FIGURE 3-2. Specific impulse and exhaust velocity of an ideal rocket at optimum nozzle expansion as functions of the absolute chamber temperature T, and the molecular mass 9W for several values of k and p\/p2-
with an increase of the pressure ratio; this ratio increases when the value of the chamber pressure p¡ increases or when the exit pressure p2 decreases, corresponding to high altitude designs. The small influence of ^-values is fortuitous because low molecular masses are found in diatomic or monatomic gases, which have the higher values of k.
For comparing specific impulse values from one rocket system to another or for evaluating the influence of various design parameters, the value of the pressure ratio must be standardized. A chamber pressure of 1000 psia (6.894 MPa) and an exit pressure of 1 atm (0.1013 MPa) are generally in use today.
For optimum expansion p2 = Pi and the effective exhaust velocity c (Eq. 216) and the ideal rocket exhaust velocity are related, namely
and c can be substituted for v2 in Eqs. 3-15 and 3-16. For a fixed nozzle exit area ratio, and constant chamber pressure, this optimum condition occurs only at a particular altitude where the ambient pressure happens to be equal to the nozzle exhaust pressure p2. At all other altitudes c / v2.
The maximum theoretical value of the nozzle outlet velocity is reached with an infinite expansion (exhausting into a vacuum).
This maximum theoretical exhaust velocity is finite, even though the pressure ratio is infinite, because it represents the finite thermal energy content of the fluid. Such an expansion does not happen, because, among other things, the temperature of many of the working medium species will fall below their liquefaction or the freezing points; thus they cease to be a gas and no longer contribute to the gas expansion.
Example 3-2. A rocket operates at sea level (p = 0.1013 MPa) with a chamber pressure of pi = 2.068 MPa or 300 psia, a chamber temperature of Tx = 2222 K, and a propel-lant consumption of m = 1 kg/sec. (Let k = 1.30, R = 345.7 J/kg-K). Show graphically the variation of A, v, V, and M, with respect to pressure along the nozzle. Calculate the ideal thrust and the ideal specific impulse.
SOLUTION. Select a series of pressure values and calculate for each pressure the corresponding values of v, V, and A. A sample calculation is given below. The initial specific volume V\ is calculated from the equation of state of a perfect gas, Eq. 3^1:
Vx = RTJpi = 345.7 x 2222/(2.068 x 106) = 0.3714 m3/kg
In an isentropic flow at a point of intermediate pressure, say at px = 1.379 MPa or 200 psi, the specific volume and the temperature are, from Eq. 3-7,
Vx = V1(pl/px)l/k = 0.3714(2.068/1.379)I/I3 = 0.5072 m3/kg Tx - Tl(px/p]f~,)/k = 2222(1.379/2.068)°'38/l'3 = 2023 K
The calculation of the velocity follows from Eq. 3-16:
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