I am designing a quarter turn ball control valve.  I have system inputs for pressure ranging from 350-750 psi and a flowrate of 52 gpm.  It has also been given that due to another valve downstream, the backpressure at my valve's outlet will be 150 psi.  With these values it seems I automatically have a pressure drop for the valve.  What I am not understanding is, how can the outlet pressure of my valve be forced to be 150 psi when based on the valves internal geometry, is likely to be something else? In other words how can I just assign a pressure drop to my valve given inlet and backpressure? Thanks for any and all input!

With 750 PSI in and 150 PSI out, (Please specify Absolute unit or gage units) This valve will have some form of critical pressure drop. Probably caviation, possibly flashing.  Choked flow almost guaranteed.   

Since the flow is a liquid, you'd better know the critical pressure, vapor pressure, temperature, viscosity, density, and the Fl of your trim should be very very close to 1.0.  1.0 would occur if the valve trim exhibits pure frictional adiabatic flow and would correspond to the flow through something like a cigarette filter, where energy is lost through shear and there is no acceleration of the fluid.   

More than likely you will show cavitation at this kind of pressure drop. It is also likely that you are designing the tortuous trim to avoid cavitation.  

The Masoneilan equation listed above will not apply as it does not address critical drops. Use ISA equations as appropriate.(ISA Handbook of Control Valves).  Les Driskell wrote many books with regard to valve analytics and theory.  Search Amazon and Stock up your library.
 

You did not specify which liquid you are designing for.  Chemically pure fluids cavitate with damage likely.  Water is probably worst since it is dense, has a lot of surface tension, and releases a lot of energy as it condenses from a vapor back to a liquid at very precisely defined conditions.  Hydrocarbons are usually a mixture of chemicals with similar but not identical vapor pressures, so when the cavitation bubbles form and collapse they tend to cushion each other. Damage is thus minimized.

One last thing: It is not impossible but it is difficult to design this sort of trim into a ball valve.  Globe anticaviation valves are a pretty mature concept and available from almost every major control valve company.  Only you or your superiors know your situation and whether is is justifiable to re-invent the wheel with this product. 

If the downstream valve modulates to hold a constant pressure it would correspond to your specifications.  
If the downstream valve is in a constant position, then 150 psi would occur at one set of service conditions.  Other properties of the fluid staying unchanged, your pressure would vary with the square of the flowrate.  

ManyMany valve data sheets are filled out by lazy or unthinking engineers who specify, for example,  100 psi upstream and 50 psi downstream at three widely varied flowrates.

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