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Mathematical analysis of dome geometry

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  • david s
    replied
    Re: Mathematical analysis of dome geometry

    This also applies to small ovens. The dome and flame from the active fire are much closer to the pizza. My own oven has the interior height at only 10 and a half inches. It still cooks much the same 90 sec to 2min pizzas. The larger the oven, the larger the fire so I guess it all evens out.

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  • Neil2
    replied
    Re: Mathematical analysis of dome geometry

    It is not too complicated.

    An elliptical oven has a lower dome than a hemisphere of the same diameter. The heat from the ceiling bricks is closer to the top of the pizza.

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  • mklingles
    replied
    Re: Mathematical analysis of dome geometry

    For sure the Convection is a dominant factor in heating the oven and distributing the heat within the oven. For cooking, I think the radiation factor dominates. However, I don't know that to be the case.

    I did talk to one physics professor about handing this off to have a student do some more work on it, but haven't even followed up on that. (To busy cooking pizzas, building wood shed, etc).

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  • benguilford
    replied
    Re: Mathematical analysis of dome geometry

    mkingles, as an engineer I salute you sir! I liked your quote from a while back 'The maths keeps me out of trouble'

    Surely there must be an engineer on this Forum with access to some professional thermal modelling software, three dimensional modelling of radiated heat is fairly complicated, and it doesn't take the effect of convection into account - even in a sealed dome there will be a surface temperature differential between the floor and the walls, which would set up a circulating air flow within the oven. Would this be considered negligible in this situation?

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  • mklingles
    replied
    Re: Mathematical analysis of dome geometry

    I'm working on adding the door opening to the calculation. Maybe a week or so, I'll get the next round of math done.

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  • Lburou
    replied
    Re: Mathematical analysis of dome geometry

    I thought we had a good beginning for this discussion....Are we done? Ready to read more

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  • mklingles
    replied
    Re: Mathematical analysis of dome geometry

    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).
    What Neil2 is describing here is the behavior of photons that are reflected off of the surface. For that fraction of the inferred radiation that is not absorbed and re-admitted the reflective properties would apply. This article at wikipedia has a good description of specular reflection from curved mirrors. Wikipedia - Curved Mirrors

    I have not been able to find a reliable source of data on the amount of energy absorbed by fire brick vs the amount reflected. I believe from the numbers I have found that ~80% of the energy is absorbed and only 20% reflected.

    Also the roughness of the firebrick will greatly distort any reflected "image

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  • Neil2
    replied
    Re: Mathematical analysis of dome geometry

    To sum it up then, for pizza cooking:

    For smaller ovens a hemispherical shape will do fine and provide the opportunity for a somewhat higher door (maintaining the 63% door/dome ratio).

    For a larger oven, say 40 inches or more, an elliptical or low dome shape might be more efficient for cooking pizza.

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  • texassourdough
    replied
    Re: Mathematical analysis of dome geometry

    Thanks, mklingles!

    Would be interesting to do a conical oven also - easy shape to specify. I think you will find all of the options are very similar for the no fire case.

    I believe the biggest advantage of the lower oven for pizza lies in the flame being closer to the pie - and wrapping more over the top of the pie. If we assume the flames are say 16 inches "tall" they will reach an inch or two further across a "low" dome compared to a taller one and will be several inches closer. We are all familiar with the fact that the side of a pizza cooks faster than the "dark" side (which is receives less flame radiation and more black body refractory radiation from a cooler source than the flame). As a note, for small flames there would be little difference.

    Thanks for bringing up the topic and for your calculations!
    Jay

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  • Neil2
    replied
    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).

    Leave a comment:


  • lwood
    replied
    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.

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  • stoveup
    replied
    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!

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  • mklingles
    replied
    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)

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  • mklingles
    replied
    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.

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  • Wiley
    replied
    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
    Last edited by Wiley; 01-31-2011, 10:21 AM.

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