How do I select the right motor and gear ratio for a mid-drive e-Bike?

[evbkezz]

TL;DR Summary

Need help finding the correct motor and gear configuration for a 250W mid-drive.

Hey all!

I am working on my first ebike build and posted a question about friction drives earlier. Now, I have decided to go with a Mid-drive. I made calculations and found out the torque and rpm required for the build. But I am having difficulty find the right motor and gear ratio. I am trying to build something like the Brose S Mag, the size of the drive is so small but the power it delivers is insane. It delivers a torque of 90Nm. They said they use a planetary gear box. As per my understanding, they have used a high rpm motor and geared it to achieve the required torque and RPM. The place I live has a legal power limit of 250W and speed limit of 25Kmh for ebikes.

These are the specs of my bike:
GVW= 105kg
Wheel radius = 330mm
Max Speed = 25kmhI calculated the air resistance using the formula:

RA = rhoCdAV^2/2 where rho is density of air, Cd is the drag coefficient(0.88), A is area of contact in m^2(0.4) and v is velocity in m/s(6.94)
RA = 10.38NI calculated rolling resistance using the formula:

RR = GVW*Crr where GVW is total vehicle weight(105Kgs) and Crr is rolling resistance of tire on asphalt road(0.004)
RR = 4.12NI calculated grade resistance using the formula:

RG = GVW*sin theta where sin theta = 0.196 for 20% gradeablity
RR = 202.09N

The acceleration force required for 25kmh in 5 secs is 145.74N

The torque required when considering grade resistance is 83.6Nm
The torque required on flat roads is 37.15Nm
The RPM required for 25kmh with a wheel dia of 26" is 240 after including a fudge factor of 20%

I should now select a motor that can achieve the above figures and the rated power has to be below 250W and size of motor should be really small. I think using a high rpm motor with planetary gear reduction can help me achieve these numbers. I am planning to use a 36V 10A controller so the peak power can be till 306W assuming a 85% efficiency of the motor.
Idk what motor should I choose and what gear ratio I should use. Will RC motors work as they provide very high rpms and are small too?

A user on the other forum told me that I would have to design a gearbox such the peak power of the motor is delivered at 60RPM since the bicycle drivetrains are built for the average human rpm which is around 60. But when I look for motors online, manufactures aren't giving the peak power inform. How do I find the rpm and torque at the peak power? I should build a gear to reduce it to 60 RPM? What should be the ideal torque or RPM values before reduction? If I reduce the RPM to 60 will still be able to achieve 25Kmh?
Should I design it such that it exceeds the 83.6Nm torque at the highest gear?

Please do let me know if my calculations are right.

Thanks

 

[anorlunda]

Why a gear? A BLDC motor plus its digital controller work over a large range of RPM including zero. Electric motorcycles and scooters use them. They come in many sizes, and they are relatively affordable.

Edit: incorrect paragraph removed.

 

[evbkezz]

Why a gear? A BLDC motor plus its digital controller work over a large range of RPM including zero. Electric motorcycles and scooters use them. They come in many sizes, and they are relatively affordable.

Edit: incorrect paragraph removed.

Gear because the small BLDC motors don't provide enough torque.