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As Friday was virtually wind-free, or at least as calm as it gets,
it seemed a good time to do the power requirement measurements on my
latest car. The speed range is not as wide as I would have liked; but
the data should be satisfactory for most purposes.
The data was fitted to a third degree polynomial with the data
for 80, 70, 60, and 50 MPH as boundary conditions. The resulting
function was within my reaction time for the other data points; so
I judged the results good to two significant figures.
The calculations file, Lotus 123 format, is in ~drew/grand_am.wk1.
-------
Mass (curb): carpoint.msn.com/Vip/Specifications/Pontiac/Grand%20Am/1993
2756 lb
Crew &
cargo: 150 lb approx
Fuel: 45 lb approx
Total: 2951 lb
Speed Time Rough Poly Accel Accel Drag Power
(MPH) (sec) diff calc (MPH/sec) (grav) (lbf) (W)
------------------------------------------------------------------------
80 7.6 80 -0.81462 -0.03713 -109.585 -17500
75 13.6 6 75.26804 -0.76272 -0.03476 -102.603 -15400
70 20.8 7.2 70 -0.70066 -0.03194 -94.2551 -13200
65 28 7.2 65.17786 -0.63885 -0.02912 -85.9406 -11200
60 36.6 8.6 60 -0.56535 -0.02577 -76.0533 -9100
55 47.1 10.5 54.53279 -0.47610 -0.02170 -64.0464 -7000
50 57.6 10.5 50 -0.38737 -0.01765 -52.1108 -5200
Speed vector: 80 70 60 50
Time matrix: 1 1 1 1
7.6 20.8 36.6 57.6
57.76 432.64 1339.56 3317.76
438.976 8998.912 49027.89 191102.9
Inverse: 2.290999 -0.21251 0.006008 -0.00005
-2.08755 0.367957 -0.01326 0.000130
0.946291 -0.18643 0.008937 -0.00010
-0.14973 0.030991 -0.00168 0.000025
Polynom constants:86.44175 -0.88062 0.004350 -8.0E-07
Deriv constants: -0.88062 0.008701 -2.4E-06
-------------
As expected, the power requirement, at least for one-minute miles,
is much less than that of an _Impala_'s 25 kW.
As for overall efficiency, I've measured 30 miles per gallon under
conditions of a mixture of 70 MPH average freeway and variable surface
speeds. Taking 2.3 gallons per hour as the fuel consumption under
standard freeway operations, and 115000 BTU/gallon of fuel, we get
264500 BTU/hour, which is about 77.5 kW. The efficiency, fuel to
propulsion, works out at around 17%. I seem to recall higher efficiencies
with earlier vehicles.
I'll post results for the station wagon when I manage to get data
for it.
28 responses total.
Probably doesn't matter, but doing a least-squares fit of your polynomial to all the data points, rather than an exact fit to just four, would distribute the error and provide a more "efficient" parameter estimation. Also, if you do an orthogonal polynomial fitting, you can test what order of polynomial is sufficient.
Thanks. I'll try it soon as I can find some information on least squares and orthogonal fitting.
Matlab does least-squares, an area wherein Lotus 123 seems to be deficient. I shall have to see about writing an add-on... Coefficients are slightly different, but not grossly so. I expect the final results to be very much the same.
In really interesting cases, fitting a third order polynomial to four points, can give really wild fits. The most important point is to use *all the data*. An estimator that uses all the data is called an efficient estimator, and always has the smallest variances of the parameter estimators. You could illustrate this point by comparing the sum of squared deviations for the remaining three points when an exact fit is made to four points, with the sum of squared deviations for a least squares estimator using all seven points.
I've got data for the other car now; and I've redone the calculations for the
first one. ~drew/cardrag.wk1 does not include code for doing the least-squares
fit, which I did in MatLab using the POLYFIT function, since I don't know how
to get Lotus 123 to do that yet.
Grand Am:
Mass (curb): carpoint.msn.com/Vip/Specifications/Pontiac/Grand%20Am/1993
2756 lb
Crew &
cargo: 150 lb approx
Fuel: 45 lb approx
Total: 2951 lb
Speed Time Poly Accel Accel Drag Power kW*hr
(MPH) (sec) calc (MPH/sec) (grav) (lbf) (W) /mile
------------------------------------------------------------------------
80 7.6 80.00630 -0.86589 -0.03947 116.5 18624 0.233
75 13.6 75.07199 -0.78044 -0.03557 105.0 15736 0.210
70 20.8 69.78537 -0.69032 -0.03146 92.9 12991 0.186
65 28 65.09877 -0.61376 -0.02797 82.6 10726 0.165
60 36.6 60.15105 -0.54008 -0.02462 72.7 8712 0.145
55 47.1 54.83992 -0.47636 -0.02171 64.1 7044 0.128
50 57.6 50.04656 -0.44146 -0.02012 59.4 5934 0.119
Polynom constants:87.04021 -0.98765 0.008507 -4.4E-05
Deriv constants: -0.98765 0.017014 -1.3E-04
========================================================================
Safari (wagon):
Mass (curb): carpoint.msn.com/Vip/Overview/Pontiac/Safari/1988.asp
4182 lb
Crew &
cargo: 150 lb approx
Fuel: 30 lb approx
Total: 4362 lb
Speed Time Poly Accel Accel Drag Power kW*hr
(MPH) (sec) calc (MPH/sec) (grav) (lbf) (W) /mile
------------------------------------------------------------------------
85 3.7 85.25442 -1.42615 -0.06501 283.6 48174 0.567
80 8.1 79.40030 -1.23836 -0.05645 246.2 39370 0.492
74 12.5 74.32544 -1.07195 -0.04886 213.2 31523 0.426
70 16.8 70.02861 -0.92997 -0.04239 184.9 25870 0.370
65 22.7 65.03736 -0.76838 -0.03502 152.8 19848 0.305
60 30 59.99955 -0.62164 -0.02833 123.6 14822 0.247
55 39 54.92147 -0.52172 -0.02378 103.7 11403 0.207
50 48.6 50.03282 -0.51370 -0.02341 102.1 10207 0.204
Polynom constants:90.84928 -1.60061 0.024596 -0.00018
Deriv constants: -1.60061 0.049193 -5.5E-04
As expected from rough fuel economy measurements, it takes about twice as much
energy, mile for mile and MPH for MPH, to run the station wagon as it does
the smaller car. A station wagon type hull must somehow be better for drag
than a sedan, since somehow it can do one-minute miles on less than 15 bkW
of power, as compared to 25 kW for the car I first did this experiment with
('81 _Impala_). I'm getting about the same fuel economy, but I've also
typically run the station wagon around 70 MPH instead of 60; and the station
wagon *is* a bit more massive - irrelevant on the highway, but a disadvantage
on surface streets.
Re the odd data point in the station wagon test: I was a bit late in calling
off the 75 MPH mark.
Data (and analyses) like these should be inlcuded in car evaluations!
Hmm. Wonder what the figures would look like for my Econoline. It has all the aerodynamic design of a boxcar. I don't have the tools to do the curve fitting, though.
It took this mathematical lackey a while to figure out what the heck you were doing but I think I got it. I don't know how valuable this sort of data would be as there are a lot of variables. Tire type and pressure would be a significant one that doesn't seem to be addressed here. Same goes for how well the wheels are aligned. As for it being useful for purchasing a car, the numbers can be easily muddied by options such as roof-racks, side mirrors, wheel and body trim options, additional suspension components increasing drag under the car, the mechanical drag added by going from 2WD to 4WD, etc. Your drag figures for the station wagon seem a little high or the figures for the sedan are a little low. I don't see how the drag of the wagon can be over twice that of the sedan! Two sedans are more efficient than one station wagon? It just doesn't make sense!
It's not a full sized sedan. The '93 _Grand Am_ is a smaller car than in previous years. In fact, a sedan that I tested over a decade ago was *less* efficient than the current wagon - 25 kW for one-minute miles, compared with less than 15 kW for the wagon. As for the variables, yes, that's a problem. Still, I consider this information better than none at all. Re #7: I intend to see about teaching Lotus 123 to do curve fitting (have to look it up myself first). You might want to collect your data during the next convenient calm period. You'll need a stretch of highway that is both straight and level; and a tape recorder, along with a sound card and Windows Sound Recorder to play it into, will be extremely helpful.
There will always be a lot of variables in any comnparison of two cars, even of identical models. Yet, cars are always being compared! In fact, the comparisons that are made are seldom as precise as this, in part because they average driving conditions over some distance. That is both the strength and weakness of what drew is doing. His measurements give instantaneous variables, and would have to be also done up and down hill (for example), etc. Of course, they also don't measure engine performance - just the power losses. But those are what the engine makes up for, so this helps separate the contributions of the active (engine) and passive (vehicle) components.
Re #7: It's not that hard to do a least-squares fit with a calculator. I did my first one with a BASIC program I wrote given an algorithm on the blackboard, IIRC. If I was interested, I could probably derive the algorithm now (I didn't know calculus then). Re #9: Instead of a tape recorder, a camcorder might be better. Just aim it at the speedo, mention significant events for the soundtrack and do the timing by frame-counting on your VCR at home. (This won't work if your speedometer is a bit sticky, like mine.) If you can also see the outside in the FOV, the camcorder also allows you to associate acceleration anomalies with the terrain. Re #10: Once power demand is determined, it's not much of a jump to convert fuel consumption into an overall drivetrain efficiency. This can be very illuminating. One of the complications not mentioned thus far is that crosswinds will increase vehicle drag in both directions along a road. This might, or might not, make a significant change in the numbers derived. The obvious way to control for it is to test on a calm day.
What's still missing is a fuel consumption rate meter.
That would be nice to have. I considered getting one, but they were scarce; and the one I did see in an auto parts store (calibrated in "miles per gallon at 35 MPH and 55 MPH), as well as "car computers" in catalogs, were a bit pricey for me at the time. And I'm still not sure that *any* of them used an actual fuel flow sensor, rather than a simple vacuum sensor or ignition timing hookup.
Flat roads are scarce in this part of Minnesota, but if I find one I may try it. I just have an idle curiousity as to how the figures would compare to the Grand Am's.
I have uploaded ~drew/cardrag2.wk1, which now contains least-square code. Once you have substituted your own data, and stretched or shrunk the column to fit the number of points, you will need to execute a macro to get Lotus 123 to do the matrix operations. The macro is activated by typing alt-M, and will recalc the whole spreadsheet afterward. A derivation of the least squares method can be found in ~drew/leastsq.doc (MS Word format). I am confident that it is correct as the numbers it spits out seem to agree with MATLAB's results.
You could compensate for flatness and wind by running the test in both directions. (Is the most efficient shape that of a falling rain drop?)
A falling rain drop looks like a pancake, somewhat thickened around the periphery. It looks *nothing* like the metaphorical "teardrop". In fact, it takes the shape that has nearly the maximum possible drag!
Interesting! I didn't know that. So what shape has the least amount of drag?
Something like the *proverbial* raindrop! In regard to the shape of a real raindrop, consider that the maximum pressure due to its fall velocity occurs at its 'nose'. Since the drop is deformable, the nose gets pushed in, and the result is the deformed pancake. This shape and air turbulence leads to the shape continually and rapidly deforming. The drops you see fall so fast you can't see all the bizarre shapes they take. Surface tension is fighting these shape changes, so very small drops approach spheres and the shapes get more irregular as the size increases, until the irregular motions of the drops lead them to break up.
Most people assume the most aerodynamic shape looks like a rocket, with a very sharp, tapered nose. The Concorde is another example of this shape. However, rockets and the Concorde are both made to fly at *supersonic* velocities. The most efficient shape at subsonic speeds has a rounded nose and a tapered tail -- as rcurl says, much like a teardrop. Look at a 747; blunt, rounded nose at the front, long, tapered tailcone at the rear. The GM EV-1 is also a pretty good example; it has the lowest drag coefficent of any production car, IIRC. Somewhere around 0.19. (It has only 19% the drag of a flat plate the same size as its frontal area.) For comparison, most pickup trucks have drag coefficients around 0.75.
Re #10: Considering that a falling raindrop is somewhere between spherical (for the really small ones) to hamburger-shaped (for the larger ones)... I'd guess not.
On drag coefficients, it should be noted that it is necessary to specify just what dimensions you are counting as the "area" for the purpose. #20 specified the frontal area. However, airfoils generally have drag - and lift - coefficients listed based on the area of the top of the wing, ie, the large surface. The drag coefficient - or rather, that part of it that is not induced by lift, is largely dependent on the Renolds number, another dimensionless quantity based partly on the size of the object. Like with drag coefficients, the Renolds number must be given in context of which dimensions of the object are used as its size.
Very true. I was under the impression (possibly wrong) that for automobiles the quoted drag coefficient is usually the frontal area. You need to know both to get any idea of how "aerodynamic" the object really is. For example, a 1960's VW bus has a better drag coefficent than a Beetle of the same era -- the bus has a rounded nose and flat sides, and the Beetle has a flat windshield and wide fenders that stick out into the airstream. The Beetle's frontal area is so much smaller, though, that the total drag ends up being a lot less. On flat ground the Beetle will do 80 with roughly the same amount of power it takes to get the bus to do 60.
That's Reynolds number. The lift is at right angles to the drag. By definition, drag and lift are separate, orthogonal, forces. One can't say lift contributes to drag, therefore. Of course, the whole profile contributes to both drag and lift. The "projected frontal area" convention for a drag coefficient is just that - a convention to define the drag coeff.
Escort Wagon
Mass (curb): autos.msn.com/Vip/Specifications.aspx?modelid=1064&src=vip
2513 lb
Crew &
cargo: 160 lb approx
Fuel: 60 lb approx
Total: 2733 lb
Speed Time Poly Accel Accel Drag Power kW*hr
(MPH) (sec) calc (MPH/sec) (grav) (lbf) (W) /mile
------------------------------------------------------------------------
80 7.87 79.76637 -1.15239 -0.05253 143.6 22954 0.287
75 11.7 75.43812 -1.10717 -0.05047 137.9 20675 0.276
70 16.8 69.95688 -1.04123 -0.04746 129.7 18148 0.259
65 22 64.73187 -0.96724 -0.04409 120.5 15654 0.241
60 27.2 59.90930 -0.88644 -0.04040 110.4 13243 0.221
55 32.7 55.28575 -0.79357 -0.03617 98.9 10867 0.198
50 40.1 49.91168 -0.65658 -0.02993 81.8 8174 0.163
Polynom constants: 89.16584 -1.23369 0.004669 0.000042
Deriv constants: -1.23369 0.009338 1.3E-04
This car turns out to be, on rough average, about 40% harder to push
than the Grand Am was, which explains the lower fuel economy. In fact,
the Escort's engine is actually more efficient at producing power, since
it only consumes 20% more fuel over a given distance (25 MPH versus 30
MPG) instead of the full 40% extra.
Curiously, the Escort wagon wins big over the Safari wagon at higher
speeds, but loses most of this advantage around the 55 MPH mark:
Speed Escort Safari Escort's
(MPH) (kW*hr/mile) (kW*hr/mile) Savings
-----------------------------------------------
80 .289 .492 42%
75 .278 .426 35%
70 .261 .370 30%
65 .243 .305 21%
60 .222 .247 11%
55 .199 .207 4.3% (!)
50 .165 .204 20%
At around 55 MPH, there is less than a 5 percent difference between the
hull drag of an Escort and a Safari wagon. At 50 MPH (and presumably
below), the advantage of lower drag reappears.
Maybe some day I'll try to get figures for my '94 Honda Civic hatchback and my '82 Volkswagen Vanagon Westfalia. The Vanagon drag figures ought to be amusing.
_Windstar_ class minivan:
For this one I had on hand a video camera.
Test conducted 2007:05:20; temperature 63 degrees F, barometric
pressure 30 inches of mercury, for an air density of 1.219 kg/m^3.
Intellicast reported wind from the north at 11 MPH, which I judged
small enough compared to the test speed range, or at least as light
as it usually gets.
Beforehand I had determined that the speedometer and odometer both
read about 15% short. The other vehicles that I tested had accurate gauges.
The highway in the test location runs along course 71 degrees (ENE) and
251 degrees (WSW). For this test I managed to get data for each direction.
Mass
Curb mass 3711.4 lb
Fuel 75
Self 150
Misc 50
Total: 3936.4 lb
Polynomial approximation: V = sum( Ak t^k ); a = sum( k Ak t^k )
East-Northeast run:
Speed Speed Time Poly Accel Accel Drag Power kW*hr
(MPH, raw) (correct) (sec) calc (MPH/sec) (grav) (lbf) (W) /mile
-----------------------------------------------------------------------------
77.5 89.13 16.0 88.82 -1.577 -0.0719 282.99 49985 0.56274
75 86.25 18.0 85.72 -1.521 -0.0694 273.01 46539 0.54289
70 80.50 21.4 80.71 -1.427 -0.0650 256.04 41094 0.50914
65 74.75 25.4 75.23 -1.316 -0.0600 236.06 35313 0.46942
60 69.00 29.6 69.95 -1.199 -0.0546 215.09 29917 0.42771
55 63.25 35.5 63.36 -1.034 -0.0472 185.62 23387 0.36910
50 57.50 42.7 56.63 -0.834 -0.0380 149.64 16852 0.29756
45 51.75 50.3 51.10 -0.622 -0.0284 111.65 11345 0.22202
40 46.00 59.4 46.59 -0.369 -0.0168 66.14 6128 0.13153
West-Southwest run:
Speed Speed Time Poly Accel Accel Drag Power kW*hr
(MPH, raw) (correct) (sec) calc (MPH/sec) (grav) (lbf) (W) /mile
-----------------------------------------------------------------------------
76.5 87.98 14.5 87.78 -1.381 -0.0630 247.80 43255 0.49276
75 86.25 16.0 85.74 -1.348 -0.0614 241.89 41238 0.48100
70 80.50 19.8 80.77 -1.264 -0.0576 226.90 36443 0.45119
65 74.75 24.6 74.96 -1.159 -0.0528 207.96 30997 0.41354
60 69.00 29.2 69.86 -1.058 -0.0482 189.81 26367 0.37744
55 63.25 35.8 63.35 -0.913 -0.0416 163.76 20631 0.32564
50 57.50 43.5 56.98 -0.743 -0.0339 133.36 15110 0.26518
45 51.75 52.8 51.02 -0.538 -0.0245 96.62 9803 0.19213
40 46.00 63.5 46.52 -0.303 -0.0138 54.34 5026 0.10805
I don't have figures for actual fuel economy yet - getting them requires
filling the tank - *twice*. The behavior of the fuel gauge suggests
a rate of about 19 MPG on surface streets and *maybe* 25 MPG on
nonstop highways. I plan to get better instrumentation at some point.
25 MPG at 75 MPH is 3 gallons per hour, or about 100 kW. 20 MPG would
be 3.75 gallons per hour, or about 130 kW. This implies a total drivetrain
efficiency ranging from 23% to 31%. While still not exactly stellar (the
ideal for a 9:1 compression ratio air standard otto cycle is 58.5%), it's
significantly better than those of the _Escort_ and _Grand Am_.
These test results and other analysis will henceforth appear at
http://drew4096.livejournal.com.
You know, looking back at this, I'm rather shocked that the Escort Wagon could only manage 25 mpg. It's not that big a car.
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