Mathematical analysis of dome geometry

[mklingles]

Re: Mathematical analysis of dome geometry

I'll add barrel vault to the list. Warning - it will take some time as I only get a few hours a week to play.

[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.

Now do the analysis for a barrel vault, please.

I think itll be the same.

[Tscarborough]

Re: Mathematical analysis of dome geometry

I know it will.

[brickie in oz]

Re: Mathematical analysis of dome geometry

Ok, Ill shut up now...

[GianniFocaccia]

Re: Mathematical analysis of dome geometry

Thanks for the analysis, Michael. I put this study right up there with PizzaBob's heat flow analysis. In real terms, if correct, what this does is help remove some of the variables in oven design and construction, ie: kebwi's and drseward's compound angle indispensible tool. Relying on the fact that the dominant property in WFO-cooking is radiative, this would indicate there is no penalty for constructing a truly hemispherical dome, which is easier and more straightforward to build than a spherical cap, which is quite popular, given the number of builds with full-length soldiers.

[Neil2]

Re: Mathematical analysis of dome geometry

If they are both spherical, dome height will not matter much. If it is elliptical you will find a low dome spreads the heat better.

[mklingles]

Re: Mathematical analysis of dome geometry

It's true the elliptical dome does appear to spread the height a bit better, although I don't quite have the apples to apples comparison yet.

This shows a 42" hemispherical dome (h=21), a 42" spherical section (h=18) and a 42" ellipsoidal dome (h=15.5). The ellipsoidal dome does have slightly more even heat across the center section, and a bit more intensity for being a bit lower. I'm not sure the data indicates it's a significant difference.

I'll compare a 15.5 inch hemispherical section to the ellipsoidal dome next week.

[Wiley]

Re: Mathematical analysis of dome geometry

All in all one can probably model a theoretical WFO of any given construction. And I like what you have done, However, IMHO assumptions beyond broad generalizations are open to question. Still a fun exercise, and the graphs are cool in the fact they show no theoretical benefit from the lower dome shape. Empirical results may show something different. And of course getting followers of one type of WFO to give up preconceived ideas about one style being better than another for any stated purpose strictly on the basis of theoretical data/analysis might prove difficult.

Some imperical data would be nice as I read of people firing their WFOs for hours and I wonder at what point are they just wasting wood? Certainly there is an optimal time to fully saturate a given WFO. Burning more wood in an effort to increase that saturation is probably a futile exercise. I don't think it's a perfect anology but if one likens it to a battery one can change the battery until it is fully charged no more. Further charging results in waste of charging energy as well as a degradation of the battery (damage to electrolite). I don't know if over firing a WFO causes damage.

This is probably just an issue of someone learning their WFO, still I cringe when I read of someone firing one of our domes for hours to get it to pizza temperature.

Bests,
Wiley

[Lburou]

Re: Mathematical analysis of dome geometry

All in all one can probably model a theoretical WFO of any given construction. And I like what you have done, However, IMHO assumptions beyond broad generalizations are open to question. Still a fun exercise, and the graphs are cool in the fact they show no theoretical benefit from the lower dome shape. Empirical results may show something different. And of course getting followers of one type of WFO to give up preconceived ideas about one style being better than another for any stated purpose strictly on the basis of theoretical data/analysis might prove difficult. Some use insulated oven doors, some do not.

Some imperical data would be nice as I read of people firing their WFOs for hours and I wonder at what point are they just wasting wood? Certainly there is an optimal time to fully saturate a given WFO. Burning more wood in an effort to increase that saturation is probably a futile exercise. I don't think it's a perfect analogy but if one likens it to a battery one can change the battery until it is fully charged no more. Further charging results in waste of charging energy as well as a degradation of the battery (damage to electrolyte). I don't know if over firing a WFO causes damage.

This is probably just an issue of someone learning their WFO, still I cringe when I read of someone firing one of our domes for hours to get it to pizza temperature.

Bests,
Wiley

Points well taken Wiley

We see a LOT of member reports about time to a clear dome, and oven temperatures after 24, 48 and 72 hours. But we are most certainly comparing apples to oranges. Several of our best and most active members here have barrel ovens. Some have pompeii ovens built from early designs with varying insulation in the hearth and dome.

We can compare apples to apples if each reporting member would include pertinent details of their build in their signature. Much like board members on car forums do (they will tell you everything about their car and what they did to make it faster). I propose we do the same here at FB.com. It would tell us a lot about the performance enhancements each finds successful.

Below, in my signature, you see my attempt to start the trend
I cite the FornoBravo pompeii plan V2.0 and note extra insulation under the oven floor, and any other departure from the plans.

[Wiley]

Re: Mathematical analysis of dome geometry

Well, in the clear light of morning my analogy is poor. A battery is designed to work with a nominal voltage, 12 volts. Additional current will not turn that battery into one with significantly greater voltage. Our ovens are not designed to a nominal temperature. The temperature which we cook is below a nominal or maximum temp possible to obtain with wood as the fuel.

Because we are working within a temperature range, exceeding that range is not efficient regarding use of fuel (obviously). However, when I fire to bake bread I do so for a set time. I then rake out the fire and coals and close the oven up in an attempt to make the heat density constant throughout the WFO ("normalize"). If I have misjudged and fired for too long the normalized temp upon opening is too high and I have to wait until the oven cools to bake. The heat lost in order to achieve the new temp is wasted, both in time (to fire and to cool) as well as wood consumed. Of course this is all part of learning one's WFO. The point about how long one must fire their WFO is related to the wood (type and quanity) and how fast the bricks can absorb the heat. I am just amazed in the variance in heat up times. One would think the differences in bricks and wood (BTUs) would not be such that one WFO requires one hour and another twice that.

It would be interesting to hear if anyone has any possible reasons why two similarly sized "mature" WFOs would be so different. (I use the term "mature" here to mean after dryout and curing etc.)
Bests,
Wiley

[mklingles]

Re: Mathematical analysis of dome geometry

Perhaps I should have lead with this overview. I was too excited to get my results up, and I didn?t want to take the time to write this up well enough. In the following I explain my understanding of the various mechanisms of heat transfer that take place within the oven throughout the heating and cooking process. I believe that my analysis has some merit to understanding the effect of the shape of the dome on the cooking process. My analysis is not useful for determining the shape of the dome and oven opening on the efficiency of the heating process.

There are 3 mechanisms of heat transfer: convection, conduction and radiation. In a wood fired oven, each varies in importance at different times during the cooking process. Convection is the process of heat transfer through the motion of fluids. The heat is transferred when a group of hot particles moves from one place to another. In our case the hot combustion gases traveling around the oven. Conduction is the process of heat transfer from a hotter region to a cooler region through physical collisions. During conduction the heat is transferred by moving energy from one group of particles to a neighboring group of particles. In a brick oven conduction occurs when hot gas collides with cold bricks. Conduction also occurs as heat travels from the hot surface of the brick into the cooler center of the brick. Radiation is the process of heat transfer in which hot particles emit infrared light also known as electromagnetic radiation. Each photon of emitted light caries away an increment of energy, and the body emitting the radiation becomes incrementally cooler. Also as radiation strikes a body the body absorbs the energy becoming hotter.

When the fire is first built and the oven is being heated, combustion creates a hot gas. The heat is transferred through the motion of the hot gas to the surface of the dome (convection). The hot gas in contact with the dome transfers heat through particles colliding with the dome (conduction). During this phase the shape of the dome and the size of the oven opening and the position of the fire in the oven are dominant factors in how efficiently the energy of the fire is transferred into the dome. There is also some amount of radiation emitted from the hot gases and the coals. This radiation contributes to the heating of the dome and the floor. As the oven dome heats up the intensity and energy of the black body radiation emitted by the dome increases. The floor is primarily heated from receiving radiation from the oven dome, the hot flue gas and the goals. It is hard for convection currents to heat the floor, because the cool outside air is drawn in across the floor to feed the fire. The analysis I am doing does not apply to this phase of the process.

During pizza cooking the fire provides a continued source of heat input to the dome, while the dome?s radiation cooks the top of the food. The bottom of the pizza is cooked through conduction from the floor. The hot coals also emit radiation. Since they coals are hotter than the dome the emit more intense and higher energy radiation. The hot flue gas is also emitting radiation but as a gas it is much less dense then fire brick or coals, and so emits proportionally less radiation energy.
My analysis is focused on the effect of the dome shape on the radiation energy delivered to the top of the pizza during cooking. My next step will be to include the door opening in the calculation to show determine the relative difference in radiation caused by the missing dome bricks where the door is. This is one place where a lower dome will have an effect, because lower domes have smaller doors.

[mklingles]

Re: Mathematical analysis of dome geometry

@Wiley: I would be vary surprised if like ovens under like conditions took different amounts of time to heat. I suspect that fire building and mantenance play a part. Based on how my wood is stacked in the oven at full burn, I can either watch flames flow directly out and up the chimney, or I can watch the flame rise to the top of the dome and flow down the sides before exiting.

As for emperical data on how long is long enough, I've seem some data posted on heating times from people with thermocouples. On sunday when I cooked a leg of lamb I heated the oven for about an hour or a bit more. IR thermometer gave me a dome surface temp of about 800. Thermcouple 1/2" in from the dome surface gave a temp of 750ish, and thermocouple 4" in from the dome surface read only 170ish. Plenty good to roast a leg of lamb or cook a few pizzas. If I was going to cook pizza for a large party I prefer to see the outside temperature get up to closer to 350. Otherwise the oven cools to quickly.

@everyone - I agree there is no substitute for cooking in your own oven. This theory stuff just keeps me out of trouble - and may be helpful to people who have yet to build their ovens. People who feel like they need to know more before they make decisions in concrete (if you will)

[stoveup]

Re: Mathematical analysis of dome geometry

This is a fascinating and thought provoking thread! It also begs many more questions, as any carefull exploration should. A couple of the questions that occur to me as I read are:

1. What would the heat distribution for a parabolic dome look like? Would the sweet spot in the center be smaller or larger for a given diameter at the base?

2. For cooking nothing but pizzas, would an even thinner dome - like 2" - make sense?

Thanks to mklingles for sharing his investigation. More!

[lwood]

Re: Mathematical analysis of dome geometry

Thanks mklingles for sharing. There have been many hypothetical discussions around this subject and this at least puts some mathematical modeling on it. You obviously know your stuff about heat transfer, so please go for it. I am looking to confirm my theories about low domes and my next oven build. Just had my oven less than a year now and planning the next one. May have to wait till it falls down before I can shut it down for a rebuild. It's getting so popular.

Ultimately it comes down to using your oven and finding out about heat management, and the sweet spots in your oven. I think it's very difficult to characterize much with the all the different variations of peoples ovens on this site. But given "your model" as the standard and the results of your evaluation I think people will be much better informed as to what direction they should go when building their oven. Maybe you could get some professor to pick it up as doctoral thesis.

I would like to know what the effect of diameter is on heat transfer and a mapping of the iso-therms on the oven floor. While your at it....haha. My theory is that as the diameter increases the low dome becomes more of a factor. At a steady state oven...ie 750F, 100% heat soaked oven "radiation" is the major heating component, on the top of the pizza and "conduction" the major component on the bottom. Since radiation intensity is a function of distance from the dome to the floor, it makes sense that a lower dome would provide more intensity than a pure hemisphere. It gets more intensified in the bigger ovens greater than 40" as the dome gets farther and farther away from the floor. In pizza cooking, this is a bad thing. I have a 60" oven with a 24" dome height and a 15" door height. That's about 40% of the oven radius. Just prove it to yourself by raising the pizza up close to the dome and see how fast it chars the top of the pizza. But for making bread I think the higher dome has it's merits. A higher dome would be better for bread baking because the less radiation on the top would give you more even baking from conduction. Maybe there are other factors as well. My tops of bread tend to brown faster than the bottoms. Of course this is all my speculation and opinion. I would like for you to prove me wrong. This is a very interesting discussion so please keep it up.

[Neil2]

Re: Mathematical analysis of dome geometry

"What would the heat distribution for a parabolic dome look like? "

Mathmatically:

- A parabola will focus radiant heat at a single point somewhere above the floor.
- A hemisphere will focus at a single point on the floor (center)
- An ellipse will focus on two points on the floor (or in three dimensions, a ring).