The slow ride: geeking out on energy

When I'm biking, I often have a chance to ponder things.  One thing I like to ponder is the amount of energy used to transport humans and things around.   My electric bike is a great way to explore that geeky subject.

Today I did an experiment.  I was riding my Big Dummy with a typical load, for a gross vehicle weight of around 300 lbs.  I only used the electric assist to "take the edge off" the hills.  I didn't use it to increase my speed (except on hills), or to accelerate from stops.  I was able to get my energy consumption for the round trip down to 6.3 watt hours per mile.  In other words, my round trip consumed 90 watt-hours.  Of course, I invested more of my own energy in the ride, because the ride was longer than usual, about 13 minutes extra.  But I didn't feel substantially more worn out or tired, because of using the electric on the hills.

For perspective, if I kept energy use at this level, with my single 10-lb LiFePO4 battery (48V/10Ah = 480 watt hours), I could ride 76 miles!  I typically use more like 16 watt hours/mile, which would reduce the range to 30 miles.  

The only real difference in the ride is the speed.  Normally I average around 18 mph for the ~15-18 mile ride (depending on route).  Today I averaged 13 mph.  Wow, that seems slow!  On my road bike, I would have averaged more like 17 mph on this route.  Why such a difference?

I did a little mapping of my route using bikely, an online bike route mapping tool.  It can show an elevation profile of the ride.  I was amazed to find out that during my round trip of 17 miles, I do a total of 1000 feet of climbing (gross, not net)!  That's more than when I lived in a Canyon in the mountains of Utah.  The difference is that here, it is not nearly as obvious, because that 1,000 feet comes as a series of small ups and downs (some of them quite steep).  So by the end of the ride into the office, I've only gained a net of 50 feet, even though I pedaled up 550 feet of hills (and 450 more feet of climb for the ride home, with a net loss of 50 feet). 

No wonder I was so slow on the cargo bike.  I'm hauling at least 60-70 pounds extra compared to the road bike, with the Xtracycle, all my gear, the heavy gearhub, the battery, etc.  Carrying 60-70 pounds up 1,000 feet is not a trivial energy investment.   This was very instructive - I suggest that readers try it for their own routes, they might be surprised.

Now when people ask me whether I will "wean myself" from my electric assist, I will answer them: "let's see you pedal a loaded cargo bike up a 17-mile, 1,000 foot climb every day without electric assist."  In reality, I would simply not do this ride every day (or more than 1-2 times per week) without electric.  I would just be too wiped out to have the energy for everything else that needs energy in my life (like Cycle 9, and my full-time day job as a professor, or kids). 

All of this generated some additional thoughts.  When I show people the batteries for electric kits, they often ask me, "does it really make up for its own weight?"  The answer is emphatically yes for shorter rides like this.  On a normal day, I will do my 17 mile 1,000 foot climb with a 300+ pound bike at an average of 17 mph (riding faster than most folks could do on a road bike for this route).  This definitely pays its own way.  But, I wonder, how much climbing could I do, before I would I reach the "break even" point, where further climbing would just be me dragging the battery + motor up the hill.  Here's a rough estimate:

The 48V/10Ah battery weighs 10 lbs, or 4.5 kg.  It holds 480 watt hours, which is about 1.72 million Joules.  We'll use Joules to figure out how much energy it takes to lift a 300 pound bike up a hill.  Neglecting friction for the moment, the energy used in climbing is given by m g h, where m=mass of bike, g=gravitational acceleration, and h is height.  So for my bike on this climb we have U (potential energy in metric units) = 136 kg * 9.8 m/s^2 * 304 m = 405 kJ of energy needed to lift the bike up those hills.  If we figure the hub motor is only 80% efficient, then we used 506 kJ of energy, about 1/3 of the battery capacity.   I'll address friction losses later.

Now we can estimate for just hauling the battery and motor up the hills: the motor is another 8 lbs, or 3.6 kg.  With controller and wiring added in, we'll call it an even 20 lbs, or 9 kg.  So, U = 9 kg * 9.9 m/s^2 * 304 m = 26,812 Joules, or accounting for 80% efficiency, 33.5 kJ.  In other words, the energy used by the motor system to lift itself up the hills consumes only 2% of the energy it holds in the battery.  We can approximate how far the system could lift itself before I would start having to lift it.  Here, U = 1.72MJ * 0.8 (80% efficiency) = 1.37MJ = 9 kg * 9.8 m/s^2 * h.  We want to solve for h, so: h = 1.37 MJ/(88.2 kg m/s^2) = 15 kM - yes, 15,568 meters.  This system most definitely pays for itself - even if I'm climbing mount Everest.  How high can it carry the whole bike and I, if I were too lazy to pedal?  h = 1.37 MJ/(1332 kg m/s^2) = 1,208 M, or about 3,374 feet.  Not bad - that's a pretty big mountain.

So there we have it - the electric system pays for itself in spades where climbing is involved.

But what about friction?

So far, I'm having trouble finding good numbers for friction on a bike.  I know this: The frictional losses due to wind resistance go up with the square of the velocity (v^2).  That's why when I increase my average speed from today's 13 mph to my regular 17 mph - only a 4 mph difference, or about 25% - my energy usage goes up so much.  I am burning up much more energy on friction - but I am also contributing less total leg power, because I have less total time spent pedaling. 

 Here's a rough estimate: we know that all the potential energy gained in the 1,000 ft of climbing is burned up as friction on the way down the hills (I don't use the brakes much since the hills are short).  Typically on the downhills I am traveling closer to 20mph average.  So, we can get a very rough estimate of total friction, by starting with U = 405 MJ = downhill frictional losses for the downhill parts, and figuring out how much I loose on the uphills.  If my speed on the uphills is 1/2 that on the downhills (average of 10 mph), then the total wind resistance losses are 1/4 of that on the downhills.  Plus, I'm sure there are some additional losses for bearings in the bike and static interaction of tire to road, which we'll add in as a fudge factor of 100 kJ.  So we have U = (405 kJ (downhill total) + 405 kJ * 0.25 (uphill wind) + 100 kJ (fudge)) / 0.8 (80% efficient) = 757 kJ = 210 watt hours.  Yay - that is about what I actually end up using, except on days that I travel really fast, or have particularly heavy loads.  So that means my estimate of frictional losses on the uphill parts, of about 205 kJ total, is in the ballpark.  That's about 57 watt hours, or 8 wh/mile burnt on friction when at low speeds (total input combining motor + legs - efficiency losses of both).

That's way geeky.  But, hey, it is nice to know where the energy goes, and more importantly, that my electric bike system really does carry its own weight (and then some).  It all boils down to this:

1. If I want to save energy, I go slowly.  It makes a very big difference.

2. At 10 mph, I'm burning around 6-8 wh/mile on friction (electric + pedaling), and at 20 mph it is around 25-30 wh/mile (electric + pedaling).  At 30 mph, this rises to 60-70 wh/mile.

3. The electric system will carry itself, and me along with it, up some very big hills, as long as I don't burn up its energy on friction by going fast

4. It is fun to ride a loaded cargo bike on a 17-mile, 1,000 ft hilly commute, often passing roadies and watching in my rear view mirror as they pedal really hard to try to keep up (and knowing that I'm being super energy efficient in the process)!