So tonight was an interesting night. Had to do some calculations to get the right pulley size for my car. Now, if you are not familiar with supercharging applications, here is a quick intro.
You get different types of superchargers, namely roots, centrifugal and twin scroll, vane and positive charge superchargers. Go here for a more detailed explenation --
http://en.wikipedia.org/wiki/Supercharger
For my car, we are using a centrifugal type supercharger, which is belt driven. It uses the engine crank to turn the supercharger via a pulley wheel. The centrifugal supercharger works the same as a turbo charger, but as stated before, it is turned by a pulley system instead of exhaust gases, as is done with a turbo charger.
To increase boost on a centrifugal supercharger, one needs to increase the impeller speed of the charger. This is done by decreasing the size of the pulley wheel on the supercharger.
Now, this is the important part. Just because 3.2" pulley wheel on one supercharger makes 9psi, does not mean the same size pulley wheel will make 9psi on another supercharger. Different superchargers have different step ups and max rpm's. The same can be said for using two identical superchargers with identical pulleys, but installing them in two different cars. The Crank Pulley is the important factor here. Car A may have a crank size of 4.5" and car B has a crank size of 5.6". Due to this, the impellar on the two identical superchargers will be spinning at different speeds, thus creating different boost.
Alright... so now that we have that out the way. Onto the fun stuff.
The Supercharger I have is a vortech V3sci charger. Its max RPM is 53000, which produces 17psi boost and 1050 cfm.
Obviously I do not want to run 17psi. I want only 9psi. Therefore, I need to do some equations to calculate the correct impeller RPM needed to create 9psi.
Different cars run different demand of air to operate. This is know as the engine displacement. This number is needed for us to calculate the correct PSI required at certain impeller speeds.
To calculate this displacement for the S54b32 M3 engine - we need to use the following equation. ED=(SxB²x0.785)xN
S= stroke B= Bore N= number of cylinders
This equation is then calculated as follows
ED = (91.0x87.5²x0.785) x 6 = 3.28 litres (remember my bore is 87.5 not 87)
we need this calculation in Cubic inches.
ED = (3.58 x 3.43² x0.785) x 6 = 198.48 c.i
To calculate engine demand in cfm, we first need to calculate engine displacement in cubic feet. To do this we take the displacement in cu. and divide it by 1728. In the case of my engine which has a displacement of 198.48 c.i. the calculation is: 198.48 divided by 1728 = 0.1148
0.1148 x max RPM, in this case 8000 = 918.88
we now divide this number by the number of strokes. In this case 2.
918.88 divided by 2 = 459.44.
This is the Engine demand in CFM for the e46 M3 engine @ 100% volumetric efficiency .
Now that we have this number, we can start calculating the PSI we get from different % of the supercharger CFM.
Remember, my supercharger has a maximum CFM of 1050.
100% of engine is 8000rpm = 459.44
100% of supercharger is 53000rpm = 1050
therefore 8000rpm= 100% x 100% = 100% = 1050CFM
7200rpm = 90%x90%=81% of 1050 =945 CFM
6400rpm = 80%x80%= 64% of 1050 = 672 CFM
5600rpm= 70%x70% = 49% of 1050 = 514 CFM
You can calculate this all the way to 1000rpm.
Basically, this equation gives you the CFM number that the supercharger is making at the desired % you are running it at.
Now we need to use this formula to get the boost.
Supercharger air supply divided by engine demand = total atmospheres.
Total atmospheres minus 1 atmosphere x 14.7 = boost.
We also need to use this impeller speed calculator on Vortech's website --
http://www.vortechsuperchargers.com/page.php?id=33511
We will then match the impeller speed in the below equations to the ones calculated in the above calculator to get the pulley size you require. It needs some guess work to get to where you want to be, but here is the correct equation for 9psi using -
Supercharger air supply divided by engine demand = total atmospheres. Total atmospheres minus 1 atmosphere x 14.7 = boost.
85% - 85x85 = 72.25% of 1050 which = 758.62 CFM
758.63 / 459.44 = 1.65 atmospheres. We subtract 1 atmosphere. Thus we are left with 0.65 atmospheres. We multiply this by 14.7 = 9.5lbs (PSI)
Great. We have the PSI we want. We then take the 85% of 1050, and calculate it in RPM.
85% of 53 000rpm is 45 050rpm.
Now that we have the RPM we are looking for, using vortech's impeller calculator, we can find the correct pulley size.
The e46 M3 has a crank pulley of 4.5". so, to get an impeller speed of 45 050rpm, we need to have a supercharger pulley of 2.72"
And that my friends... is how you calculate boost for a supercharger.
Here are my calculations on paper...
Basically, this first equation just shows that if I run this Supercharger at 100% efficiency, I will blow it above 7000Rpm