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AeroGav

TWR in a winged vessel..

Question

There's been a lot written on this board about optimal TWR for rockets.  Math got involved,  some of which went over my head,  but the consensus seemed to be that 2 was a good number.

What I'd like to know is if the same Math can be used to prove the "best" TWR for a spaceplane in closed cycle mode.      

Now before everyone heads for the hills,  try not get too freaked out by the fact it has wings.

Wings generate lift, in exchange for creating a bit of extra drag and extra mass.    If you angle the wings up at 5 degrees relative to the fuselage, tweak the tailplane/canard so the plane holds a nose angle between half a degree and one degree nose up when  SAS is set to prograde,  your lift/drag ratio is going to be more or less constant throughout the closed cycle part of your flight.

Another way to look at it, is that since we have wings, and wings are counteracting gravity, we only have to be concerned with drag.  

So, whilst in a rocket, we are concerned about our margin of Thrust over Gravity,  in the airplane,  we are concerned about the margin of Thrust to Drag.    The greater the margin in favour of Thrust,  the lower our losses, but adding extra engines to increase TWR adds dry mass, and means we have less delta V to start with.        

Of course, this comparison is only valid if the airplane is generating enough lift to support itself  while building velocity.   You could set the wings at a lower angle, or zoom climb to a high altitude on jet power before going closed cycle,  and temporarily get lower drag, but it won't be sustainable as you'll soon fall back into thicker atmosphere and be worse off.      That is why it is best to chase optimal "lift:drag" ratio rather than absolute lowest possible drag.

 

 

Let's say it weighs 30 tons.  In Kerbin gravity, that works out to a gravity force of 300kn if you round things up  a bit.

Do we need 300kn lift ?  Well, not quite.     Let's say we're going at 1400 m/s -  that's actually two thirds of orbital velocity, meaning that orbital freefall is already going to be cancelling most of gravity.    We only need to get enough lift to make up the difference, and stop the airplane descending.    As our velocity continues to increase, our apparent weight decreases.   Since we're holding a constant AoA at constant lift/drag,   our lift now exceeds weight,  and we drift upwards, until the thinner air causes lift to no longer exceed weight.   At this new higher altitude however, drag is less, and so on.   As you can see,   this means it gets easier and easier to accelerate as time goes on,   and fuel burnoff is not the only factor at play.

At this point , some of you will be saying "so what" because,  with RAPIERs in closed cycle mode,  your TWR on this phase of flight is going to be so high as to make  such optimisations pretty irrelevant.

However,  this past week or so,  I've been building craft for rescaled Kerbin.      I'm currently using a rescale factor of 3.2 which raises orbital velocity from 2200 m/s (mach 7) to 4200 m/s (mach 14).      This makes things more than twice as difficult for a spaceplane.

Stock, you can get 1600 m/s air breathing,  and so theoretically only need 600 m/s delta V in closed cycle mode to make orbit.   

With the rescaled system,  even if you manage 1600 m/s air breathing, you need another 2600 delta V to make orbit - more than four times as much.        I don't think there's any chance of packing enough fuel in to do that with a 305m/s specific impulse RAPIER -  800 m/s NERVs are probably your only option.      However, they have much worse TWR -  3 tons for 60kn, instead of 180kn for 2 tons.     Pure chemical rocket engines are better still - the Dart manages RAPIER thrust levels for just one ton,  and is far from the highest TWR engine available.

Compounding the issue,   at the start of your closed cycle burn,  you are nowhere near orbital velocity on rescaled Kerbin, thus are still feeling the full effects of gravity.  This holds you deeper in the atmosphere,  with greater drag losses, for longer.

Thus,  this "optimal" spaceplane TWR becomes such a vexed question.

Starion 2 - A working prototype

dwWE4bL.png

oBFDF8L.png

I can SSTO with the rescaled Kerbin but such a craft ends up with little fuel left over in a very low orbit.      So the Starion 2 was a mod of the original SSTO that dumps its jet engines at flameout to get extra delta V.        Two whiplashes, three nervs.

Takeoff weight 47 Tons.   Upper stage mass (after Whiplash stage  separation) 38 tons

Spoiler

Why Whiplashes instead of RAPIERs?    Cost.        Whiplashes fade pretty badly after 1100 m/s , and you can expect another 300 m/s top speed with RAPIERs in air breathing mode, but Whiplash are only 2200 Kredits vs 6000 for the Rapier,  and can lift more payload due to better low speed performance.    I did contemplate carrying a single RAPIER to orbit,  and use two droppable Panthers to boost us to mach 2.8,  where the single RAPIER gets ramjet-happy and will boost us to a top speed 300 m/s or so higher than the pair of whiplashes would have managed.  But that only takes 10% or so off our closed cycle burn requirement to get to orbit , hardly worth the mass penalty this RAPIER would bring.   And dumping it is uneconomic !

Mass breakdown  of the upper stage

Stage 2 Mass 38.6t

Liquid fuel        19 t
Nerv engines       9t
Payload            6.1t       (cockpit, crew cabins, docking gear, docking fuel)
Aerodynamic bits   4.5t       (wings, control surfaces, intakes and cones)


Flight Logs

I took some screenies on the way up.   Due to the AeroData GUI being open,  they give some insight into how drag losses changed as the flight progressed :

Time	Velocity Weight	Alt	L/D     Drag
(sec)	(m/s)	 (kN)	(M)	(ratio)	(kN)
					
752     1491	 345	32340   3.7     73
827     1674     328    33435   3.7     80
972     2186	 296	38824   3.6     64
1102	2723	 264	47842   3.6     28
1174	3114	 248	50712   3.5     26
1304	3904	 218	61815   3.4     12

Imgur album of the ascent shots from which this data was transcribed - 

https://imgur.com/a/xNSfY

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On 11/8/2017 at 11:17 PM, AeroGav said:

There's been a lot written on this board about optimal TWR for rockets.  Math got involved,  some of which went over my head,  but the consensus seemed to be that 2 was a good number.

What I'd like to know is if the same Math can be used to prove the "best" TWR for a spaceplane in closed cycle mode.

It is a cool question. Snark made a good comment regarding vertical rocket ascent over here. It made me think what the analogue vertical ascent argument would imply for a winged craft climbing steadily at a fixed angle above the horizon.

Recapitulation that TWR = 2 is the optimal climb thrust.

Spoiler

 

The original argument considers a rocket climbing vertically. The craft is flying at a constant speed with constant mass, constant gravity, constant air density and constant thrust. We would like to know how much fuel it uses during the climb.

If we climb vertical at 'terminal velocity', S = 1, with 2 g of thrust, TWR = 2, then one g counters gravity and the other is used to counteract the aerodynamic drag. Let us say that it takes one time unit to perform the climb, and let us say that fuel cost is measured in units such that thrusting at one g for one time unit is equal to one fuel cost unit. The total fuel cost of this 'optimal' climb is 2 cost units, TWR / S = 2.

The idea is that any other TWR, whether less than or greater than 2, will make the climb more expensive.

If we double the speed, S = 2, the aerodynamic drag is four times as much (2 squared). We can do the climb twice as fast, i.e. in half a time unit, but we need a TWR of 5 to keep the speed up, so the total fuel cost ends being more, TWR / S = 2.5. If we increase speed and TWR even more, then the climb becomes yet more expensive. At S = 3 we need a TWR of 10, TWR = 1 + S2, to keep the speed up, and the total fuel cost is then TWR / S = 3.3. Still, there could be other reasons why you would want to design a TWR 10 missile, and you might decide to accept the 67% increase to the climb fuel cost.

Likewise, if we halve the speed, S = 0.5, the aerodynamic drag is four times less. It takes twice as long to do the climb, but we can get by with a TWR of 1.25. The total fuel cost this time is TWR / S = 2.5. Still, If we do not want to bring heavy powerful engines for the climb, we may decide to bring an extra 25% of climb fuel instead.

 

Consider lift-to-drag (L/D) ratio and vary the climb angle.

If we have wings, we may fly less vertical and at a lower TWR. As far as the relative simple model is concerned, the climb fuel cost is worse than the straight up TWR 2 climb. However, if we want to save on TWR, wings are a good alternative to just naively reducing TWR.

Spoiler

 

The four forces experienced by our climbing rocket plane is Thrust, Lift, Drag and Weight.

TLDW-7.jpg

 

We assume that the rocket plane has a known (constant) Lift-to-Drag ratio during climb. Let us consider some example solutions, and compare them to the vertical climb situation. The vertical climb situation corresponds to a L/D ratio of zero.


Example, L/D ratio 2 (modest).

Say a modest amount of wings and fins give the craft a L/D ratio of 2. With this craft, pushing the nose over to a 50 degree climb attitude, we can perform the climb for a cost of 2.5, but this time with a TWR just shy of 1.1. So, even with less thrust, TWR = 1.1 < 1.25, this (winged) craft will fly a bit faster, S = 0.57 > 0.5, and be able to climb for a cost comparable to a TWR 1.25 vertical rocket ascent.

 

Example, L/D ratio 7 (good).
A craft with a L/D ratio of 7 can make a steady 35 degree climb for a cost of 3.5, but it can do so (at S = 0.34) with a TWR of 0.7. The steady climb is sort of expensive, but it can be performed with quite a bit less power available.

 

I think this answer is interesting information, but I also realize that possibly the real questions are still unanswered. The above arguments are mostly concerned with 'slow' flying planes, and a lot of players are far more interested in fast space planes.

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On 12/11/2017 at 10:50 PM, FleshJeb said:

Thanks for the info! Is it true that if you put a Shock Cone on the back end of a Rapier and offset it inside that it reduces the rear drag on the engines? Those are about 5% of my total drag, thus far.

It does, but you have to show aero data on each individual part to know if the drag of rapier + shock cone is less than rapier alone.   ATM it is, but the drag from empty attach nodes on size 1 engines does seem to have been tweaked downward recently, so some of the less pointy nose cones actually make drag worse.    Next question is whether it's worth the mass penalty..

On 12/11/2017 at 10:50 PM, FleshJeb said:

  My only concern is that wings have a slightly worse mass fraction than tanks. (83% instead of 89%) This may not end up being significant. Hmm, I JUST rediscovered the MechJeb window that shows total drag and gravity losses..

On rescaled Kerbin,  i think it's worth paying a bit more mass to get better lift/drag (since that feeds back into less nuke engine required, therefore less mass) but on stock scale, if you're doing Matt Lowne type missions to Moho etc. things are different.   On big Kerbin however, most of the stage 2 delta V is spent within the atmosphere.

On 12/11/2017 at 10:50 PM, FleshJeb said:

I noticed your trim flaps. :wink: I just haven't found a need to configure any on mine yet.

Mine needs it because it has a lot of wing, therefore it needs a nose down nudge to get it to fly level in the speedrun,  without zooming too high for the jet engines.       With smaller wings you might just leave on prograde and forget it.     Also I find having the ability to nudge the nose up or down helps to dampen out the porpoising you can get when flying on prograde lock.

 

On 12/11/2017 at 10:50 PM, FleshJeb said:

FYI: I took the fuel out of our drop stages and compared the cost:

  • You go from 107,621 - 101,771 = 5,850
  • I go from 70,885 - 66,830 = 4,055. Tanks are still pretty cheap.

Oxidizer is very cheap it appears,   I don't know what "liquid fuel" is in game,  judging by the cost,  three legged llama milk.

Starion could do with some optimising - the first stage is under loaded (could carry a heavier second stage without increasing # of jet engines) and swapping one nerv for a terrier and some ox, perhaps.      If I get time i'll take another look at it.    Right now.  at least the less efficient characteristics stack up in a way that reduce time to orbit and make it easier to use.   

I've not tried going anywhere in the rescaled system (imagine delta V for interplanetary travel has gone up by just as much?) but if you did refuel it in orbit,  you could just use the Starion to take the crew direct to whatever base is the final destination.   It's got nerv engines,  and doesn't have that much more dry mass than a dedicated exo-atmospheric shuttle to be worth transferring the crew onto one, i'd have thought.    It's got a pretty hefty  TWR up in orbit too, with 3 nervs, so could land propulsively on quite a few airless bodies.

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And as i follow this discussion... i bite in keyboard how my Planes don't like space...

Great Designs people, i wish to see this beauties in RL!

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On 11/10/2017 at 2:14 AM, AeroGav said:
On 11/9/2017 at 7:49 PM, Laie said:

When I was dabbling in LF-only designs, I found that 0.3g was about the least acceleration that I could still get to orbit, eventually(1). Most of that was spent fighting drag and gravity, [...]

The mistake is pitching to 20 degrees above prograde.

You're missing the point: this wasn't a mistake but necessary. The low TWR forced all sorts of compromises, to the degree of not fully utilizing the airbreathers for speed because I needed upwards momentum.

Still, for the plane in question it was a good solution. While the low-thrust, single nuke ascent wasted a lot of dV,  adding the mass of another nuke would have cost even more. Or to put it the other way round: removing one nuke bought me a lot of dV, and the inefficient ascent didn't waste all of that.

Ideal? Certainly not. But better and hence "more ideal" than increasing TWR by adding another engine.

On 11/9/2017 at 10:09 PM, Spricigo said:

A good sum up of the advantages of a bit of extra TWR at the right moment. The hard part is, with all those variable, figure out when the right moment is happening.

Well that's easy: you need the thrust at about the time when your airbreathers shut down or a little before, say, when they're positively starving. Required TWR then keeps decreasing the closer you get to a stable orbit.

The better question is, how much of it do you need? This thread is exploring just how much TWR would be "ideal" and from my experience I'd guess that number to be pretty low, certainly less than 1 and possibly as small as 0.4, largely depending on how much extra engine mass would be needed to increase it.

If your plane is based on Rapiers, you have more than enough thrust at your hands. ISP may be less than ideal but the mass is already bought and paid for;  I'd say that in most cases you should just use it rather than slap on extra aerospikes or terriers or whatnot. Of these, only use as many as your mission requires (often that's not much at all) and run the rapiers as long as necessary to close the gap until the OMS can handle it alone.

Edited by Laie

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2 hours ago, Laie said:

Well that's easy: you need the thrust at about the time when your airbreathers shut down or a little before, say, when they're positively starving. Required TWR then keeps decreasing the closer you get to a stable orbit.

No quite. The question is not when TWR is necessary (as in 'not going to space without it') but when is advantageous, to which extent and how much of it takes.

And it complicate quickly if people are looking for different advantages.

 

 

 

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2 hours ago, Spricigo said:

The question is not when TWR is necessary (as in 'not going to space without it') but when is advantageous, to which extent and how much of it takes.

Hmmmm. I possibly should state my assumptions. Which I thought were so obvious that I needn't mention them, but then again that's how the worst misconceptions play out.

So, I belive that

  1. the most efficient way to proceed to orbit after airbreather shutdown is "surface prograde" all the way, for minimum drag and maximum oberth.
  2. at the time rocketry takes over, "prograde" should be nearly level but with at least a hint of upwards. IMO a strong hint is better, but for this particular argument it doesn't really matter wether it's ten degrees or one tenth of a degree.

On such a trajectory, you need the mostest (rocket) thrust at the time the airbreathers give up, and ever less as you proceed to orbit. I don't see how having a higher-than-necessary TWR would provide any benefit along the way.

---

You can make do with a lower TWR if you're on a steeper ascent. Question now becomes, how steep can it be without under-utilizing the airbreathers?

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3 hours ago, Laie said:

I don't see how having a higher-than-necessary TWR would provide any benefit along the way.

That is straight forward: reduced gravity losses. But it usualy comes with added mass and worse Isp so the question of how much of an advantage it really is.

Also a shorter time to orbit is nice since you could use that free time to something else.

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8 hours ago, Spricigo said:

That is straight forward: reduced gravity losses. But it usualy comes with added mass and worse Isp so the question of how much of an advantage it really is.

Also a shorter time to orbit is nice since you could use that free time to something else.

Nope, and nope. By assuming a prograde-only ascent I'm taking $directionOfProgradeAtAirbreatherShutdown as a given -- it should be possible to determine the required TWR mathematically from that (and velocity). Gravity losses also follow from these inputs. Having a higher-than-necessary TWR is not going to help anymore.

Also, time-to-orbit is very nearly a moot point. All the energy-efficient ascents make you reach orbital velocity in the atmosphere with an AP somewhere on the far side, and require a lengthy coast at physics warp.

But of course, that only means that $doPaAS is something worth looking into. I already alluded to the question in my last post: leaving the breathable air at a steeper angle will reduce the TWR you need to follow through. It will also reduce time-to-orbit (a little), and drag losses (because you "circularize" at, say, 50km rather than 35). This comes at the price of higher gravity losses and, probably more important, means you're not getting as much airspeed from your jets as you could on a shallower trajectory.

@AeroGavI think, but cannot prove, that "maximum airspeed" is the wrong benchmark for how well you utilized your airbreathers. What you really want is not speed but something like (waves hands) "momentum you can take with you". A mere 1200m/s with a significant climb rate may be more useful for getting to space than 1600m/s in level flight. But alas, not only am I unable to provide a proper name, I also can't tell you how to make it comparable.

Anyway, the engine question isn't strictly either-or. The Rapiers are already there, their dry mass bought and paid for, so the TWR is basically free. ISP notwithstanding, utilizing them to some degree is bound to be better than not using them at all. You don't need to run them all the way to orbit -- a 500m/s push will already help a great deal.

Edited by Laie

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7 hours ago, Laie said:

Nope, and nope. By assuming a prograde-only ascent I'm taking $directionOfProgradeAtAirbreatherShutdown as a given -- it should be possible to determine the required TWR mathematically from that (and velocity). Gravity losses also follow from these inputs. Having a higher-than-necessary TWR is not going to help anymore.

Gravity losses have nothing to do with the direction your thrust points (that are cosine losses). 

The vessel is falling, if the velocity is not enough to miss the ground then you'll have gravity losses. If that's happening for a longer time gravity losses will increase.

7 hours ago, Laie said:

Also, time-to-orbit is very nearly a moot point. All the energy-efficient ascents make you reach orbital velocity in the atmosphere with an AP somewhere on the far side, and require a lengthy coast at physics warp.

So what? It takes less time with higher TWR. It's even more noticeable when you take in consideration the time is saved from the the flighting phase that you do in lower warp or no warp.

 

 

 

 

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