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 initial-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 initial-mass-sensitive term:

Then I make the assumption that this term consists only of two thingsâ€”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 initial-mass-dependent. Or if you have a winged horizontally-launched rocket on Earth, then the wings and landing gear are initial-mass-dependent. So alter this assumption according to your needs.

Then I multiply the expression by one in the form of the acceleration due to gravity (g, 9.81 m/s^2) over itself, which is then distributed across the terms of the numerator and denominator in order to convert masses to weights, which are forces.

This will be useful since many parameters of interest are thrust/weight ratios, which are non-dimensional ratios of forces. Youâ€™ll see why I did this in just a second.

Next 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, Iâ€™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.

Next I’ll need the initial weight of the stage. What should that be? Well, letâ€™s assume that the stage has to have some initial overall thrust-to-weight ratio. That initial thrust/weight had better be greater than one if itâ€™s sitting on the surface of the Earth and meant to launch! If not then the vehicleâ€™s not going to accelerate. So Iâ€™ll make the assumption that the overall thrust-to-weight ratio of the stage at engine ignition is something that weâ€™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 initial weight of the stage.

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 canâ€™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 didnâ€™t. It only depended on the stageâ€™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. Isnâ€™t that amazing? If youâ€™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 isnâ€™t such a bad assumption, then the expression would just be:

This actually wouldnâ€™t be a bad place to stop, assuming that you knew the initial and vacuum values of engine thrust-to-weight. But if you didnâ€™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 Iâ€™m trying to do launch vehicle stage calculations, because they’re not always at vacuum or sea-level.

When you use the expression this way, you can â€œhitâ€ 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 youâ€™re designing a vehicle that has other initial-mass-sensitive terms, like landing gear or wings or landing struts or whatever, donâ€™t forget to tack them onto the end of this expression so that their effect makes its way back into the payload fraction calculation.

Let’s calculate phi for two different rocket cases. Suppose we were trying to compare the performance of a pair of upper stages intended to inject a payload from low Earth orbit onto a trans-Mars injection trajectory, and we needed that injection stage to have an overall thrust/weight ratio at ignition of 0.5. That would imply that the stage would impart an acceleration of 0.5g to any crew that might be onboard when the engines start, but that acceleration would increase as the stage consumed its propellant. Let’s also assume a thrust-structure-factor of 0.003.

The RL10 family of engines have various vacuum T/W ratios. The A4 variant has a vacuum T/W of 60.3, the B2 has 37.3, and the RL-60 engine variant has a projected vacuum T/W of 59.1. Let’s assume a vacuum T/W of 50 for the LH2/LOX engine:

Then consider a nuclear thermal reactor, with 15,000 lbf thrust and an individual engine weight of 5,000 lbm, yielding a vacuum T/W of 3:

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.

#### Latest posts by Kirk Sorensen (see all)

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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.

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Thanks for the great posts Kirk. I was wondering how you calculated/found the thrust structure factor of .003 in the example table?

Also, along those same lines, where would I look to find the ullage factors and MERs that you use in the other posts? You mentioned one of the tables was from langley, but I was hoping to have something that I could source. I tried doing a google search but to no avail. Are there books with all of these tables?

Thanks!