The price of knowing the flow rate

A flowmeter that works by obstructing the pipe is the commonest instrument in process engineering. Squeeze the flow, measure the pressure difference the squeeze produces, and infer the flow rate from Bernoulli and continuity: two equations, one unknown, and no moving parts.

The equation is the same for every such device. What is not the same is what the squeeze costs, and the range is startling. A Venturi and an orifice plate can be sized to give an identical reading from an identical flow, and the pressure one of them takes away permanently — the pressure a pump has to supply forever, for as long as the meter is installed — differs between them by an order of magnitude.

The difference is not viscosity. It is not the finish of the machining. It is the shape of the passage after the throat, and it is computable exactly.

The reading, which is the easy half #

Continuity says the speed at a throat of area ratio σ is V₁/σ. Bernoulli between the pipe and the throat — a steady, inviscid, incompressible streamline, which is exactly where the equation holds — says the pressure difference is

$$\Delta p = \tfrac{1}{2}\rho V_1^2\left(\frac{1}{\sigma^2} - 1\right)$$

so measuring Δp gives V₁ and therefore the flow rate. That is the whole principle, and it works: the relation is exact, contains nothing empirical, and is why a device with no moving parts can measure a flow to a fraction of a per cent.

The complication is σ. For a Venturi it is the throat’s own area ratio, because the passage is smooth and the flow fills it. For an orifice plate it is not: the jet through the hole contracts to a vena contracta narrower than the hole, and the narrowest section of the flow is smaller than the narrowest section of the pipe.

Working the two definitions through — the true flow rate against the flow rate the ideal formula would infer from the same reading — gives the discharge coefficient:

$$C_d = C_c\sqrt{\frac{1-\beta^4}{1-C_c^2\beta^4}}$$

with β the diameter ratio and C_c the contraction coefficient of the hole.

The calibration constant of the world’s commonest flowmeter is a contraction coefficient. That is worth stating plainly, because C_d is presented everywhere as an empirical correction — a number from a standard, tabulated against Reynolds number and tapping arrangement, with a stated uncertainty. Most of it is not empirical at all. It is π/(π+2) with a velocity-of-approach factor, and the tables are recording the difference between a two-dimensional slot and a round hole in a plate of finite thickness with a boundary layer on it.

The price, which is the interesting half #

Now the part the reading does not show.

Downstream of the throat, a Venturi opens out gradually. The flow decelerates, its kinetic energy turns back into pressure, and in an ideal fluid all of it comes back — the pressure at the outlet equals the pressure at the inlet, and the meter has cost nothing.

Downstream of an orifice plate, the jet is simply abandoned. It expands into the full pipe as a sudden enlargement, and a sudden enlargement destroys the Borda–Carnot head:

$$\Delta p_{\text{lost}} = \tfrac{1}{2}\rho (V_{vc} - V_1)^2$$

which is the exact result the internal-flow ladder derives from momentum and energy alone, with no viscosity in it anywhere.

That the loss is exactly Borda–Carnot’s is not an analogy. The site computes the permanent loss from the momentum balance across the expansion and compares it with ½ρ(V_vc − V₁)²; they agree to 10⁻⁹ relative, which is the residual of the arithmetic rather than of the physics.

The claim this refutes #

The usual account of why an orifice plate is wasteful invokes turbulence: the jet breaks up, eddies form in the corner, and the eddies dissipate energy into heat.

Every clause of that is true and the conclusion drawn from it is wrong. The eddies do dissipate the energy — energy has to become heat somewhere, and viscosity at small scales is what finally does it. But they do not decide how much. That is fixed by a momentum balance across the expansion, in which viscosity does not appear, before any eddy exists. It is the same amount whatever the fluid, whatever the Reynolds number and whatever the plate is made of.

The test that makes this concrete is the Venturi. Both devices are equally turbulent downstream — the Reynolds numbers are the same, the fluid is the same — and one of them loses nothing. What differs is not the turbulence but whether the flow was decelerated against a wall that could push back or released into a space where nothing was pushing.

The trade, which is a good one and is not obvious #

Reading the loss curve backwards gives the design rule.

A narrow orifice — small β — produces a large reading and destroys nearly all of it. A wide orifice produces a small reading and destroys less of a smaller quantity. So the trade is between signal and running cost, and it is a steep one: at β = 0.3 the meter costs about 90 per cent of a large reading, at β = 0.7 about 54 per cent of a small one.

A Venturi escapes the trade entirely, at the price of being twenty times the size and forty times the cost, because a diffuser that recovers pressure has to open out at about seven degrees included angle or the boundary layer in it separates and the recovery is lost. A Venturi is a metre of carefully machined passage where an orifice plate is a disc of steel with a hole in it, clamped between two flanges.

The whole difference between the two instruments is the length available to decelerate in. That is the same statement as the diffuser essay’s, as the sudden-expansion rung’s, and as the reason a car’s radiator duct is longer than it looks.

The same meter, sized differently #

Sizing is the one decision an engineer makes about an orifice plate, and the arithmetic above settles it before any catalogue is opened.

A wide plate is a poor instrument and a cheap one to run; a narrow plate is a good instrument and an expensive one. The pair of curves is the whole of the negotiation, and it is worth noticing that neither end of it is safe: too wide and the reading is lost in the noise of the transmitter, too narrow and the meter is a permanent throttle costing pumping power that dwarfs the instrument’s capital cost within a year.

There is a family resemblance here to the branching rung’s cost function, where a pumping cost and a capital cost are traded against each other and the optimum is a stationary point of the sum. The difference is that Murray’s law has a clean answer and meter sizing does not: the transmitter’s resolution is not a fluid-mechanical quantity, so the optimum depends on the instrument as much as on the flow.

What the model leaves out, and how much it matters #

Three things, and the second is the one that makes the standards thick.

A real Venturi is not free. Friction along its walls and the small separation at the diffuser’s throat cost 5 to 15 per cent of the reading in practice. This site computes none of it: the model’s statement is that an ideal diffuser recovers everything, and the honest form of that statement is that everything a real Venturi loses lies outside this calculation.

The contraction coefficient used here is two-dimensional. π/(π+2) is Kirchhoff’s exact answer for a slot in a plane wall. A meter’s hole is round, and the axisymmetric problem has no closed form. The measured value for a sharp-edged round orifice is 0.61 to 0.62, and the agreement to within a per cent is a fact rather than a proof.

The tapping positions matter and are not in the model. A meter reads the difference between two holes in the pipe wall, and where those holes are — one diameter upstream and half a diameter down, at the flanges, or in the corner — changes the reading by several per cent, because the pressure recovers along the pipe and the vena contracta sits somewhere between the plate and half a diameter downstream. The international standards for orifice metering are largely a catalogue of tapping arrangements with their own discharge coefficients, and that is what remains genuinely empirical about the device.

Compressibility, if the fluid is a gas. Everything here assumes constant density. A gas accelerating into a throat expands, and the correction is an expansibility factor of a per cent or two at ordinary pressure ratios and a great deal more near choking, which is a different subject with a different limit.

The instrument that has to be paid for twice #

There is a general point here that belongs to the whole applied field, and the orifice plate is its cleanest example.

A measurement of a flow costs energy, permanently, because measuring the flow means interfering with it. The reading is a pressure difference, and a pressure difference across an obstruction is a force on the obstruction, and a force on an obstruction in a moving stream is a power the pump must supply. It is not an installation cost or a one-off; it is a fraction of the pumping bill for as long as the instrument is in the line.

Choosing a meter is therefore a choice about what to spend: capital on a Venturi, or electricity on an orifice plate, for the life of the plant. This is a computation rather than a preference, and both sides of it are in this essay.

The other half of the same idea — that the instrument changes the answer as well as costing for it — is the next rung, where the interference is not a pressure loss but a systematic error in the number itself.

Three other ways to measure a flow, and what each pays with #

The differential-pressure meter is one family among several, and the comparison sharpens what its particular price is.

A turbine meter puts a rotor in the stream and counts revolutions. It costs a pressure drop too — the rotor is extracting the power that spins it, plus bearing friction — but the loss is small, because the rotor is nearly free-running and the flow is barely obstructed. What it costs instead is moving parts in the fluid: bearings that wear, a rotor that fouls, and a reading that drifts as either happens.

A vortex meter puts a bluff body in the stream and counts the shedding frequency, which is proportional to the velocity through the Strouhal number. It has no moving parts and a modest loss, and it pays in range: below a Reynolds number of about ten thousand the shedding is not regular enough to count, so a vortex meter simply stops working at low flow rather than reading badly.

An electromagnetic meter puts nothing in the stream at all — it measures the voltage a conducting fluid generates crossing a magnetic field — so its pressure loss is exactly zero. It pays by requiring the fluid to conduct, which rules out hydrocarbons and most gases entirely.

The pattern is worth naming, because it is the same one this whole field keeps producing. Every instrument pays in some currency, and the differential-pressure meter’s virtue is that its currency is pressure, which is the one thing every installation has a budget for and can calculate. Nothing in the family needs calibration against a standard, nothing wears, and the physics is two equations — which is why a device with a seventy-per-cent running cost has outlasted several generations of cleverer ones.

Who found it, and when #

Giovanni Battista Venturi described the effect in 1797 and Clemens Herschel turned it into a meter in 1887, patenting the tube that carries Venturi’s name and selling it for water-supply metering. The orifice plate is older as a device and younger as an instrument: holes in plates were used to apportion water in the Roman and Islamic irrigation systems, and the modern metering standard dates from the 1930s, when the American and German gas industries independently discovered that everybody’s coefficients disagreed and set about measuring them properly.

The Borda–Carnot loss that explains the difference between the two predates both meters, and is covered in its own rung of the internal-flow ladder. What this essay adds is the observation that the two devices are the same equation with different downstream halves, and that the difference between them was settled in 1766 by an argument about a pipe that does not narrow at all.

The reading a plant actually gets #

One more practical point, because it decides how the instrument is specified.

The relation between flow and pressure difference is a square root, and a square root has an awkward property: it is steep where the signal is small. A meter sized for a maximum flow reads a quarter of its full-scale pressure difference at half flow, and a hundredth at a tenth flow. So the usable range of a differential-pressure meter is about three to one in flow — below that the reading is buried in the transmitter’s own noise and in the zero drift.

That is why process plants with wide turndown fit two meters in parallel with different bores, or choose a technology with a linear characteristic instead. It is a property of Bernoulli’s equation rather than of the instrument, and no amount of transmitter quality repairs it: the information is simply not in the pressure difference at low flow.

The square root has a second consequence, and it costs money #

The turndown problem above is the well-known cost of a square-root characteristic. There is a second one, it is less often noticed, and it produces a systematic error rather than merely a noisy reading.

A differential-pressure meter infers the flow as $Q \propto \sqrt{\Delta p}$. If the flow is steady, that is exact. If it pulsates — a reciprocating compressor upstream, a positive-displacement pump, a control valve hunting — then the instrument does not compute the square root of each instantaneous reading and average the results. It averages the pressure difference, in the tapping lines and in the transmitter’s own damping, and takes the square root of that.

Those are different numbers, because a square root is concave. Write the flow as a mean plus a fluctuation of relative amplitude $a$; then the pressure difference, being quadratic, has a mean inflated by the fluctuation,

$$\langle \Delta p\rangle \propto \bar Q^2\left(1 + \tfrac12 a^2\right),$$

and the flow inferred from it is $\bar Q\sqrt{1+\tfrac12 a^2}$, which is too high. The error is about $a^2/4$: six per cent for a flow swinging by half its mean, and over twenty for one swinging from nothing to twice it.

It is always in the same direction. A pulsating flow through a differential-pressure meter always reads high, never low, so it is a bias rather than a scatter and it does not average out over a shift or a year. On a custody-transfer meter selling gas by the cubic metre, twenty per cent is not a measurement problem.

The remedies are all about the pulsation rather than about the meter: put a receiver or a length of pipe between the source and the instrument, move the meter away from the compressor, or measure fast enough to square-root the instantaneous signal instead of the damped one. The one thing that does not help is a better transmitter, since a more accurate average of the wrong quantity is still the wrong quantity.

And it is worth setting beside the essay’s other result. The permanent loss is exact and cannot be polished away; the pulsation error is exact and cannot be calibrated away. Both are consequences of the same two equations, and neither is a defect of the instrument.

Where the ladder goes next #

An orifice plate takes a fixed fraction of the pressure and gives an honest reading. The next rung is about the meter that gives a dishonest one — not through any fault of its own, but because a body in a bounded stream is closer to its own reflections than to infinity, and the correction is an infinite series of images with a closed form.