Rider's contribution to power in pedal-assist mode

[Pluto]

To propel a bike at a constant speed, power needs to be developed at the rear hub to overcome the forces of friction, wind and gravity. Adding these forces, multiplying by the bike speed, and allowing for the mechanical efficiency of the drive-train forwards to the pedals, the power developed at the crankshaft can be computed from well-known equations (see, for example: https://support.strava.com/hc/en-us/articles/216917107-How-Strava-Calculates-Power).
In a pedal-assist bike, the question arises: how much of this power is supplied by the rider?
The following graphs give examples for two different masses (rider+bike+gear) and two different gradients. In the legend, Crr is the coefficient of rolling resistance for a properly inflated 700x45 tire, and CdA is the combined value of the air-drag coefficient and the frontal area for a rider in an upright position. The bike is taken to be a Specialized Como, for which the motor support in the ECO, SPORT and TURBO modes is quoted to be 35, 75 and 100%, respectively. To get the rider’s contribution to the total power developed at the crankshaft, pick a speed, then go up to the particular power mode of interest, and then read across to the power.
For example, in the 200-lb case at 24 km/h in ECO mode on the flat, the rider is putting out 120 W. With the motor OFF, the rider would need to develop 162 W.
Another example: in the260-lb case at 20 km/h in TURBO mode on a 5% hill, the rider is putting out 214 W. With the motor OFF, the rider would need to develop 428 W.

 

[Bruce Arnold]

Thanks for the data.

For comparison, it would be interesting to see how many watt-hours that same Specialized bike would consume under the same conditions, given maybe a 5 mile stretch of road.

From your graph, it appears that the big difference is between No Assist and Eco. The others add incremental value, but at what cost in battery drain?

 

[Pluto]

Thanks for your reply. I agree that your test would provide some very useful information; I wonder why the bike companies don't do it?

To make a stab at answering your question, let's consider a flat 10K course and a rider who maintains a power output of 100 W, irrespective of the power mode she's in. The total mass is not very important on the flat as rolling resistance is much less than wind resistance, but let's use the graph for the 200-lb case. In ECO, SPORT and TURBO, the motor's mechanical power at the crankshaft would be 35, 75 and 100 W, respectively, and the associated bike speeds would be 22.4, 24.8 and 26 km/h, ie, the times for the 10K ride would be 0.45, 0.40 and 0.38 h, respectively. Thus, the mechanical energies provided by the motor-drive unit for the 3 modes are 16, 30 and 38 Wh, respectively. We cannot go any further without knowing the electrical input- to mechanical output- power efficiency for various output powers and rotational speeds. I understand that, generally, over a certain load range, the efficiency of a dc motor increases and that a maximum efficiency of about 80% is not unreasonable. But I don't know how this relates to the Brose 1.2 motor-drive unit in the Specialized Como.

We could make a guess at the motor efficiency by assuming that, under the ECO conditions stated above, for example, the battery was fully drained after 80 km, ie, after 3.6 hours of riding. Guessing wildly at 65 Wh (500 mA at 36 V for 3.6 h) for the energy consumed by the control electronics and lights, the electrical energy supplied to the motor would be 395 Wh. The mechanical energy would be 128 Wh (16Whx3.6/0.45). Thus, the motor-drive efficiency would be 128/395=32%.

I know I haven't answered your question, but that's enough speculation for now!

 

[Pluto]

To answer Bruce's question, please see the chart kindly provided by Specialized's ebike centre in Switzlerland. They also told me that the lights, display, controls draw about 30 W, and that the motor itself runs at about 75-80% efficiency. The lower overall efficiency that I estimated is due to the gear/drive unit.

 

[Bruce Arnold]

Thanks, that's interesting. Looking at the graph at the 16mph line: going from Eco to Turbo gives you roughly an equivalent amount of boost as going from None to Eco. Looking at the second chart, you get less than half the mileage. I know that involves all kinds of interpolation and unanswered questions, but it sure makes you think that Turbo isn't worth it unless you're doing a fairly short ride and time is of the essence.