Mathematical analysis of dome geometry

[mklingles]

I have been working on some calculations to determine the radiant heat pattern on the floor of the oven. This should shed some light on the effect of different dome shapes (e.g. high vs low, circular cross section or parabolic, tall soldier or no solider).

My calculations are ignoring any coals or fire, and are focused just on the pattern of heat radiated from the firebrick in the dome. I assume the entire dome is a uniform temperature. I assume the fire brick is radiating as a black body (Black body - Wikipedia, the free encyclopedia). The black body assumption simplifies the calculation and I believe is also correct. If anyone knows the absorption / reflection coefficients for fire brick at ~800degF please post. I believe the absorptance of fire brick is around 0.8. This means about 80% of the radiation that hits the firebrick is absorbed, and the other 20% is reflected. The absorbed radiation is re-radiated as black body radiation. So, I'm ignoring the ~20% of the energy that is reflecting off the surfaces inside the oven.

Briefly I'll discuss the calculation then show a few preliminary results. If anyone wants to discuss the math or review the calculation let me know and I'll start a separate thread.

Black body radiation is emitted from a surface uniformly in all directions. Therefore the energy of the radiation at a distance D from the surface is proportional to 1/D^2. So, to calculate the intensity of the radiation at a point on the floor of the oven you just have to add up 1/D^2 for every D formed between a point on the dome and the point on the floor. Then repeat this for every point on the floor. For those who suffered through multi-dimensional calculus you may recall this is a surface integral.

To get started I have not yet included the door opening in the oven. That is I assume the dome is complete and covers the space where the dome is. This is simplifies the math. Now that I have the calculations working, I will eventually go back and re-calculate with the door opening.

So, far I have just worked out the answer for two cases: 1) a hemispherical dome with 42" diameter and therefore a height 21", and 2) a spherical cap (Spherical cap - Wikipedia, the free encyclopedia) dome with a floor diameter of 42" and a height of 18" (this matches my oven)

The 2 images below show the intensity of radiation on the floor of the dome. The conclusion is that the difference between a 21" height and a 18" height is very negligible (I did all this math for that !?!?)





The final picture shows a cross section plot across the two images above. I think the most interesting thing I've learned is that the intensity is a bit more then double at the edges of the oven. Also in a 42" oven there's a good 20" middle section with nearly uniform radiation, but beyond that the intensity starts to increase quickly. Also here you clearly see that 18" vs 21" height didn't make much difference.

[Faith In Virginia]

Re: Mathematical analysis of dome geometry

Nice start to your calculations. It's cool to see that the radiation pattern on the floor from one height to another is not significant.

I would like to see the affects of heat absorption (food) to the equation. The 18" to 21" domes have different surface areas. So what affect if any would that have on the food?

I would love to be able to help with this mathematical analysis but that is not my strong suit. But I will look forward to your future posts. I find it quite interesting and you explain things well. Thanks

[brickie in oz]

Re: Mathematical analysis of dome geometry

Who cares as long as the oven works and can cook stuff?

[brokencookie]

Re: Mathematical analysis of dome geometry

Who cares as long as the oven works and can cook stuff?

Turn over a home made oven and many times you will find an engineer. Playing with the math just extends the amount of enjoyment they get from the oven. The rest of us have our curiosity buttons pushed and we get to have fun watching all of the data run by. Some of the studys even lead to real world improvments to future builds. Sit back , relax and let the math boys have fun

[Faith In Virginia]

Re: Mathematical analysis of dome geometry

I care and I find it interesting. If you don't like what your watchi'n on dat TV...turn the channel.

[brickie in oz]

Re: Mathematical analysis of dome geometry

Here are my scientific findings on wood fired ovens.

If its too cold, add fuel.
If its too hot, wait a while.

[RTflorida]

Re: Mathematical analysis of dome geometry

Not my cup of tea either. My one and only question - wouldn't the composition of the firebrick or refractory material have a huge impact? Considering there are literally hundreds of manufacturers, all with several different grades......no two ovens are alike.
I don't know much about this, but I do know different firebricks/refractories have different absorption rates, so assuming a .8 absorption rate is just that - an assumption.

One other question - how does any of this make me cook better? Really, cooking with fire is pretty basic. There are alot of specs that go into any gas or electric oven......none of which can help someone become a better cook.

RT

[mklingles]

Re: Mathematical analysis of dome geometry

@Brickie Yea the need to answer questions is more like a disease then a good thing for some of us scientist / engineer types.

@RTflorida. Will it make you cook better? No. But, it keeps me out of trouble . Yes certainly lots of variation in the bricks used. This kind of analysis mostly lets us try to answer the question does the dome shape matter. After you've all ready built your oven none of this matters.

@Faith. I'll compare 36" and 42" dome. I don't have the plots ready, but the intensity of radiation in the center of the oven is very similar (assuming both are hemispherical). The biggest difference from the surface area of the larger dome will be how long it holds it's heat, but both will cook the same in the center portion of the dome. Studying how the pizza takes heat from the floor of the oven would be interesting, but that's conductive heat transfer not radiative. (Which is harder to model).

[mklingles]

Re: Mathematical analysis of dome geometry

Here's the cross section comparison for a 42" hemispherical dome (21" height) and a 36" hemispherical dome (18" height"). The intensity at the center of the floor is the same for both domes. Because of the symmetry it would be the same at the center for any size hemispherical dome.

[Lburou]

Re: Mathematical analysis of dome geometry

mklingles, you have peaked my interest with these data. The calculations suggest a flaw in the asserted conclusion that a shorter dome has superior pizza cooking performance to a traditional hemispherical dome.....The calculations also show little or no difference in floor temperature!!

The shorter dome is more difficult to build, must have a shorter door opening, and in general limits ease of use for anything but pizza and maybe some breads. If there is no REAL difference in the temperature pattern of the two designs, then your data has definite substantive significance.

My goal is an oven to cook pizza, bread and roasts. The taller door is important to us. I plan a 39.25 inch oven floor and a 22-23 inch dome and a door 63% of the dome in height. Any thoughts? Would your calculations differ with the shape of the oven dome I propose?

A calculation, or any research, is more valid when the variables within the calculation are controlled and equal when comparing one unit (oven height in this case) to another. Controlling the variables makes the data comparable from one condition to another. Generalizing the data to our oven builds could open the flood gates of debate about the best performing oven designs.

I'm cutting bricks for my dome....And, very interested in these data.

[Polo]

Re: Mathematical analysis of dome geometry

mklingles, you have peaked my interest with these data. The calculations suggest a flaw in the asserted conclusion that a shorter dome has superior pizza cooking performance to a traditional hemispherical dome.....The calculations also show little or no difference in floor temperature!!

The OP's first post states that his figures do not take a fire or live coals into account. It seems that would negate the findings in regard to pizza cooking, which is done with a fire and live coals. As a matter of fact, I would think that the whole benefit from a low dome would result from the reflective nature of the dome during pizza cooking, which of course includes a fire and hot coals.

The mathematics are interesting, but real practice may produce different results.

[mklingles]

Re: Mathematical analysis of dome geometry

@Lburou: your dimensions are close to the 42" w/ 21". It's going to take me some time to get things coded up for other then spherical surfaces.

From everything I've read here, I've concluded that dome geometry isn't particularly important. Nobody has posted that they built an XYZ dome and it cooks poorly. I'm doing the math to back up the "dome geometry doesn't matter" conclusion.

Insulation matters. Door opening for proper air flow matters.

[mklingles]

Re: Mathematical analysis of dome geometry

@Polo. Certainly this is an analysis of only one factor in how the oven works for cooking. However, I believe it is the dominant factor in the effect of the dome shape on the cooking process.

[Lburou]

Re: Mathematical analysis of dome geometry

The OP's first post states that his figures do not take a fire or live coals into account. It seems that would negate the findings in regard to pizza cooking, which is done with a fire and live coals. As a matter of fact, I would think that the whole benefit from a low dome would result from the reflective nature of the dome during pizza cooking, which of course includes a fire and hot coals.

The mathematics are interesting, but real practice may produce different results.

Polo, I agree. But, that fact does not negate the questions the data raise, or deny the significance of the data. Therein lies the primordial clash between theory and practice. Sorting truth and fiction is what the scientific method does. Calculations are part of the scientific method. I worked in Military acquisition a few years and, believe me, I understand the balance between theory and practice.

@Lburou: your dimensions are close to the 42" w/ 21". It's going to take me some time to get things coded up for other then spherical surfaces.

From everything I've read here, I've concluded that dome geometry isn't particularly important. Nobody has posted that they built an XYZ dome and it cooks poorly. I'm doing the math to back up the "dome geometry doesn't matter" conclusion.

Insulation matters. Door opening for proper air flow matters.

Thanks for the thoughts mklingles, I feel better about my plan to raise the dome height now. It will also increase the mass of the dome by 28 more half bricks.

[Tscarborough]

Re: Mathematical analysis of dome geometry

Now do the analysis for a barrel vault, please.