First, consider the case with negligible auxiliary loads (no air conditioning).
For a Civic-sized engine (1.8 liters), this US DOE worksheet estimates about 0.3 US gallon/hour fuel consumption at idle.
Here is a conservative starter calculation:
- The Civic starter is rated at 1.0 kW (83A$\times$12V). A 3 second start therefore produces 3 kJ. Assume an additional 25% in battery internal dissipation that must be replaced.
- As you note, this energy must be replenished by the ICE (internal combustion engine). Max ICE efficiency is only 30%. The incremental efficiency, which is what matters for this small additional load, is no doubt higher, but I’ll use 25% as a conservative estimate.
Alternator efficiency is not great either; I’ll use a conservative 50%.
With these values, it requires 3.0 kJ$\times$(1.25 / 0.25 / 0.5) = 30 kJ worth of fuel to recharge the battery (Note the overall charging efficiency is only 10%!).
Now, the energy density of gasoline is 120 MJ per US gallon (42.4 MJ/kg), so the amount of fuel required to recharge the battery, including all the inefficiencies, is 30 kJ $\div$ 120 MJ/gal = 0.00025 US gallon.
So, the “crossover” idle time in this case, above which it is more efficient to stop and restart, is 0.00025 gal $\div$ 0.3 gal/hour $\approxeq$ 8.3 $\times10^{-4}$ hours, or about 3 seconds.
Now suppose an air conditioner (PDF) is consuming 1 kW of electrical power.
- With the engine running, the A/C requires (via the alternator) an additional engine fuel consumption equivalent to 1 kW / 0.5 / 0.25 = 8 kW, or 29 MJ/hour, or 0.24 gal/hour of gasoline. For a duration $t$, the total fuel consumption with the engine running is (0.3 + 0.24)$t$ = 0.54$t$ (with $t$ in hours).
- With the engine stopped, the A/C still consumes 1 kW, or 3.6 MJ per hour. With that low 10% charging efficiency, it requires 36 MJ worth of fuel (or 0.3 gal) to recharge an hour’s worth of A/C operation. Adding in the starter contribution, the total fuel requirement is 0.00025 + 0.3$t$ (with $t$ again in hours).
Equating these two new fuel requirements, the crossover time with the A/C on increases, but only to about 4 seconds.
Although the battery charging efficiency is low, the waste of the idling fuel consumption dominates the calculation.
Note that I don’t have a reference for the 25% battery re-charge inefficiency.
Unfortunately, that’s an important number when running an A/C, since it reduces the advantage of shutting off the engine. At some high load level (in the neighborhood or 4 kW) that disadvantage outweighs the advantage of turning off the engine.
Further (experimental) data to confirm the above estimates can be found here: http://www.iwilltry.org/b/projects/how-many-seconds-of-idling-is-equivalent-to-starting-your-engine/
In my case it consumes about the same amount of fuel as 7 seconds of idling. However, the additional fuel consumption observed seems almost entirely due to a faster idle speed setting for the first 20 seconds after starting. Any good driver would start moving within 1-2 seconds after starting, which would effectively eliminate the fast idle losses. If you can begin extracting useful work from your engine within 1 second after starting the engine then it appears starting the engine consumes fuel equivalent to about 0.2 seconds of idling.
What you are talking about is called a combined cycle engine. They are commonplace in stationary power generation, i.e. utility-scale electricity generation. There has even been some talk of combined cycle engines in cars.
As pointed out in the answer by dmckee, the reason this hasn't been widely applied in cars is that no one has demonstrated an economically competitive combined-cycle car. I promise you, if such a thing can pay for itself in gas savings then it will eventually be built and sold, unless some better technology makes it irrelevant.
In general there are many reasonable ideas that are physically permissible but economically or technically difficult or nonviable. You are effectively suggesting to add a steam engine to a car, which is quite a difficult proposal. I'd suggest that a hybrid gas-electric car is more economical than what you suggest, and even they have had a hard time catching on. In electric power generation it matters much less that the combined cycle engine has a larger sunk cost than a normal engine, is heavier, etc., so the economic balance works out.
Bringing the question back to physics, no matter what you use for heat scavenging, your engine including all of its "subengines" cannot exceed the Carnot efficiency corresponding to the largest temperature difference in the engine. Adding additional heat engines will help to approach the Carnot limit. In order to beat Carnot, you can't use heat as an intermediate step between chemical energy (fuel) and mechanical work.
Best Answer
During electrolysis, there is over-voltage on the electrodes, meaning that electrolysis doesn't start until a certain voltage is applied (which is NOT zero volts). Since circuit power is given by P=IV, you will note that there is a substantial loss during electrolysis because that over-voltage does NOT produce any products. This means that the efficiency of the overall process is low from the start. That efficiency can be improved somewhat by using exotic metal electrodes (e.g., platinum), but it can't be driven to 100%.
Note that the efficiency of your school experiment can be improved somewhat if you use the energy from the wind turbine to directly charge a battery, which will then be used to drive the electric car. This simply means that the talk around hydrogen as a "renewable" energy source has been "hyped" to a certain degree. Given the relatively low efficiency of hydrogen generation via electrolysis, and the difficulty of storing enough hydrogen in a tank to drive a car over a reasonable range, there are probably better technologies to use for transportation than hydrogen fuel cells.