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Old 06-15-2006, 07:15 AM   #1
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A First Order Equation for Estimating Drag Coefficients

These are some of my thoughts and notes to give myself some understanding of the topic of how to design an economical car, how to estimate their drag coefficients, and some of the design considerations. I think you'll find it interesting. If you are bored, skip the math and look at the graphs. If you are mathematically inclined, check over my figures.

Have a look at the following diagram.



Designing an automobile is an exercise in juggling various priorities. Note that it is possible to design a shape that gets a drag coefficient far below modern cars - the streamlined body, teardrop or icecream cone. Most cars have a Cd of around 0.35. The streamlined body has 11% the drag of a typical automobile, and at highway speeds, at least 60% of the work of an automobile is fighting this drag.

There are numerous ergonomic reasons why the teardrop shape is not optimal ergonomically. It's harder to manufacture, consumers aren't used to it, handling will not be optimal because the wheels will be required to be shorter and narrower than most cars, the trunk space will be reduced, the rear seats will have less room, etc.

However, this is a site about saving fuel. Mother nature has provided us with the teardrop shape; we have to work around that. One thing to note is that with a rake of greater than 11 degrees for the rear cone part of the icecream cone shape, there is just as much drag if not more than if it was just severed with a 90 degree bend. (Such as done with the VW 1L car).



We know that drag is primarily a function of the largest area of turbulence. Minimize turbulence completely and you are left with the a Cd of near zero - 0.04. If you have a streamlined shape up until the end (as in the above car), how big the area of that end of the car will determine largely what Cd the car has.

As such, I thought it would be useful to postulate a formula for roughly working out drag coefficients in such cars and our own modifications, relative to vehicle length so that we can get a rough idea of the optimization process and why the cars are built this way.

The first thing to do is realize that we can approximate the car as a choice between two extremes. The first of those is the half sphere. The second is the teardrop, the streamlined body, which is approximately a half sphere and a cone with 11 degree taper. (Obviously this is an approximation, but quite close.)


So, if we say that the drag should be approximately the same as the final area where we cut off the icecream cone shape, and we know that the drag has to lie between two extremes, 0.42 for the half sphere and 0.04 for the streamlined body, we can propose a formula.

Cd = 0.04 + (Af/Ahs) * 0.38

Where
Cd = drag coefficient
Af = final area at rear of car
Ahs = area of half circle (or cross sectional area of car)

Note that when Af = Ahs, we get a Cd of 0.42, which is the same as the hemisphere. And when Af = 0, we get a Cd of 0.04, as we would expect from the full streamlined body.

Next we want to get Cd in terms of l, or length from the half circle part of the car to the rear, as indicated in the above diagram. This will enable us to graph Cd versus l, give us an indication why car manufacturers make the tradeoffs they do, and give us an indication for our own car modifications.

First of all, we know that Af corresponds to the following formula for area of a circle:

Af = pi * r2 ^2
Where r2 is the radius of the final portion of the car.

If r1 = r-r2,

And since tan (theta) = opp/adj
tan theta = r1/l
therefore r1 = l * tan (theta)

therefore r2 = r - r1 = r - l * tan (theta).

Now, Af = pi * r2 ^2
= pi * (r- l * tan (theta))^2

So, Cd = 0.04 + (Af / Ahs) *0.38
= 0.04 + [(pi * (r- l * tan (theta))^2)/Ahs] * 0.38

But since Ahs = pi * r^2,

Cd = 0.04 + [(pi * (r- l * tan (theta))^2)/(pi * r^2)] * 0.38

Cd = 0.04 + [(r- l * tan (theta))^2 / (r^2)] * 0.38

So now we have a formula for drag in terms of length and radius at the widest point. Note that the above formula can be simplified in terms of (l/r) ratio. Also note that the formula will only apply with theta <=11 degrees.

If we adopt theta = 11 degrees, l/r = 5.14 gives us the full streamlined body, and l/r = 0 gives the half circle. We can now graph the result to give us a handy curve by which we know when it is appropriate to "cut off" our streamlined body.



Here are the same figures, but plotted against with the y-axis as logarithmic scale to give a better indication of where the diminishing returns are - around l/r = 4.



So, now we have a much better grasp on how far to extend the tail of our car in order to achieve a particular coefficient of drag, or how much we sacrifice by cutting that tail short. And note that the constants in the formula will vary from car to car, depending on how streamlined the front is, etc, but it's still good to get an overall picture of how the "boattail" thing works and the design considerations involved.
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Old 06-15-2006, 07:57 AM   #2
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A couple of curious things:

1) The half-sphere has a better Cd than a full sphere. It seems counterintuitive. The full sphere looks more streamlined.

2) The long cylinder has a better Cd than a short cylinder. I've spoken to a lot of diesel F350 owners, and most of them have an extended cab and claim 22 mpg, a lot better than the sub-20 mpg that I get. Could the longer truck cab have a better Cd and get better mileage?
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Old 06-15-2006, 08:27 AM   #3
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Quote:
Originally Posted by Sludgy
A couple of curious things:

1) The half-sphere has a better Cd than a full sphere. It seems counterintuitive. The full sphere looks more streamlined.
Yes, it is highly counterintuitive. Which is why it's very important to know if we are to maximize our fuel economy. Basically the idea I'm trying to get across here is that we can get a first order approximation at the potential a given car shape has for minimum drag (after underbody etc has been smoothed) by identifying the cross-sectional area of the car at the point where it departs from a certain angle (unless of course the front is really, really boxy).

I live for the day when they start to design cars that ARE streamlined instead of LOOKING streamlined to the average person. I suspect we are approaching an era when we will see this. We have two things in our favor: high gas prices (going higher), and instantaneous mpg meters that are trivially easy to make. This is the difference to the 70s - the best thing they had back then was a vacuum guage. The instantaneous mpg meter enables cross-car competition, which enables the early adopters to bypass marketing bs such as "the tornado" or "brockie's crystals" and vote with their pocketbook for cars with good fuel economy.

Because let's face it; car companies won't respond with economical cars until it starts hurting their bottom line. They have every incentive to keep up the status quo, to sell something that the public thinks is economical but is woefully inefficient. The buyer pays the fuel bill, not the auto maker. Once that situation changes... and it is already... the momentum will shift. Car companies will face bankruptcy and the one that does something right will be rewarded.

Of course, this needs sustained high oil prices, hopefully over a period of greater than 3 years (the time it takes an auto maker to take a car from conception to market).

Quote:
2) The long cylinder has a better Cd than a short cylinder. I've spoken to a lot of diesel F350 owners, and most of them have an extended cab and claim 22 mpg, a lot better than the sub-20 mpg that I get. Could the longer truck cab have a better Cd and get better mileage?
Yes, quite possibly. Of course, it would be even better with a boat tail, or if the cab started to dip earlier.

One other thing I need to check is the 11 degree figure. I'm not 100% sure that it's correct - it needs verification. However, if everything else is correct the shape of the graph will not change, just the x axis will be shorter. For example, if 20 degrees is the correct angle for the fastback, then we only need 2.74 times the radius at the point of maximum cross-sectional area in order to reach a full point at the rear. I'm not sure where I heard about the 11 degree figure - I think it was somewhere on this site.
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Old 06-15-2006, 02:41 PM   #4
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Mira, this is all excellent stuff. As yet I haven't done any aero mods to my car, but it is obviously the step to providing large leaps in fuel efficiency. I have front and rear wheel skirts designed, but not built.

I wasn't certain if I wanted to do aero mods at first, but I am really looking forward to testing some once I get my SuperMID. I'm hoping to build the aero mods over the next month. I will probably start this weekend.

Hoping to get the MID in August. Depending how long it takes to calibrate it will decide when I get to start testing things.
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Old 06-15-2006, 05:28 PM   #5
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Quote:
Originally Posted by 95metro
Mira, this is all excellent stuff. As yet I haven't done any aero mods to my car, but it is obviously the step to providing large leaps in fuel efficiency. I have front and rear wheel skirts designed, but not built.
Thanks. I haven't done any aero mods either, but am very keen to. Obviously they will have to wait until I can measure instantaneous fuel economy. It's good that there are plenty of scientifically minded people who desire to have a control to measure their experiments against and an immediate, accurate method of testing.

One thing that I think needs to be established is a proper metric (or set of metrics) for fuel economy increases. At the moment it seems that mpg is used. E.g. something is worth 1-2mpg. Depending on whether you get 25mpg now or 50mpg will determine whether that fuel economy increase is large or small. But maybe this is asking too much.
Quote:
I wasn't certain if I wanted to do aero mods at first, but I am really looking forward to testing some once I get my SuperMID. I'm hoping to build the aero mods over the next month. I will probably start this weekend.
I'm sure you'll keep us updated! Look forward to seeing the results.
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