Brake Means Effective Pressure

The following notes are in the SI, system international, or metric system for simplicity.  This also reduces the need for unnecessary and confusing constants in the derived equations.  The only concession will be to include the power in horsepower.

When an engine is tested in the laboratory it is generally connected to some form of "brake", or dynamometer.  As the slang name implies this device maintains a constant engine speed independent of the throttle position and the power produced by the engine.  The outside casing of the dynamometer is allowed to rotate freely and is restrained by a load cell at the end of an arm of known length.  This load cell behaves like an electronic spring balance, and after calibration outputs a force value (F) in Newtons.  The distance between the centreline of the dyno. and the load cell is the length of the torque arm (L).

The Brake Torque of the engine can then be written:

Tb =  F x L (1)

The idea of torque can be illustrated by a see-saw with different masses at different distances from the pivot.  The same torque is produced by a small mass m acting at a distance 2d from the pivot point as a larger mass 2m acting at a distance d.  Thus the see-saw is in balance.  In a vehicle the torque is the ability of the engine to turn the wheels , (similar to the ability of the engine to turn the dyno.), at a particular speed.

The work done by any mechanism is defined as the force exerted multiplied by the distance over which it is applied.  When pushing a weight along the floor a distance d using a force F, the work done is equal to F x d.  The work done by the engine per revolution is the force F times the distance traveled by the force in one revolution of a circle radius L.

Work per revolution = 2 x Pi x F x L = 2 x Pi x Tb ( From eqtn 1) (2)

Power is the rate of doing work, i.e. how fast the weight was pushed across the floor.  For the engine the Brake Power is given by:

Pwrb= ( Work per Revolution) x (Revolutions per second) = 2 x Pi x Tb x rpm / 60 = Pi x Tb x rpm / 30 Watts = Pi x Tb x rpm / 30000 kW (3) = (Pi x Tb x rpm / 30000) / 0.7457 Hp

The brake specific fuel consumption, bsfc, is then defined as:

bsfc = (fuel consumption rate) / (Power output) = M.f / Pwrb kg/kWh (4)

In order to understand bmep we must consider indicated mean effective pressure (imep) and make some comparisons with the theoretical ideal Otto cycle which the two-stroke cycle engine aspires to emulate.

An ideal Otto cycle is shown in Fig. 1 and can be described as follows : From position 1 at exhaust port closure the piston ideally compresses the gas in the cylinder, which has the properties of air, with no heat loss to the walls, (isentropically), to top dead centre (TDC). The gas is compressed according to the equation:

P x V ^ Gamma = Constant (5) where gamma is the ratio of the specific heats of the gas ( a number which is constant for a particular gas at a particular temperature).  Gamma is 1.4 for air at atmospheric temperature, for mixtures of air and fuel or mixtures of air and exhaust gas or pure exhaust gas gamma is less than 1.4. For pure exhaust gas at: 1000 Deg K  Gamma approx = 1.31, 500 Deg K  Gamma approx = 1.35

Heat is then added to the gas "instantaneously" at TDC and the cylinder pressure rises to a peak still at TDC.  The gas is then allowed to expand, again with no heat transfer, according to equation (5) until the exhaust port opens.  When the exhaust port opens the heat is rejected "instantaneously".  There is no gas exchange during the cycle, heat is only added and extracted from the gas, so the cycle does not have the open cycle of a two-stroke cycle engine, (the time when the exhaust and transfer ports are open).  The ideal Otto cycle is compared to a two-stroke cycle in Fig. 3 and the effects of non-ideal processes can be identified.   The pressure history can be presented in several different ways which are used to look at specific areas of the engine cycle. As well as the overall shape of the cycle ,several important parameters can be noted from Figs. 3 and 4, such as the cylinder pressure at exhaust port opening and closing, peak cycle pressure and location and if several consecutive cycles are plotted on the same axes cycle to cycle variation can be observed.

The coefficients of compression and expansion are the actual values of gamma measured from the real expansion and compression processes. These are difficult to determine from pressure - crankangle, or pressure - volume graphs, but my be determined from the gradient (steepness) of the straight lines if the data is plotted on as log P versus log V as shown in Fig. 5.  Mathematically this can be shown:

From eqtn 5 P x V ^ Gamma = constant
Log ( P x V ^ Gamma) = Log (constant)
Log P + Gamma  x Log V =  Konstant
Log P = - Gamma  x Log V + Konstant

Compare with y  =  m . X  + c

Where m is the gradient and c is the intercept, (where the line would cross the x axis if extended).

As stated above the real cycle does not follow the ideal and so the real coefficients of compression and expansion differ from 1.4, (ideal for pure air), due to temperature effects and gas composition.  The inefficiency of the burning, or heat addition process, can also be seen in Figs. 3 to 5 as the real cycle does not achieve the same peak cylinder pressure as the ideal and the peak is not at TDC.  The use of in-cylinder pressure traces and other data which can be calculated from it can help to give an understanding of the combustion processes happening within the cylinder and hopefully help develop an optimised combustion chamber, ignition timing and observe detonation.

The ideal work done during the cycle is the work done on the top of the piston during one complete revolution, (two-stroke). In this case the force F is created by the gas pressure p times the area of the piston A, the distance is x moved by the piston.  If the motion of the piston is separated into very small increments, dx, then the total work produced per cycle is the sum of all the work done during these small movements.  (I'm avoiding the use of words like calculus and integration)

Work produced per cycle = Sum of all  [ F x dx ] = Sum of all  [ p x A x dx] = Sum of all  [ p x dV ]

Where dV is the small change in cylinder volume caused when the piston moves a distance dx.  The imep or indicated mean effective pressure is the average or mean pressure in the cylinder, which when multiplied by the cylinder swept volume would produce the same work out of the cycle as the real pressure.  The ideal work from the cycle is the area enclosed by the loop as shown in Fig. 6 and is the same area as the shaded area calculated from the imep.  The use of the word indicated is historical and comes from the fact that the devices used to measure cylinder pressure were called indicators, and the graphs produced were called indicator diagrams.

The imep is the ideal average pressure which the engine can output.  Friction of bearings etc. and the pumping losses of the crankcase are not taken into account, it is calculated from the cylinder pressure only.  The bmep or brake mean effective pressure can be calculated from the measured torque and is related to the imep by the following equation.

bmep =  imep - fmep - pmep

Where fmep is the friction mean effective pressure and pmep is the pumping mean effective pressure.  All are calculated in the same way as imep ie. the pressure times swept volume which gives the required work per cycle used for friction or pumping.  Bmep is a useful comparitor of engine performance and state of tune, although, care must be taken to consider the rpm at which the engine is operating to determine the power potential.  Different engines can be compared directly as swept volume has been removed from the power or torque figures to calculate bmep.  The relevant equations are shown below along with a comparison of several engines tuned by Factory Pipe.

bmep = Power / ( Swept Volume  x revolutions per second)
bmep = 2  x  Pi  x Torque  /  (Swept Volume)