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This is a very important equation, as it allows us to calculate pressure during a burn. Pressure itself is an important value to know for things such as making sure the motor is strong enough to handle the pressure during a burn and knowing the shape of a motor's thrust curve. Also, knowing the pressure allows us to find the burn rate at a given time.
Something to note is that A_b / A_t is also known as "Kn". Once you know Kn, it is trivial to calculate pressure during the burn.
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The inner area starts off by expanding (increasing burn area), but then it starts to get cancelled out when the inner circle meets the outer wall. This means that the pressure/thrust increase for a little at the start of the burn, but then decreases overtime as there is less exposed propellant area to burn. Something to note is that the Moon Burner shape in particular is not used very much. Not because it creates a bad thrust curve, but because it is very difficult to implement in practice. For example, it is difficult to create a nozzle in which the propellant doesn't cover up the throat.
Now that we've gone over the burning of real propellant shapes, we can come back to our question of what burn rate lets us find. Well, it lets us find Kn again! This means that there is a cycle a variables we can find during a burn:
By knowing the Kn at a point of a burn, we can find the pressure. The pressure then allows us to find the burn rate, and the burn rate lets us find the new Kn. This is how openMotor works; it uses the calculated burn rate over a short time to find a new burn area / Kn, continuing the cycle until the propellant is depleted.
In sum, our model takes in propellant geometry, propellant properties, and throat size, goes through the openMotor program with certain assumptions, and then spits out Kn, pressure, and thrust as a function of time for the propellant.
Now that we've talked about modeling propellant grains, it's time to look at some more aspects about designing the grain shape.
First, when designing a grain shape, it is important to make sure its opening is larger than the nozzle throat. Otherwise, the grain will start to act like a nozzle. To avoid this, but still pack as much propellant as possible, it is advisable to "step" the cores. This means that the grain is made up of many sections, and each one has a larger core than the previous one.
Another important thing to keep in mind when creating propellants is erosive burning. This is when a number of factors such as pressure, mass flux (amount of exhaust per time per area that flows through a point along the motor), and propellant strength lead to chunks of propellant tearing off. This causes burn area to spike, meaning the pressure spikes. If there isn't a margin for erosive burning, then a pressure spike could be too much for the motor to handle. Erosive burning can be seen on pressure and thrust curves where there is a large spike that quickly resolves.
When designing/modeling for erosivity, you have to determine the mass flux the propellant can handle without erosive burning, which is typically around 2lb/(in^2-s). If you want to give the rocket a "kick", you can also exceed the limit by a small amount. Finally, it is important to keep in mind that it depends on grain geometry.