Question 7 of 7: Flow Measurement

Question 7: Flow Measurement (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) Given. Bernoulli's equation and continuity applied across a restriction of known geometry. Find. How the pressure drop relates to the flow rate. Combining $\tfrac12\rho V_1^2+p_1=\tfrac12\rho V_2^2+p_2$ with continuity $A_1V_1=A_2V_2$ gives a flow rate that varies with the square root of the pressure drop, i.e. the pressure drop varies with the square of the flow rate: $$Q=C_d A_2\sqrt{\frac{2\,\Delta p}{\rho\left[1-(A_2/A_1)^2\right]}}\quad\Longleftrightarrow\quad \Delta p\propto Q^2.$$ For example, doubling the flow rate through a fixed restriction quadruples the pressure drop across it ($\Delta p_2/\Delta p_1=(Q_2/Q_1)^2=\boxed{4}$ for $Q_2=2Q_1$) — a non-linear (square-law) relationship that the instrument's readout must account for.

b) Differential-pressure flow meters based on this same Bernoulli/continuity principle include the orifice plate, the Venturi meter, the flow nozzle, the Pitot tube (and Pitot-static/averaging Pitot-array meters), and the Dall tube/elbow meter — all of them create or exploit a restriction or curvature that produces a velocity-dependent pressure difference, differing mainly in permanent pressure loss, accuracy, turndown ratio, and installation/straight-run requirements.

c) The calculation assumes steady, fully-developed, single-phase flow with known (and effectively constant over the small temperature rise) fluid density and specific heat; that essentially all of the heater's power goes into raising the fluid's temperature, with negligible heat loss to the pipe wall or surroundings; that the heated fluid is fully and uniformly mixed by the time it reaches the downstream thermometer (no thermal stratification across the pipe cross-section); and that the heater's power input itself is accurately known and stable.

d) To justify those assumptions in practice: thermally insulate the pipe section between the heater and the downstream thermometer to minimize heat loss to the environment; install a static mixer, or allow enough straight downstream run, so the heated fluid is genuinely well-mixed (uniform temperature across the cross-section) before it reaches the sensor; use a calibrated heater with a known, electronically controlled and monitored power input rather than an assumed nominal value; and verify (or directly measure, e.g. with an independent density/composition sensor) the fluid's density and specific heat rather than relying on tabulated nominal properties, particularly if the fluid's composition can vary in service.

e) Given. A Wheatstone bridge with three fixed resistors $R$ and the heated sensor as the fourth arm; a power op-amp senses the two bridge midpoint voltages and drives the bridge's common excitation node until it balances. Find. The relationship between the resulting output voltage and the fluid speed. With the inverting input tapping the sensor-side midpoint and the non-inverting input tapping the fixed-resistor-side midpoint, balance requires the two midpoints to sit at the same potential: $$V_{out}\frac{R_{sensor}}{R+R_{sensor}}=V_{out}\frac{R}{2R} \;\Longrightarrow\;R_{sensor}=R\quad\text{at every operating point.}$$ The feedback loop is therefore a constant-resistance (constant-temperature) servo: whatever electrical power the amplifier must deliver to hold the sensor exactly at $R_{sensor}=R$ is exactly the power being carried away by forced convection from the flow. Forced-convection heat loss from a heated element follows King's Law ($Nu=A'+B'\sqrt{Re}$, i.e. heat loss rising with the square root of fluid speed), so the electrical power $V_{out}^2/R$ — and hence the output voltage — is related to fluid speed $U$ by $$\boxed{V_{out}^2=A+B\sqrt{U}}\qquad\Longleftrightarrow\qquad V_{out}=\sqrt{R\left(A+B\sqrt{U}\right)},$$ a known, monotonically increasing but non-linear function of speed that can be inverted point-by-point (or via a lookup/calibration curve) to recover $U$ from a measured $V_{out}$.