Power density 2

It can be said that the output power of a reciprocating combustion engine is quite roughly proportional to the product of the piston displacement times the speed. The piston displacement should be quite coarsely proportional to the overall volume and hence also to the mass of the engine. Since bigger units rev slower, their power density will be lower than with small engines. Just see the values in the attached table: The engines of the car and the motorbike are somewhat too slow, but this is because mass and volume are crucial here and not so much longevity. With industrial engines for permanent full load duty the speeds would appear realistic.

Also the data from the rating plate of the big ship engine are only roughly approximated and not precisely hit by this calculation model, but at least the chosen factors and exponents seem to point into the right direction across 5 orders of magnitude.

Now with electric machinery we observe the opposite: When doubling the dimensions – length, width and height – of the machine we end up with 8 times the mass and volume and 4 times the cross sections. The latter applies to the space available in slots for accommodating magnet wire as well as to the core cross section. Now, for one, the cross section of the magnet wire and with it its current carrying capability can be enhanced by a factor of 4, while quadrupling the core cross section simultaneously allows for quadrupling the voltage across the ends of the respective winding. So the power rating that can be allocated to said winding will go up by a factor of 16, while mass and volume of the respective device have increased only by a factor of 8. So the size and volume increase with an exponent of 3/4 with the power rating of an electrical machine.

So with electrical plant the power density will increase with size, while in combustion engines it decreases.

The break-even point is just around the size of an electric car, and indeed, the electric motors considered here are about the same size as the conventional engines.

But this also means that splitting up the power rating across 2 or 4 motors, while saving the differential gear, brings with it additional weight and volume.

Feel free to play around a bit with the data! Suggestions for improvement welcome.