## Nozzles and Diffusers
| Device | Function | Ideal Model | Efficiency |
| ------------ | --------------------------- | ------------------------------------------------------------------------------------------------------------------------ | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ |
| **Diffuser** | P↑, v↓, h↑<br>KE → Enthalpy | $\left( h + \frac{v^{2}}{2} \right)_\text{in}=\left( h+\frac{v^{2}}{2} \right)_\text{out}
lt;br>$s_\text{out}=s_\text{in}$ | ${} \eta _\text{d} = \frac{(h_\text{in}-h_\text{out})_\text{rev}}{(h_\text{in}-h_\text{out})_\text{act}} = \frac{(v^{2}_\text{in}-v^{2}_\text{out})_\text{rev}}{(v^{2}_\text{in}-v^{2}_\text{out})_\text{act}} {}$ |
| **Nozzle** | P↓, v↑, h↓<br>Enthalpy → KE | $\left( h + \frac{v^{2}}{2} \right)_\text{in}=\left( h+\frac{v^{2}}{2} \right)_\text{out}lt;br>$s_\text{out}=s_\text{in}$ | ${} \eta _\text{ n}= \frac{(h_\text{in}-h_\text{out})_\text{act}}{(h_\text{in}-h_\text{out})_\text{rev}} = \frac{(v^{2}_\text{in}-v^{2}_\text{out})_\text{act}}{(v^{2}_\text{in}-v^{2}_\text{out})_\text{rev}} {}$ |
| **Throttle** | P↓,T↓, V↑<br>Pressure drop | $h_{\text{in}} = h_{\text{out}}$ | N/A |
### Typical Assumptions:
- Steady
- PE negligible
- Adiabatic
- No shaft work
### 1st Law
$\cancel{ \frac{dE_{\text{cv}}}{dt} }=\cancel{ \dot{Q} }-\cancel{ \dot{W} }+\sum_{\text{in}}\dot{m}_{\text{in}}\left( h+\frac{v^{2}}{2}+\cancel{ gz } \right)_{\text{in}}-\sum_{\text{out}}\dot{m}_{\text{out}}\left( h+\frac{v^{2}}{2}+\cancel{ gz } \right)_{\text{out}}$
$\left( h+\frac{v^{2}}{2} \right)_\text{in}=\left( h+\frac{v^{2}}{2} \right)_\text{out}$
$h_\text{in}-h_\text{out}=\frac{v^{2}_\text{out}}{2}-\frac{v^{2}_\text{in}}{2}$
### 2nd Law
$\cancel{ \frac{dS_{\text{cv}}}{dt} }=\cancel{ \frac{\dot{Q}}{T_{\text{s}}} }+\dot{S}_{\text{gen}}+\sum_{\text{in}} \dot{m}_{\text{in}}S_{\text{in}} - \sum_{\text{out}} \dot{m}_{\text{out}} S_{\text{out}}$
$s_\text{out}-s_\text{in}=\frac{\dot{S}_\text{gen}}{\dot{m}}$
**Reversible:** $s_\text{out}=s_\text{in}$
![[Pasted image 20250425092804.png|275]]
$h_{1}-h_{2} < h_{1} - h_{2R}$
- Nozzle is analogous to a positive shaft work machine
- KE increase is smaller for an irreversible nozzle
![[Pasted image 20250425092826.png|275]]
$h_{1}-h_{2} < h_{2R}-h_{1}$
- Diffuser is analogous to a negative shaft work machine
- Higher KE expenditure needed to get to the same pressure for an irreversible nozzle
### Efficiency
For defining efficiency, we use the same inlet state and outlet pressure, but not necessarily inlet velocity
**Nozzle Efficiency**
$\eta _\text{ nozzle}= \frac{(h_\text{in}-h_\text{out})_\text{actual}}{(h_\text{in}-h_\text{out})_\text{rev}} = \frac{(v^{2}_\text{in}-v^{2}_\text{out})_\text{actual}}{(v^{2}_\text{in}-v^{2}_\text{out})_\text{rev}}$
**Diffuser Efficiency
$\eta _\text{ diff} = \frac{(h_\text{in}-h_\text{out})_\text{rev}}{(h_\text{in}-h_\text{out})_\text{actual}} = \frac{(v^{2}_\text{in}-v^{2}_\text{out})_\text{rev}}{(v^{2}_\text{in}-v^{2}_\text{out})_\text{actual}}$
**Diffuser Pressure Recovery Factor**
$C_{p} = \frac{P_\text{out}-P_\text{in}}{\frac{1}{2}\rho _\text{in}v^{2}_\text{in}}$