What is the relation between exhaust velocity and specific impulse?
The relation between exhaust velocity and specific exhaust is one of direct
proportionality.
We can define thrust as follows:
F = m * [Ve + Ae * (Pe – Pa)]
Where :
(1)
F = Thrust
m = propellant mass flow rate
Ve = propellant exhaust velocity
Ae = nozzle exit area
Pe = gas pressure at nozzle exit
Pa = ambient pressure
It can thus be deduced that the thrust of any rocket engine on earth increases as
it gains altitude, attaining a maximum when there is no more atmosphere ( when
Pa = 0 ).
The effective exhaust velocity, C, can then be defined as:
C ≡ Ve + Ae * (Pe – Pa)
(2)
Which yields, when put into equation (1) :
F=m*C
(3)
The engine specific impulse is found with the following equation:
Isp = F / (m * g)
Where :
(4)
Isp = Engine specific impulse
F = Engine thrust
m = Propellant mass flow rate
g = gravity
The engine specific impulse thus depends not only on the propellant weight flow
rate (mass * gravity yields weight), but also on the engine thrust, which itself
depends on the exhaust velocity of the fuel. By combining equations (3) and (4),
we get:
Isp = m * C / (m * g)
(5)
Simplifying the equation, we get :
Isp = C / g
(6)
This gives the expected result, that the engine specific impulse is directly
proportional to the effective exhaust velocity. By examining the definition of the
effective exhaust velocity in equation (2), the link between the propellant
exhaust velocity, Ve and the engine specific impulse becomes readily apparent. It
can thus be concluded that one way to increase the specific impulse of an
engine, and thus its effectiveness, is to increase the exhaust velocity of its
propellant.