The Efficiencies of Real Engines — Question 6

Question 6

Compressor work must be subtracted from turbine work in a gas-turbine cycle

In an ideal Brayton model, compression 1 to 2 and expansion 3 to 4 are reversible adiabatic, with common pressure ratio $r_p=9$ and $\gamma=1.40$. Heat addition 2 to 3 and rejection 4 to 1 occur at constant pressure. Use $T_1=300$ K, $T_3=1200$ K, $c_p=1.00$ kJ/(kg K), and $a=r_p^{(\gamma-1)/\gamma}$.

1. Find temperatures $T_2=aT_1$, $T_4=T_3/a$, compressor and turbine specific works, and net specific work.

2. Derive efficiency and the range of $a>1$ needed for positive net work.

Question 6 - Solution

Compressor work must be subtracted from turbine work in a gas-turbine cycle

(a) Method and calculation

$$a=9^{2/7}\approx1.87344,\quad T_2\approx562.033\,\mathrm K,\quad T_4\approx640.532\,\mathrm K.$$$$w_c=c_p(T_2-T_1)\approx262.033,\quad w_t=c_p(T_3-T_4)\approx559.468\quad\mathrm{kJ/kg}.$$$$w_{\mathrm{net}}=w_t-w_c\approx297.435\,\mathrm{kJ/kg}.$$

(b) Interpretation and further calculation

$$q_h=c_p(T_3-aT_1),\quad w_{\mathrm{net}}=c_p(a-1)(T_3/a-T_1).$$$$\eta=w_{\mathrm{net}}/q_h=1-1/a\approx.466224,\quad w_{\mathrm{net}}>0\ \Leftrightarrow\ 1<a<T_3/T_1=4.$$A turbine can deliver substantial gross work while the compressor consumes much of it.

Physical check

At the upper limit, the temperature available for external heat addition vanishes along with net work.