numerical solution of partial differential equations
(PDEs). Some of the material in this section is adapted from Ref. [6, 8].

F.7.2.1 Introduction
A simple and clear description of the FDM is given using the following two-
dimensional equation which is typical of the two-dimensional heat conduction
problem [6, 8].
2 2
ð
ð
∂ Tx, yÞ ∂ Tx, yÞ
+ ¼ qx, yÞ (F.55)
ð
∂x 2 ∂y 2
The boundary condition has the form.
Tx, yÞ ¼ gx, yÞ, a x b and c y d (F.56)
ð
ð
In the FDM, derivatives are approximated by differences. For example, for a function
of one variable, the approximations have the form (as given in Ref. [6, 8])
2
df xðÞ fx + ΔxÞ fxðÞ d fxðÞ fx + ΔxÞ 2fxðÞ + fx ΔxÞ
ð
ð
ð
¼ , and ¼ (F.57)
dx Δx dx 2 ð ΔxÞ 2
F.7.2.2 Formulation of grids and nodes [8]
With the choice of integers n and m, define the step sizes.
ð b aÞ ð d cÞ
Δx ¼ and Δy ¼ (F.58)
n m
The partitioning of the intervals [a, b] and [c, d] enables the formation of grids with
vertical and horizontal lines and the definition of nodes as the intersection of the
grid lines. Thus,

x i ¼ a + iΔx and y j ¼ c + jΔy; i ¼ 0,1,2,…,n; j ¼ 0,1,2,…,m (F.59)
An example of grids and nodes for n¼m¼4 is illustrated in Fig. F.1.
The intersections of the grid lines x¼x i and y¼y j are the nodes or the mesh
points (i, j) of the finite difference grids.

F.7.2.3 FDM solution of the two-dimensional heat conduction
problem [8]
The two-dimensional heat conduction equation is given in Eq. (F.55) with boundary
conditions in Eq. (F.56). The second partial derivatives of temperature T(x, y)
are approximated as follows [8].
2
∂ Tx i , y j Tx i +1 , y j 2Tx i , y j + Tx i 1 , y j
(F.60)
∂x 2 ð ΔxÞ 2

2
∂ Tx i , y j Tx i , y j +1 2Tx i , y j + Tx i , y j 1
(F.61)
∂y 2 ð ΔyÞ 2
The heat conduction Eq. (F.56) at node (x i ,y j ) is then written in discrete form as.