Here is an interesting site:
motorgeek.com :: Turbo/Engine Flow Calculations and Maps
Just be sure to plug in some realalistic numbers.
To get manifold pressure as we know it you have to add the environmental air pressure to the boost pressure. And 0.5 pressure loss through an inter-cooler is a bit optomistic.
THe program is for airflow requirements and does not take into acount riction with rpm orthe power needed to drive the supercharger. While it doesn't have anything for detonation limits the heat rise in the supercharger is interesting as are the intercooler and water injection sections.
OF course nothing there considers size, cost, engineering time and other factors which decide wither something is done or not.
And remember in WW II no computers. You had anywhere from a couple of men to a couple of hundred working at desks with sliderules, adding machines, and pencil and paper to do complex calculations.
To get back to you question you have three basic choices.
1. A. Use a 2 speed gear box on the existing engine supercharger.
Advantages=quick, cheap (both in engineering effort and manufacture) , minimal impact on engine size or weight so minimal (or no) problems installing in existing airframes. better take-off, low altitude (6-7000ft?) performance and climb.
Disadvantages= does nothing to improve perfomance at any hight above what the 81-99 engiens can already do. Some slight increase in intial cost and maitainence.
B. design larger capacity, higher effecincy 2 speed supercharger.
Advantages= only a bit more expensive in engineering erfort and manufacture. Only a bit heavier and larger in volume so that installing in exisrting airframes should still be easy.
Disadvantages= may only raise critical height (full throttle) by about 2,000ft, may do nothing for engine at lower altitudes compared to option A.
2. use a 2 speed, 2 stage supercharger.
A. Use a set-up like the Merlin. both impellors on a common shaft, always turning at the same speed. aftercooler not required but strongly recommended.
Advantages= compact for a two stage supercharger. will boost performance into the low 20,000ft area, higher with aftercooler. cheapest 2 stage in terms of intial layout and set up.
Disadvantages=size and weight compared to option 1. may require a fair amount of deveopment time in impellor and diffuser design to "match" the superchargers. it will impact engine installationin existing airframes but probably not to an extent that prevents use.
B. Use set up Like U.S. Navy. 1st stage (auxilary stage) supercharger has it's own 2 speed gearbox with clutch whihle engine has a single speed drive. Auxilery supercharger is dis-engaged for take off/low altitude to save power. Lowgear is engaged for medium altitude and high gear used for high altitude. Intercooler used when 1st stage is engaged.
Advantages= good high altitude performance with no real engine penelities at low altitudes, being able to vary the speeds of the 1st impellor may make supercharger m"matching" easier?
Disadvantages=weight and bulk are starting to get even bigger. engineering time could be getting rather large and the intercooler if aircooled is no longer universal but specific to each air frame.
C. As in B but using a hydralic drive to the first stage. This is what Allsion was working on. Allison also skipped the intercooler and substituted water injection instead. Trade voume of intercooler for smaller but denser ADI tank.
3. Use a turbo-charger. Already being done.
Advantages= Great high altitude performance
Disadvantages= weight and bulk of installation make a tremendous impact on airframe. Maintainence issues for part of the war and perhaps an allocation of resources problem.
So as a bright young spark which way do you go? quick and cheap with limited performance or long and expensive with high performance.