reservoirs in series and Jordan’s form

We consider 4 identical reservoirs connected in series. It is good example to understand the Jordan form of the underlying system. Water circulates with a diluted substance that is mixed instantly into each reservoir. The flow between the reservoirs is constant ¨b¨ so the volumes are preserved. There is also a deposition coefficient "sigma".

# Identical reservoirs in series

# example of Jordan form

# x[i]: specie in reservoir i [gr]

# V= volume [lt]

# b= volume flux [lt/seg]

# sigma = deposition coeff

# f = entry concentration [gr/lt]

# First reservoir...

x1'= -sigma*x1-b*x1/V+f*(1+cos(omega*t))*b

init x[1]=1

# and the other reservoirs...

%[2..4]

x[j]' = b*x[j-1]/V-sigma*x[j]-b*x[j]/V

init x[j]=0

# Parameters

param V=1, sigma=0.1

param b=1, f=0, omega=2

@ TOTAL=10

done

We model the evolution for a periodic entrance of the substance. Initially, the first reservoir has a concentration 1 and the others 0.

This type of models are more generally called “compartimental models”, used in physiology, chemistry, ecology and biochemistry for instance.