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A heating element should be operated below its maximum surface load (power per surface area.) For a given element, one might be tempted to say the surface load is the power given by Ohm's law divided by surface area. However, my gut tells me that's wrong. It's more probably the power density from the Stefan-Boltzmann law. 1.) Is that correct?

Clearly, the temperature of an element radiating into "empty space" will be less than the temperature of an element very near other black-body radiators (which have been heated by the element) - or even it's own structure (which may not be a straight wire.) In these cases, I suspect the surrounding black body matter will effectively "back heat" the element. Thus, the shape of the element and the arrangement of bodies in proximity will affect the surface load.

Obviously, this is non-trivial to calculate, but when using the radiative heat transfer equation for heat transfer between a resistive heating element at temperature, T1, and other black-body radiators at temperature, T2, (with appropriate constants, k1,k2.) ...

Power/Area1 = k1*(T1)^4-k2*(T2)^4 ...

I believe the temperature of the two bodies must be such that the radiative heat transfer balances the power of the resistive load, i.e.:

(V1)^2/(R1*Area1) = k1*(T1)^4-k2*(T2)^4

2) Is that sound, and is the term, k1*(T1)^4, the surface load of the element? (It may be much higher than the resistive power divided by the surface area!)

3) How is the surface load for a given element (in a given geometry and operational state, i.e. temperature,) measured in practice?

4) Manufacturers specify maximum surface loads for elements of different shapes and for different applications, yet the elements are made of the same material. Why does the recommendation vary depending upon shape and application?

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Maximum watt density of a heater, all other things being equal, is primarily determined by the technology of the heater. A swaged ceramic type is capable of a much higher watt density than a humble mica heater because the internal nichrome element is held much closer in temperature to the 'skin' of the heater.

If he internal element runs too hot for its diameter, life is compromised, sometimes dramatically. It's probably determined in a fairly empirical manner- heaters have been around for a very long time.

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I am not into thermodynamics very much but I think I understood your question.

You have to understand that ordinary engineering calculations are based on an meaningful ambient temparature. So, they never have absolute energy levels, but as that common ambient temparature doesn't varies much, some simplifications are inevitable.

1) They are the same, ohmic power is the additional entropy:

(V1)^2/(R1*Area1) = k1*(T1)^4-k2*(T2)^4

2) Yes, explained in the first paragraph.

3) I don't know. I guess that first it is simulated, the critical sections are determined and then get some scalar readings from a few points in the real experiment to assure the simulation.

4)

In these cases, I suspect the surrounding black body matter will effectively "back heat" the element. Thus, the shape of the element and the arrangement of bodies in proximity will affect the surface load.

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