Developer - Lead Beta Tester/Producer/German Efficiency Expert
Joined: Thu Mar 10, 2011 11:00 pm
Location: Erzhausen, Germany
Cars: I owned a Twingo... totally bad-ass!
I have thought about this for a while and put quite some effort into this post. So here we go. First,
let us try to establish a footing for the terms torque and power. Because there are so many factors to
consider, I will focus just on stroke and bore and nothing else, keep that in mind while reading the following.
Also, if you are not familiar with engines, I recommend taking a look at reference  which has a very nice
illustration of the basic working principle in the form an animated figure.
Power is generated by adiabatically  compressing a fuel/air mixture, combusting it and then letting
it adiabatically (isentropically? ) expand. The power output is generated by the difference between
a) the energy required for the compression, and b) the energy generated by the heated expanding gas in the
cylinder during/after the combustion process  during the so-called power stroke. How much energy
can be extracted from the expanding gas depends on how much the gas is allowed to expand. The first steps
of the expansion release more energy than the final steps of the expansion (or at least that seems logical to
me... but don't quote me on that ).
Torque is produced by the turning of the crank... or more precisely: the force pushing down the piston
acts on the crank, which has an offset to the axis formed by the connection rods, and thus gives rise
to torque. The torque generated at any given moment (roughly) is the force on the piston times the
orthogonal offset of the crank to the axis.
M = F * r
Edit: this is for the orthogonal case only, else it's the orthogonal projection of the force vector which
is considered, i.e. M = r x F as Deus ex Machina pointed out correctly.
Easier than it sounds, the maximum offset from the connection rod axis is given by stroke/2. The
maximum torque is generated about half-way through the expansion phase as the crank becomes
orthogonal to the axis.
torque maximum = force on piston * stroke/2
This will not be maximum effective torque... which will be closer to the maximum torque divided by
the square-root of 2.
The following two simple examples will help us understand the dependencies and get them right.
Example a) bore increase
We have one cylinder with inner radius r, i.e. a bore of 2r. The stroke is s. If we now just increase
r (bore/2) to R (BORE/2) and leave all other parameters the same (while scaling the amount of fuel
injected), how changes the power output and torque of the engine?
We established that the power is created by the expanding gas. The difference in the amount of
gas is given by the difference in engine displacement. The engine displacement is given by:
pi * r^2 * s,
and the new displacement is
pi * R^2 * s.
As everything else is the same, except for the amount of fuel (as stated before), we have a linear
dependence between the force on the piston and the engine displacement. Thus, the force on the
piston of the engine increases (to first order) by a factor R^2 / r^2. The torque of the engine
depends on the force on the piston if the stroke is the same in both cases. Thus the engine torque
also increases by a factor R^2 / r^2. This scaling factor can be rewritten as
R^2 / r^2 = (BORE/2)^2 / (bore/2)^2 = BORE^2 / bore^2
Summary of example a)
Increasing the bore and just the bore, both power and torque scale approximately linearly with the
Example b) stroke increase
We have one cylinder with inner radius r, or a bore of 2r. The stroke is s and is increased to S.
All other parameters (also the amount of fuel!) is left the same, how changes the power output and
torque of the engine?
The initial volume from where the gas expansion starts is the same as before. The force on the
piston is the same as before as well. What is different is the lever that produces the torque and the
amount the gas is allowed to expand before the power stroke ends. The difference in lever length
yields an increase in maximum torque from
M = F * s/2 to M = F * S/2
which is an increase by a factor of S/s. Simply a linear increase.
The extended stroke is more tricky. It makes the engine more efficient as more energy is extracted
from the expanding gas in total. Compared to the engine with stroke s, a force acts on the piston for
the original path + the additional path. The dependence here is not linear if the initial expansion is worth
more than the later stages of the expansion - this would mean it's "less than linear", and probably depends
on the gamma factor mentioned in references  and . The actual dependence can be extracted
when knowing and understanding these references... (which I admit that I currently don't, even though
I should) and having some values to put in from known engines. Without this, it's just guessing... so let's
say it is 0.7, which would be "less than linear", which would be 1.0, but more than a square root
dependence, which is 0.5.
By making this wild guess of 0.7, the increase in power output would be a factor of
For example, by increasing the stroke from 50mm to 75mm, the power would increase by roughly 33%,
instead of 50% which would be the case for a linear dependence.
Also note that I did not speak of compression, which is important in this case because keeping
everything constant while increasing the stroke actually increases the compression value.
Summary of example b)
Increasing the stroke and just the stroke increases the torque of an engine linearly with the stroke,
while it increases its power output less than linearly.
Comment: this dependency can be determined by our engine experts as well, by just comparing
engines with equal bore but different stroke (although there are many more parameters...)
So much for that... the discussion is on you now