User:AERO.Olivarez/sandbox

Performance
One of the most common measures of rocket performance is its specific impulse, which is defined as the thrust per sea-level weight flow rate (per second) of propellant consumption :

$$I_\mathrm{sp} $$ = $$\ T/m_\mathrm{e}g_\mathrm{0} $$

When rearranging the equation such that thrust is calculated as a result of the other factors, we have:

$$ T = -I_\mathrm{sp}g_\mathrm{0} * dm/dt$$

These equations show that a higher specific impulse means a more efficient rocket engine, capable of burning for longer periods of time. However, this also means a high specific impulse engine yields a lower amount of thrust compared to engines with a lower specific impulse rating and the same mass flow rate. 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 when the vehicle is no longer heavily influenced by the planet's gravity. A slow burning, long lasting engine would be more optimal for the long segment of an interplanetary trip.

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

$$ TWR = T/mg_\mathrm{0}$$

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

∆t = $$ I_\mathrm{sp}g_\mathrm{0}/T * (m_\mathrm{0} - m_\mathrm{f}) $$

where $$m_\mathrm{0}$$ and $$m_\mathrm{f}$$ 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

$$ h_\mathrm{bo} = I_\mathrm{sp}g_\mathrm{0}/m_\mathrm{e} * (m_\mathrm{f} * ln(m_\mathrm{f}/m_\mathrm{0}) + m_\mathrm{0} - m_\mathrm{f}) $$

$$ v_\mathrm{bo} = I_\mathrm{sp}g_\mathrm{0} * m_\mathrm{0}/m_\mathrm{f} - g_\mathrm{0}/m_\mathrm{e} * (m_\mathrm{0} - m_\mathrm{f}) $$

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 : 1. Partition the problem calculations into however many stages the rocket system is comprised of. 2. Calculate the initial and final mass for each individual stage. 3. 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. 4. Repeat the previous two steps until the burnout time and/or velocity has been calculated for the final stage. It is important to note that the burnout time does not define the end of the rocket stage's motion, as the vehicle will still have a velocity that will allow it to coast upward for a brief amount of time until the acceleration of the planet's gravity gradually changes it to a downward direction. The velocity and altitude of the rocket after burnout can be easily modeled using the basic physics equations of motion.

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

Ƞ = $$ (m_\mathrm{E} + m_\mathrm{p} + m_\mathrm{PL})/(m_\mathrm{E} + m_\mathrm{PL}) $$

Where $$ m_\mathrm{E} $$ is the empty mass of the stage, $$ m_\mathrm{p} $$ is the mass of the propellant, and $$ m_\mathrm{PL} $$ 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

ε = $$ m_\mathrm{E}/(m_\mathrm{E}+m_\mathrm{P}) $$

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.

λ = $$ m_\mathrm{PL}/(m_\mathrm{E} + m_\mathrm{P}) $$

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

Ƞ=(1+λ)/(ε+λ)

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

$$ m_\mathrm{p} = I_\mathrm{tot} / (g*I_\mathrm{sp} ) $$

where g is the gravity constant of the planet (which is earth in most cases). 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 comprised of 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 of the fact that 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

O/F = $$ m_\mathrm{ox}/m_\mathrm{fuel} $$

Where $$ m_\mathrm{ox} $$ is the mass of the oxidizer and $$ m_\mathrm{fuel} $$ 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.

Optimal staging and Restricted Staging
Optimal

The ultimate goal of optimal staging is to maximize the payload ratio (see ratios under performance), meaning the largest amount of payload is carried up to the required burnout velocity using the least amount of non-payload mass, which is comprised of everything else. Here are a few quick rules and guidelines to follow in order to reach optimal staging : 1. Initial stages should have lower Isp, and later/final stages should have higher Isp. 2. The stages with the lower Isp should contribute more ΔV. 3. The next stage is always a smaller size than the previous stage. 4. Similar stages should provide similar ΔV. The Payload ratio can be calculated for each individual stage, and when multiplied together in sequence, will yield the overall payload ratio of the entire system. It is important to note that when computing payload ratio for individual stages, the payload includes the mass of all the stages after the current one. The overall payload ratio is

λ = $$ \prod_{i = 1}^n $$ λi

Where n is the number of stages the rocket system is comprised of. Similar stages yielding the same payload ratio simplify this equation, however that is seldom the ideal solution for maximizing payload ratio, and ΔV requirements may have to be partitioned unevenly as suggested in guideline tips 1 and 2 from above. Two common methods of determining this perfect ΔV partition between stages are either a technical algorithm that generates an analytical solution that can be implemented by a program, or simple trial and error. For the trial and error approach, it is best to begin with the final stage, calculating the initial mass which becomes the payload for the previous stage. From there it is easy to progress all the way down to the initial stage in the same manner, sizing all the stages of the rocket system.

Restricted

Restricted rocket staging is based on the simplified assumption that each of the stages of the rocket system have the same specific impulse, structural ratio, and payload ratio, the only difference being the total mass of each increasing stage is less than that of the previous stage. Although this assumption may not be the ideal approach to yielding an efficient or optimal system, it greatly simplifies the equations for determining the burnout velocities, burnout times, burnout altitudes, and mass of each stage. This would make for a better approach to a conceptual design in a situation where a basic understanding of the system behavior is preferential to a detailed, accurate design. One important concept to understand when undergoing restricted rocket staging, is how the burnout velocity is affected by the number of stages that split up the rocket system. Increasing the number of stages for a rocket while keeping the specific impulse, payload ratios and structural ratios constant will always yield a higher burnout velocity than the same systems that use fewer stages. However, the law of diminishing returns is evident in that each increment in number of stages gives less of an improvement in burnout velocity than the previous increment. The burnout velocity gradually converges towards an asymptotic value as the number of stages increases towards a very high number, as shown in the figure below. In addition to diminishing returns in burnout velocity improvement, the main reason why real world rockets seldom use more than three stages is because of increase of weight and complexity in the system for each added stage, ultimately yielding a higher cost for deployment.

Tandem Vs Parallel Staging Design
A rocket system that implements tandem staging means that each individual stage runs in order one after the other. The rocket breaks free from and discards the previous stage, then begins burning through the next in stage straight succession. On the other hand, a rocket that implements parallel staging has two or more different stages that are active at the same time. For example, the space shuttle rocket has two side boosters that burn simultaneously. Upon launch, the boosters ignite, and at the end of the stage, the two boosters are discarded while the main rocket tank is kept for another stage. Most quantitative approaches to the design of the rocket system's performance are focused on tandem staging, but the approach can be easily modified to include parallel staging. To begin with, the different stages of the rocket should be clearly defined. Continuing with the previous example, the end of the first stage which is sometimes referred to as 'stage 0', can be defined as when the side boosters separate from the main rocket. From there, the final mass of stage one can be considered the sum of the empty mass of stage one, the mass of stage two (the main rocket and the remaining unburned fuel) and the mass of the payload.