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I would like to know how I can estimate/calculate the temperature of a heated filament. For example, the filament (rhenium) is 0.2mm in diameter and heated with a voltage of 6.4V and a current of 2.72A (i.e. power=17.4W.)

My second question: How does the temperature of the filament depend on the filament current?

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    \$\begingroup\$ Consider a disappearing filament pyrometer. \$\endgroup\$
    – jonk
    Jun 29, 2020 at 8:43
  • \$\begingroup\$ Please update your question to include more information as follows: will the filament be in open air, or a liquid or what? Second, how long is the filament? Third, is the filament glowing, and if so, subjectively what color? Third, what are you using the filament for, if you can say. \$\endgroup\$
    – mkeith
    Jun 29, 2020 at 20:15

3 Answers 3

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A simple way to measure the temperature of incandescent surface is to compare the colour of the filament of a known lamp.

Adjust the current through the lamp until its filament disappears into the sample.

Approximations of the tungsten filament temperature may by accurate enough from calculations. Calibration and temperature profile of a tungsten filament lamp suggests approx 2% accuracy.

The open access paper "Analytical expressions for thermophysical properties of solid and liquid tungsten relevant for fusion applications" https://doi.org/10.1016/j.nme.2017.08.002 is a goto source of information.

https://en.wikipedia.org/wiki/Disappearing-filament_pyrometer enter image description here

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  • \$\begingroup\$ Damn. @jonk beat me to it. \$\endgroup\$
    – D Duck
    Jun 29, 2020 at 19:30
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    \$\begingroup\$ No problem. It's nice to see someone using a disappearing filament pyro. Side note: women are better able as females can more finely discriminate among color variations. This isn't just a sick joke. It's true. One of the places I saw this in regular action (aside from using these pyros) was in the case of photographic color-checking back in the day when photolabs converted 35mm film into pictures. The test kit to evaluate if someone could be trained as a color checker was almost never passed by males (I never found one in years trying), only females. Almost all color checkers were women. \$\endgroup\$
    – jonk
    Jun 30, 2020 at 9:10
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It sounds like you want to estimate the temperature of the filament before you even build it. That is, you want to calculate the temperature without making any actual measurements.

The difficulty we have with questions like this is that you really want to solve a heat transfer problem, not an electrical problem. You know how much power you are dumping into the filament and you know that almost all of this power will be converted to heat.

To know the temperature of the filament you need to model the flow of heat away from the filament to the surroundings. You need to know the thermal conductivity of the electrical connections to the filament and of any physical supports for the filament. You need to know about the airflow and ambient air temperature. At high temperatures, or in a vacuum, you would also need to calculate the power that is lost through radiation to the surroundings. I can't help you with any of that...you might need a mechanical engineer or a physicist.

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  • \$\begingroup\$ Actually, people who design hot wire cutters have rules of thumb for this. However, the length of the filament needs to be known/specified. \$\endgroup\$
    – mkeith
    Jun 29, 2020 at 20:12
  • \$\begingroup\$ Obviously in the hot wire cutter case, the temp is based on heat transfer in air. When actually cutting it cools down somewhat due to heat tansfer to the work piece. \$\endgroup\$
    – mkeith
    Jun 29, 2020 at 20:13
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The method you use will depend on the accuracy you want. As you've used the word 'estimate' in your question title, I presume that you don't need standards lab accuracy. Adding an accuracy specification, or adding the purpose of the equipment to your question, may get you more relevant and focused answers.

The simplest way to estimate is to measure the resistance when cold, and at temperature. This depends on having a graph of resistance versus temperature.

This data is for pure rhenium from Kaye and Laby at the npl.co.uk via the wayback machine

temp K  resistivity (10^-8 ohm.m)

  78.2   2.62
 273.2  17.2  
 373.2  24.9
 573.2  39.7
 973.2  63.5
1473.2  84.4  

There will be errors from the purity of the element, and the fact that the measured resistance is an average of the cool bits of element near the mounts, and the hot bits in the middle. There may be a steady drift in the diameter and so resistance of the element due to degradation of the element, though this should be recoverable by a new cold measurement. There may be a drift in the resistivity of the element material, due to reaction with the atmosphere it is operated in.

The main alternatives are infrared thermometry, and spectroscopy. The latter may be possible fairly cheaply with a diffraction grating and a camera.

For the roughest and dirtiest way, look at the colour of the filament, and interpolate from colour description versus temperature. You will find many tables of colour versus temperature online, generally from artisans who weld, operate kilns, heat-treat metals. Spectroscopy is just a way to instrument this process and make it objective rather than subjective.

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    \$\begingroup\$ @Mattes see my additions to the answer. Given that you already have V and I at temperature, you need only make one cold resistance measurement, and then interpolate from the figures I've provided. \$\endgroup\$
    – Neil_UK
    Jun 29, 2020 at 9:05
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    \$\begingroup\$ Dear Neil, thank you again. The full truth is: I got a power supply for that filament, which directly displays me the voltage I apply together with the current flowing. I turned the voltage from 0 to 7 V and received values for the current between 0 and 3 A. Of course if I plot this, I got the ~R dependency. I didn't measure the resistiance at cold temperature yet, good idea. If I understood you correctly, I can plot the cold resistance value and one of my hot values and then interpolate between those with your data? Thank you! \$\endgroup\$
    – Mattes
    Jun 29, 2020 at 11:28
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    \$\begingroup\$ @Mattes Many customers tried (and failed) to use extremely high precision current drive, in the false belief that doing so would mean they were holding the right emitter temperature. Those customers would replace their USD80k emitter heads every two months or so. Other customers used rigorous methods, including regular use of disappearing filament pyrometry as well as other more expensive, regularly applied calibrations, to maintain a true knowledge of temperature. Said customers replaced their emitter heads every two to three years. Ten degrees too hot meant rapid degradation. So it mattered. \$\endgroup\$
    – jonk
    Jun 29, 2020 at 17:28
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    \$\begingroup\$ @mkeith So, for example, say you get 2 years of life if you can hold \$2550\:\text{K}\$. You may only get 2 month's life if you let that drift up to \$2570\:\text{K}\$. You'd get better beam intensity and faster lithography write times (nice to get work done faster) but you'd also pay for it in lifetime. Too low, and write times are too slow. So there was always a push towards higher temperatures to get the speed. But there was a sweet spot that gave you the "best value" approach. And this was very hard to hold over time. People spent a lot of money on this, you can be sure. \$\endgroup\$
    – jonk
    Jun 29, 2020 at 20:31
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    \$\begingroup\$ @mkeith I love it when I see tables with values like "\$1473.2\:\text{K}\$!" Do you have any idea how expensive it is to get accuracy of 50ppm Kelvin at that temperature looking at an extended and oddly shaped object from a distance and with uncertain emissivity, likely through various materials like quartz? If only current could replace that work for you!! (Lots of expensive temperature solutions and consulting would have no business, I guess.) We can place it into a situation, though, where the emissivity is near 1 (integrating sphere.) Which is helpful at times. \$\endgroup\$
    – jonk
    Jun 29, 2020 at 20:38

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