Rocket:First approximations

First approximations for our rocket

This page summarizes the important numbers about rockets launched from balloon or from aircraft, mainly what minimum vehicle mass we can expect. The two first sections about Delta V, combined with the famous rocket equation, serve as basis to evaluate the mass ratio of the rocket. More study is required to derive a minimum mass estimation from this, done in following secitons. This information is used as a first approximation for the design of other systems, like the aircraft and its engines. Information about particular systems of rocket engines can be found in the rocket engines page.

Delta V requirements

The delta V we will require to achieve orbit depends on several factors:

• rocket launch altitude, the higher it is, the less delta V is required to face atmospheric drag and climb against gravity (gravity drag) and actually start the useful horizontal delta V. The planned release altitude is 30km ±5km, so that leaves at least 100km to climb, probably even 160km if we want to be able to do 9 orbits, because of aerodynamic drag on upper atmosphere for such a small satellite.
  • Atmospheric drag should be very low at this altitude, although it scales badly with small vehicles as explained at many places, in particular in this paper . The paper also models the delta V losses and states that the difference between ground launch and a subsonic 40'000 ft (12km) launch is 1.5km/s, for a 77kg/20cm diameter vehicle, which is huge! (see page 46). It still has to be calculated in our conditions. If we need to pitch up after rocket release this drag will increase.
  • Gravity drag is about 1100m/s to 1500m/s with ground launch, and it is said here that Orbital's Pegasus system has its delta V reduced by approximately 750m/s with the aircraft cruise launch (40'000 feet, 500mph).
• rocket launch velocity, which is close to the velocity of the aircraft at rocket release. If the aircraft can do its release with the same velocity vector than the rocket flight path's beginning, this speed will be completely used by the rocket; if the aircraft needs to release the rocket and then after a few seconds the rocket fires, the z component of the velocity vector is probably lost.
• high-altitude flight strategy, a choice that has to be made between subsonic flight with large wingspan and reduced thrust or supersonic flight with high power engines.
  • Choice 1 (subsonic) has a planned horizontal release velocity of 270m/s ±20m/s (0.9M) and a quite small, 5° ± 5° vertical release velocity.
  • Choice 2 (supersonic) has a planned horizontal release velocity still unknown, possibly 1.5M ±0.5M but has the advantage of giving a higher release altitude and vertical velocity component, likely consistent with the rocket flight path initial vector.
• launch latitude, important for ground launches, the latitude determines the speed given by Earth's rotation. We have to keep in mind that the aircraft velocity is relative to the centre of the Earth and not to the surface. We don't know where we will be able to launch it yet, so let's take a conservative value for now, 300m/s. At equator, it is 465m/s.
• final velocity of the rocket, which is probably the same than the satellite velocity, except if we decide to catapult it from the rocket somehow, it is the main part of the delta V. At an altitude of 200km, the orbital speed is 7.8 km/s, and it doesn't change much for an altitude ±100km.

Our delta V will then be (if numbers are not explained above, they are just guessed until they are correct):

• For choice 1: 7800 (orbital speed) + 70 (atmospheric loss) + 700 (gravity loss) - 270 (release velocity) - 300 (earth rotation) = 8000m/s
• For choice 2: 7800 (orbital speed) + 100 (atmospheric loss) + 550 (gravity loss) - 400 (release velocity) - 300 (earth rotation) = 7750m/s

Delta V achievement

Once we have a realistic estimation of the required delta V for our rocket, we can start thinking about the characteristics of the rocket by itself and of its engine.

The rocket equation tells us that , where m₀ is the initial total mass including propellant, m₁ is the final total mass (dry rocket mass), and v_e is the effective exhaust velocity. The given delta V doesn't take into account gravity and other forces acting on the vehicle.

• the exhaust velocity of rocket engine's produced gas is closely related to the Isp of the engine, as we can see here: . For the duet LOX/E85 we chose, max Isp should be between the max Isp of RP-1 (353s) and max Isp of 92.5% alcohol (338s), so around 344 ± 6s. Isp is very important and we should aim and actual Isp approaching 20s from the max Isp. Max Isp is vacuum Isp with the most efficient rocket engine, the sea level Isp is generally around 10% less, and since we will use a cheap design, we may not have the optimal nozzle and fuel combustion efficiency.
• thrust is related to this exhaust velocity in the following way . It is not directly related to the delta V, but it is related to the mass of the payload that has to be accelerated. For example, Ion engines provide very high Isp and v_e but they don't provide enough thrust so they can be used to launch something from Earth to space.
• the dry mass of the rocket, once the fuel and consequently Isp have been fixed, is the most important factor. The lighter the rocket compared to the fuel, the more efficient it will be in term of delta V. As the mass ratio will be fixed by the rocket equation, it also means that the lighter the rocket, the many times less fuel will be required and the smaller the rocket. To make a light rocket, high quality materials, thus expensive, should be used. This is studied in the next section. For the N-prize, the payload mass is nearly insignificant compared to the rocket mass, which is good for the delta V.
  • the mass of the engine itself is closely related to the type fuel delivery system used (what kind of pump or tank pressurization) and to the thrust it produces. The thrust-to-weight ratio depends mostly on chamber pressure, which depends on the capabilities of the fuel pumps system and the efficiency of the regenerative cooling of the chamber and nozzle. Also studied below.

As an example, if the delta V is 9.0km/s, and that we have a 340s Isp, the required mass ratio of propellant is 1-e^(-9.0/3.3) = 93.4%, which is very hard to achieve. SpaceX has done it better than 95% (see spacefellowship forum thread on SSTO) for a first stage, so it doesn't include the payload, fairing, and so on. The Haas 2c rocket from ARCA (Romania) is also very lightweight and aims to be an SSTO demonstrator. Their engine has an very nice 110:1 thrust-to-mass ratio, the rocket having a 96.9% mass of fuel. The engine supports only 3 minutes of operation, will it be enough to reach orbital speed?

With a 8.0km/s delta V, assuming we have a higher release velocity, the same rocket engine and still only one rocket stage, it becomes 91.1%, still quite hard. A two stage rocket can do this, see the examples in the rocket equation Wikipedia page, but if we need to build two rocket engines' systems, we can be sure that we will exceed the money requirement.

So the main question becomes: on what subsystems can we found cheap alternatives to decrease the mass of the rocket?

Systems design for minimum mass

The mass ratio evaluated above does not tell us what mass we should expect from the rocket. It depends on the minimum mass all systems can be integrated: engine, tanks, structure, avionics, communications and other controls. Again, the paper addresses this subject.

Rocket engine

Since we will have to design our own rocket engine to meet the cost requirement, we can design an engine for any thrust. Since a more powerful or heavier engine will require a much more heavier rocket to achieve orbit because of the mass ratio, an even more powerful engine is required and this is an iterative process. A parameter that we should estimate in the first place is the thrust-to-weight ratio for the engine, which depends on choices made for the engine design. For example the wall's thickness and material are linked to the cooling method, material price and manufacturing capability; mass of accessories like pumps and pipes are linked to the pumping method. Once the rest of the rocket's mass has been evaluated, the iterations can be made to have a possible engine mass and final vehicle mass. The thrust-to-weight ratio for the engine doesn't scale linearly either, but in our advantage here . A ratio of more than 100 is common in small launchers' engines .

The overall thrust-to-weight ratio (for the vehicle) will also be required, as it will determine what will be the initial acceleration and the trajectory. To avoid atmospheric drag losses and minimize gravity drag losses, and from what we could find in the literature , a ratio of 2.0 is common for small vehicles. Another source states the opposite, that since we're not launching in vertical direction, we don't need as much ratio, and 1.25 is enough. It is, if losses are not taken into account.

Fuel tanks

Fuel tanks' mass depends on the type of fuel delivery system used: by tank pressurization or by pumping. When using pressurized tanks to force the fuel into the engine, the tanks have to sustain the pressure and must have thicker walls. When pumping fuel, the tank has to prevent it to leak into vacuum. For kerosene-like fuel, it can be very simple. For the cryogenic oxidizer, it's more complicated due to metal weakening with large temperature differential and to the evaporation losses related to poor insulation, thus requiring pressurization anyway. Tanks are the largest part of rockets, they may be used as a structural part too. That requires tanks to be strong enough to overcome the dynamic pressure stresses on the vehicle.

Given the budget, material for the structure and tanks will be aluminium, not titanium and probably not composite materials either.

TODO: Weight_tank = f_Al(volume, internal pressure)

Structure

If tanks are strong enough, they can be used as part of the structure of the rocket. If they are not, and between them in any case, an external skeleton has to be build that will undergo mechanical forces that apply to the vehicle. The max Q, maximum dynamic pressure that the vehicle has to withstand, is reduced when launching from altitude than from ground, so the structure may slightly get lighter. Bending stress is also important as rockets turn or undergo shear winds.

Even if we launch from a high altitude, the vehicle should be as aerodynamic as possible. The fairing and the rocket's body must be pretty smooth. If we actually use the tanks as structural components, we may not need an extra outer layer of metal that wraps around the entire rocket, a skirt between them will be enough. The fairing requires a jettison mechanism, the equipment bay requires structure too.

Avionics and attitude control actuators

Avionics, including sensors, but excluding actuators and wiring, is a fixed-mass package. Contrary to tanks or structure, it doesn't change when building the more powerful or larger vehicle.

estimated in 1999 that avionics could fit in 50g and 220g for batteries, but it used differential thrust throttling on several engines instead of gimbals or RCS. We will probably have only one engine, and attitude control actuators are mandatory for the rocket's first stage at least. Recent rockets, like ESA's Vega and SpaceX Falcon-1's second stage's use electromechanical, or electromagnetic, actuators. They replace the usual hydraulic actuators (jacks) that require pressurized hydraulic fluid and all their accessories: pumps, tanks, valves and so on. Batteries would have to be upgraded for this use, but off-the-shelf servomotors may be up to this function.

Other parts of avionics include the main computing board, the sensors and the wiring/connectors. The computing board can be very small and light-weight but all connectors will take more space and with the wiring will probably be heavier. Sensors mass approximation can be made using readily available components, see the sensors page.

Other elements

Engine's gimbal or RCS, valves, pressurization system (should count as engine mass), communication (should count as avionics), fairing, orbit insertion mechanism, and so on.

Minimum mass evaluation

A program has been created to make the iterations explained above. The code is coming soon online. The algorithm is the following:

start loop (
   compute vehicle wet mass: mass ratio * previous iteration dry mass
   compute propellant mass from the difference of wet and dry mass
   compute thrust required for the vehicle from the overall thrust-to-weight ratio
   compute fuel and oxidizer mass using the stoichiometric ratio and propellant mass
   compute fuel and oxidizer volumes from their mass and density
   /* now compute the mass of all subsystems */
   compute tanks properties from propellants volumes and rocket length-to-diameter ratio
   compute engine mass from the engine's thrust-to-weight ratio
   estimate thrust vector actuators and battery mass from engine's thrust
   estimate wiring mass from the length of the vehicle
   estimate structure mass from the size of the vehicle
   /* prepare next loop */
   set next dry mass as the sum of all subsystems listed above PLUS
       fixed mass systems' mass: payload, avionics board, sensors, main battery
) loop n times

An initial dry mass has to be set arbitrarily, the closer it is from the final value, the less iterations are required to converge to it, so it's not really important. The same iteration can be done on wet mass instead of dry mass, the result is the same.

Our first approximation

Input parameters:

• Mass ratio: 15.2. This is the value for a 9km/s Delta V and a constant Isp of 340s (propellants are 93.4% of the rocket's mass)
• Length-to-diameter ratio (tanks): 10
• Engine thrust-to-weight ratio: 100.
• Overall thrust-to-weight ratio: 1.25
• Payload mass: 0.0199 kg. (N-Prize oriented)
• Avionics fixed mass: 0.06 kg
• Sensors fixed mass: 0.05 kg
• Oxidizer: LOX, with 2mm thick aluminium tanks
• Fuel: E85, with 0.5mm plastic tanks (pumping, no pressurization)
• Structure is full aluminium, even skirts and fairing.

Results:

• DRY MASS: 28.3533 kg, WET mass: 430.969 kg
• Tanks size (length is rocket length excluding engine and fairing, diameter is the rocket's diameter): 3.80296 x 0.380296 m
• Engine thrust: 5279.38 N, engine mass: 5.38712 kg
• Final thrust-to-weight-ratio: 19 G
• Actuators mass: 0.879896 kg, wiring mass: 0.266207 kg, structure mass: 5.79303 kg
• LOX tank info:
  • diameter: 0.380296 m, length: 2.30944 m
  • thickness: 0.002 m, volume: 0.244837 m^3, mass: 14.8995 kg
• E85 tank info:
  • diameter: 0.380296 m, length: 1.49351 m
  • thickness: 0.0005 m, volume: 0.158336 m^3, mass: 0.847568 kg

So here we are. 431kg to carry up to 30km altitude. The numbers do not include any margin for error, meaning that if we count on some LOX venting, we need a larger LOX tank, and again much more mass at the end. Some structural parts like skirts and tanks outer skins (if required) could easily be made of composite material. That could reduce the structure mass probably by 20% (see the payload graph below, a decrease in 1kg of fixed mass is approximately balanced by a decrease of 50 kg in wet mass).

A multi-stage rocket would be lighter, but more complex to build.

Effects of parameter changes

The variation of one parameter is represented in the graphs below, demonstrating the way they affect vehicle's dry and wet mass. Take care about the scales of the dry (red) and wet (green) curves, they are absolutely not the same, however linear in both cases.

These first graphs represent the effect of the payload mass. They use the same data, but the second graph is zoomed-in for very small payloads. For higher values, the graph seems to be linear. At these small values, we can see that there is a slight non-linearity of the vehicle dry and wet masses. The fairing mass was not estimated in our model based on the payload mass, so in reality this is a bit worse. This graph can also apply to any fixed-mass subsystem, like the avionics controller.

The first graph above represents the vehicle mass evolution as function of the mass ratio (wet mass / dry mass). This ratio is given by the Isp (propellant efficiency) and the required Delta V. The result is consistent with what is explained in the mass ratio Wikipedia's page.

The second graph highlights the importance of the engine's thrust-to-weight ratio, fixed by engine's design.

The first graph here depicts the effect of initial acceleration (overall thrust-to-weight ratio) on the vehicle's mass. It related to the rocket's trajectory and release velocity. A higher overall thrust-to-weight ratio gives a more important initial impulse (the acceleration at ignition). In our air-to-orbit scenarios, fortunately, we benefit from the aircraft release speed, and the ratio can be lower than 1.3.

A possible issue with SSTO vehicles is the final acceleration. Since the engine is sized to propel the whole engine and thus a mass much larger than the payload's, when the propellants have been consumed the thrust-to-weight ratio is very important. The second graph above shows that even for a small initial acceleration, the final acceleration, if no throttling is implemented, easily reaches 20 G's. We can see on the graph that the final acceleration varies linearly with initial acceleration, while the engine's thrust required to do so is at least quadratic.