A pneumatic control valve is used to adjust the flow rate of a petroleum fraction (sp gr = 0.9) that is used as fuel in a cracking furnace. A centrifugal pump is used to supply the fuel, and an orifice meter/differential pressure transmitter is used to monitor flow rate. The nominal fuel rate to the furnace is 320gal/mm. Select an equal percentage valve that will be satisfactory to operate this system. Use the following data (all pressures in psi; all flow rates in gal/mm):

(a) Pump characteristic (discharge pressure): P = (1 – 2.44 x 10–6 q2) Pde, where Pde is the pump discharge pressure when the pump dead ended (no flow).

(b) Pressure drop across orifice: ∆P0 = 1.953 x 10–4 q2

(c) Pressure drop across the furnace burners: ∆Pb = 40

(d) R for the valve: 50

(e) Operating region of interest: 250 < q < 350. This design attempt should minimize pumping costs by keeping the pump capacity (related to Pde.) as low as possible. In no case should ∆Pv/∆Ps, be greater than 0.33 at the nominal flow rate. Show, by means of a plot of the installed valve characteristic (q vs. ℓ), just how linear the final design is.