Formula for TWR?

[Jovus]

I'm trying to figure out how to turn TWR through the flight of a stage into a function for programming purposes.

Clearly TWR as a function of time must follow the form F(t, y) = Ae^rt + Be^sy Note that I'm talking about a single stage; there are discontinuities upon staging. Also, I'm assuming instantaneous ignition and a fixed throttle.

t is time, y is altitude, taking the launchpad as zero (for ease of calculation).

The coefficient A is clearly an expression of the throttle, so let's set it aside.

The B term is likewise a scalar relating to the I^sp at sea level.

The s term is fairly easy. It's a coefficient to altitude, intended to represent change in I_sp with thinner atmosphere. But I'm still having some trouble with understanding how to use it; how does the atmosphere decay? It's clearly ln(something) depending on the celestial body, which means we could rewrite the second term as Bsy

Similarly, the r term is related to the mass-flow rate of the engine(s), or, if you like, the mass-loss rate of the rocket.

F is in Newtons. t is in seconds. y is in feet. This means r must be in Newtons/second and s must be in Newton-seconds. And at this point I'm lost. How do I go about giving specific values to my r and s terms?

 

For example, let's say I have a rocket that at F(0,0) = 300kN. Therefore 300kN = A + B. Easy, but unhelpful.

What else do I know about this rocket? I know the mass-flow rate. I know the specific impulse at sea level and in a vacuum. How do I derive the rest?

For background (and because perhaps there's an easier way to do this), I'm trying to take a given, arbitrary rocket and calculate, to a rough approximation, a zero-lift ascent from Earth (or another celestial body, but let's skip that).

 

Thanks to anyone who can help.

[Theysen]

TWR with Earth gravity as reference is

TWR  = Thrust / ( mass of the ship at given time * g_o)

with g_o ~ 9.81 m/s².

 

So when you know the fuel drain rate and the wet mass m₀ of your rocket you can integrate the mass from start to end or by adding a time based expression like this:

 

TWR(t) = Thrust / [(mass₀ - fuelflow * t) * g_o)].

 

 

Edited June 30, 2016 by Theysen

[Gaarst]

TWR is given by dividing thrust by weight (duh).

Thrust is constant (as you assumed) and is equal to: F = 9.81 * Isp * MFR. Isp is your specific impulse in seconds, MFR is the mass flow rate.

Weight is given by: W = 9.81 * M(t).
M(t) is the total mass of the rocket, equal to M = m0 - MFR*t, with m0 the wet mass of your rocket.

So, TWR = Isp* MFR / (m0 - MFR*t)