In this article, we'll have a look at the maths which sets out the performance limits of naturally aspirated four stroke engines. At the bottom of the page, we'll have a look at how you can quickly work out what state of tune your NASP engine is in in relation to it's maximum possible horsepower potential.

Bore. It's all that matters.

If I told you that, no matter what you do, for a NASP the maximum power you can make is soley dependent on the bore diameter, you'll most likely say "Rubbish".

But it is true. For any naturally aspirate four stroke engine running on regular gasoline, the maximum power possible is given by the following equation

$$ Pmax = 9.694 \times 10^{-3} \cdot Bore^2 \cdot Cylinders$$

Where bore is in millimeters.

Let's have a look at how that was derived.

Torque and capacity

All engines are limited in the torque they can produce, by the amount of physical volume they can fill with combustable mixture. That mixture has a certain amount of energy that can be released and turned into useful work, the rest going into heating up the coolant, oil, exhaust and bearings. NASP engines are only ever going to achieve a few percent over 100% volumetric effiency, so the amount of torque produced by a given capacity is finite, with a relatively small range of values over a wide range of engines. This is of course relating to Specific Torque (lbft/litre) which is also directly linked to Brake Mean Effective Pressure (BMEP).

In a NASP engine, this figure will typically be between 60-85 lbft/litre (148-210psi BMEP) for road going engines. The current production car record holder is the Ferrari 458 with 88lbft/litre (217psi). Even in top flight motorsport, the basic physics dictates that even breaching 90lbft/litre (222psi) is a tough job!

From this we are able to derive a maximum torque figure to be expected from any given engine, simply the capacity multiplied by the maximum specific torque. Let's try that out...

Specific Torque (lbft/litre)
Capacity (cc)

The importance of stroke

So, we can see that the overall torque potential from any engine is dependant on the swept volume but that is of course a combination of both bore and stroke. So what was that about only the bore mattering?

Well, capacity might dictate the torque, but power is the product of torque and RPM, and stroke has a pronounced effect on the maximum speed of an engine, with longer strokes dictating lower maximum RPM

The reason for this is piston speeds. For single engine cycle, a longer stroke moves the piston a longer distance so for any given RPM, the piston in a longer stroke engine will be travelling faster in order to move that increased distance in the same time period (1 revolution)

The equation for mean piston speed is:

$$MPS = 2 \cdot \bigg({RPM \over 60}\bigg) \cdot \bigg({Stroke \over 1000}\bigg)$$

RPM
Stroke (mm)

A major limiting factor here is how fast gasoline burns on the combustion stroke. Spin the engine too fast, and the piston will move away from TDC faster than the pressure increases, leading to a dramatic drop off in torque.Another consideration is the component stresses on the pistons, rods and wrist pins. Current materials and manufacturing technologies are only capable of producing components that can sustain a maximum of about 25 m/s reliably.

It is entirely possible to go above this, but component stresses get so high that longevity is seriously impacted. Engines operating above this speed are subjected to extremely short service lives with a rebuild interval measured in minutes (drag engines, basically).

For any engine that is expected to last more than a few minutes, let alone a hundred thousand miles as a road engine, the piston speeds should be kept below about 25 m/s. This means we must alter the equation to give us the maximum RPM possible for a given stroke.

$$RPMmax = 30,000 \cdot \bigg({MPS \over Stroke}\bigg)$$

Stroke

At this point, if you drive a Honda and have chucked a couple of known stroke lengths in, you'll appreciate just how highly strung they are from the factory!

Combining the two maximums

So now we have a maximum torque figure, and a maximum RPM figure. Torque and RPM create power, so we can figure out a maximum power figure for our engines geometry. We will take the maximum specific torque to be 90lbft/litre.

The full equation for the performance limit is:

$$Pmax = {{90 \over 1000}\cdot { \pi \cdot Bore^2 \cdot Stroke \cdot Cyls \over 4000} \cdot 30,000 \cdot {25 \over Stroke} \over 5252}$$

A bit nasty looking, but luckily this simplifies down to:

$$Pmax = 10.09\times10^{-3} \cdot Bore^2 \cdot Cylinders$$

Which is very close to the equation I gave in the opening statement. You'll notice that stroke has been removed from the equation, this is because an increase in stroke will give an increase in torque potential, but also reduces the maximum RPM by a proportionate amount, ergo cancelling itself out.

The last thing to consider is that an engine will not produce maximum torque at maximum RPM, so we must reduce these figures sufficiently to give us a realistic maximum level. A reasonable factor to use is 97% of both maximums, which when put into the above equations and simplified down, gives us our final power limit equation:

$$Pmax = 9.694 \times 10^{-3} \cdot Bore^2 \cdot Cylinders$$

Here's the inputs to test that out.

Bore (mm)
Cylinders

Now, bear mind that this does not mean that any given engine can actually reach it's potential maximum, this merely outlines what that maximum is, dicated by the bottom end geometry. This does leave one question though... how tuned is your car?