Determine the values of the gain of the process and of the time constant.

EX40HC – Tutorial 4 Dynamics of second order processesProblem 1.Consider a process made of two tanks in series. The outlet volumetric flow rates from each tank are proportional to the level in each tank, i.e. 𝐹𝐹1(𝑡𝑡)=𝛼𝛼1⋅ℎ1(𝑡𝑡) and𝐹𝐹2(𝑡𝑡)=𝛼𝛼2⋅ℎ2(𝑡𝑡). The flow rates 𝐹𝐹1, 𝐹𝐹2 and 𝐹𝐹𝑓𝑓 are in m3/s. a) Find the general form of the dynamic response of the level in tank 1, ℎ1(𝑡𝑡), as a function of the feed flow rate 𝐹𝐹𝑓𝑓(𝑡𝑡) and, using Laplace transforms, the corresponding transfer function. Comment on the order of the dynamic response of the process. Determine the values of the gain of the process and of the time constant. (Answers:𝐾𝐾𝑝𝑝1=100𝑠𝑠𝑚𝑚2, 𝜏𝜏𝑝𝑝1=100𝑠𝑠) b) Find the general form of the dynamic response of the level in tank 2, ℎ2(𝑡𝑡), as a function of liquid level in tank 1, ℎ1(𝑡𝑡), and, using Laplace transforms, the corresponding transfer function. Comment on the order of the dynamic response of the process. Determine the values of the gain of the process and of the time constant. (Answers:𝐾𝐾𝑝𝑝2=1𝑚𝑚𝑚𝑚, 𝜏𝜏𝑝𝑝2=100𝑠𝑠) c) Using Laplace transforms, determine the transfer function for the dynamic response of the level in tank 2, ℎ2(𝑡𝑡), as a function of the feed flow rate to tank 1, 𝐹𝐹𝑓𝑓(𝑡𝑡), and comment on the order of the dynamic response of the process. Determine the values of the natural period of oscillation, of the damping factor and of the gain of the process. (Answers:𝐾𝐾𝑝𝑝′=100𝑠𝑠𝑚𝑚2, 𝜏𝜏=100𝑠𝑠, 𝜁𝜁=1)