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

Question

I tried posting this question on the Physics side of StackExchange but didn't get any response after a week, presumably because it was lost within the broadness of physics in general, so I'm hoping that the focus on space-related topics here might garner more attention. This is a math-centric question though, so if there are more suitable places to ask, I'm open for suggestions.

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 rocket performance metrics will be calculated as outputs. I'm aiming for a degree of realism, but being simplified, the model as it is now will neglect a lot of the complexities behind the engineering and nuclear physics of the ship design itself.

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. Is there an analytic solution to figuring out how much fusion fuel is left when the inert prop is depleted?

2. Is there some fundamental relation between the masses of the two propellants and ship (whether it is at a const thrust power, accel, or some other condition) such that I can 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 the simultaneous depletion 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, as the depletion of the inert propellant simplifies the problem to conventional case of a single propellant. However, if anyone could resolve #2/3 either analytically or numerically, that would be tremendously appreciated.

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

Edit: Organic Marble pointed out that the "constant accel" condition may be inaccurate, as vehicle mass has to be assumed first in order to derive the propellant MFR as a final output, whereas engines should be calculating thrust from the MFR as an input. Further elaborations on this matter is welcome.

Answer 1

(Partial answer)

I noticed between the info in the question, additional info in comments, and an assumption of 0.0 delta thrust at 0.0 flowrate, there was enough information to draw a exponential curve for the throttle.

Delta Thrust (N)	Inert Prop Flowrate (kg/s)
0	0
4146140	4.18
10141140	20.0
13956140	37.6

Using this throttle characteristic curve, I wrote a simple simulation. At each time step, the "engine controller" calculates the throttle setting (i.e. mass flowrate) needed to provide enough delta thrust to achieve the desired constant acceleration.

The burn duration to use up all the inert prop is close to that stated in the question. So is the total delta-v, but that's a softball, since it's at constant acceleration.

For the parameters I calculated, the only significant difference I saw was in the inert prop mass flowrate at cutoff - I had almost twice what you did. This could be because of inaccuracies in fitting the throttle characteristic curve, or something else.

  Cutoff Parameters 
  
 Time:   8758.00000      sec
  
   Performance 
  
 Inert mdot      7.74639273      kg/s
 Thrust          7306489.00      N
 Acceleration    49.0025787      m/s^2
 Velocity        429159.062      m/s

So to answer #1

Is there an analytic solution to figuring out how much fusion fuel is left when the inert prop is depleted?

Yes (well, sorta..). Since the fusion fuel flowrate is constant, you just multiply that flowrate by the time to depletion of the inert propellant, and subtract the result from the starting fuel mass.

To partially answer #2, you can achieve simultaneous depletion by setting a different (lower) constant acceleration.

   Cutoff Parameters 
  
 Time:   498257.000      sec
  
   Performance 
  
 Inert mdot      3.23462412E-02  kg/s
 Thrust          1247464.75      N
 Acceleration    12.4745159      m/s^2
 Velocity        6226965.00      m/s

For #3, those answers can be derived by playing around with the sim.

Simulation equations

$$ T_d = (A_d M_t) - T_f $$ $$ \dot m_i = (\frac {T_d} {K_1})^{K_2} $$ $$ M_i = M_i - \dot m_i\Delta t $$ $$ M_f = M_f - \dot m_f\Delta t $$ $$ M_t = M_i + M_f + M_s $$ $$ T_t = T_d + T_f $$ $$ A_a = \frac {T_t} { M_t} $$ $$ V = V + A_a\Delta t $$

• $ A_a $ Actual acceleration
• $ A_d $ Desired acceleration
• $K_1 $, $ K_2$ Throttle characteristic coefficients from curvefit
• $ M_f $ Mass of fusion propellant
• $ M_i $ Mass of inert propellant
• $ M_s $ Mass of structure (aka dry mass)
• $ M_t $ Total vehicle mass
• $ \dot m_f $ Flowrate of fusion propellant
• $ \dot m_i $ Flowrate of inert propellant
• $ T_d $ Desired thrust from inert propellant
• $ T_f $ Thrust from fusion propellant
• $ T_t $ Total thrust
• $ V $ Velocity
• $ \Delta t $ Simulation time step