The volume of a hemispherical storage tank, as a function of its height, is given by V(t) =pi*R*h^2(t) - (pi/3)*h^3(t), where R is the radius of the hemispherical tank.The following physical information is available for the tank: R =2m; c =1.5 m^(5/2) /min. The steady state feed flowrate to the tank is F0 = 1.35 m^3/min.
(a) Derive the dynamic mass balance that describes height of gasoline in the tank as a function of time.
(b) Linearize this model about a steady state, h (you will need to nd this numerical value), and convert the model to deviation variable form.
(c) Construct a transfer function model of the process.
(d) Obtain a plot of the approximate (transfer function) model response to a step change in F0 from its initial value to a value of 1.8 m3/min.
(e) Obtain the corresponding plot for the nonlinear model to the same step change. Discuss their similarities and dierences, and comment on the suitability of the linear model in approximating the nonlinear system.