Thermodynamics: Pressure and temperature from turbine

[Henry Stonebury]

Homework Statement

A turbine is receiving air from a combuster inside of an aircraft engine. At the inlet of the turbine I know that
T1 = 1273 K and P1 = 549 KPa, and the velocity of the air is essentially 0.
The turbine is assumed to be ideal, so its efficiency is exactly 1.

Also: R = 287 J/KgK, Cv = 716 J/KgK, and Cp = Cv + R = 1003 J/KgK

What I am trying to find is the temperature and the pressure at the outlet(T2, P2).

Homework Equations

The energy balance equation for control volumes:
d(Ecv)/dt = Qdot - Wdot + mdot*[(h1 - h2) + (V1^2 - V2^2)/2 + g(z1 - z2)]

Change of Entropy for ideal gas:
deltaS = Cp*ln(T2/T1) - R*ln(P2/P1)

and of course the Ideal Gas law, if it is useful here:
Pv = RT, where v is specific volume

The Attempt at a Solution

First, I considered the energy balance equation. Energy is not changing in this process so d(Ecv)/dt = 0. Also, kinetic and potential energy are not changing so their respective terms are also made to be 0. There is no heat transfer into or out of the turbine so the Qdot term is also 0.

I am left with:

0 = -Wdot + mdot(h1 - h2)

Rearranging:

-Wdot/mdot = h2 - h1

h2 - h1 can be replaced with Cp(T2 - T1), so

-Wdot/mdot = Cp(T2 - T1)

At this point I start to become unsure of myself, as I am not completely sure what to do with the work term here. I decided to move on and look at the entropy to see if I could get further.

I assumed that this process is isentropic, so deltaS should be 0. The reason why I chose to assume this is because of the fact that the efficiency = 1 for this turbine.

so using the entropy equation for ideal gas:

0 = Cp*ln(T2/T1) - R*ln(P2/P1)

I get:

T2/T1 = (P2/P1)^(R/Cp)

This is where I hit a dead end. I have two unknowns in this equation, so I would need another equation in order to solve for both of them. I thought of using the Ideal Gas law here, but I realized that using it would add an extra unknown: specific volume.

So all I am left with is that work term in the energy balance equation. In my textbook I found that:

Wdot/mdot = int(vdP)

so I decided to try it. I replaced v with RT/P from the ideal gas law and tried integrating like so:

int( RT/P dP ) = int( Cp dT)
int( R/P dP ) = int( Cp/T dT)

and to my amazement this gave me: R*ln(P2/P1) = Cp*ln(T2/T1), which is of course equal to my entropy equation from before.

So now I am hopelessly lost, and hopefully somebody can shed some light on what I am doing wrong here.

 

[SteamKing]

Homework Statement

A turbine is receiving air from a combuster inside of an aircraft engine. At the inlet of the turbine I know that
T1 = 1273 K and P1 = 549 KPa, and the velocity of the air is essentially 0.
The turbine is assumed to be ideal, so its efficiency is exactly 1.

Also: R = 287 J/KgK, Cv = 716 J/KgK, and Cp = Cv + R = 1003 J/KgK

What I am trying to find is the temperature and the pressure at the outlet(T2, P2).

Homework Equations

The energy balance equation for control volumes:
d(Ecv)/dt = Qdot - Wdot + mdot*[(h1 - h2) + (V1^2 - V2^2)/2 + g(z1 - z2)]

Change of Entropy for ideal gas:
deltaS = Cp*ln(T2/T1) - R*ln(P2/P1)

and of course the Ideal Gas law, if it is useful here:
Pv = RT, where v is specific volume

The Attempt at a Solution

First, I considered the energy balance equation. Energy is not changing in this process so d(Ecv)/dt = 0. Also, kinetic and potential energy are not changing so their respective terms are also made to be 0. There is no heat transfer into or out of the turbine so the Qdot term is also 0.

I am left with:

0 = -Wdot + mdot(h1 - h2)

Rearranging:

-Wdot/mdot = h2 - h1

h2 - h1 can be replaced with Cp(T2 - T1), so

-Wdot/mdot = Cp(T2 - T1)

At this point I start to become unsure of myself, as I am not completely sure what to do with the work term here. I decided to move on and look at the entropy to see if I could get further.

I assumed that this process is isentropic, so deltaS should be 0. The reason why I chose to assume this is because of the fact that the efficiency = 1 for this turbine.

so using the entropy equation for ideal gas:

0 = Cp*ln(T2/T1) - R*ln(P2/P1)

I get:

T2/T1 = (P2/P1)^(R/Cp)

This is where I hit a dead end. I have two unknowns in this equation, so I would need another equation in order to solve for both of them. I thought of using the Ideal Gas law here, but I realized that using it would add an extra unknown: specific volume.

So all I am left with is that work term in the energy balance equation. In my textbook I found that:

Wdot/mdot = int(vdP)

so I decided to try it. I replaced v with RT/P from the ideal gas law and tried integrating like so:

int( RT/P dP ) = int( Cp dT)
int( R/P dP ) = int( Cp/T dT)

and to my amazement this gave me: R*ln(P2/P1) = Cp*ln(T2/T1), which is of course equal to my entropy equation from before.

So now I am hopelessly lost, and hopefully somebody can shed some light on what I am doing wrong here.

You're given P1 as the gas comes out of the combuster and enters the turbine. What is P2 at the exit of the turbine? What should it be?

 

[Henry Stonebury]

P2 and T2 are the temperature and pressure at the exit of the turbine, and in this problem they are both unknowns.

I do however know what their values are supposed to be: P2 = 293KPa, T2 = 1063K
I just do not know how to get there with what I have.

 

[SteamKing]

P2 and T2 are the temperature and pressure at the exit of the turbine, and in this problem they are both unknowns.

I do however know what their values are supposed to be: P2 = 293KPa, T2 = 1063K
I just do not know how to get there with what I have.

You must have been given some more information about the turbine which you have overlooked.

Unless there is something specific happening with this turbine, the exhaust pressure is usually taken to be atmospheric pressure.

What is the problem statement given to you verbatim?