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Mass fraction

In aerospace engineering, the mass fraction is an important measure of a rocket's efficiency. For a given target orbit, a rocket's mass fraction is the portion of the rocket's pre-launch mass (fully fueled) that does not reach orbit. In the cases of a single stage to orbit vehicle the mass fraction is simply the fuel mass divided by the mass of the full spaceship, but with a rocket employing staging, which is the vast majority of them, the mass fraction is higher because parts of the rocket itself are dropped off enroute. Mass fractions are typically around 0.8 to 0.9, with lower numbers being better.

For example, the complete Space Shuttle system has:

  • weight at liftoff: 4,500,000 lb (2,040,000 kg)
  • weight at end of mission: 230,000 lb (104,000 kg), and
  • maximum cargo to orbit: 63,500 lb (28,800 kg)

Given these numbers, the mass fraction is 1 - (293,500 / 4,500,000) = 0.935 or perhaps a little less because of the fuel brought to orbit for use when returning: this may not have been counted as cargo, in which case the figure 293,500 should be a little higher.

A lower mass fraction for the rocket means that it uses fuel efficiently, reserving a larger portion of its mass as payload. Without the benefit of staging, SSTO designs are typically designed for mass fractions around 0.9. Staging increases the mass fraction, which is one of the reasons SSTO's appear difficult to build.

For individual stages, however, a higher mass fraction is better, meaning that there is less non-propellent mass.

The mass fraction plays an important role in the rocket equation:

Δv = - veln(mf / m0)

Where mf / m0 is the ratio of final mass to initial mass (i.e., one minus the mass fraction), Δv is the change in the vehicle's velocity as a result of the fuel burn and ve is the effective exhaust velocity (assuming a perfectly efficient nozzle).

There are two different ways in which the term specific impulse is used. In one,

ve = Isp

When this is called specific impulse, the units might be given as newton seconds per kilogram (1 N新/kg = 1 m/s) or as lbf新/lb (1 lbf新/lb = 32.174 ft/s) or as kgf新/kg (1 kgf新/kg). However, because of the usage below, many just avoid the term specific impulse with this meaning.

Since those pounds or kilograms were often just canceled out, even though one was a unit of mass and the other of force, with the result called merely seconds, the formula has also been rewritten by the insertion of a gn factor to make the dimensional analysis result in units of seconds:

ve = gnIsp

where Isp is the fuel's specific impulse in seconds and gn is the standard acceleration of gravity (note that this is not the local acceleration of gravity).

In theory, a mass fraction of less than 0.5 would allow a spacecraft to carry enough fuel to orbit that it could make an entirely powered landing, avoiding the need for extensive aerobraking, and the difficulties associated with the intense heat generated.

See also Payload fraction.

12-19-2008 14:25:18
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