Fluid mechanics & piping systems

1. Viscosity temperature dependence
2. Bernoulli pressure & energy line
3. Pump power & motor sizing
4. Minor loss fitting K-factor
5. Pipe velocity & diameter sizing
6. Darcy-Weisbach friction loss
7. Compressed air pressure drop
8. Hydraulic diameter (non-circular)
9. Natural gas piping sizing
10. Pipe packing (circles in circle)
11. Linear thermal pipe expansion
12. Tank fill volume & dipstick table
13. Flow rate unit converter

$$\mu(T)=\mu_0\left(\frac{T}{T_0}\right)^{3/2}\frac{T_0+S}{T+S}\qquad \nu=\frac{\mu}{\rho}$$

Sutherland's law for gases ($S$ in K: air S≈110 K); Andrade $\mu=Ae^{B/T}$ for liquids. Source: Engineering ToolBox.

$$p_1+\frac{1}{2}\rho v_1^2+\rho g z_1=p_2+\frac{1}{2}\rho v_2^2+\rho g z_2$$

Solves for the missing state variable along an inviscid streamline. Source: Engineering ToolBox (apps.engineeringtoolbox.com).

$$P_{\text{hyd}}=\rho g Q H,\quad P_{\text{shaft}}=\frac{P_{\text{hyd}}}{\eta_{\text{pump}}},\quad P_{\text{elec}}=\frac{P_{\text{shaft}}}{\eta_{\text{motor}}}$$

$H$ = total dynamic head (m). Source: Engineering ToolBox (apps.engineeringtoolbox.com).

$$h_L=K\left(\frac{v^2}{2g}\right),\qquad \Delta P=K\left(\frac{1}{2}\rho v^2\right)$$

Typical K-values: 90° elbow 0.3–1.5, tee 0.5–2.0, gate valve (open) 0.2–0.5, globe valve 3.5–10. Source: Engineering ToolBox.

$$v=\frac{4Q}{\pi D^2},\qquad D=\sqrt{\frac{4Q}{\pi v}}$$

Recommended: liquids 1–3 m/s, gases 10–20 m/s, steam 25–40 m/s. Source: Engineering ToolBox.

$$\frac{1}{\sqrt{f}}=-2\log_{10}\!\left(\frac{\varepsilon/D}{3.7}+\frac{2.51}{Re\,\sqrt{f}}\right),\quad h_f=f\frac{L}{D}\frac{v^2}{2g}$$

Colebrook solved iteratively (10 rounds from Swamee-Jain seed). Source: Engineering ToolBox.

$$\Delta P=\frac{c\cdot L\cdot Q^{1.85}}{p\cdot D^{5}}$$

$c\approx 7.57\times10^{-4}$ for US units (psi, ft, cfm). Accounts for gauge pressure. Source: Engineering ToolBox.

$$D_h=\frac{4A}{P_{\text{wetted}}},\qquad Re_{D_h}=\frac{v D_h}{\nu}$$

For a full circular pipe $D_h=D$. Source: Engineering ToolBox.

$$Q=0.623\left(\frac{d^5(p_1^2-p_2^2)}{S\cdot L\cdot\mu^{0.15}}\right)^{0.574}$$

Low-pressure natural gas (<7 kPa). $d$ in mm, $p$ in kPa, $L$ in m, $S$ = specific gravity (≈0.6), $\mu\approx1.1\times10^{-5}$ Pa·s. Source: Engineering ToolBox.

$$N\approx0.907\left(\frac{D}{d}\right)^2-0.98$$

Approximate maximum count of equal small circles inside a larger circle (conduit fill). Source: Engineering ToolBox.

$$\Delta L=\alpha\cdot L\cdot(T_{\text{max}}-T_{\text{install}}),\quad L_{\text{loop}}=K\sqrt{D\cdot\Delta L}$$

Steel $\alpha\approx12\times10^{-6}/$°C, PVC $\alpha\approx80\times10^{-6}/$°C. $K\approx3$ for U-loops. Source: Engineering ToolBox.

$$V(h)=L\left[R^2\arccos\!\left(\frac{R-h}{R}\right)-(R-h)\sqrt{2Rh-h^2}\right]$$

Partial-fill volume for flat-ended horizontal cylinders. Source: 1728 Software Systems.

$$Q=\frac{V}{t},\qquad \dot{m}=\rho Q$$

Converts between volumetric and mass flow rate units. Source: 1728 Software Systems.