Calculations for a detailed refutation of "EER"

Introduction

In this diatribe I shall pursue detailed computations of various aspects of EER, or, as the original author puts it, "electronic energy repository". I will note that I am quite naive about various aspects of molecular interrelationships (e.g., I could not begin to compute hydrogen bonding or nanometer covalent effects between materials), and do not intend any devices mentioned in this diatribe to be constructable, except in theory. Any reasonable critiques can be E-mailed to me at ewill3@earthlink.net. Flames and spam will of course be ignored.

Construction of a theoretical D-sized atomic capacitative device

EER depends, in its heart of hearts, upon capacitance, which for the purposes of this computation is defined as the storage of energy by means of separating two conductors apart by a dielectric. The grooving mechanism mentioned by feerguy is not practical; I shall instead endeavor to attempt to compute the capacitance and energy storage capabilities of a D-cell(LR20)-sized sandwich-based device. It is not at all clear to me whether such a device would actually function as predicted; a far more likely happenstance is that ionic effects will dominate. In short, instead of a capacitor I may very well be building a battery -- and were I to replace the TiO₂ with NaCl and one of the stacks of Cu with Zn, I would indeed be doing such. Such is what Count Alessandro Volta did long long ago, albeit with far thicker discs.

The dimensions.

A common D-cell (or LR20 cell) is generally cylindrical, with a positive nub at the top and the surrounding can being negative, usually contacted in the battery holder at the very bottom. We will assume a can of height 61.5 mm, 34.2 mm diameter. Midstate Batteries (http://www.midstatebattery.com/alka.htm) specifies a target energy of 16500 milliAmpHours, or 59.4 kiloJoules. Our hypothetical device must be able to store at least this amount of energy or we're dead in the water. (The weight of Midstate Batteries' device is 134 grams; the explosive energy of an equivalent mass of TNT would be 616.4 kJ. Were we to somehow be able to convert those nitrogen bonds into electricity we could have more efficient batteries. Of course TNT as a battery would make people very nervous. It turns out, however, that 134gm of kerosene would have 2 megaJoules, although the form factor would have to be bigger since kerosene isn't as dense as copper or titanium. There are hints of an electrical generation device being developed that runs on hydrocarbons.)

The materials.

It's a capacitor, so we want a good conductor and a good dielectric. I will use copper as the conductor and titanium dioxide as the dielectric, with a dielectric constant of 110 (according to the ASI Instruments Inc. Dielectric Constant Reference Guide). Surprisingly, distilled water might be even better, or at least comparable, as it has a dielectric constant of 88 at freezing to 80 at room temperature. However, there are issues if the unit is opened -- literally.

All data for physical items are courtesy of www.webelements.com.

Packing the materials.

At the atomic level things get mildly interesting for copper (₂₉Cu^63.546). The Cu-Cu bond distance in metallic copper is actually not twice the van der Waals radius of 140 pm (doubled, that would yield 280 pm), but turns out to be 255.6 pm. One would expect this if familiar with QM theory so it's not really a problem; this means the Cu radius is 127.8 pm. I'll use 128 pm in my diagrams for simplicity; it doesn't really matter.

Strangely, titanium (₂₂Ti^47.867) has no van der Waals radius in www.webelements.com. Why this is so I cannot say. However, since TiO₂ is what I'm using the covalent radius is probably more useful anyway. That is an empirically-determined 136 pm. The covalent radius of oxygen (₈O^15.9994) is 73 pm.

All these considerations lead to the naive but rather pretty atomic packing diagrammed in Figure 1.

Figure 1. A possible packing for a highly theoretical atomic capacitor.

I have no idea whether the atoms would actually be so neat in their arrangements, should someone actually manage to fabricate such a device. The copper, for instance, would want to bond with the oxygen, forming cupric oxide. Such considerations will probably tend to nullify the effectiveness of this device. I strongly suspect, absent further info, that the dielectric will change its bond angle as it absorbs energy, acting as a sort of strange electronic sponge. While this does not affect my naive computations any it may lead to some interesting mechanical effects in a real such device.

For my computations I'll use 500 pm for the thickness of the electrical conductor and 400 pm for the dielectric thickness.

Area

I've already mentioned that the radius of the can is 17.1 mm. The nubbin is not specified; I'll assume about 2 mm radius. This nubbin allows for the electrons to flow into the copper sandwich stacks from the central conductor. The area of course is π × (r2² - r1²) which works out to be 217.09 square millimeters. This area is of course per disc side.

Figure 2. Disc size for the capacitor.

Since a sandwich is 1800 pm (we need both sides of the cap) we can calculate an approximate number of sandwiches in a can of 61.5 mm. This is of course 6.15 × 10^-2 ÷ (1.8 × 10^-9) = 34,166,666 discs. Because a disc side is 217.09 square mm and there are two capacitors (connected end-to-end; ultimately they'll be wired in parallel) we get a total cap area of 14,834,000,000 mm² or 14,834 m².

Not bad for a can of mostly metal, although unfolding it might be a pain.

Capacitance

Capacitance is straightforward. Given the dielectric constant, the distance between the plates, and the area of the plates, the capacitance in pF is 8.8543 × K × A ÷ d. K is 110. A is 14,834 m². d is 400 pm. Therefore, we get 36,120 farads.

For a cap, that's very beefy, especially in that size factor. For a battery, well, see below.

Energy.

The energy of a capacitor is half the square of the voltage times the capacitance. Since a common D-cell is 1.5 volts we get an energy of 0.5 × 1.5 × 1.5 × 36,120 F = 40635 J. A bit short, and it's far easier to construct the battery than this device.

The GM EV1 -- A vehicle whose time has come and gone

I was thinking of getting a EV1 at one point. The EV1 was an interesting vehicle; it is a pure electric and a prototype for the ideas espoused by feerguy9. The initial model was powered by lead-acid batteries; a subsequent rev substituted nickel metal hydride. Sadly (or maybe out of necessity!) the vehicle is no longer available. I shall analyze this vehicle and attempt to address some of the salient points in feerguy9's article regarding the trip to the corner grocer.

General specifications.

All specs are taken from http://www.gmev.com and reprised here as required. Because this is a scientific analysis all units are SI as much as possible; fortunately GM did many of the conversions for us on specs page (size) and this specs page (power).

Quantity	Value
Length	2.512m
Width	1.766m
Height	1.281m
Curb weight (lead-acid)	1400 kg
Curb weight (nickel metal hydride)	1320 kg
Battery pack weight (lead-acid)	1310 lbs (595 kg)
Battery pack weight (nickel metal hydride)	1147 lbs (520 kg)
Battery energy (lead-acid)	18.7 kWh (67.32 MJ) @ 312V
Battery energy (nickel metal hydride)	26.4 kWh (95.04 MJ) @ 343V
Engine Power	102 kW
Engine Torque	150 Nm
Drive ratio	10:946:1
0-60 mph Acceleration	< 9 seconds
60-0 mph Braking	49 m
Drag coefficient	0.19

General problems with pure battery-driven electric vehicles

It is clear that the battery packs in this vehicle are very heavy; one is carrying half a metric tonne of weight simply to move an inefficient pack around. If one were to carry an equivalent amount of gasoline (by weight) one would have to have an extremely large fuel tank as the density of gas is about 0.8 kg per liter. That liter of gas, however, can hold 35 MJ. 600 liters of gasoline would weigh 480 kg and hold a whopping 21 GJ, although the gas tank would probably have to be carted around on a following trailer.

Compared to that the nickel metal hydrides are pathetic dead weight.

Since the EV1 is not using any special technology it's clear that any battery-driven car will have problems. The new hybrids, however, most likely have tiny battery packs compared to the 595 kg being lugged around by the EV1. Frankly, I'd have to look.

Of course a full-gasoline is carrying around dead weight: a single lead-acid battery which is provided in lieu of a very old-time crank which was long since been discarded. Since the crank was made out of metal the weight is presumably about the same. But even if it weren't it's more convenient with the battery.

Highway travel of the EV1

The maximum rated distance for the NiMH battery pack is 130 miles. This is 210 km. Since 100 miles consumes 30 kWh on the highway it's not quite clear how this value is arrived at; the battery pack only holds 26.4 kWh.

To get the EV1 moving to 60 mph (26.8224 m/s), neglecting the 0.19 drag factor, one needs 1/2 × mass × v² or 480 kJ -- about 8 dry cells. Fortunately the EV1's battery pack is bigger than our LR20, and the 8 dry cells are probably better employed in a portable boom box anyway.

So now we travel that 210 km, completely draining the NiHM battery pack. This suggests that the energy consumption, once an EV1 is up to speed, will be (95 MJ - 480 kJ) ÷ 210 = 450 kJ/km. At the lower speed of 30 mph we can probably halve that value, assuming laminar air flow; however, the startup cost is 120 kJ, and that startup is done from every stoplight. As a total guess (the EV1 spec page does not mention the actual amount at all) one might be able to reclaim 20% of that at most from the brakes.

The trip is specified as a quarter mile. Stoplights are not included in the spec, which leads to some nasty guesswork. 0.25 mi = 0.40 km. Startup cost is 240 kJ, assuming no stoplights. Travel cost is 225 kJ/km or 90 kJ for the trip. Stopping might reclaim 48 kJ if we're very lucky. Therefore, the total energy consumption (with reclamation) would be 282 kJ -- which would be 0.29% of the NiMH pack.

The Wearing of the Cells

A number of variables ensue when one throws in solar cells. First off is the weight, but one might be able to partially compensate by simply replacing the roof with glass (a solar cell is essentially silicon dioxide -- glass). There is the issue that the roof might be heavier, though, and each kg of weight adds 179.8602 J to our startup costs for that grocery trip (we have to start up twice). Fortunately, drag isn't an issue if we do this right.

Now we have an additional energy source. Maybe. A naive calculation suggests that we might be able to get 4.436 m² but one must remember that the weight and height are at the maximum points of the vehicle, which is a teardrop shape. Also, the window is slanted, which makes it useless for this purpose, although one might do something with the dashboard in a pinch. As it is, I estimate about 50% usable area, which gives us 2.218 m². The insolation is 1.369 kW/m², but we can only count on 20% of that; the rest is waste heat.

So the car sits there for an hour or so while Ms. Jones shops. Every second could pour in 605 J into the battery pack. 3600 seconds -- 2.178 megaJoules. Nice gain but wouldn't we be better off just plugging the vehicle into a solar-powered charging station? Especially if the charging station puts the solar cells on the roof, which not only gives us a nice source of power but also keeps the car nice and cool by blocking sunlight. (Of course this requires foresight by the property owner, or maybe incentives. And then there's the issue that building this structure could double his parking if he simply does two floors of parking garage instead of one floor garage, one floor solar cells.)

Of course 2.178 MJ = 2.2% of our NiMH battery pack.

Another computation

General theory.

I've since discovered -- or perhaps thought about -- another method of proving the absolute maximum energy stored in a capacitor cannot exceed the energy stored in gasoline, and it's absurdly simple.

Briefly, the dry cell uses chemical means to store energy, unbreaking and breaking bonds in the process of pushing electrons around. As the bonds are used up the battery eventually runs out of "juice", and has to be recharged, either electrically (reversing the reaction; lead-acid batteries in particular are easily recharged, and other batteries such as nickle metal hydrides or the older nickel cadmiums are as well), or by replacing the spent materials with fresh somehow. (In a spud battery one would simply replace the spud and the two electrodes, for example.)

How does a capacitor store energy? I'm not entirely sure, but I have a fairly straightforward theory, based solely on the observations that distilled water (H₂O) is a surprisingly good dielectric, and titanium dioxide (TiO₂) has a shape that's reminiscent of water as well. The dielectric molecule bends in the electric field; if it bends too much the molecule breaks. Put that baldly it sounds slightly silly, and the actual bending would probably be adjustments in the electron probability clouds/bonds anyway, but that suggests that, if the voltage get too high in a cap, the dielectric will fail -- which is more or less what happens anyway.

So, given that, when does a cap fail? When the bond energy is exceeded. how much bond energy is there? Well, the bond enthalpies for many compounds are very well known; the energy for TiO₂ in particular is given as 672.4 kJ/mol^-1. This means that the entire molecule can store 1.3448 MJ/mol^-1, and, since TiO₂ is 79.867 g/mol, one gets 16.838 MJ/kg.

That's less than half of gasoline's 36-45 MJ/kg. If there is a better dielectric than titanium dioxide, I'm not aware of it.

There is the issue that gasoline "cheats" -- part of the energy is taken from the very air, which is not counted in gasoline's 36-45 MJ/kg figure. This may be an issue in such applications as moon buggies and Mars rovers, but not for terristrial vehicles.