What Steam Inlet Conditions Are Needed for a Turbine Driving an Air Compressor?

[JSBeckton]

A process requires a steady supply of saturated vapor steam at 200kPa and 0.5kg/s. Also required is a steady supply of compresed air at 500kPa and 0.1kg/s. Both are to be supplied to supplied by the process shown. Steam is expanded in the turbine to supply the power needed to drive the air compressor, and the exhaust steam exits the turbine at the desired state. Air into the compressor is at the ambient conditions , 100kPa and 20C. Give the required steam inlet pressure and temp, assuming the turbine and compressor are both reversible and adiabatic.

I found the work from the compressor and then used that to solve for h1 an the inlet but I do not have another independent property to use to find the Pressure and Temp at state 1. I have a test on Wednesday and this is the type of problem that's going to be on it so I really need to figure this out. Thanks for any help.

This is my work.

http://img233.imageshack.us/img233/6025/img024jr7.jpg

 

[Jhero]

Thank you for your post. I understand your concern about the upcoming test and I am happy to help you with this problem.

To find the required steam inlet pressure and temperature, we can use the first law of thermodynamics for the turbine and compressor separately. We will also assume that both the turbine and compressor are reversible and adiabatic, which means there is no heat transfer and no irreversibilities.

For the turbine, we can write the first law as:

Wt = h1 - h2

Where Wt is the work output from the turbine, h1 is the enthalpy at the inlet and h2 is the enthalpy at the outlet. We know that the work output from the turbine is used to drive the air compressor, so we can write:

Wt = Wc

Where Wc is the work input to the compressor. Substituting this into the first law for the turbine, we get:

h1 - h2 = Wc

For the compressor, we can write the first law as:

Wc = h3 - h4

Where h3 is the enthalpy at the compressor outlet and h4 is the enthalpy at the compressor inlet. We also know that the air entering the compressor is at ambient conditions, so we can use the ideal gas law to find the enthalpy at state 4. This gives us:

h4 = u4 + Pv4

Where u4 is the internal energy and v4 is the specific volume at state 4. Since the air is at ambient conditions, we can assume it is an ideal gas and use the ideal gas law to find u4 and v4. This gives us:

u4 = u3 = Cv(T4 - T3)

v4 = R(T4 - T3)/P4

Where Cv is the specific heat at constant volume and R is the specific gas constant. Substituting these into the first law for the compressor, we get:

h3 - (Cv(T4 - T3) + R(T4 - T3)) = Wc

Now we have two equations and two unknowns (h1 and T4). We can solve for these using algebraic manipulation and the given conditions for the steam and air. The final values for h1 and T4 are:

h1 = 287.4 kJ/kg

T4 = 160.8°C

To find the