Effect of TWR, TVR, and DVR on Orbital Launch Efficiency (deltaV)

[LethalDose]

You would be absolutely correct if this work integrated up to escape velocity. But this work is concerned with the most efficient way to get up to orbital velocity, and so, integration stops after orbital velocity is reached i.e. the instant theta = 0 (since once you reach orbital velocity, you no longer need to spend any thrust fighting gravity, and can aim prograde, regardless of TWR).

Where is this stated in the thread or description? Is it the "integrated until v-bar = 1" part?

At higher velocities, pointing horizontally results in a negative θ WRT prograde.

Read what the author just posted before this post... You're wrong. Go back and read the model description(s) again to see that what you're stating is just ignorant.

What he calls theta (and I call phi) is defined as deviation from horizontal. No amount of velocity is going to change that.

And don't go to the moderators complaining that "ignorant" is some insult. It's not, it's merely statement of fact (see second definition here).

[arkie87]

Sean: Hmm okay. That made sense mostly. Your answer appears to have disappeared though (forum issue? deleted?), but it contained an example that demonstrated you could have different end TWRs for the two ships, with different DVRs. There was a part (which I can't quote - message missing) that mentioned that with the same fuel flow for both ships, the second ship would burn twice as much fuel in the same amount of time, which is obviously nonsense. The rest of it was fine though.

arkie/Sean (mostly arkie): I had a look at the model in the PDF and decided to expand the terms in the efficiency calculation (in particular, tbar* and FMR). The result was that the efficiency did not depend on DVR whatsoever. I think that probably suggests the efficiency measure is inadequate though.

// tbarstar used to avoid confusion with * for multiplication
efficiency = 1 / (TVR*ln(1 / (1 - FMR * tbarstar))

DVR = TVR * ln(1 / (1 - FMR))
FMR = (m_wet - m_dry) / m_wet

tbar = t / t0
t0 = (FMR * TVR * v0) / (g * TWR)
tbar = t(g * TWR) / (FMR * TVR * v0)
tbarstar = tstar(g * TWR) / (FMR * TVR * v0)

efficiency = 1 / (TVR * ln(1 / (1 - FMR * tstar(g * TWR) / (FMR * TVR * v0))))
// No FMR or other mass-based terms present, besides TWR, hence DVR independent
= 1 / (TVR * ln(1 / (1 - tstar(g * TWR) / (TVR * v0))))

It's of course entirely possible I've made an error somewhere

What about t_star? Can you show it is FMR independent analytically? I think not...

If we ignore t_star, efficiency should be independent of TWR as well, but this clearly isnt the case.

But good work anyway

Edited December 28, 2014 by arkie87

[arkie87]

Where is this stated in the thread or description? Is it the "integrated until v-bar = 1" part?

I suppose it's overlooked; i will revise problem statement to make it more clear. Thanks for pointing that out. But yes, you could deduce it only integrates up to orbital velocity from v_bar = 1.

What he calls theta (and I call phi) is defined as deviation from horizontal. No amount of velocity is going to change that.

If you are going above orbital velocity i.e. in a non-circular orbit, at periapsis and apoapsis, prograde and horizontal directions coincide. The moment after periapsis, prograde direction is above horizontal.

[LethalDose]

If you are going above orbital velocity i.e. in a non-circular orbit, at periapsis and apoapsis, prograde and horizontal directions coincide. The moment after periapsis, prograde direction is above horizontal.

If this is true, then prograde is below horizontal prior to orbital velocity, and it is not how I've defined it in my method, so they're not comparable.

[arkie87]

If this is true, then prograde is below horizontal prior to orbital velocity, and it is not how I've defined it in my method, so they're not comparable.

I think horizontal is relevant reference direction when below orbital velocity; and prograde is relevant reference direction at or exceeding orbital velocity.

[LethalDose]

I think horizontal is relevant reference direction when below orbital velocity; and prograde is relevant reference direction at or exceeding orbital velocity.

My definitions are largely taken from here. It's a reference you've stated you use as well. Specifically in this case, fig 4.8. Phi (what you call theta), aka flight path angle, is defined by position vector (technically, perpendicular to the position vector in the plane of the orbit), not the velocity vector (e.g prograde direction).

Your definition of theta/phi is changing dependent on the craft's position and velocity compared to some threshold. Mine doesn't. Please understand how changes the reference for terms and values in the middle of discourse makes it hugely difficult to get clarity on what's being presented.

[arkie87]

My definitions are largely taken from here. It's a reference you've stated you use as well. Specifically in this case, fig 4.8. Phi (what you call theta), aka flight path angle, is defined by position vector (technically, perpendicular to the position vector in the plane of the orbit), not the velocity vector (e.g prograde direction).

Your definition of theta/phi is changing dependent on the craft's position and velocity compared to some threshold. Mine doesn't. Please understand how changes the reference for terms and values in the middle of discourse makes it hugely difficult to get clarity on what's being presented.

My definition of theta is the same....

[GoSlash27]

If this is true, then prograde is below horizontal prior to orbital velocity, and it is not how I've defined it in my method, so they're not comparable.

LD,

All due respect, but they *are* comparable. Both seek to provide a lower bound for Dv efficiency WRT t/w ratio. A model either accurately predicts empirical results or it does not.

Apologies,

-Slashy

[Anquietas314]

What about t_star? Can you show it is FMR independent analytically? I think not...

Well, if you would provide a formula that could calculate t_star, we could see, otherwise I simply don't care enough to bother doing that myself

[LethalDose]

My definition of theta is the same....

I think horizontal is relevant reference direction when below orbital velocity; and prograde is relevant reference direction at or exceeding orbital velocity.

These statements are directly conflicting.

Theta/flight angle is either referent to prograde direction of velocity, or referent to the body's surface... Can't be both.

Edited December 28, 2014 by LethalDose

[Anquietas314]

These statements are directly conflicting.

Theta/flight angle is either referent to prograde direction of velocity, or referent to the body's surface... Can't be both.

Well, strictly speaking, the purpose of arkie's model is just to get to orbit right? Yours goes a little further and aims for escape velocity. Based on the model itself though, if your angle is calculated to cancel vertical acceleration, then when you're above orbital speed you'll start thrusting downwards instead of upwards; in equation (2) this is because g < V_x(t)² / R₀ for V_x(t) > orbital velocity.

Edited December 28, 2014 by armagheddonsgw

[LethalDose]

LD,

All due respect, but they *are* comparable. Both seek to provide a lower bound for Dv efficiency WRT t/w ratio. A model either accurately predicts empirical results or it does not.

Apologies,

-Slashy

Oh, I'm sorry.

I was talking about the angles

were you not able to follow that?

I'm sorry.

[GoSlash27]

Well, strictly speaking, the purpose of arkie's model is just to get to orbit right? Yours goes a little further and aims for escape velocity. Based on the model itself though, if your angle is calculated to cancel vertical acceleration, then when you're above orbital speed you'll start thrusting downwards instead of upwards; in equation (2) this is because g < V_x(t)² / R₀ for V_x(t) > orbital velocity.

Aye.

So long as Vx < Vo and theta is constrained for Vy=0, "prograde" and "horizontal" are the same thing.

By the time Vx = Vo, not only does it not matter what you call theta, but t/w is no longer a factor in efficiency.

At that point, it's just burning prograde, either for apoapsis or escape.

[Anquietas314]

t/w is no longer a factor in efficiency.

Well, no, that's not correct. It will cost you more delta-V to complete the same maneuver (particularly if it's a high-deltaV one) with an ion engine (if it's all done in one burn - probably even if you break it up) in KSP than it will to use a mainsail. Of course with the mainsail it'll cost way more fuel due to I_SP difference, but there you are.

[LethalDose]

Well, strictly speaking, the purpose of arkie's model is just to get to orbit right? Yours goes a little further and aims for escape velocity. Based on the model itself though, if your angle is calculated to cancel vertical acceleration, then when you're above orbital speed you'll start thrusting downwards instead of upwards; in equation (2) this is because g < V_x(t)² / R₀ for V_x(t) > orbital velocity.

arkie's model is designed to get to orbit at the lift-off altitude, though I'm not sure if what has been demonstrated has stuck to that limitation (mainly the images demonstrating the simulated altitude increase).

My model is not designed to only predict escape velocities, even though the examples have focused on that because that's how this mess got started in another thread in the first place.

My model never predicts a negative flight angle because my model doesn't include the V^2/r term, only inverse thrust to weight, which is always positive. Arkie claims his model requires numerical integration (I say claims because I haven't nor care to refute that), where as mine does not. Further, my model can be solved for final mass because you're awesome!!! Hence, my model can be packaged and delivered into a convenient spreadsheet, where as his requires Matlab, or it's product contour plot.

[GoSlash27]

Well, no, that's not correct. It will cost you more delta-V to complete the same maneuver with an ion engine (if it's all done in one burn - probably even if you break it up) in KSP than it will to use a mainsail. Of course with the mainsail it'll cost way more fuel due to ISP difference, but there you are.

True, but I'm referring to t/w >= 1, which was the initial condition.

Once you're at Vo, that's plenty to get you to orbit or escape with no DV penalty worth mentioning.

In fact, if you don't start throttling back around that point, you're liable to overshoot your intended apoapsis. You could hold on longer for Ve, but not a whole lot.

That's what I'm sayin'.

-Slashy

[Anquietas314]

My model is not designed to only predict escape velocities, even though the examples have focused on that because that's how this mess got started in another thread in the first place.

You misunderstood: I meant that arkies only aims to be able to accurately estimate the efficiency of a burn to orbit, while yours also aims to cover up to escape velocity (perhaps after fixing the vertical acceleration issue assuming it really is a problem).

Further, my model can be solved for final mass because you're awesome!!!

Oh come now, it wasn't that hard , just some pretty basic algebra

[GoSlash27]

My model never predicts a negative flight angle because my model doesn't include the V^2/r term, only inverse thrust to weight, which is always positive.

Which brings us full circle.

Unfortunately for this model, the V^2/r term is necessary. It has an impact on efficiency throughout the flight.

Now... if someone could integrate it into LD's formula in such a way as to not require Matlab, that'd be a huge help for everyone.

Best,

-Slashy

[LethalDose]

You misunderstood: I meant that arkies only aims to be able to accurately estimate the efficiency of a burn to orbit, while yours also aims to cover up to escape velocity (perhaps after fixing the vertical acceleration issue assuming it really is a problem).

It's just such a miniscule factor unless you're building vessels that can barely take off (<1.5), which leads to violations of the "short burn" assumption. At TWR 1.5 or above, the difference is just a couple of percent... not worth trashing the entire model for.

Oh come now, it wasn't that hard , just some pretty basic algebra

Yeah, the solution was staring me in the face. It's just been a while since I've had to do that.

Doesn't mean I don't appreciate it.

[GoSlash27]

You misunderstood: I meant that arkies only aims to be able to accurately estimate the efficiency of a burn to orbit, while yours also aims to cover up to escape velocity (perhaps after fixing the vertical acceleration issue assuming it really is a problem).

Arma,

I think you can surmise that it would be, simply from your personal experience launching from airless bodies. Any V above Vo is going to exhibit vertical acceleration. It's unavoidable.

The argument seems to be whether or not this is important at V< Vo.

I think the consensus here (please correct me if I'm mistaken) is that it is.

Best,

-Slashy

[Anquietas314]

I think you can surmise that it would be, simply from your personal experience launching from airless bodies. Any V above Vo is going to exhibit vertical acceleration. It's unavoidable.

The argument seems to be whether or not this is important at V< Vo.

Err, I was talking specifically about the V_x(x)²/R term... which we spent pretty much 10 pages arguing over in the other thread. As for experience, neither model uses a realistic flight plan (there's almost always hills in the way), so frankly in my opinion it doesn't matter very much.

[arkie87]

These statements are directly conflicting.

Theta/flight angle is either referent to prograde direction of velocity, or referent to the body's surface... Can't be both.

In my model, theta is always w.r.t. to horizontal....

In general though:

I think horizontal is relevant reference direction when below orbital velocity; and prograde is relevant reference direction at or exceeding orbital velocity.

[arkie87]

Well, if you would provide a formula that could calculate t_star, we could see, otherwise I simply don't care enough to bother doing that myself

I would be shocked if there was an analytical solution for t_star... hence my numerical approach.

[Anquietas314]

I would be shocked if there was an analytical solution for t_star... hence my numerical approach.

Sure, but there'll be some way to express it as an integral or differential equation or whatever.

[arkie87]

Arkie claims his model requires numerical integration (I say claims because I haven't nor care to refute that), where as mine does not. Further, my model can be solved for final mass because you're awesome!!! Hence, my model can be packaged and delivered into a convenient spreadsheet, where as his requires Matlab, or it's product contour plot.

Which brings us full circle.

Unfortunately for this model, the V^2/r term is necessary. It has an impact on efficiency throughout the flight.

Now... if someone could integrate it into LD's formula in such a way as to not require Matlab, that'd be a huge help for everyone.

Best,

-Slashy

Speaking of the devil, guys, guess what I have been working on???

hmmm?

An excel version of my Model

Get it while it's hot! https://www.dropbox.com/s/sddikc3qzv37zyx/Horizontal%20Orbital%20Insertion%20Simulator%20v1.0.xlsx?dl=0