Temperature Gradients of Independent Radiating Bodies within A Mixture.

[Kmenex]

I am interested in describing the heat transfer and temperature gradients within a mixture of independent radiating bodies. States like this occur in many systems and i have been having a bit of trouble modeling this simple scenario. An example of a system is charcoals within a BBQ, and also glass beads within a water bath, or even nano scale hetero junctions.


Of particular interest to me at this time is to start with the simple model of charcoals.
A quick note is that this is probably a "suspension" and not a mixture because i am only considering one type of body and it surroundings are the fluid air. Charcoal will be called bricks.

First of all, I will start the model where the bricks are all at the same temperature and that temperature will be high enough that we can consider the bricks already ignited. This is the initial state. In dynamics we can favorably assume that heat transfer between bricks will reach an equilibrium with temperature gradients throughout the volume.

Second, the shape of the mixture is very important in determining how heat is transferred from the system and how temperature varies within the volume of bricks. Thus we can naturally assume that the "hottest" bricks are the ones which are closest to a large mass of radiating bricks.
Because of this the model will be confined to the geometry of a cylinder which is packed full of bricks and open on both ends. The rigid wall of the cylinder will be a moderately conductive material such as sand, but not metal.

Third, loss of surface area on the coals due to destructive reaction will not be considered at this time even though it plays a critical role in determining how heat flows within the model... This is for simplicity as i feel allowing the bricks to vary their mass in time will make things very complex, too complex for the application i have.

Now that these three conditions of the model have been stated let us consider more about the phenomena taking place.

To my knowledge the release of potential energy within charcoal is accomplished by a combustion reaction. Some how the coal is able to burn with a very limited supply of oxygen. I assume that the charcoal works by having a large volume of slow combustion material rather then a small volume of fast combustion. When you blow on a brick it seems to get hotter and decompose faster, and my intuition tells me that this is because you are "changing" The environment around the coal to create conditions favorable for faster combustion (remove the waste, bring in the oxygen).

So in this cylinder i have the hypothesis of what will happen.

The first is that the center of the cylinder volume will have the hottest bricks due to being surrounded by the most radiating bodies. However this could prove false because the amount of oxygen reaching the center might be different enough from some other position within the mixture which will have a higher temperature.

The second is that the ends of the cylinder will be hottest due to having an environment with more oxygen.

Any help with this?


-Kyle

(p.s after this then i move onto modeling with air blowing in from one end.

 

[mikeph]

I think you should start with a lot more simplifications than you are currently using.

Ultimately I think it will boil down to a model where rate of combustion is proportional to (a) concentration of reactants multiplied together (oxygen density, coal), and (b) some exponent of the temperature.

This is what the combustion models I know of are based on. The theory I studied involved mixtures of gaseous reactants only though, so you immediately have a big complication wit modelling the combustion between a solid and gas, ie. on a surface. Try to model an infinite wall of coal burning in 100% pure oxygen on one side? If you model this accurately, maybe you have a research paper you can publish!

 

[Kmenex]

MikeyW,

Thanks for the Reply.

Yes, what you suggest is truly what is needed and i figured that the model i was proposing was somewhat capable of being set to simple parameters, I.e make the radius of the cylinder very small. I was trying to short-cut the problem by ... basically outsourcing it to someone else... LOL... However on further reflection i see that my model assumes that information about individual coal bricks is readily obtainable. This is not the case, as my experience has proven, and in truth the only articles / examples i can find concerning similar scenarios involves heat and temp gradients in gases, like you had researched (was that university level thermodynamics by chance?).

Quick note, when i say coal here i mean CHARCOAL and not the more oily coal used in industrial production of energy.

IN pursuit of a simpler model i think it would be practical to first consider a single coal brick burning in various environments, I.E air or oxygen as you propose.. Here is the main problem i have, i don't know if the combustion rate near the center of the mass of the coal if different then upon the surface, and i have no way to measure as my thermometer capable of 1000C measurement is broken somehow.. So in essence what is needed is information about how heat flows within an individual brick. Hell, i could do with a chemical description of the coal first since. I guess in these scenarios it is best to bring model and experiment together.

One other thing, when it comes to individual masses of char-coal the form of the mass is important in determining the combustion/heat flow. Most bricks come in a oval type shape.

Well, enough rambling, i best do some measurements and slap-dash together some equations. I will need help here and i hope that no one minds if i continue to submit questions and comments concerning this topic.

-Kyle

 

[mikeph]

Yeah, it was a late undergrad/early graduate course on the mathematics of combustion theory, as I understand the field is very upcoming and exciting with many of the important results being derived only a few tens of years ago. Arrhenius' law is a good starting point for modelling the combustion, but like you said, it is a good idea to see how a single piece of coal will burn. My instinct is that it will only burn on the surface, and there will be some sort of air flow which sees the burnt mixture rise in a column above the coal, and unburnt oxygen being drawn up from below, so basically a flow of gaseous fuel into the flame region from one side, and exiting on the other side. Then the flame region is responsible for (a) lowering the concentration of oxygen in the flow and (b) causing a dramatic rise in temperature. I don't know how the charcoal will actually burn, though- perhaps you could exploit the idea that a lot of oxygen reacts with a tiny bit of carbon to "assume" the charcoal is unchanged over a reasonable time period?

I think a physical understanding of where the combustion happens is key- and I don't know this!

When we learned it, the flame region was a well-defined region over which the temperature gradient was big, and we used asymptotic analysis to treat it as a boundary layer where all the meaningful chemistry happened. The geometry of the problem was crucial to understanding this, so I think the first thing is to work out where exactly the flame is in the coal.