OK, it is probably a Bad Idea to get into the middle of a Scott and Dean argument, but I had an idea when I read the post above about using the built-in mileage meter.
Here's my theory:
The built in mileage meter is computed from vehicle speed and injector duty cycle. Now injector duty cycle is calculated from among other things, feedback from the oxygen sensor. I think what the hydrogen does is combine with some of the available oxygen in the cylinder and by doing this, the oxygen sensor gives a too-rich signal. The ECU compensates by leaning out the mixture. From this the engine computer calculates a better mileage.
In addition, the hydrogen is burning in the cylinder and creating heat. This does contribute to the power of the engine and may produce more power. Now, if the system was like Scott says, and there is just a little bottle of H2 from some other source, I would say the vehicle probably does get better mileage. So these claims I believe.
Now for the smoke and mirrors:
All these guys claim there is "extra" power being generated by the alternator -- this is not true it will only produce as much power as is being demanded. What there is with the alternator is extra capacity -- the ability to produce extra energy, and that is where fallacy with these systems lie. They are deliberately confusing the concept of power generation capacity with actual demanded power.
These systems use power generated by the alternator to produce the H2 (or Brown's gas or whatever the fuck they want to call it.) If you draw more electrical power from the alternator, it requires more mechanical power to drive. The problem with this is that the alternator is not 100% efficient at this conversion. and some of the mechanical power required to drive it is lost as heat. This is wasted energy, or an increase in entropy if you are a Physics geek. (Whatever you do to increase entropy, you don't get back.) Wasted energy as heat has to come from somewhere -- and where does energy come from in an engine? -- fuel.
So even though your engine is getting better specific power output, you are demanding more power from it by running the alternator harder.
Was that long enough?
C/N: the things actually lean out mixtures, but take more power from the alternator, thus decreasing actual mileage.
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