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Just kill it off already
- Thread starter Hank42
- Start date
Perhaps you missed my original supposition: Like with many Tesla things, you start with a technology that the rich will pay a lot for (like high-performance electric acceleration) to develop the technology. Then, once proven, you wrap economies of scale into it so it is affordable for all. I'm sure it will be the Lexus and MBZs going to the ski lodges first, then it will come down to the more affordable market.
seenhear
Member
OK I did the (or some) math (very quickly so feel free to check and correct me, but be kind I'm just doing this for fun) to figure out roughly how much thrust these cold-air rocket boosters would need to have; I then kept going to answer some other questions too ...
I made a lot of assumptions, and in the process of so doing, really realized how unrealistic this whole endeavor probably is. I highly doubt Elon/Tesla will bring it to fruition. But regardless, the first principles math was fun and fairly simple, so here it is.
So I assumed constant acceleration. I also, therefore, must de facto assume that the rocket boost nozzles produce constant force for the duration of the acceleration. This is actually NOT a good assumption. They are driven by a pressure tank and the delta-P from tank to atmosphere reduces as the "fuel" (compressed air) is spent. This is a fairly linear reduction in force from max to essentially zero thrust as you use up the tank. Also, there's the non-linear thermal effects. The nozzle will get extremely cold as thrust is produced. This will effectively reduce the amount of thrust the nozzle can produce, the colder it gets. If you've ever used a can of compressed air, you know that if you hold the trigger for more than a few seconds, the velocity of the air coming out goes almost to zero, even though the canister is still quite full. And you can barely hold the canister because it's so cold. This might be countered by heating elements on the nozzle, I'm not sure how viable such a solution is, or how well it could keep up with the heat transfer rate of a rapidly exhausting gas nozzle. Or you could prime them by heating them up ahead of time. This would imply the system is not always ready though, so not useful in emergency situations, only for the track/drag strip. Regardless I ignored this effect for this first-pass "first principles" estimation of the nozzle forces. But this, combined with the safety questions of a 10+ gallon 10kpsi pressure tank in the back seat of a 4500lb sports car capable of insane speeds, makes me think this whole proposition is highly unlikely.
I made a lot of assumptions, and in the process of so doing, really realized how unrealistic this whole endeavor probably is. I highly doubt Elon/Tesla will bring it to fruition. But regardless, the first principles math was fun and fairly simple, so here it is.
Code:
We have two acceleration situations/claims for the Roadster2: Without rocket boost assist is 2.0 seconds to 60mph, and with rocket boost is 1.1 seconds to 60mph.
To simplify things, I assume constant acceleration in all cases.
This is a reasonable assumption for such a short acceleration.
The effects of non-linear acceleration would have minimal impact on the distance and time calculations result. But... See my comment at the end about this assumption.
Anyway, math:
KE = .5*m*V^2 is the energy (Joules, kg-m^2/sec^2) to get an object going from zero to a given speed, regardless of acceleration.
For 60mph and a 4500lb car, this works out to 734130 Joules. (I'm not showing the arithmetic and units conversion; GIYF).
Energy is also Force * distance. So the force required to accelerate that object to that speed is equal to the energy / the distance travelled.
Distance travelled from zero to some speed = 0.5*a*(time) where "a" is the acceleration and time is how long it takes to get to that speed.
To accelerate an object from 0-60mph in 2.0 seconds is 13.4 m/sec^2, and to do it in 1.1 seconds is 24.4 m/sec^2.
So distances travelled in the two situations are:
d(1.1) = .5*(24.4)*(1.1)^2 = 14.76 meters, and
d(2.0) = .5*(13.4)*(2.0)^2 = 26.8 meters.
The energy required to get the car to 60mph is the same regardless of how quickly it's done (remember this is "energy" not "power").
So to get the Force required for these two acceleration claims we divide the energy by the distance:
F(1.1) = 734130 J / 14.762 m = 49731 Newtons = 11,180 pounds of force (EV force + rocket force)
F(2.0) = 734130 J / 26.8 m = 27392 Newtons = 6,158 pounds of force (EV force alone)
So to estimate the required rocket boost force to get the 4500lb Roadster to improve it's 0-60mph time from 2sec to 1.1sec, we subtract those two Force values:
F(boost) = F(1.1)-F(2.0) = 11180lbf - 6158lbf = 5022 lbf of rocket boost.
So if we assume the 10 thruster setup Elon mentioned allows for all of the possible/available compressed air to be used for forward acceleration, that means the system has a total of about 5,022 lbf of available boost (maybe a bit more to account for losses?). This would align on a first-principles level (Elon's favorite way of working out engineering claims) with Elon's suggestion that the car might even be able to "fly" since the car weighs 4500lb. If they were able to vector all the thrust downward the car could momentarily levitate or jump (the more I think about it the more I think Elon has been on a mission to create a real-life K.I.T.T. )
If we assume 3 out of 10 thrusters point backwards for acceleration boost, that's 5,022/3 = 1674lbf of thrust from each nozzle. (and keep in mind this is assumed constant force which a cold-gas nozzle will never be able to produce, so we'd need much more to start)
That doesn't seem like a lot to me, when trying to control a 4500lb vehicle sliding out of control over ice. For just adding some interesting handling enhancements? Sure could be kinda cool and interesting. But as a mechanism to enhance safety in a sudden loss of traction scenario? Unlikely, IMO.
Cold air rockets are not super efficient. The best ones have a low specific impulse (I_sp) in the range of 50-75 seconds for Nitrogen at 0C which is a close approximation for air (which is mostly Nitrogen). Let's give SpaceX engineers the benefit of the doubt and assign 75seconds as their cold-air-thrusters' specific impulse value.
Thrust = g*I_sp * (m-dot) where m-dot is the mass flow rate of the gas through the nozzle, and g is gravity (9.8m/s^2)
So, if we go with the calculated thrust per nozzle above of 1674lbf or 7446.3N and I_s of 75 sec, we get
m_dot = 7446.3N/(9.8m/s^2)/(75sec) = 10.13 kg/sec mass flow rate at each of the three acceleration nozzles. Again, I'm assuming that a 1.1sec 0-60 is using the full capability of the system (why wouldn't it be?) so that means all available compressed gas is diverted through the three acceleration nozzles pointing backwards. So the max flow rate of the system is probably 3x10.13 kg/sec or 30.4 kg/sec.
OK so they won't want to store more than the air necessary as it's costly to other performance (mass) of the vehicle (see lazy part above). So let's assume that to do a 1.1 second 0-60mph run, that uses up their full reserves of compressed gas boost. So that means the car will carry about 30.4*1.1 = 33.4 kg of compressed air (just under 74lb).
It's been noted (on Joe Rogan again I think) that the composite over-wrapped pressure vessel (COPV) would carry the gas at about 10,000psi. Applying the ideal gas law, we get a volume for the 74lb of compressed air as 2500 cubic inches, or about 10.8 US gallons, or about 41 Litres. That's a pretty damn big vessel holding a crap-load of potential energy at 10kpsi, sitting right behind you. AKA a literal bomb in your back seat. I highly doubt this will be US DOT / NHTSA safe. Then again, hydrogen cars exist, so…
Yeah. Unfortunately my bet is the EV 718 will be more like $100K.A basic Macan EV configuration is mid-90's and I don't think the 718 will be cheaper.
seenhear
Member
OK figuring out additional downforce required assuming that tire-to-road friction is the limiting factor is pretty straight forward. Honestly it took longer to write out than to just do the math.
First of all we do know that tire-pavement friction is the limiting factor when accelerating below 2sec 0-60mph. This is why many validation tests of such claims required prepped surfaces, special tires, etc., and I think I recall Elon even stating as much. What we don't know is how much more the Roadster2's plaid drivetrain could accelerate the car if traction limits were fixed. So we will assume that the plaid drivetrain has ample power & torque and that traction is the only limitation in getting to a 1.1 second 0-60mph sprint.
So, assumptions: we have to assume that the coefficient of static friction "mu" (Greek letter μ) is constant here or the analysis gets too complicated for this. It's not a bad assumption; μ won't change much with increased downforce, but tires do deform under load so a real simulation has to account for that. This is first principles, not a simulation.
And, math:
Definitions:
m = mass of vehicle (4500lb)
g = gravity = 9.81 m/sec^2
a = horizontal acceleration of the vehicle
a11 = acceleration for 1.1 second 0-60mph = 24.4 m/sec^2
a20 = accel. for 2.0 second 0-60mph = 13.4 m/sec^2
Fn = normal force = m*g + Fboost
Fn11 = normal (downward) force on ground for the 1.1 second (boosted) acceleration case
Fn20 = m*g = normal force for the 2.0 second acceleration case (no boost)
Fboost = extra downward force from thruster boostFn11 = Fn20 + Fboost
Fa = m*a = acceleration Force (that which pushes the car forward, accelerating it to 60mph)
where "a" is either a11 or a20.
Static Friction Force, Fs = μ*Fn
Force balance:
The force accelerating the vehicle is equal and opposite to the static friction force of the tires on the ground:
Fa = Fs
m*a = μ*Fn
2.0 second situation:
m*a20 = μ*Fn20
m*a20 = μ*m*g
μ = a20/g -- there's our "constant" coefficient of static friction!
1.1 second situation:
Fa11 = Fs11
m*a11 = μ*Fn11
Fn11 = (m*a11)/μ … but we now know μ so…
Fn11 = (m*a11)*(g/a20) … rearranging gives us
Fn11 = Fn20 * (a11/a20) … and
Fn11 = m*g + Fboost … so
Fboost = Fn11 - m*g
Fboost = Fn20*(a11/a20) - m*g = Fn20*(a11/a20) - Fn20
Fboost = Fn20*(a11/a20 - 1)
Fboost = 0.821*(Fn20)
Fboost = 0.821*(m*g) = 16,437.8N = 3,695.4 lb
So assuming a constant μ, and assuming that tire friction is the only limiting factor in getting this car to accelerate more quickly (which implies we are also assuming the drivetrain has ample capacity to provide the additional required torque) we can say that a ball park downforce increase of 3,700 lb would provide the additional traction required to allow a 4,500lb car to improve its 0-60mph time from 2.0s to 1.1s.
3,700 lb is significantly less than the 5,022 lb of horizontal boost the car would require to get the same 0-60 improvement.
What I have not bothered to wrap my head around is how the plaid drivetrain would respond when given an extra "push" in the horizontal boost case. Could it have enough power to still put down max torque at higher speeds? Who knows.
Also, the optimal solution is probably with the boosters pointed at a downward angle less than 90deg. This way there's additional traction for the tires and additional horizontal thrust.
But I still say that a huge 10kpsi pressure bomb in the back seat is not something I want in my car, and again I'm assuming constant force from the nozzles which isn't possible so you'd have to have a starting thrust much higher than I've calculated in order to average close to what I've calculated. Also I haven't heard Elon or anyone else address the thermodynamic limitations of max thrust for 1.1 seconds through a cold-gas nozzle. I guess the nozzles will have massive heat sinks on them, with electric heating elements imbedded. Again the whole thing seems super unlikely, but fun to think about.
oops. here you go:
I didn't check your math or parameters, however, I do question your initial assumption that the thrusters would be used for 1st order acceleration. Is there enough electrical thrust - f(motor torque and wheel traction)? I'm guessing that wheel traction is the limiting factor and, just adding down-force would be enough to increase the wheel traction. I don't have time to try to compute the increase in static friction that might be needed though.
- I probably shouldn't try to post during my short lunch break between meetings.
good point. However, I that a lot of people still drive around with gasoline bombs under their seats. Racers often also have tanks of N2O as well.
I guess, like so many things: This could be done. Whether it should be done is different.
FWIW hybrid 911 unveiling is on the 28th. I went to a Porsche dealer yesterday to tell them about it and get sales contact info. Taycan is alright but still hard to pay 2-3x a Plaid for a slower car. We’re really happy with our Cybertruck. Maybe we’ll throw a hybrid 911 turbo next to it. Damn it Elon.
seenhear
Member
Gasoline isn't as explosive as movies make it out to be; it's highly flammable, and will make a fireball, but won't have the same explosive energy as movies imply. Gasoline VAPOR is explosive, but usually the amount of vapor available isn't nearly as large.
However if a 10kpsi cannister fails in your back seat, you are dead. No question. If your gasoline tank explodes, you will suffer serious burns and maybe die. If your air pressure tank explodes they will struggle to identify the pieces of your body.
Both are very, very bad. One is a bit worse. :-/
Hmm,
I do agree about the issue with 10kpsi tanks. Back in the 1990's and early 2000's, folks were always coming up with the air car that could be fueled with a tire compressor. In addition to the cost to compress the air, they always missed this 'little' safety issue. Its also a big limiting factor to hydrogen as a passenger-vehicle fuel.
Thanks for going through the math! You saved me a lot of time which I don't have right now.
so choice between dead in a microsecond or burn for a while first. Not much of a set of choices.
I do agree about the issue with 10kpsi tanks. Back in the 1990's and early 2000's, folks were always coming up with the air car that could be fueled with a tire compressor. In addition to the cost to compress the air, they always missed this 'little' safety issue. Its also a big limiting factor to hydrogen as a passenger-vehicle fuel.
Thanks for going through the math! You saved me a lot of time which I don't have right now.
seenhear
Member
That and the time and energy to compress the air. Elon never spoke about this beyond saying an electric compressor would draw from the man HV battery. It would take a LONG time to compress 75lb of air to 10kpsi
wrt being full of hot air, no one knows more than Elon.
No, the choice is dead in microseconds vs probably live with minimal injury, but small chance of burning to death.
The nitrous tanks referred to earlier are much smaller than 10 gallons and only store 1000 psi. Even then, when they fail, the results are catastrophic
Who knows? One of Musk's good points is that he has the guts to try and see if they can come up with a design that will make regulators comfortable instead of just saying it is hard and not trying.
Like him or not: He fails, he succeeds. He make friends, he makes enemies. We're actually lucky to have a person like him around.
Strong disagree. The world would be a better place if he didn’t have any influence.
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