I. M. Appelpolscher has acquired considerable knowledge of pneumatic control valves over the years. In particular, he knows that the flow characteristic of a “quick-opening” valve is f = √ℓ which is one reason they also are called “square root” valves. Appelpolscher reasons that, if there is a need for a quick-opening valve, there may be a need for a slow-opening valve as well. He conjectures that it probably would have the flow characteristic f = √ℓ2 and decides to build and test such a valve.

(a) Plot f versus ℓ for his “square” valve in the same manner as was used in figure. Does your plot resemble any other known valve characteristic? Explain.

(b) How does the gain of Appelpolscher’s valve (i.e., change in f versus change in lift) compare numerically with linear and quick-opening valves at values of ℓ = 0, 05, and 1. Do these values correspond to your expectations? Explain.

(c) If Appelpolscher wants a valve that will provide a maximum flow rate of 1024gpm with a pressure drop across the valve of 64 psi using a liquid of specific gravity = 1, what would be the Cv of the valve?

(d) For a pneumatic valve, the lift is related to the air pressure signal p applied to the topworks, which varies between 3 and 15 psig. Develop an expression for ℓ in terms of p.

(e) With the particular Cv you calculated In part (c) and the lift expression in part (d), write out the relation between flow rate q and the two valve “inputs”: (1) the air pressure signal p and (2) the liquid pressure drop developed across the value NP,. Check your relation to make sure it gives reasonable values of q for various chokes of p and