Question 9.6: An Actual Gas-Turbine Cycle Assuming a compressor efficiency...

The Brayton cycle discussed in Example 9–5 is reconsidered. For specified turbine and compressor efficiencies, the back work ratio, the thermal efficiency, and the turbine exit temperature are to be determined.

Analysis (a) The T-s diagram of the cycle is shown in Fig. 9–37. The actual compressor work and turbine work are determined by using the definitions of compressor and turbine efficiencies, Eqs. 9–19 and 9–20:

\eta_{C}=\frac{w_{s}}{w_{a}} \cong \frac{h_{2 s}-h_{1}}{h_{2 a}-h_{1}}

\eta_{T}=\frac{w_{a}}{w_{s}} \cong \frac{h_{3}-h_{4 a}}{h_{3}-h_{4 s}}

Compressor:    w_{\text {comp, in }}=\frac{w_{s}}{\eta_{C}}=\frac{244.16 kJ / kg }{0.80}=305.20 kJ / kg

Turbine:    w_{\text {turb,out }}=\eta_{T} w_{s}=(0.85)(606.60 kJ / kg )=515.61 kJ / kg

Thus,

r_{ bw }=\frac{w_{\text {comp,in }}}{w_{\text {turb,out }}}=\frac{305.20 kJ / kg }{515.61 kJ / kg }=0.592

That is, the compressor is now consuming 59.2 percent of the work produced by the turbine (up from 40.3 percent). This increase is due to the irreversibilities that occur within the compressor and the turbine.
(b) In this case, air leaves the compressor at a higher temperature and enthalpy, which are determined to be

\begin{aligned} w_{\text {comp }, \text { in }}=h_{2 a}-h_{1} \rightarrow h_{2 a} &=h_{1}+w_{\text {comp,in }} \\ &=300.19+305.20 \\ &=605.39 kJ / kg \quad\left(\text { and } T_{2 a}=598 K \right) \end{aligned}

Thus,

\begin{aligned} q_{\text {in }} &=h_{3}-h_{2 a}=1395.97-605.39=790.58 kJ / kg \\ w_{\text {net }} &=w_{\text {out }}-w_{\text {in }}=515.61-305.20=210.41 kJ / kg \end{aligned}

and

\eta_{ th }=\frac{w_{\text {net }}}{q_{\text {in }}}=\frac{210.41 kJ / kg }{790.58 kJ / kg }=0.266 \text { or } 26.6 \%

That is, the irreversibilities occurring within the turbine and compressor caused the thermal efficiency of the gas turbine cycle to drop from 42.6 to 26.6 percent. This example shows how sensitive the performance of a gas-turbine power plant is to the efficiencies of the compressor and the turbine. In fact, gas-turbine efficiencies did not reach competitive values until significant improvements were made in the design of gas turbines and compressors.
(c) The air temperature at the turbine exit is determined from an energy balance on the turbine:

\begin{aligned} w_{\text {turb,out }}=h_{3}-h_{4 a} \rightarrow h_{4 a} &=h_{3}-w_{\text {turb,out }} \\ &=1395.97-515.61 \\ &=880.36 kJ / kg \end{aligned}

Then, from Table A–17,

T_{4 a}=853 K

Discussion The temperature at turbine exit is considerably higher than that at the compressor exit \left(T_{2 a}=598 K \right) , which suggests the use of regeneration to reduce fuel cost.