## Exhaust Velocity

The effective exhaust velocity as defined by Eq. 2-6 applies to all rockets that thermodynamically expand hot gas in a nozzle and, indeed, to all mass expulsion systems. From Eq. 2-14 and for constant propellant mass flow this can be modified to c = v2 + {p2-p3)A2/m (2-16)

Equation 2-6 shows that c can be determined from thrust and propellant flow measurements. When p2 = Pi, the effective exhaust velocity c is equal to the average actual exhaust velocity of the propellant gases v2. When p2 / p3 then c Â± v2. The second term of the right-hand side of Eq. 2-16 is usually small in relation to v2; thus the effective exhaust velocity is usually close in value to the actual exhaust velocity. When c = v2 the thrust (from Eq. 2-14) can be rewritten as

The characteristic velocity has been used frequently in the rocket propulsion literature. Its symbol c*, pronounced "cee-star," is defined as c*=p]A,/m (2-18)

The characteristic velocity c* is used in comparing the relative performance of different chemical rocket propulsion system designs and propellants; it is easily determined from measured data of m, pu and A,. It relates to the efficiency of the combustion and is essentially independent of nozzle characteristics.

However, the specific impulse Is and the effective exhaust velocity c are functions of the nozzle geometry, such as the nozzle area ratio A2/At, as shown in Chapter 3. Some values of Is and c* are given in Tables 5-4 and 5-5.

Example 2-2. The following measurements were made in a sea level test of a solid propellant rocket motor:

Burn duration 40 sec

Initial mass before test 1210 kg

Mass of rocket motor after test 215 kg

Average thrust 62,250 N

Chamber pressure 7.00 MPa

Nozzle exit pressure 0.070 MPa

Nozzle throat diameter 0.0855 m

### Nozzle exit diameter 0.2703 m

Determine m, v2, c*, c, and Is at sea level, and c and Is at 1000 and 25,000 m altitude. Assume an invariant thrust and mass flow rate and negligible short start and stop transients.

SOLUTION. The mass flow rate m is determined from the total propellant used (initial motor mass - final motor mass) and the burn time.

The nozzle areas at the throat and exit are

A, = tzD2/4 = n x 0.08 5 52/4 = 0.00574 m2 A2 = JZD2/4 = tzx 0.27032/4 = 0.0574 m2

Equation 2-14 is to be solved for v2, the actual average exhaust velocity.

= 62,250/24.9 - (0.070 - 0.1013)106 x 0.0574/24.9 = 2572 m/ sec

The characteristic velocity and effective exhaust velocity are found from Eqs. 2-6 and 218 for sea level conditions.

c* = plAt/m = 7.00 x 106 x 0.00574/24.9 = 1613 m/ sec Is = F/rhgQ = 62,250/(24.9 x 9.81) = 255 sec c = Isg0 = 255 x 9.81 = 2500 m/ sec

For altitudes of 1000 and 25,000 m the ambient pressure (see Appendix 2) is 0.0898 and 0.00255 MPa. From Eq. 2-16 the altitude values of c can be obtained.

At 1000 m altitude, c = 2572 + (0.070 - 0.0898) x 106 x 0.0 574/24.9 = 2 527 m/ sec Is = 2527/9.81 = 258 sec

At 25,000 m altitude, c = 2572 + (0.070 - 0.00 2 5 5) x 106 x 0.0 574/24.9 = 2727 m/ sec Is = 2727/9.80 = 278 sec