How Quickly Do the Exhaust Gases Coming off a Fire Cool Down?

Prince Rilian

While trying to deduce the answer to a physics-related (and safety-related) question, I ran across an apparent paradox. I am trying to figure out how quickly the hot gases rising off of a fire cool down. From what I had heard from atmospheric science:
1. Air is a lousy conductor of heat.
2. Air parcels that are of different temperatures will not want to mix. They will want to stay separate.

Applying these rules would mean that the exhaust gases rising up from a fire would remain quite hot for some time. However, that does not seem to be in agreement with the observations that I had made of the temperature changes that occur as hot exhaust gases rise up from a small sized fire (in an 8 in x 8 in fire pit). The exhaust gases cool down within seconds, and there is no risk of receiving any burns if you were to place your hand a foot above the nearest flame (but it is still quite warm here). And at a distance of four feet above the fire, the exhaust gases are practically at ambient air temperature.

It looks like there is something that I had missed with my knowledge of physics here. Can anyone resolve this apparent paradox?

Darwin123

Your first hypothesis is correct. Air is a lousy conductor of heat (i.e., entropy). However, there are processes other than heat conduction that lower the temperature of the air in flames.
Your second hypothesis is incorrect. There is no rule in physics that forces air parcels of different temperatures to stay separate. There are two types of diffusion that mix gases in a parcel of air with the ambient air. There is molecular diffusion and turbulent diffusion. Molecular diffusion is rather small, working on the smallest scales of length and time. However, turbulent diffusion can be very large in a flame.
Turbulent diffusion is when random motions on a large length scale causes air to mix. Turbulent diffusion enhances the molecular diffusion. By breaking hot air parcels into smaller air parcels, turbulence increases the rate of heat transfer. Random motions, also called turbulence, increase with the difference in temperature between the air parcel and the ambient air. So turbulence diffusion in and near flames is actually very large.
Most theoretical studies of flames emphasize the role of turbulence. Because of turbulence, air parcels at different temperatures do "want" to mix. Because of turbulent diffusion, hot air parcels in your experiment probably didn't last very long. Cool air from the ambient environment replaced the hot air in the air parcel. Therefore, the temperature of the air coming from the flame cooled down much faster than your model predicted.
Adiabatic expansion is another reason the temperature in your experiment may have gone down a little faster than you thought. Convection carries hot air parcels upwards. Because of the drop in air pressure with height, the air parcel expands as it rises. There will be a drop in air temperature due to the expansion of the parcel. This is called adiabatic expansion. It may be small on the distance scale of your experiment, but it is there. However, adiabatic expansion is probably very small compared to turbulent diffusion.
Here is a link to a study where the experimenters also found that ignoring turbulence causes an underestimate to the rate of cooling of air parcels just above the flame.
http://fire.nist.gov/bfrlpubs/fire82/PDF/f82010.pdf
You may find the following quotes from this study reassuring.
Page 3
“(See the Appendix for a discussion of the e f f e c t of neglecting turbulent transport. Above the flame region (AT+-0) neglect of turbulence w i l l lead t o underestimates of the flux while in the flame the e r r o r s appear to be in the opposite direction.)”
Pages 23-24
“For the present any underestimate or overestimate of Q will be reflected in M, i.e, 1 - x should be increased or decreased. Hence a and (3 will vary by the square root of the change but m(z) and H(z) are functions of ci and f3 to the second power. The change will therefore be directly reflected in m and H. If for example the lack of accounting for turbulence leads to a the flame t i p H should be is correct then somewhere 15% overestimate of H then in figure 5 below reduced by 15%. If George's [Zl] contention above the flame tip the curves would have to be increased by 15%, this decreasing and increasing of the mean result, the mean result will overestimate in the flame region, pass through the 'correct value' and then begin to underestimate in the plume region and therefore on a height-averaged basis yield something close to valid results.”

Turbulence continues to be a fascinating topic in physics. Here is a link an a quote from a Wikipedia article on turbulent diffusion.
http://en.wikipedia.org/wiki/Turbulent_diffusion
“Turbulent diffusion flames
Using planar laser-induced fluorescence (PLIF) and particle image velocimetry (PIV) processes, there has been on-going research on the effects of turbulent diffusion in flames. Main areas of study include combustion systems in gas burners used for power generation and chemical reactions in jet diffusion flames involving methane (CH4), hydrogen (H2) and nitrogen (N2). Additionally, double-pulse Rayleigh temperature imaging has been used to correlate extinction and ignition sites with changes in temperature and the mixing of chemicals in flames.”

Prince Rilian

I should mention that my "second hypothesis"; i.e., that air parcels of different temperatures are unwilling to mix, has a corallary with weather fronts. I am going to need some more explaination here...

Andre

That's a matter of scale. On a small meter type scale like a great bonfire the turbulent air is mixing quickly. On a weather frontal scale in 100s of km, air masses remain isolated.

Darwin123

The scaling of air parcels is often governed by the Peclet number. In terms of the turbulence that we are talking about, the scaling of air parcels is controlled by the turbulent Peclet number.
The Peclet number determines the strength of advection processes relative to the diffusion process. If the Peclet number is large, then the importance of advection current is hypothesized to be bigger than the importance of the diffusion.
One can roughly say that an air parcel won’t mix rapidly with the ambient air if the Peclet number is large.
One definition of turbulent Peclet number is given as follows:
P_e=UL/D,
where P_e is the turbulent Peclet number, U is the advection velocity, D is the turbulent diffusion diffusivity and L is the characteristic length scale for the measurement. One can claim that the length scale of an air parcel under consideration is roughly the diameter of the air parcel. Of course, the decision of what determine a characteristic length scale is part of the art of scaling laws.
If,
P_e>>1,
then one can basically assume that the air parcel doesn’t mix very fast with the surrounding air. I hypothesize that this is the condition that underlies your second hypothesis.
However, if,
P_e<<1,
then that air parcel will mix with surrounding air rapidly.
For synoptic length scales, such as are used in meteorology, P_e>1. Therefore, the air parcels retain their integrity over a fairly long time. However, I believe that your flames were measured on a microscale where P_e<1. Therefore, your air parcels won’t keep their integrity on a very long time scale. Turbulent diffusion will dilute any hot air parcel in your experiment in a very short time.

Here is a Wikipedia link on the Peclet number.
http://en.wikipedia.org/wiki/P%C3%A9clet_number
“The Péclet number is a dimensionless number relevant in the study of transport phenomena in fluid flows. It is named after the French physicist Jean Claude Eugène Péclet. It is defined to be the ratio of the rate of advection of a physical quantity by the flow to the rate of diffusion of the same quantity driven by an appropriate gradient.”

256bits

Radiation to the environment plays a large part of the cooling of the hot gases. The amount of radiation is proportional to the fourth power of temperature as per the Stephan-Boltzmann law. You will note how quickly an open hot gas flame can lower in temperature in the thousands of degrees, and the spectrum of radiation can shift from mainly visible ( the flame) to infrared, before heat conduction and convection begin to play a more predominate role.

Prince Rilian

So this would mean that, for objects radiating an appreciative amount of light in the visible spectrum, that such objects cool primarily by radiation, and for lower temperatures, the cooling is done mainly through conduction and convection?

russ_watters

I doubt that. Because air is a lousy conductor of heat, the air can be quite hot without burning you for quite some time. For example, I've noticed that I can't touch the hood above my range, when cooking. Similarly, the air coming out of a hairdryer can be several hundred degrees without burning you quickly. Ehh, it doesn't even have to center around visible if there is enough surface area. I'm not sure if there is a tight cutoff, but I know for un-insulated steam piping at 300F, you need to consider both the convection and radiation. Radiation may even still be higher at that temp.

Prince Rilian

Well, those were my observations. And I believe that the reason that the flame hood above the stove becomes so warm is that 1) the flames are hotter, 2) the flame hood is one foot closer to the fire, and 3) the flame hood is in one fixed place and therefore has quite a long time to be heated up by any fires below it.