PDC / 12 / 1

Modelling, linearization, and simulation of two interacting tanks

Consider the process consisting of two interacting liquid tanks in the figure. The volumetric flow

rate into tank 1 is 0F , the vol. flow rate from tank 1 to tank 2 is 1F , and the vol. flow rate from

tank 2 is 2F . The height of the liquid level

is 1h in tank 1 and 2h in tank 2. Both tanks

have the same cross-sectional area A . The

flow rates 1F and 2F depend on the liquid

levels according to

211 hhF , 22 hF

where is a constant parameter.

a) Derive a dynamic model for the process consisting of two coupled first-order differential

equations with the liquid levels as dependent (output) variables and the incoming flow rate

0F as independent (input) variable. This means that the differential equations have the

general form

),,(d

d0211

1 Fhhft

h , ),,(

d

d0212

2 Fhhft

h

where 1f and 2f are the (nonlinear) functions to be determined. Note that all arguments 1h ,

2h , and 0F need not appear in both functions. Assume that the liquid density is constant.

b) Linearize the two differential equations and the given constitutive relationship for 2F at a

steady-state ),,( 021 Fhh . This will introduce -variables that denote the deviation from

the corresponding steady-state values.

c) Assume that 0.5A m and 2 m2.5/h and that the process initially is at the steady-state

defined by 0 2F m3/h. From this, the steady state-values of the other variables can also be

calculated. Determine the linearized model using these numerical data.

d) Simulate both for the nonlinear and the linearized model how 1h , 2h , and 2F change as

functions of time when 0F is changed (i) stepwise (i.e. immediately) from 2 m3/h to 2.5

m3/h, (ii) stepwise from 2 m

3/h to 1.5 m

3/h, (iii) sinusoidally with the average value 2 m

3/h,

amplitude 1 m3/h and frequency 2 cycles/h. Simulate all cases for 10 hours. The simulations

are most easily done with SIMULINK or MATLAB and its ODE solver ode45.

e) Use MATLABs plot command to plot how 1h , 2h , and 2F change as functions of time for

the tree cases. Plot 1h , 2h , and 2F in different diagrams, but the nonlinear and linear

simulations in the same diagram for each variable (to make comparisons easy). Note that we

want plots of 1h , 2h , and 2F even if -variables have been used in the linear simulations.

f) Based on these plots, do you think the linearized model is a good or less good approximation of the nonlinear system? Or is it not so clear? Please motivate! Consider, for

example, whether there are remaining errors (after 10h) in some variables for step changes in

the linearized model (as compared to the result with the nonlinear model).