ABSTRACT

Continuous stop-and-go traffic patterns, found ubiquitously inside city environments, represents a significant waste of fuel resources for modern automobiles. One way to recover this wasted energy is to add regenerative breaking to the car, but this requires some form of hybrid-electric system at the minimum. With suitable civil engineering, another alternative is possible, whereby a vehicle's exchange between gravitational potential energy and its kinetic energy can provide and store energy necessary for convenient and rapid inner-city transportation, while still requiring a minimum of fuel.

Introduction

Continuous stop-and-go traffic patterns are a significant waste of fuel for modern automobiles. One way to recover this wasted energy is to add regenerative braking to the vehicle, but this requires some form of electric-hybrid at the minimum. Efficiency is expected to fall between 25% and 50% due to declining generator efficiency as its shaft angular velocity decreases and other factors (indeed, 31% is estimated based on existing implementations [1]). Control circuit complexity and batteries tends to add weight to the vehicle, adding both to its cost and subtracting from its maneuverability in emergency situations. Additionally, there are lingering issues about the environmental impact of increased utilization of high-density lithium-ion batteries, as they use materials significantly different from the traditional lead-acid battery design.

Another approach is to exploit the storage and reclamation of gravitational potential energy through a novel road system design. Conversion from gravitational potential energy (GPE) to kinetic energy (KE), or vice versa, is not a thermodynamic process, and therefore should be able to exceed the Carnot 50% efficiency limit with proper road and vehicle co-design. However, efficiencies will never total 100% due to the various sources of friction, such as rolling resistance and aerodynamic drag. For this reason, prime movers, their fuel systems, and even traditional brakes are still required in the vehicle. The goal is not their elimination, but rather, the reduction of the use of heat-dissipating braking, engine braking, and excessive vehicle complexities, all of which contributes to fuel waste in an urban environment.

The Basic Physics of the Problem

Let's assume that the type of vehicle that we are designing our new road system for is the average house-hold car, which tends to have a mass of 1364kg (approximately 3000 pounds of weight on Earth) with modern safety regulations in place. The average U.S. speed limit inside most city streets is 13 m/s (30MPH). We will take the average length of a city block to be 213m (700 ft) [2].

Therefore, it is quite common for a vehicle to start from dead-stop and accelerate until reaching 13 m/s velocity along a stretch of road, traveling for a while, then stopping again (e.g., to allow for other traffic, or encountering a stop sign, a red traffic light, etc.). We can therefore take the distance traveled between stops to be 213m.

Some road vehicles are performance sports cars, capable of doing 0-100km/h in 4 seconds or so. Others aren't quite so powerful, ultimately taking as long as 12 seconds. Most cars have at least 150HP engines now-a-days, so let's say that the average car can accelerate 0-100km/h in 8 seconds under full throttle. When traveling inside an urban area, it's rare to go full-throttle, but in the lower gears, the car still accelerates very fast. It is reasonable to expect that those with high-powered vehicles will tend to use half- or even quarter-throttle, while those with leaner engines will tend to stomp on it just to keep up with traffic. Thus, we stick with an 8-second average acceleration time:

1. a_acc = dv/dt = (100km/h - 0km/h) / 8s =

   = (28m/s - 0m/s) / 8s = 3.5m/s^2

From this, we can see how much forward thrust a car requires to go from 0 to 30MPH in 8 seconds:

2. F_acc = m * a_acc = (1364kg)(6.3m/s^2) = 8.6kN (about 1900lbs)

As we can see, our car requires a fair amount of forward thrust in order to get moving. This requires a certain amount of energy investment, which as we know, comes from the petrol or diesel in the fuel tank. To determine how much energy is actually used, we first need to find out precisely when the vehicle reaches 13m/s. Let's assume t=0 when the car is at rest. Then:

3. v = a_acc * t_0-13

   t_0-13 = v / a_acc = (13m/s) / (3.5m/s^2) = 3.7s

Now we can determine total energy consumed during the acceleration period:

4. W_acc = F_acc * s_acc

   = F_acc * ( v(3.7s) - v(0s) ) = (8.6kN) * ( 13m/s - 0 ) = 110kJ

   CHECK: KE = (mv^2)/2 = (1364kg * (13m/s)^2)/2 = 110kJ

   It matches, disregarding rounding errors in insignificant digits.

Thus, to accelerate a car of 3000 lbs to a speed of 30MPH on a theoretically perfect, flat, frictionless surface, it requires 110kJ minimum equivalent of fuel, give or take due to significant digit issues.

Let's have some numbers fun, just to show you how much energy this actually is. Let's calculate how far the car actually travels from the red light to when it's moving at 13 m/s.

5. s_acc = v * t_acc = 13m/s * 3.7s = 48m (about 159 ft)

Now, assuming perfect energy conversion, to accomplish this goal requires an engine of a size much smaller than what's in most cars today:

6. P_acc = W_acc / t_acc = 110kJ / 8s = 13.8kW (about 18HP)

That's right -- a meager 18HP is all that is required to accelerate to the desired speed. Now look at the engine inside your car -- I'm willing to bet that you have at least 10x that power (and, consequently, 10x the fuel consumption). For additional comparison, let's note that the average household in North America consumes more or less 1kWh of energy [3], every month. Hence, even this hypothetical 18HP idealized car, to travel a meager 159 feet, consumes enough energy to run close to 13 households for a single month. Or, putting it another way, it can power a single household for more or less 13 months. [e1]

After accelerating to its final velocity the vehicle will need to travel for an estimated 117m, assuming that there are no points of congestion from cars trying to go into or out of driveways and the like. Then, somewhere in the last leg of the remaining 48m, it'll need to brake somehow to slow down to a stop, thus dissipating the 110kJ of energy that had previously been invested into the vehicle. Imagine those same 13 households' worth of heat -- from air conditioners, from dryers, from stoves, over the course of an hour, all dissipated in another four seconds. That's what your friction-based brakes do every time you step on them.

The Physics of a Possible Solution

It is known in the science of Physics that objects with altitude possess something called gravitational potential energy (GPE). The general formula for GPE is:

7. W_gpe = mgh

where:

W_gpe = the energy/work of the object, measured in Joules

m = mass of the object in kilograms

g = acceleration due to gravity (constant: 9.81m/s^2)

h = the height of the object's lowest surface, in meters.

As you drop an object, the object accelerates due to gravity at the rate of g. This acceleration is through non-thermodynamic means -- hence, the conversion from GPE to kinetic energy (KE) is non-thermodynamic as well. Indeed, barring aerodynamic losses and assuming the object never reaches its terminal velocity on the way down, the efficiency of energy conversion is near 100%.

Thus, suppose we want to accelerate a 1364kg mass to 13m/s by somehow "dropping" it. This dropping can be done via a road that is some height 'h':

8. h = W_gpe / mg = (110kJ) / (1364kg)(9.81m/s^2) = 8.2m

Hence, a hill that is approximately 8.2m (27 ft) tall, with sufficient slope and lack of rolling resistance, ought to be enough to accelerate the vehicle to 30MPH under nothing but the sheer force of gravity.

This works both ways as well. Because the conversion of energy can go from KE to GPE with equal ease, it follows that decelerating can be done by simply proceeding to drive the vehicle up an incline of 8.2m vertical. This will naturally result in the car slowing eventually to a stop, somewhat shy (due to unavoidable rolling and aerodynamic resistances) of the apex of the hill. Hence, we get braking and reclamation of a larger portion of the acceleration energy, virtually for free. The only time energy from the car's engine is required is to overcome the rolling and aerodynamic losses in between these two events.

Since most people don't like the idea of riding a roller coaster to work, the 8.2m altitude change should be done over a reasonable span of land. For example, if we span the 8.2m drop across an 82m horizontal swath of land, we have a hypotenuse of 82.4m. The force on the vehicle will be approximately 1300N. Because the vehicle has a smaller force providing it thrust, it follows that it will take longer to accelerate -- in this case, close to 24 seconds.

To compete with our traditional acceleration characteristics, let's see what would be required of the slope's span. First, we need the weight of the vehicle, in N:

10. W = mg = (1364kg)(9.81m/s^2) = 13.4kN

If we were to drop the car off a cliff, this means the car would accelerate with a force of 13.4kN (about 3000 lbs, as initially set in the statement of the problem).

Next, we need to determine the required span of the hill:

11. F_t = W * sin(theta)

    sin(theta) = F_t / W

    theta = arcsin( F_t / W )

    theta = arcsin( 8.6kN / 13.4kN )

    theta ~ 40 degrees (the angle of the hill against sealevel is 40 degrees)

    tan(theta) = 8.2m / span

    span = 8.2m / tan(theta)

    span = 9.8m

So, as you can see, this is going to be a pretty steep hill. Since we have 213m between stops, we can certainly widen this out a bit to make travel more comfortable. For example, if an 8s acceleration time is desired, then a 21m span may be desirable.

Regardless of the span chosen, from a dead stop at the top of the hill to the bottom of the hill, the vehicle will be moving about 30MPH in the forward direction. At that point, the engine of the vehicle can kick in to overcome rolling resistance for the flat portion of the block. Then, as the next intersection approaches, the vehicle will ride up the slope (also of 8.2m plumb-line height) to find itself coming naturally, gently, quietly, and efficiently to a stop at or near the crest.

Thus, we see a system of locomotion whereby fuel is consumed only to maintain forward momentum, and the heat-dissipating brakes are used only in cases of utmost urgency. This represents the highest possible efficiency in providing motive power known to the author. However, realizing this system of transportation requires some changes. At a minimum, with our existing technology investment remaining the same, civil engineering practices will need to be changed.

Changes to Civil Engineering

On a flat street system, one of the causes for stop-and-go action is the fact that people need to pull in and out of their driveways. This is a specific case of the more general problem of cross-flow traffic. To prevent cross-flow traffic problems, traffic should be optimized for unidirectional flow patterns.

For example, if we eliminate the idea of driveways as we currently know them and go instead with subterranean parking garages, such that homes or offices sit on top of these garages, then we can realize an off-ramp system, similar to those used on freeways, where the vehicle would peel off of the normal flow of traffic, and follow an inclined switchback into inlet side of the parking garage.

To leave the facilities, your vehicle would exit out the outlet side, where it'll accelerate down a declining switchback, ultimately to peak concurrently with the adjacent road. Since merging traffic is another source of road congestion in some areas, the actual merge would be delayed until the next physical road intersection.

Traffic lights would round-robin at an intersection, intelligently skipping past lanes that are known not to contain pending vehicles. Only one or two cars per lane (not per direction or per street) are allowed to proceed at a given time, thus effectively multiplexing vehicles going in the same direction via on-ramps and their merging street. This is how the state of California manages congestion on its freeways due to merging of vehicles in multiple lanes from on-ramps. It largely works on that scale, so it is expected to also work on this smaller scale.

Changes to Vehicle Designs

With the majority of the vehicle's energy coming from the transfer of GPE to KE and vice versa, the need for beefy engines is significantly reduced. Indeed, for mass-transportation systems like buses, a heavily loaded vehicle will see significantly improved "performance" due to its increased weight. Therefore, researching engines that are sized just right to overcome basic rolling resistance at the best possible efficiency may become a priority. [e2]

Additionally, the use of smaller, thinner, higher-pressure tires may prevail due to their reduced rolling resistance. The increased air pressure requirement increases tire wall rigidity, and therefore helps to reduce deformation losses. This is why racing bicycles have very thin tires with pressures often exceeding 120psi.

In addition to the use of smaller engines, more eco-friendly fuels such as ethanol may be used. Ethanol contains 67% the energy content of gasoline [5], thus would require a larger fuel tank to travel a given distance. But, with a smaller engine, capable of overcoming rolling resistance when used in its specified parameters, less fuel would be consumed. Therefore, fuel tank size may remain the same, or even decrease, thus potentially making the vehicle less expensive further.

I should point out that this document has assumed the point of view of improving fuel economies for a 4-wheeled, air-inflated, rubber-wheeled automotive vehicles. The physics described in this document is universally applied to all forms of transportation which rely on tracks, roads, or other rolling systems. Indeed, rail-based systems, due to their uncompressable wheels and outstanding opportunities for aerodynamic refinement, represent the single highest efficiency land-based transportation system in existance.

Changes to Lifestyle

Many technocratic, utopian, and futurist plans all recognize the need for automated, rail-based transportation systems. Rail is highly effective because the wheels are rigid -- being made of pure metal, there is no deformation losses of any kind. If electricity is used to power the rail system, especially via the rails themselves, then the shape of the cars used in the rail system can be specifically optimized for aerodynamic drag in a much tighter form factor than if you also have to house a generator/engine combination. Electric power also allows for the elimination of engine idle losses when power to the wheels aren't needed (the mechanical equivalent of "standby power" in many home electronics). Computers can effectively reduce, if not prevent, congestion by carefully tracking vehicle locations relative to a "big picture" map of the track system and where all cars are currently located.

The problem with rail-based systems is that they don't offer the passenger total transportation freedom like a privately-owned car does. The passenger would end up waiting for a train that happens to be headed "their way" to arrive according to a schedule that is not theirs.

The principles of unidirectional travel, the extensive use of on- and off-ramps to ameliorate traffic congestion, use of GPE to accelerate and decelerate smoothly, safely, and quietly, plus the use of computers to determine the best car routing to minimize congestion can all be put to use in an elevator-like "urban rail transportation system." Imagine the following scenario as a way of explaining how such a system would work.

You're getting ready to go to work, and instead of a driveway, you have a small-scale train platform. You log into the system somehow and request a transport from your current location to your office. A train car arrives within five minutes. The train cars used in this urban setting are about the size of a taxi cab in bird's eye surface area, and about the half the height of a traditional city bus, and can carry up to about 4 comfortably. The car that arrives was specifically and individually routed to your location on demand, in response to you requesting the transport. Then, when you board the vehicle, you are identified by smart-card or similar mechanism, and the train car is off.

Transfers to other transporation centers (including other train platforms if required) are automatically determined and scheduled at the time you filed your transportation request. An estimated time to arrival is provided for your convenience and is routinely updated on the fly, for your ability to contact your boss should you end up being unexpectedly late as a result of an unforeseen source of congestion.

In a sense, this technology would be an enabling technology to allowing a user to "book a travel" to any location necessary, because:

1) Urban areas would rely on slow-speed, small-car-sized train systems where the cars can be individually provisioned for personal use on an as-needed basis. Since most of a car's energy will be dissipated upon reaching the crest of a hill, the noise level of a train passing through a residential area should be no worse than a fleet of Toyota Prius vehicles.

2) Transfers to higher-speed, higher-capacity rail-based transportation systems can be made and scheduled for the long-haul to more distant locations.

3) At the time service is requested, a full route and transportation schedule is computed, and is regularly updated throughout the trip in real-time. Paths can be determined in a manner analogous to call-routing through switches in asynchronous transfer mode (ATM) networks. Congestion notification and rerouting systems can be implemented in a manner not unlike explicit congestion notification systems used in these same ATM networks.

Conclusion

Making use of gravitational potential energy conversion could provide legacy vehicles with an important method for maximizing their fuel economy, especially if their engines are upgraded to more fuel economical units and urban centers have their road system suitably modified to accomodate this mode of operation.

But, perhaps more importantly, it provides the enabling technology for more advancements in the area of aerodynamic drag research, rolling resistance optimization, and the eventual change-over to a fully automated, rail-based, urban transportation system that potentially can retain the level of comfort and personalized transportation freedom that currently only manually-driven cars can provide.

End Notes

[e1]	This is hardly what I'd call efficient, when you consider a human can jog that distance with an average power output of a quarter to third of a horse power in comparable times [4].

[e2]

These engines must still be able to pull the vehicle from a dead-stop at the bottom of a hill to the top though, just in case an accident requires the vehicle to come to a stop prematurely. It's assumed that doing so will be far more inefficient, since the engine is being operated outside its normal parameters. Hopefully, this mode of operation will tend to be statistically insignificant.

Works Cited

[1]	"Regenerative Breaking." 2006 Jun 25. Wikipedia. 2006 Jul 11. <http://en.wikipedia.org/wiki/Regenerative_braking>

[2]	"Cudahy, California." 2006 Jun 24. Wikipedia. 2006 Jul 11. <http://en.wikipedia.org/wiki/Cudahy,_California>

[3]	"12 Steps." n.d. 2006 Jul 12. Greenpeace International. <http://www.greenpeace.org/international/campaigns/climate-change/take_action/12_steps>

[4]	"Human-Powered Transport." 2006 Jul 11. Wikipedia. 2006 Jul 11. <http://en.wikipedia.org/wiki/Human_powered_vehicle>

[5]	"Alternate Source of Energy." n.d. 2006 Jul 11. <http://www.vigyanprasar.com/comcom/definition.htm>