Jökull - 01.12.1966, Page 49

container I
container 2
container 3
V f' V 2 V f 3
hl ql h2 q2 h 3
r2’c2
Fig. 3
The lumped capacitor-resistor chain.
f[ = surface area of container i
hj = water level in container i (measured
from an equilibrium pos.)
q0= output of the lst container (pumped
flow)
q, = flow from container i + 1 to con-
tainer i
Cj = conductance between container i + 1
and i, Cj = 1 /rt
rj = resistance between container i + f
and i.
We assume that the system is linear; that is,
the coefficients fj and r; are independent of
the h/s. In this case the equations for the i-th
container are simplv
qS = ci (hi + l -hi)
For a system of n containers we have 2n such
linear equations, which can be solved with
elementary methods if the parameters are given.
In the present context we are interested in
the response of the system to an output flow
q0 from the Ist container. We will assume that
the system is in equilibrium at a certain water
level until t = 0 when the output q0 is initiat-
ed. Since we have a linear system, the output
in terms of the water level of the first con-
tainer can be written as the following convolu-
tion integral (we assume that the containers are
so deep that the water level nowhere drops to
zero);
hl(t)= / gi(t-u) q0(u) du (3)
0
wliere q0(t) is the given output flow, and gt(t)
is the impulse response of the system mea-
sured as the response of the first container.
Equation (3) is very much simplified if we
take the Laplace-transform of both sides,
Hr(s) = Gx(s) Q0(s), (4)
where G-^s) is the driving point system func-
tion. On the basis of simple algebraic opera-
tions (see Bodvarsson, 1966) we can show that
G, (s) can be developed into a continued frac-
tion
G] (s) = —*—
flS + 1 (5)
+J_______
f2s + . . .
Since we have assumed a linear system, the
driving point function will be a rational func-
tion, and the development (5) is easily obtain-
ed. Hence, if we can derive G, (s) from experi-
mental data, we will be able to deduce the
lumped parameters of the system.
(3) EXPLORATION, DESIGN
AND OPERATION POLICIES
(3.1) General remarks. The design of geo-
thermal plants should aim at the optimum ex-
ploitation of the heat sources available. This
statement will appear quite obvious, but we
should, of course, realize that there may be
no agreement as to the measure of optimality.
JÖKULL 203