Calculating effective Isp or: how I learned to stop worrying and love ion engines

[iHateLadders]

I don't see much agreement between topics on Isp in KSP. The equations often don't even agree on the dimensions of the value. So, here goes another thread on the subject.

I have a contraption which places a balanced rotating horizontal beam on top a free spinning washer just off the ground. It gets torque from ion engines near the end points. It has connection points at the end points which can release objects(kerbals or vehicles).

I'm trying to determine its fuel use efficiency as a function of its rotational frequency(which increases with diminishing returns as it picks up speed even in vacuum planets). The goal is to find out when it becomes too costly to keep building up additional rotational speed relative to other solutions. I have run the following test with Bill. The test was to strap him to an external seat and launch him into mun orbit and definitely not into the ground. The results:

leaver arm = 45 m

rotational frequency at launch(measured) = 5/4 Hz

velocity(calculated) = cf = r*2pi*f = 45 *2pi *5/4 = 350 m/s

xenon units spent = 900 units

and lastly, he did survive. He had to use about 2.4 units of his personal fuel to get orbital after his initial launch. Yay Bill! You live to see another experiment.

So, what I'd like to find is the effective Isp of this set up or alternatively I'd like to be able to solve for the deltaV of a simple ion vessel spending equal amounts of xenon(900 units), supposing the vessel was idealized to just the ion engine, 4 6x1 solar arrays, a kerbal, a seat, and 3 xenon tanks for a total mass of .7 t.

I appreciate any help with explaining how to solve the two questions above. Ideally, if I could understand the Isp equations, I could determine how to optimize this solution. Some obvious questions concern increasing the radius or adding more torque. The limiting factor on torque is rotational frequency because a single revolution must be slow enough for human reflexes to aim the release in the right direction. The initial purpose of this 'accelerator' was to overcome TWR issues by building up speed perpendicular to the pull of gravity while relying on the device to maintain elevation until release. This permitted less idle time for ships fighting gravity while also trying to speed up. I'm trying to figure out what ranges of different variables make this solution more viable.

Edited June 3, 2014 by iHateLadders

[cantab]

The dimensions of specific impulse are seconds. The dimensions of effective exhaust velocity are metres per second. The two are related by the conversion factor 9.8 m/s², ie Earth surface gravity.

If I remember rightly KSP imposes a hard limit on angular velocity. Meanwhile the ion engines spinning up your beam will have the same specific impulse as always, but I'm not sure how meaningful it is.

[allmhuran]

This is awesome and needs to be a video. And I found this sentence particularly amusing:

The test was to strap him to an external seat and launch him into mun orbit and definitely not into the ground.

Edited June 3, 2014 by allmhuran

[iHateLadders]

I tried fraps but it wouldn't give me anything more than 30 second clips. I'd be better off just taking pictures of it. This is definitely not an excuse to cover up massive amounts of reloading to keep my favorite test subject alive.

[Yasmy]

Let's do the simple vessel example:

900 units of Xenon weighs 0.09 t. Initial mass: 0.7 t, Final mass: 0.61 t

g0 = 9.82 m/s^2 (KSP reference acceleration. Not the local g or Earth's 9.81 m/s^2.)

Isp = 4200 s

dV = Isp * g0 * ln (0.7 / 0.61) = 5676 m/s

Though for your hypothetical vessel I get 0.73 t, without a kerbal.

Kerbals in a command seat weigh 0.09375. Thus:

dV = 4200 * 9.82 * ln ((0.73 + 0.09375)/(0.64 + 0.09375) m/s = 4772 m/s.

Edited June 3, 2014 by Yasmy

[Dangerous_Beans]

I think this is right:

dV = isp * g0 * ln(Mi / Me)

dV being our change in velocity: we know that you went from 0 to 350m/s, so dV is 350.

we can ignore the mass of the launching contraption, as we are interested in the fuel efficiency of using it to fire payloads into orbit. so the final mass is Bill on his own, so 0.09375 according to Yasmy above (sounds about right to me), and the initial mass is Bill + the xenon used to accelerate him.

so that give us 350 = Isp * 9.81 * ln (0.18375/0.09375). reorganise: Isp = 350/(9.81*ln(0.18375/0.09375)) = 53.0s

so bugger all.

sadly mass drivers (http://en.wikipedia.org/wiki/Mass_driver) don't seem to work in kerbal at the moment.

edit: think my calculation was slightly off

Edited June 3, 2014 by Dangerous_Beans

[Streetwind]

It should also be mentioned that the sample craft has a Mun TWR of about 1.46, so it could in fact launch Bill as well - using far less fuel.

In fact, you could mount only one xenon tank instead of three, saving almost a third of the weight and still have more than enough dV:

dV = 4200 * 9.82 * ln(0.58375 / 0.51375) = ~5268 m/s

TWR = 2 / 0.58375 / 1.66 = ~2.064

This thing could probably take off from the Mun, travel to Eve, land on Gilly, take off from Gilly and return to Kerbin...

Edited June 3, 2014 by Streetwind

[Red Iron Crown]

I approve of the OP's title because Dr. Strangelove.

I haven't really heard any disagreement on Isp or the rocket equation here, as cantab mentions the only real confusion is between Isp and effective exhaust velocity. The rocket equation works just fine with ion engines, as do the different dV calculator mods.

[iHateLadders]

Really appreciate all the help, guys.

The effective Isp now makes more sense. It was the change in mass part of the equation that confused me because I was only counting the poor kerbal and not the fuel used.

With that formula, I've been able to ...formulate the range of dimensions where this contraption is more or less efficient at putting things in the orbit. Small things only require a small amount of thrust so using this huge rotational arm and spending a whole lot of time getting it up to speed isn't optimal. However, I've found large objects do much better. The following is a second test:

leaver arm = 45 m

additional length to CoM of vessel connected to end of leaver arm = 8 m

rotational frequency at launch(measured) = 73/60 Hz

velocity(calculated) = cf = r*2pi*f = (45 + 8) *2pi *73/60 = 405 m/s

xenon units spent = 2100 units

vessel mass = 12 t

The effective Isp of this test was given by dangerous beans(thank you sir) as Isp = 405 /(9.81*ln(12.21/12)) = 2380s

And the comparison to the dV of the ideal case is given by yasmy and streetwind(again, thanks guys) as dV = 4200 * 9.82 * ln(12 / 12.21) = 715 m/s

The notable gain here is from the fact that even for larger payloads, the rotation does eventually get up to about the same frequency as for lighter loads. So it seems that there is a sweet spot of using rotational arms of mass and torque proportional to the payload mass. I'll have to fiddle with R or matlab to see if I can find what that precise optimal solution is.

The one thing that I'd like to also calculate is how much dV from the ideal case is lost to idling, working against gravity by thrusting normal to the surface rather than horizontal. Whats tripping me up is the range of TWR I could choose from for this hypothetical vessel. I still think there may be some range of vessels where it is more expensive to launch them normally, I need to figure out what that range is.

Edited June 3, 2014 by iHateLadders

[iHateLadders]

I approve of the OP's title because Dr. Strangelove.

I was also considering

MIEN KERBAL! I CAN FLY!

or

Where's Bill?

Edited June 3, 2014 by iHateLadders

[FleshJeb]

This is a really fascinating exercise. Please post some pics when you can.

I'd really like to see if it can be scaled up enough to fling processed fuel off of the smaller moons.

[MarvinKitFox]

My approach on this..

Why waste Xenon?

Slap a stack of ASAS reaction wheels on the hub of your contraption, insert a power source. (rtg is fine, 1 rtg powers 2 asas)

Spin that sucker up, release.

Repeat as often as needed.

Effective ISP = some higher order of infinity.

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Oh, you are interested in this from a theoretical viewpoint?

If you can source a massless, frictionless pivot and release system, then the energy imparted to the system is portioned out between the need to accelerate the not-massless engine, the not-massless fuel, and the not-massless launchee. Mounting your fuel on the hub where it almost doesn't matter, and your engine as far out as you can(for leverage), you just have to share energy between the engine(250 kg) and the Kerbal(93kg, + seat = 50kg), so with engine and launchee on same lever arm, you could achieve deltav of (ISP * fuelmass/393), effective ISP = ISP*93/(393)=993.

Note that increasing your "payload" mass will almost linear increase your ISP, diminishing returns up to ISP of the engine.

Note that final delta-v is *linear* to fuel mass expended! No mass fraction involved here, because we cheat physics by putting the fuel mass out of the movement system!

Yes, assuming infinite material strength in your components and no game engine limit to rotation speed, you can achieve almost infinite delta-v.

It fun, exploiting the loopholes in Kerbal Fizzics to defeat Grabbity like this.

[Jesrad]

Mind the Mighty Fuel Fraction !

When devising the maximum deltaV from a tank/engine combination, go straight for the Ff. Ff is the Mass fraction of fuel of your stage. KSP's Xenon tanks have a Ff of 0.58, this means that, no matter your efforts, any given stage on your ship can NEVER have a Ff bigger than 0.58, and will likely range 0.3 to 0.5 tops (meaning 30 to 50% of the total mass of the stage is Xenon).

This saves you boring maths by telling you straight ahead what your max deltaV will be.

See this curve ? High Ff is hard to reach.

A Ff of 0.5 means you have to stop halfway up this curve.

With 0.5 Ff on a ion stage: dv max = 4200 * 9.81 * ln(2) = 28559 m/s

With a more realistic and practical 0.25 Ff: dv max = 4200 * 9.81 * ln(1.33333) = 11853 m/s

And that should be enough for any purpose.

Edited June 4, 2014 by Jesrad

[Spark Plug]

I have a contraption which places a balanced rotating horizontal beam on top a free spinning washer just off the ground. It gets torque from ion engines near the end points. It has connection points at the end points which can release objects(kerbals or vehicles).

. . .

I tried fraps but it wouldn't give me anything more than 30 second clips. I'd be better off just taking pictures of it.

If you don't mind, I'd like to see even a single screenshot of your contraption. It piques my curiosity. Plus, I'm a visual kind of guy and would understand these equations better with a diagram of sorts.

[iHateLadders]

My approach on this..

Why waste Xenon?

Slap a stack of ASAS reaction wheels on the hub of your contraption, insert a power source. (rtg is fine, 1 rtg powers 2 asas)

Spin that sucker up, release.

Repeat as often as needed.

Effective ISP = some higher order of infinity.

Ya, I have a ton of ASAS on the center of the wheel. It certainly improves my base efficiency tests. However, I have found that after an initial increase in rotation, it drops off almost entirely. So what I do is begin with using ASAS and then switch over to ion torque.

The washer I use is part of infernal robotics. I don't know if it implements friction or if some other game mechanic is effectively doing it but I have found that ASAS seems limited. Maybe I just need moar ASAS. Either way, you are correct. All my base tests can be improved by ASAS.

Mind the Mighty Fuel Fraction !

When devising the maximum deltaV from a tank/engine combination, go straight for the Ff. Ff is the Mass fraction of fuel of your stage. KSP's Xenon tanks have a Ff of 0.58, this means that, no matter your efforts, any given stage on your ship can NEVER have a Ff bigger than 0.58, and will likely range 0.3 to 0.5 tops (meaning 30 to 50% of the total mass of the stage is Xenon).

And that should be enough for any purpose.

Does this apply to a set up where the fuel is not stored on the thing being moved? What I mean is, I have xenon on the base of my Kerbal AcceleratorTM that does not not add to the mass of anything rotating on the arm. This works because Xenon has Flow Mode = Everywhere, like electric charge. So while my first test had a mass of one kerbal + the arm being rotated, the total xenon available was like 15,000 units and nothing stopped it from being 150,000 units except for fps drop due to part count.

Either way though, that is an awesome graph.

Edited June 4, 2014 by iHateLadders

[iHateLadders]

Here is what it looks like for those asking for pictures: http://imgur.com/a/UzY0H