Bicycle Energy

by David S. Lawyer mailto:dave@lafn.org More bicycle articles by David S. Lawyer

February 2008 (original June 2007) minor changes March,July 2010 COPYRIGHT: (c) 2007-8 by David S. Lawyer. Feel free to make copies but commercial use of this work is prohibited. For example, you can't (except to an insignificant degree) combine it with advertising on the Internet. Please let me know of any errors or suggestions for improvement.
The fossil fuel energy required to travel on a typical bicycle is compared to that of a typical automobile. Based on the mechanical energy used (by calculating the forces needed for steady motion) the bicycle is roughly 10 times more efficient. But since the auto must brake from a higher velocity it uses relatively more braking energy than a bicycle so the ratio of 10 becomes about 13.5. But while an auto engine is rated about 50% higher in thermal efficiency than the "human engine", in actual urban use the auto engine is about 40% lower in thermal efficiency and about 10% of fuel is burned during idling. Result: The bicycle is about 25 times more efficient than the auto. But when one considers that for every Calorie of food eaten, it takes perhaps 15 Calories of fuel to grow, transport, and cook that food, then the bicycle is only about 1.7 time more energy efficient than the auto. Due to low annual miles bicycles are ridden, the energy cost of making bicycles per mile of use is quite high and estimating this is still a "to-do" item. In addition if one counts the energy cost of the extra time it takes to travel by bicycle, it's not clear that the bicycle is any more energy-efficient than the auto. But if one counts the energy expended by bicycling as the energy cost of healthy exercise and doesn't count it as an input to transportation, then the bicycle would be extremely energy-efficient.

1. Is this Still a "Work in Progress"?

2. Do Bicycles Save Energy?

3. Compare to an Automobile

4. Compare Vehicle Resistances

5. Braking Losses

6. Summary Thus Far and Roadmap

7. Efficiency of the Human "Engine" vs the Automotive Engine

8. Compare to a Small Car?

9. Fuel Required to Make Food

10. Biased and Unbiased Ways of Comparing a Bicycle to an Auto

11. Energy to Create and Maintain Humans, Bicycles, and Autos

12. Energy Value of Time Spent Riding a Bicycle

13. Exceptional Cases Favoring the Bicycle

14. Tentative Conclusions

15. Appendix: Vehicle Resistance and Drag

16. Appendix: Miscellaneous

17. Appendix: Fuel to Make Food

18. Appendix: Human Efficiency of Pedalling

19. Appendix: Automobile Efficiency

20. Appendix: Braking Energy

21. Appendix: Automobile Occupancy

22. Bicycle at Constant Power


1. Is this Still a "Work in Progress"?

It's been "in progress" for so long that it's about time to consider it finished with some desirable future improvements that may never get done by the original author. Also needed is a full discussion of the energy required to create and maintain human beings which is part of the problem of the energy value of human time and labor (which is far from being solved by anyone). Theres also the question of how much energy it takes to manufacture and maintain bicycles and autos. Also to do are estimates re the energy it takes to manufacture gasoline. Thus there are "to-do" notations in this article for these and other possible improvements. For author's notes on previous changes, see the "Comments" in the html source code.

2. Do Bicycles Save Energy?

Most people think that bicycles are very energy-efficient and thus save a lot of energy. They are only partly right: While the bicycle may not be very energy-efficient, it usually saves energy anyway. The Internet is loaded with erroneous statements that claim that the bicycle is much more energy-efficient than the automobile. Actually, when one considers that for every Calorie of food eaten it takes roughly ten or more Calories of fuel to create, distribute, and cook that food, then a bicycle is seemingly not much more energy efficient than the auto. But bicycles do save energy because they encourage less miles of travel since they are much slower than the auto. They also may save energy since much of the energy people expend riding them might be otherwise expended anyway for recreational exercise. Let's examine the details.

3. Compare to an Automobile

Let's compare the energy-intensity of a typical automobile to a typical bicycle when it's used for non-recreational purposes. Energy intensity is just the energy used per person-kilometer (or per passenger-mile). We will estimate the ratio of energy intensities (EI): how many times more energy intensive is the automobile than a bicycle? Is EIauto/EIbicycle = 4, 10, etc.

Using a dimensionless ratio like this means that it doesn't matter what units of energy intensity are used (miles or kilometers for distance; joules, Calories, BTUs, or foot-pounds for energy, etc.) since the units cancel when the ratio is formed. Of course, the units of energy intensity must be the same for the bicycle and the auto before dividing them to get a ratio.

The values of this ratio depend on what type of energy we are talking about. There's the mechanical energy (force times distance) used to move a person in a vehicle which results in a ratio of about 10 which becomes 13.5 after considering braking losses.

Then due to the human "engine" being more efficient than the auto, the auto uses about 25 times more Calories (of food or gasoline) than a bicycle. But considering that it takes (on average) perhaps 15 Calories of fossil fuel to create a Calorie of food that we put in our mouths, then the ratio in terms of initial fossil fuel becomes 25/15 or 1 2/3 (5/3).

To-do: estimate energy it takes to make gasoline.

Note that energy-intensity is the inverse of energy-efficiency. Thus the energy-intensity ratio we are finding (the auto over a bicycle, is identical to the ratio of energy efficiency of the bicycle over an auto. For example, saying that the auto is two times more energy intensive than the bicycle is the same as saying the bicycle is two times more energy efficient than the auto.

Now for the long discussion of how these rough numerical estimates presented above were obtained. We will make comparisons and find the ratios shown above (and other possible ratios) using various assumptions which will result in a wide variation of results. To repeat more precisely: The task ahead is to estimate how many times more energy efficient a bicycle is per person-mile of travel (not counting the energy required to make the vehicles or to construct highways).

4. Compare Vehicle Resistances

4.1 Introduction

A first step is to estimate how much more force is required to move a person in an automobile as compared to a person on a bicycle. This force is called vehicle resistance. It's the force required to keep the vehicle moving at steady speed on a level road. If you were pushing a coasting vehicle at steady speed on a level road (with the engine off or the bicyclist not pumping on the pedals) then the vehicle resistance is just the force you would apply with your hands to the vehicle as you pushed it. This first step is comparing the mechanical energy used in "cruising" at steady speed on level ground without any braking.

If you've ever pushed an auto by hand and also pushed a bicycle, you know it's a lot harder (takes much more force) to push an auto than a bicycle. Of course, when pushing a bicycle, it's assumed that there's a person on it which makes it harder to push than just a bicycle alone. But even so, the auto requires a lot more force.

Total vehicle resistance (or just "vehicle resistance") has two components: it's the sum of the rolling resistance and aerodynamic drag. Rolling resistance is just the force needed to push the vehicle forward at very low speed. Aerodynamic drag is the wind force due to the forward motion thru still air, mostly against the front of the vehicle. It you've every stuck your hand out of an auto window and felt the wind force against it you've felt aerodynamic drag.

Rolling resistance for autos and bicycles (and other rubber-tire vehicles) is just equal to the weight of the vehicle times a constant, known as the rolling resistance coefficient (or specific resistance) which depends on the tire design, inflation pressure, and road surface roughness. Aerodynamic drag is approximately equal to a constant times the square of the velocity. This constant depends on the shape and size of the vehicle. So the vehicle resistance force is thus just F = aw + bv^2 where a and b are constants, w is the vehicle weight, and v is velocity. The first term, aw, is rolling resistance and the second term, bv^2 is aerodynamic drag.

Total vehicle resistance is also the mechanical energy required to travel a unit distance (energy-intensity). For example 4 pounds of resistance force would result in 4 foot-pounds of energy expended in traversing one foot. Other units of force may be used such as Newtons, which is equal to Newton-meters per meter or joules/meter or KJ/km (since a Newton-meter is a joule).

We want to determine how much more force is required to push a person in an auto as compared to a person on a bicycle. Does it take 10 times as much force (or energy)? So we are seeking the ratio of forces, the auto resistance per person divided by the bicycle resistance (assuming only one person on the bicycle).

4.2 Rolling Resistance Comparison

At very slow speed where there is little wind resistance (aerodynamic drag), mechanical energy used per unit distance is mostly rolling resistance. Thus it's approximately the force with which you would push the auto or bicycle to keep it rolling slowly at a steady speed on an level road or path.

From the Appendix Rolling Resistance we see that for properly inflated tires, the specific rolling resistance of both a typical bicycle and automobile is roughly 1% (or 10 grams-force per kg of vehicle weight). But we need to know the resistance per person and also need to know the vehicle weights to convert specific resistance into the actual resistance force. So let's estimate that Weight of Loaded Bike is 220 lb. and that Weight of Loaded Auto is 3750 lb. with 1.25 persons ( Automobile Occupancy) in it (on average). The rolling resistance is just 1% of the vehicle weight resulting in 2.2 lb./person for the bicycle and 30 lb./person for the automobile. Thus the automobile uses nearly 14 times as much mechanical energy per person as the bicycle does at very slow speed (30/2.2 = 13.64 --about 14)

4.3 Aerodynamic Drag Comparison

But wait, the above 14 to 1 advantage of the bicycle is only at very slow speed. Aerodynamic Drag (the wind force one feels as they ride a bicycle) is directly proportional to the square of the velocity. Double the speed, and aerodynamic drag goes up four times. And the aerodynamic drag must be added to rolling resistance to get the total resistance and energy used.

Unfortunately bicycles have high aerodynamic drag. For a bicycle with the rider leaning only a little forward, the aerodynamic drag is about the same as for an automobile which can carry more people than the bicycle. See the Appendix Aerodynamic Drag. Thus for an auto transporting 1.25 persons, the aerodynamic drag per person is only 80% of the bicycle moving at the same speed. Thus, at the same speed, the auto is a little better in aero drag than the bicycle. But since autos usually move faster than bicycles the aero drag is usually much worse.

4.4 Total Resistance

From Total Resistance in the Appendix the resistance force per person for typical autos and bicycles is:

Automobile: F = 30  +  1.322 v^2  pounds-force (v in deca-miles/hour) 
Bicycle:    F =  2.2 + 1.653 v^2    "      "    "  "   "    "    "

If we go fast enough the aerodynamic term dominates (since it's proportional to v^2) and we can find a speed where the automobile resistance drops below that of the bicycle. This speed turns out to be over 90 mi/hr which is much higher than it is feasible to go on a bicycle. So for all practical purposes we may claim that the auto has much more resistance than the bicycle. At 11.6 miles/hour (v = 1.16 deca miles/hour) F about doubles for a bicycle (due to high aero-drag) but only increases about 5% for an auto.

4.5 Bicycle vs. Auto at Various Speeds

A typical bicycle speed on the level is about 20 km/hr or 12.5 mi/hr (See Average Bicycle Speed) At this speed resistance/person more than doubles, going from 2.2 lb. (at very slow speed) to 4.7 lb. resulting in 0.156 horsepower or about 500 kcal/hr of food energy consumption. See Power Cruising at 20 km/hr (12.5 mi/hr). While the bicycle went from 2.2 lb. to 4.7 lb of resistance, the auto only goes up from 30 lb. to 32 lb at 20 km/hr. So now the ratio of resistance is 32/4.7 or nearly 7 to 1 instead of 14 to 1 making the bicycle only 7 times more energy-efficient.

In the above, we compared the bicycle and the auto at the same speed, but we all know that the auto usually goes a lot faster so let's compare a bicycle at 20 km/hr (12.5 mi/hr) to an auto at quintuple the speed: 100 km/hr (62.4 mi/hr). Then the aero drag of the auto goes up almost 23 fold to 64 lb giving a total drag of 94 lb. The bicycle is still at 4.7 lb. so the auto now has 20 times the resistance of the bicycle. For a comparison where the auto goes at 60 km/hr (37.3 mi/hr), (triple the speed of the bicycle) the auto has about 10 times the resistance.

Which ratio should one use 7, 10, or 20 as found above or some other number? If we were comparing the use of a typical bicycle to an auto for a short trip on urban streets, then the ratio of 10 seems a reasonable compromise.

4.6 Compare at Same Speed and Load Factor?

One can object to the above ratio of 10 and claim that we are comparing apples to oranges. Shouldn't we compare bicycles and autos at the same speeds and the same load factors (percent of seats occupied = occupancy/capacity)? A 5-seat auto will use about 1/4 the mechanical energy per person if all 5 seats are occupied (instead of just 1.25 seats occupied). The result is that the 7 ratio (for identical speeds) is reduced to 1.75. Thus under identical conditions (100% of seats occupied and same speed) the auto uses about 75% more mechanical energy per person than the bicycle.

But since the typical real "conditions" today are far from identical, this conclusion is not valid for the real world unless we want to consider a situation say where car-pooling was mandatory, auto speed limits were set to say 15 mi/hr to save energy, and perhaps picking up hitchhikers was mandatory also to increase auto occupancy. However, if speed limits were this low, one could design a lightweight auto for low speed with a tiny engine which would use significantly less mechanical energy per person than the bicycle.

4.7 What About Grades?

While the bicycle has perhaps a 10-fold mechanical energy advantage on level ground, in going up a steep grade it has almost a 14-fold advantage, since an auto weighs about 14 times as much per passenger. Even if the grade isn't steep, the 10-fold advantage increases. Shouldn't rolling resistance include grades to account for this? No, it shouldn't since the energy used for going up a grade may be recovered (at least partially) when going downhill after climbing the grade (what goes up must come down). However, if one uses brakes when descending, then some of the energy used in climbing the grade is wasted in braking energy, but that will be accounted for in the next section on braking. So thus no explicit calculation will be made for, energy used due to climbing grades (but rolling resistance and aerodynamic drag used when climbing grades will of course be accounted for).

5. Braking Losses

Automobiles waste relatively more of their energy in braking than do bicycles. Automobiles not only brake with friction brakes like a bicycle but use their engines for braking. Except that a hybrid auto will recover some of the braking energy by charging its batteries. An auto enters a mild braking mode as soon as one takes ones foot off the gas pedal, unless one coasts by either putting the transmission in neutral or depressing the clutch.

The energy loss in braking is kinetic energy, which is just half of the mass times the velocity squared. If an auto brakes to a stop from 25 miles/hour it will thus loose 4 times the energy that it would lose if it braked starting from 12.5 miles/hour (the speed used in this article for bicycle comparison).

But a bicycle may need to stop even more often than an auto. This is because signals are often set so that automobile traffic can pass through a a number of signals on a stretch of road without stopping. But at bicycle speeds a bicycle will have to stop at many more signals than an auto. Autos can travel on freeways non-stop without braking (unless there's congestion) but bicycles aren't allowed there. Nevertheless, due to the higher speed of the auto, a much larger percentage of it's energy is lost in braking than the bicycle.

For braking, an urban auto expends about 1/3 of it's energy as compares to a guesstimated 10% for a bicycle. See Appendix: Braking Energy Thus only 2/3 rds of the auto's energy is available for useful work while 90% of the bicycle energy is available for this. This also includes energy used for braking on downgrades. The ratio of 90% to 2/3 rds is 1.35 thus increasing the bicycle's 10-fold advantage to 13.5-fold.

6. Summary Thus Far and Roadmap

So using the 13.5-fold mechanical energy efficiency advantage of the bicycle there are still three other considerations that drastically modify this value. First is the energy required to create and maintain the vehicle (including its engine, which for a bicycle is a human being). Second is the ratio of the Calories of fossil-fuel needed to produce a Calorie of food. Third is the energy value of peoples' time: since it takes more time to bicycle, there is less time to spend doing other things in life that could save energy. Fourth is the energy-efficiency of the internal combustion engine used in autos (including energy wasted idling) as compared to the energy-efficiency of humans, which we'll tackle next.

7. Efficiency of the Human "Engine" vs the Automotive Engine

7.1 Introduction

Since it takes about 13.5 times as much mechanical energy to move a bicycle rider a mile as a person in an auto, does it also take 13.5 times as many fuel Calories (where the fuel is either gasoline or food)? Gasoline has about 31,500 Calories per gallon (see Heat Content of Gasoline.

From the Appendix, Human Efficiency of Pedalling, we find that a human being is about 20% efficient in turning Calories of food into the useful work (energy) of pedalling. An automobile engine at best is about 30% efficient (50% better than the bicycle). See Appendix: Automobile Efficiency Except for two major problems: idling losses and lower auto efficiency at partial engine loading. Next we'll look at the later problem

7.2 Decrease in Auto Engine Efficiency at Low Torque

Automobile engines are so overpowered that their efficiency for urban driving (lower speeds than highway driving) is only around 12%. Near 30% efficiency is obtained only when high-powered autos are rapidly accelerating or climbing up steep grades. The bicycle rider efficiency can drop too, say to 17% if the bicyclist pedals move too fast, etc. (selects a very non-optimal gear) but it's not as severe a degradation as for the auto. So we have the strange situation where the maximal efficiency of the auto engine is 50% higher (better) than the bicyclist but in actual use it's about 40% lower (worse).

The reason that estimating 12% for automobile engine efficiency may be reasonable is that while going uphill or accelerating gives better efficiency, efficiency below 12% happens when an auto is driven downhill, when slowing down, and sometimes just cruising on level ground. If some cases the efficiency is negative when the engine is braking but still consuming fuel. Some engines can decrease or shut off fuel when braking with the engine and hybrids can recover some of the braking energy by using an electric motor as a generator.

Since the engine efficiency needs to be weighted by fuel flow rates, the weightings on these low efficiency conditions is quite low which gives more weight to the higher efficiencies. The previous Braking section estimated braking losses but didn't account for the fuel used during braking which is accounted for by lowering the average efficiency which results in an estimated overall 12% efficiency (for a non-hybrid), neglecting idling losses which will be considered next.

7.3 Idling

About 10% of gasoline energy is wasted in idling for urban automobiles. See the book: Transportation & Energy, by David Lloyd Greene. Eno Transportation Foundation, Inc. 1996. See p. 140.

You may think that a bicycle does idle in the sense the the rider is burning food energy even when stopped but so is a motorist when stopped. This energy will be accounted for later on when the value of a persons time is considered. The energy value of ones time, of course, includes the calories burned while at rest (when the bicycle or auto is stopped). The bicycle rider will seemingly use more caloric energy when stopped than an automobile rider since the bicyclist is recovering from exercise but this additional energy is assigned to the propulsion of the bicycle.

7.4 Direct Fuel Efficiency Ratio is 25

The 12% engine efficiency of the auto is reduced by the 10% losses in idling to a little under 11% (12% x .9 = 10.8%). Thus comparing the 20% efficiency of the bicycle "engine" (a person) to the 11% overall thermal efficiency of the auto results in a 20/11 fold advantage in efficiency for the bicycle. The bicycle's 13.5 fold advantage over the auto thus increases to 25-fold (13.5 x (20%/10.8%)).

8. Compare to a Small Car?

This article originally assumed the weight of the auto plus passengers was assumed to be only 3000 lb. which implies a small car. The current revision assumes a 3750 lb. medium sized car is being used since it represents the average car. But since a bicycle is small, shouldn't one compare it to an energy efficient small car? However, the bicycle used for comparison in this article is not the most efficient either: it's a typical bicycle used for utility purposes and is a compromise between a road bicycle and a mountain bicycle.

In the future, as gasoline prices increase, it's likely that there will be a shift to more efficient autos but for bicycles the trend may be in the opposite direction as less athletic people take up bicycling for utility purposes. Such people will be more concerned about bicycle comfort with a resulting shift away from the more efficient (but less comfortable to ride) road bicycles. This would support the comparison of a "compromise" bicycle to a small auto while this article compares a compromise bicycle to a medium sized auto.

Note that per Transportation Energy Data Book, the curb weight of a typical small auto was 3,286 lb. in 2006 (it was about 2,800 lb. in the 1980s). The 3000 lb. estimate would be for 2,765 lb. of auto with 235 lb. for the 1.25 passengers at 188 lb. each (including clothing and luggage). Thus the 3000 lb. auto plus passengers is one of the more efficient cars in the small car class. Since it weighs 80% of the 3750 lb. average auto, its rolling resistance will be 20% lower. It's aerodynamic drag will also be lower too, but not 20% lower since there are economies of scale in aerodynamic drag (doubling the weight of the auto doesn't double its aero drag). However at 60 km/hr only 38% of the resistance force is aero drag so the total resistance will still be almost 20% lower.

Furthermore, the engine efficiency will be significantly better for the small car since the engine will be more often operated at closer to its maximal efficiency operating points on its "brake specific fuel consumption map". So if the energy-intensity is 30% lower overall, then it's 17.5 times more energy-intensive than the bicycle (70% of 25 = 17.5).

This estimate can be checked by comparing the reported miles/gallon for an efficient small car to that for the average car. The average car should get 70% of the mi/gal of the small car if the above estimation is correct.

9. Fuel Required to Make Food

9.1 15 Calories of Fuel per Calorie of Food

It's not at all clear how much fossil fuel it takes to make the food that contains the energy that the bicyclist uses. Three estimates are that for every Calorie of food we eat, it takes either 10, 17, or 23 Calories of fossil fuel to create that food (including the energy used to transport it, process it, cook it, etc.) See Appendix: Fuel to Make Food So if we were to use a figure of 15 (approximate average of 10, 17, and 23) then the 25-fold advantage of the bicycle over the auto is reduced to 5/3-fold.

9.2 Better Energy-Efficiency in Food Production

In order to make the bicycle much more energy-efficient than the auto, we need to significantly lower the energy that it takes to create and transport food. That's much easier said than done. See Food, Energy, and Society.

One way to help do this is to eat less (or no) meat that is obtained by feeding animal (become a vegetarian). An exception is that eating meat from animals which grazed on natural range land doesn't consume much fossil fuel. If we reduced our energy inputs (such as fertilizers and pesticides) that are used to grow food, it would take far more land to feed the expanding world population and we just don't have this land. In fact we are already using land for agriculture that shouldn't have been used for agriculture resulting in excessive loss of topsoil and depletion of underground water supplies by irrigating land for food. So what is needed is a substantial decrease in population (negative population growth). Then we could use less energy to grow food in spite of the resulting lower yields per hectare (or acre). And then bicycle transportation would be much more energy-efficient.

10. Biased and Unbiased Ways of Comparing a Bicycle to an Auto

10.1 Introduction

If one searches the Internet, they will find numerous analyses claiming that a bicycle is several times more energy-efficient than the automobile. These analyses are likely wrong. Why are the results of other studies so biased favoring the bicycle? It's not likely due to any conspiracy or attempts to deceive. Instead, the reason is likely due to the nature of the food energy supplied for bicycle use. There's no way to directly measure food energy used on bicycle trips and even if there were you still wouldn't know how many Calories of fossil fuel it took to make that food. Let's first consider mechanical energy used on trips and the consumption of food Calories.

10.2 Ideal Bicycle Trip vs. Typical Auto Trip is Biased Favoring Bicycle

Making such a comparison is often done with no mention (or even understanding) of the bias. To estimate food Calories consumed on a bicycle, a person in excellent health pedals on a stationary "bicycle" and the oxygen consumption is measured. This enables calculating how much food energy is used to travel on a high quality, well maintained bicycle on a level road at constant speed by a well conditioned person.

Then this ideal bicycle efficiency is compared to actual automobile efficiency which is far from ideal. The energy used is based on measurement of the amount of liquid fuel consumed by cars in the real world. This auto fuel consumption is rather high due to poor auto maintenance: tires underinflated, brakes dragging, wheels misaligned and numerous other mechanical defects. In addition, autos waste energy braking and operating under conditions of low thermal engine efficiency. An extreme example of such low thermal efficiency is when the engine is being used as a brake. This efficiency is negative due to the engine using fuel (positive power input) while the car is braking (slowing with negative power output). Thus, comparing of a "real world" auto trip to an ideal bicycle on a level road will result in a bias which will strongly favor the bicycle. This is one reason why most other studies are biased.

10.3 Ideal Bicycle vs. Ideal Auto is Also Biased

Here "ideal" means the the vehicle is in perfect condition with the tires fully inflated. Also, they only travel on level ground and never do any braking. The weight of the vehicles is only typical so they are not fully "ideal". The comparison is made with the "ideal bicycle" an the "ideal auto", both moving at constant speed on a level road. However this will be very biased in favor of the auto since braking is ignored because the percentage of energy wasted in braking is much higher for the auto. Also, the auto is apt to be not as well maintained as the bicycle since bicyclists can more easily sense problems than motorists. For example, a bicyclist will likely detect brakes dragging while a motorist usually doesn't.

The initial part of this article compared the ideal utility bicycle to the ideal auto, comparing a typical bicycle and a typical auto in good condition, with tires properly inflated, etc. They both moved at constant speed on level ground. But later these results were corrected to account for the relatively higher efficiency of the human engine at low loads and the relatively higher waste of energy in automobile braking. These two effects taken together make the auto twice as energy intensive than it would otherwise be in an ideal auto vs. ideal bicycle comparison with no braking.

10.4 Real Bicycle vs. Real Auto

Another way to compare them is to compare fuel consumption under real conditions. This is easy to determine for the auto since one just checks gasoline sales and estimated vehicle miles which has been done by the U.S. Dept. of Energy. But for a bicycle it's not so easy to do it right and it's not done in this article although a rough estimate is made later on: Example of Real Bicycle vs. Real Auto

One needs to evaluate a bicycle used for utilitarian purposes (shopping, commuting to work, etc.). Such a typical utility bicycle may have tires underinflated, brakes slightly dragging, chain and suspension (if any) not well lubricated etc. with a saddle, frame, and suspension that absorbs (wastes) energy going over bumps.

Then one needs to consider the effect of hills. Just accounting for the energy lost in braking going downhill is not enough: Since the bicycle will be going much faster downhill, the resulting increased aerodynamic drag should be accounted for (and owing to the v-squared law, it quadruples if the velocity is doubled).

Also when putting high torque on the wheels (when climbing hills and accelerating), the tires flex, greatly increasing the rolling resistance. See Tire resistance due to torque. The aerodynamic effect of natural winds must be considered and a sidewind can be almost as detrimental as a headwind. See Effect of Natural Winds.

10.5 Example of Real Bicycle vs. Real Auto

Real Auto Energy-Intensity

The Dept. of Energy estimates 5489 BTU/automobile-mile. See Auto Energy Intensity. But for comparison to a bicycle, it needs to be increased to account for the higher energy intensity of urban driving, say to 6500 BTU/auto-mi (4.26 MJ/km). But this article estimated that autos with 1.25 persons in them use 25 times more energy per person-mi (or person-km) than the bicycle. So it estimates 31.25 (25 x 1.25) times more energy per auto-mile than per bicycle-mile. This implies that a bicycle gets 6500/31.25 = 208 BTU/bicycle-mi. If this figure is reasonable, it will imply that the 25 times estimate is approximately correct.

Now to check to see is 208 BTU/bicycle-mile is reasonable. This is BTU of food energy so to get mechanical energy, assuming the rider is 20% efficient in turning food energy into mechanical energy, we get 41.6 BTU-work/bicycle-mi (27.3 kJ/km) or 6.1 pounds of resistance force on the bicycle.

This 6.1 pounds may be reasonable since we had 4.5 pounds resistance at a steady 20 km/hr (12.5 mi/hr) under ideal conditions? The 4.5 is lower than 6.1 since it's under ideal conditions: fully inflated tires, no natural wind, no grades, etc. Thus the the 25-fold efficiency advantage of the bicycle derived by comparing the ideal auto and bicycle using a physics approach (and correcting for braking and engine efficiency) is also about the same as when a real auto is compared to a real bicycle using statistical data for the auto and the physics approach for the bicycle. Note that this 25-fold figure is for calorific valve of food energy at bicyclist's mouth and gasoline energy put into the gas tank at the gas station.

Let's look at the effect of a steep grade for a bicycle. Suppose there is a 4% upgrade on part of a trip. This adds to the 1% rolling resistance plus say another 1% for the flexing of the tire due to the high torque. So the energy expended going up the grade is over 3 times what it would be on the level. Specific resistance: 4.5/200 = 2.25% (on level); 4% + 1% + 2.25% (on upgrade). But no energy is used on the downgrade (its only dissipated in the brakes and in high aerodynamic drag). The result is about 61% more energy being used on the portion of the trip consisting of up and downgrades (equivalent drag = 7.25 lb. = 1.61 x 4.5). This example also helps show why the 6.1 lb resistance derived above may be reasonable for real bicycles on real trips since it may be more than 6.1 lb. on some uphill-downhill trips.

11. Energy to Create and Maintain Humans, Bicycles, and Autos

11.1 Statistical Comparisons

One way to estimate this is to find the cost of this and assume that energy is directly proportional to cost. For bicycles we could estimate this by dividing Bicycle Shop Revenue by the Bicycle Passenger-miles = 6.2 billion/yr. For 2001 this results in $5.1-billion/6.2-billion-Pass-Mi = 82 cents per bicycle-mile. To this one should add say 18 cents per mile for the cost of the bicycle rider's own labor in maintaining a bicycle.

Since costs exclude fuel energy (which has already been compared with the auto) and will also exclude insurance costs which will be included in accident cost (to-do), the auto costs (less fuel and insurance) might be roughly 20 cents/pass-mile (25 cents/mile for the auto itself) due to 1.25 occupancy).

Thus this result shows that the manufacturing and maintenance cost of a bicycle to be about 4 times as much as the auto per passenger mile. Since a bicycle weighs roughly 30 lbs. and a small auto roughly 3000 lb., the auto weighs about 100 times as much as the bicycle, yet statistics show it only costs 1/4 as much to buy and operate per passenger-mile. How is this possible?

For one, there is a lot more labor cost in each pound of bicycle. Since Miles per Bicycle-Year=62 while automobiles have Miles per Car-Year=12,000 each automobile is driven almost 200 (12,000/62) times as many miles per year as a bicycle. So while an auto may cost 50 times what a bicycle costs, one gets 250 (200 x 1.25) times the use out of it (in pass-mi), a 5 to 1 benefit/cost ratio for the auto as compared to a bicycle. If we assume that the auto lasts twice as long as the bicycle (See Bicycle Life Expectancy = 8.5 years Automobile Life Expectancy = 17 years and thus the initial expense is amortized over twice as long for the auto, the bicycle costs ten times as much in amortization (or depreciation) as the auto per passenger-mile.

Another way to look at this is that Miles per Auto-Lifetime = 200,000 while Miles per Bicycle-lifetime = 525. Thus an auto provides about 1.25 x 200 = 250 thousand passenger-miles during it's lifetime since in urban areas there are 1.25 persons/auto occupancy. This is nearly 500 times more than a bicycle, so if the auto costs 50 times more than the bicycle to purchase it costs ten times as much for the amortization (depreciation) of the initial cost of the bicycle per passenger-mile.

Just how much more energy of using a bicycle is caused by the production, distribution, and maintenance of the bicycle? to-do

11.2 Does it Really Cost a Lot More to Buy and Maintain a Bicycle?

To the above title should be added "and thus does it take a lot more energy per person-mile to buy and maintain a bicycle as compared to an auto? The answer today is that it does. This is mainly because many people are not riding their bicycles much which is mainly due to the major use of bicycles being recreational.

Many people purchase a bicycle because they want it for exercise and recreational riding. Then they may get more involved in other activities and/or their work so they don't have much time to ride it. They find that it's not as much fun as they thought, traffic makes it less enjoyable, and riding some bicycles may not be too comfortable due to the saddle and the need for the rider to lean forward. Other activities, such as walking, make it easier and safer to carry on a conversation while exercising. And then there's the sense of danger as cars pass by at close range. All of this results in many people riding their bicycle far less than they expected to and results in much a higher cost per mile (including energy use).

As far as how long bicycles last, one would expect bicycles to last more years than automobiles since they take a longer time to become obsolete. After all, the human engine that powers them doesn't change like automobile engines do. But the statistic of 8 1/2 years for the life of a bicycle is likely too low since it's based on the number of bicycles people say they own in surveys and might not include ones that people may own but are seldom (if ever) ridden (and may not be rideable). Thus 8 1/2 years may be how long a bicycle lasts in rideable condition.

But even if they do last 10 years or so, that's well under 1000 miles of lifetime for a bicycle as compared to an automobile's 200,000 miles. So there are real problems with bicycle longevity. Although a bicycle is otherwise OK, it may be discarded since repair parts are not available. to-do

So using the statistics of presented above is not really a fair comparison for the use of a bicycle for utility purposes. But it's important because it shows that present use of bicycles is quite costly and inefficient and that currently the bicycle uses excessive energy per passenger-mile for manufacture and maintenance.

11.3 How Much Should it Cost per Mile to Buy and Maintain a Bicycle?

to-do

12. Energy Value of Time Spent Riding a Bicycle

12.1 Introduction

Riding a bicycle somewhere usually takes a lot more time than going there by auto. What is the energy value of the additional time it takes to get somewhere by bicycle? It will depend on the person and the type of trip. For example, a highly skilled person may have a high value of time implying a high energy cost of time. Even utility uses of a bicycle may have some recreational component which should tend to decrease the energy cost of time.

12.2 Labor Accounting Method

While riding a bicycle at times may be recreation, at other times it may also be work. Often it's both. It depends on the circumstances. For example, riding where there are no autos with nice scenery like in the mountains, parks or shorelines, is likely to be recreational, especially if 2 bicyclist can ride side-by-side and converse with each other. But riding in heavy traffic alone may be more like work. Commuting to work on a bicycle might be considered part of ones job and the energy allocated to this would be about the same as the energy allocated to labor at a job. This energy of labor can be quite high. See the article Human Energy Accounting

Some of this high labor energy has already been accounted for by charging 10-15 calories of fossil fuel for every calorie one eats. But this article shows that about 100 calories of fuel are used by each person in society for every calorie one eats. This includes not only food but housing, clothing, health care, etc. But workers require the labor of other people to serve them, such as government workers, health care workers, retail sales workers, maintenance workers, etc. The energy these service workers use is allocated to production workers (often indirectly via other service workers) and ultimately allocated to what the production workers produce. So products that we purchase took a lot more energy to make than most people think (including published studies using mathematics and matrix algebra which are erroneous due to their failure to take human labor energy into account). All this can't be explained well in just a few paragraphs and it gets more complicated. For a production worker, the energy cost of labor may be as much as 1000 times the energy spent on the job (for a white-collar worker).

What does this mean for a bicyclist?

12.3 Opportunity Cost Method

Introduction

One way to look at it is by what an economist might call "opportunity cost". If the person spent the extra time it takes to travel by bicycle in some other way, would this save energy? It all depends on what the alternative activity is and the skills of the person in that activity.

For example, if the person spent the time educating themselves, their better knowledge might help them (and others whom they influence) both save energy and perhaps vote more rationally (which should save energy). Of course, if the person spent the saved time on an activity that uses a lot of energy, like recreation flying of a private airplane, then more energy would be consumed. But even in this case, one might argue that in rare cases, the diversion of piloting an airplane would help solve emotional problems that would have cost even more in energy consumption were they not resolved. One can think up all kinds of other examples that either save energy or waste more energy.

It seems that there is in fact, on average, a tradeoff between energy and time: by spending more time one can save energy. For example take shopping. If one spends more time at shopping (perhaps with the help of the Internet) one can purchase goods that use less energy and perhaps reduce the transportation energy used to obtain them. Or another possibility is to spend time repairing something instead of buying new. Or one could work on a job where the human labor saves energy and the tradeoff between energy and time is known as the "marginal rate of substitution" in economics (theory of production).

Finding a value

In the real world the tradeoff between energy and time is determined by one's value of time. For example, if a firm needed to increase production and there are two choices to achieve the same result: 1. hire a new employee; or 2. use more energy; they should pick the choice that is the least costly. The wage rate where the two choices are tied would give a time value to energy. So for the case of energy from gasoline where the wage rate is $10/hr. and gasoline cost $5/gallon, the tradeoff is 2 gallons of gasoline for 1 hour of time: the firm would break even if it paid for an extra hour of work to save 2 gallons of gasoline.

Now while a typical bicycle rider rides at 12.5 mi/hr. on the level, due to grades and stops the average speed is more like 10 mi/hr. So this means that for the above tradeoff, the time to go 10 miles is worth 2 gallons of gasoline, or the bicycle gets only 5 miles for 1 gallon (5 mi/gal) of time-value gasoline. The same energy-cost of time must be applied to the automobile. If the automobile averages 25 mi/hr then it gets 12.5 pass-mi/gal of time-value gasoline.

What happens to relative energy-efficiency in this case? Well, suppose the auto gets 25 pass-mi/gallon (of real gasoline) and the bicycle gets the equivalent of 50 pass-mi/gallon (it's twice as energy-efficient). Then add to these the time-value gasoline mi/gal resulting in 8 1/3 pass-mi/gal for the auto vs only 4 6/ll pass-mi/gal for the bicycle. The auto is thus nearly twice as energy-efficient when the energy cost of time is considered in this example. Note that the way one adds mi/gal numbers is to first take the inverse (find gal/mi), add the two inverses, and then take the inverse of this sum resulting in the total mi/gal.

12.4 Embodied Energy Method

Introduction

This method looks at a person as containing the energy it took to create and develop that person as well as the energy used to maintain that person. It includes the energy required for development both before s/he was born and part of the energy consumed during childhood. A person requires a lot more than just food, such as clothing, shelter, health care, and some degree of government. These all require energy and contribute both to development energy and to maintenance energy. All of this this energy becomes embodied in the person and is expended during the person's awake lifetime. The person's embodied energy is depreciated over time and use something like an automobile is depreciated both over time and miles driven. When the person dies, they should have used up all of their embodied energy.

For example, one sleeps at night and burns sugar and fat for energy to keep breathing, circulate blood, etc. (basal metabolism). The shelter where the person sleeps is also energy-depreciating during the night and the energy used for shelter heating, depreciation, and basal metabolism is allocated to the sleeper. But this energy depreciation is allocated during waking hours. In addition the developmental energy, including energy used for education is also depreciated during waking hours.

Is the rate of such energy-depreciation constant over time or does it depend on what activity the person is doing? It strongly depends on what one is doing. When doing mental work, one is depreciating the energy used in the past for one's education so the energy-depreciation rate is increased. When one is doing heavy physical work, the body is using more of its physical capabilities and the energy-depreciation rate should be higher, just like the depreciation rate on an automobile engine increases when more load is put on it.

There is a major difference between an automobile and a person regarding deterioration. If an automobile engine isn't used much, it doesn't deteriorate like the human engine does when it's not used much. So physical exercise is needed to maintain the human body. If the bicycle rider is getting enough exercise from non-bicycling activities, it seems reasonable to charge a higher rate of energy-depreciation to the bicyclist.

But in the opposite case where the bicyclist needs the exercise for health, the depreciation may even be negative. Some (or all) of the physical energy used while bicycling improves the persons health and results in a longer life expectancy. This energy becomes embedded in the person and is depreciated later on.

Finding a value

to-do

13. Exceptional Cases Favoring the Bicycle

There are a number of exceptional circumstances where a bicycle is likely to be much more energy-efficient than the automobile.

13.1 Very Short Trips

If the trip length is short, say only a few city blocks, the bicycle usually has a lot of advantages. A conventional automobile trip often requires warming up the engine which runs at a fast idle speed while backing out of a driveway, etc. During the warm-up period, it is both much more polluting and has higher fuel consumption.

Furthermore, the trips time may even be faster on a bicycle for such trips since it may not be feasible to park the auto right at the trip origin or destination. For example on a trip from an apartment to a store, the auto driver must first walk to the parking area where his auto is parked. Also, at the store destination he may need to park some distance away from the entrance door of the store. A bicycle, being much smaller, is more feasible to park close to the doors of both the apartment and and the store, thereby avoiding most of the walking time consumed by the auto driver.

Thus bicycles would be ideal for use where people lived very close to work, shopping, etc. See my article on travel reduction Travel Less.

13.2 Short-cuts Impassable to Autos

Sometimes there are shortcuts a bicycle can take that are much shorter than if one went by auto. If so, this advantage is mainly significant for short trips. By riding a bicycle on a path, pedestrian walkway, or carrying it along a stairway or across railroad tracks, much time may be saved in some cases. Some cases (like crossing railroad tracks where there is no crossing or riding on pedestrian walkways) may be of questionable legality. Again, the trip time may be shorter than for an auto trip.

13.3 Bypassing Heavy Auto Congestion

Where automobile traffic is heavily congested, a bicyclist may ofter pass up the autos by riding on the far right-hand side of the roadway or by sometimes riding on the sidewalk. Such a trip may not only be faster by bicycle but will use a lot less energy since most autos have their engines running at idle in the traffic jam and waste energy while going nowhere.

13.4 Special Cases Favoring the Auto

In the special cases described above the bicycle may be a few times more energy-efficient than the auto. But is there also a converse to this? Are there special cases where the auto will be more energy-efficient?

One such case is icy or snow-covered roads. A bicyclist may try to travel over them but it's dangerous. An accident by the bicyclist will have a high energy consumption due to all the medical facilities and people involved. Another case in favor of the auto is muddy roads. Still another case unfavorable to the bicyclist will be high winds, especially headwinds where forward progress is very slow and in some cases not feasible.

14. Tentative Conclusions

Using a bicycle under ordinary conditions, one can travel about 25 times as far on a Calorie of food than on a Calorie of gasoline (by automobile). Unfortunately, it took perhaps 15 Calories of fuel to create that Calorie of food, so the bicycle is only about 5/3 as energy-efficient per person-mile of travel if one doesn't count the energy cost of the extra time it takes to travel by bicycle.

It takes about 15 times more mechanical energy to transport one by auto as compared to a bicycle. Even though automobile engines are at best about 50% more efficient than a human bicyclist pedalling, the bicyclist is typically about 2/3 more efficient than the auto engine in converting food/fuel into useful work, due to the failure of autos to fully utilize their higher efficiency. So considering only the mechanical and engine efficiencies (including the human "engine") the bicycle is about 25 times more energy-efficient. But the high fossil fuel energy cost of making food reduces the 25-times advantage to less than 2-times. Counting the energy cost of the additional time it takes to get somewhere by bicycle may result in the overall energy-efficiency of the bicycle being worse than the auto

Then there is the high energy cost of making and maintaining bicycles and autos with quantitative estimates still in the "to-do" state and the situation is complex since currently the bicycle energy is high due to very poor utilization of most bicycles purchased. This might make the above "less than 2-times" advantage of the bicycle go even lower (to roughly one ??). In other words, the bicycle may not save any energy at all ??

But a bicycle does save energy in many cases, since it limits the distance people can travel and some (or all) of the energy used for bicycling might be used anyway for the exercise needed for health. To make it many times more more energy efficient (before counting the energy cost of time) requires greatly reducing the energy required to create and transport food. See Better Energy-Efficiency in Food Production as well as getting better utilization from new bicycles purchased.

15. Appendix: Vehicle Resistance and Drag

15.1 Appendix: Aerodynamic Drag

Definitions

This is the wind force against the bicycle (and rider). It's normally measured in still air where there are no natural winds blowing. For the drag of a bicycle rider (and bicycle), instead of determining a drag coefficient and frontal area, an "effective drag area" = "drag area" is often used. This is the frontal area which would give the correct drag force (using the standard drag formula) if the drag coefficient were assumed to be one. Thus the "drag area" is defined as the product of the actual drag coefficient and the actual frontal area. The "drag areas" reported below are thus not the actual frontal areas of the bicycle rider (and bicycle). They may be somewhat larger than the actual frontal area since it's claimed (see Hoerner, p. 3-14 below) that the actual drag coefficient ranges from 1.0 to 1.3.

Hoerner's Book

Hoerner, Sighard F., "Fluid Dynamic Drag", book published by the author, 1965.

Per p. 3-14, the "drag area" of a typical standing man facing the wind is 9 ft^2 and per p. 12-10 it's about 5.5 to 6.1 ft^2 for a motorcycle. These 3 values in square meters are 0.84 (standing man), 0.51, and 0.57 A typical bicycle has higher drag than a motorcycle since the motorcycle rider is in a low seated position, but bicycle drag should be less than for a standing person. Taking an average of 0.57 and 0.84 gives 0.70 m^2.

Pivit's Article

Pivit, Rainer, "Measuring Aerodynamic Drag" (bicycles), published in Radfahren 2/1990, pp. 47 - 49. Translated from German by Damon Rinard. Measuring Bicycle Aerodynamic Drag.

Near the end of this article the effective drag area is reported to be about 0.35 m^2 for a racing bicycle and 0.71 m^2 for an all terrain bicycle. This compares with about 0.6 m^2 for the typical automobile. For comparing the bicycle to the auto I'll assume that they both have about the same aerodynamic drag with a drag area of 0.6 m^2.

Comparing Pivit and Hoerner

These above data by Pivit are roughly the same as reported by Hoerner's book where automobile drag is shown on p. 12-7. Since Hoerner covers old autos of the 1930's including futuristic designs, assume a drag coefficient of 0.4 and a frontal area of 1.5 m^2 (16 ft^2) resulting in .6 m^2, the same as per Pivit. Hoerner shows a small streamlined 1935 "Auto-Union" (Germany; Today it's Audi) motorcar with frontal area 17 ft^2 having a drag coefficient of 0.44. What was the best case of the 1930's was not much worse than the typical auto today.

Aero Drag Force Hoerner's air density, p is for an air pressure of 760 mm mercury (sea level) and 15 deg C (59 deg F) temperature.

So what is the formula for the aerodynamic drag force F given the velocity v and effective area A? It's just F = 1/2 pAv^2 where p is the mass density of air. See Hoerner p. 1-9, equations (3) and (5) Using p = 0.00238 'lb' sec^2 / ft^4 per p. 1-10 of Hoerner (just above equation 16) and a drag area of 0.6 m^2, one obtains
F = 1.653 v^2 pounds-force for both the bicycle and the auto where v is velocity in deca mi/hr.

Effect of Natural Winds

The natural winds is much more than just the increase in the frontal headwind (riding against the wind). Sidewinds are more detrimental than one would think at first glance. For example, for an automobile going at 60 mi/hr, a 20 mi/hr sidewind (perpendicular to the direction of travel) can increase the aerodynamic drag in the forward direction by 28%. See the Russian article: "Aerodinamicheskie ispytaniya avtomobilya GAZ-24" in "Avtomobil'naya promyshlennost'" No. 3, 1969, pp. 15-17. This shows an increase in the drag coefficient from 0.325 to 0.375 when the direction of air flow is changed to 20 degree off straight-ahead (for the GAZ-21 auto). This effect will exist both for and automobile and a bicycle but there may be no experimental results for a bicycle.

15.2 Rolling Resistance

Introduction

Rolling resistance (RR) is given here as parts per-thousand which is 1000 times the rolling-resistance coefficient (RRC). RR's for inflated rubber tires range from 3 (racing bicycle on smooth surface) to 20 (old bias-ply automobile tires).

Bicycle Rolling Resistance

A major problem is that bicyclists interested in rolling resistance are usually racers. So most of the data available is for racing bicycles which fall into the category of high-end "road bicycles". But we want to compare a typical bicycle to the typical auto.

Wilson, David "Bicycling Science:" (book) 2004. p. 162, RR is 6 (a misprint showed the RRC as 0.06 which should be 0.006). At the end of the Pivit article values of 3.2 and 3.5 are given for a road bicycle on a smooth plastic-coated floor. It's suggested that for "rough bitumen" (the typical road surface ??) these values should be multiplied by 2 resulting in RR's 6.4 and 7.

Rolling resistance is Very dependent on tire inflation pressure. See Rolling Resistance of Bike Tires. These graphs show (for various bicycle tires) how the rolling resistance varies with tire pressure. Single wheels were tested while rolling on a large smooth steel drum. A specified load (50 kg. or 30 kg.) was applied to push the wheel down on the drum. To get RR from these curves you divide the grams (gm) of resistance (shown on the vertical axis of the graph on the Internet) by the load in kg.

Inspecting these curves shows that for a mountain bike with 1.75 in. wide tires, even at the high pressure of 70 psi the RR is about 17. This is about 3 times as much as that of racing bike tires inflated to pressures well over 100 psi. So for a typical bicycle tire properly inflated, I will use a RR of 10 for this article, which turns out to be the same as used for automobile tires.

Automobile Rolling Resistance

The 2006: Transportation Research Board Special Report 286 is Tires and Passenger Vehicle Fuel Economy ... Per p. 17, RR ranges from 7 to 14 for new tires. Tires on new cars averaged 9.1 while replacement tires averaged 11.2. So using 10 seems reasonable.

The book "Mechanics of Pneumatic Tires", U.S. National Bureau of Standards Monograph 122, 1971, has a number of significant sections. In the section "Straight Line Rolling Experiments" pp. 553-630 by van Eldik Thieme, H.C.A., it's claimed on p. 601 that when high torque is applied to the wheels, the rolling resistance may double. Such torque deforms the tire which consumes energy. This effect will significantly increase rolling resistance at high speeds or steep grades for both the auto and bicycle. Since it may affect bicycles and autos roughly the same, it's not accounted for in this article, but needs to be accounted for in a more exact study.

15.3 Total Resistance: Bicycle and Auto

For level terrain with no wind and using the results above, we can write a formula for the total resistance in force (F) per person for the typical automobile and bicycle, assuming there are 1.25 persons in a 3750 pound auto (including weight of the passengers) and that the bicycle with rider weighs 220 pounds. The resistance per person for the auto is just 80% of the auto's resistance. Rolling resistance for both cases is 1%. The bicycle Aerodynamic Drag Formula is used.

Automobile: F = 30  +  1.322 v^2  pounds-force (v in deca-miles/hour) 
Bicycle:    F =  2.2 + 1.653 v^2    "      "    "  "   "    "    "

For example, for the bicycle at 10 mi/hr (1 deca-mi/hr) F = 2.2 + 1.653 = 3.853 lb. From the 2.2 and 30 numbers, we see that at very low speeds, the bicycle is almost 14 times more energy-efficient for mechanical energy. At 11.5 mi/hr the bicycle rolling resistance of 2.2 lb. is equal to its aerodynamic drag and total resistance has doubled. This doesn't happen to the auto until 47.6 mi/hr. If we equate the two formulas above and solve for v, we find the speed where the resistance per person is equal (the auto is no more energy-efficient than the bicycle). It's over 90 mi/hr (much faster than it's feasible to ride a bicycle).

16. Appendix: Miscellaneous

16.1 Weight of Loaded Bike is 220 lb.

Per the U.S. Center for Disease Control (CDC) the weight of people 2002 has been increasing over time. In the U.S. it's (as of 2002) 191 lb. for men and 166 lb. for women (average 178.5 lb). It seems that these weights are without clothing and excluded pregnant women. With a 32 lb. bicycle, 6 lb. of clothing (including shoes, stuff in pockets, and possible helmet, raingear) and 3.5 lb. of luggage the result is 220 lb. for bicycle with rider.

16.2 Weight of Loaded Auto is 3750 lb.

Using the tables in Transportation Energy Data Book (Edition 26), Chapter 4: Light Vehicles and Characteristics : "Sales Weighted Curb Weight of New Domestic and Import Cars by Size Class ..." and then weighting the size classes by market share of small, midsize, and large cars (and station wagons too) using the table "Light Vehicle Market Shares by Size Class" results in an overall average curb weight of 3,552 pounds for 2006. "Curb weight" includes the weight of a full tank of gasoline. Then add the weight of 1.25 persons in it using the same estimates as for bicycles as above, giving 1.25 x 188 = 235 lb. of passenger weight resulting in 3787 lb. of vehicle weight. Let's round this to 3750 lb. (the gas tank may not be full).

16.3 Heat Content of Gasoline

Automotive gasoline has a high heat of combustion of 125k BTU/gallon reported in Appendix B - Transportation Energy Data Book. Click on "Heat Content of Various Fuels". 125k BTU/gallon is equivalent to 31,500 Calories/gallon

16.4 Automobile Energy Intensity

For the latest estimates see Transportation Energy Data Book, Ch. 2. Examine the Excel table: "Energy Intensities of Highway Passenger Modes". In 2004 it was 5489 BTU/automobile-mile. This is for both city and inter-city driving so the actual value for city driving will be higher but for inter-city driving (or non-congested freeway driving) it should be lower.

16.5 Average Bicycle Speed

This article uses 20 km/hr or 12.5 mi/hr as the average bicycling speed on level ground. It is for typical bicycles ridden for utility purposes and includes mountain bicycles. There doesn't seem to be any hard data on this and the people who keep track of their speeds usually report higher speeds since they ride road bicycles and are not typical riders.

A Chapter 19: Bicycles Highway Capacity Manual (for 2000) seems to use 24 km/hr. One way to estimate speed is to Google: bicycle "average speed". Then ignore all racing (and the like) and realize that some average speeds will be low due to uphill sections. Also consider that people who go to the trouble to record their speeds and report them on the Internet may tend to make better speed than the average utility cyclist.

In Bicycle Racing (Tour de France Amitabha Mukerjee writes regarding bicycle racing speeds of 50 km/hr:" 50 kmph? Most of us are lucky if we make 20 km/hr on a bicycle, and then we surely can't keep it up for hours, let alone days."

16.6 Power Used at 20 km/hr (12.5 mi/hr)

The power used for the assumed speed of 12.5 mi/hr (estimated above) and and force of 4.7 pounds (found using Total Resistance formula) is just the force times the velocity (12.5 x 4.5 lb-mi/hr). After converting to other units, the power is 0.156 horsepower which is about 75% of a manpower (0.2 horsepower). It's also 117 watts or 100 Calories/hr of mechanical energy. But since a person must eat about 5 Calories of food to produce 1 Calorie of physical work (20% efficiency), the bicyclist is burning 500 kcal/hr (Calories/hr) of direct food energy. This is about 5 times the energy a person uses while sitting still (100 Cal/hr) and about 4 times the energy used at a desk job (125 Cal/hr).

500 kcal/hr at 12.5 mi/hr is 40 kcal/mi (500/12.5).

16.7 Bicycle Passenger-miles = 6.2 billion/yr

For 2001 the passenger-miles of bicycles is estimated at 6.2 billion as reported by the Bureau of Transportation Statistics (BTS) based on data from the National Household Travel Survey: BTS Daily Travel by Walking and Bicycling (from NHTS 2001)

16.8 Bicycle Shop Revenue

For 2002 bicycle shop revenue is estimated at $5.2 billion per the National Bicycle Dealers Association's Industry Overview 2006. Linear interpolation would estimate $5.1 billion for 2001.

16.9 Number of Bicycles in the U.S. = 100 million

From the 2001 National Household Travel Survey as reported by the Bureau of Transportation Statistics (BTS) in BTS | Table A-2 - Mean Number of Drivers, Vehicles, and Bicycles per Household There are 0.86 bicycles per household. But how many households are there? Unable to find this number, I noted that (in the same report) there are 1.90 personal use vehicles per household (cars, SUVs, etc.). Then per Transportation Energy Data Book, edition 26, Table 4.1: Summary Statistics for Cars, and Table 4.2: Summary Statistics for Two-Axle, Four-Tire Trucks (includes SUVs) there were about 221.8 million personal use vehicles in 2001. So dividing this figure by 1.9 of such vehicles per household results in 116.7 million households in the U.S. at that time. Multiplying this by 0.86 bicycles per household gives 100 million full size bicycles.

16.10 Bicycle Sales = 12 million/yr.

Per the National Bicycle Dealers Association Industry Overview 2006 bicycle sales 1990-2006 in the U.S. ranged from 11-14 million with an average of roughly 12 million (for bicycles with tire size 20" or over).

16.11 Bicycle Life Expectancy = 8.5 years

Dividing Bicycle Sales = 12 million/yr. into Number of Bicycles in the U.S. = 100 million results in a typical bicycle life expectancy of 8.5 years.

16.12 Automobile Life Expectancy = 17 years

Per Transportation Energy Data Book edition 26, Table 3.8: Car Scrappage and Survival Rates, the median life of an auto is 16.9 years (for a the 1990 model year). Hand calculation of the mean gives about the same (17.1 years).

16.13 Miles per Bicycle-Year=62

Dividing Number of Bicycles in the U.S. = 100 million into Bicycle Passenger-miles = 6.2 billion results in bicycles being ridden only 62 miles/year on average.

16.14 Miles per Auto-year = 12,000

From Transportation Energy Data Book, edition 26 (based on the 2001 NHTS) in Table 8.9: Average Annual Miles per Household Vehicle by Vehicle Age, it shows 11,100 miles (for all vehicles: of all ages). Another method is is divide 1628 billion car-miles in 2001 (Table 3.4: Shares of Highway Vehicle-Miles Traveled by Vehicle Type) by the 137.6 million cars (Table 3.3: U.S. Cars and Trucks in Use). The result is 11,830 miles per year. The higher number implies that cars are driven more miles than SUV's and other non-car personal transportation motor vehicles. For 2005 the figure is 12,370 miles per year.

16.15 Miles per Bicycle-lifetime = 525

One way to estimate this is to multiply Miles per Bicycle-Year=62 by Bicycle Life Expectancy = 8.5 years resulting in 527 miles. Another method using direct statistical data is to assume that bicycles are in a steady state and that the yearly sales of bicycles is the same as the yearly disposal of bicycles. One may assign the miles per bicycle year to these disposed bicycles. This way all miles are assigned to bicycles without double counting, etc. Thus to get miles per lifetime one divides Bicycle Passenger-miles = 6.2 billion/yr by Bicycle Sales = 12 million/yr. resulting in 517 miles, the same as above (slight difference due to rounding).

16.16 Miles per Auto-lifetime = 200,000

This will use the same methods as for finding Miles per bicycle lifetime = 525 using data from: Transportation Energy Data Book, edition 26 (U.S. Dept. of Energy). The first estimate multiplies Miles per Auto-year = 12,000 by (ref id="auto_life_expectancy" name="Automobile Life Expectancy = 17 years"), resulting in 204,000 miles.

To get the second estimate, divide 1689 billion car-miles in 2005 (Table 3.4: Shares of Highway Vehicle-Miles Traveled by Vehicle Type) by the 7.67 million cars sold (Table 4.5: New Retail Car Sales in the United States) to get 220,000 miles. However per this table there has been a decline in car sales from the mid-1980's so the number of cars scrapped in 2005 is likely greater than the number sold resulting in 220,000 being too high of an estimate. Thus a lower and rounded estimate of 200,000 is used.

17. Appendix: Fuel to Make Food

17.1 Introduction

It's often claimed that for every Calorie of food one eats, it took (on average) about 10 Calories of fuel energy to grow, harvest, process, transport, and cook that food. The source of this 10 ratio is often not cited. Is it really 10? A couple of sources for it seem to be based on research in the 1970s by Eric Hirst and David Pimentel. After some preliminaries we'll look at these studies.

17.2 Preliminaries: Food Calories, 1970 and 2000

Some studies will claim that a certain percentage (e.g. 12%) of total energy consumption (of mostly fossil fuel) in the U.S is used to produce food. But we would like to use this 12% figure to find out how many Calories of fuel it takes to make a Calorie of food. To do this we need to estimate how many food Calories are consumed in the U.S. each year and divide this by fossil fuel Calories consumed to obtain the amount of food Calories eaten as a percentage of the total energy consumed. For example, if food Calories are 2% of fossil fuel Calories, then if 12% of our fossil fuel is used to produce food, it means that it took 6 ( 12%/2% ) Calories of fuel to produce each Calorie of food

We'll estimate these values for both 1970 and 2000. Let's start with looking up total energy consumed in those years. Per Minerals Yearbook (U.S. Bureau of Mines), 1971, p. 23, table 8: "Gross consumption of energy resources ..." we find 67.2 quadrillion BTU (aka a "quad") of energy consumed in 1970. Per Table 1.3 Energy Consumption by Source, 1949-2005 from "Annual Energy Review" of the Energy Information Administration (U.S.) it was 98.98 quads in 2000 (67.84 quads in 1970 but we'll use the 67.2 quads per above).

Now, to find food Calories. Per Statistical Abstract of the U.S., 1970 population was 203 million in 1970. The amount of Calories eaten per day was estimated by surveys reported in: Centers for Disease Control and Prevention. (Feb. 6, 2004) "Trends in Intake of Energy and Macronutrients -- United States, 1971-2000." MMRW Weekly 53(04):80-82. Available at CDC and at http://www.cdc.gov/mmwr/preview/mmwrhtml/mm5304a3.htm.

For 1971 for people between the ages of 20 and 74, the average man ate 2450 Calories (same as kcal) and the average woman ate 1542 Calories. These average to about 2000 Calories/day. For the year 2000 the average was about 2250 (people are eating more). But Since the population figure includes children and very old people (outside of the 20-74 age range) who eat less food than average, the 2000 Calories/day figure has been reduced by me to 1800 Calories/day. This was done by assuming that 80 % of the population consumes 2000 Calories/day with 20% (mostly children and very old people) consuming on average 1000 Calories/day.

Using the 1800 Calories/day estimate, the food energy eaten in 1970 was: 203 million-persons x 1800 Calories/person-day x 365 days/year x 3.968 Calories/BTU = 0.47 quadrillion-BTU/year of food. In 2000 it's 281 million-persons x 2025 Calories/person-day x 365 days/year x 3.968 Calories/BTU = 0.82 quadrillion-BTU/year.

The 0.47 quadrillion BTU for 1970 is only 0.70% of the 67.2 quadrillion BTUs consumed in 1970. For the year 2000, the 0.83 quads is 0.83% of the 99.0 quads consumed in 2000. Thus (neglecting the energy it takes to make food) food Calories are well under 1% of our energy consumption:


        Food Calories as a Percent of Total Energy Consumption
        1970:  0.70%        2000: 0.82%           

17.3 Eric Hirst's Study, 1974

"Food Related Energy Requirements" by Eric Hirst in Science, 12 April 1974 (vol. 184 No. 4133) pp. 134-138. See abstract at Food-Related Energy Requirements by Hirst, 1974.

Hirst claims that about 12% of total energy consumed by the U.S. is used to create food (including it's distribution and preparation). But we found in Preliminaries: Food Calories, 1970 and 2000 (last paragraph) that in 1970 food Calories consumed represented only 0.70% of the total energy used in the U.S. Thus for every Calorie of food eaten, it took 17 Calories of energy to created it (12/0.7 = 17). This is significantly higher than the figure of 10 commonly given.

Why is this so high? For one, Hirst used input-output analysis using Leontief matrices. I can find no satisfactory explanation of the Leontief matrix on the Internet but see Leontief-Matrix (DOC) and Simple Leontief Model. This method accounts for both direct and indirect input from all sectors of the economy to the food industries. If properly formulated, it doesn't miss a thing. It is equivalent to summing an infinite number of components (where the sums converge, of course, to a finite value). Hirst discusses the practical limitations of this method, and why it's not possible to "properly formulate" it as mentioned above.

Later investigations didn't use the sophisticated input-output analysis used by Hirst and thus may have missed a lot of components needed to produce food. Thus the low ?? value of 10 Calories-fuel per Calorie of food. To resolve this issue (Is 10 a good estimate or should it be higher?) requires redoing input-output analysis but it's not easy to do due to the huge amount of imported goods into the United States for which we don't have much data on the energy it took to produce them, etc.

17.4 David Pimentel's Study

See the book: "Food, Energy and Society" by David Pimentel and Marcia Pimentel (editors). University of Colorado Press. 1996. (Third edition to be published in late 2007 by CRC Press).

On p. 8 and again on p. 290 it's claimed that about 17% of fossil fuel energy used in the US goes into food production. But since about 0.75% of such fossil fuel energy is represented by food Calories, there is thus about 23 Calories of fuels consumed to create each Calorie of food eaten. 0.75% represents a compromise between the 0.70% figure for 1970 and the 0.83% figure for 2000. See Preliminaries: Food Calories, 1970 and 2000 (last paragraph).

This value of 23 Calories-fuel per Calorie-food is even higher than the 17 obtained by Hirst. Why it is so high hasn't yet been researched by the author yet.

18. Appendix: Human Efficiency of Pedalling

18.1 Net vs. Gross Efficiency

Gross efficiency is the physical work output divided by the caloric energy from the food burned (metabolized) during the exercise. Net efficiency is similar except that the energy burned while one is resting (to breath, pump blood, etc.) is subtracted from the caloric energy of the total food burned. Thus the net case only considers how much extra food Calories you would have to eat to perform the exercise. The energy one uses while resting is known as basal metabolism. For the calculating of efficiency, this article uses net efficiency.

18.2 Measuring Energy Used by a Bicyclist

The way the caloric energy of food consumed is estimated is to measure the oxygen consumption of the bicyclist by measuring the oxygen content of exhaled air. Then assume that the energy is obtained by the oxidation of sugar (burning sugar), write the chemical formula of the reaction, and from it determine how much sugar was burned. Since we know the Calories in sugar, the calories of energy burned is found. The sugar the body uses for energy is not the same as the ordinary sugar that we eat. But it's not quite the simple as all this since people often burn some fat as well. But that can be accounted for in a related manner by measuring the amount of carbon dioxide exhaled (in addition to oxygen).

Measurements are usually started only after the person has exercised a bit, since at the start of exercising one doesn't use much oxygen since energy is used at first which doesn't require any oxygen (called anaerobic).

18.3 S. Dickinson's measurement, 1929

See: The efficiency of bicycle pedalling as affected by speed and load by Dickinson, S. in J. Physiol. (1929) 67: 242-255. Net efficiency (after subtracting basal metabolism) is shown in Fig. 1 (p. 246). At optimal cadence it shows 22% efficiency.

Is 22% efficiency too low?

Since this is net efficiency, one might expect it to be higher but the following three factors in the experiment may have tended to make it lower:

Energy debt method

Measurement of oxygen consumption started as soon as pedalling started. Thus, at first, the oxygen consumption is low and an energy debt in incurred which is paid back after pedalling ceases (recovery). This recovery oxygen consumption was thus measured. In all cases basal metabolism oxygen consumption was subtracted. Could there be "interest" assessed by the body on energy debt, resulting in a higher payback? This would decrease efficiency.

Assumed no fat burned

It was assumed that no fat energy was burned by the bicyclist. Fat has about 4.7 kcal/literO2 instead of 5.10 kcal/literO2. Since no measurements of carbon dioxide were made, the percentage of energy from fat could not be determined. This tends to make input food Calories higher than the actual amount if some Calories are from fat.

Is optimal cadence 33 RPM?

The cadence for maximum efficiency was 0.9 sec per leg movement (of 1/2 revolution) or 33 1/3 RPM. This seems too low as it's reported by Cycling Science-1996: Optimal Cadence that for recreation cyclists 50-60 RPM is optimal to minimize energy. The article doesn't state RPM but only shows the time in seconds for a "leg movement". So if a leg movement was actually one revolution then 66 2/3 RPM would have been the optimal cadence reported by this article. But the maximal cadence reported is a 0.27 sec foot movement which would then amount to 222 RPM and this cadence is almost impossible to do and get a lot of power out. So it seems likely that one leg movement was actually 1/2 revolution as claimed in the article. But a possible mistake could happen if a metronome was set up for one beat every foot stroke and the rider (erroneously made a whole revolution for each beat).

18.4 Daniel's measurement

See Determinants of metabolic cost during submaximal cycling by J. McDaniel, J. L. Durstine, G. A. Hand, and J. C. Martin. J Appl Physiol 93: 823-828, 2002. This doesn't seem to mention efficiency, except for so called "delta efficiency" which is the marginal efficiency (the resulting incremental increase in work output divided by the incremental increase in Calories burned to get that increase in output). If we used marginal efficiency for autos we would grossly overstate the actual efficiency. All is not lost since inspecting the first graph indicates that the gross efficiency is roughly 20%. Since this experiment was done with "trained" bicycle riders, it doesn't represent the efficiency of the typical bicycle rider which is likely about 10% less (say 18% gross efficiency).

18.5 Sidossis's measurement, 1992

See Load and velocity of contraction influence gross and delta mechanical efficiency by Sidossis, L. S., Horowitz, J. F., and Coyle, E. F. (1992). Int. J. Sports Med. 13: 407-411.

Gross efficiency is shown in Fig. 5 (p. 409) for cadences of 60, 80 and 100 rpm at 50-90% of VO2max. This is for cases where the athletic bicyclist is pedalling with at least half his maximum continuous effort. The results show a maximum of about 21% gross efficiency at 60 rpm for VO2max over 50%. For 100 rpm efficiency is from 20.6% at 90% of VO2max to only 19% at 50% of VO2max.

Non-athletes

So for athletes, it looks like the net efficiency is about 23% if they pedal at near optimal cadence (RPM). But what about the typical bicycle rider that uses a bicycle for utility purposes? For running, the efficiency of non-athletes was estimated to be about 5-7% lower on p. 368 of: R Margaria, P Cerretelli, P Aghemo, G Sassi "Energy cost of running". Journal of Applied Physiology (1963). 18:367-370. Regarding why the non-athletes showed such good performance it's stated on p. 369: "because running is a common activity of all young people" This implies that the non-athletes were young people in good physical shape and likely more conditioned than the typical bicycle rider. Thus it would seem reasonable to assume that non-athletes are about 10% lower in efficiency than athletes resulting in a net efficiency of about 21% for non-athletic bicyclists. But since they are likely to be significantly off of the optimal cadence (and to keep efficiency a round value) assume 20% for net efficiency of non-athletes.

19. Appendix: Automobile Efficiency

It's reported various places on the Internet that the thermal efficiency of the automobile engine is about 30% and this is just a little higher than what the author learned from his "senior project" at the University of California at Los Angeles in 1957 where he measured automobile engine efficiency under various operating conditions (with a specially designed carburetor).

But while the thermal efficiency may be 30% under optimal conditions (which means high torque and not too high RPM) under actual conditions the efficiency is often much less. See Cruise Efficiency which reports only a 10% or less efficiency at cruising. However this seems to be too low.

For a 150 horsepower Ford V-8 of the mid 1960s (a gas guzzler) the efficiency was 14% when cruising at 30 mi/hr (source: data I plotted given me by an ex-engineer from the Ford Motor Co.). Inspection of Slide 1 (Engine performance map on p. 10) shows a maximum efficiency of 28% (275 gm/kwh) at a torque of about 60% of maximum at around 2500 rpm. The efficiency drops to about 15% at about 20% of maximum torque. This is for a 2-liter 4 cylinder spark-ignition engine.

Now while averaged over time, the efficiency could be somewhat under 10%, to obtain average efficiency it needs to be weighted by fuel flow and this flow is high during periods of higher efficiency at torques above about 30% of maximum torque. So until surveys can be conducted which will find the actual value of average efficiency the author believes that for urban driving where the load on the engine is low (resulting in worse efficiency) a 12% efficiency would be a reasonable value to use. This includes negative efficiency when the engine is being used to brake the auto, although modern autos can shut down the fuel feed when this happens. It doesn't include the 0% efficiency that occurs at idling when the auto is temporarily stopped at signals and stop signs. Small engines will do better than this 12% estimate and large engines worse.

20. Appendix: Braking Energy

About 1/3 of the mechanical energy supplied to a car's wheels is dissipated in braking. See Automobile Braking Energy, Acceleration and Speed in City Traffic SAE (Society of Automotive Engineers) technical paper series: 800795, by Paul Wasielewski et. al. June 1980. The abstract points out that about half of the mechanical energy is used for acceleration and 2/3 of this is dissipated in braking for city driving. Assuming a bicycle uses about 10% of its mechanical energy for braking (including going downhill), then 90% of a bicycle's energy is available for overcoming resistance while only 66 2/3% (2/3) of an auto's is so available. The ratio of these two percentages is 0.9/(2/3) = 1.35. So braking will increase the energy ratio of 10 by 35% resulting in an new energy ratio of 13.5.

This 1.35 ratio implies that for braking-idling energy use, a bicycle uses 10% of its energy and an auto 40%. Such percentages imply that at bicycle uses 90% (100% -10%) of its mechanical energy for overcoming resistance while an auto only uses 60% (100% - 40%) for this. Are this figures correct? Well the ratio 90%/60% = 1.5, the same as above so it's correct. One could write simple formulas for this but the above should suffice.

Note that we are not saying that the 1.35 ratio implies that the auto uses 35% more mechanical energy for braking-idling than the bicycle. Not at all. Since before considering braking-idling the auto used 10 times more energy, let the auto use 10 units of energy to the bicycle's 1 unit of energy. Then multiply the 10 units of auto energy by 1/3 resulting is 3 1/3 (or 3.33...) more units of energy for braking the auto and multiply the bicycle's 1 unit of energy by 10% resulting in 0.1 unit of energy for braking the bicycle. Thus we have the auto using 33.3 times as much energy for braking-idling as the bicycle (3.33/0.1 = 33.3) So it's not 35% more as in the first sentence of this paragraph but 3,333% more mechanical energy used in braking-idling the auto!

21. Appendix: Automobile Occupancy

This is the average number of people per automobile. It's the passenger-miles divided by vehicle-miles. Thus a long trip counts for more since it produces more passenger-miles than a short trip. And since longer trips tend to have more passengers, it results in higher occupancies than would be be case if we defined occupancy as the average number of people per trip.

Averages are found by surveys conducted every several years by the National Household Transportation Survey of the Federal Highway Administration. Some results are reported in "Transportation Energy Data Book" (edition 25: Figure 8.1 "Average Vehicle Occupancy by Vehicle Type ..." which for 2001 shows 1.58 persons/auto (excludes SUVs and vans).

This figure seems too high. If you were to count occupancy during the morning or afternoon peak travel in an urban region, you would get only about 1.2 or less. But then you are missing the higher occupancies that happen at night, on weekends, and for long trips. Passengers includes children. Reported occupancy includes case where the driver is just taking someone else somewhere (taxi-like service) and the trip is not providing useful transportation service for the driver. For this case, the occupancy is counted as 2 when it is effectively only 1.

So what to use for comparison with a bicycle? Since a bicycle is not feasible for long distance travel (except possibly for extended recreational trips) it should be compared to shorter trips where the occupancy is lower. After decreasing estimated occupancy to account for "taxi-service" an occupancy value of 1.25 persons/auto will be used in this article for comparison purposes.

22. Bicycle at Constant Power

When a bicyclist encounters a grade or headwind, more energy is used since the resistive force against the bicycle becomes greater. But it doesn't increase as much as one might expect since since due to the increased resistance force the bicyclist slows down, thereby reducing aerodynamic drag. Can we calculate what this amounts to? Yes, approximately. A bicycle rider is likely to ride while exerting constant power P. Since power is force times velocity
P = FV or F = P/V then to obtain more force on the wheel rim the rider just deceases the velocity of the bicycle by shifting to a lower gear (but maintains about the same leg velocity in pedalling).

Now the bicycle resistance force is:
F = a + bv^2 or P = FV = av + bv^3 = constant. For a grade change, a will change so we need to find how F changes with a change in a (the derivative of F with respect to a. We have:
dF/da = 1 + 2bv%v/%a where % is used for the partial derivative symbol (with power P held fixed). To find the partial derivative we use implicit partial differentiation of P = av + bv^3
a%v/%a + 3bv^2%v/%a = 0 which when solved for %/v%a gives:
%v/%a = -v/<3bv^2 + a when when applied to the equation for dF/da results in:
dF/da = 1 - <2bv^2) / (3bv^2 + a) At a = 0 this becomes 1/3 which means that while if maintaining the same speed the resistive force would increase at the same rate as the increase in grade, in reality it only increases 1/3 as much due to the "slowing down" effect of the grade.

So for the typical bicyclist F = 2 + 1.653 v^2 with v = 1.1 deca mi/hr (a = 2, b = 1.653) resulting in dF/da = 0.50. This means that the increase in grade only results in half of the force increase one would expect if the same velocity was maintained.

One can repeat a similar analysis for headwinds. Let a headwind have velocity u resulting in a wind velocity of v + u. Then F = a + b(v + u)^2. Proceeding in a manner similar to the above for grades we obtain:
dF/du = 2b(v + u)(1 - 2bv(v + u) / (a + b[v+u][3v + u] ) This is less than the 2b(v + u) value when velocity v is maintained. For a headwind u of 1.1 deca mi/hr and the same values of a and b as above, dF/du = 2b(v + u)0.555 . So again, the increase in aerodynamic drag is only a little over half what it would be if steady velocity were maintained when facing a headwind.

Another way to get the same results is to start by solving the cubic equation P = av + bv^3 for v by using handbook formulas. This is a little more complicated but it enables finding the values of velocity and force instead of just derivatives as was found above. While the slowing down for grades or headwinds helps save energy, the force is still higher than normal and it takes more time due to slower velocity which also has an energy cost.