**Transcribed Image Text: **1 2 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 Fo, the vol. flow rate from tank 1 to tank 2 is F, and the vol. flow rate from tank 2 is F. The height of the liquid level is h in tank 1 and h, in tank 2. Both tanks Fo- have the same cross-sectional area A. The flow rates F and F, depend on the liquid levels according to Fath-h, F2 - Bha where ß 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 F, as independent (input) variable. This means that the differential equations have the general form dh dh, th,hy, Fo), == /2ch, h2, FO) d: dt where fi and f, are the (nonlinear) functions to be determined. Note that all argumentsh, hy, and F, 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 at a steady-state (,,,F). This will introduce “ A-variables" that denote the deviation from the corresponding steady-state values. c) Assume that A=0.5 m and 3=2 m/h and that the process initially is at the steady-state defined by 5o = 2 m/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 lincarized model how h, hy, and F, change as functions of time when Fo is changed (1) stepwise (i.e. “immediately") from 2 m/h to 2.5 m/h, (ii) stepwise from 2 m/h to 1.5 m/h, (iii) sinusoidally with the average value 2 m/h, amplitude 1 mºh and frequency 2 cycles/h. Simulate all cases for 10 hours. The simulations are most easily done with SIMULINK O MATLAB and its ODE solver ode15. e) Use MATLAB's plot command to plot how h, hy, and F, change as functions of time for the tree cases. Plot hey, hey, and F, 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 hy, hy, and F, even if A-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).

**More** **Transcribed Image Text: **1 2 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 Fo, the vol. flow rate from tank 1 to tank 2 is F, and the vol. flow rate from tank 2 is F. The height of the liquid level is h in tank 1 and h, in tank 2. Both tanks Fo- have the same cross-sectional area A. The flow rates F and F, depend on the liquid levels according to Fath-h, F2 - Bha where ß 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 F, as independent (input) variable. This means that the differential equations have the general form dh dh, th,hy, Fo), == /2ch, h2, FO) d: dt where fi and f, are the (nonlinear) functions to be determined. Note that all argumentsh, hy, and F, 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 at a steady-state (,,,F). This will introduce “ A-variables" that denote the deviation from the corresponding steady-state values. c) Assume that A=0.5 m and 3=2 m/h and that the process initially is at the steady-state defined by 5o = 2 m/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 lincarized model how h, hy, and F, change as functions of time when Fo is changed (1) stepwise (i.e. “immediately") from 2 m/h to 2.5 m/h, (ii) stepwise from 2 m/h to 1.5 m/h, (iii) sinusoidally with the average value 2 m/h, amplitude 1 mºh and frequency 2 cycles/h. Simulate all cases for 10 hours. The simulations are most easily done with SIMULINK O MATLAB and its ODE solver ode15. e) Use MATLAB's plot command to plot how h, hy, and F, change as functions of time for the tree cases. Plot hey, hey, and F, 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 hy, hy, and F, even if A-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).