Please answer it to the point in one hour , doo not attach anything irralevant

The following is a simplified schematic of part of a wastewater treatment plant (WWTP). It is comprised of two sections: Section \#1 (Pre-treatment) and section \#2 (Anaerobic digestion). The whole plant can be considered an isothermal process. The density will be constant throughout the process, too. In the pre-treatment process (tank \#1) several impurities (mostly inorganic matter) are removed. The volumetric flow rate Ff1 of these impurities has been modelled as a small constant fraction T of the volumetric flow rate FIN of raw water into the WWTP and is removed from the plant for revalorisation. The rest Fout,1 flows to section \#2. Fout,1 does not follow Bernouilli's principle but can be changed using the device D1 to allow h1 to re-adjust when there are variations in FIN while preventing the tank from overflowing. The cross-section area A of the tank is constant. Section \#2 consists of a CSTR where the anaerobic digestion of organic matter in the water flow takes place. The mass fraction xin of organic matter in the incoming flow rate is constant. The degradation rate of organic matter is proportional (with proportionality constant k[s1] ) to the mass of organic matter in the tank. To obtain the degradation rate you must multiply k times the volume of water in the tank times the mass fraction x of organic matter in the tank. As not all the organic matter is degraded, there is a constant fraction R of the volumetric flow rate Fout,2 recycled back into the CSTR. Bernouilli's principle applies to the CSTR. The cross-section area A is the same as that for tank #1 and the proportionality constant between volumetric flow rate out and square root of the height of liquid in the tank is . The system is working at known steady state, and at a given instant (t=0) a light rain event occurs, modifying FIN within manageable limits for the WWTP. Answer the following questions while stating any further assumptions you need: a) Write the differential equation for the overall mass conservation of water in tank \#1, linearise it, and apply the Laplace Transform to it (zero initial conditions). Leave your final answer in the format H1(s)=i=1n1gi(s)vi(s), where H1(s) is the Laplace transform of the height of liquid in the tank, n is the number of variables in the ODE, gi(s) are transfer functions, and vi(s) is the Laplace Transform of any variables in the ODE other than H1(s). Hint: In this case, n=3. [10 marks] b) Write the differential equation for the overall mass balance in the CSTR, linearise it and apply the Laplace transform to it to obtain a transfer function between Fout,1(s) and H1(s). [10 marks] c) Write the differential equation for the mass balance of organic matter in the CSTR. Make sure that your equation is dimensionally consistent. Linearise the equation and apply the Laplace Transform to leave it in the same format as a). [10 marks] Assume now that a control loop can be implemented to keep the level of liquid in tank \#1 constant when the light rain event happens by manipulating the device D1. d) Assume the light rain event can be modelled as a step change of magnitude M in FIN. What will be the value of h2(t) when t. [15 marks] e) If tank \#2 can accommodate a height of liquid h2=h2,st+(1R)5Mh2,st(1r) before it overflows. Will it overflow? (subscript st means steady state prior to light rain event) [15 marks]