PX285 - J6 - pitot pipe

\begin{matrix} P_{f} + ρ_{f} \frac{u_{f}^{2}}{2} + ρ_{f} g z_{0} = P_{0} + ρ_{f} g z_{0} \\ P_{0} - P_{f} = ρ_{f} \frac{u_{f}^{2}}{2} \\ (1) & ∴ u_{f} = \sqrt{\frac{2 (P_{0} - P_{f})}{ρ_{f}}} \end{matrix}

\begin{matrix} P_{A} = P_{f} + ρ_{f} g z^{'} + ρ_{m} g Δ z \\ P_{B} = P_{0} + ρ_{f} g (z^{'} + Δ z) \end{matrix}

\begin{aligned} P_{0} - P_{f} & = ρ_{f} g z^{'} + ρ_{m} g Δ z - ρ_{f} g z^{'} + ρ_{f} g Δ z \\ = g Δ z (ρ_{m} - ρ_{f}) \end{aligned}

∴ u_{f} = \sqrt{2 g Δ z (\frac{ρ_{m}}{ρ_{f}} - 1)} \approx \sqrt{2 g Δ z (\frac{ρ_{m}}{ρ_{f}})}

where, the approximation is for $ρ_{m} ≫ ρ_{f}$