Rocket math for custom fuel tanks in RSS

[Halo_003]

Hi, this may or may not be the right place. I'm making my own fuel tanks for use in RSS, just for fun really, but I'm trying to work out the math behind it. So I have it set up in 2 stages, both LH2/LOX fueled, at a 6:1 mixture ratio. I'm assuming 95% of the volume in each stage is available volume for fuel, which is sort of a guess. I'm using 381s ASL, 455s VAC isp.

So each tank is a 6' radius cylinder. Stage 1 is 140' height, stage 2 is 40' height. I figured out volumes etc, and worked all of this out in Excel, I just want a math check to see if I'm doing this right.

Assumptions:

1. ISP @ VAC: 455s
2. ISP @ SL: 381s
3. 95% volume of each cylinder is available for fuel storage. (referred to in Excel as Efficiency)
4. Stage 1 height: 140'
5. Stage 2 height: 40'
6. Stage radii: 6' (both)
7. Went from liters to kgs of fuel by multiplying by density of O2 and H2 (1.1417kg/L O2, 0.07078kg/L H2).
8. Used volume ratio of SS ET to find amount in liters of each fuel per stage. (72.81% volume is H2, 27.19% volume is O2)

 

The +5000kg, +10000kg, etc relates to payload capacity. Am I on the right track here? To simplify the math some I just used SL ISP for Stage 1, and VAC ISP for Stage 2.

So this would result in a maximum payload of less than 10,000kg, or am I missing something? If yes, how could I modify it to reach a 20,000kg payload to LEO?

 

*SS kg/L refers to the weight to volume ratio of the space shuttle ET just as a sanity check.

Edited February 18, 2016 by Halo_003

[UmbralRaptor]

Without direct access to the spreadsheet, it's difficult to see if there are errors, though the SL Isp for its full burn is rather pessimistic. I'd suggest looking up the Delta IV Medium's performance for a sanity check if nothing else. (Also in part because it seems to have double the payload fraction?)

[A Fuzzy Velociraptor]

I can't say I know what RSS is in this context or at all so some of what I'll say may not make sense or apply for your application.

However, there are a couple issues with your system estimations I can see. I am not sure about the validity of saying that 95% percent of the cylinder available for propellant, though since I don't think you are trying to actually design something it is probably fine. Also I don't know where or how you have calculated your structural mass fractions but since your system uses a LH2/LOX you should anticipate your structural mass to be no less than 10% the mass of your system and likely closer to 12-15%.

Assuming you plugged the stuff into the equation correctly (be sure to take into account the mass of the second stage in your first stage delta V estimation) your delta V estimations should be accurate. So yes, your payload would be less than 10kt. Increasing the payload capacity of your vehicle would actually be fairly easy to do. Vary the ratio of the mass of your lower stage and upper stage until you hit a peak. This can be done pretty easily through creating a plot in excel or running a program through matlab/fortran etc. However, increasing your capacity to 20kt from there may be quite a bit harder than just optimizing your system. You will likely need to increase the mass of your vehicle or the number of stages of your vehicle including adding parallel staging.

___________________________________

I went ahead a wrote a quick program for you that will work with matlab at least. By increasing your ratio so that your first stage is about 83% of the total mass of the system not including your payload and your second stage to about 17% the total mass you can get your payload up to 10kt with a total mass of 220kt.

I added in the assumptions of your structural mass being 10% for the first stage and 15% for the second stage. I also added in your ISP values though I accidentally used 386 instead of 381.

Program:

x=[1:100];
y=zeros(1,100);
c=1;
% Mass Fractions
nuL=0.1;
nuU=0.15;
% Effective Velocity
EVL=386*9.81;
EVU=455*9.81;
% Total Rocket Mass
Mt = 200000;
% Payload Mass
Mp = 10000;
for c=1:100
    % Lower Stage Mass
    Ml=Mt*x(c)/100;
    % Upper Stage Mass
    Mu=Mt*(100-x(c))/100;
    % Delta U estimation
    delU = EVL*log((Ml+Mu+Mp)/(nuL*Ml+Mu+Mp))+EVU*log((Mu+Mp)/(Mu*nuU+Mp));
    y(c) = delU;
end
plot(x,y)
title('Delta U Estimation');
xlabel('Percentage First Stage of Total Vehicle Mass');
ylabel('Delta U')
grid on

[Halo_003]

I should have put that in, RSS is Real Solar System for KSP.  I would have attached the Excel file but I'm not sure how frankly.

25 minutes ago, A Fuzzy Velociraptor said:

I can't say I know what RSS is in this context or at all so some of what I'll say may not make sense or apply for your application.

However, there are a couple issues with your system estimations I can see. I am not sure about the validity of saying that 95% percent of the cylinder available for propellant, though since I don't think you are trying to actually design something it is probably fine. Also I don't know where or how you have calculated your structural mass fractions but since your system uses a LH2/LOX you should anticipate your structural mass to be no less than 10% the mass of your system and likely closer to 12-15%.

Assuming you plugged the stuff into the equation correctly (be sure to take into account the mass of the second stage in your first stage delta V estimation) your delta V estimations should be accurate. So yes, your payload would be less than 10kt. Increasing the payload capacity of your vehicle would actually be fairly easy to do. Vary the ratio of the mass of your lower stage and upper stage until you hit a peak. This can be done pretty easily through creating a plot in excel or running a program through matlab/fortran etc. However, increasing your capacity to 20kt from there may be quite a bit harder than just optimizing your system. You will likely need to increase the mass of your vehicle or the number of stages of your vehicle including adding parallel staging.

___________________________________

I went ahead a wrote a quick program for you that will work with matlab at least. By increasing your ratio so that your first stage is about 83% of the total mass of the system not including your payload and your second stage to about 17% the total mass you can get your payload up to 10kt with a total mass of 220kt.

*snip*

Very interesting! I have a few questions though:

 

How would I run the program you made for me? (Thank you!) I have Matlab but haven't ever used it. Or, how would you do it in Excel?

If for I wanted to compare switching the first stage to run on LOX/Methane, how would I figure out the amount of CH4/LOX? Is there an easy way to do that?

Why does changing the mass ratio of the first to second stage affect it so much? Currently 21.3% is assigned to second stage, and the remaining 78.7% is the first stage.

Edited February 18, 2016 by Halo_003

[A Fuzzy Velociraptor]

Open matlab and you can either just paste it in the command window or create a function file, save and hit F5 or the run button. If you wanted it in excel create a column from 1-99 and then another column from 99-1 using a desired starting mass determine the mass of the stages for each case. Add in the other factors like isp and structural mass ratio then just drop in the delU equation (though change variables for the respective excel boxes) and drag it down. Then you can either plot it or search for the highest delta V value.

If you want to determine rough ratios for different propellants try to find the stoichiometric mass ratio. If you want more specific numbers use a program called NASA CEA though it isnt very user friendly if you dont know what youre doing. CEA will also give you your ISP values. Your structural mass ratio will vary as you change in size and as your propellant densities change. Denser propellants will have lower structural mass ratios as will larger vehicles. While it is possible to get your structural mass ratios quite low I would keep them around 10 percent for more things you will be looking at and especially at the level you are looking at.

As for why the graph behaves like that I cant say with certainty but some of the difference will be caused by our using different values, especially in the structural mass regime. As the structural mass and payload in your upper section increases the graph will tend to shift towards a larger first stage. As your structural mass decreases it will shift towards a larger second stage. The varying ISPs will also affect your system quite a bit. Unfortunatley here there are a lot of variables that all affect how the graph looks that it is sort of hard to give a proper explanation of how exactly it gets that shape.

[A Fuzzy Velociraptor]

Looking at your original problem you have about a 12ish percent mass ratio on your first stage and about a 6 percent mass ratio on your second. If you were to plot that you would probably find that your  graph would be left shifted quite a bit wanting a substantially smaller first stage. A bunch of variables were changed when I wrote the program so a direct comparison isnt particularly valid.

Thing wouldn't let me edit the previous post.

Edited February 18, 2016 by A Fuzzy Velociraptor