Calculating Rocket Thrust and Fuel For Specific Altitude

[[email protected]]

How do I calculate the amount of thrust and fuel required for a given engine to reach a specific altitude at a specific speed?

I'm early in the career mode, and I've got the "Test RT-10 "Hammer" Solid Fuel Booster in Flight Over Kerbin" mission.  I started using trial and error with a new configuration - constantly reverting -- but then I realized I should be able to just work out the math for this and build it once, yes?? 

I spent the day learning all about calculating Delta V, Isp, etc.. and have learned a bunch - and I feel a new passion rising in me.  BUT - I feel I'm in the same place, because I can't seem to figure out the math I should be using.  We can calculate Delta V at Sea Level - or at Vacuum - but what about in between.  Also, since I'm really only using solid rockets at this point, I need to reduce the thrust of the rocket -- but wouldn't I need to recalculate the engine's Isp when I do that?  The flow changes, along with the pressure changing throughout flight (if we've reach a vacuum, we've gone too far!)

So... Regardless of whether I'm doing the mission right or not (I'm sure people will say to use an engine that I can tweak in-flight) - I still want to know what steps I would go through to calculate the resources I would need to make the trip.  (I mean in real-life there are no do-overs, right?)

Any help is much appreciated!

Thanks!

-Dave

[GoSlash27]

I'm afraid there's no simple mathematical formula for that problem. Too many interdependent variables. IRL they would model it using a simulation. In your case... sandbox mode.

I can give you a quick data point from experience, though: 1,800 m/sec DV will get you roughly 27km altitude and 850 m/sec on a nominal gravity turn.

Best,
-Slashy

[MaxL_1023]

The main issue is air drag. You can never be sure how much you will run into and it varies according to nonlinear relationships involving altitude and speed, among other things. 

What I used to do was use a somewhat overpowered liquid engine, so I could reach the minimum speed before the minimum altitude. Then, I would throttle down until I was very slowly accelerating, reaching the target speed at the bottom of the altitude range. Then, trigger your Hammer (or hit the test button). If you have to activate it through the staging sequence, you could have two as radial boosters and use your liquid core to lift them up. It is not efficient for launching (obviously) but if all you want to do is test them it should work. Just make sure to use the Swivel so you have throttling as well as thrust vectoring. 

It would be almost impossible to do with pure solid rockets - an iterative approach would be as effective as anything considering the complexity of calculations. 

[bewing]

Keep in mind that the engine that you test doesn't have to have any fuel in it. In fact, it's a lot easier if it doesn't. At that point, all you have to do is calculate (or simulate) how much of a booster you need to get your payload (the command pod plus parachutes plus empty hammer) to altitude x at speed y. You can approximate this answer, assuming an instantaneous burn at sea level. You try to make it be a little bit too much -- because it's pretty easy to tweak it down if it's too much, but you're stucco if you're a little short. To tweak it down, you can add a little mass or a little drag (another parachute maybe? some fins?) -- or you can tweak the thrust limiter on your SRB. Using the thrust limiter almost always reduces the total punch you get from an SRB by just a little -- so it helps you make fine adjustments downward. But when I do those types of contracts, I almost always do them with a pure SRB design.

I think it almost always works best on these to do a pure vertical launch, because you get the most money if you land back at KSC anyway.

Edited December 31, 2016 by bewing

[aeroz2011]

So I got curious about this too and decided to do an experiment. Long story short I've got a close (less than 10% error) approximation for calculating thrust by altitude. I tried it based on different thrust limits and engines. 

One important note here is that the lower the "efficiency" (thrust limit) the closer the the equation you get. Also, around 13 km altitude, the data and equation are essentially the same.

Without further ado, the equation:

 T(h) ~ lim% * T_max + 3 * ln(h/35)

where lim% is thrust limit as a percent (i.e. A 48 thrust limit is 0.48 in the equation) and h is altitude in km. T_max is the max thrust for the motor/engine. 

If you were looking to account for liquid motors' throttle, you would add that in as another "efficiency" type term.

The imgur link below shows the data vs the calculated values. Black is data, green is math.

http://imgur.com/ksXG3Du

[Spricigo]

15 hours ago, aeroz2011 said:

So I got curious about this too and decided to do an experiment. Long story short I've got a close (less than 10% error) approximation for calculating thrust by altitude. I tried it based on different thrust limits and engines. 

One important note here is that the lower the "efficiency" (thrust limit) the closer the the equation you get. Also, around 13 km altitude, the data and equation are essentially the same.

Without further ado, the equation:

 T(h) ~ lim% * T_max + 3 * ln(h/35)

where lim% is thrust limit as a percent (i.e. A 48 thrust limit is 0.48 in the equation) and h is altitude in km. T_max is the max thrust for the motor/engine. 

If you were looking to account for liquid motors' throttle, you would add that in as another "efficiency" type term.

The imgur link below shows the data vs the calculated values. Black is data, green is math.

http://imgur.com/ksXG3Du

I didn’t check your math and have no idea how do you come up with the equation(mostly because I'm lazy). However I suggest to check it in a airless body. The reasoning its that lower thrust results in lower speeds and thus lower drag, since you get closer results with lower thrust there is reason to suspect drag is causing the difference.

I suspect that the actual formula(even ignoring drag) will have at least a ln and will need some calculus for deduction. Its really something I'm not willing to delve into, the "how useful is"/"how difficult to do" ratio its not high enough for me.

[[email protected]]

16 hours ago, aeroz2011 said:

So I got curious about this too and decided to do an experiment. Long story short I've got a close (less than 10% error) approximation for calculating thrust by altitude. I tried it based on different thrust limits and engines. 

One important note here is that the lower the "efficiency" (thrust limit) the closer the the equation you get. Also, around 13 km altitude, the data and equation are essentially the same.

Without further ado, the equation:

 T(h) ~ lim% * T_max + 3 * ln(h/35)

where lim% is thrust limit as a percent (i.e. A 48 thrust limit is 0.48 in the equation) and h is altitude in km. T_max is the max thrust for the motor/engine. 

If you were looking to account for liquid motors' throttle, you would add that in as another "efficiency" type term.

The imgur link below shows the data vs the calculated values. Black is data, green is math.

http://imgur.com/ksXG3Du

Thank You aeroz2011!!  I will try it the next time I fly!  Regarding Spricigo's comment, that makes sense since it would be a logarithmic inefficiency based on surface area up to the "max q" - then a fall as atmosphere decreases (very quickly).  Thanks you guys!!

[aeroz2011]

58 minutes ago, Spricigo said:

I didn’t check your math and have no idea how do you come up with the equation(mostly because I'm lazy). However I suggest to check it in a airless body. The reasoning its that lower thrust results in lower speeds and thus lower drag, since you get closer results with lower thrust there is reason to suspect drag is causing the difference.

I suspect that the actual formula(even ignoring drag) will have at least a ln and will need some calculus for deduction. Its really something I'm not willing to delve into, the "how useful is"/"how difficult to do" ratio its not high enough for me.

What I did has nothing to do with drag or velocity produced - it's only a measurement of thrust relative to altitude - from 0 to [effectively] vacuum. 

Drag doesn't affect thrust. 

Thrust limit is a percentage, so if you set the limit to 40 on the Thumper, your max thrust will be 120kN. 

Edited January 23, 2017 by aeroz2011

[Gaarst]

@aeroz2011 as @Spricigo said, I'd like to see where your formula comes from. I've done the experiment and found something very different.

 

I find that the thrust for a given engine is given by , using the pressure P as a parameter.

Taking measurements of Kerbin's atmosphere, you can roughly express pressure as a function of altitude only, which gives 

In both cases A is a constant specific to the engine that can be calculated using Isp figures given in the VAB: 

This is accurate to about 1% in low atmosphere, a bit more in higher atmosphere. I believe that aeroz2011 derived his formula for a specific engine, which means it won't work for all engines (I get almost constant thrust using a SRB for example).

[aeroz2011]

19 minutes ago, Gaarst said:

@aeroz2011 as @Spricigo said, I'd like to see where your formula comes from. I've done the experiment and found something very different.

 

I find that the thrust for a given engine is given by , using the pressure P as a parameter.

Taking measurements of Kerbin's atmosphere, you can roughly express pressure as a function of altitude only, which gives 

In both cases A is a constant specific to the engine that can be calculated using Isp figures given in the VAB: 

This is accurate to about 1% in low atmosphere, a bit more in higher atmosphere. I believe that aeroz2011 derived his formula for a specific engine, which means it won't work for all engines (I get almost constant thrust using a SRB for example).

If you see the data I took in the image, I got the data from watching the thrust values reported by the game at different altitudes (I paused it) for two engines (the BACC - 100% and 50%, + and x respectively; and the RT-10, 100% and 85%, diamond and triangle) but didn't get around to checking for the liquid motors. My bet is that it's probably close because it worked for both others. It probably holds - close - for all engines/motors with a F_asl/F_vac ~ 0.8.

The equation literally came from checking values based on my supposition that the equation was a constant related to max thrust allowed with a natural log addition (subtraction, really) that deminished with altitude. 

That said, while I like you're equation (frankly more than my own), there must be a more generic form of it because that varies using an experimental value and pressure (ASL, Kerbin) related. 

[Spricigo]

3 hours ago, aeroz2011 said:

What I did has nothing to do with drag or velocity produced - it's only a measurement of thrust relative to altitude - from 0 to [effectively] vacuum. 

Drag doesn't affect thrust. 

Thrust limit is a percentage, so if you set the limit to 40 on the Thumper, your max thrust will be 120kN. 

As Said in my early post, I'm lazy.  I did a speedreading and jumped to the conclusion the formula was used to determine speed at given altitude since that was what OP asked.  Well,  looks like proper reading is not as overrated as I thought. 

That said thrust(/Isp) varies with pressure according with the atmosphereCurve.  It look in the configs like that:

atmosphereCurve
{
  key = 0 300
  key = 1 280
  key = 9 0.001
}

And to really see the curve you want something like that:

If someone manage to extract a formula from all that,  nice.  I'm sure I won't, mostly because it still is some steps away from an util result (and these steps involves calculus) 

 

[Gaarst]

3 hours ago, aeroz2011 said:

If you see the data I took in the image, I got the data from watching the thrust values reported by the game at different altitudes (I paused it) for two engines (the BACC - 100% and 50%, + and x respectively; and the RT-10, 100% and 85%, diamond and triangle) but didn't get around to checking for the liquid motors. My bet is that it's probably close because it worked for both others. It probably holds - close - for all engines/motors with a F_asl/F_vac ~ 0.8.

The equation literally came from checking values based on my supposition that the equation was a constant related to max thrust allowed with a natural log addition (subtraction, really) that deminished with altitude. 

That said, while I like you're equation (frankly more than my own), there must be a more generic form of it because that varies using an experimental value and pressure (ASL, Kerbin) related. 

The first formula is accurate regardless of the body you are standing on. Using direct measurements of pressure, I get accuracies of a thousandth of a percent.

The third formula, defining A, is expressed in terms of Kerbin asl parameters since it's the ones that are available in the VAB (vac thrust, vac Isp and asl Isp), but the constant A remains the same wherever you go, for a given engine.

The second formula is indeed only valid on Kerbin because it uses an approximation of pressure as a function of altitude. In fact, pressure varies only with density and temperature but can reasonably be approximated as a function of altitude, as I did. The function I used to express P(h) is extrapolated from experimental data on Kerbin and is only valid there.

Edited January 24, 2017 by Gaarst

[purpletarget]

You can use thrust and ISP at either Vacc or ASL to determine the max flow rate for the engine. The ISP runs linear from the ASL (atmo 1) to Vacc (Atmo 0) ratings... and therefore, so will the thrust difference between the ASL and Vacc values.  The pressure curve for Kerbin (and other atmo bodies) is generally available on the wiki. 

Others have mentioned that there's no easy deterministic formula for this....so I'd probably look at using a iterative approach with a spreadsheet. But, if you have a particular height and speed in mind, you can use basic STD formulas to determine things like what average acceleration you would need to make it to x alt at y speed, and that can lead you back to your TWR calculations to make sure you have the right engine and fuel load for the job. Chances are, your drag losses will be fairly minimal, and you can get close just allowing for gravity losses over your estimated time, unless you're flying a pancake.