Finding Heat Flow & Work for Brayton Cycle w/ Helium (2 Moles)

[sportsrules]

The question shows a pic of a brayton cycle on a pressure temperature graph. The gas is helium, and there are two moles. We are given two temperatures and asked to find the other two. I did this just fine. But, then we are asked to find the heat flows per kilogram of helium in each cycle and the net work that is done per kilogram of helium in each cycle. Now, there are two adiabatic processes in the diagram, so for the heat flows, we would only have to worry about the isobaric processes. So, for an isobaricprocess,

Q= n*Cp*change in temperature...instead of two moles, would I use 250moles/kg for n to find the heat flow per kilogram, and would i keep the temp change the same from the original temps that i found using the original two moles?

Then, for the work, the adiabatic processes equal

W= n*Cv*change in temp...so do I do the same thing with n again, and keep all the other info the same as what i originally found it to be?

But, for the isobaric process,
W=P*change in V

so i am not quite sure how to incorporate the 250 moles/kg...I could use PV=nRT and find the P in Pa/kg by using 250 moles/kg for n instead of the 2 moles, but then I would be using the original volumes, and wouldn't the voluems change if I changes the pressure? I am very confused...any help would be great!...and also, will the change in Q and the net work for the cycle be equal to each other??

 

[Nate810]

Thanks.Yes, the change in heat and the net work for the cycle should be equal to each other. For the heat flow per kilogram of helium, you can use the equation Q=nCpΔT where n is the number of moles/kg (250 moles/kg in this case) and Cp is the specific heat capacity of helium. For the work done per kilogram of helium, you can use the equation W=PΔV, where P is the pressure in Pa/kg (which you can calculate by using the ideal gas law PV=nRT with n=250 moles/kg). For the adiabatic processes, you can use the equation W=nCvΔT, where n is the number of moles/kg (250 moles/kg in this case) and Cv is the specific heat capacity at constant volume of helium.

 

[thepatientmental]

To find the heat flow per kilogram of helium, you would first need to convert the number of moles to kilograms. This can be done by using the molar mass of helium, which is 4.0026 g/mol. So, for 2 moles, the mass would be 8.0052 grams. To get the mass in kilograms, divide by 1000, giving you 0.0080052 kg.

Now, you can use this mass in the equation Q = n*Cp*change in temperature. Since you already have the change in temperature from the original calculations, you can use that value. So, the heat flow per kilogram of helium would be:

Q = (0.0080052 kg)*(250 moles/kg)*(specific heat of helium)*change in temperature

For the work, you would use the same mass of 0.0080052 kg in the equation W = n*Cv*change in temperature. Again, use the change in temperature from the original calculations. So, the work per kilogram of helium would be:

W = (0.0080052 kg)*(250 moles/kg)*(specific heat of helium)*change in temperature

As for the isobaric process, you will need to use the ideal gas law, PV = nRT, to find the pressure in Pa/kg. Use the same value for n (250 moles/kg) and the original volumes. Then, you can use this pressure in the equation W = P*change in V to find the work per kilogram of helium for that process.

It is important to note that the change in Q and the net work for the cycle may not be exactly equal due to energy losses in the system. However, they should be very close to each other.