Answer to Question #157920 in Physics for satyapal singh

Question

Two storage tanks with a tapering circular cross-sectional area are used to intermittently supply a water-soluble binder fluid to a mixer, in an ibuprofen tablet manufacturing process. The noninteracting tanks are arranged in series, as shown in Figure Q1. The effluent volumetric flowrates (m3hr−1 ) are related to the fluid height in each tank by the following linear relationships: 𝑓1 (𝑡) = 𝐶v1 × ℎ1 𝑓2 (𝑡) = 𝐶v2 × ℎ2 Where 𝐶v1and 𝐶v2 are the valve discharge coefficients of the first and second valves respectively. The inlet volumetric flowrate is 𝑓𝑖 . The relevant parameters of the system are as follows: 𝐷1 = Top diameter of the first tank = 0.915m 𝑥1 = Base diameter of the first tank = 0.305m 𝑦1 = Vertical height of the first tank = 10𝑥1 Figure Q1: Non-interacting tanks in series 𝐷2 = Top diameter of the second tank = 0.725 𝐷1 𝑥2 = Base diameter of the second tank = 0.725 𝑥1 𝑦2 = Vertical height of the second tank = 10𝑥2 𝐶v1 = Valve discharge coefficient = 0.366 m3 . m−1 . hr−1 𝐶v2 = Valve discharge coefficient = 0.183 m3 . m−1 . hr−1 1.) Develop a linearised dynamic model for the system. 2.) Obtain the transfer function (in deviation variable form) between the liquid level in the second tank and the volumetric flowrate into the first tank. 3.) If the initial steady state height in tank 1 is ℎ̅ 1 = 1.83m and that in tank 2 is and ℎ̅ 2 = 1.11m. Plot the open loop time response of the process for a unit step change in the inlet flowrate for the following valve discharge coefficients values: 𝐶v2 ∈ {0.01, 0.10, 1.00}. Assume all other parameters remain as originally defined. Explain the plots you obtained. 4.) If the initial steady state height in tank 1 is ℎ̅ 1 = 1.83m and that in tank 2 is and ℎ̅ 2 = 1.11m, fit a proportional controller to the system and plot the closed loop time response of the process for a unit step change in the inlet flowrate. Vary the value of the controller proportional gain, 𝐾C three times. Assume that all other parameters remain as originally defined. Explain the plots you obtained. 5.) Using the same steady state conditions as in (4) and the original valve discharge coefficients, replace the proportional controller in (4) with a proportional-integral controller. Fix the controller proportional gain, 𝐾C and vary the integral gain, 𝐾I three times. Plot all three responses and discuss the significance of your results. 6.) A sensor-transmitter modelled by the following first order transfer function: 𝐺m(𝑠) = 0.4 5𝑠 + 1 is fitted to a closed loop. Replace the proportional integral controller with a proportional integral derivative (PID) controller and tune it using the continuous cycling ZieglerNichols method.

Answer 1

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