# Calculating total burn time for a rocket under constant acceleration with two propellant consumption rates

I'm trying to create a simplified model of hypothetical fusion-powered thrusters for a sci-fi setting (on excel), such that upon entering ship mass and operating parameters, all the common performance metrics will be calculated as outputs.

I'll provide all the work I have done further below, but I want to elaborate on my question first to provide context. Essentially, I'm having trouble figuring out how to calculate the total time a constant acceleration thruster can burn because I have two types of propellant being consumed at different rates: the fusion fuel, and inert propellant (i.e. water). This is because I wanted a high exhaust velocity thruster that could inject extra mass into the exhaust to increase thrust/accel when needed. I have kept the fusion fuel's mass flow rate constant to maintain a constant thrust power at a fixed specific power, so I know that exhaust velocity and thrust will go down as the mass of the ship decreases in order to maintain the target constant acceleration.

With a deltaV calculated, I assumed that the total burn time was just deltaV divided by the acceleration, as it should give me the time it takes to reach that velocity, and by definition of deltaV, should be when it has expended its full propellant load. However, a spreadsheet iteration check proved otherwise, and after a couple days of fruitless thinking later, have led me here.

Anyway, here are my numbers for the ship.

Pure fusion means the ship is only utilizing the fusion products as reaction mass. Therefore the additional propellant is untouched and is treated as essentially the ship's dry mass. The following calcs were done with that in mind.

Overall, I didn't see any issue with this set of results. Since I'm not varying the fusion fuel's mass flow rate, the thrust power remains the same, and by extension the Ve and thrust. As this entails a constant thrust case, the acceleration should increase to account for the ship getting lighter, which I am pleased to see being reflected here. The two time calcs for expending all the fusion fuel also match as expected.

Then comes the constant acceleration case. I kept the same thrust power to emulate the scenario of simply injecting more reaction mass without changing the fusion mdot. Here, I figured that since acceleration is known (being chosen by the pilot), I could calculate the variable thrust instead, and get Ve from there, along with everything else.

Overall, a lot of things still seem to be working. Exhaust velocity, thrust, additional prop flow rate, and specific impulse follow the expected trends. But the two highlighted values of delta V and the burn time are ones I'm suspicious about, since when I tried iterating on a spreadsheet, that burn time was at a point when the ship still had enough of both types of propellant to use. Delta V started off as the calculated value at t=0, but confusingly started to increase for a bit before going down as expected. It did approach zero around when the ship's total mass equaled its dry mass, but the inert propellant mass had gone negative, so that doesn't make realistic sense. Ve, Trust, and Fmdot maintained their proper trends.

To wrap it up, I suspect that something isn't being properly reflected in my calculations, probably the change in mass over time, but technically the ideal rocket equation is independent of time. I also think that the mass ratio doesn't distinguish between the two propellants, but I'm not sure if that matters, or how I can account for it.

All I can say is that for a constant Fmdot in a constant accel case, the propellant flow rate is decreasing with time. I'd like to be able to calculate when the propellant is expended, as there are many resulting scenarios I want to consider:

1. How much fusion fuel is left when the inert prop is depleted?

2. Is there some fundamental relation between the starting masses of the two propellants/ship (whether it is at a const thrust power, accel, or some other condition) such that I can computationally find a set of parameters to deplete both around the same time?

3. And while I'd rather not have transient Fmdots and thrust powers, but as an extension of #2, is there a way to achieve that by steadily increasing the Fmdot to bump up the thrust power, reducing prop MFR so that both are used up at the same time?

I hope that the solution is simple and I'm just overlooking it, or misunderstanding something. If it's not, then I hope that the smarter folks out here might be able to lend me a hand. Resolving #1 would be more than enough for me, but if anyone could resolve #2/3, that would be tremendously appreciated.

And finally a thank you to those patient enough to look through all of this in the first place.

• Some of this might be addressed in this paper and this one. Note that the reaction in those papers is aneutronic D - ${}^3$He which is a practical choice in that much less shielding is needed, and embrittlement is significantly reduced. I'm not sure which reaction you have chosen. The mass of fuel in your model seems very high, but perhaps you have reasons for that. Jul 12, 2021 at 11:48
• @garyp thank you for the resources. The nature of the fusion reaction is certainly open to adjustments in the future. My current preoccupation is with the more fundamental aspects of my propulsion model, namely trying to understand its performance metrics. As the magnitude of many of those metrics imply, I'm trying to make a "torchship", aka a class of spacecraft with extremely high performance as to overcome a lot of the conventional limitations of space travel (i.e. low acceleration and long travel times). Jul 13, 2021 at 3:40
• If you haven't already, you might look at the 100 Year Starship project, a DARPA initiative. The same group that is proposing D-${}^3$He rocket propulsion has [outlined a trip](support.psatellite.com/papers/starship_100_year_paper.pdf) to $\alpha$-Centauri that arrives there in 500 years. Disclaimer: I'm a co-author. Jul 13, 2021 at 11:25