Multistage rocket
A multistage rocket or step rocket is a launch vehicle that uses two or more rocket stages, each of which contains its own engines and propellant. A tandem or serial stage is mounted on top of another stage; a parallel stage is attached alongside another stage. The result is effectively two or more rockets stacked on top of or attached next to each other. Two-stage rockets are quite common, but rockets with as many as five separate stages have been successfully launched.
By jettisoning stages when they run out of propellant, the mass of the remaining rocket is decreased. Each successive stage can also be optimized for its specific operating conditions, such as decreased atmospheric pressure at higher altitudes. This staging allows the thrust of the remaining stages to more easily accelerate the rocket to its final velocity and height.
In serial or tandem staging schemes, the first stage is at the bottom and is usually the largest, the second stage and subsequent upper stages are above it, usually decreasing in size. In parallel staging schemes solid or liquid rocket boosters are used to assist with launch. These are sometimes referred to as "stage 0". In the typical case, the first-stage and booster engines fire to propel the entire rocket upwards. When the boosters run out of fuel, they are detached from the rest of the rocket and fall away. The first stage then burns to completion and falls off. This leaves a smaller rocket, with the second stage on the bottom, which then fires. Known in rocketry circles as staging, this process is repeated until the desired final velocity is achieved. In some cases with serial staging, the upper stage ignites before the separation—the interstage ring is designed with this in mind, and the thrust is used to help positively separate the two vehicles.
Only multistage rockets have reached orbital speed. Single-stage-to-orbit designs are sought, but have not yet been demonstrated on Earth.
Performance
Multi-stage rockets overcome a limitation imposed by the laws of physics on the velocity change achievable by a rocket stage. The limit depends on the fueled-to-dry mass ratio and on the effective exhaust velocity of the engine. This relation is given by the classical rocket equation:where:
The delta v required to reach low Earth orbit requires a wet to dry mass ratio larger than has been achieved in a single rocket stage. The multistage rocket overcomes this limit by splitting the delta-v into fractions. As each lower stage drops off and the succeeding stage fires, the rest of the rocket is still traveling near the burnout speed. Each lower stage's dry mass includes the propellant in the upper stages, and each succeeding upper stage has reduced its dry mass by discarding the useless dry mass of the spent lower stages.
A further advantage is that each stage can use a different type of rocket engine, each tuned for its particular operating conditions. Thus the lower-stage engines are designed for use at atmospheric pressure, while the upper stages can use engines suited to near vacuum conditions. Lower stages tend to require more structure than upper as they need to bear their own weight plus that of the stages above them. Optimizing the structure of each stage decreases the weight of the total vehicle and provides further advantage.
The advantage of staging comes at the cost of the lower stages lifting engines which are not yet being used, as well as making the entire rocket more complex and harder to build than a single stage. In addition, each staging event is a possible point of launch failure, due to separation failure, ignition failure, or stage collision. Nevertheless, the savings are so great that every rocket ever used to deliver a payload into orbit has had staging of some sort.
One of the most common measures of rocket efficiency is its specific impulse, which is defined as the thrust per flow rate of propellant consumption:
When rearranging the equation such that thrust is calculated as a result of the other factors, we have:
These equations show that a higher specific impulse means a more efficient rocket engine, capable of burning for longer periods of time. In terms of staging, the initial rocket stages usually have a lower specific impulse rating, trading efficiency for superior thrust in order to quickly push the rocket into higher altitudes. Later stages of the rocket usually have a higher specific impulse rating because the vehicle is further outside the atmosphere and the exhaust gas does not need to expand against as much atmospheric pressure.
When selecting the ideal rocket engine to use as an initial stage for a launch vehicle, a useful performance metric to examine is the thrust-to-weight ratio, and is calculated by the equation:
The common thrust-to-weight ratio of a launch vehicle is within the range of 1.3 to 2.0.
Another performance metric to keep in mind when designing each rocket stage in a mission is the burn time, which is the amount of time the rocket engine will last before it has exhausted all of its propellant. For most non-final stages, thrust and specific impulse can be assumed constant, which allows the equation for burn time to be written as:
Where and are the initial and final masses of the rocket stage respectively. In conjunction with the burnout time, the burnout height and velocity are obtained using the same values, and are found by these two equations:
When dealing with the problem of calculating the total burnout velocity or time for the entire rocket system, the general procedure for doing so is as follows:
- Partition the problem calculations into however many stages the rocket system comprises.
- Calculate the initial and final mass for each individual stage.
- Calculate the burnout velocity, and sum it with the initial velocity for each individual stage. Assuming each stage occurs immediately after the previous, the burnout velocity becomes the initial velocity for the following stage.
- Repeat the previous two steps until the burnout time and/or velocity has been calculated for the final stage.
When comparing one rocket with another, it is impractical to directly compare the rocket's certain trait with the same trait of another because their individual attributes are often not independent of one another. For this reason, dimensionless ratios have been designed to enable a more meaningful comparison between rockets. The first is the initial to final mass ratio, which is the ratio between the rocket stage's full initial mass and the rocket stage's final mass once all of its fuel has been consumed. The equation for this ratio is:
Where is the empty mass of the stage, is the mass of the propellant, and is the mass of the payload.
The second dimensionless performance quantity is the structural ratio, which is the ratio between the empty mass of the stage, and the combined empty mass and propellant mass as shown in this equation:
The last major dimensionless performance quantity is the payload ratio, which is the ratio between the payload mass and the combined mass of the empty rocket stage and the propellant:
After comparing the three equations for the dimensionless quantities, it is easy to see that they are not independent of each other, and in fact, the initial to final mass ratio can be rewritten in terms of structural ratio and payload ratio:
These performance ratios can also be used as references for how efficient a rocket system will be when performing optimizations and comparing varying configurations for a mission.
Component selection and sizing
For initial sizing, the rocket equations can be used to derive the amount of propellant needed for the rocket based on the specific impulse of the engine and the total impulse required in N·s. The equation is:where g is the gravity constant of Earth. This also enables the volume of storage required for the fuel to be calculated if the density of the fuel is known, which is almost always the case when designing the rocket stage. The volume is yielded when dividing the mass of the propellant by its density. Asides from the fuel required, the mass of the rocket structure itself must also be determined, which requires taking into account the mass of the required thrusters, electronics, instruments, power equipment, etc. These are known quantities for typical off the shelf hardware that should be considered in the mid to late stages of the design, but for preliminary and conceptual design, a simpler approach can be taken. Assuming one engine for a rocket stage provides all of the total impulse for that particular segment, a mass fraction can be used to determine the mass of the system. The mass of the stage transfer hardware such as initiators and safe-and-arm devices are very small by comparison and can be considered negligible.
For modern day solid rocket motors, it is a safe and reasonable assumption to say that 91 to 94 percent of the total mass is fuel. It is also important to note there is a small percentage of "residual" propellant that will be left stuck and unusable inside the tank, and should also be taken into consideration when determining amount of fuel for the rocket. A common initial estimate for this residual propellant is five percent. With this ratio and the mass of the propellant calculated, the mass of the empty rocket weight can be determined.
Sizing rockets using a liquid bipropellant requires a slightly more involved approach because there are two separate tanks that are required: one for the fuel, and one for the oxidizer. The ratio of these two quantities is known as the mixture ratio, and is defined by the equation:
Where is the mass of the oxidizer and is the mass of the fuel. This mixture ratio not only governs the size of each tank, but also the specific impulse of the rocket. Determining the ideal mixture ratio is a balance of compromises between various aspects of the rocket being designed, and can vary depending on the type of fuel and oxidizer combination being used. For example, a mixture ratio of a bipropellant could be adjusted such that it may not have the optimal specific impulse, but will result in fuel tanks of equal size. This would yield simpler and cheaper manufacturing, packing, configuring, and integrating of the fuel systems with the rest of the rocket, and can become a benefit that could outweigh the drawbacks of a less efficient specific impulse rating. But suppose the defining constraint for the launch system is volume, and a low density fuel is required such as hydrogen. This example would be solved by using an oxidizer-rich mixture ratio, reducing efficiency and specific impulse rating, but will meet a smaller tank volume requirement.