Fluid Mechanics

Reynolds

$$\begin{array}{c}\mathrm{Re}=\frac{\rho vL}{\mu }=\frac{vL}{\nu }\end{array}$$

• $L$ is characteristic length
• $\rho$ is density
• $v$ is velocity
• $\mu$ is dynamic viscosity

Bernouli’s

• General form between to points in units of head (meters)
• Only applies to incompressible fluids!!!

$$\frac{P_1}{\rho g}+\frac{v_1^2}{2g}+h_1=\frac{P_2}{\rho g}+\frac{v_2^2}{2g}+h_2+h_f$$

Venturi

• Specific case of Bernoulli’s used to measure flow rate $Q$
• Needs differential static pressure, cross sectional area, and density

$$Q=A_2\sqrt{\frac{2(P_1-P_2)}{\rho \left(1-{\left(\frac{A_2}{A_1}\right)}^2\right)}}$$

Darcy-Weisbach

$$\begin{array}{c}{H}_f=f\cdot \frac{L}{D}\cdot \frac{v^2}{2g}\end{array}$$

Where:

• $H_f$ = head loss due to friction (m)
• $f$ = Darcy friction factor (dimensionless)
  • This is obtained from a moody diagram, derived experimentally, which is varied by $R_e$ and surface roughness $k$.
• $L$ = length of the pipe (m)
• $D$ = diameter of the pipe (m)
• $v$ = average velocity of the fluid (m/s)
• $g$ = acceleration due to gravity (9.81 m/s²)

Cavitation

• Cryogenic liquids
  • Unique effects that are more dangerous than typical fluids
  • Because ${\rho }_{gas,cold}<<{\rho }_{gas,warm}$, more liquid has to evolve to equalize pressure
  • Because $K_{liq,cold}<<K_{liq,warm}$ greater temp gradients form in liquid phase