The Science and Design of the Hybrid Rocket Engine

The science and design of the hybrid rocket engine Epub edition

Richard M. Newlands

Copyright

All of this work is copyright 2017 Richard M. Newlands apart from illustrations credited at the end of the book.

All rights reserved: no part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without the prior permission of the copyright owner.

ISBN: 978-0-244-90291-9

Dedications

For all the rocketeers who taught me rocketry, especially the late John Stewart, John Bonsor, and John Pitfield. And for all the gentlemen at Southampton University who taught me rocket science.

Thanks to my friends within our rocketry society Aspirespace, the United Kingdom Rocketry Association, and the British Interplanetary Society, who I bounced ideas off. Particular thanks to Dr. Martin Heywood who reviewed the book and provided many useful comments, and Dave Edgar who checked my thermophysics.

Thanks to Jim M. for his rocket testing knowledge and advice.

Thanks to Helen for her support, and book writing and editorial experience.

About the author

Rick ‘the Rocketeer’ Newlands has been interested in Space travel since childhood.

He graduated with an Honours degree in Aeronautics and Astronautics from Southampton University.

He is currently chairman of the Aspirespace rocket engineering society (www.aspirespace.org.uk), a member of the Scottish Aeronautics and Rocketry Association, a member of the British Interplanetary Society technical committee (www.bis-space.com), and a founder member and council member of the United Kingdom Rocketry Association (www.ukra.org.uk).

He’s also a technical advisor for several Space projects.

The way to discover whether you know something is to try to explain it to someone else.

Disclaimer

The author has strived to maintain the accuracy of information presented herein, but neither the publisher nor the author accept any responsibility for accident or injury arising from following the information given within this book.

Introduction

Man must rise above the Earth - to the top of the atmosphere and beyond - for only in that way will he fully understand the world in which he lives.- Socrates.

This book will concentrate on hybrid rocket engines utilising Nitrous oxide as their oxidiser, though the same principles apply to any oxidiser. I will discuss both small hobbyist ‘HPR’-sized hybrids, as well as one capable of launching one person into Space.

First off, although many refer to the whole craft as a rocket, it’s actually just the engine that is the rocket (some types of rocket engines are also known as rocket motors). The rest of the craft can be called the rocket vehicle.

The right toolkit

If you want to plumb in a new kitchen sink, you’ll need the right tools for the job. You’ll need spanners (wrenches) to screw some of the plumbing joints together, and a blowtorch to braze the remaining joints together. If you don’t have the correct tools, you’ll never be able to do the job.

For rocketry, the correct and necessary tools are a vivid imagination to be able to picture in your mind what you need to build before you build it, and most essentially: mathematics, science, and engineering. (And plumbing!)

Chapter 1:  A recap of science and maths

Rocket science, as the name suggests, involves many of the sciences: physics, chemistry, thermodynamics, mechanics, and stress analysis, (and also plumbing!). Also required is mathematics: primarily algebra.

So to begin, let’s review the core science and mathematics required to design a hybrid engine, some of which you may already know.

Calculus

You may wonder why we’re starting with one of the hardest concepts in mathematics that you learnt in high school. Actually, the core ideas of calculus are simple; your main problem at school - apart from the confusingly arcane language used to describe it - was that nobody bothered to tell you what calculus was for.

Isaac Newton co-invented a lot of the methods of calculus because he needed a tool to analyse and predict the motion of falling objects and orbiting planets. That’s calculus’s main use in rocketry: analysing the rocket vehicle’s trajectory as affected by the gravity of the Earth, as we’ll do in chapter 9.

Differentiation

Let’s start with a high school physics example, the equation:

which allows us to calculate the acceleration a of an object from the change in its velocity (V - U) over some measured time interval t.

We can also write this as:

where the Greek letter (delta) means ‘a finite change in’. ( V occurs a lot in rocketry.)

Though nobody bothered to tell you in school, these equations are actually specialised cases of differential calculus, the specialisation being that the acceleration is constant over the measured time interval, i.e. the velocity ramped up or down linearly with time (the velocity was a straight rising or falling line on a velocity-versus-time graph).

If the acceleration wasn’t constant due to the velocity changing in a complex way with time, then we might want to know the instantaneous acceleration of our rocket vehicle at some particular time. We can get this (extremely closely) by using a vanishingly small time interval for our t, (but not quite zero).

This is written:

The ‘d’ signifies ‘very small change in’.

so:

which can also be written as:

This is the general form used in physics, and is described as ‘The time rate of change of velocity is acceleration’ where ‘time rate of change’ means in plain English: ‘divided by the infinitesimally small change in time’.

The mathematician Leibniz invented the notation:

however, Isaac Newton used a different symbology which he called fluxion notation to denote the same thing: he’d put a dot above the variable, so he’d write:

In rocketry, both notations are used.

Integration

Newton also needed to perform the reverse process, starting with acceleration and calculating the resulting change in velocity that would occur. [And so, apparently, did the ancient Babylonians, who used a similar method of calculus to predict the motions of the planets.]

Again, high school physics gave us specialised equations that again assumed that the acceleration was constant over the time interval:

or:

Integration, also called integral calculus, extends this special case to deal with acceleration that changes over time, which is what generally happens in rocketry as the thrust and mass of the vehicle change with time which changes the vehicle’s acceleration with time.

We write: ‘Velocity is the time integral of acceleration’:

The curly

shape denotes integration.

We integrate from the start time of our finite time interval (perhaps engine ignition) to whatever is the end time of our time interval (perhaps engine burn-out). These time limits on the integration region are shown at the top and bottom of the:

symbol.

We’ll add together a whole chunk of little instantaneous accelerations ‘a’ over the finite time interval of interest, here shown as a series of narrow coloured rectangles of width t and height ‘a’:

(We use a rectangle with an assumed constant acceleration across the width of the rectangle. The rectangle has area: height a times width Δt which from equation 1.6 = V).

It turns out that the answer we’re aiming for (the change in velocity of our rocket vehicle that occurs at the end of our specified time interval) is numerically equal to the area of all the little rectangles shown here added together. This is known as ‘the area under the curve’ as it is the area under the thick black acceleration curve shown here:

As you can imagine, the smaller we make each time interval t, the narrower each rectangle will get (and so we’ll need ever more rectangles) but the closer we’ll get to the actual area under the thick black curve. The steppy ‘roof’ made of the tops of all the little rectangles will fit closer to the black curve the narrower the rectangles get.

In the theoretical limit of an infinite number of infinitely narrow rectangles, the area will match exactly the right answer. Remember that a vanishingly narrow (but not quite zero) t is written ‘dt’ hence the dt to the right of the curly integration symbol above.

Integration can be performed algebraically, but it’s easier to let a computer do it, as we do in chapter 9. There we’ll see that the time interval dt doesn’t need to be vanishingly small to get a reasonably accurate answer.

Forces

A force is a push or pull; generally in rocketry, one that tries to move something.

Forces are measured in Newtons, named after Isaac Newton. (If anyone tries to tell you that Thrust force is measured in kilograms, they’re very wrong! I’m alarmed that British T.V. science presenters describe forces in kilos.)

When you describe a force, you must always indicate in what direction the force pushes. The direction is very important, because if I push you backwards instead of pushing you forwards, you’ll move in a totally different direction. So always conider what direction the force is pushing (or pulling).

To show the direction of a force we often draw the force as an arrow, which is called a force vector. The shaft of the arrow shows the direction, and this direction is called the line of action of the force.

[A vector is any quantity where direction is as important as its magnitude.]

Forces add vectorially (force arrows are added nose-to-tail).

Weight and mass

A rocket vehicle has weight, and it also has mass. These are related but are not the same. Mass indicates the number of atoms something contains, which won’t change unless you cut it, and is measured in kilograms. 5 kilos of spacecraft remains the same whether they're at ground-level, or out in Space. Gravity doesn’t change mass.

Weight and gravity

Earth’s gravity pulls every atom (so that's everything with mass) down to the ground. This pull that holds things firmly on the ground is the force we call weight.

On the surface of our planet Earth, gravity has a strength of around 9.81 metres per second per second.

Isaac Newton devised a law to describe the effect of gravity: he realised that the gravity of the Earth (or any other planet or star) varies with an inverse square relationship with altitude.

where, for the Earth: = 398,600,500 km³s² is the Earth’s gravitational constant, RE is the mean radius of the Earth (6378.14 kilometres), and h is the height you’re at above the Earth’s surface in kilometres.

Although somewhat irrelevant in everyday life on the ground, in Space, the difference between mass and weight is very important.

Specific quantities

Quite often, it’s easier to deal with quantities per kilogram, such as energy per kilogram. Per-kilogram quantities are known as mass-specific quantities, shortened to ‘specific’. They’re usually denoted by lower-case letters, so if energy is E then specific energy is e.

The rocket principle

It was realised, by Isaac Newton and others, that forces never occur on their own, they always come in pairs. If you create a force, then another force will automatically appear.

Newton referred to the force that you make, the action, and he called the force that then automatically occurs, the reaction.

Newton also discovered that the reaction has a strength exactly equal to the action, and furthermore, it pushes in exactly the opposite direction to the action.

This was all encapsulated in his 3rd Law of motion, sometimes referred to as the rocket principle.

How it works

The rocket consists of a combustion chamber, a large open bottle full of very high pressure gas. You usually heat up the gas in a rocket by burning something, which is termed combustion, hence the name.

The hole at the end of the bottle is termed the nozzle; the narrowest part of which is the nozzle throat.

If one were to project the circle of the throat onto the inner forward wall of the combustion chamber, we could call this circle the action area because the high internal gas pressure pushing on this area generates the majority of the rocket’s thrust force.

The internal gas pressure is, of course, pushing forcefully all over the inner walls of the bottle, but everywhere except the action area it cancels out.

For example, with the rocket lying horizontally as shown here, the pressure is pushing equally on the upper wall as on the lower wall, so no net upper-lower force is generated. All that happens is that the walls get stretched. But opposite the action area there is just a hole, so a net thrust force prevails. The downside (which we could term the ‘reaction’) is that the precious high pressure gas squirts out the hole so has to be continually replenished. And yet as we’ll see shortly, the nature of this outflow is very important for maintaining the internal gas pressure.

The rocket designer tailors the size of the throat, and hence the action area, to achieve the amount of thrust required for the Space mission: thrust force equals gas pressure multiplied by action area. Varying the size of the hole doesn’t affect the engine’s efficiency.

On the end of the combustion chamber there resides the nozzle, of which the throat is the narrowest part. Its function will be described in chapter 4.

Newtons’s other laws

Newton’s 1st law: An object at rest will remain at rest unless acted on by an unbalanced force. An object in motion continues in motion with the same speed and in the same direction unless acted upon by an unbalanced force. [‘Unbalanced’ here means that if several forces are acting on an object, their combined effect - the resultant force - is not zero.]

This law is often called the Law of inertia because inertia is the property associated with mass that resists acceleration. Kick a three-tonne satellite in Space and it’ll hurt because it resists your attempt to accelerate it.

Newton’s 2nd law: Newton originally expressed this law as The time rate of change of momentum of an object equals the force applied to it divided by its mass. This derivation is of more use to rocketeers (as explained in chapter 4).

Or descriptively, the thrust force F (in Newtons) equals the differentiation with time t (seconds) of mass m (kilograms) times velocity V (metres per second), where the quantity m times V is called momentum.

Now in the case of an unchanging mass, then m can be taken out of the brackets, and the equation collapses into the familiar high school physics form:

Or Force equals mass times acceleration because as we saw earlier, the rate of change with time t of

velocity V is simply acceleration:

However, with a rocket, the velocity of the gas mass exiting the rocket nozzle is more or less constant as we’ll see in chapter 4, but as this gas mass is continually being lost by the engine out the nozzle, then the remaining propellant mass m (and hence the mass of the entire rocket vehicle) is changing with time, so:

i.e. velocity times the rate that the mass is changing with time in kg/second.

We explore this equation more fully in chapter 4, where we use Newton’s fluxion notation (see earlier) :

[Note that the Law of conservation of momentum says that momentum is conserved: it can be redistributed, but its overall value won’t change.]

Gas inertia and thrust

Interestingly, Newton’s 1st and 2nd laws tell us that if the high-pressure gas within the combustion chamber had no inertia, it would squirt itself out of the chamber instantly. With zero gas residence time within the chamber there would be no high pressure in the chamber. It’s the gas’s inertia and the ensuing finite rate of increase of gas momentum down the nozzle that sustains the pressure difference between inside the chamber and outside: the gas’s flow down the nozzle, and its decreasing pressure drop down the nozzle, support each other in a reciprocal arrangement. No gas inertia would mean no pressure difference and hence no thrust, so the gas inertia is a vital part of the functioning of a rocket.

The kinetic theory of gasses

Daniel Bernoulli developed the first convincing theory of how gasses exert a pressure. He proposed that a gas consists of a large number of particles (which we now know as atoms) which cause pressure by bombarding the walls of their container.

Generally, gas atoms stick together in pairs. For example, it’s unusual to find single atoms of oxygen (O) they generally come in twos, which is written O2.

A clump of joined atoms is called a molecule, so O2 is an oxygen molecule. However, molecules bounce off the walls just as single atoms do, so molecules still cause pressure in the same way.

A gas’s molecules are free to move about at high speed. All the tiny molecules bounce off one another like rubber balls. This is called the kinetic theory of gas. (Kine is an ancient Greek word for movement.)

If the gas is inside a container, such as a box, the gas molecules also impact the inside walls.

When a molecule hits the wall, it may not change speed, but it does change direction (it rebounds) so its velocity changes, because velocity is a vector (direction is important). Therefore its momentum (its mass times its velocity) changes. By Newton’s 3rd law, this momentum change exerts a force on the wall.

Each bounce off a wall creates a really tiny force on the wall. The billions of bounces from the billions of tiny molecules inside add up to what feels like a large steady force, but spread over all of the inside wall. When a force is spread all over a wall like this, we call it a pressure.

To develop an understanding of gas pressure, science proposed several simplifying assumptions, which although artificial do not detract from the overall picture of what a gas is; the ensuing theoretical model is useful.

The molecules are treated as structureless particles (are point-like object of ‘zero’ size).

The molecules exert no force on each other unless they collide.

Newton’s laws of motion govern the dynamics of the moving molecules, but gravity has a negligible effect.

All molecular collisions are elastic, which means that no energy is lost in the collisions.

The molecules are in ceaseless random motion.

It may surprise you that extremely tiny molecules can exert the enormous pressure of the atmosphere (see later) upon you, however the atoms are moving far faster than any bullet: at hundreds or thousands of metres per second. Also, there are an awful lot of them: around 10 to the power 25 (25 zeros) air molecules strike your hand every second.

Components

Let’s look more closely at a gas molecule bouncing off one of the inner walls of the combustion chamber. Remember that pressure is the ceaseless bombardment of the walls by millions of molecules every second.

Typically, the molecule hits the wall at an angle, it follows the line drawn in black in the diagram below then hits the wall (on the left in the diagram). Because it’s flying along at an upwards angle, it’s moving to the left but also upwards, both at the same time.

By drawing a right-angled triangle (called a vector diagram) we can show the part of the molecule’s movement that acts only to the left (line number 1) and also the part of its movement that goes only upwards (line number 2).

We’ve resolved the molecule’s movement into two components. A component is that fraction of the whole that acts in the particular direction that you’re interested in.

Notice that the molecule’s movement upwards (line 2) just slides along the wall, it can’t cause a bounce. It’s only the molecule’s movement to the left (line 1), hitting the wall straight-on, that can cause the molecule to bounce off the wall.

So the force that the molecule’s bounce makes on the wall only depends on that part of its motion that hits the wall straight-on (called normal to the wall). And so, pressure always pushes normal to a wall. When a pressure P acts upon a wall of area A then the resulting force on the wall is equal to the pressure times the wall area A that the pressure acts upon:

Pressure within a fluid

But that’s when the molecules hit a wall. When the molecules are still far from a wall, they bounce in all directions off the molecules around them, so the pressure within the gas (not at a wall) pushes in all directions at once.

But what exactly is the meaning of pressure within a gas? When pressure acts on a wall, that’s easy to understand in terms of molecules colliding with the wall. But what is the pressure at some point within the gas? What we really have to understand is that we’re talking about a ‘force-field’ of pressure acting in all radially inward directions on a sphere of gas of very small - but not zero - size. Wide enough to contain perhaps 100 molecules.

But what we really mean is not that there’s some fictitious force pushing in on the sphere of gas. Actually, it’s all to do with collisions: there’s a momentum exchange between the particles outside our imaginary spherical boundary, and those inside. So really they’re internal stresses rather than forces at the boundary.

Mixtures of gasses

Mixtures of gasses such as air behave just like a gas of one species. So we can treat the gas within the combustion chamber - which is a mixture of many different gasses - just as if it were a single gas, and refer to it as ‘the combustion chamber gas’.

Temperature

Most chemical rockets burn propellants to achieve hot combustion chamber gas.

Heat causes the gas molecules inside the combustion chamber to vibrate, rotate, and fly around at high speed. The more heat you put in, the faster the