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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?
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
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.
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.
