Temperature Calibration

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The Basics

A man with one watch, know what time it is. A man with two watches are never really sure.

This goes for temperature measurements as well, in my collection I have 5 ( mobile ) sources of temperature, only randomly they show the same. Time for some poor-man's temperature calibration :-)

First, some basic facts that are easy overlooked:

  • All measurements are wrong
  • Nothing measures the temperature of the air/surroundings. Everything measures its own temperature.
  • Different sensors and mounting means time lag when temperature in the surroundings are changing.


Y = Ax + B

Good old Y = Ax + B. We meet her first when we were 8 or 9 years old, and she has been with us ever since. Sometimes we see her and admire her, sometimes we hate her, most of the time we don't notice her, but she is always there.

Y = Ax + B is a very crude and linear error correction, but for this method, we have nothing better as we can only obtain 2 data points we can somewhat trust. It has some drawbacks, naturally, one of the most obvious is non-linearity of the item we are correcting, we simply have no idea the response of the sensor is linear, exponential ( to X degrees ) or shaped like 80's beloved TV-animal, Flipper.

Other issues can arise, lets say we have a measurement at value "0" and see an offset of "+5", so B = +5. But then we measure at value "100" and read "105", would that make A = (100/105) = 0.95 or is it only an offset so A = 1.00 ? I simply don't know, nobody does, as it would depend on the situation, experience and access to a 3rd measuring point witch we don't have, hence this is the poor-man's method. Good thing is the poor man can't afford 2 watches.


Known Points

As you may have guessed, we are going to calibrate our thermometer at 0 C and 100 C, as those are household items that can be re-created to some level of confidence.

There are many tutorials out there for making a proper ice bath, as the 0 C calibration is called. Google it :-) Important notes are again settling time of sensor and stirring the icy water to make sure it is at 0 C.

100 C is much easier to achieve. boil a pot of water, an electric kettle will be sub-optimal, as it switches off when boiling. Be sure not to touch the bottom or sides of the pot with your sensor and be sure to let the sensor stabilize in the boiling water.


Results

From the above 2 points, I got a boiling temperature at 97.4 C and a freezing point at +1.2 C. Thus the correction of the thermocouple becomes TC cor = (97.4/100)x +(1.2) => TC cor = ( 0.974 * measured C ) - 1.2 C

I gather results from my 5 meters:

  • "Sanofi External" and "Sanofi Internal" being a cheep indoor/outdoor sensor from one company "Sanofi" - actually that is a medical company and the meter is so cheep it is a OEM + sticker = give_this_to_your_most_worthless_clients.
  • "Biltema internal" - again, cheep meter from Biltema, no external measurements as I busted the external sensor some time ago :-)
  • Blurry Duck - she needs no introduction to veteran readers
  • Biltema DMM - this being a Biltema DMM 1000p with its delivered Type K thermocouple - this is the one we are "calibrating".

I measured a few times, with at least a few minutes between measurements. The naming speaks for itself, "Deviation" is ( max - min ) / 2, the distribution is somewhat symmetrical so why make it complex.

All 5 sensors were placed within 20 cm of each other with no air-draft felt, they *should* read pretty similar...

Temperature calibration results.png

From this, a graph was born.

Wow much lines, such colours - yes, well the important bits are this: red = average of all but TC and TC corrected. The brown one at the top is the Thermocouple before correction applied, is is clearly an outlier, but when corrected ( thick black ) it becomes more centred in the data-set and follows the average computed from the 4 other measurements. The 4 are presumed to be NTC's of dubious past and quality, but I do find some conciliation in the fact that they on average agree with our freshly non-traceable calibrated thermocouple.

Temperature calibration graph.png


Uncertainty

Oh accuracy, thou are a heartless bitch - that is why we invented uncertainty, the frisky next-door entity we work on and apply when accuracy is being a PITA.

A complete budget for the uncertanty of all the above is rather pointless, as we have no specs. of any sort for anything but the DMM + thermocouple. We can however stick to one decimal point and see the specs for the DMM + TC says "+/-3% + 5 counts" - the resolution of the display in the range used is 0.0 C but what the range really is, apart from within 0 and 100 C, and if the 3% is of the reading or the range, we don't know. Nor do we have any info on long term stability of the meter and so on.

To sum it up: we don't know shit, yo!.

Time to make assumptions then:

  • Home-made ice-bath: +/- 0.1 or +/- 0.2 C
  • Home-made 100 C: depends on air-pressure and purity of water, again I would say +/- 0.1 or +/- 0.2 C
  • DMM TC: lets say best case the 3% is measured value, and the 5 counts = +/-0.5 C

Warning: Math ahead.

At 0 C, we would be determining the "B" offset of Y = Ax + B with an accuracy of 0.5 C from the DMM and 0.2 C from the ice-bath on top of that, we have the 0.1 C resolution of the meter.

Combined and with an assumed square distribution, this gives us a total uncertainty of:

Sqr( (0.25/sqr(3))^2 + (0.1/sqr(3)^2 + (0.05/sqr(3)^2 ) = 0.16 C, rounded to 0.2 C due to display resolution.

Notice how the end result is less than half of the +/- 0.5 C, this is due to the conversion between rectangular distribution to normal/Gaussian distribution, this also goes from 100% ( rectangular ) to a confidence level of the result at 95%. Same goes for at the boiling point, as the estimates of contributions are the same ( how lucky was that!?) we get a result of 0.2 C as well.