Polytropic process where n is not given

[canadiansmith]

Hi I really need help figuring out this problem. A balloon behaves such that the pressure inside is proportional to the diameter squared. It contains 2 Kg of ammonia at 0 degrees celcius and 60% quality. The balloon and ammonia are now heated so that a final pressure of 600 kPa is reached. Considering the ammonia as a control mass, find the amount of work done in the process.
the answer is 117.19 KJ
i was able to find the initial volume from the quality and mass
I got Vinitial = 0.3483
I can't find a way to get a value for n in the polytropic process equation
W = (P2V2-P1V1)/ 1-n

 

[bbbeard]

Hi I really need help figuring out this problem. A balloon behaves such that the pressure inside is proportional to the diameter squared. It contains 2 Kg of ammonia at 0 degrees celcius and 60% quality. The balloon and ammonia are now heated so that a final pressure of 600 kPa is reached. Considering the ammonia as a control mass, find the amount of work done in the process.
the answer is 117.19 KJ
i was able to find the initial volume from the quality and mass
I got Vinitial = 0.3483
I can't find a way to get a value for n in the polytropic process equation
W = (P2V2-P1V1)/ 1-n

If the pressure is proportional to the square of the diameter, then the diameter is proportional to the square root of the pressure. Now the volume is proportional to what power of the diameter? ... does this help?

 

[canadiansmith]

Well not really. The question you asked is exactly the thing I am having problems with. I found that V=(4/3)pi(d/2)^3 but I am not sure if what I am trying to do is to get Volume to the power of something to equal pressure or some random constant.

 

[bbbeard]

Well not really. The question you asked is exactly the thing I am having problems with. I found that V=(4/3)pi(d/2)^3 but I am not sure if what I am trying to do is to get Volume to the power of something to equal pressure or some random constant.

You're almost there. You are trying to find an n for which PVⁿ = constant. So if

V=(4/3)pi(d/2)^3

and

d \propto sqrt(P) → d = d1(P/P1)^0.5

then

V=(4/3)pi(d1(P/P1)^0.5/2)^3

Now collect the P and the V on the same side of the equation, and put it in the form PVⁿ = constant. What is n?

 

[canadiansmith]

after simplifying I got (p^1.5)V = 3.9116*10^-5 which gives me 0.3483^n = 9.1052*10^-8
so n would be 15.37?

I got V1 = 0.3483
P1= 429.6

 

[canadiansmith]

Thank you so much for your help. I figured out what I was doing wrong. I got n = -2/3 now. Which gave me my relation. Thanks again

 

[bbbeard]

Good luck!

BBB