Help with thermodynamics problem

[Deathcrush]

Homework Statement

Hi I would like you guys to help me with this problem

Air is to be compressed steadily and isentropically from 1 atm to 25 atm by a two-stage compressor. To minimize the total compression work, the intermediate pressure between the two stages must be
(a) 3 atm (b) 5 atm (c) 8 atm (d) 10 atm (e) 13 atm

Homework Equations

(a) 3 atm (b) 5 atm (c) 8 atm (d) 10 atm (e) 13 atm

P=sqrt(P1*P2)

The Attempt at a Solution

I found that this pressure can be calculated with the square root of the product of those two pressures, however, i want to know where that equation comes from

thanks!

 

[Deathcrush]

cmon, help me, please, how do you obtain that equation?

 

[Andrew Mason]

Homework Statement

Hi I would like you guys to help me with this problem

Air is to be compressed steadily and isentropically from 1 atm to 25 atm by a two-stage compressor. To minimize the total compression work, the intermediate pressure between the two stages must be
(a) 3 atm (b) 5 atm (c) 8 atm (d) 10 atm (e) 13 atm

Homework Equations

(a) 3 atm (b) 5 atm (c) 8 atm (d) 10 atm (e) 13 atm

P=sqrt(P1*P2)
!

If there is any savings of work by doing it in two stages, the heat from the first compression has to be removed. So compress it to 1/5 of original volume (adiabatic), remove the heat and then adiabatically compress it further to 1/5 of this compressed state.

You should be able to show that if you do it any other way (eg. compress 1/2 then 1/12.5, or vice versa) you will end up doing more work. Write out the mathematical expression for work for the first compression to V1 + work for the second compression to Vf. You have to use the adiabatic condition (T and V) for each compression with a reduction in temperature to initial temperature after the first compression. When is the work a minimum?

AM

 

[Deathcrush]

yeah, well, what i am interested in , is in finding how can I get that equation listed above, I had thought in deriving the equation for the sum of work in each phase, and finding the minimum value, however i can't set the whole thing to get that equation. Can I consider the air as an ideal gas? or how else can I get an equation P to integrate

 

[Andrew Mason]

yeah, well, what i am interested in , is in finding how can I get that equation listed above, I had thought in deriving the equation for the sum of work in each phase, and finding the minimum value, however i can't set the whole thing to get that equation. Can I consider the air as an ideal gas? or how else can I get an equation P to integrate

Use the adiabatic condition for the first stage from V₀ to V₁:

$$T_0V_0^{(\gamma-1)} = T_1V_1^{(\gamma-1)}$$

Since it is adiabatic, the work done is equal to the change in internal energy: $W = nC_v\Delta T$

Do the same for the second stage. Then show that the minimum value for total Work is when V2/V1 = V1/V0 (ie. $V_1 = \sqrt{V_2V_0}$).

AM

 

[Deathcrush]

I think I'm getting it, I would thank you if you could recommend me some reading about this