Bernoulli's Equation: Problem Solving

A Venturi meter is essential for measuring fluid flow rates in pipelines. It utilizes the relationship between fluid velocity and pressure described by Bernoulli's equation. When installed in a sewage system, the Venturi meter accurately determines the wastewater flow rate by measuring pressure differences.

The first step is to compute the cross-sectional areas of the pipe and the Venturi throat to analyze the pressure difference indicated by the pressure gauge. Next, the continuity equation is applied, which states that the volumetric flow rate (Q) remains constant along the pipeline for an incompressible fluid:

A₁ and A₂ are the pipe and throat cross-sectional areas, and V₁ and V₂ are the fluid velocities at these sections. By rearranging the equation, the velocity at the throat can be expressed as:

Thus, the velocities V₁ and V₂ can be calculated for a given flow rate.

Bernoulli's equation relates pressure and velocity in an ideal fluid, given by:

Here, P₁ and P₂ are the pressures in the pipe and throat and ⍴ are the fluid's density. The pressure difference (P₁ - P₂) can be computed by substituting the velocities V₁ and V₂ into this equation. This pressure drop (ΔP), indicated by the pressure gauge, reflects the increase in velocity as the wastewater flows through the narrower throat, consistent with energy conservation.