PX285 - J4 - venturi pipe

considering a horizontal pipe of varying cross section

from bernoulli's principle:

\begin{matrix} P_{1} + ρ \frac{u_{1}^{2}}{2} = P_{2} + ρ \frac{u_{2}^{2}}{2} \\ (P_{1} - P_{2}) = \frac{ρ}{2} (u_{2}^{2} - u_{1}^{2}) \\ = \frac{ρ}{2} u_{1}^{2} (\frac{u_{2}^{2}}{u_{1}^{2}} - 1) \end{matrix}

from mass conservation:

u_{1} A_{1} = u_{2} A_{2} ⟹ \frac{u_{1}}{u_{2}} = \frac{A_{2}}{A_{1}}

\begin{matrix} (1) & ∴ P_{2} - P_{2} = \frac{ρ}{2} u_{1}^{2} (\frac{A_{1}^{2}}{A_{2}^{2}} - 1) \end{matrix}

using a monometer - a bent pipe of constant cross section filled with mercury, with density, $ρ_{m}$ , such that $ρ_{m} ≫ ρ$

\begin{matrix} P_{1} = P_{2} + ρ_{m} g Δ z \\ P_{1} - P_{2} = ρ_{m} g Δ z \end{matrix}

substituting this into equation $(1) :$

u_{1}^{2} = [2 (\frac{ρ_{m}}{ρ}) g] Δ z {(\frac{A_{1}^{2}}{A_{2}^{2}} - 1)}^{- 1}

where, all but $z$ are constants