The Rankine cycle (Figure 16.9) is a model suitable for the heat engines widely used in power stations which employ a steam turbine. The cycle involves a working fluid which undergoes a phase change, and consequently allows large amounts of heat transfer without great changes in pressure, and for this reason it is also useful in refrigerator applications. There are four stages: compressor, boiler, turbine, condenser. The compressor raises the pressure of a liquid at approximately constant volume. Because not much volume change takes place, not much work needs to be supplied. This also raises the temperature slightly. Then the liquid is heated in a boiler at constant pressure. Its temperature rises to the boiling point, and then stays constant as large amounts of heat are supplied and there is a large increase in volume. Usually a complete phase change takes place. After this the fluid (now vapour) passes through a turbine which it drives in an approximately adiabatic expansion. Finally, the fluid is cooled at constant pressure by heat exchange so that its volume falls again. In early devices, condensation began in the turbine, but because water droplets can damage the turbine blades, modern power stations avoid this by using a high operating temperature at the input to the turbine, and also a further stage called reheating.
(i) Consider a Rankine cycle with water as the working fluid, as shown in Figure 16.9. Show that the work extracted per unit mass of water is $w=\left(h_c-h_d\right)-\left(h_b-h_a\right)$ and the heat input is $q_{\mathrm{in}}=h_c-h_b$, where $h$ is specific enthalpy.
(ii) Given that the pressure limits are $10 \mathrm{kPa}$ and $5 \mathrm{MPa}$, and the temperature limits are $45^{\circ} \mathrm{C}$ and $400^{\circ} \mathrm{C}$, find the thermal efficiency defined by $\eta=w / q_{\text {in }}$. [Use the $h s$ chart shown in Figure 16.7]
(iii) Calculate the flow rate required for an output power of $1 \mathrm{MW}$.
(iv) Calculate the volume of water vapour entering the turbine per second.
Figure 16.9 can't copy