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. | ||||||||