Question 5 of 7: Bernoulli-Based Flow Meters (Orifice, Venturi, Flume)

Question 5: Bernoulli-Based Flow Meters (Orifice, Venturi, Flume) (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

a) The constant $k$ absorbs everything about the meter's own geometry and the fluid's properties that is fixed for a given installation: the throat/orifice flow area (or the area ratio $\beta=d/D$ between the restriction and the approach pipe), the discharge coefficient $C_d$ (an empirical correction for real effects Bernoulli's ideal-flow derivation ignores — friction, flow contraction past a sharp edge, non-uniform velocity profile), the fluid density $\rho$, the approach- velocity correction $1/\sqrt{1-\beta^4}$, and, for compressible fluids, an expansibility factor $Y$. Combined, $$k=\frac{C_dA_{throat}Y}{\sqrt{1-\beta^4}}\sqrt{\frac{2}{\rho}},$$ a single calibration constant that need only be determined once (by design calculation or a wet calibration) for a given meter and fluid.

b) The reduction from the full Bernoulli/continuity system to $Q=k\sqrt{\Delta p}$ assumes: steady flow (no significant unsteadiness beyond what $C_d$ already absorbs); incompressible, single-phase, constant-density fluid across the meter (or a compressibility correction folded into $Y$); a fully developed, known velocity profile at the upstream tap (undisturbed by nearby fittings/bends); negligible elevation change between the two pressure taps (or an elevation term already folded into $\Delta p$); and no other energy addition or removal (no pump, turbine, or significant heat transfer) between the two tap locations.

c) A differential-pressure gauge measures $\Delta p$ directly, as one quantity spanned over exactly the (typically small) range the meter needs, so its error is set by ITS OWN span-referenced accuracy. Subtracting two separately measured ABSOLUTE pressures instead computes $\Delta p=P_1-P_2$, where $P_1$ and $P_2$ are both large numbers close in value to each other; each absolute-pressure transducer's error is a fixed percentage of ITS OWN full-scale range (which must cover the full line pressure, not just the small difference), so the two absolute errors do NOT cancel in the subtraction and can easily be comparable to, or larger than, the true (small) $\Delta p$ itself. This near-equal-numbers subtraction badly degrades resolution and SNR compared to a single differential gauge purpose-built for the small $\Delta p$ span.

d) Because $Q=k\sqrt{\Delta p}$ is a NONLINEAR (concave, square-root) function of $\Delta p$, its time-average behaves differently from a linear quantity's average. For a fluctuating flow, the TRUE mean flow rate is the time-average of the instantaneous $k\sqrt{\Delta p(t)}$, but computing flow from the time-AVERAGED pressure instead gives $k\sqrt{\overline{\Delta p}}$ — and by Jensen's inequality for a concave function, $\overline{\sqrt{\Delta p}}\le\sqrt{\overline{\Delta p}}$ always, with equality only for perfectly steady (non-fluctuating) flow. A small illustrative check (pressure swinging between 1 and 9, same units, mean $=5$): the true mean of $\sqrt{\Delta p}$ is $\tfrac12(\sqrt1+\sqrt9)=2.000$, while $\sqrt{\overline{\Delta p}}= \sqrt5=2.236$ — a $\sim$12% OVER-estimate of the true mean flow from using the averaged pressure. Using an averaged $\Delta p$ reading therefore systematically OVER-states the true average flow whenever the flow pulsates; the fix is either to sample $\Delta p$ fast enough to compute $\sqrt{\Delta p(t)}$ and average THAT, or to apply a known pulsation-dependent correction factor.