# Computing Flight Characteristics for Different Propellant Grain Geometries?

Consider the following image:

I am interested in mathematically accounting for some of these different geometries but have had little success so far in finding how this can be done. The first thought I had was to characterize the mass burn rate as it changes over time, which I believe can be accomplished by the following, if we just consider Case 1:

The rate of change of the fuel mass would be dependent on the mass burn rate $$\dot{m}_{fuel}$$: $$\dot{m}_{fuel} = \rho \dot{V}$$

And now, $$\dot{V}$$ can be represented as:

$$\dot{V} = z\pi((r + \Delta t\dot{r})^2-r^2) = z\pi(2r\dot{r} + \Delta t^2\dot{r}^2)$$

Therefore $$\dot{m}_{fuel} = \rho z\pi(2r\dot{r} + \Delta t^2 \dot{r}^2)$$ Or, if you will: $$\dot{m}_{fuel} = \rho z\pi(2r\delta r + \delta r^2)$$

There are a few obvious issues I have with this. First, I am not aware of how I can integrate according to $$\delta r$$ when we have the $$\delta r^2$$ term. Second, calculating $$\dot{m}_{fuel}$$ will be essential in computing thrust for a given geometry. Since that is what I want to study, being a function of grain geometry, I am not sure there is another way to calculate this.

What are your thoughts? Is there a way to work this or do you have alternative suggestions?

• nakka-rocketry.net/burnrate.html#Intro Oct 21, 2023 at 18:20
• I reviewed the page beforehand, and maybe I'm missing something, but my problem with the article your posted is that I need to calculate pressure from burn rate, rather than the other way around, in order to determine thrust as a function of grain geometry...at least, I think that's the case, isn't it? I may be able to make an assumption for pressure, which I can use to then find burn rate, and with that, thrust, but I'm not sure what kinds of assumptions we can even make with this. Oct 21, 2023 at 18:38
• If you look at the the thrust curves, they are all explainable by just looking at the surface area of the propellant exposed to combustion and how it changes over time. You probably can just assume combustion consumes a constant depth of propellant over unit time, and look how the surface are exposed to consumption changes over time. Oct 22, 2023 at 0:15