“The calculation are for a Hemishpere and not a Sphere: That is where the 1/2 came in the formula above. An HX pot will probably be more efficient with material as it is in the plane of the flame spread than the hemisphere as you don’t know how well the flames will hug the surface. My 2 cents.”
Well, I am not sure about the heat transfer through a hemispherical part of this pot design. You will find that heating the pot is only the first phase of heat transfer. The actual goal is to let the heat, warm the water so you can cook/warm up stuff in the water.
As such, a minor exercise in overall physics will show that a sphere has the least surface area for the internal volume. Rather, you can think of the water as totally surrounding the heat source (like the proverbial Dyson Sphere, but on a smaller scale.) I believe they have the mathematics of a hemispherical bottomed pot backwards.
Actually, you would want an inverted sphere around a heat source to gather as much energy as possible from the heat source. Since this is not really practical, an inverted hemisphere is used as an example.
This is constrained by having a gas (generally speaking, everything must be a gas before it can be burned) burn releasing byproducts: CO2, H2O (primarily.) These are hot gases that rise due to gravity/displacement. Soo, a perfect heat transfer would work well. Unfortunately, the trapped air in an inverted hemisphere prevents further entrance by rising gasses from the source.Simply, the heated gas puddles allowing heated gas to flow around the pot. I was getting about a 20%reduction in heat transfer when tested.
In the opposite case, where you have a hemisphere pointed down, but, this allows a smaller volume of heat to be transferred to the water. Surface area vs heat transfer is all opposed to maximum heating effect of the water.
The actual heating effect of smaller surfaces conducting the same heat to the larger internal volume (water in this case) means the outer skin needs to be hotter (10Cal/4sqin or 10Cal/10sqin.) Hence, there is greater opportunity for heat loss due to simple radiation as delta of ambient radiative heat loss. (Generally, ambient radiative heat loss means garbage in equals garbage out, or, GIGO. Heating one side increases radiative heat to the lower ambient side.)
So, if neither a inverted hemisphere nor a hemisphere work well for heat transfer to our water, then the answer lies in a plane as the mostest heat absorption. Ofcourse, allowing heated gasses to transfer the heat a little time to work means a slight ripple in the plane will work. Or, the basis for heat absorption of a heat exchanger.
You can optimize these effects (gas flow, heat flow, heat absorption, etc) by having the depth of each ripple correspond to the average heated gas flow, preventing puddling, but increasing surface area to heat absorption. Of course, the ideal distance changes with the amount of heat generated from the heat source.
In conclusion, a hemispherical heat gatherer, whether inverted or not, does not work the best for heat transfer from our average cooking stove. The compromise is a flat bottom with a resistance to gas flow along the bottom. Roughly speaking, when you can touch the top while heating water, you are doing pretty good.
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