Thermodynamics turbocharger question

[thorx440]

Homework Statement

A 10 L truck engine has a volumetric efficiency of 100%. It has a turbocharger which increased the mass flow of air and its pressure by a factor of 2 when the engine is running at 4300 rpm. Find the turbocharger rpm, efficiency, and the temperature of the air leaving the turbo.

Inlet conditions are 1 atm and 80 degrees F. The AFR is 98% and therefore the volume occupied by fuel is negligible. Also, the turbocharger can be assumed to be adiabatic

(Also given compressor map not shown here)

The attempt at a solution
So I first used the ideal gas law to find the density of air at the given conditions. Using this density, I then used the volumetric equation to find the mass flow rate through the turbo, and doubled that value since it is increased by a factor of two. Now by looking at the compressor map, I have the corrected air flow, as well as the pressure ratio, which allowed me to find the rpm as well as the compressor efficiency. This is where I got stuck. Assuming an isentropic turbo, I was able to find the ideal exiting temperature of the air using the T2 = T1(P2/P1)^(k-1/k) equation. Using that value, I then plugged it back into the efficiency equation and found the actual exiting temperature of the air. Now I have the inlet and outlet temps of the air in the compressor, as well as the turbo rpm speed, but know nothing about the turbine efficiency or temperatures. Is there a relationship between the turbine and compressor that I am missing? Or is there another way to find the temperature of the turbine?

I am assuming that I need to find the outlet temperature of the turbine, and then go about finding the efficiency of the turbine. With the efficiency of the compressor and the turbine, I can then find the efficiency of the whole turbocharger. I am just not seeing how to find any information on just the turbine.

Any help?

 

[Davidthefat]

Can't you calculate the turbine inlet temperature assuming the exhaust from the engine is adiabatic using the adiabatic flame temperature knowing the AFR? Using AFR, you can get the molar ratio and balance the equation and solve for the T adiabatic using a reference temperature and the enthalpies of the reactants can be looked up and use the Cp(T_ad-T_ref) relation for enthalpy of each product. 1st law of thermo in effect. Steady, so the dt term is gone. Left with mass flow rate * enthalpy terms. Instead of mass, you can do molar relation instead.

Knowing that, you know the work requirement of the compressor and you can calculate what the outlet temp needs to be for the turbine to get your work out. Then calculate efficiency.