Page 276 - Dynamics and Control of Nuclear Reactors

278 APPENDIX D Laplace transforms and transfer functions

D.8 Laplace transforms and partial differential equations
Partial differential equations are those that have more than one independent variable.
For reactor engineering applications, the independent variables are time and position.
Applications include one, two, or three position variables. The most common and
most easily solved are one-dimensional models.
Recall that Laplace transformation reduces an ordinary differential equation to an
algebraic equation. Laplace transformation (with respect to time) of a one-
dimensional partial differential equation results in an ordinary differential equation
with position as the independent variable. Laplace transformation of multi-
dimensional models eliminates the time derivative term, but the resulting model
is still a partial differential equation in the position variables. The solution of the
Laplace-transformed equation provides a transfer function. The space- and
frequency-dependent frequency response may be obtained by substituting jω for s
in the transfer function.
An example illustrates the procedure. Consider the following simple, one-
dimensional partial differential equation:
∂u ∂u
+ + bu x, tÞ ¼ 0 (D.31)
ð
∂t ∂x
Variable u(x, t) is a function of position (x) and time (t). The initial condition is
assumed to be zero. That is, u (x, 0)¼0. Laplace transformation of Eq. (D.31) yields
dU
+ s + bÞU ¼ 0 (D.32)
ð
dx
where U(x, s) is the Laplace transform of u(x, t).
The solution of Eq. (D.32) is
ð
Ux, sÞ ¼ U 0 e s + bÞx (D.33)
ð
The frequency response for a specified value of a x is given by (setting s¼jω)
ð
Ux, ωÞ bx jωx
¼ e e (D.34)
U 0
The magnitude and phase angle of the frequency response function at (x, ω) are
given by
Magnitude of Ux, ωÞ ¼ e bx (D.35)
ð

ð
Phase angle of Ux, ωÞ ¼ ω xÞ (D.36)
ð
Note that the magnitude of U(x, ω) is not a function of frequency, ω; but changes
with variable, x. The phase angle, for a given value of ω, changes as a linear
function of x.