# Payload fraction derivation for vehicle with split delta-V (case #2)

Consider a vehicle carrying a payload that undertakes a first , then drops off that payload and undertakes a second in the same overall vehicle configuration (tanks, engines, payload handling, etc.). It carries the propellant for both maneuvers, but only on the first maneuver does it have the added mass of the payload. This situation might be representative of:

1. a reusable lunar lander, based in lunar orbit, fully fueled by propellant delivered from the Earth and loaded with a payload from Earth, which then lands and unloads its payload, then returns to lunar orbit with nearly all its propellant expended.

2. a space tug that departs for geosynchronous orbit carrying a satellite, then returning to a low-Earth orbit for refueling and reloading.

Consider the mass of propellant ( ) for the first and the mass of the propellant ( ) for the second to be distinct amounts, carried in common tankage.

First, define the mass conditions at the beginning and end of :

(1) Alternatively, and just as importantly, the conditions bracketing can be described in terms of an initial mass: This expression can be conveniently rearranged to yield the propellant mass consumed by the vehicle in : In a similar manner, we define the mass conditions at the beginning and end of :

(2) We can also express the conditions bracketing in another way, in terms of initial mass:    The propellant mass consumed by the vehicle in can also be expressed in a manner analogous to :  Now we are positioned to calculate the total propellant load: substituting the definitions for and  collecting terms and simplifying  Now let us define the vehicle’s “dry” mass entirely in terms of initial-mass-sensitive ( ), propellant-mass-sensitive ( ), and payload-mass-sensitive ( ) mass terms. This is a substantial simplification, but it should do for now.  substituting the definition of the vehicle’s mass in we collect terms related to the initial mass on the left hand side    now substituting and collecting terms further simplifying With all terms relating only to initial mass and payload mass, a general expression for payload fraction can at last be defined: We can compare this to our previous expression for payload fraction by assuming that and simplifying the result. and see that for the same assumptions they are identical. A bit more insight can be obtained by remembering that the final mass fraction (FMF) is simply the inverse of the mass ratio, and that the propellant mass fraction (PMF) is one minus the final mass fraction:   Remember that this is just for the case where and thus .

The following two tabs change content below. #### Kirk Sorensen

MS, nuclear engineering, University of Tennessee, 2014, Flibe Energy, president, 2011-present, Teledyne Brown Engineering, chief nuclear technologist, 2010-2011, NASA Marshall Space Flight Center, aerospace engineer, 2000-2010, MS, aerospace engineering, Georgia Tech, 1999  1. TheRadicalModerate says: