Example Problem with Complete Solution

9A-1 : Optimal Compressor Outlet Pressure for the Ideal Brayton Power Cycle
5 pts
The compressor inlet temperature for an ideal Brayton cycle is T1 and the turbine inlet temperature is T3. Using a cold air-standard analysis, show that the temperature T2 at the compressor exit that maximizes the net work developed per unit mass of air flow is T2 = (T1 T3)1/2.
                   
Read : The compressor ratio corresponding to the maximum net work per unit mass of air flow is: Start with this equation.
Make all of the usual assumptions for the standard Brayton Cycle. 
Make special use of the fact that heat capacities and heat capacity ratio are all constant.
Take advantage of the fact that the compressor is isentropic in the ideal Brayton Cycle.
Given: An ideal Brayton Cycle is analyzed on a cold air-standard basis. 
Find : Show that the compressor exit temperature that maximizes net work per unit mass of air flow is given by T2 = (T1 T3)1/2.
 
Diagram :
Assumptions : - Each component is an open system operating at steady-state.
- The turbine and compressor are isentropic.
- There are no pressure drops for flow through the heat exchangers.
- Kinetic and potential energy changes are negligible.
- The working fluid is air modeled as an ideal gas.
- The specific heat CP and the specific heat ratio γ are constant.
Equations / Data / Solve :
 
The compressor ratio corresponding to the maximum net work per unit mass of air flow is:   Eqn 1
Because the compressor is isentropic, we can use the following PVT relationship for isentropic processes:   Eqn 2
Set Eqn 2 equal to Eqn 1 to get: Eqn 3
Simplify Eqn 3 algebraically to get: Eqn 4
Finally, solve for T2 : Eqn 5
Verify : The assumptions made in the solution of this problem cannot be verified with the given information.
 
Answers : The optimal temperature of the compressor effluent in a Brayton Cycle is the geometric average of the temperatures of the compressor and turbine feed streams.
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