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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.
