The very basis of rocket propulsion, and in turn solid rocketry, relies heavily on various physical theories and mathematical formulas. It is what allows the rocket to launch off of the ground and maintain a high velocity for a prolonged period of time.
Understanding the basics behind these concepts allows us to adapt our rockets to the highest efficiency and ensure their safety and stability.
Newton's Third Law
Newton's Third Law dictates every action has an opposite and equal reaction. In other terms, if Object A exerts a force on Object B, Object B will exert the same amount of force back onto Object A.
Rocket propulsion utilizes this concept to boost the rocket upwards-- by pushing itself against the ground with highly pressurized gas (or water, in terms of liquid propulsion), the ground, in turn, pushes against the rocket with an equal and opposite force, thus lifting it off the ground. Using this law, we are able to decipher two important factors when launching: the mass of the rocket and the force it applies on the ground-- for the initial launch to occur, the propulsion system must be capable of creating a downward force greater than the rocket's mass multiplied by the gravitational acceleration (9.81 m/s). [ Frocket > m*g ]
Conservation of Momentum
Conservation of Momentum notes that the total momentum of a system always remains constant. The equation associated with this concept is:
mAvA + mBvB = mAvA' + mBvB'
The left-hand side describes the initial momentum of two objects, [ Note: momentum(p) = mass(m) * velocity(v) ] and the right-hand side describes the final momentum.
Applying this to rocket propulsion, visualize the two objects being the rocket and the propellant inside the rocket. When the propellant burns, it becomes tiny particles of pressurized gas shot out of the back of the rocket.
If the rocket is initially at rest, when this occurs, the rocket's momentum will compensate for that downward velocity by pushing itself upwards as a result, allowing the rocket to maintain flight for a longer period of time. To improve the rocket's trajectory, we must consider the velocity of the gas as it exits the rocket and the amount of propellant in the rocket.
Components of a Solid Propulsion System
Taking these concepts into account, the solid propulsion system must address and optimize its features to these conditions. As a result, we have this:
A. Forward Closure - caps the motor to prevent the propellant from exiting the wrong end, into the rest of the rocket
B. Case - encases the motor, closing it off from the rest of the rocket
C. Liner - maintains the shape of the propellant
D. Propellant - fuels the motor and burns to become pressurized gas
E. Nozzle - adjusts the efficiency of the rocket
The nozzle, in particular, is a key part of this system. Though fully functional without it, the rocket would have a dramatic drop in efficiency without this piece, as it is what adjusts the pressure of the gas-- improving the thrust and speed of the rocket without needing to heavily alter the propellant or any other features.
There are two parts of the nozzle to take note of: the converging section (A.) known as the nozzle throat and the diverging section (B.) known as the nozzle divergence.
A. The nozzle throat chokes the flow of gas through the nozzle, thus pressurizing the exhaust. Due to this high pressure, the velocity of the motor is greatly increased.
B. The nozzle divergence generates thrust by expanding the exhaust to ambient pressure. This subsection controls the shape and angle of the gas as it exits, preventing it from shooting out in all directions and constricting its range
Without this piece, all the gas created by the propellant would exit the motor without much pressure, and would not generate enough force to support the rocket's full weight and propel it upwards.
Calculating Thrust
Thrust is the rocket's propulsive force, and is what allows it to launch itself off of the ground. To calculate thrust we use this formula:
T = Cf (t) * At * P0(t)
- T = Thrust
- Cf (t) = Thrust Coefficient (approximately constant)
- At = Throat Area (m²)
- P0(t) = Chamber Pressure (Pa)
The thrust coefficient stays relatively constant and the throat area may be measured however, the chamber pressure, the force the walls of the rocket exert onto the space within it, is still unknown. Therefore, we must use another formula:
P(t) = (Kn * c* * ρp * a)1/(1 - n)
- P(t) = Chamber Pressure (Pa)
- Kn = Propellant Burn Area to Throat Area Ratio (AB/At)
- c* = Characteristic Velocity (found experimentally)
- ρp = Propellant Density
- a = Propellant Ballistic Coefficient
- 1/(1 - n) = Propellant Ballistic Exponent
In particular, (c*ρpa)1/(1 - n) becomes approximately constant while Kn varies with time, thus making calculations much simpler.
Essentially, Kn affects Chamber Pressure which then affects Thrust.
*Characteristic velocity may also be found through this formula:
All variables are the same and mp is the mass of the propellant.
Specific Impulse
Specific impulse determines the impulse per weight of propellant. It may also be found to determine the efficiency of a rocket, as shown here:
Isp = Cf * c*/g
Isp = Specific Impulse
Cf = Thrust Coefficient
c* = Characteristic Velocity
g = Gravitational Acceleration (9.81 m/s)
These fundamentals form the basis of Solid Rocketry. It is important that you refer back to these formulas when designing parts or formulating propellants in the future to ensure the safety of your team and the success of your project.
What is a rocket?
The rocket is by far the coolest application of Newton’s Third Law. As you all may already know, the law states that for every action, there is an equal and opposite reaction. Some examples of this include throwing something when you are on ice (a classic example portrayed in physics problems), Baymax’s rocket fist, or perhaps even super high-speed exhaust. For the case of rocket propulsion, this can be done with a liquid propulsion system or a solid propulsion system.
Now what is a solid propulsion system?
Let’s say that you mix aluminum powder, ammonium perchlorate, HTPB, etc. to make a solid mixture of rocket propellant.
You will have this ball of propellant.
Let’s say you light the propellant on fire. What would happen?
It would probably explode like this animation right here.
We have no use for an uncontrolled explosion so let’s say you add a liner for insulation, a case, and a forward closure to seal off one side of the case… and you light it on fire. What would happen?
You see that exhaust has been directed towards a single direction, generating thrust for our makeshift rocket.
Now we want to exhaust the case at faster speeds so that we can get more thrust out of our rocket. How do we do this?
Let’s say we add a nozzle that has a converging section that pressurizes the exhaust and shoots it out of the motor with higher velocities and a diverging section in order to generate thrust expanding exhaust to ambient pressure… and you light it on fire.
This is a solid rocket motor in a nutshell. As you can see in this picture, we have all the components that we just talked about. We have the nozzle, forward closure, liner, case, and propellant. You may be asking yourself, why is there a hole in the middle of the propellant? Well, this is one example of a propellant grain, called a BATES grain. There are many more grains that alter the thrust curve of a solid rocket. You can imagine how certain grains have more surface area to burn initially and gradually mellow out over time. We can take advantage of these different grains by stacking them as well.