The profile a meter cannot see

A differential-pressure flowmeter measures one number and reports another. The number it measures is a pressure difference across a contraction; the number it reports is a volume flow rate.

This collection has an essay on what that measurement costs in permanent pressure and another on the instrument’s own presence in the answer. This one is about the step in between: what has to be known to get from the pressure to the flow.

The two halves of the inference #

The pressure difference across a contraction follows from an energy balance between two sections, and that balance is written in terms of the mean of the cube of the velocity rather than the cube of the mean, because kinetic energy is quadratic and the velocity is not uniform. Similarly the momentum balance needs the mean of the square. So two profile-dependent numbers enter:

$$\alpha = \frac{\langle u^3\rangle}{\langle u\rangle^3}, \qquad \beta = \frac{\langle u^2\rangle}{\langle u\rangle^2},$$

both exactly one for a uniform profile.

A meter’s calibration assumes the uniform values, which is why it is calibrated on long straight pipe with a fully developed turbulent profile, where $\alpha$ is 1.058 and $\beta$ is 1.020 and the assumption is nearly true.

It is worth being clear that neither coefficient is an approximation or a fudge. They are exact consequences of integrating a quadratic and a cubic quantity over a section where the velocity varies, and they would appear in the algebra of anybody who wrote the balances carefully. What makes them easy to lose is that both are one for the uniform profile that a textbook derivation assumes, so they never appear in the derivation at all.

The same two numbers turn up elsewhere in this subject under different names. In open-channel hydraulics they are the Coriolis and Boussinesq coefficients and are carried explicitly, because a river’s velocity profile is never close to uniform. In pipe flow they are usually dropped, because a turbulent profile is close to uniform — and the habit of dropping them survives into the cases where it is not.

The values, which are not close for every profile #

A fully developed laminar profile — a parabola — has $\alpha$ exactly two and $\beta$ exactly four thirds. Not approximately: the integrals of a parabola are rational numbers and the quadrature returns them to six figures.

A turbulent profile has $\alpha$ near 1.06, and it falls towards one as the profile gets fuller: 1.031 at a one-tenth power law against 1.058 at a one-seventh.

And a disturbed profile can have $\alpha$ anywhere. The annular jet used here — the shape a bend or a partly open valve leaves behind — has $\alpha = 3.16$.

At a contraction ratio of 0.6 those become errors in the reported flow rate of 0.44 per cent for the turbulent profile, 7.7 for the laminar one and 17.6 for the annular jet.

And two profiles the meter cannot distinguish #

The coefficients are two integrals of the profile, so two profiles sharing both of them give the same reading, and the family of such profiles is enormous.

Here is one member of it: a distortion of the fully developed turbulent profile, built by solving two conditions for two parameters so that both coefficients match to nine figures. It differs from its target by forty-eight per cent of the mean velocity across the pipe, and the flow rates the meter infers from the two agree to six figures.

The difference between them integrates to zero — both carry the same flow — and so do the two combinations the meter reads. What is left is a function with two sign changes and an amplitude of half the mean velocity, and it is invisible to the instrument.

The construction is worth a note, because a single matched pair could be a fluke. The distortion has two free parameters — an amplitude and a radial wavenumber — and there are two conditions, so the match is a two-by-two Newton solve rather than a search. That the solve converges means the family of matched profiles is not empty; that it converges for a range of starting points means the family is not a single point either.

The physical reading is more useful than the arithmetic. The two coefficients are averages weighted by $u^2$ and $u^3$, both of which emphasise the fast core and both of which are almost blind to the region near the wall where the velocity is small. So a redistribution that moves fluid between the core and the annulus, keeping the total flow and both weighted averages fixed, is available — and that is exactly the kind of redistribution a fitting produces.

Why the standards ask for straight pipe #

This is the reason a metering standard specifies tens of diameters of straight pipe upstream of a meter rather than specifying a correction.

The correction that would be needed is a function of two coefficients the instrument cannot measure — and even if they were known, two profiles with the same coefficients read the same anyway, so the correction would not identify the profile. The only available remedy is to wait until the profile is the one the calibration assumed.

That is a stronger statement than “a disturbed profile causes an error”. It says the error cannot be corrected from the instrument’s own reading, at any level of sophistication, because the reading does not contain the information.

Why the laminar case is the awkward one #

The laminar number deserves separating out, because it is both the largest ordinary error and the one most likely to be met unexpectedly.

A kinetic-energy coefficient of exactly two means the approach term in the energy balance is twice what the calibration assumes, and at a moderate contraction ratio that is nearly eight per cent in the flow rate. Eight per cent is an order of magnitude beyond the uncertainty a metering standard claims and beyond anything a user would tolerate.

The awkward part is when it happens. A meter sized for a design flow and then run at a small fraction of it — a plant at part load, a pipeline in low demand, a laboratory rig at the bottom of its range — can drop below the transition Reynolds number in the approach pipe without anything announcing it. The meter goes on reading; the reading is wrong by eight per cent; and the transition between the two regimes is not gradual, because the profile changes character rather than deforming.

Viscous fluids make it routine rather than exceptional. Oil, syrup, polymer solution, a slurry: all of them can be laminar in a pipe at flow rates that would be firmly turbulent in water, and all of them are metered. A discharge coefficient quoted as a function of Reynolds number does capture part of this, which is why such curves fall away sharply at low Reynolds number — that fall is partly this effect being fitted rather than derived.

The contraction ratio changes everything #

There is one lever, and it is geometric rather than numerical.

A meter that squeezes the flow hard makes the throat velocity so much larger than the approach velocity that the approach’s shape stops mattering: the energy balance is dominated by the throat term, where the profile is nearly uniform because the contraction has accelerated it. At a contraction ratio of 0.3 a fully laminar approach costs 0.41 per cent; at 0.75 it costs 26.7.

A factor of sixty-five, from geometry alone. Which is why a metering standard specifies the contraction ratio as tightly as it specifies the pipe length, and why the two requirements are not independent: a meter with a small ratio needs less straight pipe than one with a large ratio, because it is less sensitive to what the straight pipe is for.

How long the pipe has to be #

The standards’ pipe lengths are worth reading in the light of all this, because they look arbitrary and are not.

A disturbed profile relaxes back to the fully developed one over a length that depends on what disturbed it: ten diameters after a gentle bend, forty after two bends in perpendicular planes, eighty or more after a partly closed valve. Those numbers are measured rather than derived, and they are quoted as requirements rather than as corrections precisely because the correction is not available.

What is derived is the tolerance. A standard states a permissible additional uncertainty — half a per cent, typically — and the pipe length is whatever is needed to bring the profile close enough that the coefficient error is inside it. Working backwards through the numbers on this page, half a per cent at a contraction ratio of 0.6 corresponds to a kinetic-energy coefficient within about 0.07 of the fully developed value, which is a fairly tight requirement on the profile.

That also explains the flow conditioner. A perforated plate or a tube bundle placed upstream does not restore the fully developed profile; it produces a different profile, closer to uniform, whose coefficients are near one. Since the calibration assumes coefficients near 1.06, a conditioner that over-corrects towards uniformity introduces an error of its own — which is why conditioners are themselves specified and calibrated rather than simply inserted.

The general shape of it #

The pattern is the one this collection keeps meeting.

The pressure difference is a force balance and is exact. It does not care what the fluid is, how the velocity is distributed, or whether the flow is laminar or turbulent — the same statement that makes a capillary viscometer’s stress exact.

The flow rate inferred from it is not, because getting there requires two integrals of a profile, and what an exact constraint reaches is what lies in its own span.

And the freedom left over is measurable rather than vague. Forty-eight per cent of the mean velocity, in a profile the instrument declares identical to its calibration.

What would actually determine the profile #

It is worth asking what measurement would close the gap, since the argument so far has only said that this one does not.

The profile is a function, so determining it needs data of the same kind: a traverse. A pitot traverse across the pipe, or an ultrasonic meter with several chords, or a laser measurement at enough radii — each supplies a set of point values rather than two integrals, and from enough of them the profile follows.

That is exactly what a multi-path ultrasonic meter does, and it is why such meters are less sensitive to installation than a differential-pressure meter is. Four chordal paths give four weighted averages rather than two, and the weights are chosen so that the combination is a quadrature rule for the flow rate — which means the meter is integrating the profile rather than assuming it.

A Coriolis meter closes the gap differently, by measuring the mass flow directly through the force a moving fluid exerts on an oscillating tube. That is a measurement of the quantity itself rather than an inference from a related one, which is why it is insensitive to the profile altogether, and it is why such meters are used where the approach conditions cannot be controlled.

The general rule is worth stating. A measurement that infers a functional of a profile from a different functional of the same profile needs the two to be related, and the relation is the assumption. Measuring more functionals, or measuring the wanted one directly, is the only way out — and both are what the expensive instruments do.

Where else two integrals stand in for a profile #

The structure of this measurement is not peculiar to flowmeters, and three other instances in this collection make the family clear.

A capillary viscometer reads a pressure drop, which is exact, and a flow rate, which is one integral of the profile, and needs a wall shear rate, which is a derivative at a point. The gap is closed by measuring the derivative of a curve through several runs.

The momentum integral of a boundary layer reads two functionals of an assumed profile and returns a drag, and the two functionals are almost all a smooth family has to offer — which is why it works, and why it cannot find a separation point.

A wall model in a pipe integrates an assumed near-wall profile across the radius, and the answer is dominated by the constant that sits in the outer part rather than by the shape it was written to describe.

In every case a total is being formed from a profile by a weighted integral. The weight decides which features of the profile are visible, and the instrument sees precisely as much of the profile as its weights allow. The number of weights is the number of things that can be determined, and no amount of accuracy in reading each weight adds another one.

Which gives the question to ask of any indirect measurement, and it takes ten seconds. How many independent functionals of the unknown does this instrument return, and how many numbers is it being asked for? A meter that returns one and is asked for one — a Coriolis meter, a wake survey’s drag — is exact. A meter that returns two and is asked for one, through a relation that needs a third, is carrying an assumption whose size is worth computing.

What the discharge coefficient is really absorbing #

It is worth returning to the coefficient itself, because the argument above has treated it as a fixed calibration and it is more interesting than that.

A discharge coefficient is a single number that converts an ideal inference into a real one, and it is absorbing several distinct things at once: the vena contracta downstream of an orifice, the pressure recovery between the tappings, the boundary layer on the plate, the edge sharpness, and — this page’s subject — the approach profile’s kinetic-energy coefficient.

Those have different dependences. The contraction is nearly geometric and nearly constant. The boundary-layer terms depend on Reynolds number and fall away as it rises. The approach profile’s contribution depends on Reynolds number too, and it does something the others do not: it changes character at transition rather than varying smoothly, because a laminar profile and a turbulent one are different shapes rather than the same shape at different strengths.

That is visible in every published discharge-coefficient curve as a sharp change at low Reynolds number, and it is why standards restrict their coefficient tables to Reynolds numbers above a stated minimum. The restriction is not a statement about the accuracy of the fit below it; it is a statement that below it the coefficient is absorbing a different profile.

Which is the last thing worth carrying from this page. A number quoted without the regime it belongs to is a number that will be used outside it, and a coefficient that absorbs a profile is a coefficient that changes when the profile does.

What a wet-calibrated meter is worth #

One last practical point, because it decides how much of this matters in a given installation.

A meter that has been wet-calibrated in place, against a reference, on the actual pipe with the actual fittings upstream, has had its coefficient fitted to the profile it will see. The approach profile’s contribution is then absorbed into the calibration rather than being an error, and the argument on this page becomes a statement about how much that calibration is doing rather than about how wrong the meter is.

What survives is the transferability. That calibration belongs to that installation at that flow rate. Move the meter, change the upstream fittings, run it at a different Reynolds number, or let the approach flow transition to laminar, and the coefficient it was given is a coefficient for a profile that is no longer there.

Which turns the question into a familiar one. A fitted parameter is a record of the conditions it was fitted at, and this collection has just made the same point about a mechanism whose constant has to be re-tuned at every operating point. The remedy is identical too: check the parameter’s stability across conditions, and treat a drift as a measurement of what the model is absorbing.

The permanent loss, and why it behaves differently #

There is a companion quantity worth mentioning, because it is the one metering property that does not depend on the approach profile.

The permanent pressure loss of an orifice or Venturi — the price of knowing the flow rate — follows from a control-volume momentum balance across the expansion downstream, and momentum balances do not ask what the profile was. It is a Borda–Carnot loss with no viscosity in it, and the approach profile’s coefficients enter it only through the second-order term.

So the two headline numbers of a metering device behave oppositely. The reading depends on a profile the instrument cannot see; the loss follows from a balance that does not care. That is the same division as everywhere else on this page — a total that is exact and an inference that is not — and it is worth knowing which of the two a given quantity is.

It also gives a diagnostic. A meter whose measured permanent loss matches the prediction and whose reading does not is telling somebody that the geometry is right and the approach is not, which is a more specific complaint than “the installation is bad” and points at the pipe rather than at the plate.

What is not claimed #

The discharge coefficient absorbs more than this. A real meter’s coefficient is a fit that includes the vena contracta, the pressure-tapping positions, the edge sharpness and the boundary layer on the plate, and it is a function of Reynolds number. What is computed here is one term of that fit — the approach profile’s kinetic-energy coefficient — isolated so that its size can be seen.

The throat profile is assumed uniform. A strong contraction does make it nearly so, and the assumption is what makes the sharper meter more forgiving; a real throat has a boundary layer and a core and its own coefficient near 1.02.

Compressibility is absent. Everything here is incompressible, so nothing describes a gas meter at a large pressure ratio, where an expansibility factor enters and carries its own assumptions.

And swirl is not modelled. A bend leaves rotation as well as a distorted axial profile, and swirl changes the pressure distribution across the pipe as well as the coefficients — which is why standards specify flow straighteners in addition to pipe length, and why the numbers here are a lower bound on what a disturbed approach can cost.