Proof that engine frictional losses grow as cube root of number of cylinders

Proof that: "Wall Friction Losses Grow as the Cube Root of the Number of Cylinders" by L. Van Warren

For a "square" engine where bore equals stroke, we seek a relation between contact surface area (friction) and engine displacement (volume) as the number of cylinders increases.

If we make the preliminary assignments that:

n	=	the number of cylinders in the engine
S_n	=	contact surface area of cylinder walls
		for an engine with n cylinders
V	=	some fixed engine displacement or volume
d_n	=	cylinder diameter
		for an engine with n cylinders

and note that the volume rule is given by:

Then the relationship between frictional losses, volume and cylinder count is given by:

where n is the number of cylinders and c is a constant.

Thus the frictional loss for an engine fixed volume with n cylinders grows as the cube root of the number of cylinders.