How Is Work and Heat Transfer Calculated in a Polytropic Process for Helium Gas?

[dillonmhudson]

Helium gas initially at 2 bar, 200K undergoes a polytropic process, with n=k, to a final pressure of 14 bar. Determine the work and heat transfer for the process, each in kJ per kg of helium. Assume ideal gas behavior.

Ok so I have the following:
P1=200kPa
T1=200K
R_He=2.0769kJ/kg*K
Cp=5.1926
Cv=3.1156
k=Cp/Cv=1.6666

P2=1400kPa

I know that
PV^k=const

My trouble is in finding either the mass or the volume of the helium.

Thanks for any help.

 

[Andrew Mason]

Helium gas initially at 2 bar, 200K undergoes a polytropic process, with n=k, to a final pressure of 14 bar. Determine the work and heat transfer for the process, each in kJ per kg of helium. Assume ideal gas behavior.

Ok so I have the following:
P1=200kPa
T1=200K
R_He=2.0769kJ/kg*K
Cp=5.1926
Cv=3.1156
k=Cp/Cv=1.6666

P2=1400kPa

I know that
PV^k=const

My trouble is in finding either the mass or the volume of the helium.

Thanks for any help.

You do not need to know the mass or volume. You only need to know the proportional change in volume. From that you can determine the change in temperature.

Use the fact that [itex]P_iV_i^k = P_fV_f^k[/itex] to find the relative change in volume. Then use the ideal gas law to determine the change in temperature. That will enable you to find the change in internal energy.

Since k = Cp/Cv, what kind of process is this? What does that tell you about the heat transferred during the process?

Apply the first law of thermodynamics to determine the work from the change in internal energy and the heat flow.

AM

 

[dillonmhudson]

Since k = Cp/Cv, what kind of process is this? What does that tell you about the heat transferred during the process?

Are you asking me or are you getting me to think?
It is a polytropic process. I'm not sure what you are getting at.

Can you help me solve for the difference/proportional increase in the volume? Not really sure how to do that either.

Thanks.

 

[dillonmhudson]

Nevermind, I got it.

 

[rwisz]

In order to solve for the work and heat transfer for this process, we need to know the specific volume (v) of the helium gas at both the initial and final pressures. This can be calculated using the ideal gas law, PV = mRT, where m is the mass of helium, R is the gas constant, and T is the temperature.

Since we are assuming ideal gas behavior, we can use the equation P1V1 = P2V2 to solve for the specific volume at the final pressure:

V2 = (P1V1)/(P2) = (200 kPa)(1 m^3/kg)/(1400 kPa) = 0.143 m^3/kg

Now that we have the specific volume at the final pressure, we can use the polytropic process equation PV^n = const to solve for the specific volume at the initial pressure:

V1 = (P2V2^n)/(P1)^(n-1) = (1400 kPa)(0.143 m^3/kg)^1.6666/(200 kPa)^0.6666 = 0.118 m^3/kg

Next, we can use the ideal gas law again to calculate the mass of helium:

m = PV/RT = (200 kPa)(0.118 m^3/kg)/(2.0769 kJ/kg*K)(200 K) = 0.0018 kg

Now that we have the mass of helium, we can calculate the work and heat transfer for the process:

Work = PdV = P2(V2-V1) = (1400 kPa)(0.143 m^3/kg - 0.118 m^3/kg) = 3.5 kJ/kg

Heat transfer = mCp(T2-T1) = (0.0018 kg)(5.1926 kJ/kg*K)(1400 K - 200 K) = 10.5 kJ/kg

Therefore, the work for the process is 3.5 kJ/kg and the heat transfer is 10.5 kJ/kg. I hope this helps to clarify the process for solving this problem.