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A typical valve filter regulator to limit the pressure as needed (actuator supplier will advise), and to allow draining of any condensates.

Don't remove the counter weights, there used for balancing the forces. Your actuator is small, unless you require extreme opening or closing speeds, 1/4" is okay. You might want to use an air filter and provide a tee and a liquid drain valve on the airline as a precaution should you have high...

Carry out you measurements as described and record the data. The world wide web has countless resources to better understand the test.

Try Bernoulli's equation. Many valve, flowmeter suppliers have sizing software that will help.

DC valve solenoids cannot be actuated by 4-20 ma circuitry.

You'll have to pull the pump up and test it under discharge pressure control.

You need to know the fluid, it's pressure, and temperature, vapor pressure, pipe velocity and valve characteristics (globe, etc. and pressure drop). Then you grab a mechanical engineering hand book and get busy. Flow meters less that 20-50 pipe diameters down stream of a valve are not...

Your desired flow depends on the head differential, and the pump curves. Get your process designer involved.

You have piping design problem. Good luck!

Is your VFD filtered to insure 60 Hz AC wave forms?

Unless you know the temperatures of the sample being measured, the numbers are meaningless.

You are only calculating the "relative gain." If it is low, the heat is gently increased and if high the heat is changed abruptly. If PB is too small the heat increases abruptly, when properly adjusted the heat increases gradually without overheating.

Traditionally, set high PB (or low gain), and gradually decrease the PB (or increase gain) it as needed for tighter control. Set the maths aside.

Please include the sample temperature of the respective measurement.

Page 23 is the trouble shooting guide for a related model. You may just need new pumps.

Spend some time in the control room to understand what the operators have to deal with.

Stop all motion, where there is risk of injury.

Depends on the level of redundancy you need for safety.

Sounds like you may be dealing with load changes on the turbine, the Gt experts will jump in soon

Sensor can fail. Are these RTD or T/C's.