Question

Question 1 Two non-interacting tanks are connected in series as shown in the following figure. At steady state the input flow rate is 900 Limin. The valve coefficients areG = G = 1 and G = 0.5. The cross- sectional areas of Tank 1 and Tank 2 are 2 m2 and 1 m', respectively. Fi 900L/min hi h2 Figure 1 Two non-interacting tank system i. Develop the mathematical models for this system ii. Find the transfer function models in standard forms that describe this process. What should the units of Cı, C2 and C3 be? Evaluate all parameters and steady state values with units for the constants and variables used in your models. What are the orders, gains, time constant or characteristic time, damping coefficient of the system? ii. In Simulink show the block diagram of the process. At time of 5 units of time, suddenly the inlet flowrate changes from 900 Lmin to 450 /min In the same graph plot the responses of the liquid levels in each tank to the step change in the inlet flowrate.

USING LAPLACE TRANSFORMS

Answer 1

For tank 1 F_i - F_1 - F_3 = A dh_1/dt F_i - c_3 h_1 = A dh_1/dt -(1) at steady state F_3 - c_3 h_3 - c_1 h_3 = A dh_S/dt = 0 F' - c_3 H_1 - c_1 H_1 = A dH_1/dt Applying laplace F'(S) - c_3 H_1(S) - c_1 H_1(s) = A S H_1(1) 1 - H_1(S)/F_1(S) (c_3 + c_1) = ASH_1(1)/F'(S) 1 = H_1(S)/F'(S) (AS + c_3 + c_1) H'(3)/F'(1) = 1/AS + c_3 + c_1 = 1/25 + 1.5 = 1.5/4/3 s + 1 (i) from eq (1) 2d h_1/dt \ F_i - 1.5 h_1 dh_1/dt = 0.5 Fi - 0.75h. tank 2 F_1 - F_2 = A_2 dh_2/dt c_1 h_1 - c_2 h_2 = 1dh_2/dt dh_2/dt = h_1 - h_2 0 = h_1 S - h_2 S d H_2/dt = H_1 - H_2 S H_2(S) = H_1 (S) - H_2 (S) S H_2 (S)/H_1 (S) = 1 - H_2 (S)/H_1 (S) (S + 1) H_2 (S)/H_1 (S) = 1 H_2 (S) /H_1(S) = 1/S + 1 (ii) Transfer functions H_1(S)/F_i (S) = 1.5/4/3 S + 1