Paper Rockets

A textbook on minimum cost parametric design and various launch architectures.

 Copyright (c)  2008  Bob Steinke.

  Permission is granted to copy, distribute and/or modify this document

  under the terms of the GNU Free Documentation License, Version 1.2

  or any later version published by the Free Software Foundation;

  with no Invariant Sections, no Front-Cover Texts, and no Back-Cover

  Texts.  A copy of the license is included in the section entitled "GNU

  Free Documentation License".

GNU Free Documentation License

 fdl-1.2.txt

Chapters:

Intro

This chapter talks about how the launch vehicle design space hasn’t really been very well explored, and the exploration that has been done has generally used the misguided metric of minimum GLOW instead of minimum cost, and significant cost improvements over the current state of practice are probably achievable.

This chapter will also state that this book explicitly avoids discussing the technical details of how to build real rockets, and instead assumes that you can build components with certain parameters, for example a LOX/kerosene engine with an Isp of X and thrust to weight of Y, and goes from there to examine different architectures built out of those components.

Minimum Cost Design

This chapter lays out the ideas behind minimum cost design and does a simple example from start to finish.

My idea for the example is a sounding rocket on the moon.  Your mission is to deliver a fixed amount of payload to a fixed altitude.  You fix everything about the architecture like propellant type and chamber pressure, and all of the parametric parameters like thrust to weight and tankage factor.

You only vary one independent parameter, the engine thrust.  Other dependent parameters will vary such as the engine weight, or the amount of propellant needed to get to the required altitude.  You have the simplest possible trajectory straight up, straight down, and no drag.  You are basically balancing gravity losses against engine dry mass.

The example then comes up with an equation for GLOW as a function of engine thrust and takes the derivative of that function to find the design point for minimum GLOW, and then does the same with an equation for cost as a function of engine thrust.  Then you point out things like how engines cost more per pound than tanks and propellants and how that drives you to a different design.

Then the chapter goes through the limitations of minimum cost design such as how the results are only as good as your assumptions, and you sometimes need to make simplifying assumptions to make the equations tractable.  But it can be useful for things like sensitivity analysis to see which parameters are most important to optimize even when the exact values are not 100% accurate.

Here’s a start at the example:

The rocket consists of an engine with a thrust to mass of 500 N/kg, Isp of 3000 Ns/kg and tankage with a mass of 0.1 kg/liter of contained volume holding a propellant with a density of 1.0 kg/liter.  If we select thrust T and propellant mass P then the engine will burn for:

Total impulse = Isp * propellant mass = 3000 * P Ns

Burn time = total impulse / thrust = (3000 * P) / T s

Mass of the rocket = thrust / thrust to weight + propellant mass(as a function of time) + (propellant mass / density) * tankage factor = (T / 500) + (P(t)) + ((P / 1.0) * 0.1)

Acceleration = (thrust – gravity) / mass of the rocket

Burnout velocity = acceleration * burn time

Altitude achieved = …

Then you do some substitution to find propellant mass for a fixed thrust and altitude achieved.  Then you do some more substitution to come up with a closed form differentiable equation for GLOW as a function of thrust.

The Simulator

This is where all the hard work is.  The simulator is a set of equations and parameters that allow the student to quickly fly rockets on paper.  There will be equations for rocket construction and equations for trajectory simulation.  We will almost certainly have a computer program to help crunch the numbers, but we make it clear that the simulator is the equations and the point is to have the student become intimately familiar with the equations and able to work from first principles, not just plug numbers into a computer program.

Since this is a textbook the equations will probably have simplifications to make the homeworks tractable while still maintaining some semblance of reality.  One example of this would be treating atmospheric density as a simple exponential function of altitude.

Here’s a start.

A launch vehicle consists of the following parts.  Each type of part is followed by the parameters that describe that part:

• Engine – mass, thrust, thrust to mass, propellant inlet pressure, chamber pressure, specific impulse
• Tankage- mass, volume, pressure, tankage figure of merit (mass per pressure times volume)
• Propellant – mass, volume, density
• Structure
• Etc.

Engine thrust to mass:

Tankage figure of merit:

A chapter on each of the launch vehicle architectures

This will follow the progression of Jon Goff’s articles on orbital access methodologies, although it might have extra chapters covering other methodologies that Jon doesn’t cover like expendable rockets and ground launch SSTO.  It’s good to have those for comparison even if they don’t represent the near future of low-cost orbital access.

Each chapter will have homeworks that will generally take the following progression:

1)      Build a rocket, see how it flies

2)      Build a rocket to some fixed performance goal

3)      Calculate the minimum cost design point of some parameter for a fixed performance goal