Rocket

Physics

Operation

In all rockets, the exhaust is formed from propellants carried within the rocket prior to use.

Rocket thrust is due to the rocket engine, which propels the rocket forwards by exhausting the propellant rearwards at extreme high speed.

In a closed chamber, the pressures are equal in each direction and no acceleration occurs. If an opening is provided at the bottom of the chamber then the pressure is no longer acting on that side. The remaining pressures give a resultant thrust on the side opposite the opening; as well as permitting exhaust to escape. Using a nozzle increases the forces further, in fact multiplies the thrust as a function of the area ratio of the nozzle, since the pressures also act on the nozzle. As a side effect the pressures act on the exhaust in the opposite direction and accelerate this to very high speeds (in accordance with Newton's Third Law).

If propellant gas is continuously added to the chamber then this disequilibrium of pressures can be maintained for as long as propellant remains.

It turns out (from conservation of momentum) that the speed of the exhaust of a rocket determines how much momentum increase is created for a given amount of propellant, and this is termed a rocket's specific impulse.

As the remaining propellant decreases, the vehicle's becomes lighter and acceleration tends to increase until eventually it runs out of propellant, and this means that much of the speed change occurs towards the end of the burn when the vehicle is much lighter.

Forces on a rocket in flight

The general study of the forces on a rocket or other spacecraft is called astrodynamics.

Flying rockets are primarily affected by the following:

• Thrust from the engine(s)
• Gravity from celestial bodies
• Drag if moving in the atmosphere
• Lift; usually relatively small effect except for rocket-powered aircraft

In addition, the inertia/centrifugal pseudo-force can be significant due to the path of the rocket around the center of a celestial body; when high enough speeds in the right direction and altitude are achieved a stable orbit or escape velocity is obtained.

During a rocket launch, there is a point of maximum aerodynamic drag called Max Q. This determines the minimum aerodynamic strength of the vehicle.

These forces, with a stabilizing tail present will, unless deliberate control efforts are made, to naturally cause the vehicle to follow a trajectory termed a gravity turn, and this trajectory is often used at least during the initial part of a launch. This means that the vehicle can maintain low or even zero angle of attack. This minimizes transverse stress on the launch vehicle; allowing for a weaker, and thus lighter, launch vehicle.

Net thrust

The thrust of a rocket is often deliberately varied over a flight, to provide a way to control the airspeed of the vehicle so as to minimize aerodynamic losses but also so as to limit g-forces that would otherwise occur during the flight as the propellant mass decreases, which could damage the vehicle, crew or payload.

Below is an approximate equation for calculating the gross thrust of a rocket:

where:

propellant flow (kg/s or lb/s)

jet velocity at nozzle exit plane (m/s or s)

flow area at nozzle exit plane (m² or ft²)

static pressure at nozzle exit plane (Pa or lb/ft²)

ambient (or atmospheric) pressure (Pa or lb/ft²)

Since, unlike a jet engine, a conventional rocket motor lacks an air intake, there is no 'ram drag' to deduct from the gross thrust. Consequently the net thrust of a rocket motor is equal to the gross thrust.

The term represents the momentum thrust, which remains constant at a given throttle setting, whereas the term represents the pressure thrust term. At full throttle, the net thrust of a rocket motor improves slightly with increasing altitude, because the reducing atmospheric pressure increases the pressure thrust term.

Specific impulse

As can be seen from the thrust equation the effective speed of the exhaust, Ve, has a large impact on the amount of thrust produced from a particular quantity of fuel burnt per second. The thrust-seconds (impulse) per unit of propellant is called Specific Impulse (Isp) or effective exhaust velocity and this is one of the most important figures that describes a rocket's performance.

Vacuum Isp

Due to the specific impulse varying with pressure, a quantity that is easy to compare and calculate with is useful. Because rockets choke at the throat, and because the supersonic exhaust prevents external pressure influences travelling upstream, it turns out that the pressure at the exit is ideally exactly proportional to the propellant flow , provided the mixture ratios and combustion efficiencies are maintained. It is thus quite usual to rearrange the above equation slightly:

and so define the vacuum Isp to be:

V_evac = C_fc ^*

Where:

the speed of sound constant at the throat

the thrust coefficient constant of the nozzle (typically between 0.8 and 1.9)

And hence:

Delta-v (rocket equation)

The delta-v capacity of a rocket is the theoretical total change in velocity that a rocket can achieve without any external interference (without air drag or gravity or other forces).

The delta-v that a rocket vehicle can provide can be calculated from the Tsiolkovsky rocket equation:

where:

m₀ is the initial total mass, including propellant, in kg (or lb)

m₁ is the final total mass in kg (or lb)

v_e is the effective exhaust velocity in m/s or (ft/s) or

is the delta-v in m/s (or ft/s)

Delta-v can also be calculated for a particular manoeuvre; for example the delta-v to launch from the surface of the Earth to Low earth orbit is about 9.7 km/s, which leaves the vehicle with a sideways speed of about 7.8 km/s at an altitude of around 200 km. In this manoeuvre about 1.9 km/s is lost in air drag, gravity drag and gaining altitude.

Mass ratios

Mass ratio is the ratio between the initial fuelled mass and the mass after the 'burn'. Everything else being equal, a high mass ratio is desirable for good performance, since it indicates that the rocket is lightweight and hence performs better, for essentially the same reasons that low weight is desirable in sports cars.

Rockets as a group have the highest thrust-to-weight ratio of any type of engine; and this helps vehicles achieve high mass ratios, which improves the performance of flights. The higher this ratio, the less engine mass is needed to be carried and permits the carrying of even more propellant, this enormously improves performance.

Achievable mass ratios are highly dependent on many factors such as propellant type, the design of engine the vehicle uses, structural safety margins and construction techniques.

Staging

Often, the required velocity (delta-v) for a mission is unattainable by any single rocket because the propellant, tankage, structure, guidance, valves and engines and so on, take a particular minimum percentage of take-off mass.

The mass ratios that can be achieved with a single set of fixed rocket engines and tankage varies depends on acceleration required, construction materials, tank layout, engine type and propellants used, but for example the first stage of the Saturn V, carrying the weight of the upper stages, was able to achieve a mass ratio of about 10.

This problem is frequently solved by staging � the rocket sheds excess weight (usually empty tankage and associated engines) during launch to reduce its weight and effectively increase its mass ratio. Staging is either serial where the rockets light after the previous stage has fallen away, or parallel, where rockets are burning together and then detach when they burn out.

Typically, the acceleration of a rocket increases with time (if the thrust stays the same) as the weight of the rocket decreases as propellant is burned. Discontinuities in acceleration will occur when stages burn out, often starting at a lower acceleration with each new stage firing.

Energy efficiency

Rocket launch vehicles take-off with a great deal of flames, noise and drama, and it might seem obvious that they are grievously inefficient. However while they are far from perfect, their energy efficiency is not as bad as might be supposed.

The energy density of rocket propellant is around 1/3 that of conventional hydrocarbon fuels; the bulk of the mass is in the form of (often relatively inexpensive) oxidiser. Nevertheless, at take-off the rocket has a great deal of energy in the form of fuel and oxidiser stored within the vehicle, and it is of course desirable that as much of the energy stored in the propellant ends up as kinetic or potential energy of the body of the rocket as possible.

Energy from the fuel is lost in air drag and gravity drag and is used to gain altitude. However, much of the lost energy ends up in the exhaust.

100% efficiency within the engine (η_c) would mean that all of the heat energy of the combustion products is converted into kinetic energy of the jet. This is not possible, but the high expansion ratio nozzles that can be used with rockets come surprisingly close: when the nozzle expands the gas, the gas is cooled and accelerated, and an energy efficiency of up to 70% can be achieved. Most of the rest is heat energy in the exhaust that is not recovered. This compares very well with other engine designs. The high efficiency is a consequence of the fact that rocket combustion can be performed at very high temperatures and the gas is finally released at much lower temperatures, and so giving good Carnot efficiency.

However, engine efficiency is not the whole story. In common with many jet-based engines, but particularly in rockets due to their high and typically fixed exhaust speeds, rocket vehicles are extremely inefficient at low speeds irrespective of the engine efficiency. The problem is that at low speeds, the exhaust carries away a huge amount of kinetic energy rearward. This phenomenon is termed propulsive efficiency (η_p).

However, as speeds rise, the resultant exhaust speed goes down, and the overall vehicle energetic efficiency rises, reaching a peak of around 100% of the engine efficiency when the vehicle is travelling exactly at the same speed that the exhaust is emitted. In this case the exhaust would ideally stop dead in space behind the moving vehicle, taking away zero energy, and from conservation of energy, all the energy would end up in the vehicle. The efficiency then drops off again at even higher speeds as the exhaust ends up travelling forwards behind the vehicle.

From these principles it can be shown that the propulsive efficiency η_p for a rocket moving at speed u with an exhaust velocity c is:

And the overall energy efficiency η is:

η = η_pη_c

Since the energy ultimately comes from fuel, these joint considerations mean that rockets are mainly useful when a very high speed is required, such as ICBMs or orbital launch, and they are rarely if ever used for general aviation. For example, from the equation, with an η_c of 0.7, a rocket flying at Mach 0.85 (which most aircraft cruise at) with an exhaust velocity of Mach 10, would have a predicted overall energy efficiency of 5.9%, whereas a conventional, modern, air breathing jet engine achieves closer to 30% or more efficiency. Thus a rocket would need about 5x more energy; and allowing for the ~3x lower specific energy of rocket propellant than conventional air fuel, roughly 15x more mass of propellant would need to be carried for the same journey.

Thus jet engines which have a better match between speed and jet exhaust speed such as turbofans (in spite of their worse η_c) dominate for subsonic and supersonic atmospheric use while rockets work best at hypersonic speeds. On the other hand rockets do also see many short-range relatively low speed military applications where their low-speed inefficiency is outweighed by their extremely high thrust and hence high accelerations.

Safety, reliability and accidents

Rockets are not inherently highly dangerous. In military usage quite adequate reliability is obtained.

Because of the enormous chemical energy in all useful rocket propellants (greater energy per weight than explosives, but lower than gasoline), accidents can and have happened. The number of people injured or killed is usually small because of the great care typically taken, but this record is not perfect.