Q4: A liquid transport system involving three tanks with the cross-sectional areas of A. A and As, is shown in Figure 2. fo() h(t) A 1 N ha(t) A2 h(t) Ag f(t) 121) > Figure 2: Liquid transport system with three tanks and three valves. The variable, fo(t), refers to the volumetric flowrate of liquid at the inlet of the first tank. The volumetric flowrates through three valves, f(t). [2(t) and 13(6), vary in time and are given by fi(t) hat) 120) (ho (1) - h3(0) x kg 13(0) hs) x ks where hi(t), hy(t) and h() are liquid levels (heights) in the tanks and ki ky and ky are the valve resistance coefficients in the unit of ms"). Answer the following questions for given tank system: (a) Write the unsteady mass balance for each tank and combine them to obtain a system of ODEs for the liquid transport system. The density can be assumed to be constant. f marks] (b) Determine the eigenvalues for the system with assuming that fo is a constant (not a func- tion of time), using the characteristic equation for the following constants: A 8ma, A2 = 5 m2, A3 = 10 m, ki = 0.1 m?s-, k2 = 0.025 ms-1 and ks = 0.05 ms-1 (8 marks) (c) Determine the eigenvectors using Gaussian elimination. (8 marks (d) State the general solution for the system of ODES. [2 marks] (e) For the initial steady state case of fo = 500 Ls 1, find the particular solution [8 marks