## Shaft Work Machines
| Device | Function | Ideal Model | Efficiency |
| -------------- | ------------------------------------------ | ------------------------------------------------------------------------------------------------------------------------------------ | ----------------------------------------------------------------------------------------------------- |
| **Turbine** | P↓, T↓, h↓: $W_\text{out}
lt;br>Extract work | Steady, Adiabatic, reversible<br>$\dot{W}_{\text{rev}} = \dot{m}(h_{\text{in}} - h_{\text{out,rev}})lt;br>$s _\text{in}=s_\text{out}$ | ${} \eta_t = \frac{\dot{W}_{\text{act}}}{\dot{W}_{\text{rev}}} = \frac{h_{1}-h_{2}}{h_{1}-h_{2r}} {}$ |
| **Compressor** | P↑, T↑, h↑: $W_\text{in}lt;br>Apply work | $\dot{W}_{\text{rev}} = \dot{m}(h_{\text{out,rev}} - h_{\text{in}})lt;br>$s _\text{in}=s_\text{out}$ | ${} \eta_c = \frac{\dot{W}_{\text{rev}}}{\dot{W}_{\text{act}}}=\frac{h_{2r}-h_{1}}{h_{2}-h_{1}}$ |
## Ideal Gas Shaft Work
1st Law:
$-\frac{\dot{W}}{\dot{m}}=h_\text{out}-h_\text{in} = c_{p}(T_{2}-T_{1})$
2nd Law:
$\frac{\dot{S}_\text{gen}}{\dot{m}}=s_\text{out}-s_\text{in}=c_{p}\ln \frac{v_{2}}{v_{1}}+c_{v}\ln \frac{P_{1}}{P_{2}}$
For reversible case, $s_{2}=s_{1}$
Then 2nd law becomes: $P_{1}v_{1}^{\gamma}=P_{2}v_{2}^{\gamma}$
$\gamma = \frac{c_{p}}{c_{v}}$
$Pv = RT$
Benchmarks:
$\eta^{+}=\frac{\dot{W}_\text{actual}}{\dot{W}_\text{rev}}=\frac{(h_{2}-h_{1})_\text{actual}}{(h_{2}-h_{1})_\text{rev}}$
$\eta^{-}=\frac{1}{\eta^{+}}$
$R = c_{p}-c_{v}$
$-\frac{\dot{W}}{\dot{m}}=\frac{c_{p}}{R}\left( P_{2}v_{2}-P_{1}V_{1}\right)=\frac{c_{p}}{c_{p}-c_{v}}(P_{2}v_{2}-P_{1}v_{1})=\frac{\gamma}{\gamma-1}(P_{2}v_{2}-P_{1}v_{1}) = \frac{\gamma}{\gamma-1}\left( P_{2}\left( \frac{P_{1}}{P_{2}} \right)^{1/\gamma}v_{1}-P_{1}v_{1} \right)$
$\boxed{-\frac{\dot{W}}{\dot{m}}=\frac{\gamma}{\gamma-1}P_{1}v_{1}\left( \left( \frac{P_{2}}{P_{1}} \right)^{(\gamma-1)/\gamma}-1 \right)}$
1. Using isentropic properties, solve for the reversible state ($h_{2R}$)
2. Calculate $\dot{m}$ using $\frac{\dot{W}}{\dot{m}}=\eta(h_{1}-h_{2R})$
3. Caculate $h_{2}$ using $\frac{\dot{W}}{\dot{m}}=h_{1}-h_{2}$
4. Determine final state using $h_{2}$ and $P_{2}$
5. Calculate entropy generation rate with $S_\text{gen}=\dot{m}(s_{2}-s_{1})$
## Incompressible Shaft Work
$h = u + Pv=cT+Pv$
$-\frac{\dot{W}}{\dot{m}}=h_\text{out}-h_\text{in}=c(T_\text{out}-T_\text{in})+v(P_\text{out}-P_\text{in})$
Reversible case ($s_\text{out}-s_\text{in}$)
$-\left( \frac{\dot{W}}{\dot{m}} \right)_\text{rev}=v(P_\text{out}-P_\text{in})$
$\dot{W}_\text{rev}=\dot{V}(P_\text{in}-P_\text{out})$
$\eta^{+}=\frac{\dot{W}_\text{actual}}{\dot{W}_\text{rev}}=\frac{c(T_\text{in}-T_\text{out})+v(P_\text{in}-P_\text{out})}{v(P_\text{in}-P_\text{out})}$
$\eta^{-}=\frac{1}{\eta^{+}}$