Bernoulli Equation and Flow Measurement

Venturi meter flow measurement

A venturimeter is a differential-pressure flow meter inserted in a pipe. In the direction of flow it has three parts: a converging section, a throat of smaller area, and a diverging section. Pressure tappings are placed just upstream of the converging cone and at the throat. Its operation combines the continuity equation with Bernoulli's equation. For steady incompressible flow in a horizontal pipe with negligible losses, continuity gives V_2 = V_1 (A_1/A_2), and Bernoulli's equation gives p_1 - p_2 = \frac{\rho}{2}(V_2^2 - V_1^2). Eliminating V_1 gives the working formula Q = A_2 \sqrt{\dfrac{2(p_1-p_2)/\rho}{1-(A_2/A_1)^2}}. In practice, a discharge coefficient C_d < 1 multiplies this result to account for real losses. The device is designed for turbulent flow at high Reynolds number, where losses are small and the head relation is reliable; it is not suited to very low Reynolds number flow. Because the throat velocity is high, the throat pressure is low. This low pressure can draw a liquid upward from a reservoir through a connecting pipe, and the height to which the liquid can be lifted is limited by the suction head, roughly |p_{throat}|/\rho g measured from the reservoir surface. To find the pressure drop between an upstream section and the throat, compute both areas and velocities from Q and the diameters, then apply \Delta p = \frac{\rho}{2}(V_{th}^2 - V_1^2) and convert pascals to kilopascals by dividing by 1000. The same Bernoulli-continuity pair applies to a concentric reducer in a horizontal line. The density is obtained from specific weight by \rho = \gamma/g, the velocity at the smaller section rises by the factor (d_1/d_2)^2, and the mass flow rate is \dot m = \rho A V.

• converging cone — the inlet cone whose area decreases in the flow direction, so the fluid accelerates through it.
• throat — the minimum-area section, where velocity is highest and pressure is lowest.
• discharge coefficient — the ratio of actual to ideal discharge, always taken below one for real meters.

Pitot-static tube velocity measurement

A pitot tube measures stagnation pressure, the pressure at a point where the fluid is brought to rest. Along a streamline, p_0 = p + \frac{1}{2}\rho V^2, so the difference between stagnation pressure and static pressure gives the local velocity. A pitot probe is a single tube with an open tip facing the flow. A piezometer is a flush wall tap that senses static pressure without disturbing the flow. A pitot-static probe combines both in one assembly: an inner tube with an upstream-facing tip senses stagnation pressure, while side holes on the outer tube sense static pressure. When the two pressures are read on a manometer, the reading gives \Delta p = (SG_m - SG_f)\rho g h, where SG_m is the specific gravity of the manometer fluid and SG_f that of the flowing fluid. The velocity is then V = \sqrt{2\Delta p/\rho}. If a pitot tube and a piezometer are tapped into the same horizontal pipe, the difference between their column heights equals the velocity head V^2/2g, so V = \sqrt{2gh} with h the differential column height. This relation follows from Bernoulli's equation along the centreline streamline. The theoretical pitot velocity is V_{th} = \sqrt{2gh}. A velocity coefficient C_v, defined as actual velocity divided by theoretical velocity, corrects for real effects, so the actual centreline velocity is C_v V_{th}. If the mean velocity in the pipe is a known fraction of the centreline velocity, compute V_{mean} = Q/A and divide by that fraction times V_{th} to obtain C_v.

• stagnation pressure — the pressure at a point where the flow velocity is reduced to zero.
• velocity head — the term V^2/2g, the height equivalent of the kinetic energy per unit weight.
• piezometer — a flush or open tube at a wall that reads static pressure as a column height.