Payload Fraction Calculation for Reusable Vehicles
Mar 1st, 2010 by Kirk Sorensen
This is my last update to the payload fraction calculation, I promise.
When I was learning how to use mass-estimating relationships (MERs) at Georgia Tech, our focus was on reusable launch vehicles, and most of our MERs came from NASA Langley, where my professor had once worked. When it came to much of the reusability aspects of the spacecraft, the MER tended to depend on the entry or landing mass of the spacecraft rather than the gross mass or the propellant mass, as I have previously defined.
Systems like thermal protection, wings, aerosurfaces, and landing gear all tended to scale with entry or landing mass. So to capture these effects in order to help calculate performance of reusable vehicles, I introduce another non-dimensional term. I use the letter epsilon to describe the entry-mass-sensitive mass items divided by the entry mass. The difference between the entry mass and the dry mass depends mostly on whether or not the payload is intended to return with the spacecraft, so I include a jettison factor (fjett) whose value is one if all the payload is jettisoned and zero if none of the payload is jettisoned before entry.
I employ the now-familiar substitution for propellant mass:
And simplify:
Which as you can see is the same as the previous expressions if epsilon is zero.
A simple way to understand this expression is to think about it in terms of one. One is the most payload fraction you could have–if all of your spacecraft is payload. But some fraction has to be propellant, according to the rocket equation. So you start out with your final mass fraction (FMF). From that point, which will always be less than one, you subtract lambda times the propellant mass fraction. Then you subtract your phi term, which depends mostly on your engines. Finally you subtract epsilon times your final mass fraction. If you have anything left over, you have a payload fraction. If you were going to jettison your payload before reentry, then the denominator gets a little smaller than one and your payload fraction improves a bit. But it can never improve a payload fraction that is less than zero.
If your payload fraction is less than zero, then you had better go change something to clean things up. You better use a better Isp to improve final mass fraction, or better tankage or propellants to improve lambda, or better engines to improve phi, or better TPS or wings or landing gear to improve epsilon. Because if the numerator of the payload fraction is less than zero, you’ve got no reason to build your rocket.
Dear Mr Sorenson,
Thank-you for this very informative series! I’ve passed it on to a friend of mine who is doing some studies on various propellant combinations for a given mission, and I think I’ll pass it on to scott Lowther of UpShip.com and Winchell Chung of the “Atomic Rockets” website!
A small problem is that the public-access computers here at Bexleyheath Library can’t show your equations, and I have to use a local cyber-cafe…which is closed as the two prpprietors take a holiday, meaning I can’t really appreciate the last episode…GRRR!
I’d imagine that most people reading this will have this data already, but how about a table of exhaust velocities, structural factors, etc, just to spoon-feed me?
Thanks Again,
Grif Ingram
Grif, that Scott Lowther web site is Up-Ship.com.
Kirk, could you put some numbers in so we can see a comparison between feasible and non-feasible SSTO designs.
Bob Clark
Fascinating reading! Most of it is way above my pay grade, but I can appreciate the equations.
Grif Ingram, thanks for offering to clue me in to this blog. However I already stumbled over it.
I agree with Winch above – excellent stuff. It tends to confirm my impression that absent major materials advances, SSTO is a nonstarter, requiring extreme designs that are marginal to work at all, let alone with the robustness for operational use.
My guess is that the first fully reusable orbital craft will be three stages to orbit, on extended SpaceShipTwo lines – airbreathing first stage, spaceplane second stage, and the orbiter, allowing each stage to be more conservative in requirements and engineering margins.