Hello From the Frozen Tundra of Wisconsin!

SI W25765 North view Road Waukesha, Wisconsin 53188 and Bob
Stadowicz New Berlin Wisconsin 53146

We have been reading Gas Engine Magazine for years and have seen
many questions about engine power and how it is measured or
calculated, most recently in the January, 1997 issue. There
certainly seem to be a lot of questions and confusion about this
basic concept in our hobby. Working for an engine manufacturer for
many years has allowed us to become familiar with the subject. We
would like to try to explain how horsepower, torque, and mean
piston speed are calculated. We will only talk about brake
horsepower. This is the nameplate power available at the flywheel
for useful work, and would be the power measured by a brake or
dynamometer. Friction and other losses are real, but do not have to
be considered to calculate brake horsepower. Some mathematics are
absolutely necessary but we will try to keep it to a minimum.

We have to start with the concept of Brake, Mean, Effective
Pressure, or BMEP, which is measured in pounds per square inch
(psi). Power is generated by an engine by the burning of a fuel in
the confined space of the cylinder. As the fuel burns, it generates
a high pressure, which forces the piston downward. This movement is
converted into a rotary motion of the crankshaft by the connecting
rod. BMEP is that theoretical constant pressure which can be
imagined as being exerted on the piston during each power stroke of
an engine, which will produce the same brake power output as does
the varying pressure of the engine’s real operation. To state
this in a simpler way, an engine can be considered as an air motor,
where a constant pressure input results in a constant power output.
By using the concept of BMEP, the power output of different engines
can be compared on a basis which is independent of size or
speed.

Consider a large and small engine operating with the same BMEP.
The same pressure acting on a larger piston area and through a
longer stroke will result in a larger power output proportional to
the larger displacement.

BMEP values can also be used as an indicator of how hard the
engine is working. A small engine with a BMEP of 100 psi is working
four times harder than a large engine with only a BMEP of 25 psi,
even though the larger engine may develop more horsepower because
its displacement is much larger than the smaller engine.

Three fundamental concepts of physics are required to understand
the relationship between BMEP and output horsepower of an
engine:

1. Pressure x Area = Force

2. Force x Distance = Work

3. Work per Unit Time = Power

Starting with BMEP, the theoretical constant pressure working
against the piston throughout its power stroke, we want to finish
wi+th the output horsepower of the engine. The resultant force of
the pressure acting against the piston is equal to the pressure
times the piston’s area.

Force = Pressure x Area = lb/sq in x sq in = lb

The work output of a piston is equal to the force acting on it
times the distance through which it acts which is the stroke of the
engine:

Work = Force x Distance = (lb/sq in x sq in) x in = lb x in =
in-lb

Combining the piston area times the stroke results in the cubic
inch displacement of the cylinder:

Work = lb/sq in x (sq in x in) = lb/sq in x cubic inch =
Pressure x Displacement

Therefore, the work of a cylinder equals pressure (BMEP) times
displacement. To obtain power from work, a time factor must be
introduced. The appropriate time factor is engine speed in
revolutions per minute Rev/min or RPM. In a four cycle engine there
is a power stroke, or work output stroke, every second revolution.
Therefore, multiplying work by the revolutions per minute speed of
the engine times gives us the work per unit time which we need for
power:

Power = Work per Unit Time = Pressure x Displacement x RPM x

Power = BMEP x Displacement x RPM x

Power = (psi) x (cubic in) x Rev/min x = lb x in/min = in –
lb/min

Note that ‘revolutions’ do not have units, and that this
example is for a four stroke engine. If we are talking about a two
stroke engine, which has one power stroke for every engine
revolution , the ” factor would be ‘1.’ This
explains why a 2 stroke engine has half the calculated BMEP value
for the same power output as the same size and speed 4 stroke
engine.

The final step is to convert the (in-lb)/min value into familiar
horsepower. By definition, one horsepower equals 550 ft –
lb/second, or 33,000 ft – lb/minute, or 396,000 in -lb/minute. This
is simply changing units, knowing that 60 seconds = 1 minute and 12
inches = 1 foot. The final formula is found by multiplying power by
the equivalent 1 bhp = 396,000 in – lb/minute:

Power =   BMEP/2 x Displacement x RPM/min in – lb
x 1 Bhp 396,000 in-lb/run

Power = Bhp = (BMEP x Displacement x RPM)/792,000

As an aside, remember that the displacement of a single cylinder
engine is equal to piston area times stroke of Displacement = (pi x
D ) x L/4. Where D is the diameter of the piston in inches, L is
the stroke in inches, D2 means D x D, and pi = 3.1416 is a constant
used to calculate the area of a circle. Then, Displacement = 0.7854
D2 x L. If the BMEP is now taken as the 65 psi average from page 3
of the January, 1997 issue, the formula for power for a four
stroke, single cylinder engine becomes:

Bhp =(65 x 0.7854 x D)²/792,000 x (L x RPM)/15514 =
D² x L x RPM

This is very close to the formula for horsepower given in the
January 1997 issue on page 3. The difference in the numerical value
of the constant may be due to rounding off of the numbers used in
the calculation.

If the original horsepower formula is rearranged:

BMEP = (792,000  x  Bhp) / (Displacement x RPM)
psi

These equations to calculate Bhp and BMEP are the same no matter
what fuel is burned in the engine. To use these equations to
calculate Bhp or BMEP we must know three of the four values of BHP,
Displacement, RPM, and BMEP. Displacement is usually known, since
we can calculate it from the bore and stroke of the engine. Note
that total displacement of an engine with more than one cylinder is
simply the displacement of one cylinder times the number of
cylinders. The total engine displacement should be used in the
formulas. RPM is easily measured. Bhp and BMEP are not so easily
measured. Of the two, Bhp is usually known from measurement on a
brake or from the manufacturer’s nameplate. BMEP can be
measured, but it requires a careful measurement of the varying
pressure during the entire power stroke, and ‘mathematical
averaging’ of the results. Old timers in the engine industry
(and steam engine men) may remember taking ‘indicator
cards’ from larger engines. Nowadays, electronics and sensors
make direct measurements of BMEP easy.

While old hit and miss engines and many smaller engines never
had BMEP measured, most of these will have a rated Bhp, as measured
or otherwise determined by their manufacturer listed on their
nameplate. We can then calculate a BMEP value for these engines and
use it to compare how hard these old engines worked. For a normal,
naturally aspirated engine, the maximum possible BMEP is about 110
psi. This is the highest mean pressure that can be generated by an
engine working with normal atmosphere intake pressure. Engines
which are supercharged or turbocharged can develop much higher
cylinder and, therefore, BMEP pressures. In this case, the highest
BMEP is limited by strength or heat limits of the engine and not
atmospheric pressure.

Now, an engine will not generate any brake, or useful,
horsepower, unless a load (opposing force) is put on it. This leads
to the concept of torque. Torque is the capability of an engine to
do work, while power is the rate at which the engine does this
work. A simple example is that of a tractor pulling a load. The
torque developed by the engine will determine if the tractor is
capable of pulling the load, while the power developed by the
engine will determine how fast the load can be pulled.

Torque is defined as a force acting at a distance. Note the
difference between torque and work. Torque is only a twisting
effort there is no movement while work is a force moving through a
distance.

Torque = Force x Distance = lb x in = lb – in

These are the same units as work, but are expressed as the
opposite to indicate that there is a difference. The fact that
these two quantities have the same basic units means that we can
calculate a torque for an engine which is doing useful work that
is, an engine with a Bhp value.

If you resist the turning of the flywheel of an engine you must
exert a force on it. That force will be applied at some distance
from the center of the flywheel. Together that force (f) and the
distance (d) will form a torque value. Now if the torque (Force x
Distance or f x d) effort that we are applying to the flywheel does
not stop the flywheel, it will move with the flywheel. This moving
force is now work. For one revolution of the flywheel.

Work = Force x Distance = f x d x 2 x pi

Pi (pi = 3.1416) is the same constant used before and is used to
calculate the distance around a circlethe flywheel in this case.
The distance traveled by the opposing force, f, during one flywheel
revolution is d x 2 x pi. Remember that:

Power = Work per Unit Time

Again, the appropriate time factor is engine speed in RPM.

Power = f x d x 2 x pi x RPM = (f x d) x 2 x 3.1416 x RPM.

 Power = Torque x 6.2832 x RPM in-lb/min

Note that the RPM factor above is not multiplied by as in the
earlier power equation, because the resisting effort (torque) acts
during all the revolutions of the crankshaft whether or not that
revolution has a power stroke. If we now use the same definition of
a horsepower that we used earlier we get:

Power =Torque x 6.2832 x RPM in- lb/min x 1 Bhp / 396,000
in-lb/min

Power = Bhp = (Torque x RPM) / 63025 Or, by rearranging:

Torque = 63025 x Bhp /RPM with torque in lb – in

Torque = 5252 x Bhp /RPM with torque in lb – ft

The final thing we want to talk about is Mean Piston Speed MPS
usually expressed in ft/min. During engine operation a piston is
always changing speed and direction, and actually comes to a stop
twice during every revolution of the crankshaft. We can calculate a
mean, or average, piston speed. The distance that a piston travels
during each revolution is twice the stroke:

Distance/Rev = 2 x Stoke inches/Rev

The distance traveled in one minute is the distance per
revolution times the revolutions per minute, or RPM.

MPS = Distance/min = 2 x Stroke x RPM inches/min

or, if we divide by 12 inches per foot:

MPS = (Stroke x RPM)/6  feet/min

The materials that engine pistons, rings, and cylinders are made
out of limit the maximum piston speed to about 2000 ft/min. At
higher piston speeds wear is very rapid and higher forces can lead
to early failure. This relative limit helps to explain why small
engines and short stroke engines usually operate at higher RPM than
large or long stroke engines.

How can we use BMEP and MPS to help us understand engines? Gary
is a collector of Briggs & Stratton engines and has an
extensive collection. Some old models always seemed to be worn out
when found, while other models, equally old, were usually found in
very good condition. We wondered why this was. Could it be how they
were used or was it something in the design of the engine itself?
We compiled a list of many old Briggs & Stratton models.* This
list, all of which are 4 stroke engines, follows. From the
available horsepower, speed, bore and stroke information, we
calculated displacement, torque, BMEP, and MPS. These values are
included on the list.

Looking at this list shows some interesting things. For example,
we can see that before about 1950, only four of the models on our
list, the models, N, R, Z, and ZZ, had BMEP values above 50 psi. On
the model H the BMEP was as low as 22.7 psi. After 1950 the BMEP
values took a large jump to as high as 96.2 psi for the model 19.
This last model is heavily loaded at its rated horsepower, running
at about 87% of the maximum possible BMEP allowed by the
atmosphere. Some of the lightly loaded models run at only about 20%
of the maximum possible BMEP at their rated horsepower. This can
help explain why some engines are usually found worn out! Modern
engines tend to have high BMEP values to give the highest
horsepower from the smallest engine.

If we now also look at the MPS of these engines we can see that
some are working very hard indeed. For example, the model 23 has a
BMEP at its rating of 8.25 BHP of 88.5 psi which is about 80% of
the maximum and, at the same time, an MPS at its 3200 RPM rated
speed of 1733 ft/min which is also quite high. We would expect to
find most examples of this model to be in worn out condition.
Compare this to a model FC, which has a BMEP at rated load of only
24-7 psi and a rated speed of MPS of 675 ft/min. These engines are
much more likely to be found in good shape.

As another example, compare the model K, rated at 3 Bhp at 2700
RPM, with the model 9, rated at 3.1 Bhp at 3200 RPM. The MPS of the
model K is actually higher at 2700 RPM for this engine than the
model 9 at 3200 RPM, because of the longer stroke of the model K.
Since both values of MPS are well below 2000 ft/min this should not
be significant. Look at the BMEP values. The model K is operating
at 45.5 psi while the model 9, at an almost identical power rating,
is running at 86.2 psi. This value of BMEP is almost twice as high
as that of the model K for the same power. We would expect to find
many more worn out model 9 engines.

We hope our information will help other collectors understand
their hobby better and maybe shed some light on why certain engines
seem to wear out quickly. Over the last several decades major
advances have been made in engine design, materials, and
lubrication, but the basic physical concepts and formulas remain
unchanged.

See chart ‘Briggs & Stratton Gasoline Engines,’ on
the next page, which is part of this article.