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The input is a scaling factor between 1 and 0 that is multiplied with the area of the valve. Do note that for most valves, putting the valve half open does not mean the throughput area is half the total area, so you might want to link that in an equation. $$\frac{dM_{gas}}{dt} = u(t)A_{valve}\rho V_f$$ $$0\leq u(t) \leq 1$$ Which also instantly leads to an ...

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Typically, the output of the controller is the input to the valve. Depending on the sophistication of the valve model, it could be either the voltage or the valve opening in % or mm (or similar).

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A pilot operated valve (often unfortunately generalized as a "solenoid valve") uses a small control solenoid valve and the pressure differential across the valve to open a diaphragm that allows flow to pass. This is a very common design because it allows for a large flow to be controlled with very little energy. This YouTube video has a good ...

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