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you need more specifics for a general response, mean while you might consult Considine's Process Instruments and Control Handbook, Section 9, with a half-dozen designs to choose from

Work details for on battery pack as a separate pid, then on the master show the valve takeoff's (A through Y) that refer to the detail of each pack.

T/C's don't normally fail high, they may drift over time due to inappropriate material selection. When they fail abruptly it is typically the result of loss of continuity. RTD's can fail high depending on the details of the sensor failure.

sensors do fail, so a bit of trouble shooting may be needed.

www.isa.org/certification is a good start,

There are countless texts dealing with process control. That is a good starting point.

To calculate the pressure, you need to know what is down stream.

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.