## Pipe Flow
Also known as Bournoulli, energy, or head equation. Variant of the first law
$\frac{P_{2} - P_{1}}{\rho g} + \alpha \frac{v_{2}^{2} - v_{1}^{2}}{2g} + z_{2} - z_{1} =h_{f} - h_{p}$
$\alpha=\left\{{\begin{array}{l l}{2}&{{\mathrm{ laminar}}}\\ {1.06}&{{\mathrm{turbulent}}}\\ {1}&{{\mathrm{uniform}}}\end{array}}\right.$
$h_{p} = \text{Pump Power} = \frac{\lvert \dot{W}_{p} \rvert}{\dot{m}_{p}g}$
| Term to Drop | Condition |
| ------------------------------- | --------------------------------- |
| $\frac{P}{\rho g}
lt;br> | Ambient / free surface / outlet |
| ${} \alpha \frac{v^{2}}{2g} {}$ | Slow flow / Cross-section is wide |
| ${} z_{2}-z_{1} {}$ | no height change |
| $h_{f}$ | No losses |
### Losses
$f \frac{L}{D}$ term is major losses, $\sum K_{i}$ term is minor losses
$
h_{f} = \left( f \frac{L}{D} + \sum K_{i} \right) \frac{V^{2}}{2g}
$
### Friction Factor
For laminar flow, we have
$
f = \frac{64}{\mathrm{Re}}
$
For turbulent flow, we can use the colebrook equation:
$\frac{1}{f^{1/2}}=\,-2.0\,\log\left(\frac{\epsilon/d}{3.7}+\frac{2.51}{\mathrm{Re}_{d}f^{1/2}}\right)$
or Haaland's Formula, an approximation to 2% accuracy
$f\!=\!\!\left[-1.8\log_{_{10}}\!\left(\frac{6.9}{\mathrm{Re}_{{D}}}\!+\!\left(\frac{\varepsilon\,/\,D}{3.7}\right)^{\!1.11}\right)\right]^{\!-2}$
### Moody Chart
![[Pasted image 20250303015200.png]]
### Pipe Flow Pressure Drop
1. Use known flow rate to calculate Reynolds Number
2. Identify whether flow is laminar or turbulent
3. Determine friction factor
4. Use $h_{f}$ to determine friction head loss
5. Sum minor loss coefficients
6. Determine total pressure drop using head equation
### Pipe Chains
1. Net flow into any junction must be zero (KCL)
2. Net head loss around any closed loop must be zero (KVL)
3. Head losses must satisfy Moody and minor losses (V=IR)
### Pipe Entrance Length
For Laminar Flow:
$
\frac{L_{e}}{d} \approx 0.06 \mathrm{Re}_{d}
$
For turbulent flow, we have
$
\frac{L_{e}}{d} \approx 4.4 \mathrm{Re}_{d}^{1/6}
$
![[Pasted image 20250302224518.png|500]]
### CDA references
![[Pasted image 20250310221213.png|525]] ![[Pasted image 20250310221727.png|525]]