Next, let's explore the question of whether or not this conclusion that increased flows lead to a predictable decrease in the response time applies to larger systems than the one discussed above. To do this, we consider a system of two reservoirs in a connected and disconnected state, as shown in Figure 2.10 below.
Figure 2.10. Experiment with a
two-reservoir system in a connected and disconnected state. Note that
the response time of the two reservoirs is the same in the connected
state and this connected response time is shorter than either of the
reservoirs in the disconnected state. Graphically, the response time
is the length of time needed to accomplish 63% of the ultimate change
required to bring the reservoir into a steady state condition. For
the disconnected reservoirs, that steady state is characterized by
completely empty reservoirs.
Figure 2.10 shows the effect of connecting two reservoirs. first, study the two reservoirs W1 and W2 in their disconnected state. They each have draining flows, and given the rate constants, k1 and k2, we can say that their response times are 10 and 2 time units. The graph shows how these two reservoirs approach their steady states at different rates, according to what their resonse times are; both reservoirs will eventually end up with a value of 0 since there are no inflows in this case. Figure 2.10 shows what happens when these reservoirs are connected to each other. This would be like having two tanks of water with draining hoses of different diamters; in one case, the hoses just pour the water out onto the ground, while in the other case, they pour water into the neighboring tank. Based on our study of the single reservoir with two outflows, we might assume that that the response time for this connected system would be given by:
.
We test this hypothesis by running the model and we observe that,
sure enough, the response time is 1.67 time units. The response time
can be graphically determined by finding the length of time required
for the reservoir to accomplish 63% of the eventual change it makes
on its way to the steady state. So, we can conclude that connected
systems have shorter response times than disconected systems, and the
response time of the connected system turns out to be shorter than
the shortest response time of wither reservoir in the disconnected
state. This may lead us to wonder if more complex systems are in
general more responsive than simple systems. Before accepting this
notion,we clearly need to investigate more complex systems.
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