So far, we've represented this system in the form of a drawing and
in the form of a STELLA model; one additional way to represent this
system is in the form of a set of equations. In fact, this is what
STELLA does behind the scenes. Here are the equations used to
represent the first system - the one with negative feedback:
Faucet = 1.0 {l/sec} (1)
Drain = 0.1 * Water Tub {/sec *l} (2)
INIT Water Tub = 10 {l} (3)
Water Tub(t) = Water Tub(t-DT)+ (Faucet-Drain)*DT {l + l/sec *
sec} (4)
Just to show that there is nothing too difficult about these
equations, let's look at them a little more closely. Equations 1-3
are the starting conditions for the system; Eq. 1 specifies that the
Faucet flow rate is constant at 1 liter/second; Eq. 2 specifies that
the Drain flow rate is 0.1 multiplied by the number of liters in the
water tub at any given time (t); Eq. 3 simply fixes the starting
volume of water in tub to be 10 liters.
So at the very start, at time t=0, the value for the drain flow
will be 1 liter/second, exactly the same as the faucet flow rate. The
final equation is used by STELLA to keep track of how much water is
in the tub throughout time. As the program runs, time changes from 0
to some specified value in time steps (DT) that are also specified by
the user, so Water Tub(t) is the amount of water a particular time,
Water Tub(t-DT) is the amount in the tub in the previous time step,
and (Faucet-Drain)*DT is the difference in the inflow and outflow
rates over the time step. You can see that the program is basically
doing some accounting - adding material and subtracting material in
small increments.
Another way of writing Eq. 4 is slightly more abstract, but familiar to mathematicians:
where F represents the faucet flow rate while kW represents the
drain flow rate. This is called a differential equation; it is
discussed more fully in Appendix A, at the end of this chapter, where
you can see what is involved in solving the equation without the use
of a computer.
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