We have been reading Gas Engine Magazine for years and have seen many questions about engine power and how it is measured or calculated. There certainly seem to be a lot of questions and confusion about such a basic concept to 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 it 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.

## BMEP — Brake Mean Effective Pressure

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

Consider a large and a 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 100psi is working four times harder than a large engine with only a BMEP of 25psi 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:

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

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

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

Therefore, the work of a cylinder equals pressure (BMEP) times displacement.

A time factor must be introduced to obtain power from work. The appropriate time factor is engine speed in revolutions per minute — rev/min or rpm. In a 4-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 1/2 gives us the work per unit time that we need for power:

Note that “revolutions” do not have units and that this example is for a 4-stroke engine. If we were talking about a 2-stroke engine, which has one power stroke for every engine revolution, the 1/2 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 550ft-lb/second, or 33,000ft-lb/minute, or 396,000in-lb/minute. Here, we simply change units, knowing that 60 seconds = 1 minute and 12 inches = 1 foot. The final formula is found by multiplying power by the equivalent 1bhp = 396,000in-lb/minute:

As an aside, remember that the displacement of a single-cylinder engine is equal to piston area times stroke or:

Where D is the diameter of the piston in inches, L is the stroke in inches, D² means D x D, and π (pi) = 3.1416, which is a constant used to calculate the area of a circle. Then, Displacement = 0.7854 x D² x L. If the BMEP is now taken as the 65psi average from the January 1997 issue (see Page 25), the formula for power for a 4-stroke, single-cylinder engine becomes:

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

If the original horsepower formula is rearranged:

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 cyclinder 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) may remember taking “indicator cards” from larger engines. Nowadays, electronics and sensors make it easier to take the direct measurement of BMEP.

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 110psi. This is the highest mean pressure that can be generated by an engine working with normal atmospheric intake pressure. Engines that are supercharged or turbocharged can develop much higher cylinder pressure, and therefore, BMEP pressure. 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, not 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.

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

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

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

Note that the rpm factor above is not multiplied by 1/2 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:

Or, by rearranging:

And:

## MPS — Mean Piston Speed

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 = 2 x stroke inches

rev rev

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

Or, if we divide by 12 inches per foot:

The materials used to create engine pistons, rings, and cylinders limit the maximum piston speed to about 2,000 ft/min. At higher piston speeds, wear is rapid and higher forces can lead to early failure. This relative limit helps explain why small engines and short-stroke engines usually operate at higher rpms than larger 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 were found worn out, which other models, equally as old, were discovered to be in 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 briggs-stratton-gasoline-engines, all of which are 4-stroke engines. From the available horsepower, speed, bore, and stroke information, we calculated displacement, torque, BMEP, and MPS.

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 50psi. On the model H, the BMEP was as low as 22.7psi. This last model is heavily loaded at its rated horsepower running at about 87 percent of the maximum possible BMEP allowed by the atmosphere. Some of the lightly loaded models run at only about 20 percent 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 hard indeed. For example, the model 23 has a BMEP at its rating of 8.25 Bhp of 88.5psi — which is about 80 percent of the maximum — and, at the same time, a MPS at its 3,200rpm rated speed of 1,733 ft/min — which is also quite high. We would expect to find most examples of this model to be in worn condition. Compare this to a model FC, which has a BMEP at rated load of only 24.7psi and a rated speed 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 2,700rpm, with the model 9 rated at 3.1 Bhp at 3,200rpm. The MPS of the model K is actually higher at 2,700rpm than the model 9 at 3,200rpm because of the longer stroke of the model K. Since both values of MPS are well below 2,000 ft/min, this should not be significant. Look at the BMEP values. The model K is operating at 45.5psi while the model 9, at an almost identical power rating, is running at 86.2psi. 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 and maybe shed some light on why certain engines seem to wear out more 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.

*If you have further questions on this topic, you can reach the authors via phone: Gary Pegelow, (262) 719-0234, or Bob Stachowicz, (262) 349-9828.*