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Control Valve transfer functions assistance. Any help is appreciated.

Control Valve transfer functions assistance. Any help is appreciated.

Control Valve transfer functions assistance. Any help is appreciated.

(OP)
I have an effluent treatment plant that brings pH of effluent to between pH 6-9 through multiple tanks. The flow coming out of the last tank must have a pH of 6-9. It is followed by a pump and then 2 valves figure included). One valve goes to a river, which is why pH must be close to neutral, and the other is a valve to recycle the effluent back to the beginning of the process for further treatment if the pH does not meet spec. I would like to use a on/off controller for the valve that goes to the river to close the valve completely if the pH transmitter at tank 4 outlet reads an unacceptable pH (anything less than 6 or greater than 9). The recycle valve will always be closed, but I would like to use a controller to open it if the flow coming out of the pump is less than 15000 kg/hr (minimum flow to avoid deadheading the pump). This decrease in flow happens as a result of the river valve closing due to pH not being correct. I would like the recycle valve to only open enough to allow a flow of 15000 kg/hr to make sure the pump has enough head to pump it back up to the beginning of the process. I have made a basic block diagram to demonstrate what I mean. I am struggling with the transfer functions for the valves and transmitters. They don't have to be very rigorous models, this is just a uni class project, but our professor struggled with the online teaching and as a result my group is struggling through this. The density of the fluid is assumed constant at 1000 kg/m^3 and the pressure drop across the recycle valve can be no more than 3-4 psi to maintain enough head to pump up the fluid, by my calculations. The valves can be considered linear, I believe. Any help would be appreciated and thank you in advance.

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