Acceleration on a level road is the same energy-wise (in energy consumed per mile of travel) as climbing a hill at constant speed if the acceleration in g's is equal to the slope of the hill, for example an acceleration of 0.02g is equivalent to climbing a 2% grade. Similarly, decelerating on a level road is the same as coasting down a hill at constant speed. Coasting at constant speed of course will only happen if the downslope is just right; too steep and you will accelerate, too shallow and you will slow. In the glide phase of P&G the time taken to fall from say 44 to 22mph defines the deceleration rate: it would be 1.0g if the time was one second, and .02g if the time taken was 50sec. My HCH takes about 60 seconds on a level road to drop from 44 to 22, so I should be able to coast downhill at constant speed on a downslope of 1/60 or .0167. The problem I have is there are very few level roads around here, especially ones that I can use to dink around at 22mph. So I don't have a strong feeling for the accuracy of that 60 seconds. I do have some pieces of road that I have determined from GPS data have a slope of about .017 and indeed I can coast at nearly constant speed on them.

So where is all this leading? Well, I have used my Trip A meter to measure fuel consumption while climbing long hills of various slope and I've found that my fuel consumption is something like

FC = B(1+C*S) gallons per mile

where B and C are constants and S is the slope (.02 for a 2% grade). B is the fuel consumption you get at constant speed on the level, and C is a number somewhere between 30 and 50. Now let's imagine a P&G session where on a level road we pulse from 22 to 44mph in P seconds and then glide from 44 to 22mph in G seconds. The slope (S) in the equation can be replaced by acceleration (1/P), so the fuel consumed in pulse is B(1+C/P) multiplied by the distance traveled in pulse. The average speed is 33mph, so the distance traveled is 33P/3600. Fuel consumed in glide is zero and the distance traveled is 33G/3600. Taking total fuel used divided by total distance traveled we get

FC = B(P+C)/(P+G) gpm

You can play with this result but as long as C is numerically less than G you'll get lower FC with very small values of P - meaning hard acceleration is better. You have no choice with G - you get whatever your tire pressure dictates. Finding C is a little tricky because some of the hill climbs I made were assisted by the battery pack. If I started with low SOC I got values of C near the high end of the 30-50 range mentioned above.

I would dearly love to hear what others get for G - the number of seconds to glide from 44 to 22mph, and how it depends on tire pressure. You should go both directions to average out wind and slope (if there is any).

Dave (your friendly geezer physicist)

That jives with the results of the Pulse & Glide marathon in the Prius. The best laps worked out as higher average FE with the engine on (harder acceleration) but more time with the engine off.

2001 Prius Photos       2005 Prius Photos
'97 Prius Simulation Prius inner workings

Define 'hard' acceleration. To the marathon men, it meant an instantaneous MPG of 20 - 30 at 30 miles per hour. Most people would call that gentle to moderate.

The OP's equations do not take into account varying engine efficiencies at different power demands.

R2-E2, 2G Prius.
Highway/City/Husband/Wife MPG: 56.5, as of 12/2005, 26K miles

Jac Nasser, Ford President: "We are planning to launch a hybrid version of
this car [P2000] within this year [1998]. We will also make FCEV available in
2004."

"hard" acceleration means something less than 50 sec to get from 22 to 44mph (that would be a=.02g), but the equation says that you'd get even better performance to do it in 2 sec (which would get you from 0 to 60 in about 5 seconds). I can't imagine smoothly going to a glide after that kind of neck-jerking, tho. Just letting the engine idle down before you kill it would probably take a few seconds.

I think the varying engine efficiency is accounted for in my hill-climbing experiments which led to the equation involving C, the number which lies between 30 and 50 for a HCH.

Decelerating from 44 to 22 miles per hour depends no how fast I was going and the slope of the hill if any. Drafting behind larger vehicles will result in longer glides as well.

I've been wondering about the pulse and glide.If the gasoline motor didn't have any internal friction then pulse and glide wouldn't work any better than just keeping a constant very slow speed that would be equivalent to the average speed of the pulse and glide run.I'm just thinking out loud,and just thinking in terms of using it with a gasoline only engine.The same situation applies to the hybrids,but it is easier for me to see it with just a gas motor.
I'm guessing it works because keeping a constant 40 miles per hour-say 2000 rpms with an auto trans Corolla costs more in internal friction losses than accelerating to 60 miles per hour from 20 miles per hour-shutting down the gasoline motor and gliding down to 20 miles per hour..Yes,a rev at 60 mpg-3000 rpms-cost more internal friction than a rev at 40 miles per hour-a 2000rpms rev because the piston-rings are moving faster,and the higher pressure in the CC should press the rings into the cyl walls a bit harder.Still,I think the frictional loses from a gasoline motor would be less with pulse and glide.It would probably work-some-not shutting the motor off since auto trans do tend to let the motor drop waaay down when you let off the gas.There probably isn't any practical way to do an actual shut down of the motor with a modern vehicle.Part of the advantage of hybrids is that they actually shut the gasoline motor off when you let off the gas(well-the toyotas will at some speeds-te Hondas will at least let the RPMS drop waaay down-I think).
I might have this backwards,so I'm open to correction.Thanks.Charlie

Dave,

As a Prius marathon man and an HCH hypermiler with some 38,000 miles in my HCH, I can attest that a harder acceleration is better for FE in the P&G mode in the HCH. As to the rest of your post, I barely passed 9th grade algebra, and never took Alg II, and I don't even know what Trig or Calc are.

The problem I have with a hard acceleration pulse mode, is that the IMA is in assist mode draining the battery pack. In glide mode, the battery is not getting recharged. It does not take long before the battery pack is empty.

Further, it seems to me (a history major, not a physicist), that much of the increased ICE efficiency would come from using the energy stored in the battery pack without replacing it.

In my mind, the only sustainable P&G in the HCH is to ensure that during pulse you are also charging the battery pack. This means a very gradual acceleration with a whole lot of gas pedal control, watching the IMA display to make sure you have green lights, not none or blue. I have found P&G in this manner in my HCH is good for FE of 113 mpg over a closed course and the battery back pack stays charged. That's good enough for me.

Bob

P&G has always seemed rather intuitive to me once I heard about it -- so maybe I have it all wrong

Pulse: Use the ICE at it's most optimal power output range
Glide: Engine off, so no idle waste, no partial power drag

Ever wonder about theoretical maximum MPG ? For the Prius:
According to Wayne Brown, on level ground with no wind at 35 miles per hour, steady state energy demands are 3.5 - 4 kW. One gallon of summer gasoline has 36.3 kWh of energy, and the Prius atkinson ICE at it's best turns about 38% into mechanical energy, or 13.8 kWh. Five to 10% is lost in the transmission, leaving 12.4 - 13.1 kWh.

Computing the two extremes of the range:

12.4 kWh/Gallon / 4kW * 35 miles per hour = 108 mpg.
13.1 kWh/Gallon / 3.5kW * 35 miles per hour = 131 mpg.

The best leg of the marathon was about 120 mpg

R2-E2, 2G Prius.
Highway/City/Husband/Wife MPG: 56.5, as of 12/2005, 26K miles

Jac Nasser, Ford President: "We are planning to launch a hybrid version of
this car [P2000] within this year [1998]. We will also make FCEV available in
2004."

And our average was just 110 mpg. My leg was just 108.5 mpg if I remember correctly.

It only goes to show that we left room for others to improve on our efforts.

[quote=EricGo]According to Wayne Brown, on level ground with no wind at 35 miles per hour, steady state energy consumption is 3.5 - 4 kW.

This energy consumption should define the B factor, before you start pulse and glide. Doing P&G will improve mileage further possibly getting you to 150mpg, depending on the other factors.

RJB: I hear your complaint about depleting the pack, but didn't Xcel & Tbaleno find that it stops depleting when it gets low enough? My problem with P&G in HCH is you never get into lean burn.

Dave (wish I could figure out how to show quotes )