In a paint mixing plant, two tanks supply fluids to a mixing cistern. The height, h, of the fluid in the cistern is dependent upon the difference between the input mass flow rate, q, and the output flow rate, qe. A nonlinear differential equation describing this dependency is given by (Schiop, 2010)

where A = cross-sectional area of the cistern, A_{e} = cross-sectional area of the exit pipe, g = acceleration due to gravity, and ρ = liquid density.

a. Linearize the nonlinear equation about the equilibrium point (h_{0}, q_{0}) and find the transfer function relating the output cistern fluid level, H(s), to the input mass flow rate, Q(s).

b. The color of the liquid in the cistern can be kept constant by adjusting the input flow rate, q, assuming the input flow’s color is specifically controlled. Assuming an average height, hav, of the liquid in the cistern, the following equation relates the net flow of color to the cistern to the color in the cistern.

where e_{1} = fractional part of flow representing color into the cistern, and e = fractional part of the cistern representing color in the cistern. Assume that the flow out of the cistern is constant and use the relationship, q_{e} = ρA_{e}√2gh_{av }, along with the given equation above to find the transfer function, E(s)/Q(s), that relates the color in the cistern to the input flow rate.