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This may be anecdotal. Playing in the kitchen I realized the frying pan comes with both a flat, and a concave bottom. So here's the question -

Given two pans made of brass, one has a concave base & the other a flat base, which of the two would use heat more efficiently? I would believe the latter because the flat base means any heat must necessarily travel through the cooking medium, and the recipe before reaching open air.

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The Wok


is a design that gets the maximum efficient use of energy for cooking and needs a minimum of heat source.

A flat bottom equivalent pan transfers the heat to the food from the bottom plane, and the walls are just containers radiating most heat away, since the food is at the bottom. A Wok focuses the heat from the walls to the food being cooked ( think of a parabola and its focal point). The walls partake in the cooking process effectively giving more area contact to the food than the flat bottom one.

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But the walls in a vertical plane present the least surface to radiate from ... – Everyone Sep 29 '12 at 4:24
The thing is where is the radiation focused? A vertical cylinder has a focus line going up. There is little food there unless one is boiling a stew, usually it is the first ten centimeters where the food is cooking. The heat from the walls heats the air. – anna v Sep 29 '12 at 4:28

I guess that the concave pan would heat more efficiency because the hot air on the flat surface can't move quickly in the horizontal direction. The curved shape is more 'aerodynamic' than the flat shape. The contact force of the metal surface on the hot air would drive the hot air horizontally. Thus, the hot air would be driven in the direction of cold metal.

Let us suppose that the heat source was small and localized near the center of the pan. Cold air is denser than hot air. So the hot air starts rising just below the center of the pan. The hot air from the localized heat source starts moving straight up toward the bottom of the pan due to buoyancy forces. There would be little if any component of horizontal velocity.

Suppose the bottom is flat. Look at parcel of hot air that is just off-center. When a parcel of hot air touched the metal, the contact force of the metal stops the hot air. There is no force to drive it horizontally other than the cold air coming in from the side. So the hot air will just stay there longer. The hot air will pile up. The metal will increase in temperature at the point of contact. The temperature will be nearly the same on both sides of the metal. So there will be little conduction.

Work will be done on the air rather than on the metal pan. Most of the enthalpy of the flame will apply work to the hot air, moving it horizontally.

Suppose the bottom is curved. Look at a parcel of air that starts out just off-center. It rises to meet the metal. The contact force of the metal pushes the air horizontally as the air rises. So the hot air is pushed to a cold part of the pan. The difference in temperature is much larger at the edges of the pan. So the heat conduction is larger at the edges.

I don't think the analogy between a wok and an optical lens is valid. The system can be characterized more a matter of fluid flow and heat transfer. The wok (curved surface) spreads the hot air horizontally over the metal. The flat surface makes the hot air move into regions of cold air. The flat surface deposits the energy in the cold air rather than the pan.

The flame does work on either the atmosphere or conducts heat to the pan. There is a certain amount of energy that is applied to either the atmosphere or the pan. In the case of the flat pan, the flame does work on the atmosphere. In the case of the curved pan, there is less work done on the atmosphere. So more heat is conducted through the metal.

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I think a diagram would be helpful. – Floris Nov 11 '14 at 6:58

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