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 Anders Larsson April 15, 1999 07:33

Hot rod in transverse air flow

Hi everyone!

I would like to know if there exists a simple (analytical) and accurate formula for the heat loss of (or heat flux from) a hot cylinder in a transverse air flow as a function of the air flow speed. Typical dimensions: rod diameter 10 mm, air flow speed 0-100 m/s, rod temperature 5000-10000 K. Thanks for any hints.

 Patrick Godon April 16, 1999 10:05

Re: Hot rod in transverse air flow

Hi there. I am just going to give a first hint which is what I think of the problem (as a Physcist). First if I look at the 'burning' rod in the vacuum (simplification) it will emit as a black body and the power radiated (per unit area) is just sigma*T**4, where sigma is given by sigma=a*c/4, a is the Stefan-Boltzmann Constant and c is the speed of light (so sigma=5.67/10**5 erg/cm**2 /sec/deg**4). Now in this range of temperatures (5-10x1000K) the peak of the spectrum is in the optical mainly (roughly like the surface of the sun at 6000K). The air (and the atmosphere) is transparent to optical light, so the medium in this range of frequencies is optically thin. For a part of the spectrum (in the ultra-violet mainly) the air is optically thick (all this depends of course on the size of the problem you are looking at). Therefore, once you introduce air in your problem you will have some of it heated due to radiation. Now, if you look only at a standing medium (v=0), for sure you will have the air heated nearby the rod and convective current going upwards. Once you have a wind (v.ne.0) the rod will loose heat also by conduction at a higher rate. So to conclude it does not seems that you have a simple analytical answer to the problem. The simplest estimate is first the emission assuming a black body, however once you have a wind the rod will probably not be isothermal anymore, and it will cool not only due to radiation emission but also direct contact with the air (condution). There is also an important unknown: the rate of heating of the rod (or is the rod just supposed to cool down with time?). I hope this give you a fist 'shot' at the problem. Cheers, PG.

 Anders Larsson April 19, 1999 04:47

Re: Hot rod in transverse air flow

Thanks Patrick. Maybe it would be appropriate to be more precise about my problem. I am modelling a free burning arc in air (e.g. a lightning channel during its continuing current phase) when attached to a moving object (e.g. an aircraft). Regarding the radiation losses, black-body radiation is not applicable; the main radiation losses are due to retardation and recombination radiation. Further, the radiation losses are small compared to thermal conduction ones for temperatures below 10000 K.

Often, a good approximation of an arc channel is to assume that it consists of a hot and impermeable core with a certain radius where the energy losses due to thermal conduction dominates. A transverse air flow will increase these losses and I am looking for some estimate of this increase.

The power input into the channel is given by the lightning continuing current (of the order of 100 A) giving a power input per unit length of the order of 10e5 W/m. But a decaying channel, i.e. without any current, is also of interest.

I have also solved the conservation equations for mass, momentum and energy in 1D (rotational symmetry) but introducing a transverse air flow would destroy this symmetry...

 Patrick Godon April 19, 1999 09:28

Re: Hot rod in transverse air flow

Yes, if the heating of the rod is due to a current then the situation is a bit different than what I first thought. The excitation of the electrons and their recombination has to be taken into account. If you assume that only the thermal conduction is important for the energy loss, then that's the only thing that needs to be resolved. In one dimension did you use some kind of mixing length theory for the convection? You could use maybe the same for a non-axisymmetric situation where you would change the parameters of the problem to fit the new situation. e.g. you could look at two extreme cases where you have no wind at all in one situation and the other where the wind would be axisymmetric somehow; and then average between the two to get some input parameters for a more realistic case. For sure the problem has no simple analytical exact solution, but an approximation to the problem (negelcting second order terms) must have a simpler form that could be resolve analytically. Cheers, PG.

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