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Power estimation
© 2017 Mattia Piron. All rights reserved.
JANUARY 2nd, 2019
In the absence of a power test bench, it is possible to estimate the power of a vehicle by means of some practical tests. During an acceleration test, a vehicle goes from a state of zero energy, to one in which it has gained energy: kinetic (the speed) and potential (the altitude difference). Some of the energy will instead be lost through friction.
First approximation top
Suppose we perform an acceleration test on a flat road (zero potential energy), at the end of the test we will have reached a certain velocity v. The accumulated energy will be:
U = 0.5*m*v2
- U = kinetic energy;
- m = total mass of vehicle.
Regarding the units of measurement, I recommend to always use those of the S.I.. The average power used during the test is calculated by dividing the energy by the time:
P = 0.5*m*v2/t
- t = time taken to perform the acceleration.
For example, if I have a car that at the time of the test weighs 1600 kg (of course, including the driver) and accelerates from 0 to 100 km/h in 5 seconds, I calculate:
Second approximation top
Add inertia. In addition to accelerating the vehicle, the engine must also rotate the gearbox, wheels and any transmission shafts. Suppose we only consider the wheels, their kinetic energy at the end of the test will be:
Uwheel = 0.5*I*ω2
- Uwheel = kinetic energy of wheel;
- I = moment of inertia of the wheel;
- ω = rotation speed.
Consider the wheels as if they were uniform cylinders. With this simplification, their moment of inertia is:
I = mwheel*r2/2
- mwheel = mass of one wheel;
- r = wheel radius.
The rotation speed ω will depend on the forward speed and the wheel radius according to the relationship:
ω = v/r
By substituting, I find:
Uwheel = 0.25*mruota*v2
And the total energy will be:
Uwheels, car = mwheelv2 , Uwheels, bike = 0.5 * mwheelv2
This energy must be added to the kinetic energy of the entire vehicle, and the total energy must be divided by the time to get the power:
Pcar = v2 (0.5 m + mwheel)/t , Pbike = v2/2 (m + mwheel)/t
Consider the vehicle from the previous example, and assume that each wheel weighs 20 kg, then:
Third approximation top
Let us now consider the dissipative effects. The most important is aerodynamic drag, a force that opposes the motion of the vehicle:
FD = 0.5 ρ CD A v2
- ρ = air density;
- CD = drag coefficient;
- A = frontal area;
- v = speed;
The work dissipated by this force will be:
We do not know the relationship between velocity and time. At low speeds, the acceleration will be high, and the closer we get to the maximum speed, the lower the acceleration will be, until it is zero at the maximum speed. Thus, the function v(t) will be a decreasing curve, with a horizontal asymptote at maximum speed. Since we do not know this function, we approximate the velocity with a linear function, knowing that we are making a mistake:
v(t) = (v2 - v1)/(t2 - t1) * t + v1
- v1 = speed at the start of the acceleration test, zero if starting from stop;
- v2 = speed at the end of the acceleration test;
- t1 = time at the start of the acceleration test;
- t2 = time at the end of the acceleration test.
I thus find the work of the dissipative force. The general expression is:
If the acceleration test is with a standing start:
We add a correction coefficient, to account for the fact that v(t) is not linear:
C = 2.5 * (v/vmax)2 - 3.3 * (v/vmax) + 2.5
Where vmax is the maximum speed of the vehicle. We probably don't know the maximum speed, in that case we can consider C = 1.45, the error we make is small. Now, this energy must be added to the previous two. As before, dividing by the time I find the power. Consider again the previous example, let's assume that the car under test has a coefficient CD = 0.3, a frontal area equal to 2.6 m2 and that the air density is equal to 1.2 kg/m3:
Further Considerations top
It may also happen that the test is carried out on a road that is not flat, but uphill or downhill. In this case, the potential energy term should be added:
Uh = m * g * Δh
Where Δh is the difference in altitude between the finish and the start. It is important to consider the time required to change gears, and the total number of gear changes. This time should then be subtracted from the total test time. If the car in the example above gets to 60 km/h in second gear, and the gear change time is 0.4 seconds, then the time should not be considered 5 seconds, but 4.6. The power thus increases from 177 hp to 192 hp, a considerable difference. A power calculation made in this way is always an approximation, and much depends on the driver's ability to make the most of the vehicle's power. In addition, especially if you are testing a very powerful vehicle, any skids, wheelies (in the case of motorcycles) and friction will only increase the total time, reducing the value of the calculated power. For this reason, I recommend that you consider tests that have an arrival speed of at least 2/3 of the maximum speed, in order to "dilute" the losses in the very early stages of the test.
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