# Altitude reached by a rocket

1. Jan 13, 2014

### strive

Hi

I have written a set of equations for rocket flight (below) as i need to know the final altitude at various configurations (fuel mass/payload/exhaust mass flow/thrust).

I would ask if someone can please verify if this is correct (i have compared the results to results gained from Tsiolkovsky equation but it is difficult to approximate varying Isp with altitude and to include drag). Although the results are near.
I am calculating iteratively because i need to know the following parameters at 1 second intervals: vehicle velocity, acceleration, altitude and mass.

I know i should be using natural logarithm for mass ratios, but at 100 steps per second i postulated the error is negligible (a max error of 2% is allowed).
The equations in question are blue, the rest is just the programs structure (it's matlab/octave syntax).

Atmospheric conditions for each iteration are derived by a large number of equations which i have already verified thus i have not included them.

m0=1100; % initial rocket mass [kg/s]
masso=500; % initial oxidator mass [kg]
massouto=6; % oxidator consumption [kg/s]
massoutg=1.5; % fuel consumption [kg/s]
massout=massouto+massoutg; % total propellant consumption [kg/s]

flightdur=(masso)/massouto % time of powered flight
t=0.01; % timestep

Rmax=0.826; % max vessel radius [m]
A=pi*Rmax^2; % max vessel cross-section [m2]
R=287; % air gas constant [J/kgK]
h0=0; % starting altitude [m]
T0=288.15; % temperature at 0m isa [K]
v0=0; % starting velocity [m/s]
g=9.81; % starting gravitational acceleration [m/s^2]

Cd1=0.015; % subsonic drag coefficient [/]
Cd2=0.04; % supersonic drag coefficient [/]

Xtotal=flightdur/t; % total number of timesteps [/]
% START CONDITIONS--------------------------------------------------------------
m=m0;
s=h0;
v=v0;

% TIMESTEP EQUATION BLOCK-------------------------------------------------------
for i=1:Xtotal
TIME=t*i;

%%% CALCULATION OF ATMOSPHERIC CONDITIONS (also defines vout)
(vout % exhaust gas velocity [m/s])

F=vout*massout; % generated thrust [N]
ro=p/(R*T); % ambient density [kg/m^3]

if v<340
Fd=A*v^2*ro*Cd1/2; % subsonic flight drag [N]
else
Fd=A*v^2*ro*Cd2/2; % supersonic drag [N]
end

Fu=F-Fd; % useful thrust [N]

dm=(massout*t); % mass change in timestep X [kg]
ma=m-(dm/2); % average mass during timestep X [kg]
a=(Fu/ma)-g; % acceleration in timestep X [m/s^2]
s=(a*(t^2)/2)+v*t+s; % altitude in timestep X [m]
v=(a*t)+v; % velocity at the end of timestep X [m/s]
m=m-dm; % vehicle mass at the end of timestep X [kg]

end

%%% COASTING PERIOD--------------------------------------------------------
ii=0;

while v>0

ii=ii+1;
TIMEcoasting=ii*t;

%%% CALCULATION OF ATMOSPHERIC CONDITIONS

ro=p/(R*T); % ambient density [kg/m^3]

if v<340
Fd=A*v^2*ro*Cd1/2; % subsonic flight drag [N]
else
Fd=A*v^2*ro*Cd2/2; % supersonic drag [N]
end

a=-(Fd/massempty)-g; % acceleration in timestep X [m/s^2]
s=(a*(t^2)/2)+v*t+s; % altitude in timestep X [m]
v=(a*t)+v; % velocity at the end of timestep X [m/s]

end

2. Jan 13, 2014

### Filip Larsen

Perhaps you can use OpenRocket [1] to check your results. You may have to select "model rocketry" parameters for this to be accurate.

[1] http://openrocket.sourceforge.net

3. Jan 13, 2014

### cjl

Those drag coefficients look perhaps an order of magnitude too low to me, and you'll lose a significant amount of accuracy by only having a subsonic and supersonic value, instead of a CD vs mach curve. If you want a more realistic aerodynamic simulation, RASaero is pretty good at estimating rocket drag all the way up to pretty high supersonic speed.

http://www.rasaero.com/

4. Jan 13, 2014

### strive

Thank you both.

I will test both programs.

I have checked the NACA RM A53D02 (from the RASAero site) now and it realy seems the coefficients are way too low (i migth have used the planform area drag coefficients instead of frontal area drag coefficients).

By the way, can you perhaps guide me to some condensed reading on supersonic aerodynamics? I seem to be unable to find any »to the point« books.

5. Jan 15, 2014

### bigfooted

Modern Compressible Flow: With Historical Perspective - J.D. Anderson
Elements of Gasdynamics - Liepmann and Roshko