Working out temperatures on an ideal brayton cycle.

[beeohbee]

Homework Statement

An ideal Brayton cycle has a pressure ratio of 15 and can be analysed using air standard cycle assumptions. The gas temperature is 300K at the compressor inlet and 1500K at the turbine inlet.

The compressor and turbine can be considered to be isentropic.
For an isentropic process,$$pv^{\gamma}$$ = constant.

Work out the temperature at the end of the compressor.

Homework Equations

$$P_{1}$$ $$V_{1}^{\gamma}$$ = $$P_{2}$$$$V_{2}^ {\gamma}$$

$$P_{1}$$ $$V_{1}$$ = $$R T_{1}$$

$$P_{2}$$ $$V_{2}$$ = $$R T_{2}$$

The Attempt at a Solution

I've sketched the cycle on a t-s property diagram to illustrate the problem.


From the relevant equations, I have that:

$$T_{2} = T_{1}(\frac{P_{2}}{P_{1}})^{\gamma - 1/ \gamma}$$

Which gives me, $$T_{2} = 300.15^{\gamma - 1/ \gamma}$$

I have $$T_{1}$$ = 300 and the pressure ratio = 15 but I see no way to complete the equation as I don't know the heat capacity ratio, $${\gamma}$$ or a way to work it out.

 

[beeohbee]

I've now found that the heat capacity ratio is a constant and for this question is 1.4. I don't know why this wasn't mentioned anywhere in the question or my textbooks but a lot of googling came up with the answer in the end.

By using gamma=1.4, T2 comes out as 650K. :)

 

[drivenbyfate]

Yea I was reading this and thinking the exact thing...the heat capacity ratio is 1.4 for air...easy after that

 

[A87]

hey do you guys know what the equations for efficiency, work done and losses are within the brayton cycle. I am trying to do a backwards model starting from the size of electrical generator I need and thus working towards what capacity gas turbine, pressure ratio and inlet temp I should have

Thanks

 

[DeeNos]

I would suggest using the ideal gas law to find the heat capacity ratio, {\gamma}. The ideal gas law states that PV=nRT, where P is pressure, V is volume, n is the number of moles of gas, R is the gas constant, and T is temperature. Rearranging this equation, we can find the heat capacity ratio, {\gamma}, by using the equation {\gamma} = CP/CV, where CP is the specific heat at constant pressure and CV is the specific heat at constant volume. By using the ideal gas law and known values for P, V, and T, we can solve for {\gamma} and then use it in the equation T_{2} = T_{1}(\frac{P_{2}}{P_{1}})^{\gamma - 1/ \gamma} to find the temperature at the end of the compressor. Additionally, we can also use the specific heat values for air to find the heat capacity ratio and then use it in the equation. This will provide a more accurate solution for the temperature at the end of the compressor. It is important to use all available information and equations to accurately solve for the desired temperature.