Surge tanks are commonly used in chemical plants to act as buffer to dampen disturbances. Consider a two non-interacting tank system as shown in Figure 2. F, h2 F2 Figure 2 The flow head relationship for the valve in tank 1 is proportional to the square root of the height , F, = kivh. The flow head relationship for the valve in tank 2 is proportional to the square of the height F2 = ka(hz)?. The following data are given the areas for both tanks are equal, Az = Az = 2 ft2. The steady-state inlet flow rate is Fis = 10 ft/min and the valve coefficients are kz = 5 fti/min and ka = 10 ft/min. = a a) dhe and dhe State an Develop dynamic model for both tanks to obtain assumption that you made. dt dt b) Calculate the steady state height for both tanks, his and has c) dt Linearise the model developed in part (a) to obtain an and and where (0) represents deviation variables. Express the linearised model in state-space model: dx' dt = Ax' + Bu' y' = Cx where x' = [x xi] = {hi hil is the vector for state variables, the input variable u' is Fi and the output variables y' = (hi hil. (Note: you are expected to give numerical values for matrices A, B and C). d) Assume that the process is initially at steady state. Find the transfer function H(s) F(s) using Laplace transform. Hence, show that, for a step change of magnitude M in the inlet flow rate (i.e., Fi(t) = M), the height of tank 1 is given as: hy(t) = 1.6M (1 - ) + 4 = eT The state-space model in part (c) can be converted to the transfer function form. Consider the general state-space model: dx' - Ax' + Bu' dt y' = Cx' By taking Laplace transform on the above state-space model, show that the expression to convert from state-space model to transfer function is given as: Y'(s) = C(sl - A)-1B U'(s) where Y'(s) and U'(s) are Laplace transform of output y' and input u' respectively and I is an identity matrix with the same dimension as matrix A. Assume that the process is initially at steady state