# [Physics] scaling of motor power

electricityheat-enginethermodynamics

For car engines, the cylinder volume is often associated with the engine power, which suggests scaling of the power as $L^3$ where L is the linear size. Consider a system consisting of a motor and its energy supply, e.g., an internal combustion engine with a fuel tank, or an electric motor with a battery, or a steam turbine with a nuclear reactor etc. Now we scale the system geometrically, up or down by some factor, using same materials, and assume that in the scaled system nothing breaks or burns through, so it can function. What should we expect for the scaling of the output mechanical power? Would it be close to $L^3$? Can an example be constructed where the scaling is very different from $L^3$?

If you take some example engine cylinder of capacity $C$ then the fuel/air mixture that fills it has to flow in through the inlet valves, and because air has a non-zero viscosity this takes time. If you now double the cylinder size (i.e. raise its volume to $8C$) but don't change the valve size the air/fuel flow rate wouldn't be adequate to fill the cylinder in the time available between cycles. So you double the valve size as well, but of course this only increases the valve area, $A$, by a factor of four to $4A$, and the ratio of valve area ($\approx$ flow rate) to cylinder volume has fallen by a factor of two. This means power won't scale with $L^3$. In practice engine manufacturers combine increased cylinder volume with changes to the valve design, and not simply size. Even so it's hard to maintain efficiency over a large range of cylinder sizes, which is why large engines tend to have more cylinders rather than the same number of bigger cylinders.