Flow Measurement Devices

Bernoulli's equation is the basis for many practical flow measurement instruments. By measuring the pressure difference between two points, the velocity or flow rate can be inferred.

Venturi Meter

A Venturi meter is a converging-diverging tube inserted in a pipe. Flow accelerates at the narrowest section (the throat), dropping the pressure. Applying Bernoulli and continuity between the inlet ($A_1$) and throat ($A_2$):

$Q = A_2\sqrt{\frac{2\Delta P}{\rho\left(1 - \left(\dfrac{A_2}{A_1}\right)^2\right)}}$

where $\Delta P = P_1 - P_2$ is the measured pressure difference.

The throat-to-inlet area ratio $A_2/A_1$ is typically 0.25–0.75. Smaller ratios give a larger $\Delta P$ signal but higher permanent pressure loss.

In practice a discharge coefficient $C_d \approx 0.98$ accounts for viscous losses, but we omit it here for clarity.

Pitot Tube

A Pitot tube points directly into the flow. It measures the stagnation pressure $P_0$ at its tip (where the fluid is brought to rest) against the static pressure $P_s$ from a side port. Bernoulli gives:

$v = \sqrt{\frac{2(P_0 - P_s)}{\rho}}$

Pitot tubes are used on aircraft to measure airspeed. The pressure difference for a 100 m/s aircraft in air ($\rho = 1.225$ kg/m³) is about 6 kPa.

Your Task

Implement:

• venturi_flow(A1, A2, delta_P, rho) — volumetric flow rate $Q$ (m³/s)
• pitot_velocity(P_stagnation, P_static, rho) — flow velocity $v$ (m/s)