In the last post, I attempted to calculate a basic expression for the propellant-mass-sensitive term (lambda) and in this one I will attempt to do the same thing for the gross-mass-sensitive term (phi). In so doing, I will hopefully be able to show how a number of key factors in the rocket design affect the payload fraction. We begin by reiterating the definition of the gross-mass-sensitive term:

Then I make the assumption that this term consists only of two things—the engines and the thrust structure. This assumption will have to be modified for different designs. For instance, if you have a vertically-launched rocket that is taking off from landing legs, like a moon lander, then those landing legs are gross-mass-dependent. Or if you have a winged horizontally-launched rocket on Earth, then the wings and landing gear are gross-mass-dependent. So alter this assumption according to your needs.

Then I commit a sin against the SI system of units by switching from a mass ratio to a weight ratio. This is done to get everything in terms of forces rather than masses. You’ll see why I did this in just a second.

Now I make the assumption that all of the engines on the stage are the same kind of engine, and assume that I can multiply the number of engines (n) by the individual weight of the engine (W_{engine}). I also assume that the thrust structure weight is proportional to the total thrust that the thrust structure will ever feel, which is the total vacuum thrust. This proportionality factor between the weight of the thrust structure and the vacuum thrust I call f_{TSW}.

Now, in an effort to get the vacuum thrust in terms of the engine weight, I’m going to replace vacuum thrust with the total number of the engines, multiplied by the individual engine weight, multiplied by the vacuum thrust-to-weight ratio of the engine. That should give me a substitute value for the total vacuum thrust.

Now I can group some terms and simplify things a bit.

I need the gross weight of the vehicle. What should that be? Well, let’s assume that the vehicle has to have some initial overall thrust-to-weight ratio. If it’s sitting on the surface of the Earth and it’s meant to launch, then that value had better be greater than one! Actually, it better be a bit more than that or the vehicle’s not going to accelerate. So I’m going to assume that the overall vehicle thrust-to-weight ratio at liftoff is something that we’re going to specify in the design, and that by knowing that value and the initial thrust (not the vacuum thrust) of all the engines, we can calculate the gross weight of the vehicle.

Now something really important happens. The number of engines (n) is sitting in both the numerator and the denominator and cancels out of the expression altogether. I can’t tell you how happy I was to find this result when I first tried this derivation! I had had this hunch that phi would depend on the number of engines, but it turns out that it didn’t. It only depended on the vehicle’s initial thrust-to-weight ratio and the initial and vacuum thrust-to-weight ratios of the engine. And the thrust structure factor of course. But that cancellation means that phi becomes something that can be calculated for each engine type rather than for the number of engines.

Three values of thrust-to-weight and the thrust structure factor give you phi. Isn’t that amazing? If you’re doing a calculation for a rocket stage that operates entirely in space, then the initial thrust-to-weight value for the engine IS the same as the vacuum value, and you only need two thrust-to-weight values.

If you assumed that your thrust structure factor was negligible, which often isn’t such a bad assumption, then the expression would just be:

This actually wouldn’t be a bad place to stop, assuming that you knew the initial and vacuum values of engine thrust-to-weight. But if you didn’t know initial engine thrust-to-weight you can calculate it, or thrust-to-weight at the desired altitude (which would determine the back pressure) by using this variation of the expression. I find this version especially useful when I’m trying to do air-launch calculations, because the ambient pressure is neither vacuum nor sea-level.

When you use the expression this way, you can “hit” the engine for pressure losses in the atmosphere. If you know the vacuum thrust, and the exit area, and assuming you can calculate the ambient pressure by knowing what altitude you are at, you can figure out the initial thrust-to-weight ratio and use the expression effectively.

Now as I mentioned earlier, if you’re designing a vehicle that has other gross-mass-sensitive terms, like landing gear or wings or landing struts or whatever, don’t forget to tack them onto the end of this expression so that their effect makes its way back into the payload fraction calculation.

Like with lambda, here’s some examples to get you started:

The upper group of engines are assumed to start at sea level, and the lower group of engines are assumed to operate entirely in vacuum.

Kirk, can you plug Sea Dragon into these equations for us? That’s “abundant propellant” dejure.

No, but you can. I’ve shown you how.

Pingback: Selenian Boondocks » Blog Archive » Using Payload Fraction Expressions in an Example