Questions about rocket equations (ISP, exhaust velocity, and T/W ratios)

[PTNLemay]

I've been meaning to edit my KSP rocket parts, but I want to do so in as non-cheaty a way as I can. For now I just want to improve the exhaust velocity to make them more efficient, while keeping them relatively realistic.

My question is why does Ve = 9.81 x ISP? I mean that's an Earth-based constant, yet KSP seems to use roughly the same ISPs as real world rocket engines (around 300 for typcial liquid fuelled engines). What is Kerbin's surface acceleration exactly? And if we go to another planet with different gravitational accelerations the ISP doesn't suddenly change (so far as I know it's constant) so why use it if it's based on that specific surface gravity? Oh by the way, sorry if this thread includes multiple questions all roughly related to rocket equations, I'm pretty noob at rocket science but I'm trying to learn bit by bit.

Edited August 2, 2013 by PTNLemay

[Brenok]

Scott Manley has a video about the gravitational acceleration in ISP equatons:

Basically, the g is only there to allow people to use both imperial and metrical. It doesn't matter what the local gravity actually is.

Edited July 30, 2013 by Brenok

[PTNLemay]

Those god-damn logarithms... I can measure the exact point at which my mathematical understanding of a topic breaks down, based on when logarithms need to be introduced.

Now, about the mass ratio that he mentioned, that's needed to determine the ultimate delta V of one's rocket, but I'm confused about the masses in KSP. You need the full mass of the rocket, divided by the empty mass of the rocket (when the tanks are dry). But the fuel tanks in KSP use dimensionless values, for example the FL-T800 is 360 units of fuel and 440 units of oxidizer. We can assume those are liters, but it doesn't tell us anything about their weight. Oh, and the dry mass is 0.5... something (tons I'm guessing).

Also, what is the "power" unit for the game's engines Newtons? Watts?

[tavert]

The mass units are metric tons, thrust is kilonewtons. Whatever the volume unit is for liquid fuel, oxidizer, and solid fuel, the mass is 0.005 tons per unit for liquid fuel and oxidizer, and 0.0075 tons per unit for solid fuel. Monopropellant is 0.004 tons per unit, and Xenon is 0.0001 tons per unit.

The acceleration due to gravity at Kerbin's surface is 9.81 m/s^2, which is only possible with Kerbin's smaller radius via an unrealistically high density for whatever the planet is made of.

I need to re-check this in 0.21, but for whatever reason KSP 0.20 actually used 9.82 m/s^2 for the g0 constant of proportionality in the fuel flow equation.

Edited July 30, 2013 by tavert

[PTNLemay]

So if they're liters, that'd be... 5 kg per liter of liquid fuel, 7.5 kg for the solid fuel. Though the tanks are probably too big to only hold hundreds of liters.

Now... I did a test vehicle to make sure I understand this equation properly.

Command capsule: 0.1 tons

Fuel tank: 0.125 tons + 1.0 tons of fuel

Engine: 1.25 tons

Total mass = 2.475

Empty mass = 1.475

Mass ratio = 1.678

Engine exhaust velocity = 320 x 9.81 = 3139.2 m/s

Delta V = 1624.8 m/s

I plopped it on the ground and turned off gravity, then burned. I got a peak velocity of 566 m/s before air resistance slowed it back down. Is it normal for air resistance to slow me down this much? Or did I futz the numbers somehwere?

[metaphor]

The rocket equation is Delta-v = Ve * ln ( M_initial / M_final ) . So the exhaust velocity is what really matters. Isp is an artificial unit that is proportional to exhaust velocity by Isp = Ve / g . The units for Isp come out as seconds since it's velocity divided by acceleration. Isp is the same no matter what units you use for velocity and acceleration.

For example, if you use metric, an exhaust velocity of 4000 m/s and g=9.81 m/s^2 give you Isp = 4000/9.81 = 408 s.

If you use imperial, the same exhaust velocity of 13,123 ft/s and g = 32.2 ft/s^2, you get Isp = 13123/32.2 = 408 s.

That's why an engine's efficiency is easier measured in Isp instead of exhaust velocity, but exhaust velocity is what really matters. The extra g is just a constant of multiplication to make the units come out the same.

[metaphor]

So if they're liters, that'd be... 5 kg per liter of liquid fuel, 7.5 kg for the solid fuel. Though the tanks are probably too big to only hold hundreds of liters.

Now... I did a test vehicle to make sure I understand this equation properly.

Command capsule: 0.1 tons

Fuel tank: 0.125 tons + 1.0 tons of fuel

Engine: 1.25 tons

Total mass = 2.475

Empty mass = 1.475

Mass ratio = 1.678

Engine exhaust velocity = 320 x 9.81 = 3139.2 m/s

Delta V = 1624.8 m/s

I plopped it on the ground and turned off gravity, then burned. I got a peak velocity of 566 m/s before air resistance slowed it back down. Is it normal for air resistance to slow me down this much? Or did I futz the numbers somehwere?

That sounds right. It's normal for air resistance to slow you down so much since you're going over terminal velocity.

[PTNLemay]

Well good... now I need sleep. But tomorrow I'll definitely need some help figuring out the thrust-to-weight ratios (if anyone's willing).

Thanks for the help so far, much appreciated.

[tavert]

Air resistance can be pretty high, yeah. One way to look at it is terminal velocity at low altitudes on Kerbin is around 120 m/s. That's the speed at which losses to drag equal losses to gravity (which would be 9.8 m/s^2, if you hadn't turned it off). And drag scales as velocity squared, so you're losing over 200 m/s of delta-V per second to drag at your peak velocity.

Edited July 30, 2013 by tavert

[K^2]

He isn't going to be at sea level when he achieves peak velocity, however. And terminal velocity goes down by factor of 2.7 for every 10km on Kerbin.

In fact, I'm willing to bet that his peak velocity of 566m/s is the equilibrium for his TWR on empty tanks, which is 215kN/(9.8m/s²*1.475T) = 14.87. That would put terminal velocity at that altitude at 151m/s, meaning he reached peak velocity at 3,500m over the sea level. Still means he is wasting about 150m/s for every second the engine is running at that speed, of course. But that's what you get when your TWR is so high.

Ideal TWR for vertical ascent is 2. Since TWR tends to increase as the rocket burns fuel, typically recommended TWR at launch is about 1.8. So if you want to fly a 0.1T command capsule with a 1.125T tank, your good options are either Rockomax 48-7S, which will give you 1.54 TWR, or LV-909 which will give you 2.9 TWR. Neither is optimal, but either one will get you much, much higher. Albeit, at slower speeds.

Edit: Well, I ran a few tests. LV-T30 actually cut out at 4,800m, so I guess I shouldn't have assumed that acceleration is negligible at that point. Maximum altitude about 8,900m. With LV-909 I got cutout at 26km and apoapsis at 104km. With 48-7S the fuel ran out after the space theme kicked in and the apoapsis was at 1,000km.

Less is more, Gentlemen. If you are taking off from atmo, bigger engine is just not worth it.

Edited July 30, 2013 by K^2

[tavert]

Course not, was just throwing out some very rough numbers to say that yes, that looks ballpark about right without going into the details of the KSP drag equation and where the values of the magic-number coefficients out front come from.

Presumably TWR questions were going to come later, and this particular example was just a simple construction to check delta-V calculations rather than any indication of whether it makes sense from a design standpoint.

Since the KSP engines weigh so much, you can often end up with an overall lower mass by using a lower-than-fuel-optimal TWR.

[PTNLemay]

Okay, so 1 unit of kerbal thrust = 1 kilonewton, so if my engine has a thrust of 50 I need a vehicle that will hold itself to the ground with 50,000 N of force if I want it to stand still (anything less and it will take off, I want to do some hovering tests). F=ma, so... 50,000 = m x 9.81, so I want a mass of 5096.8 kilos. I'm guessing it's kilos and not tons, although the part weights are in tons.

Is there a way to measure the changing gravity as I move higher up in the atmo? Ideally a way that doesn't involve messy add-ons.

[tavert]

Acceleration from gravity is mu / r^2, where mu is a constant for a given planet called the gravitational parameter, and r is the distance between your craft and the center of the planet. For Kerbin, mu is 3.5316 * 10^12 m^3 / s^2. The radius of Kerbin at sea level is 600000 meters, so r equals your altitude plus 600000 meters.

[K^2]

If you just want to measure gravity in game, under the science tab you will find a "Gravitoli Detector". It measures acceleration due to gravity.

[PTNLemay]

Huh, both of those are quite useful actually. I like a quick and easy solution, but knowing the math and being able to do it myself is quite satisfying. Thanks.

I have question still but they're mostly related to familiarizing myself with the equations, more so than asking for a direct explanation (and I don't want to waste people's time asking them to just repeat themselves). I just need practice. When I feel more comfortable with the T/W mechanics I'll come back and ask about orbital mechanics.

For example what would be nice is an approximate equation (or even a spreadsheet calculator) where you can punch in your inbound speed (and vehicle mass, if the game takes that into consideration) and whatever other variables it would need, and it gives you the ideal atmospheric depth you want to aim for to drop into a nice orbit. So if I'm coming in at Kerbin at 4 km /sec, he can say "oh yes you'll want to aim for an periapsis of precisely 28.3 km". And bam! Your apoapsis comes down to a nice 80 kms.

Edited August 2, 2013 by PTNLemay

[metaphor]

Huh, both of those are quite useful actually. I like a quick and easy solution, but knowing the math and being able to do it myself is quite satisfying. Thanks.

I have question still but they're mostly related to familiarizing myself with the equations, more so than asking for a direct explanation (and I don't want to waste people's time asking them to just repeat themselves). I just need practice. When I feel more comfortable with the T/W mechanics I'll come back and ask about orbital mechanics.

For example what would be nice is an approximate equation (or even a spreadsheet calculator) where you can punch in your inbound speed (and vehicle mass, if the game takes that into consideration) and whatever other variables it would need, and it gives you the ideal atmospheric depth you want to aim for to drop into a nice orbit. So if I'm coming in at Kerbin at 4 km /sec, he can say "oh yes you'll want to aim for an periapsis of precisely 28.3 km". And bam! Your apoapsis comes down to a nice 80 kms.

Try this website.

[PTNLemay]

You guys are a good community, always so helpful. lol

Thanks again.

Edited August 2, 2013 by PTNLemay

[SciMan]

Regarding aerobraking (and depending on structural integrity, aerobreaking):

MechJeb can help with that too, and it even does it in real time. On MechJeb's "landing guidance" page, just tick the "show landing predictions" box and it will show you what your orbit will be after you pass thru the atmosphere (if your periapsis is below the top of the atmosphere, that is). If it detects that you would end up hitting the surface first, it will tell you the coordinates of the predicted landing site instead of the orbital parameters, and if you told it you want to land at a specific site (KSP launchpad, for example) it will tell you how far off target you are. Even if you end up flying the landing manually, the information that it provides is incredibly useful for planning aerobraking and pin-point accurate landings.

[K^2]

Yeah, there is not a simple formula for aerobraking, regardless of whether you want to land or establish a new orbit. This is something that has to be solved with an algorithm.

[SciMan]

Just because the math is non-trivial does not prevent the construction of a spreadsheet from good old fashioned empirical data, and with enough points on that spreadsheet you should be able to get a reasonable approximation of the equation's true output thru curve fitting or even just using linear interpolation for inputs that land between the collected data points.

For example, my typical Mun-orbit to LKO return trip for fuel tankers makes use of the atmosphere of Kerbin by setting the periapsis of the orbit to 35km altitude ASL Kerbin relative. If you go much lower than 32km Kerbin pe when returning from the mun, you will instead execute a direct re-entry and landing, and that trajectory will not exit the atmosphere.

That might be a "rule of thumb" to some, but for the spreadsheet I'm thinking of, it could represent a data point, and even if it's not a very accurate or precise one, it's still useful data.

However, the whole "curve fit to a spreadsheet based on thousands of test runs" idea does feel very much like a "brute force approach" to solving the issue. Sure it works, but I'm almost certain that there's a better way to do it. Most likely something you don't need to break out a scientific calculator for, but a standard one might be helpful (god help me if I ever have to do long division again....)

Edited August 2, 2013 by SciMan

[PTNLemay]

@ SciMan

Even if I still use a calculator, it feels less cheaty when I'm not using an in-game plug-in. I'm not opposed to MechJeb, but I want to master the game and actually accomplish a few things on stock before I start making heavy use of it. Otherwise I'll have done all my achievements with a "crutch" so to speak.