Designing rockets and the Tsiolkovsky rocket equation

[DanP]

I've been reading up about the rocket equation in Wikipedia, and I have a set of questions.

The first and more focused one:

How would the rocket equation look like for a design where some of the stages overlap? For example, the space shuttle uses the SRBs and the SSMEs together and then only SSMEs. So presumably the first stage is SSME + SRB and the second is SSME. What's the equation for this kind of design?

Wikipedia replies with an intriguing "more complicated", but I could have figured that one out on my own. It's tempting to say that one should simply add the exhaust velocities of the two components together and solve normally. However that can't be right... let's say you have three identical engines as the first and only stage... to solve the rocket equation you use the exhaust velocity of the engine, not three times the exhaust velocity (errr, right?). If I understand the equation correctly, this is because it deals with how much propellant you need to get a certain dV, and not how quickly you get to that dV (a thrust issue?).

Second - Let's assume we want to design a rocket, which is probably a true statement for many of us. We start of with a set of given specs. For example, it must be TSTO, use a Kerosene/LOX engine as the first stage and get 20 tons into LEO.

What would be the general procedure for that? Edit: I'm talking theoretical of course (not including aerodynamics, structural requirements etc.).

Presumably, you start with the rocket equation to get the required propellant mass in % total mass, then work back to find the total mass. Ok, so now you know how much propellent you need burned to give the spacecraft the required dV. Correct me if I'm wrong, but that still doesn't mean your bird will ever leave the pad, because it might be too heavy to lift off? Consider an Ion engine attempting to launch from earth - high Isp but low thrust. So the next step would be to some how calculate required thrust, but how and based on which requirements?


I would greatly appreciate any help, or advice as to where I might find the answer.


DanP

 

[Urwumpe]

Please... can't you post in default color like anybody else. Black is impossible to read here.

I can't help you if my eyes hurt after reading the first line.

 

[MajorTom]

My answer's not going to be terribly helpful here, but I'm suspecting that you're looking at a numerical solution. You've got different boosters and engines, burning different fuel types at different rates. The only thing they have in common is that they're all fixed together.

Fortunately, you have Orbiter...you can set up your own boosters, engines, ISP's, etc. using various tools such as multistage2 or similar, and test "fly" your designs and see the solution generated in real time. You can set multistage to generate "telemetry" that will serve as your test flight data (altitude, velocity, mass, etc. given in time steps.)

For sophisticated treatment of multiple engine / overlapping booster systems, where the simple rocket equation is not solve-able, I'll leave it to others to suggest techniques.

Welcome to the forum!
MT

 

[Urwumpe]

The total ISP of a stage can be calculated by remembering the thrust equation:

Thrust = Mass flow * ISP

or:

ISP = Thrust/Mass flow

If you know the total thrust and the total mass flow of the logical stage, you can calculate the total ISP.

 

[DanP]

Please... can't you post in default color like anybody else. Black is impossible to read here.

I can't help you if my eyes hurt after reading the first line.

Sorry.

Must be because I wrote that post in MS Word first? My connection timed out so I wrote the post outside IE while the page is loading.

BTW, I'm seeing all posts (mine and others) in the same colour, apparently black.

Does this look any better?

" I've been reading up about the rocket equation in Wikipedia, and I have a set of questions.

The first and more focused one:

How would the rocket equation look like for a design where some of the stages overlap? For example, the space shuttle uses the SRBs and the SSMEs together and then only SSMEs. So presumably the first stage is SSME + SRB and the second is SSME. What's the equation for this kind of design?

Wikipedia replies with an intriguing "more complicated", but I could have figured that one out on my own. It's tempting to say that one should simply add the exhaust velocities of the two components together and solve normally. However that can't be right... let's say you have three identical engines as the first and only stage... to solve the rocket equation you use the exhaust velocity of the engine, not three times the exhaust velocity (errr, right?). If I understand the equation correctly, this is because it deals with how much propellant you need to get a certain dV, and not how quickly you get to that dV (a thrust issue?).

Second - Let's assume we want to design a rocket, which is probably a true statement for many of us. We start of with a set of given specs. For example, it must be TSTO, use a Kerosene/LOX engine as the first stage and get 20 tons into LEO.

What would be the general procedure for that? Edit: I'm talking theoretical of course (not including aerodynamics, structural requirements etc.).

Presumably, you start with the rocket equation to get the required propellant mass in % total mass, then work back to find the total mass. Ok, so now you know how much propellent you need burned to give the spacecraft the required dV. Correct me if I'm wrong, but that still doesn't mean your bird will ever leave the pad, because it might be too heavy to lift off? Consider an Ion engine attempting to launch from earth - high Isp but low thrust. So the next step would be to some how calculate required thrust, but how and based on which requirements?


I would greatly appreciate any help, or advice as to where I might find the answer.


DanP "

MajorTom, thanks for the useful info. Unfortunately I'm still not THAT familiar with Orbiter but I might try test flying too. Although I guess the real-life rocket scientists must have a way to do the other than simulate-correct-simulate ?

 

[MajorTom]

MajorTom, thanks for the useful info. Unfortunately I'm still not THAT familiar with Orbiter but I might try test flying too. Although I guess the real-life rocket scientists must have a way to do the other than simulate-correct-simulate ?

Yes DanP,

You're probably right. I think Urwumpe and other senior members will have more info, but somehow I suspect this problem is not something you can easily do with pencil and paper. But maybe you can get "close" with some gross approximations.

But to get to orbit, I find little things can loom large. Like the last few kilograms of fuel might be all you need to change your trajectory from that of a missile (which will fall to earth 12000 km from its launch point) to that of a satellite (nice circular orbit). So the many variables, such as drag, trajectory (including azimuth), etc. etc. can each play an important role. In other words, crude approximations may not get you far enough to achieve a design that will actually work. This is just my intuition, however. I eagerly await somebody pointing out a "pencil and paper" technique that will provide useful info for a complex rocket design!

(I say this having not yet read the current literature on the subject. There are quite a few books on rocket propulsion that I see waiting for me at Amazon.com, but haven't read yet...but there's still time... )

MT

 

[Urwumpe]

There are aspects, you can calculate in advance. For example, you can calculate a gravity turn for the beginning of flight, so you have minimal AOA.

But finally, you can only find out the optimal open-loop guidance for the first flight phase by iterative simulations.

 

[MajorTom]

The total ISP of a stage can be calculated by remembering the thrust equation:

Thrust = Mass flow * ISP

or:

ISP = Thrust/Mass flow

If you know the total thrust and the total mass flow of the logical stage, you can calculate the total ISP.

Thanks, Urwumpe,

I believe you get us most of the way to an analytical solution (all other variables aside). I see that with "logical stages" and your rocket flight divided up into segments, with each "segment" holding as many variables (thrust, etc.) constant, you can assemble a crude analytical solution.

MT

 

[Urwumpe]

Yes, but very crude. Backtracking is a recommended strategy to get a good analytical solution: Start at orbit and go backwards to the surface. While you can't know all parameters exactly when in Orbit, together with the normal iterative way, you can refine the trajectory step by step.

 

[DanP]

Thanks a lot!

I would assume mass flow is kg propellant per sec ?

 

[Urwumpe]

Thanks a lot!

I would assume mass flow is kg propellant per sec ?

Either kg/s or the imperial equivalents. But I recommend using SI units.

 

[DanP]

We need Ve though, not Isp, right?

Mass Flow = Thrust / Isp

Dimensionally that's:

M / T = ( M * L / T^2 ) / T

which doesn't add up... it should be

M / T = (M * L / T^2 ) / (L / T)

Therefore m/sec instead of sec, therefore Ve instead of Isp?

 

[Urwumpe]

Therefore m/sec instead of sec, therefore Ve instead of Isp?

ISP in seconds are incorrect imperial units (lbm != lbf), the SI unit of specific impulse is "N * s/kg"

 

[DanP]

Aha! Thanks.

But you still need to multiply by g for that, correct?

 

[Urwumpe]

No.

When thrust is measure in N and mass flow in kg/s, the resulting ISP in N*s/kg is correct.

 

[DanP]

The Isp of different rockets often appears on the web in seconds. For example, 267.5 for the SSSRB. Is that the same figure then for N * s/kg , just different units for SI ?

For example, consider the first stage of one the DIRECT proposals, 2 * SSSRB and 2 * RS-68.

2*SRB:

Combined thrust: 29,640 kN
Isp each: 267.5, presumably N*s/kg
Therefore mass flow...
29,640,000 / 267.5 = 110,804 ~

2*RS-68:

Combined thrust: 6681 kN
Isp each: 409, presumably N*s/kg
Therefore mass flow...
6,681,000 / 409 = 16335 ~

So, total mass flow for the first stage (until SRB sep) is: 127,139 kg/s ~

We use the thrust equation again for the first stage to find out the combined Isp:

Isp = total thrust / total mass flow
= 36,321,000 / 127,139
= 286 ~ (again presumably N*s/kg)

Or... I could have the whole unit conversion thing wrong though, I don't appear to be making much sense of it for some reason :huh:

 

[Urwumpe]

When in seconds, it is imperial units.

ISP (imperial) (lbf*s/lbm) * 9.81 (m/s²)= ISP (SI) (N * s/kg)

I know that many US home pages mix the SI unit (kN for thrust and kg for mass) with the imperial seconds for specific impulse - but that just stupidity.

I would recommend Mark Wades pages. He usually works in imperial units and gives the kN thrusts only additionally - together with the mass flow data.

 

[DanP]

Thanks for the tip.

Here's what he has to say about the SRB:

Thrust (vac): 11,519.999 kN (2,589,799 lbf). Isp: 269 sec. Burn time: 124 sec. Isp(sl): 237 sec.

and the RS-68:

Thrust (vac): 3,312.755 kN (744,737 lbf). Isp: 420 sec. Burn time: 249 sec. Propellants: Lox/LH2. Isp(sl): 365 sec.

So he calls SI units of Isp seconds as well. That's strange...

 

[Urwumpe]

No, he uses imperial units as primary data Most historic stuff is in imperial units.

2589799 (lbf) / 269 (s * lbf/lbm) = 9627.5 (lbm/s)

 

[DanP]

But then the what he mentions as SI Isp must be a wrong figure. If we take his SI figures for the SRB, we get 11,520,000 N thrust and 237 so called seconds Isp.

11520000 / 237 = 48,608 ~ (kg/s)

Which should actually be ~ 4332 kg/s (1 lbm = 0.454 kg), based on the calculation you did using his imperial data.

Trust the imperial data then convert?

Edit: The SI also doesn't correspond with the conversion from imperial Isp to SI Isp you mentioned.
269 * 9.8 is not 237, nor 2370.