Calculations Showing Thruster Acceleration

[MESM]

On the 9th June Elon Musk tweeted “SpaceX option package for new Tesla Roadster will include ~10 small rocket thrusters arranged seamlessly around car. These rocket engines dramatically improve acceleration, top speed, braking & cornering.”.

On the 12th June I tried to clarify Elon’s mysterious and exciting tweet with “... is this a curtain jet {2,{3,3},2} to create a momentary pressure reduction under the car?” but he is busy, so I thought that I would do some calculations for myself to check out the options.

From physics you will remember that friction is a fraction of the downward force on the tyres. So 1 newton downward force on a rubber tyre might give 1 newton of forward drive; you would only get 0.02 newtons if you had melting ice tyres. So we can do a simple calculation to tell us that the a boring Roadster, without the SpaceX option, is already relying on more than static mass and friction. If it accelerated at 9.8 m/s for 2 seconds it would not be going at 60 mph.

Other cars have done better than this. The historic data is plotted here http://www.mbtrusts.co.uk/DropBox/Historic Data.PNG (thanks to https://www.0-60specs.com/0-60-times/ for the data). The red line on the graph is for the boring Roadster.

If you have forgotten your physics then now is a good time to look at these videos that show that 'High pressures can be used to create low pressures' - demonstrated by perfume atomisers and also here https://commons.wikimedia.org/wiki/File:VenturiFlow.png https://www.youtube.com/watch?v=AUyczZ3EiZg )

I took the boring Roadster’s performance and tried to guess the suction under the car (I assumed that the suction force increased with the square of the velocity). I get the right kind of acceleration when we have a 4% pressure reduction as it gets to 60 mph. So I take that as my baseline.

Next, I plotted the achievable accelerations at different velocities for a range of reasonable suction levels (so when I say 2% bar I mean ‘boring Roadster suction increased by 2% of one atmosphere’). Amazingly even very small pressure differences involve enormous forces. The different curves are for different levels of suction. In the most extreme case the braking power appears unsafe for many drivers (A 10% pressure reduction might be hard to create? Not everyone is a racing car driver so I guess that they will limit it to 3 G anyway).

Using a mild design (a little bit of suction for one second). I have estimated the 0 - 60 time to be 1.2 seconds. I suspect that the thruster suction gets weaker at high speed (graph here http://www.mbtrusts.co.uk/DropBox/New%20Acceleration.PNG the thrusters turn off at 0.5 seconds).

I have attached my calculations here (http://www.mbtrusts.co.uk/DropBox/2018-07-01%20Thrust%20Estimate.pdf ). It would be great if someone could check them.

Yet another Elon Musk idea that is openly derided as ridiculous but actually seems to be possible.

Have fun

MESM


Elon’s original tweet ended with the comment that “Maybe they will even allow a Tesla to fly …”. If this post correctly interprets the thrusters then it might be possible to have secondary vents and active suspension that makes the car jump!

You might build the SpaceX option Roadster using three-nozzles left and right curtain and two-nozzles front and back curtain (= 10 thrusters).

I guess that the nozzles would point away from the centre of the car and down at ~30 degrees below the

horizontal. There is a small upward thrust from the nozzles but that is insignificant because a small suction is a big

(There is another refinement that is possible here. If the car normally had rubber flap flat against the base of the car then the thruster jets would temporarily suck the rubber flaps down and increase the suction under the car. When the thrusters stop then the flaps flip up again. My calculations do not assumed the rubber flaps.)

 

[J X 3]

To begin with, you got μ = coefficient of friction of the tires so F=μN and for the base model μ has to be at least 1.44 so rather high for street legal but somewhat possible.
Then the thrusters are compressed air thusters, they take air from the COPV, and the pressure drops inside the air tank. You can't add a mass under the car to decrease pressure.
Yes they can have active air intakes under the car to add downforce but that does not require the COPV and it's not the purpose of the system. The McLaren F1 has kevlar fans under the car for example.

A few other things you might have ignored.
Downforce= 1/2 x air density x v^2 x Cl x frontal area
here Cl (lift coefficient) is for the entire vehicle. Downforce is the term we use for negative lift.
drag is same formula with Cd instead of Cl but you can ignore it at low speeds
Rolling resistance - Wikipedia
And don't forget that power is not unlimited , at some point you have more aero grip than needed so the acceleration scales with hp and torque not grip. Ofc downforce increases rolling resistance too so too much also hurts that way once you are not limited by grip.

 

[MESM]

Thank you for your comments.

I believe that my calculations assume mu<=1; let me know which bits of the pdf were confusing and I could repost something that is clearer. I am very surprised that it is possible for mu to be higher than 1 in a practical situation.

For the avoidance of doubt I am proposing that the thrusters are NOT there for thrust - they are there to create suction for the lower momentum early stage of the take-off. The suction allows the car to accelerate faster because the grip is greater. The claim is that the thrusters can make a significant difference to braking, cornering and acceleration.

The Roadsters are aiming to have a very long range so I do not believe that you could put fans under the car (like your McLaren F1? - I did not know that). We can be sure that most of the floor pan is covered in battery packs. That is why I believe that the car uses air curtains instead, because they are consistent with a flat battery-pack base.

Obviously the thrusters cannot be actual thrusters because they might entrain debris and effectively shoot people. The car can only hold a small mass of air and we have to limit the exhaust velocity so it would be very hard to get a useful impulse. My Dad says that the sound from a high velocity exhaust on a jet aircraft is sometimes the twelfth power of the airstream velocity differences; an air curtain would have much small air-stream velocity differences and so it would be less likely to deafen.

I have circumvented the use of a generic downforce calculation by estimating the unknown constant k in the k v^2 terms to match the known performance of the boring Roadster - later I add on the impact of the SpaceX option.

I think that Elon is telling us that he is not energy limited below 60 mph. Particulate lithium ion batteries have really big electrode areas so when he scaled the Roadster range to 600 plus miles he got an ocean of instant power. If I am wrong then the COPVs might have had a role in energy storage - but I do not think that that is consistent with the words in his statement.

Why not tell me what you think and cast your vote? It will be fun to compare the responses to this post with Elon's announcement.

(By the way I think that Jeremy Clarkson would approve of the 0.5 second roar at the beginning of take-off - I not sure that he likes silent take-offs)

 

[J X 3]

Thank you for your comments.

I believe that my calculations assume mu<=1; let me know which bits of the pdf were confusing and I could repost something that is clearer. I am very surprised that it is possible for mu to be higher than 1 in a practical situation.

As for tires, a dragster has μ=4 while the average tire is just under 1.
For the avoidance of doubt I am proposing that the thrusters are NOT there for thrust - they are there to create suction for the lower momentum early stage of the take-off. The suction allows the car to accelerate faster because the grip is greater. The claim is that the thrusters can make a significant difference to braking, cornering and acceleration.

The Roadsters are aiming to have a very long range so I do not believe that you could put fans under the car (like your McLaren F1? - I did not know that). We can be sure that most of the floor pan is covered in battery packs. That is why I believe that the car uses air curtains instead, because they are consistent with a flat battery-pack base.

Obviously the thrusters cannot be actual thrusters because they might entrain debris and effectively shoot people. The car can only hold a small mass of air and we have to limit the exhaust velocity so it would be very hard to get a useful impulse. My Dad says that the sound from a high velocity exhaust on a jet aircraft is sometimes the twelfth power of the airstream velocity differences; an air curtain would have much small air-stream velocity differences and so it would be less likely to deafen.

I have circumvented the use of a generic downforce calculation by estimating the unknown constant k in the k v^2 terms to match the known performance of the boring Roadster - later I add on the impact of the SpaceX option.

I think that Elon is telling us that he is not energy limited below 60 mph. Particulate lithium ion batteries have really big electrode areas so when he scaled the Roadster range to 600 plus miles he got an ocean of instant power. If I am wrong then the COPVs might have had a role in energy storage - but I do not think that that is consistent with the words in his statement.

Why not tell me what you think and cast your vote? It will be fun to compare the responses to this post with Elon's announcement.

(By the way I think that Jeremy Clarkson would approve of the 0.5 second roar at the beginning of take-off - I not sure that he likes silent take-offs)

I am not sure what confuses you, the thrusters can not create suction and explained this in the simplest way possible, you add a mass under the car, you do the very opposite and create lift.

The air that is compressed and stored in the COPV needs to be harvested from somewhere an that's likely gonna be front or under the car. Adding fans might even be required if they want to fill the COPV fast. You can look at this as a system on it's own and this can help create downforce by sucking air from under the car.

Once in the COPV, the air is at ultra high pressure and the so called thursters are simple valves that let the air out. There is NO ability to suck air from anywhere but the COPV , it's a passive system -like a balloon filled with air, you let go and the air exiting the balloon creates thrust.
The thursters can be used to create a force in any direction, if angle above horizontal there is a component that goes down and adds downforce but that has nothing to do with the air pressure under the car.
There is an entire thread on this already ,maybe take a look there.

Edit: I suppose they could have a couple of thrusters under the car at the front facing backwards to kinda fake the shape of an ideal underbody but that might be too unstable.

 

[MESM]

Exhausting the air outward in a ring creates suction behind the nozzle. The area behind the curtain nozzles is ~ 9 sq meters so the low pressure area behind the nozzle is a 9 m^2 suction patch under the car. An extra few percent pressure reduction under the car gets you a big additional grip (because 1 atms = ~100 kPa).

I believe that I showed a YouTube video of the effect in the original post

I believe that mu ~= 4 for dragsters may not be real mu (in the sense that you are right but it is an averaged stiction + enhanced stiction from a downward force). I am fairly sure that mu ~= 4 comes from fixed aerodynamic lift from a large wing above the drag racer; do you have any links? The Roadster can not afford to have a fixed aerodynamic downforce of that size because it needs to be able to cruise > 600 miles.

Do you have any figures for your theory that the thrusters provide impulse? When I do the calculations I get almost irrelevant momentum changes? (Well actually there is a lot of energy there - I found a video of a scuba cylinder with a top nozzle breaking off and the cylinder goes through a brick wall. I mean that it seems that there is not nearly enough energy to make a useful difference to the kinetic energy of a 2 tonne car)

Elon's first car was a McLaren; so you have used an excellent example.

Interesting idea about the 'ideal underbody'. I asked around and I was told about experimental aircraft wings with fine holes that released air to give smooth airflow. Personally not my favourite theory in this case.

 

[J X 3]

Exhausting the air outward in a ring creates suction behind the nozzle. The area behind the curtain nozzles is ~ 9 sq meters so the low pressure area behind the nozzle is a 9 m^2 suction patch under the car. An extra few percent pressure reduction under the car gets you a big additional grip (because 1 atms = ~100 kPa).

I believe that I showed a YouTube video of the effect in the original post

I believe that mu ~= 4 for dragsters may not be real mu (in the sense that you are right but it is an averaged stiction + enhanced stiction from a downward force). I am fairly sure that mu ~= 4 comes from fixed aerodynamic lift from a large wing above the drag racer; do you have any links? The Roadster can not afford to have a fixed aerodynamic downforce of that size because it needs to be able to cruise > 600 miles.

Do you have any figures for your theory that the thrusters provide impulse? When I do the calculations I get almost irrelevant momentum changes? (Well actually there is a lot of energy there - I found a video of a scuba cylinder with a top nozzle breaking off and the cylinder goes through a brick wall. I mean that it seems that there is not nearly enough energy to make a useful difference to the kinetic energy of a 2 tonne car)

Elon's first car was a McLaren; so you have used an excellent example.

Interesting idea about the 'ideal underbody'. I asked around and I was told about experimental aircraft wings with fine holes that released air to give smooth airflow. Personally not my favourite theory in this case.

How do you create suction with a valve that lets air out from an air tank? You don't seem to understand what the thrusters are. They don't suck air from under the car, they take air from the COPV and the air under the car is still there.Not to mention that you create lift if you point them down. A diffuser is a reverted wing , accelerates the air thus lowering the pressure. The thrusters can only add air under the car, not take any out. A skirt physical or air is tricky on road cars as you are toast if you lose seal and ofc ultra high air pressure is not wise. And a very aggressive underbody design can also lead to situations where you end up with lift and the car ends up like the Le Mans 99' Mercedes CLR-GT1.

As for tires, the coefficient of friction has nothing to do with downforce and downforce varies with the square of speed. Showed you the formula for downforce so do the math at 60mph . Something like the Koenigsegg One 1 has Cl=-0.9 and frontal area about 2 square m while a formula 1 car might be 3.5x1.4 more or less but F1 cars also have very high drag and are far from ideal designs due to strict regulations.
The Space X COPV best I could figure out is about 1.6m in height and 0.6m in diameter while being able to take 5500PSI so a lot of energy , again check the thread dedicated to the Space X option.
On diffusers maybe watch this

 

[bobinfla]

Need another voting category - "I don't have a clue what you guys are talking about, I'm just gonna hit the go pedal and see what happens".

 

[PismoPat]

Need another voting category - "I don't have a clue what you guys are talking about, I'm just gonna hit the go pedal and see what happens".

Yup. Kick the tire, light the fire, and go!