From the earlier section, the component balance for the mixing system was
In this model there are three potential inputs: the inlet concentration and the two inlet flowrates. We could solve for changes in all three, but in this example we'll restrict ourselves to changes in the diluent flowrate (Fd) and the inlet concentration(Cin). The solutions we'll be looking for are the effects of changes in the flowrate of 5% and of changes in the inlet concentration of 5%. The nominal steady-state we'll use for this solution is V=20 m3, F1=1 m3.min-1, Fd=9 m3.min-1, Cin = 50 kg.m-3 and Cout (obtainable from steady-state balance - just set derivative to zero)= 5 kg.m-3.
Prior to the analytical solution we need to look for any non-linearities in the equation. The first term (F1 Cin) is fine since, although both are possible variables, we will restrict the solution and assume F1 is constant - you should always assume that as many things as possible are constants since it makes solutions much simpler. The second term is, however non-linear: when we multiply out the term we end up with F1Cout + Fd Cout and the second term involves the product of two time varying quantities - a non-linearity.
Have a go at linearising the equation yourself before revealing the solution below:
As you can see, the equation can be rearranged into a standard first-order form and we can directly use the standard solution to a step change (NB: you solve the equation for each step in turn, NOT simultaneously). A 5% change in the diluent flowrate means that Fd changes by 0.05*9 = .45, and the solution is (remember that we are working in deviation variables - the value of the input is its change):
The steady-state gain of the process (Fd to Cout) is -0.5, so any change in Fd will result in -0.5 times that value ending up in the outlet concentration. The time constant of the process is 2 minutes - the response will be complete in about 10 minutes (5 time constants)
For changes in Cin, the time constant is still 2 so the speed of response will be the same, but the steady-state gain is 0.1 - the final change in the outlet concentration will be one tenth the change in the inlet concentration. Write out the solution to inlet changes for yourself.
The VisSim solution to the problem can be found here.