## July 2, 2013

### Rockets and Thrust Systems

We've discussed how a spacecraft is able to alter its direction, but not its position.  So now we'll discuss how spacecraft can produce thrust, a force which propels them through space.
Thrust is produced through Newton's third law of motion, which states that whenever a force is applied to an object that object applies an equal force back.  For a very simple example imagine you are sitting in a wagon throwing balls backwards.  You will be pushed forward, the harder you throw the ball the harder you'll be pushed.
So to produce thrust we need two things: some mass, or propellant, and a source of energy.  We use the energy to give the mass some velocity and throw it in one direction, which causes us to move in the other.  From Newton's second law we know that force is equal to the change in momentum per time.  Thrust then is simply the rate that we shoot propellant, mass flow rate, times how fast the propellant is ejected, its exhaust velocity:
$T&space;=&space;\dot&space;m&space;v_{ex}$
Rockets are thrust systems that carry all their propellant with them.  Jets, on the other hand, get some of their propellant from the surrounding atmosphere.  What separates different kinds of rockets is what kind of propellant they use and how they get their energy.
The simplest form of rocket is a cold gas rocket, which uses stored pressure as its source of energy.  An example is an inflated balloon with the end open, when released the air rushes out the balloon flies around.  Thrust comes from the mass stored in the balloon and the pressure energy.
So much energy is stored in the balloon when it is inflated.  Energy is a constant quality, as pressure energy is lost the balloon gains kinetic energy.  Power, on the other hand, is a time dependent quality.  Power depends on how fast the balloon releases its energy.  More power means the balloon turns its pressure into motion in less time.  High power systems produce a great deal of thrust quickly, while high energy systems produce thrust over long periods of time.
Change in momentum is called impulse.  Recall that the change in momentum is, because of Newton's laws, the same for the rocket and the propellant.  So if we know how much mass is in the balloon and how fast it leaves then we know how fast the balloon should be going.  Specific impulse is a very useful rocket term, the total impulse divided by propellant weight.  It is often used as a way of expressing how fuel efficient a rocket is, and is measured in seconds:
$I_{sp}&space;=&space;{{\Delta&space;p}\over{\Delta&space;m&space;g_0}}&space;=&space;{{v_{ex}}\over{g_0}}$
Let's say we want to figure out how fast the balloon is going when it is out of air, without air resistance.  Of course the impulse of the propellant is the same as the impulse of the rocket.  But the rocket is losing less mass, so the change in speed keeps going up.  A little calculus lets us solve this and gives the ideal rocket equation, an expression of how much the vehicle's velocity changes based on how fast propellant leaves the rocket, its mass before thrust, and its mass after:
$\Delta&space;v&space;=&space;v_{eq}&space;ln({{m_i}\over{m_f}})$
The change in velocity produced by a maneuver is called delta-v.  We can describe the amount of fuel used by a rocket, or necessary during a mission, in terms of the total change in velocity possible, the delta-v budget.
Cold gas thrusters are rarely used for primary propulsion, since they have very low Isp (around 50s) and produce little thrust.  They are sometimes used maneuvering thrusters, which require little thrust.  However, they do make a good, simple example of how rockets are described.

Chemical rockets are the most common form of rockets used.  They produce energy through the chemical reaction of two substances.  Two different propellants, fuel and oxidizer, are mixed and burn to produce energy in the form of fast moving reaction products.
A chemical reaction occurs when two different substances are brought together under the right circumstances.  Sometimes heat or a spark is needed to start the reaction, sometimes a third substance (called a catalyst) must be present.  Reactions that produce heat are called exothermic reactions, while those that require or absorb heat are called endothermic.  Exothermic reactions produce heat in the form of motion of the products of the reaction, the more energy the reaction produces the faster these products move.  So very hot reactions means very fast moving propellant.
Exhaust from the reaction travels out of the combustion chamber and through a nozzle.
Nozzles in general are fluid devices that convert fluid energy into kinetic energy.  That is some energy is stored in the fluid, in the form of internal energy or pressure, and the nozzle converts this into fluid velocity.  An immediate example is with a garden hose.  When you narrow the passage, say by half-covering the opening with your thumb, the water travels faster.
As the passage gets narrower the fluid moves faster, up to a point.  At a certain point the fluid will be moving at the local speed of sound, sonic velocity.  This point is known as the choke point, and the fluid at that point is experiencing choked flow.  Beyond the choke point the fluid is traveling at supersonic speeds, that is faster than its own speed of sound.  For supersonic flow the opposite of the normal nozzle shape is required.  Narrowing the passage slows supersonic flow, and widening the passage speeds it.  Such a design is called a converging-diverging, or dual-bell nozzle.
We want the exhaust coming from the nozzle moving as fast as is reasonable.  Along the way we reduce the pressure in the flow, a process called expansion.  If the pressure of the flow is still greater than the surrounding atmosphere then there is some fluid energy that is wasted, the exhaust could be moving faster, a condition called under-expanded flow.  Conversely if the pressure is less then a shock wave develops at the exit, which converts some velocity back into pressure, known as over-expanded flow.  Ideally expanded flow has a pressure equal to the environment.
Unless the nozzle is ideally expanded then the flow is not moving as fast as it could be.  We introduce a nozzle correction factor, to account for loss of thrust:
$\lambda&space;=&space;{{ideal}\over{real}}$
The difficulty with bell nozzles is that pressure changes with altitude while the exhaust pressure does not.  Ideal expansion is only reached for a very brief point during ascent, before that point the thrust is over-expanded and after that it's under-expanded.  The typical solution is to design for an altitude two-thirds of the way up.  However some new designs for nozzles are believed to improve on the bell nozzle by achieving ideal expansion over a much longer period of flight.
One method is a modification of the traditional bell nozzle by having vanes that can be opened and closed.  This allows a slight change in the nozzle shape, and therefore a change in the expansion ratio, the ratio of exhaust pressure to choke pressure.  However the mechanism to create vanes is complex and expensive and the gains are usually fairly minor.  Often times the added weight consumes more fuel to lift than it saves.
Plug nozzles are conventional bell nozzles, save that a plug has been placed just past the choke point.  The plug forces flow to move along the nozzle walls, instead of throughout the nozzle, and atmospheric pressure on the inside provides a sort of air wall.  Eventually the nozzle reaches a design pressure and is filled with gas, beyond which it will still suffer under-expansion, but in theory this should avoid over-expanded flow entirely.  However cooling of the plug is a serious concern.
Aerospike nozzles are similar to the plug nozzle, or a bell nozzle turned inside out.  There is no outer wall in this nozzle at all, flow is pushed by atmospheric pressure against an interior wall (the spike).  The spike itself is usually truncated and trailing vortices behind it form the rest of the taper (the aerospike).  The flow should always "exit" the spike at ambient pressure because it is ambient pressure that is acting as the outer wall.  Cooling the spike is a problem and aerospike nozzles are still a young technology, so very few space missions have made use of them.
Nozzles can also be used for stability and guidance.  Vanes can, in addition to changing the expansion ratio, also direct thrust a certain way.  Many times nozzles are gimballed, or allowed to pivot, so that their pointing can be altered.  Doing this ensures that the thrust is always pointed correctly and the vehicle moves in the correct way.  To ensure stability and for redundancy most missions use more than one engine.

However the products leave the vehicle, they are produced by a chemical reaction between fuel and oxidizer.  Chemical rockets are divided into three different categories, based on what form their propellant takes.  Solid rockets have solid propellant in a granular mixture.  Liquid rockets use separate tanks of liquid fuel and oxidizer.  Hybrid rockets use a combination of liquid oxidizer and solid fuel.
In a solid rocket the fuel and oxidizer are mixed together is a solid or gelatinous substance.  Gunpowder would be the oldest example of a solid fuel.   It's composed of potassium nitrate (the oxidizer) and a mixture of sulfur and charcoal (the fuel).  To use as a rocket fuel it is mixed with a binding agent, which holds it together in a fixed shape rather.  Modern solid rockets have exactly the same components, they are mostly fuel grain with some oxidizer mixed in, and a binding agent to hold it all together.  If necessary a curing agent is also added, depending on the binder chosen.
Since the fuel is a solid it can be formed and shaped however the designer chooses.  The shape affects how the fuel burns.  Only the fuel that is exposed burns, so the amount of propellant that is released depends on the surface area of fuel exposed.  By altering the shape of the fuel grain it is possible to alter the burn rate, and even cause it to change during flight.
Liquid rockets are a little more complicated than solid rockets.  They come in two varieties: monopropellant and bipropellant.  Monopropellants are fuels which require no oxidizer to react.  Instead the fuel breaks down when exposed to a catalyst, releasing energy that way.  Bipropellants, on the other hand, have separate fuel and oxidizer.
A monopropellant system is composed of a fuel tank, a fuel line, a valve, the catalyst and the nozzle.  When the valve opens fuel is allowed to flow through the catalyst, where it breaks down and heats up.  Monopropellants produce less thrust and have lower specific impulse than bipropellants, but their simplicity means they are often used for maneuvering thrusters.
Most liquid rockets are bipropellant systems.  This means that they have quite a few more components than the monopropellant does.  Both the fuel and oxidizer have their own storage tanks, there are separate fluid lines for both, the fluid lines lead to a combustion chamber where the two react and which opens into the nozzle.  This has many parts that can become very complicated.
First is the question of fuel and oxidizer choice.  The highest specific impulse that is possible is the reaction between hydrogen and oxygen (fluorine is actually slightly higher than oxygen, but is extremely toxic and difficult to work with) but both substances are a gas at room temperature.  In order to be kept a liquid they must be kept cryogenic, or very cold.  On the other hand RP-1 (also known as JP-8, or kerosene) is liquid at room temperature, or storable.  While cryogenic fuels may have higher Isp they must also be kept under high pressure, requiring thick tanks, and the support hardware is expensive.  Storable fuels trade specific impulse for ease of use.  Many rockets use storable RP-1 and cryogenic LOX (liquid oxygen).  Some prefer LH2 (liquid hydrogen) for its higher specific impulse and cope with the added complexity.  Conversely some use hydrogen peroxide (high test peroxide is at least 85% pure, as opposed to drug store which is 3%) as the oxidizer, since it is storable.
In addition to the problem of storing the propellant is igniting the reaction.  While hydrogen and oxygen combustion produces a great deal of heat, it doesn't happen automatically when the two meet.  A source of ignition is needed.  Propellants that do ignite immediately on contact are called hypergollic.  Hydrazine (N2H4) and nitrogen tetroxide (N2O4) are such a mixture, they also have the advantage of being storable.
We also want to make sure that the fuel and oxidizer are mixed in the correct proportion, the oxidizer-fuel ratio (o/f ratio).  There must be enough oxidizer to react with all the fuel.  Too much of either means wasted, unburned, propellant.  The exact right proportion to ensure all propellant burns is the stoichiometric ratio.  If the ratio is too far from stoichiometric the fuel won't burn at all.  For example ignition can't happen in an environment that is more than 80% hydrogen, even if the rest is oxygen.
Of course even if we do have the right proportion of fuel and oxidizer we have to make sure that they mix completely.  This means a system of injection that spreads the liquid propellant and mixes it.  Injectors ideally should atomize propellant, or spread it into very small drops.  Large globs of fuel can't fully interact with oxidizer, since the fuel in the center never sees any.  The two injection streams must also mix together in the center of the combustion chamber.
The combustion chamber itself gets extremely hot, far hotter than the melting point of any metal.  One method of injection that also deals with this is to finely spray the propellant on the chamber walls.  The liquid propellant evaporates, absorbing the heat (just as your sweat cools you).  Now gaseous fuel and oxidizer mix in the high heat environment and burn.
Hot propellant now pushes on into the nozzle.  We can't cool the nozzle in the same way so we need another method.  One is to use ablation.  An ablative coating is sprayed over the nozzle wall before launch.  This coating is one that will evaporate during thrust, taking the excess heat along with it before it reaches the nozzle walls.  This method is commonly used for solid rockets when nozzle cooling is required.
With liquid propellant we have an option called regeneration.  A regeneration cycle is one where the fluid is heated once, and then heated again in a separate process before being used for power.  In an engine this means we heat up the propellant before we burn it.  Doing so increases efficiency.  That first heating comes from running the fuel lines, not directly from the tanks to the combustion chamber, but around the nozzle.  Cool fluid moves through the hot nozzle, taking that heat away and cooling the nozzle.  Warmed propellant is injected into the combustion chamber.
Fuel lines themselves are also worth considering for a moment.  Unlike a solid rocket, we have the capacity to stop and start the combustion at will.  Fuel and oxidizer tanks are connected to valves that let through propellant in the correct proportion.  Opening or closing these valves allows on/off control of the engine.  We can also throttle the engine, by controlling exactly how much propellant it gets.  Consider the equation for thrust above, what we do is change mass flow rate without changing exhaust velocity.  The proportion of oxidizer and fuel is the same, but the amount has changed.  Greater mass flow rate means more thrust but burns through propellant faster.
Getting the propellant into the feed lines is also a concern.  Without gravity there's nothing pulling the liquid fuel into the line, it may tend to mass together in the center of the tank, or even disperse into a gas.  A method of control is to use elastic bladders that compress the fluid.  Pumps can be used to pull the fluid, with additional non-reactive fluid pushed into the tank to keep it moving.  Very commonly liquid helium is used to keep tank pressure constant, it is added to the tank as propellant is removed to push the propellant towards the feed pump.
Let's stop and consider all the parts of a liquid engine:
• fuel tanks - must keep fuel under pressure
• feed lines - throttle, stop/restart valve, regeneration
• injectors - stoichiometric mixing
• ignitor - if not hypergollic
• combustion chamber - extremely intense heat
• nozzle - intense heat
This is a lot of components to design and many things that can go wrong.  Liquid rockets are desirable for their higher Isp and throttle ability, but they are complicated and heavy.

Hybrid rockets are a combination of liquid and solid.  A hybrid rocket system uses a solid fuel grain and liquid oxidizer.  The liquid portion is designed must like a monopropellant engine.  It has a tank, feed lines, pump, valve, and throttle.  Regeneration can be used for cooling, if necessary.  The fuel grain, obviously, must be cut with a channel for the oxidizer to move through and burn, but beyond that any grain configuration is possible.
Hybrid rockets have many of the advantages of both systems.  They are much simpler than bipropellant systems, though a little more complex than solid propellant or monopropellant engines.  They can be throttled and are capable of stop/restart.  They're capable of specific impulse better than solid rockets, though lower than many liquid rockets.  The biggest drawback for hybrid rockets is simply a lack of prior work.  They're a young technology and still being developed, though increasingly popular for amateur rocketry.
For comparison monopropellant rockets get a specific impulse around 180 - 230 seconds (compare to 50s for cold gas).  Solid rockets get about 300s, bipropellants about 350 - 450s, and hybrid rockets around 325s.

Electric rockets use electrical energy to accelerate particles.  Unlike a chemical rocket there is only one propellant, which is chemically stable.  The most common choice is a heavy noble gas, with xenon being the most common.  Some electrical rockets use krypton or argon, and many early models used mercury.  Electric rockets come in two primary varieties: gridded ion engines (ion engines or GIE's) and Hall effect thrusters (Hall thrusters or HET's).  Ion engines were developed in the United States during the 1960's and are somewhat easier to analyze, though harder to build.  Hall thrusters were developed in the Soviet Union around the same time, they utilize more complicated physics, but are easier to manufacture.  Ion engines tend to have slightly higher specific impulse while Hall thrusters tend to produce slightly higher thrust.
Ion engines are composed of four parts: propellant injection, ionization chamber, grids, and neutralizer.
Neutral propellant enters the ionization chamber.  Here it is ionized, electrons are stripped away and the positive ion remains.  The method of stripping the electrons varies.  One is to use high power radio waves to excite electrons away from the nucleus.  Another is an electric arc, which pulls the electrons away.  Yet another is to use cyclotron resonance.  However it is done the ions are repelled by each other and so move towards the plates.
One plate has a positive charge and the other a negative charge.  Ions are pulled towards the negative grid, but repelled by the positive grid.  Holes of the two grids line up so that the ions, wanting to avoid the positive plate, pass through the holes.  Once through the positive grid they are pulled by the negative grid and pushed by the positive.  Their momentum keeps them moving straight through the holes in the negative grid.
Finally the electrons that were stripped away are added into the ion stream.  This is done to avoid building up a negative charge on the spacecraft.  If it became negatively charged the ions would be attracted back to the vehicle, which would slow the ion stream and so reduce the thrust of the engine.
Many GIE's actually have more than just the extraction (positive) and acceleration (negative) grids.  Many include a neutral or low negative voltage grid behind the acceleration grid to decelerate the ions and reduce ion backflow.  Four grid systems add a second positive grid, with the first as a plasma grid to reduce random ion sputtering on the extraction grid.
Hall effect thrusters accelerate the ions by entirely different means.  Crossed electric and magnetic fields produce a force parallel to both.  Below a magnetic field points from anode to cathode, or out from the thruster.  A magnetic field points in an outward circle form the center of the engine to the outer wall.  This causes a total force which tries to pull particles in a circle, orbiting the center.

When neutral gas enters this engine it collides with high energy electrons trapped by the fields, which strips away the electrons.  Nuclei are pulled along the electric field, but electrons are light enough to be trapped by the magnetic field and contained.  Nuclei are pulled along through the engine and accelerated by the electric field.  When they leave they pull electrons with them, creating a neutral plume.  Hall thrusters are smaller, lighter, and easier to build than ion engines but usually produce lower specific impulse.

Both of these systems are known as electrostatic thrusters, and are the electric rockets that are currently in use.  They produce specific impulse in the range of 3,000 - 12,000 seconds (10 times that of chemical rockets) and thrust on the order of 1 newton (chemical rockets can range anywhere from 20N for model rockets to 2,000,000N for the Space Shuttle Main Engine).  Because of their low propellant usage electrostatic thrusters are used for very long missions, but when high thrust is required chemical rockets are used exclusively.
There are newer, experimental, electric rockets known as electrodynamic rockets, which are capable of even higher specific impulse and much higher thrust.  While some of these show great promise, they also have great power requirements.  At this time no commercial space power system can meet the power requirements of the most mature electrodynamic systems.  Nuclear power plants should be capable, but fission generators have not been used for space missions except for a small number of test articles in the 1980's.

A number of more exotic proposals have arisen over the years, but I have no plan to spend class time on them.  If you are interested I suggest the additional reading: