Thermal anemometers

Amateur design report by Johan Liljencrants

A thermal anemometer uses a heated probe element that is inserted into an airstream. Air speed can then be inferred from the heating power necessary to maintain the probe at a temperature elevation. This power should be some way proportional to air speed.

In my efforts to monitor airflow in my organ trunks without disturbing its musical operations I have experimented with a number of thermal anemometers. The purpose was to hint at their relative merits regarding ease of fabrication, probe size, and time required for a measurement.

A number of experimental circuits are described, using internally and externally heated sensors, being diodes or NTC resistors to monitor temperature. With internal heating you vary the sensor operating voltage and current such that it is heated by its internal dissipation. This implies fast response, but also an issue of whether sensing is disturbed by the heating. Alternatively, with external heating the sensor is only thermally connected to a separate heater such that there is no such electrical interference. Then instead there is a delay for heat to be conducted from heater to sensor. This makes the device substantially slower and puts restrictions on the control circuit in order to maintain stability.

The thermal circuit of a probe is modeled. Essential parameters in this model are derived for several experimental probes of the various types. It was found that the classical King's law, saying power dissipation is proportional to the square root of air speed, does not hold well for the large size sensors used in the actual case. Here a direct proportionality to air speed makes for a better model fit to measured data. The model data suggests two derived parameters to characterize the probe quality and its suitable speed range. Quality depends on the probe design and material properties. Mid-range speed is critically dependent on probe element size, the higher speeds one wants to measure, the smaller the probe has to be.

1. Different circuits and sensors

1.0 Background: Internally heated transistor - 'Tranemometer'

Zeb Vance once suggested to me a link to an anemometer design, see reference below. Here is my somewhat modified version of that circuit: The temperature sensing elements are the base-emitter junctions of two probe transistors Q1, Q2. The base-emitter junction voltage is typically 0.7 Volts with a temperature coefficient near -2 mV per deg C. The lower transistor Q2 has its collector wired to its base. This one acts as a passive diode, only there to sense ambient temperature. These transistors form the left side of a bridge, the right side is resistors R1, R2, and the trimmer R3. Amplifier A1 senses the balance of the bridge. If the voltage over the Q1 junction is too high, then A1 will drive the Q1 base up. More current will pass through both transistors but Q2 is fully conducting and does not change its temperature appreciably with change in current. Having a high collector voltage, Q1 will be heated while Q2 remains essentially at ambient temperature. That heating lowers the Q1 base-emitter voltage until balance is restored. The heater and the temperature detection are inherent in the transistor itself. So A1 keeps Q1 a certain number of degrees hotter than Q2. How many depends on the trimmer setting, with this circuit typically around 5 degrees centigrade. Resistor R4 senses how much current is flowing through Q1-Q2. The (small) voltage developed over R4 by this current is amplified by A2 into the output pin 7. A2 has an offset input but otherwise simply translates the additional current needed to maintain the temperature difference between the two B-E junctions. The more current, the more heat is being removed from hot Q2. Actually A2 is not simple at all. If R9 and R10 are trimmers, you can go nuts trying to adjust them. The reason is that “input offset” in the front. As the bias changes, the gain is affected.

The original article mentions a problem with this circuit. The sensor transistors may latch up in a current rush mode, with the top Q1 fully on and current limited essentially only by the small sensing resistor R4. Then Q1 can no more hold its temperature and the bridge balancing fails. This mode is easily evoked by a minimal disturbance, e.g. like putting a scope probe in contact with the circuit. The remedy is the threshold feedback from A2 via two diodes (a transistor in the original article). If the output at A2 goes too high, essentially over some half the supply, then the feedback diodes open and A1 is quenched such that probe current is cut off again. While this safety device is in operation, the output of the circuit is in error (output no longer goes up with airspeed). Without it, however, it goes up and stays up until you turn off the circuit.  The capacitor C1 is not commented in the original article, but apparently slows operations to be in the tens of milliseconds range, preventing oscillation. Still this is much faster than the thermal time constants in Q1-Q2.

Power is supplied from a single 9V battery. The power-on indicator LED is used to offset the nominal ground and form a negative supply for the op-amps. Otherwise their inputs come too close to the negative supply, such that they do not operate.

It is hard to understand the talk about linearity in the original article until you realize it is stated for a rather small range, up to 250 ft/min, 1.27 m/s. I find from my calibrations in the 1 to 10 m/s range that the device is close to logarithmic, as seen here in the calibration for a probe with TO-18 case transistors. Its output voltage increases about equally much for each doubling of the air speed. And this is also the kind of behavior one would like to have, this largely obviates need for a range switch.

There is some freedom in dimensioning, on one hand the current sensing R4, on the other the divider R11-R9, these together define the probe 'rest' (still air) current. Additionally the A2 gain controlling R10 implies a limit on max. air speed when the feedback diodes open to cut out Q1 heating.

When switched on, the meter goes beyond full scale since Q1-Q2 are initially the same temperature. Then output creeps down as Q1 heats, takes time. For the balance trimmer R3 I use a multiturn pot, this is a very sensitive one to set. I prefer to zero the meter output at about 0.25 m/s air speed. Rather than in still air which is somewhat indeterminate because of whatever thermal convection then goes past Q1, also taking maximal time to reach equilibrium. To get readings at low air speed like 1 m/s is a matter of tens of seconds.

There remain problems with this circuit. This is the reason I have gone through all the following variant schemes. The worst objection is the setting of the balancing R3 trimmer that is extremely sensitive - when you touch it the reading moves very far out before returning to near where it was, and this takes a lot of time. Also I blame this for poor stability in calibration, several times it has differed as much as a factor of two in air speed, taking the instrument out from store. It can be questioned also on more theoretical grounds. The 'cold'  transistor Q2  is also heated to a variable degree because it conducts the governed current. This current times the 0.7 volt Q2 voltage is no negligible power. Also the Q1 base-emitter voltage depends not only on temperature but also on the controlling base current injected by A1 via R5. This gives a spurious extra voltage right at the most sensitive spot where bridge balance is sensed, actually causing a positive feedback that may harm stability.

1.1. Internally heated transistor - alternate layout

After many frustrations with the above circuit I tried the following alternate layout in a circuit that maybe is easier to understand. The base potentials of the cold reference Q2 and the hot Q1 are directly compared by the differential amplifier. Q2 is fed with a largely constant current determined by R1. The amplifier gain is basically R6/R5 resulting from its negative feedback. The problematic thing is that the servo action to maintain a temperature difference also implements a positive feedback via R4 and the base-emitter resistance inherent in Q1. This latter depends rather unpredictably on the Q1 properties. The original #1 circuit has the same problem, but with the present circuit it is easier to see this is the case. If the positive feedback is too large, then the circuit will go unstable or latch up, but this can be cured by increasing R4 or decreasing R6.

This positive feedback is probably the reason for the statement that the original circuit could be approximate linear.  In my opinion that feedback rather implies a discouragement from using this type of circuit, however clever is the idea of internal controlled heating of a transistor.

The Q1 heating current is converted into a proportional voltage by R2. This also raises the potential of the amplifier inputs enough that no special trick is necessary to hold down the amplifier negative supply rail. Further the diagram suggests a convenient arrangement of zero and calibration trimmer potentiometers, should you want to display results directly with a milliampere meter.

The four calibration curves pertain to different implementations of the transistor probes, all run in identically the same circuit except for the bias resistor R3. Two use classical style TO18 metal cased transistors (type BC107B), the other two use miniature surface mount SOT23 (type BC847B). The big difference between the red (copper) and black (iron) traces is a change in thermal conductivity in the connecting wires used. Not until late stages in my experimentation I realized the extreme importance of this feature. Theory and experiments on this sub topic of probe design follow below.

The optional R3 controls the Q1 base current at rest and thereby indirectly governs its temperature elevation. Alas, it seems you have to adjust this to match whatever current amplification factor that device happens to have. A goal may be to set output voltage U at still air to come in the 1-2 V range.

1.2 Externally heated diode bridge

This circuit remains with the principle of diode forward voltage temperature dependence, but now the hot diode is externally heated by a resistor. This diode was clamped to the heater with a tiny strip of brass sheet and also sealed to it with a drop of cyanoacrylate glue. The photo shows the probe tip cold and heated diodes. They are mounted on a flexible multiple conductor strip, retrieved from a head arm of a junked hard disk drive. The glass encapsulated 1N4448 diodes seem to have a fairly low thermal resistance, the data sheet says 0.24 K/mW including 10 mm leads.
The small voltage developed over R3 purports to govern the forward drop difference, and hence the temperature difference between the diodes. The gain of the balance sensing amplifier is by necessity moderated by the R5/R4 feedback network, together with a big slowing down capacitor C1.  There is a delay of heat transfer from the heater to the heated diode. If the servo loop gain is too high, this will make the circuit oscillate between fully on and off. The R6 heater resistor consumes more power than the bare amplifier can deliver, so an intermediate emitter follower transistor is added. The rather low input voltage bias to the amplifier from the sensing diodes necessitate D3 to increase the margin of amplifier negative supply.

The calibration appears to be better reproducible and have a larger air speed range than circuit #1. Also, the characteristic of voltage U vs. air speed is attractive. But response is very slow, and possibly somewhat oscillating.

1.3 Internally heated NTC resistor bridge

The resistance of an NTC (Negative Temperature Coefficient) resistor, often called thermistor, typically decreases to about half at a 25 degree centigrade temperature rise. This makes it a very sensitive component, much used in electronic thermometers. I used Mitsubishi type RH 16 in a common miniature form, a little bead at the end of two thin connecting leads. In the diagram the left arm of the bridge is high impedance while the right arm is low impedance. The balance sensing amplifier provides the bridge feed voltage via a buffer emitter follower to boost power. When the bridge feed voltage U goes up, then only the low impedance arm of the bridge is appreciably heated - the high impedance arm holds another thermistor and is for compensation against ambient temperature change.

This circuit is simple, reliable, and sensitive. But one slight difficulty may be to find a sufficiently low resistance thermistor such that it can be driven enough hot at high speed, given the rather low supply voltage. The alternative with R1=10k is for a very moderate temperature rise, some 5K.

Amazingly, the output sometimes oscillates a trifle with a period of a few seconds, motivating C1 to quench that. I guess this may be because the NTC chip is heated unevenly throughout its volume, and that the oscillation period relates to the time it takes for local heat to even out within the chip.

1.4 Externally heated NTC resistor bridge

Also an externally heated version was tried. The photo shows the probe with the hot NTC resistor lashed with thin copper wire to the heater resistor. The original heater resistor leads are cut off and replaced by 0.24mm wire wrap leads to reduce uncontrolled thermal leakage. The hot array is isolated from the cold reference NTC resistor by a lashing of sewing thread.

This circuit performs well, except for its inherent thermal delay between heater and sensor. This necessitates the slowing down feedback capacitor. Without it the circuit will oscillate between on and off, with it the final reading is reached after a prolonged delay, to the order of 30 seconds.

 This particular probe design is perhaps not optimal. The red markings show readings when the probe was rotated in 45 degree increments relative to the airflow direction. When the 'cold' reference thermistor is located downstream the of the hot one, then readings go much too high. It might have been better when the cold reference thermistor had protruded beyond the heated one.

I believe this is a similar principle as used for a hot ball anemometer in The Amateur Scientist, Sci. Am., Nov. 1995. I have not been able to retrieve that article right now, but ISTR that they used thermocouples rather than thermistors. However, the fairly big balls used there must make it extremely slow, maybe adequate for measuring average wind speed in meteorology.

1.5 Hot wire, internally heated

The hot wire anemometer is a classical type and appears to be the one predominantly used for professional work. An orthodox such probe is made from Wollaston wire, a thin silver wire with a platinum core (priced like US$ 500 for 8 inches of it). After soldering it to its posts under a microscope you etch away the silver to leave a sub micrometer diameter platinum wire. An exercise well beyond most amateurs.

I found a workaround to this by breaking the glass bulb off a small incandescent lamp, the type shown beside. After soldering its external connecting leads and lashing the assembly to a slender wood stick I filed a tiny notch at the bottom of the bulb (at the mark in the photo). The assembly was cautiously held in a vise and the bulb broken off, using a loose fitting tube for a lever. It was then mounted in a protective holder, fabricated from 0.2 mm brass plate.

It must be noted that without its bulb and inert atmosphere the filament can not withstand anywhere near the original lamp specification before being burned out. Be aware the lamp cold resistance is 10-20 times lower than what is given by nominal voltage and power. This particular lamp happens to have 20 ohms cold resistance. For tungsten the resistive temp coefficient is 0.0045 /K such that the bridge balance value of 22 ohms is reached at about 22 degrees centigrade temperature elevation. This is a very moderate rise, such that correction will be necessary if ambient temperature deviates appreciably from normal. Tweaking the fixed resistor values in the bridge allows for some other temperature level. On one hand R2 should be large enough to ensure the filament is never burnt out. On the other hand R2 and supply voltage limit heating current such that there is a definite maximum measurable speed.

Few lamps are constructed such that you can break off the bulb, leaving the filament intact. Another one I found is a 24 V lamp for decoration candlesticks. A 12V 5W halogen lamp was marginally successful, but since that one has a cold resistance below 1 ohm it draws considerable current and needs an additional power transistor to drive it.

Having broken the barrier of fabricating a probe, this is my favorite anemometer beyond all competition. The circuit is simple and stable and measurement time is milliseconds, shorter than any of the other alternatives by several orders of magnitude. But indeed this can make it difficult to calibrate, since it follows the rapid speed variations from any turbulence in the air stream.

One must remember that this poor man's version of a hot wire anemometer has its limitations. It still does not obey King's law, probably because the lamp filament is helically coiled, making for an outer diameter vastly larger than that of an orthodox hot wire. Also the contact between the filament and its post may be questionable with the low voltage used in this application. At one instance I have repaired a faulty probe by carefully pinching such a joint with a pliers.

2. Calibration procedures

2.1 Air speed

To get an air stream of known speed I used a vacuum cleaner, fed from a variable transformer. That way the fan speed could be set arbitrarily over some range. The cleaner hose was connected to a Venturi tube to measure flow rate and the device ended with a nozzle where the probe under test was placed centrally. To extend the measurement range I could alternate between 22, 46 and 86 mm diameter nozzles. Knowing its diameter and assuming a uniform air speed over its intake area A (m2) it is elementary to convert from flow Q (m3/s) into speed V (m/s): V = Q/A. After running for a while the fan will heat the air passing it. To avoid spurious effects from that, air was sucked from the room rather than blown out from the nozzle.

The flow meter Venturi tube has two probe holes to measure the pressure drop from input to constriction. By virtue of the Bernoulli law this drop is proportional to the square of the flow rate and was taken with a water U manometer, later with a differential pressure transducer. The tube was calibrated by measuring the time elapsed to fill a plastic bag of known volume. The Venturi expands gradually after the constriction such that pressure is partially regained. This has nothing to do with the flow measurement as such, but it reduces the throttling effect of the meter.

There are alternative ways of calibrating for air speed. The probe could be put on a motor driven trolley, or at the end of a rotating boom. Or you could compare with some calibrated reference anemometer.

2.2 Temperature

One would like to know the temperature of the probe element. For the thermistor and hot wire this is simple since the bridge balance criterion (same resistance ratio both sides) tells about their hot electrical resistance. For those the temperature can then be computed from their known cold resistance and the temperature coefficient.

For a direct measurement I used a small oil filled container, carried on a digital thermometer probe. First the anemometer circuit was left to stabilize in still air and its output voltage was recorded. The container was heated with a soldering iron and was then left to cool down slowly while its temperature was tracked by the thermometer. At intervals the probe hot element was dipped into the oil. At the point where the anemometer output then stayed at its earlier recorded value, the oil temperature equals that of the probe tip. The small paper wing in the photo was to shield the cold reference sensor from hot air rising from the oil bath.

3. Modeling probe thermal resistance

3.1 Internally heated sensor

In the internally heated probe the heater and sensor are one and the same element. Below is a thermal analogy to such a probe. It looks like an electrical circuit, but it is not. Instead there is power flowing in the conductors, not current. And there is temperature developed over its elements, not voltage. One can do this because of the definition of thermal resistance to be the temperature difference developed, divided by the power conducted. Thermal resistance is measured in K/W, degrees (Kelvin or Centigrade) per Watt. This is the thermal variant of Ohm's law.

At left is a generator that injects power P (Watts) into the circuit. C is the thermal capacity of the device (e.g. transistor) measured in Joule (Watt*seconds) per degree (Kelvin or Centigrade), and the internal temperature of  it (junction temp) is To. All temperatures are relative to ambience = 'ground'.

After an initial progressive heating the 'capacitor' is charged (=heated) to a stationary temperature. Once this has been reached you can disregard C. Then all the power will flow through the fixed  resistance Ro (chip and casing) and give rise to a surface temperature Ts. Ts depends on power P and two resistances in parallel. One of those is Rl representing heat losses like conduction in the connecting leads. The other is the variable resistance Rv that represents the cooling from the streaming air.

One basic relation between P and Ts for a wire placed normal to the flow was suggested by L V King (1914). In a simple form it reads P = Ts/Rv = A + B*V0.5 where A and B are calibration constants that depend on area and gas properties. At sufficiently high speed V we can furthermore drop the A constant to render a maximally simple representation Rv =  kRv/sqrt(V). Means e.g. half resistance at air speed 4 m/s compared to 1 m/s. This relation is said to be valid for Reynolds numbers below 40. The Reynolds number is Re = (rho)*V*D/(mu), where (rho) is the density (1.2 kg/m3) and (mu) is viscosity (1.81*10-5 Ns/m2) for air. At the maximum speed investigated here, V=25 m/s, then Re=40 is reached when the characteristic dimension D= 0.024mm, about 1 mil. Our sensors are larger than that by very far, the Reynolds number in same proportion, so we cannot expect King's law to be accurately valid for our application. In the experimental work (section 4 below), when Ro had been directly estimated, then also the exponent for V was optimized to the data. From that it appears Rv for the present bigger sensors varies more like inversely to speed rather than to the square root of this.

Then the coefficient kRv gets the dimension of  K(m/s)/mW. It may be handy to remember that Rv and kRv get the same numerical value at air speed 1 m/s (which equals 3.6 km/h, about 197 ft/min, or 2.24 miles/hour).

Here are a few examples of theoretical calibration curves using this model. They show the variation of input thermal resistance Rtot vs. air speed, a measure that mostly is proportional to the voltage U in a simple way, maybe shown on a milliampere meter after offset and scaling. They are normalized such that they all reach unit value at infinite air speed where Rv=0. Their lower limit is at Ro/(Ro+Rl), with the illustrated values of these we reach the value 1/3. The difference between curves within each figure is only kRv, which reflects the smallness of the sensor.

The two curve sets show the same thing, but the left one is on a linear scale of air speed while the right one is on a logarithmic. You will have noticed a logarithmic scale was also used in the earlier presentations of the various circuits. Since zero cannot be displayed on a log scale, the scales are broken, appended portions show the zero speed reading at extreme left.

It should be obvious that result indication with a plain analog meter on a single range is a very good option. It will then have a scale not very much deviating from a logarithmic one, like illustrated at right. Note that on the log scale all the curves have the same basic shape, only translated along the speed axis.

From some study of the background formulas, bypassing algebraic detail, but supported by these curves, I would recommend two derived key parameters to characterize a sensor:
To get a sensitive probe we would like a greater part of the supplied power P to be dissipated in Rv. There should be a substantial difference in power used in still air, as compared to that at high air speed. This requires that the internal resistance Ro should be as small as possible (sensor a good thermal conductor), and that the lead leakage Rl should be as large as possible (connecting wires poor thermal conductors). Both criteria are combined in the Q factor. Q reflects probe design and thermal properties of the materials in the sensor and its connections. It has nothing to do with the working principle or the circuit. In the example curves Q=2. I would suggest that a probe with Q less than 1 is hardly worthwhile to use. In that limiting case the still air power is half of the power at infinite speed.
A mid range Vm comes when the air cooling resistance Rv equals the parallel combination of Ro and Rl. The formula above assumes Rv to be inversely proportional to speed. Observing that the coefficient kRv is also inversely proportional to the sensor area cooled by the air stream conveys an important message. To make an anemometer for low speeds kRv should be small, i.e. the sensor have a large area. And conversely, a meter for high speeds should have a small sensor. In the example curves Vm=0.6, 1.2, 3, and 6 m/s respectively.

The total useful range of the scale, in the examples about 1:100, does not depend appreciably of Q or Vm. But the range does increase when 1/Rv has the King's law square root dependence of V rather than the direct proportionality that was found for the present probes.

We can also make a simplified direct reasoning about the relation between the internal Ro and surface Rv resistances. Let us assume a round wire of diameter D for the sensor element. The internal heat conduction resistance is on one hand proportional to heat travel distance, i.e. proportional to D. On the other hand it is inversely proportional to the surface traversed by the heat flow. Also this is proportional to D. Therefore Ro should be independent of wire diameter.

For conductivity 1/Rv the value is proportional to surface, hence to D, and additionally to some power x of air speed V. This leads to a quotient between the two resistances to be Ro/Rv = (const)*D*Vx. This ratio must be small, e.g. smaller than 1, for the probe to be useful. If large, then Ro will dominate such that the meter becomes more or less insensitive to any changes in Rv. The important consequence is that the probe size D puts a limit to the highest speed V that can be measured. The higher speed you want to measure, the smaller the probe will have to be. This same relation prevails if you assume e.g. a spherical element, where the heat traversed surface is proportional to D2 rather than D.

3.2 Externally heated sensor

External heating eliminates the problem that heating power injection may interfere with the temperature sensing. Here is what I figure is a relevant though simplified thermal analogy for an externally heated probe. The sensor is enclosed in a separate body, thermally connected to the heater by Ro which includes its internal drop. Now there is a distinction between the thermal capacity Ch of the heater and that of the sensor, Cs. The Ro-Cs circuit makes a low pass filter that slows down operation and raises a stability problem to be handled by the electrical feedback circuit. For instance, when Cs is being heated and reaches the target temperature Ts, then the circuit turns off power. Despite that,  Cs will be even more heated  from the energy already stored in Ch.

The highlight is that sensor temp Ts goes more in proportion to Rv alone, not to the sum of Ro plus Rv in parallel to Rl. Obviously one still should strive to keep Rl large, to minimize heat loss in connecting leads.

The principle of externally heating the sensor may extend the useful range for air speed measurement, but the stability criterion and its long time to adapt may be a serious drawback.

4. Experimental values for the thermal resistances

It was possible to estimate approximate numerical values for the various thermal resistances in the test probes. The starting point was then a two column tabulation of output voltage U vs. air speed V, I used Excel for the purpose. Examples of such data are the calibration graphs shown with the circuits above. Next step was to develop tables of heating voltage and current. These can be inferred from U and relevant elements in the circuit diagrams. Their product renders the powers Pexp, the values actually developed in the heated element, one value for each of the measured speeds V. Then I put up a set of three constants, the so far unknown values for Ro, Rl, and kRv. From the latter another column was computed, applying an exponential law like King's to find the different Rv for the points of the calibration. Now we are in position to compute the total input thermal resistance to the thermal analogy, namely the sum of Ro and the parallel combination of Rl and Rv. Knowing the temperature rise T, thus compute a hypothetical model power Pmod = T / { Ro + RlRv/(Rl+Rv) }.
These Pmod form an additional column, one item for each of the calibration points. An ultimate column computes their relative errors squared, relative to the actual powers as E2 = { Pmod/Pexp -1 }2, and these are summed into a cell to indicate the total mismatch between data and model. The final step was to let the Excel Solver automatically iterate the three resistance coefficients in order to minimize that mismatch. To get absolute values for the thermal resistances we must know the temperature rise T, otherwise only their relative magnitudes will result.

This came out reasonably successful with a typical RMS error E around 2%, higher for a few individual points. But the automatic iteration has a flat minimum, such that other balances between the three coefficients could give almost as good correspondence between model and data. Also these optimal solutions in some cases gave negative values for Ro which is not physically acceptable. But Ro can indeed be directly estimated. With the cooling from "infinite" air speed, i.e. Rv=0, we can separately determine Ro = T/Pinf. So the calibration tables were supplemented with such extreme points, determined from the voltage U attained when the probe forcefully cooled. That was done by  immersing and agitating it in room temperature WW naphtha. Fortunately the voltage in all cases could be supplied without overloading the amplifiers. But the sect. 1.5, 24V lamp case gave an abnormal result in that such radical cooling still gave a lower voltage than the highest air speed used. Maybe this can be explained by its helically wound filament obstructing the liquid cooler by virtue of viscosity.

Having got this relatively firm grip on the magnitude of Ro the resulting total matches between model and data were a disappointment (iterating Rl and kRv only). Next step was then to assume Rv=kRv/Vxv, where not only kRv, but also the exponent xv was iterated for a best match. According to King's law this exponent should be 0.5 for square root. But for the various probes iterations it clustered in the 0.85 - 1.15 range, plus the extremes using NTC, the bridge at 0.71 and the externally heated at 1.31. Probably also the probe surface structure has an influence. Anyway, I then selected to use xv=1 throughout, i.e. Rv=kRv/V, with no appreciable loss in model match. That deviation from ideal theory, also based on experience, may be justified as the Reynolds number with the present probes by far exceeds the range specified for King's law.

The several meters and probes rendered the following parameter values. They are approximate since measurements are not overly precise. The photo background paper has 5 mm squares.

1.1 1.1 1.1 1.1 1.2 1.3 1.4 1.5 1.5

 Cu leads
Fe leads
Cu leads
Fe leads
Ext Diode NTC bridge Ext NTC Lamp
6V 30mA
Int. res.
Ro K/mW
0,24 0,24 0,22 0,21 0,03 0,10 0,02 0,12 0,35
Loss  res.
Rl K/mW 0,09 0,13 0,07 0,55 0,12 0,24 0,19 2,18 6,90
Cool coeff.
kRv K(m/s)/mW
0,15 0,09 0,47 0,30 0,11 0,38 0,25 1,06 7,43
Temp. rise
dT K
30 39 29 32 5 5 7 20 22
E %
0,4 0,6 0,4 2,2 2,7 4,4 9,6 13,0 7,4
Rl/Ro -
0,4 0,5 0,3 2,7 4,6 2,3 11,4 19,0 19,6
Mid speed
Vmed m/s
2,4 1,1 8,9 2,0 5,1 5,2 16 9,7 22
Time const

It is encouraging the Ro values for the transistors agree reasonably with the 0.2 K/mW junction to case, stated in their data sheet.

This figure shows what indicator meter scales would look like for the various alternatives. They are derived from the actually measured voltages and are expanded such that the range is from 0 to 20m/s. The figures at left is the U voltage for the scale left end (0) in percent of that for full scale (20). With the lower quality probes this offset is rather high.

These scales differ from the earlier theoretical examples since heating power and U are not directly proportional in the bottom five cases.

I find no obvious explanation to why the scale for the NTC bridge comes so very close to an ideal logarithmic.

Here two other probes, warning examples that were not very successful:

Rugged form with transistors potted inside an aluminum tube using Araldit two component epoxi. The increased masses and thermal resistances make the settling time several minutes. Sensitivity was reduced to about half, and it turned out bad in reproducing calibration. Looks nice, but is completely useless.
12V 5W
Halogen lamp. Cold resistance less than 1 ohm requires additional power drive.

5. Conclusions

The number of different circuits and probe designs investigated were all within reach for a handy amateur to construct. They were tested in the air speed range from 0.25 to 25 m/s (0.56 to 56 MPH). It appears that none of them can compete in performance with an orthodox hot wire anemometer, using a sub micrometer wire, such that will require special equipment to implement. The best approximation to this ideal was found in incandescent lamps with the bulb broken off. These stand out with response times being magnitudes shorter than the alternatives using transistors, diodes, or NTC resistors.

King's law describes the cooling of a probe to be in proportion to square root of air speed, at Reynolds numbers below 40. The present probes are large enough to render much higher Reynolds numbers, and experiments show the cooling effect is then rather proportional to speed. This decreases the speed range over which the probe is useful. This includes the lamp filament probes where effective diameter is enlarged because of their helical structure.

The various circuits and principles are by themselves not very important to measurement range. Instead, what matters is the thermal conduction properties of the probes. A quality measure was defined as the ratio of  thermal resistance in the connecting leads divided by the internal thermal resistance from active region to cooled surface. This is inherently much better for a wire than for a lumped device like a transistor, diode, or thermistor. One transistor probe improved a spectacular way when its connecting leads were changed from copper wire into a thermally less conductive iron alloy.

A measure was developed for the mid range air speed Vm where a probe is useful. This depends on how the cooled area relates to the thermal conduction within the probe. An essential conclusion is that Vm is inversely proportional probe element size. For instance, this means that a comparatively large sensor like a TO18 transistor is suited to measure low air speeds only.

In these circuits I used the TS462 dual operational amplifier, also available in single and quadruple versions, for battery max. 10V supply. The type is not critical at all, you can use almost any general purpose op amp type, like e.g. 741. For installation in my organ I settled for the thermistor bridge which appears a best choice combining performance, simplicity, and ruggedness.

6. References

Transistor anemometer:
King's law:
King, L V (1914): On the Convection of Heat From Small Cylinders in a Stream of Fluid: Determination of the Convection Constants of Small Platinum Wires, with Applications to Hot-wire Anemometry. Proc. R. Soc. London, Vol 90, pp 563-570.
Thermal conductivity:
Resistor Temperature Dependence:

2004-07-17, -20. 2005-12-23. Major revision 2006-01-28