As was discussed in the previous post, the single stage rocket is not a very feasible way to place masses into orbit given the fact that the mass ratio between initial mass and burnout mass is not smaller than 0.1, and the Isp (specific impulse) of fuels is no higher than ~450 and the thrust to weight ratio is no bigger than 2 or so. This leads us to multistage rockets.
I will be reviewing section 8.2 of the Thompson book (Introduction to Space Dynamics). The section in the book is really short but it packs a fair amount of math. What I like about the chapter is that is shows an analytical way to optimize the mass of each stage so that the maximum velocity can be achieved. I will subdivide this section into 2 parts. The first one will show the derivation of the optimization of multi-stage rocket. The second section will show examples of multi-stage rockets.
Derivation of Optimized Multi-stage Rockets
Per Thompson, “In a multistage rocket the burnout velocity of the first stage becomes the initial velocity of the second stage, and, by casting off the empty first stage, the full burnout velocity of the second stage is available as an additional velocity to the burnout velocity of the first stage. The maximum velocity of a multi-stage rocket can then be computed as the sum of the single stage velocities, as given by:”
Thompson decides to ignore the gravity term of formula 1 in order to simplify the optimization of the rocket equations. Thus the final velocity of a multistage rocket is:
The value ui is the mass ratio of each stage, that is initial mass over mass at burnout. The problem that Thompson analyzes is, given the final velocity, which combination of mass ratios will achieve final velocity while minimizing the overall mass ratio mo1/P where mo1 is the takeoff mass of the rocket, and P is the payload mass. In order to setup the optimization problem the mass ratio is described as follows.
In formula 3, mpi is the mass of the propellant in stage i, and msi is the structure mass of stage i.
What we end up doing is now applying the lagrange multiplier optimization approach to the natural log of the mass ratio of the rocket in order to find the individual mass ratios of the stages with the constraint of the final velocity. I am not fully up to date with my understanding of the lagrange optimization approach, but check out Wikipedia for a good summary on what it is trying to do.
The lagrange equation is then:
Differentiating the formula we find:
Substituting formula 5 back into formula 2 gives us a value for the lagrange multiplier.
With this lagrange multiplier we can then solve for each stage mass ratio.
Examples of Multi-Stage Optimization
Lets solve some problems. I will start with a generic rocket example and then move on to real life examples to see where do these rockets land on when it comes to this optimization technique.
I will solve problem 2 of chapter 8 of the Thompson book. The problem reads:
“A two-stage rocket is to attain a maximum speed of 26,000 ft/sec with I1=I2=300sec and beta 1 = beta 2. Determine the mass ratio of each stage.”
We are able to find the speed of the gas by multiplying 32.2*300. Thus the speed is 9660 ft/s. Now we need to solve for lamda. However because both beta and specific impulse are the same for both stages we can simplify the equations the following way.
Thus the mass ratio of each stage should be about 3.85.
What would happen to the mass ratio if we had many stages?? The plot below shows what the mass ratio does as a function of N (number of stages) and Isp.
The previous plot shows the dramatic effect of having multiple stages. The addition of multiple stages improves the mass ratio of each stage substantially. Each stage only needs to carry just a little bit of propellant which is expected since we have many stages. Another interesting observation has to do with Isp. Notice how the benefit of Isp is valuable when the number of stages is low, but not quite as helpful when the number of stages is high. This demonstrates that Isp effects dissipate once multi-stage rockets are taken into account.
Saturn V Rocket
Ok. Now that we are done with the generic rocket, lets see where does one of the most important rockets stack up. The Saturn V rocket is the rocket that took people to the moon. It consisted of 3 stages. The specs shown below were taken from Wikipedia, so I am assuming that those are correct.
|Length||138.0 ft (42.1 m)|
|Diameter||33.0 ft (10.1 m)|
|Empty mass||287,000 lb (130,000 kg)|
|Gross mass||5,040,000 lb (2,290,000 kg)|
|Engines||5 Rocketdyne F-1|
|Thrust||7,891,000 lbf (35,100 kN) sea level|
|Specific impulse||263 seconds (2.58 km/s) sea level|
|Burn time||165 seconds|
|Length||81.5 ft (24.8 m)|
|Diameter||33.0 ft (10.1 m)|
|Empty mass||88,400 lb (40,100 kg)[note 3]|
|Gross mass||1,093,900 lb (496,200 kg)[note 3]|
|Engines||5 Rocketdyne J-2|
|Thrust||1,155,800 lbf (5,141 kN) vacuum|
|Specific impulse||421 seconds (4.13 km/s) vacuum|
|Burn time||360 seconds|
|Length||61.6 ft (18.8 m)|
|Diameter||21.7 ft (6.6 m)|
|Empty mass||29,700 lb (13,500 kg)[note 4]|
|Gross mass||271,000 lb (123,000 kg)[note 4]|
|Engines||1 Rocketdyne J-2|
|Thrust||225,000 lbf (1,000 kN) vacuum|
|Specific impulse||421 seconds (4.13 km/s) vacuum|
|Burn time||165 + 335 seconds
Given the specific impulse for each stage and the mass ratio for each stage we should be able to calculate the approximate final velocity of the rocket assuming there is no gravity using formula 2. Now the payload of the Saturn V to Low Earth Orbit was 310,000 lb. I used the following code to calculate the final velocity to low earth orbit.
%% Saturn V analysis m1_full = 5040000; m1_empty = 287000; Isp1 = 263; m2_full = 1093900; m2_empty = 88400; Isp2 = 421; m3_full = 271000; m3_empty = 29700; Isp3 = 421; P = 310000; mo1 = m1_full + m2_full + m3_full + P; mo2 = m2_full + m3_full + P; mo3 = m3_full + P; mp1 = m1_full - m1_empty; mp2 = m2_full - m2_empty; mp3 = m3_full - m3_empty; nu1 = mo1/(mo1 - mp1); nu2 = mo2/(mo2 - mp2); nu3 = mo3/(mo3 - mp3); beta1 = m1_empty/(mp1+m1_empty); beta2 = m2_empty/(mp2+m2_empty); beta3 = m3_empty/(mp3+m3_empty); vm = Isp1*32.2*log(nu1) + Isp2*32.2*log(nu2) + Isp3*32.2*log(nu3);
The final velocity comes out to 30,128 [ft/s] which is around the velocity needed to be in orbit around the earth. Amazing what a simple analysis can help you deduce!!!!!
The beta values (structural factor) are not equal for this rocket. They turn out to be 0.05,0.08,0.1 for the first, second and third stage. However from a practical perspective they are not too different.
Given the structural factor, the Isp and the number of stages we should be able to derive the optimal mass ratio for each stage and compare it with the real mass ratio of the Saturn V which is 3.42, 2.50, 1.71 for the 1st, 2nd, and 3rd stage.
I created the following Mathcad file to calculate the optimal mass ratios.
Notice that for the 1st stage the mass ratio should be 0.374, 4.809 for the second stage, and 3.546 for the third stage. This is garbage!!! It is impossible to have a mass ratio less than 1. What is greatly affecting this minimization approach is the very different Isp between the first stage and the rest of the rocket. If we were to increase the Isp of the 1st stage to say 350 we would get a mass ratio of 1.5 for the first stage, 2.975 for the second stage, 2.2 for the third stage. Notice that it almost feels like the algorithm is saying that the 1st stage should be ejected rather quickly and the rocket should rely on the second and third stages instead to reach the desired velocity. Taking this logic back to the original Saturn V, the first stage is highly inefficient when compared to the 2nd and 3rd stages. Thus the optimization is saying to not even bother building a 1st stage with such low Isp. Similarly if the 2nd and 3rd stage Isps are brought closer to the 1st stage Isp, we end up with a similar solution.
We can then summarize that this optimization approach is only good to discern which stages of a rocket should provide the most amount of work to achieve the desired velocity. Of course, the Saturn V was designed with a low Isp for the first stage. But we now know that the designers sacrificed efficiency in the 1st stage for some other criteria that was important to them.
Hopefully you have learned a little bit about multi stage rockets :). This summarizes my review of this section. Next up flight trajectory optimization!!!!