Potential point of confusion

I got a nice letter today from a man trying to help me avoid wasting my time. He had a good point, although it was based on a false assumption. This can be checked by looking at the math for the work of compression and how it is derived. Here is his letter followed by my response:

Dear Scott

I have read your blog and am afraid that I think I have found a flaw in your reasoning. In the description of your invention you say that when the booster piston moves all the way to the right the valve is opened equalising the pressure. The at least in maths it looks like the piston only has to do a small amount of work. But this is not so. The piston now has a large pressure on one side and a small one on the other. It will have to push against this pressure differential all the way as it moves right. So the mechanical work required to drive it will be great. It does not work if it has pressure on the other side either. It is easy on the compression stroke but in order to draw back to let more air in it has to do so against the full pressure difference again. However one configures it, one half of the crank rotation will have to be against the full pressure differential.

Unless I misunderstand the cycle it seems impossible for this to compress air with less work than can be extracted from it.

I write to you as I would not wish you to use up valuable time on an idea
which did not work.

N


N,

Thanks for writing. I have thought of what you're saying and here's my answer. Let me know what you think. I am happy to hear from you whether you agree or not.

The work of compressing air with a piston has 3 components.
1. intake, which is negative work; atmospheric pressure enters on its own
2. PV change work, pressure going up as volume goes down, this is the actual compression
3. delivery, after a portion of the stroke the air in the cylinder will equal tank pressure; from there till the end of the power stroke the work is done against a constant pressure so is a lot less

The way I see it is that the work done is not the difference between the two sides of the piston. If it was, then double-acting compressors would be the only possible way to compress air, especially if multi-stage.

Look at the math for compression work. There is no term in the equation for what is on the back side of the piston. It is a neutral zone, unrelated. The pressure differential against which any compressor works is the difference between initial pressure of intake and final pressure of delivery.

In the equalization engine, the piston is fighting about 2 psi. The PV curve is very short, and most of the PV diagram shows a straight line. That is the delivery portion of the stroke, which is most of the power stroke.

I went into all this in my book Compressed Air Power Secrets, 3rd edition. Believe me, I spent many agonizing weeks trying to analyze the exact question you raise. Here is something else you can think about.

A single-acting, two-stage compressor has atmospheric pressure on one side of the piston and an elevated intake pressure in the 2nd stage. The pressure differential is not related to the two sides of the piston; if it were, then there would be no advantage to multi-staging. The elevated intake pressure is negative work; it actually helps the compression stroke. There is no mention in the math of atmospheric pressure on the 2nd stage, it's like a booster.

I hope I have made my opinion clear, and if not then write again. I will post this information on the blog also.

Don't hesitate to let me know if I have made a mistake!

Scott