In this section we discuss semiempirical correction factors that have been used to estimate the test performance data from theoretical, calculated performance values. An understanding of the theoretical basis also allows correlations between several of the correction factors and estimates of the influence of several parameters, such as pressure, temperature, or specific heat ratio.
The energy conversion efficiency is defined as the ratio of the kinetic energy per unit of flow of the actual jet leaving the nozzle to the kinetic energy per unit of flow of a hypothetical ideal exhaust jet that is supplied with the same working substance at the same initial state and velocity and expands to the same exit pressure as the real nozzle. This relationship is expressed as where e denotes the energy conversion efficiency, and v2 the velocities at the nozzle inlet and exit, and cpTx and cpT2 the respective enthalpies for an ideal isentropic expansion. The subscripts a and i refer to actual and ideal conditions, respectively. For many practical applications, -» 0 and the square of the expression given in Eq. 3-16 can be used for the denominator.
The velocity correction factor is defined as the square root of the energy conversion efficiency «Je. Its value ranges between 0.85 and 0.99, with an average near 0.92. This factor is also approximately the ratio of the actual specific impulse to the ideal or theoretical specific impulse.
The discharge correction factor is defined as the ratio of the mass flow rate in a real rocket to that of an ideal rocket that expands an identical working fluid from the same initial conditions to the same exit pressure (Eq. 2-17).
The value of this discharge correction factor is usually larger than 1 (1.0 to 1.15); the actual flow is larger than the theoretical flow for the following reasons:
1. The molecular weight of the gases usually increases slightly when flowing through a nozzle, thereby changing the gas density.
2. Some heat is transferred to the nozzle walls. This lowers the temperature in the nozzle, and increases the density and mass flow slightly.
3. The specific heat and other gas properties change in an actual nozzle in such a manner as to slightly increase the value of the discharge correction factor.
4. Incomplete combustion can increase the density of the exhaust gases.
The actual thrust is usually lower than the thrust calculated for an ideal rocket and can be found by an empirical thrust correction factor £>:
Values of ÇF fall between 0.92 and 1.00 (see Eqs. 2-6 and 3-31). Because the thrust correction factor is equal to the product of the discharge correction factor and the velocity correction factor, any one can be determined if the other two are known.
Example 3-7. Design a rocket nozzle to conform to the following conditions:
Chamber pressure 20.4 atm = 2.068 MPa
Atmospheric pressure 1.0 atm
Chamber temperature 2861 K
Mean molecular mass of gases 21.87 kg/kg-mol
Ideal specific impulse 230 sec (at operating conditions)
Specific heat ratio 1.229
Desired thrust 1300 N
Determine the following: nozzle throat and exit areas, respective diameters, actual exhaust velocity, and actual specific impulse.
SOLUTION. The theoretical thrust coefficient is found from Eq. 3-30. For optimum conditions p2 — By substituting k = 1.229 and p\/p2 — 20.4, the thrust coefficient is CF = 1.405. This value can be checked by interpolation between the values of CF
obtained from Figs. 3-7 and 3-8. The throat area is found using = 0.96, which is based on test data.
A, = F/ttpCpPi) = 1300/(0.96 x 1.405 x 2.068 x 106) = 4.66 cm2
The throat diameter is then 2.43 cm. The area expansion ratio can be determined from Fig. 3-5 or Eq. 3-25 as e = 3.42. The exit area is
The exit diameter is therefore 4.50 cm. The theoretical exhaust velocity is v2 = Isgo = 230 x 9.81 = 2256 m/ sec
By selecting an empirical velocity correction factor such as 0.92 (based on prior related experience), the actual exhaust velocity will be equal to
Because the specific impulse is proportional to the exhaust velocity, its actual value can be found by multiplying the theoretical value by the velocity correction factor f„.
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