|Thermodynamics and Propulsion|
We have considered the overall performance of a rocket and seen that is directly dependent on the exit velocity of the propellant. Further, we have used the steady flow energy equation to determine the exhaust velocity using the combustion chamber conditions and the nozzle exit pressure. In this brief section, we will apply concepts from thermodynamics and fluids to relate geometrical (design) parameters for a rocket nozzle to the exhaust velocity.
We will make the following assumptions:
and this expression for an isentropic flow,
where will be used to refer to a chamber upstream of the duct. The first law for steady flow from can be written in terms of Mach number as
where it has been assumed that the velocity at is small. From these latter two equations it follows that
Conservation of mass states
All of these expressions can be combined to produce
The above equation relates the flow area, the mass flow, the Mach number and the stagnation conditions (conditions at ). It is frequently rewritten in a non-dimensional form by dividing through by the value at (where the area at is ):
This equation takes a form something like that shown in Figure 14.4.
We can use these equations to rewrite our expression for rocket thrust in terms of nozzle geometry ( ), and exit area, . From before,
Evaluating the mass flow at the throat, where ,
The other terms in the thrust equation can be written in terms of chamber and exit conditions:
We can now specify geometry ( and ) to determine , and then use with the combustion chamber conditions to determine thrust and Isp. UnifiedTP