Sunday, December 18, 2011

Specific Impulse

Specific Impulse is one of the two key method of measuring the performance of a rocket system (the other being mass ratio). Understanding Specific Impulse is one of the most important things to know to understand orbital rockets.

Just like one of the measures of the efficiency of a car is miles per hour, rockets have their own measure of efficiency, Specific Impulse (or Isp), measured in seconds.

Specific Impulse (or Isp) is the number of seconds that one unit weight of propellant will produce one unit force of thrust. The equation for calculating Isp derived from its definition is:

Since there is a unit of force in both the numerator and denominator, they cancel out to leave just seconds.

In both American Standard and Metric, the same formula applies. In metric, you can
just put the thrust force in Newtons and convert the mass of the propellant into its
weight by multiplying by the gravitational acceleration (g = 9.80665 m/s^2) because
Force = mass * acceleration:

Knowing that a propellant can provide 200 seconds of 1 pound of thrust with one
pound of propellant, one can also determine that the propellant will produce
400 seconds of 1/2 pound of thrust with one pound of propellant.

Sometimes, Metric users will just use Mass instead of Weight and so the
value of Metric Specific Impulse is off by a constant factor of 9.8. You'll
see that value used in the literature as well.

The rocket force equation is:

 Force  = the thrust of the rocket
 mdot   = the mass flow rate through the rocket
 Ve     = the exhaust velocity out of the nozzle exit
 Pe     = the pressure at the nozzle exit
 po     = the ambient operating pressure
 Ae     = the area at the nozzle exit
Since Force is part of the Isp equation, we can subsitute this force equation into the Isp equation to show some important relationships between the factors that produce the thrust of a rocket and its Isp:

 Isp = the specific impulse
 mdot = the mass flow rate
 Ve   = the exhaust velocity
 Pe   = the pressure at the nozzle exit
 po   = the ambient operating pressure
 Ae   = the area of the nozzle exit
 g    = the acceleration of gravity
As can be seen, the pressure of the ambient operating environment affects the Isp of the rocket.

The following table shows some representative values for different propellant combinations and their vacuum Isp.

But, a rocket motor behaves differently, having different Isp at different altitudes and ambient pressures. The following graph shows the Isp characteristic of a rocket through different atmospheric conditions. It is for a LOX-Isopropyl Alchohol motor with a 1.65:1 mixture ratio, a 3.12:1 expansion ratio and a chamber pressure of 250 PSI (with 90% combustion efficiency).

As can be seen, the Isp varies greatly throughout the flight regime. At sea level, the Isp is about 213 seconds, but by the time it is at 25,000 feet, the Isp has gone up to about 253 seconds. This is with no change of propellant flow characteristics or change of combustion pressure. This is solely due to the effect of the nozzle throughout its flight regime.

Therefore, since a rocket in flight spends more of its flight time at higher altitude, it will see a higher average Isp throughout its flight than its sea level value. This is an important thing to understand and exploit in designing rocket boosters.

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