Determining Compressor Pressure Ratio for Maximum Net Work
Determine the pressure ratio across the compressor of an ideal Brayton cycle for the maximum net work output per unit of mass flow if the state at the compressor inlet and the temperature at the turbine inlet are fixed. Use a cold air-standard analysis and ignore kinetic and potential energy effects. Discuss.
Known An ideal Brayton cycle operates with a specified state at the inlet to the compressor and a specified turbine inlet temperature.
Find Determine the pressure ratio across the compressor for the maximum net work output per unit of mass flow, and discuss the result.
Schematic and Given Data:
Engineering Model
1. Each component is analyzed as a control volume at steady state.
2. The turbine and compressor processes are isentropic.
3. There are no pressure drops for flow through the heat exchangers.
4. Kinetic and potential energy effects are negligible.
5. The working fluid is air modeled as an ideal gas.
6. The specific heat c_{p} and thus the specific heat ratio k are constant.
Analysis The net work of the cycle per unit of mass flow is
\frac{\dot{W}_{\text {cycle }}}{\dot{m}}=\left(h_{3}-h_{4}\right)-\left(h_{2}-h_{1}\right)
Since c_{p} is constant (assumption 6),
\frac{\dot{W}_{\text {cycle }}}{\dot{m}}=c_{p}\left[\left(T_{3}-T_{4}\right)-\left(T_{2}-T_{1}\right)\right]
Or on rearrangement
\frac{\dot{W}_{\text {cycle }}}{\dot{m}}=c_{p} T_{1}\left(\frac{T_{3}}{T_{1}}-\frac{T_{4}}{T_{3}} \frac{T_{3}}{T_{1}}-\frac{T_{2}}{T_{1}}+1\right)
Replacing the temperature ratios T_{2} / T_{1} \text { and } T_{4} / T_{3} by using Eqs. 9.23 and 9.24, respectively, gives
T_{2}=T_{1}\left(\frac{p_{2}}{p_{1}}\right)^{(k-1) / k} (9.23)
T_{4}=T_{3}\left(\frac{p_{4}}{p_{3}}\right)^{(k-1) / k}=T_{3}\left(\frac{p_{1}}{p_{2}}\right)^{(k-1) / k} (9.24)
\frac{\dot{W}_{\text {cycle }}}{\dot{m}}=c_{p} T_{1}\left[\frac{T_{3}}{T_{1}}-\frac{T_{3}}{T_{1}}\left(\frac{p_{1}}{p_{2}}\right)^{(k-1) / k}-\left(\frac{p_{2}}{p_{1}}\right)^{(k-1) / k}+1\right]
From this expression it can be concluded that for specified values of T_{1}, T_{3}, \text { and } c_{p}, the value of the net work output per unit of mass flow varies with the pressure ratio p_{2} / p_{1} only.
To determine the pressure ratio that maximizes the net work output per unit of mass flow, first form the derivative
\frac{\partial\left(\dot{W}_{\text {cycle }} / \dot{m}\right)}{\partial\left(p_{2} / p_{1}\right)}=\frac{\partial}{\partial\left(p_{2} / p_{1}\right)}\left\{c _ { p } T _ { 1 } \left[\frac{T_{3}}{T_{1}}-\frac{T_{3}}{T_{1}}\left(\frac{p_{1}}{p_{2}}\right)^{(k-1) / k}\right.\right.
\left.\left.-\left(\frac{p_{2}}{p_{1}}\right)^{(k-1) / k}+1\right]\right\}
=c_{p} T_{1}\left(\frac{k-1}{k}\right)\left[\left(\frac{T_{3}}{T_{1}}\right)\left(\frac{p_{1}}{p_{2}}\right)^{-1 / k}\left(\frac{p_{1}}{p_{2}}\right)^{2}-\left(\frac{p_{2}}{p_{1}}\right)^{-1 / k}\right]
=c_{p} T_{1}\left(\frac{k-1}{k}\right)\left[\left(\frac{T_{3}}{T_{1}}\right)\left(\frac{p_{1}}{p_{2}}\right)^{(2 k-1) / k}-\left(\frac{p_{2}}{p_{1}}\right)^{-1 / k}\right]
When the partial derivative is set to zero, the following relationship is obtained:
\frac{p_{2}}{p_{1}}=\left(\frac{T_{3}}{T_{1}}\right)^{k /[2(k-1)]} (a)
By checking the sign of the second derivative, we can verify that the net work per unit of mass flow is a maximum when this relationship is satisfied.
For gas turbines intended for transportation, it is desirable to operate near the compressor pressure ratio that yields the most work per unit of mass flow. The present example shows how the maximum net work per unit of mass flow is determined on a cold air-standard basis when the state at the compressor inlet and turbine inlet temperature are fixed.
Skills Developed
Ability to…
• complete the detailed derivation of a thermodynamic expression.
• use calculus to maximize a function.
Quick Quiz
For an ideal cold air-standard Brayton cycle with a compressor inlet temperature of 300 K and a maximum cycle temperature of 1700 K, use Eq. (a) above to find the compressor pressure ratio that maximizes the net power output per unit mass flow. Assume k = 1.4. Ans. 21. (Value agrees with Fig. 9.12.)
