If I have a tungsten heating element and I know its resistivity and dimensions of the material how do I determine what current and voltage I need to make the tungsten wire reach a certain temperature?
Do these equations have a name?
Thanks.
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1$\begingroup$ Do you know how its resistivity changes with temperature? $\endgroup$– probably_someoneCommented Feb 24, 2018 at 0:17
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$\begingroup$ Check out Joule heating $\endgroup$– pentaneCommented Feb 24, 2018 at 1:20
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$\begingroup$ yeah lets assume resistivity is a function of temperature. However I do not have the function specifically for tungsten. $\endgroup$– axawireCommented Feb 24, 2018 at 3:40
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$\begingroup$ The environment is also a factor. A tungsten element in still air surrounded by mirrors will get much hotter, for example, than the same element passing the same current under forced airflow. You need to calculate the outgoing conductive, convective, and radiative heat transfer and perform an energy balance using the dominant mode(s). $\endgroup$– ChemomechanicsCommented Feb 24, 2018 at 5:07
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2 Answers
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First, it will be easiest to first work out power through the filament: one can work out the temperature of the tungsten lamp as a function of electrical power passing through it using the Stefan–Boltzmann law, by which power output from a black body is: $$j^* = \sigma T^4$$ where $\sigma$ is the constant of proportionality or Stefan-Boltzmann constant, and $T$ is the thermodynamic temperature (absolute temperature in K) of the emitting body (Ref). We can approximate the tungsten filament as a black body (Ref). The answer quoting the Stefan-Boltzmann law here is perhaps also a useful reference (Ref).
Secondly, one can find out the current that must flow through the filament. To do this, find the electrical resistivity of Tungsten in $\Omega\cdot m$ of tungsten at the target temperature (Ref). Next, use it to calculate current through the filament via a variant of the electrical power law equation $P = VI$ (Power equals Voltage x Current) for electrical circuits: In that variant of the equation, $P$ is electrical power through the tungsten filament, $R$ is the resistance across that filament, and $I$ is the current flow through the filament:
$P = I^2R\qquad$ or $\qquad I = \sqrt{P\over{R}} $
Finally, to calculate voltage required across the tungsten filament, use Ohm's law:
$V = IR$.
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A tungsten filament (as in a light bulb) is operated in an inert atmosphere, or a vacuum, in order not to oxidize. If it matters, one could experimentally measure the tungsten wire temperature and make a closed-loop temperature control for it.
Because the heat losses are sensitive to the pressure of the surrounding gas, a common apparatus is available, mass-produced, called a 'thermocouple vacuum gauge'. The calibration curve for such a gage [see tabulation here][1] is the relationship of temperature achieved (as reported by a thermocouple) as a function of (low) air pressure. That's the response with a constant-current excitation of the filament, so it can only be directly related to pressure changes, not variations in electric heating current.
That apparatus, however, has all the connections you'd need to experiment and determine a formula that closer fits the current-versus-temperature requirement.
