If I have an electric vehicle on an incline (m=17,3oo kg) I want to figure out how much energy it would generate by regenerative braking ... and maintain a set speed.
Change in potential energy per time is as good an approach as using kinetic energy.
The answers should be the same :-).
h = change in vertical axis
m = mass
g = gravitational acceleration.
In one second power = energy numerically.
Say g = 10 (more than good enough for 'playing')
1 kph = 1000 metres in 3600 seconds.
1 kph = 0.278 m/s (or 1 m/s = 3.6 kph)
V_vertical / V_vehicle = sin(slope angle)
V_vertical = V_vehicle x sin(slope_angle)
Mass = 17300 kg
First calculate for 1 metre/second vertical rate of change to get a general feel.
Power = energy change in one second at constant V
P = mgh = 17300 x 10 x 1 = 173,000 Joule/1s
= 173 kW at 1 m/s !!!!
At your example 45 degrees and 15 kph
V_vertical = 15 x sin(45) = 10.6 kph
10.6 kph = 10.6/3.6 m/s = 2.95 m/s vertically.
So power = 2.95 x 173 kW = 510 kW.
Change g to 9.8 instead of 10 and I get 500 kW
= your result.
This tells you that under the stated conditions your motor would be trying to regenerate at about 5 x it's rated power. It is almost certain that it would not achieve this, and quite likely that it would be damaged if you ran it in that manner. 500 kW (or even 100 kW) of magic smoke would be very impressive indeed!.
It also tells you that the motor could not drive the vehicle up a 45 degrees slope at that rate. Maximum speed would be about 105 kW/500 kW x 15 kph = 3 kph
This is not a vast surprise when you consider what you are asking for.
Braking at constant V at 15 kph is equivalent to lifting a 17 ton vehicle ~= 10+ cars vertically at 3 m/S
Useful rule of thumb
Energy ~= kg x metres lifted vertically x 10 x efficiency
This is good for energy required to lift mass OR power delivered by falling masses.
Power is energy per unit time so power can be used instead of energy is rates are in meters per second rather than total metres moves.
So, for water wheel
Power = litres/second x head in metres x 10 x efficiency
Power = litres/second x head in metres x 10 / efficiency
Note that efficiency Z = power out / power in in both cases, but
Efficiency for water wheel = Power delivered / Power from water flow
Efficiency for pump = Power in outlet water flow / Power used to drive pump