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Ideal Gas, 2-Stage Compressor Design

Total power requirement for the

2-stage compressor

system:

Assume the best case for each compressor : Reversible + Adiabatic = Isentropic.
Assume the working fluid is an ideal gas at all the states that exist within the system.
Assume the working fluid has constant heat capacities.
Therefore, each compressor is a polytropic process : PVγ = C
From Lesson 8B :
From Lesson 7E :

these compressors are not isothermal.

Remember:

Subscripts "i" and "o" refer to to the inlet and outlet streams for each compressor.

So, for each compressor:

For both compressors:

Where Px is the pressure at the outlet of the 1st compressor.

Lesson
8C Blog

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Lesson 8C: page 17 of 21

The starting point for the design of our 2-stage compressor is the fact that the total shaft work requirement for the system is the sum of the work requirement for the 2 compressors.
Next, we need to make some assumptions about the nature of our two compressors.
1st we assume that each compressor is isentropic.
Next, we assume that the working fluid is an ideal gas at the temperatures and pressures that exist in the compressor system.
Finally, we assume that the heat capacity ratio, γ = CP/CV is constant as well.
This is a long list of very restrictive assumptions, so keep in mind that the result is more of a guideline than a definitive rule for the design of multi-stage compressors.
Ok, so why did we make these 3 big assumptions?
We made these 3 assumptions so we could use the equation from Lesson 8B for WS,dot/mdot, shown here.
Since we are more interested in pressures than temperatures, we want to use one of the 3 PVT relationships from Lesson 7E. This relationship only applies when our big 3 assumptions are true.
This produces a somewhat messy equation for the power requirement for each compressor.
When we combine the work requirements to obtain an equation for the total shaft work requirement of the compression system, the equation gets REALLY messy.
Remember, the goal is to determine the optimum value of PX.
Trust me here. The result will be surprisingly simple and elegant.
So, what are we going to do with this big equation in order to determine the optimal value of PX ??
Flip the page and you’ll see.

Ideal Gas, 2-Stage Compressor Design

2-stage compressor

Ch 8, Lesson C, Page 17 - Ideal Gas, 2-Stage Compressor Design