Crankshaft driven centrifugal superchargers are not driven or even augmented by exhaust thrust. Any exhaust driven supercharger is either a turbine or a turbocharger. There are various forms of compound supercharging, where different types of superchargers are combined on the same engine, most commonly when some form of crank driven supercharger is combined with a turbocharger.
Crankshaft driven superchargers are tuned for various boost levels by size and drive ratio, they do not normally have any sort of dump valve, but normally have what is called a "pop off valve", intended to vent excess pressure, usually caused by a back fire. Crankshaft driven centrifugal superchargers are somewhat similar in design to turbochargers, except they have no exhaust turbine side to drive them, as they are driven by the crankshaft through a coupling and gearbox. It is necessary to spin a centrifugal supercharger at very high speeds, often in excess of 5 times the engine speed or more, because they are not positive displacement and therefore leak inside. Usually, the drive is a combination of ball bearings and planetary gears, to allow for over speed and for slippage.
The other common form of crankshaft driven supercharger is the positive displacement type, the most common version of the positive displacement supercharger is the Roots style, commonly seen on Detroit Diesel engines (first used for this by GMC trucks). This supercharger is rarely if ever seen on anything but older Detroit Diesel engines, and race engines such as drag racing or truck and tractor pulling. The ratio between the crankshaft and the supercharger is fixed, as opposed to variable. The boost is determined by drive ratio and supercharger size. They are rarely driven any more than twice the engine speed.
A turbocharger is similar to the centrifugal crank driven supercharger, but uses an exhaust turbine inside a housing to drive the compressor side. It is on the exhaust side of a turbocharger where you find a dump valve used to control boost, it is called a waste gate. When a turbocharger reaches a certain level of boost on the compressor side, the waste gate gate is opened, allowing exhaust to bypass the turbine, and slowing the turbocharger down to produce less boost. In the case of the General Electric B series turbochargers used on the Allison in the P-38, and the P&W in the P-47, as well as on some bombers, the waste gate also controls turbocharger speed, as at very high altitudes, where there is too little air to produce full boost, the turbocharger, without the load of producing full boost, will over speed. This happens around 33,000 to 35,000 feet. The waste gate on a General Electric B series turbocharger is actually controlled by an RPM governor, and actuated by oil pressure. The P-38 and P-47 had RPM indicator lights on the instrument panel so the pilot could see if he was exceeding the RPM limit. On the P-38, pilots sometimes failed to keep the engine temperatures in the correct operating range, and if the oil got too cold it would congeal, and the governor would not work, allowing the turbocharger to over speed or over pressurize.
Most high end endurance type superchargers are coupled with some sort of intercooler, since compressing air, especially enough to make serious horsepower, heats it a great deal, leading to inefficiency and detonation. Most intercoolers are "core" type, AKA heat exchanger, as seen on the K model and later P-38, and similar in construction to a radiator or oil cooler. They can be either air to air or air to liquid. Air to air types are common in aircraft, as the cooler air of high altitudes allows them to be fairly efficient. Efficiency is determined by the amount restriction they create (pressure drop across the intercooler) and the amount they lower the air temperature. A 100% efficient intercooler (very rare) will not lose any boost, the pressure before the intercooler and the pressure after the intercooler are the same, and it will lower the air charge temperature back to the ambient air temperature, the temperature the air was before the supercharger compressed it. Properly sized intercoolers are close to 100% efficient with regards to restriction, and around 50% efficient on cooling (lowers the compressed air temperature to a point half way between the temperature of the compressed air out of the supercharger and and the ambient air temperature before it is compressed).
