What is a Kiln?
A kiln is essentially a box with some form of heating inside it - in this case electric elements - to heat up the pots in it to a high enough temperature. The elements get up to a certain temperature (typically about 1400°C) and, if perfectly insulated, the kiln and its contents would get up to the same temperature as the elements. Simple!
Unfortunately the perfectly insulated kiln doesn't exist. As the insulation heats up, heat travels to the outside of the kiln, and then heat is lost from the kiln in warming up the surrounding air. This means that the rate of heat rise will slow down as the outside of the kiln gets warmer, and ultimately the rate of heat loss will match the heat put in by the elements, and the kiln won't get any hotter. Even before the kiln stops heating up, the rate of temperature rise may slow down to a point where it is longer than we are prepared to wait, so we may upgrade to better insulation or more powerful elements.
Also, the heat from the elements is absorbed by the pots and kiln furniture, as well as the outer insulation, and the more there is to heat up, the longer it takes. If we switch back to the perfectly insulated kiln, any kiln load will eventually get to the desired temperature, but it may take a long time to get there, so to fire a bigger load we need more powerful elements.
For a simple kiln shape, and constant insulation, we can assume the outside gets to about the same temperature all over. This means that the heat loss is due to the temperature difference between the outside air and the kiln surface, and the outside area of the kiln. This means that, in terms of kiln shape, a sphere is ideal as it has the lowest ratio of area to volume, of 4.835976.
Unfortunately spherical kilns haven't caught on. Instead, kiln makers use cylindrical shapes for top loaders, and cuboid for front loaders.
For a cylindrical kiln, the most energy efficient ratio is when the diameter is the same as the height, with a ratio of about 5.54, so about 15% more heat loss. If we go to a width:height ratio of 0.5 or 2.0 the heat loss rises to 21% more than a sphere.
For a front loader, the most efficient shape is a cube, with a ratio of 6, so 24% less efficient than a sphere. Again, the further we go from this ideal, the less efficient the shape becomes. For example, if we went for sides of 2 x 1 x 0.5 then the efficiency is 54% less.
There may be good reasons to go away from the ideal ratio, such as not making a top loader too tall to reach down in to, but you need to be aware of the accompanying loss of efficiency
There are a number of approaches