40
Chapter 3
Types of Models
A
B Vessel 1
Vessel 2
Vessel 3
Figure 3.5: Three coupled mixing vessels. of this system were investigated by an impulse response experiment. The actual concentration of the pulp could not be manipulated since the experiment had to be done during normal operation. The problem was solved in the following way: A bucket of water with 2070 grams of radioactive lithium, with short half-life, was poured into the first mixing vessel at point A in Figure 3.5. The radioactivity was then measured at point B during 5 hours. This radioactivity will obviously be proportional to the concentration of lithium, after correction for the half-life and background radiation. Figure 3.6 shows the measurements. Even if the measurements are disturbed by a fair amount of noise, a clear picture of a typical time response is obtained. To correctly scale the impulse response we argue as follows: Let both the input and the output have the unit mg/liter. Then the coefficients of the impulse response will be dimensionless quantities. The total flow through the system was about 8300 liters/minute during the time of the experiment. The sudden addition of 2070 grams of lithium then corresponds to an impulse u(t) = u0 δ(t) with u0 = 2070 grams/minute = (2070/8300) ∗ 103 ≈ 250 milligrams per liter. Consequently we must divide the lithium concentration with this number to get the impulse response. This is shown in the lower plot of Figure 3.6. In this figure we also show the impulse response of the system G(s) =
1 sτ + 1
(c) The Authors and Studentlitteratur
3
for τ = 72 min
(3.1)