Posted: Mon Jul 12, 2010 7:31 pm
If pressure was not multiplied by each stage, then a paintball 3 stage compressor would be good for maybe 600 PSI instead of 4500.
Maybe my 1/2 stroke example was confusing. Let's redesign for an example. Using a low pressure example to keep from blowing out a head gasket and providing high volume instead, lets do a 2 stage compressor. Each stage is to compress it's inlet pressure 5X by compressing the volume of absolute pressure into 1/5th the volume and then the outlet valve opens. We will look at this from a power required standpoint.
First stage takes in atmospheric air. Assume no dead space such as a perfect piston fit. 1 atm in and 1 atm on the piston backside.. It works as follows. 1/2 way through the compression stroke the piston pressure is 2 atm, at 3/4 stroke the pressure is 4 atm and at 4/5ths stoke the outlet valve opens and the 1 unit of air is expelled on the remaining 1/5th of the stoke until the entirety of the volume is expelled. Force on the piston is not a difficult calculation. Most of the piston stoke delivers NO air to the outlet as the outlet valve is closed for 4/5th of the pressure stroke distance. Pressure delivered is at 5 Atm and is 1/5 the original volume.
Second stage receives this air, but to get the inlet full for the stoke, the first stage needed 5 stokes to provide this 1 volume at the higher pressure. It could have either a displacement 5 times as large, have 5 in parallel, or run 5 times faster. Note the HP requirement to provide the first stage to keep up with the input requirement of the 2nd stage.
The second stage takes in an easy cylinder full of air at 5 Atm and has 5 Atm of air on the piston backside. Intake stroke energy requirement is the same as the first stage. No pressure differential. Things change on the second stage on the compression stroke. At 1/2 way up the cylinder pressure is now 10 Atm in pressure. Please note the first stage has a 1 atm increase at this position and the second stage is pushing 5 Atm increase at 1/2 the piston travel. At 3/4 the piston travel the cylinder volume is 1/4 it's original volume at 4X the initial pressure or 20 Atm. The outlet valve is still closed. At this point in time we have 20 Atm in the cylinder and 5 Atm still in the compressor shell and piston backside. Simple math will show the crankshaft force is pushing a 15 Atm difference. At 4/5ths the piston travel, the outlet valve opens again just like the first stage and this volume of air compressed again to 1/5th its volume is delivered at 5X it's inlet pressure at 25 Atm. In this case the last 1/5th the piston stroke is delivering the cylinder of air out the valve at 25 Atm pressure.
Please note at the work portion where our perfect pump delivers the air on the last 1/5th of the piston stroke, the outlet is 25 Atm and the inlet is 5 Atm so it is pushing an increase of 20 Atm. The first stage only had a differential of 4 Atm.
Simple math will convert the absolute pressures to relative that you would read on a normal pressure gauge.
The first stage is pumping into about 30 PSI and the second stage is pumping into 360 PSI using a rounded Atm pressure of 15. For this to work the first stage needs 5X the displacement, or using matched fridges, 5 compressors in parallel feeding into one second stage. The single second stage has about the same HP requirement as the combined HP of the 5 compressors feeding into the second stage.
I hope that example works better for you. This is ignoring all leaks, dead space, compression heating, and other system losses. The real world would have to account for those factors.
Maybe my 1/2 stroke example was confusing. Let's redesign for an example. Using a low pressure example to keep from blowing out a head gasket and providing high volume instead, lets do a 2 stage compressor. Each stage is to compress it's inlet pressure 5X by compressing the volume of absolute pressure into 1/5th the volume and then the outlet valve opens. We will look at this from a power required standpoint.
First stage takes in atmospheric air. Assume no dead space such as a perfect piston fit. 1 atm in and 1 atm on the piston backside.. It works as follows. 1/2 way through the compression stroke the piston pressure is 2 atm, at 3/4 stroke the pressure is 4 atm and at 4/5ths stoke the outlet valve opens and the 1 unit of air is expelled on the remaining 1/5th of the stoke until the entirety of the volume is expelled. Force on the piston is not a difficult calculation. Most of the piston stoke delivers NO air to the outlet as the outlet valve is closed for 4/5th of the pressure stroke distance. Pressure delivered is at 5 Atm and is 1/5 the original volume.
Second stage receives this air, but to get the inlet full for the stoke, the first stage needed 5 stokes to provide this 1 volume at the higher pressure. It could have either a displacement 5 times as large, have 5 in parallel, or run 5 times faster. Note the HP requirement to provide the first stage to keep up with the input requirement of the 2nd stage.
The second stage takes in an easy cylinder full of air at 5 Atm and has 5 Atm of air on the piston backside. Intake stroke energy requirement is the same as the first stage. No pressure differential. Things change on the second stage on the compression stroke. At 1/2 way up the cylinder pressure is now 10 Atm in pressure. Please note the first stage has a 1 atm increase at this position and the second stage is pushing 5 Atm increase at 1/2 the piston travel. At 3/4 the piston travel the cylinder volume is 1/4 it's original volume at 4X the initial pressure or 20 Atm. The outlet valve is still closed. At this point in time we have 20 Atm in the cylinder and 5 Atm still in the compressor shell and piston backside. Simple math will show the crankshaft force is pushing a 15 Atm difference. At 4/5ths the piston travel, the outlet valve opens again just like the first stage and this volume of air compressed again to 1/5th its volume is delivered at 5X it's inlet pressure at 25 Atm. In this case the last 1/5th the piston stroke is delivering the cylinder of air out the valve at 25 Atm pressure.
Please note at the work portion where our perfect pump delivers the air on the last 1/5th of the piston stroke, the outlet is 25 Atm and the inlet is 5 Atm so it is pushing an increase of 20 Atm. The first stage only had a differential of 4 Atm.
Simple math will convert the absolute pressures to relative that you would read on a normal pressure gauge.
The first stage is pumping into about 30 PSI and the second stage is pumping into 360 PSI using a rounded Atm pressure of 15. For this to work the first stage needs 5X the displacement, or using matched fridges, 5 compressors in parallel feeding into one second stage. The single second stage has about the same HP requirement as the combined HP of the 5 compressors feeding into the second stage.
I hope that example works better for you. This is ignoring all leaks, dead space, compression heating, and other system losses. The real world would have to account for those factors.