[SGVLUG] OT: Hybrid efficiency (was:New Linux Lug)

[Dustin Laurence]

On Mon, Feb 20, 2006 at 03:59:35PM -0800, David Lawyer wrote:

> Of course it saves fuel.

Note specifically that I believe it is *not* the kind of driving you
advocated.  You were talking about big speed changes, which guarantees
lots of extra drag at the same average speed.  Since I happen to
remember how the drag works, I know very well that's a loss in drag
terms.

Also it's odd how you seem to think this but don't understand the
advantages of a pure hybrid being always able to run it's engine at the
most efficient place.  A hybrid sort of does what you're trying to do
here, the difference is it does it better, more safely, and without
abusing the machinery.

> As for wind resistance, you'll get better efficiency with long pulses,

No, quite the opposite, and one derivation is a simple but somewhat
tedious high-school level exercise.  For simplicity consider driving at
each of two velocities u and v for a distance s and compare it with
driving at the average speed w for a distance 2s.  We want to take the
same time to go the total distance 2s in each case, so we have uv 2s/w =
s/u + s/v --> 1/u + 1/v = 2/w or w = 2 ----- v + u

Without loss of generality we may take v > w > u (but it turns out not
to be necessary because the final answer does not depend on whether v or
u is greater).

The wind resistance goes as v^2, say a v^2 for some a.  Then if we
subtract the energy lost to the atmosphere in the second case from that
in the first, we have that driving at different speeds costs an extra
energy (we must prove this positive):

Delta = a s v^2 + a s u^2 - a (2s)w^2

or

Delta
----- = v^2 + u^2 - 2 w^2 = (v-w)(v+w) - (w-u)(w+u)
 a s

Unless my algebra is even more rusty than I think some tedious cubic
factoring finally yields

Delta   (v-u)^2 (v^2 + 4vu + u^2)
----- = -------------------------
 a s            (u+v)^2

which is guaranteed non-negative, and the only point at which it is zero
is at v = u (i.e. we didn't change speed after all).  Therefore moving
at different speeds loses more energy to wind resistance.  Also note
that it increases quadratically as the speed increases, so there is
always a speed at which this overwhelms any considerations of engine
efficiency.  I don't feel like doing more homework tonight but I will
guess that for real cars that speed is around highway speed or less.

The point is, what does save gas is essentially pumping the gas pedal
rapidly, so that the speed changes very little.  Which is what I said,
and different than your scenario where v = 2u, which loses

Delta = 13/9 a s u^2

if I can still do algebra in my head.  To actually get answers for real
cars we'd have to know how much gas the engine actually saves and the
magnitude of a, an I'm not about to try to look up stuff like that
tonight (a won't be hard to get at from basic physics, but no references
for the increased engine efficiency come to mind and it's going to be
too complex a phenomenon of the details of the engine to have a likely
obvious estimate).  Someone's calculated it for these efficiency
contests, though.

Dustin