You are interested in modelling a 10 m surge tank which holds the products exiting a plug flow reactor, as shown below. One of the purposes of the tank is to heat the reactor products prior to the next stage of the process. The sides of the surge tank are surrounded by a steam jacket. Though, due to poor design, heat is lost from the tank via heat transmitted from the surface of the product liquid to the air above the tank. The feed to the tank flows at 0.2 m^{3}/s, and you can suppose that the tank
is always full. You can suppose that the following quantities are constant:
UA (steam jacket) = 250 kJ/K/s,
UA (liquid-air interface) = 20 kJ/K/s
ρ = 1000 kg/m^{3}
, Cp = 4.2 kJ/kg/K.
a)Derive an equation which relates the outlet temperature, T, to changes in the steam temperature, TS, the air temperature, TA, and the inlet temperature, TIN. State any other assumptions made in the derivation.
b)Derive the transfer functions, T/T_{A}, T/T_{IN}, and T/T_{S}. Determine the time constant and static gain for each of these transfer functions.