The thermometer that heats itself

An aircraft carries an instrument labelled total air temperature, its reading is used to compute the static air temperature, and the static air temperature is used for the speed of sound, the Mach number, the engine’s thrust setting and the navigation computer’s true airspeed.

Every one of those numbers depends on a calibration constant that is not one, for exactly the reason a wall told nothing about its temperature does not settle at the air’s.

The instrument is the same object, made small #

A total-temperature probe is a small duct with a thermocouple in it, mounted so that the air entering is brought nearly to rest. That is a stagnation region with a solid surface in it and a sensor on the surface, which is exactly the arrangement the first rung of this ladder solves: a wall that is told nothing about its temperature, in a stream that has been slowed.

So the same expression applies with the same shape:

$$T_{\text{read}} = T_\infty\left(1 + r\,\frac{\gamma-1}{2}M^2\right),$$

with $r$ the probe’s own recovery factor. The only difference from an aircraft skin is that the designer of a probe is trying to make $r$ equal one, and the designer of a skin is not.

They get close. A well-designed shielded probe recovers 0.98 to 0.995 of the rise, which is much better than the 0.84 a flat plate in a boundary layer manages — the number the layer’s own solve returns, and the reason is geometry rather than physics: the air is slowed inside a duct where the dissipation happens over a longer path and less of the kinetic energy is left in a gradient at the sensor. The recovery factor is a design outcome, not a fluid property, which is what makes it a calibration constant.

The shortfall, and why it is not the error #

At Mach 0.85 in air at 220 K the stagnation rise is 31.8 K, and a probe recovering 0.98 of it reads 0.64 K low. That is a small number and it is not the number that matters.

What matters is what is computed from the reading. The static temperature is recovered by dividing the Mach-number factor back out:

$$T_\infty = \frac{T_{\text{read}}}{1 + r_a\,\dfrac{\gamma-1}{2}M^2},$$

and $r_a$ is the recovery factor the reduction assumes — which is not necessarily the one the probe has.

Assume a perfect probe and the reduction divides by too large a factor, so the inferred free stream is too cold. The two errors do not cancel and they do not add: the inferred temperature is out by 0.56 K where the reading was out by 0.64, which is what an inference does to two nearly-cancelling quantities of similar size.

The sensitivity is the number worth carrying, and it is computed rather than quoted: $\partial T_\infty/\partial r_a$ is $-27.8$ K per unit of $r_a$ at these conditions. So an error of 0.01 in the assumed recovery factor is 0.28 K in the reported air temperature — and 0.01 is the width of the range published probes actually span.

What that propagates into #

A third of a degree in the air temperature sounds like nothing, and it is worth following it forward rather than asserting that it matters.

The speed of sound goes as $\sqrt{T}$, so 0.3 K in 220 is 0.07 per cent in $a$, which is 0.2 m/s. The Mach number computed from a measured airspeed inherits that, so a Mach number of 0.85 moves by about 0.0006. That is genuinely negligible, and saying so is part of the argument.

The true airspeed inherits it too, at the same fraction, which is 0.2 knots in 300. Also negligible.

The engine is where it stops being negligible. A gas turbine is controlled on corrected quantities for the same reason a dimensionless group picks a machine: the corrected variables are the ones with the size and the state divided out. A turbofan’s thrust setting is scheduled on corrected fan speed, $N_1/\sqrt{\theta}$ with $\theta = T/T_{\text{ref}}$, and the corrected quantities are the whole of how a gas turbine is controlled. A 0.3 K error in $T$ is 0.07 per cent in $\sqrt{\theta}$, which moves the commanded fan speed by about the same, and on a takeoff thrust calculation those tenths accumulate against a margin that is deliberately thin.

And the recursion is the interesting part, because the reduction needs the Mach number and the Mach number needs the temperature. In practice the loop is closed by taking the Mach number from the pressures — total and static, which an airspeed indicator already has — so the temperature reduction is driven by a quantity measured a completely different way. That is good design: an error in the temperature channel does not feed back into the Mach number it is reduced with.

The two losses the arithmetic above ignores #

The expression treats the probe as adiabatic, and a real one is not, in two ways that pull the reading in the same direction.

Conduction down the stem. The sensor is attached to a support that is attached to the airframe, which is at a different temperature. Heat runs down the stem and the sensor reports something between the recovered temperature and the structure’s. The size of it depends on the stem’s length and conductivity and on the flow’s own heat transfer coefficient, and a probe designed against it is a probe with a long thin stem and a lot of internal airflow.

Radiation. The sensor exchanges with whatever it can see — the inside of its own housing, and through the inlet whatever is in front. At the temperatures of subsonic flight this is small; at the temperatures of the previous rung’s dash it is not.

Both are lumped in a real calibration into a single recovery factor quoted for the installed probe, which is why the number is installation-dependent and why it is measured in a tunnel rather than computed. The honest description of $r$ is therefore not “the recovery factor of the boundary layer in the probe” but “whatever fraction of the rise this probe on this aeroplane was found to report” — a fitted constant absorbing three different physical effects, in the same way a discharge coefficient absorbs several, and in the same way a wind tunnel’s correction absorbs the walls.

Where the whole method fails #

There is a regime in which this instrument stops working, and it is worth naming because it is where the previous rung lives.

At high supersonic speeds the stagnation temperature is high enough that the probe itself becomes a structural problem: the sensor is at 800 K at Mach 4 and 1,500 K at Mach 6, materials creep, and the probe’s own radiation is no longer small. Above that, the air behind the probe’s shock is hot enough that gamma stops being a number — the specific heats vary, some of the energy goes into vibration and dissociation rather than into temperature, and the relation between total temperature and total enthalpy is no longer the constant-$c_p$ one every expression in this essay uses.

At that point what a probe measures is not the stagnation temperature and the quantity of interest is not the stagnation temperature either. The conserved quantity is the total enthalpy, and high-speed facilities measure it by other means entirely — by an energy balance on the whole facility, or from the reservoir conditions, rather than by putting a thermometer in the stream.

So the instrument’s range is set by the same physics as its principle. It works where the air is a perfect gas and the recovery factor is a constant, and both of those fail together.

The number that is genuinely large #

Everything above has been about tenths of a degree, and it is worth ending the accounting with the case where the same arithmetic gives a number nobody would call small.

The rise is $T_\infty (\gamma-1)M^2/2$, so it grows as the square of the Mach number while the probe’s shortfall stays a fixed fraction of it. At Mach 0.85 the rise is 32 K and two per cent of it is 0.6 K. At Mach 2 the rise is 173 K and two per cent is 3.5 K. At Mach 3 it is 390 K and 7.8 K.

So a probe whose accuracy is expressed as a percentage of the rise gets steadily worse in absolute terms, exactly as the flight condition it is reporting becomes more critical. And the inference amplifies it: at Mach 3 the sensitivity to the assumed recovery factor is over a hundred kelvin per unit, so the 0.01 spread between probe designs is more than a degree in the reported air temperature.

There is a compensation and it is worth stating fairly. At high Mach numbers the reduction’s divisor is large, so a fixed fractional error in the reading produces a smaller fractional error in the inferred static temperature than it does at low speed. The two effects — a larger absolute shortfall and a larger divisor — pull opposite ways, and which wins depends on whether the probe’s error is quoted as a fraction of the reading or as a fraction of the rise.

That ambiguity is not pedantic, and it is the practical reason two specifications for the same instrument can look different by a factor of several. A probe quoted as “0.5 per cent” is a very different instrument depending on which quantity the half per cent is of, and the difference grows with the square of the Mach number.

Why the correction is applied and the airspeed one is not #

There is an asymmetry between the two air-data channels that is worth noticing, because it says something about when a correction is worth making.

A pitot-static system measures a pressure difference and reports an indicated airspeed, and the indicated airspeed is deliberately left uncorrected — it is not converted to a true airspeed on the instrument, because the quantity a wing responds to is the dynamic pressure and indicated airspeed is a measure of that. The uncorrected number is the useful one.

The temperature channel is the opposite. Nothing responds to the recovered temperature: the reading is an artefact of having put an obstacle in the flow, no component of the aircraft cares about it, and it exists solely to be converted into the static temperature that other calculations need. A measurement whose raw value has no physical consumer must be corrected, and one whose raw value is what something responds to should not be.

That is a general rule about instruments rather than a fact about aircraft, and it is a better guide than a habit of correcting everything. The question to ask of a reading is what, if anything, in the world responds to the quantity as measured — and if the answer is nothing, the reading is an intermediate value and its calibration constants matter.

The calibration, and what it is a calibration of #

Since $r$ is measured rather than computed, it is worth saying how — because the procedure explains why the number is installation-dependent and why it drifts.

A probe is calibrated in a tunnel at a known total temperature. The tunnel’s reservoir is at rest, its temperature is measured there where there is no recovery problem at all, and the flow is then expanded to a known Mach number. The probe reads something; the recovery factor is what the reading is, as a fraction of the known rise. Repeat across the Mach range and a curve results, usually flat to within a few thousandths.

Two things about that procedure decide everything downstream. The reservoir temperature is the reference, so the whole scale rests on a thermometer in still air — which is a measurement with no recovery problem, no Mach number and no compressibility, and is the reason the method works at all. And the calibration is of this probe in this mounting, because the local flow at the probe’s inlet depends on where it is on the aircraft.

The second is why the number changes when nothing about the instrument has. A probe mounted in a region of locally accelerated flow sees a higher Mach number than the aircraft’s, so the rise at its inlet is larger than the reduction assumes, and it reads high. A probe in a region of separated or retarded flow reads low. Neither is the probe’s fault, both look like a change in $r$, and both are absorbed into the installed calibration.

And it drifts because the geometry does. Erosion of the inlet lip, a partly blocked bleed hole, ice accretion on an unheated probe: each changes the internal flow, each changes the fraction of the rise recovered, and none of them changes the reading in a way that looks wrong. An instrument whose failure mode is a slowly changing calibration constant is the hardest kind to notice, and it is why total air temperature is cross-checked between probes rather than trusted singly.

What the picture cannot show #

No probe was solved. Every number here comes from the recovery expression with a stated $r$, and nothing computes what $r$ is for any geometry. Doing that would need a solve inside a duct with a sensor in it, which is a different problem from the flat-plate layer this ladder is built on.

Constant specific heats throughout. The relation between stagnation temperature and Mach number assumes a perfect gas with a fixed $\gamma$, so everything above about Mach 4 is illustrative rather than accurate.

No conduction and no radiation in the numbers. Both are named above and neither is computed. The function that produces these figures accepts them as terms and every figure here passes zero, which is stated rather than hidden.

And the probe is steady. A real one has a time constant of its own — a small version of the previous rung’s skin, measured in seconds rather than in tens of seconds — so a reading taken during a rapid climb or acceleration lags the air. That is the same convolution the previous rung wrote out, with a narrower kernel, and the reason a temperature reading is trusted less in a manoeuvre than in cruise.

The assertion behind these figures is the one that can fail and that would have caught a sign error: a probe with $r = 1$ and no losses must read the stagnation temperature exactly and infer the free stream exactly; a probe with $r < 1$ must read low by exactly $(1-r)$ of the rise; and the sensitivity expression must agree with a finite difference of the inference to four figures. It does, and getting the sign of the sensitivity wrong is the easiest mistake in this arithmetic.

Who found it, and when #

The recovery factor of a thermometer in a moving stream was studied through the 1930s and 1940s, as soon as aircraft were fast enough for the correction to exceed the instrument’s own accuracy — and the work was done in the same period and often by the same people as the flat-plate recovery calculation the first rung of this ladder reproduces.

The shielded total-temperature probe in its modern form is usually credited to work at NACA and to industrial development through the 1950s. Its design is a compromise that is worth stating because it is the whole of the instrument: air is admitted through an inlet, slowed in a diffuser, passed over a sensor and then bled out through small holes, and the bleed flow is what keeps the sensor washed — without it the recovery factor is better and the response time is unusable.

That trade is the same one the bubble wire makes and the same one every instrument makes. The quantity that improves the measurement’s accuracy degrades its speed, and where the compromise is set is a decision about what the instrument is for rather than a fact about the flow.

Where the ladder goes next #

This anchor now has the steady wall, the wall with a memory, and the instrument. What it does not have is the case the instrument exists inside.

The rung above is the probe as a transient in its own right — the same convolution as the previous rung with a kernel of a second or two, driven by a flight profile rather than by a step, and with the specific question of how much of a climb’s temperature change a probe has caught by the time it is read. That is a measurable claim: two probes of different mass on the same aeroplane disagree during a manoeuvre and agree in cruise, and the difference between them measures the kernel.

The one beside it is the reduction chain itself. Total temperature, total pressure and static pressure are three measurements and the air’s state has three unknowns, so the system is exactly determined and the errors propagate through a Jacobian that can be written down. Computing that Jacobian — which measurement’s error dominates which derived quantity, and at what Mach number the answer changes — would say where an air-data system’s accuracy actually comes from, and the answer is not the same at Mach 0.3 and Mach 0.85.