Tesla Model 3 Aerodynamics Math Explained
Dec 13, 2018
How using three graphs of aerodynamic performance of Coefficient of drag body forms helped us outline the actual performance of a stock Tesla Model 3, a Model 3 with aero appliances to lower the drag to Cd = 0.179, and a ideal body form of Cd = 0.13. The actual range of modified Tesla Model 3 was an improvement from 310 miles to 370 miles
By John A Gilkison and Phil Knox
In November of 2019 we published a blog here at EV World titled “A Tesla Model 3 with 440 miles of range?” and last week I had to take the blog down when we discover substantive math error in the calculations. I also took the supporting videos for the blog that were linked to You Tube down also for the same reason. See our new supporting blog “Tesla Model 3 Aero Potential in 3 Cd graphs “at.
While a drag reduction in the body form of any car will result in improvements in performance, increased mileage, range, and a reduction in the power needed to move the vehicle at any given speed, these improvements just would not be as much as we reported.
I would like to apologize to any of our readers for publishing this misinformation and the blame lies entirely on me as I was the one tasked with analyzing the data. Phil just was drafting up the graph of the performance curve for a Tesla Model 3 with a Cd = 0.179 which he sent to me by mail. Phil did make one error which we discovered that did not affect the outcome, and that was forgetting to translate rolling resistance horsepower into kilowatts. For that reason we decided to redo the graph and add two other new graphs for comparative purposes.
This week I received the three new graphs from Phil in the mail which were for a stock Tesla Model 3 with a Cd = 0.23, a Model 3 with an improved aero package with a Cd = 0.179, and a Tesla with a completely new body form (it would not look like a Tesla at all) with a Cd = 0.13. The idea was to graph the various iterations of drag in Cd = .005 increments.
As soon as I looked at the Cd = 0.23 graph I knew I had “screwed the pooch” (a technical engineering term from “The Right Stuff”) and my math on range and watts per mile were way too optimistic. As Phil said I had ventured off into some creative mathematics.
We thought it would be a good idea to explain of math for calculating range, watt hours per mile, and the resulting mpg-e to you all so you can check our work. Please bear in mind that we are working for ball park figures and not precise engineering figures because we just aren’t privy to data Tesla has on their cars. That said there are things that can be known and the exercise we are involved in here is indicative of what we could expect in the real world under ideal conditions. By that we mean we are assuming a windless day, mild temperatures, and a fairly level road with a two way trip to average out elevation changes, and so forth.
The first formula is HP/1.341 = kW, this is the step that Phil forgot to include with his rolling resistance line on the graph (see the linked video). This mistake simply increased the angle of the slope of the line but did not affect the outcomes for the aerodynamic drag curve of the graph which are calculated separately.
Looking at the graph you see speed along the bottom cord binned out in 5 mph increments. You see the power needed to achieve these speeds on the vertical axis of the graph. My job was to convert kilowatts into all the other needed parameters which I failed at spectacularly on my first try.
The first logical step is to convert the Capacity of the battery in kWh into the time this capacity is consumed at a given speed at the rate expressed in kW. I would express this at kWh/kW = T Lets do one example for a stock Model 3 doing 100 mph. At that velocity the car needs 38.1 kW of power to maintain that speed. Thence 75 kWh/38.1kW = 1.968 hours.
The next step is fairly simple, the Time (1.968 hours) x Velocity (100 mph) = Range 196.8 miles. Next we need to know the watt hours per mile averaged which we can get by converting Capacity from kWh into Wh by the simple expedient of multiplying it by 1,000. That is 75 kWh x 1,000 = 75,000 Wh.
This new expression of Capacity/Range = Wh per mile. That is 75,000/196.8 = 381 Watt hours per mile. Now we only have one remaining matter of interest in this exercise and that is generating the mpg-e figure for any given velocity.
For that we need a new constant and that is the energy contained in a gallon of gasoline. For this we are assuming E-10 gasoline at 32.777 kWh because that is that gasoline most of us buy. The EPA uses Regular Unleaded at 33.7 kWh but we only know of a few places you can buy this gasoline and we think it is disingenuous to use it as a metric.
Hence G-e (32,777 Wh) / (380 Wh per mile) = 86.25 mpg-e. Now we are done, that was fairly painless. Most of my headaches from crunching all this data for all the relevant speed increments for each of the three different Coefficient of drag forms simply came from trying to make sure that is had not mixed up any of the terms in each step.
Now that we explained our methodology we want to publish a simple table of our results for you given the restrictions of using the editor function embedded here at EV World for blogging.
Tesla Model 3 stock, at 65 mph, 15 kW, 325 miles of range, 231 Wh per mile, at 142 mpg-e. Please note here the published range figure for this car is at around 68 mph.
A Tesla Model 3 with a boat tail and wheel covers lowering the Cd to Cd = 0.179. At 65 mph the car requires 12.93 kW, but it has a range of 377 miles. At this velocity the car needs 198.9 Wh per mile, yielding an impressive 164.8 mpg-e figure. Tesla Model 3 with an ideal body form of Cd = 0.13 requires 11.48 kW to do 65 mph, yielding a range of 424 miles. This body form uses 176.8 Wh per mile yielding us a spectacular 185.4 mpg-e.
So there you have the performance numbers for the three different Cd packages, 0.23, 0.179, and 0.13. As you can see from my last example a Tesla Model 3 is capable of plus 400 mile ranges with a Cd = 0.13 body form. It can do over 400 miles, with the Cd = 0.179 add on appliances, but only at speeds of 60 mph and below.
We are very excited about the potential for a add on boat tail and wheel well covers to improve the performance of a stock Tesla Model 3. The graphing shows a 10 mph increase speed is possible with the same performance resulting from a Cd = 0.005 improvement. This pushes a Tesla Model 3 to somewhere around 370 miles of range on the highway. It is capable of 100 mpg-e at 100 mph. This is a 16.3% improvement over the highway performance of a stock Tesla Model 3.
Phil is working on a graph of the Cd = 0.28 body form now and I should have it in the mail by early next week. Our goal here is to outline why a Nissan Leaf and other electric cars with similar coefficients of drag perform so poorly. One notable example is the Jaguar I-Pace with only 235 miles of highway range with a 95 kWh battery pack. We think this will require a separate blog here at EV World and supporting You Tube video.
In closing I would like to comment about what a good job Tesla is doing with their electric car offerings. The low drag body form they use with a Cd = 0.23 is one of the lowest drag forms for a car on the market. This has got to be one of the “Best kept out in open secrets in automobile design ever”.
This is their magic sauce, the secret of their increased range, and higher top end speeds. Yes, Tesla has better battery tech, motor tech, and sexy design but the air doesn’t care about that. The air cares about aerodynamic drag and rolling resistance. No amount of magic motor and battery sauce can overcome the drag induced by poorly designed body forms. Good on you Tesla and Elon Musk.
We still hope to build a boat tail and wheel well covers for my Tesla Model 3 when I can get one in 2020. Playing with the length of the tail will help us tweak the performance. Our numbers are telling us we only need a 28 inch tail (from historic data) so by designing a tail from four to eight inches longer we can lower the Cd below our goal of Cd = 0.179.
This can be done with add on sections. For example it should be possible to build a 28 inch boat tail, then build a 4 or 8 inch section that simply bolts on for experimentation. This project could be really exciting and even has the potential to yield some great old fashion press demonstrations. For example a trip between two cities on one charge that is over 370 miles apart. “Or how about a 100 mph run at 100 mpg-e”, that could turn a few heads!
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