Aerodynamics Seminar # 6 - by Phil Knox
 

This post was originally written by Phil Knox (aerohead), and it first appeared on the MaxMPG group. Phil has done a lot of work educating the masses about the critical role aerodynamics play in efficiency, and has spurred many in the DIY crowd to take matters into their own hands.

This is the sixth in a series which I'm reproducing here with permission.

Go to: Aerodynamics Seminar index

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Aerodynamics Seminar # 6 - by Phil Knox

(Note: Phil wrote this preface to #6 in a separate message; I think it's worth posting along with the #6 info. - Darin)

Preface

Hello all. As my Tuesday departure deadline approaches much remains to do. I have been collating materials for the "math" section of the seminars. It looks like it will be after I return from New Mexico before I can post the formulas I wanted to share.

There is much to complete on the T-100 and after I leave here tonight,its off to HOME DEPOT for more supplies and work into the night. Hurricane Rita could drop 4-5 inches on us and I have no idea what kind of "work" conditions I will have for fabricating. It was 102-degrees today,so again I'll cook well into the night.

The formulas,once posted,will allow everyone to calculate aerodynamic loads for any vehicle,knowing a few perameters. You'll be able to estimate road load horsepower for a vehicle at various conditions.With your own mileage numbers you'll be able to calculate your brake specific fuel consumption,estimate top speeds,fuel economy,range etc.

A pocket calculator will be handy.There is no trigonometry or calculus.Its all pretty much multiplication,division,and working with percentages.We'll walk through some examples together.No tests!!!!!!!!!!!!

If you do not presently have a "baseline" for your vehicle,you'll want to get a simple notepad for your vehicle and something to write with.I record: date, odometer reading, miles driven, fuel price, fuel cost, gallons, mpg, station,and any notes about modifications to vehicle, weather,etc.

By tracking mileage I can establish what the vehicle does with no modifications.When I make a change and see different results,I can use the formulas to quantify what if anything I've accomplished. If the vehicle performance changes for no reason,I can consider a tuneup or whatever. The main thing is "write stuff down!" If you know where you are and want to investigate if a modification might make sense, you can do it on paper and see if you want to proceed.

Long ago I wrote a simple program in BASIC and witha few keystrokes,could race through some calculations. Technology asside,for the day to day world we are basically looking at odometers and fuel pump readings. If you have an electric,you'll be working with kilowatts, power transfer efficiency, motor efficiency.Then,once your at the drive wheels, your down to rolling resistance and aerodynamic drag like everything else. If anyone has any specific questions, I'll try and get something for you.

When your looking at your car,think of a sailfish or perigrine falcon. Think of bottlenose dolphin and penguins. I was in a category-5 hurricane today and thought little of it. At 150-mph the Piper Cherokee I was flying in was right at home with this kind of wind loading. All the ample radii and gently sloping surfaces of the Cherokees air frame gave the air nothing to grab hold of.

As John G. mentioned in his blog,If I encounter headwinds on my trek next week, I'll easily be driving in a Class-1 hurricane. The aero mods to the T-100 hopefully will allow me to pass without incident.

I won't worry about my home as it is also aerodynamic, a ferro-cement dome which a tornado would shy away from and tidal surge,while wet,would eventually abate and leave me relatively unscathed.

Wish me well and I'll catch up in a couple of weeks, Phil.

Seminar # 6
(Note: Where Phil refers to photos/images in his text, I intend to come back and post them once all the Seminars are online. - Darin)

Hellow all. I had hoped to post large,neat,formulas in the photo section of the site. I think everyone needs these sooner than later so here goes.

The horsepower it takes at the drivewheel of your vehicle to overcome aerodynamic drag can be estimated by the formula

HP =V/375 [ 0.00256 X Cd X A X (V squared)]

where V= speed in miles per hour, Cd is your drag coefficient,A= frontal area of your vehicle,and (V squared) is your speed times itself.

Frontal area can be approximated by multiplying your vehicles width times its height,times 0.84. Frontal area is in square feet, so as dimensions are usually given in inches,divide the result by 144 to get square feet.

After you get the result,you can obtain flywheel horsepower by dividing your HP by 0.95 which yields Brake horsepower(Bhp).

These numbers are for standard barometric pressure, no wind, dry road, no more than 0.5% grade, 60-degrees F.

If you don't have frontal area,estimate it. Plug in your drag coefficient,and speed,multiply everything inside the parenthesis together,then multiply all that by your speed again and finally,divide by 375.

Ummm... airplanes don't know what windspeed is unless they're interfacing with the ground!

Calculating Horsepower requirements - including weight
 

Understanding that HP to overcome aero/drag is helpful. I want to include vehicle weight in my calculations to fully understand the impact of weight.

I came across this, but didn't get to an understandable value. I may just be overlooking some error in my calculations.
Pmotive=1/2*Cd*A*rho*v^3+Crr1*m*g*v
from http://www.geocities.com/greglocock/

My attempt at this calculation is attached on the second tab. Please let me know if anyone can fix it or provide the conversion from Pmotive to HP (or if you have a better spreadsheet). The first tab has the calculation from this discussion (The results look correct to me).

With significant investment, I think I can lower my weight by 200 lbs. Hopefully I can get some idea of the pay back. Spending most of my time at a stead speed (70 mph), I expect less pay back than the average vehicle. Taming my driving habits has already given me a 10% improvement (from average of 27 mpg to a max of 31.5).

I also want to find [easy] ways of reducing my RPMs at highway speeds from current 2750 to about 2200, but that's a different topic...

Thanks.

Let's stick to the metric system since it seems to use it, mostly?

1/2C_d is unit-less
A in m^2
rho is density, so kg/m^3
v is m/s, so v^3 is m^3/s^3

so we have kg*m^2/s^3
1 N is kg *m / s^2 -> Nm/s
1 J is Nm -> J/s
1 W is J/s, so we are left with pure watts
1 hp is 0.7457 kW

Essentially we take the result of the first part and *10^-3 / 0.7457, or multiply by 0.01341.

I'm not sure what the second part of the equation is, the addition, but in order to add numbers, the units MUST match. Pmotive=1/2*Cd*A*rho*v^3 is normal for a flat surface. What is Crr1?

I'm no expert on this (only been in a co-pilots seat 2 times) BUT Pilots can calculate wind speed (at their altitude). The first plane I flew (piper Cherokee 180) did not have GPS and relied upon the tower to give a radar ground speed and heading, we then (re-)calculated out flight plan based on the wind speed/ direction. Plane #2 Cessna (rental) had a GPS which interfaced with the majority of the avionics and was able to display the wind speed + direction (in near real time) based on ground speed and Airspeed + Heading.

I think that's what he meant. It's analog, but it's still an interface :D

True, true

optiwatch' aero calc
 

optiwatch,sorry! I'm way late to the party.I ran your numbers for 70 mph and got 19.1 horsepower aero load at that velocity.I used 0.00238 slugs/ft cubed for density,102.67 feet per second velocity,Cd 0.32,and 25.5 ft-square for frontal area.Sorry,as yet,don't have my estimation stuff for rolling resistance with me.I don't know if your formula also simultaneously figures rolling resistance horsepower.I stuck with what I'm familiar with.Also ,apologize to members for the US Standard notation.I'll put some conversion stuff together and try and post metric units in future.Its tough living in the stone age!

Those numbers sound reasonable. I used the spreadsheet and tried to stick with metric and I got 3.4 kw @ 35 mph and 10.3 kW @ 100 mph, which translates to 4.5 and 13.8 hp. Maybe I got some conversions wrong though. :rolleyes:

slugs! *groans* :turtle:

I always thought it was v^2 not v^3, just from what I remember from my fluid mechanics class.

v^2 for force... Power is Force*Distance - which is where the extra v for v^3 comes from ;)

Aircraft Airspeed
 

Don't forget that air density and baro-pressure change as altitude increases.

For aircraft, since baro-pressure is very unreliable at higher altitudes, the standard 29.92 mm Hg is set at 18,000 feet and the call is Flight Level 1-8-0 instead of 1-seven-thousand, niner-hundred feet or lower...

Similarly, since the air is less dense at higher altitudes, the pitot tube can no longer be reliable to report the passing air molecules. Ground speed increases with the decrease in resistance, but the airspeed indicator vs. altitude has to be compared to aircraft documentation/reference , as to not induce an overspeed and consequent aircraft overstress situation. Comparisons to GPS (if equipped), should provide the most accurate velocity.

RH77

OK, plugging those in, I get 5-51kW, or 6.8-68 hp from 35-100mph. Sounds more reasonable. Graph looks nicely non linear too.

Twice the HP, Twice the Fuel Flow
 

I get twice the HP required at 70mph as compared to 55mph. I'm sure fuel flow follows that same relationship?

I guess an extra mile on a 25 mile trip to avoid 70mph traffic is almost worth it's weight in gold.

Welcome to the site, BTW (don't think I've done that yet :o)

But yeah, that extra mile at slower speeds can make the difference providing it doesn't involve the potential for stop-and-go, or added consumption beyond the baseline trip.

What I inadvertently performed in the past was to take backroads in the last 10 miles of a 30 mile trip. I attained better FE since slower speeds were involved, but I ended up using more fuel because there were 4x as many traffic lights. It was a crap-shoot, and on the best day, it still used more fuel -- but with a higher avg. FE.

Crr1 is the coefficient of rolling resistance. This is linear w.r.t. weight and speed, and becomes a small factor in the equation at speed when the air effects dominate.

BTW this spreadsheet is ONLY for cruising speeds, where weight is somewhat insignificant. Weight plays a much bigger part of the picture when you're looking at acceleration. For that, use F=ma :D

v squared vs v cubed
 

v^2 will yield force,v^3 will yield power necessary to overcome the force.

Possibly. The thing is, engine efficiency increases with load/HP, although in most cases not as fast as load/HP with speed, so in most cases it's better to drive slower in the same gear. However, if someone wants to see the full effect of the decrease in HP from 70mph to 55mph they will likely need to break up the power output into chunks to maximize efficiency.

Crr
 

Apologies for the long time to respond. Crr1 is the rolling resistance of the tires. Thank you for the detailed formula breakdown. I also looked at the updated spreadsheet boxchain provided. All this information provides a much clearer picture.

Thanks again!

That doesn't sound right. My Golf TDi gets 50 mpg at 70 mph and 63 mpg at 55 mph. That's about a 25% change...not 100%. Stan

Magic number
 

I missed the basis for 16% reduction. Is height corrected for ground clearance, sides corrected for tapers, or what?

Cheers
KB

I think it was statistically derived from comparing w x h and known frontal area values.

375
 

Three of my reference texts use the 375 value.I made a leap of faith that the Ph.Ds had checked their own work.Apologize for the trouble.

375 seems okay
 

I re-visited my books.And it looks like the 375 value is valid.
Working with the 550 lb'ft/sec value for horsepower calculation requires mph to be multiplied by 5,280 to get feet,then division by 3,600 sec/hr to get it into feet/sec..
If you already have a drag force at a given mph value,to get power,you can use this shortcut,multiplying the force by mph,then dividing by 375.
550/375 yields a constant percentage = to( feet/sec)/(mph) at any given velocity.It's always 1.466X.
Hope that helps!

Yes, your clever formula is quite correct...
 

Okay yes that helps and now i see where my confusion arose, and i agree that if you already have the Aero drag Force in unit of lbs, then you can get the Power in Horsepower by multiplying by speed in mph and dividing by 375:

1 mph x 5280 / 3600 = 1.466 ft/sec [the conversion from mph to ft/sec is 1.466].

power (in units of lb-ft/sec) / 550 = power (in units of Horsepower, HP).

So 1.466 / 550 = 1 / 375 does indeed convert mph to ft/sec and power to HP when multiplying Force in lbs times Speed in mph.

My confusion with your HP formula was three-fold: there was a factor of 1/2 missing inside the square brackets in your equation for the Aero drag "Force", and there was no conversion factor inside the brackets to convert speed in mph to ft/sec, and the constant term inside the brackets was so close to the value for the density of air.

So now i see how your formula is correct, and please allow me to expand on the derivation of this formula:

Aero drag Force = 1/2 x (air density) x V^2 x Cd x A ; air density = 0.00237 lbm/ft^3

Power = Force x Speed , so the fully expanded version of the formula for the Aero drag in Horsepower would be

HP = (V x 1.466/550) x [ 0.5 x 0.00237 x Cd x A x (V squared) x (1.466 squared)]

So the constant term inside the square brackets of your formula is really NOT the air density, but is the product of 0.5 x 0.00237 x (1.466^2) = 0.00256 , and this all simplifies down to the clever formula that you provided:

HP = (V/375) x [ 0.00256 x Cd x A x (V squared)]