T0 and p0 designate the stagnation values of the temperature and pressure. Unlike the temperature, the stagnation pressure during an adiabatic nozzle expansion remains constant only for isentropic flows. It can be computed from
The area ratio for a nozzle with isentropic flow can be expressed in terms of Mach numbers for any points x and y within the nozzle. This relationship, along with those for the ratios T/T0 and p/po, is plotted in Fig. 3-1 for Ax = A, and Mx = 1.0. Otherwise,
As can be seen from Fig. 3-1, for subsonic flow the chamber contraction ratio Ai/A, can be small, with values of 3 to 6, and the passage is convergent. There is no noticeable effect from variations of k. In solid rocket motors the chamber area A\ refers to the flow passage or port cavity in the virgin grain. With supersonic flow the nozzle section diverges and the area ratio becomes large very quickly; the area ratio is significantly influenced by the value of k. The area ratio A2/A, ranges between 15 and 30 at M = 4, depending on the value of k. On the other hand, pressure ratios depend little on k whereas temperature ratios show more variation.
The average molecular mass of a mixture of gases is the sum of all the molar fractions «, multiplied by the molecular mass of each chemical species (m,-®?,-) and then divided by the sum of all molar mass fractions. This is further elaborated upon in Chapter 5. The symbol is used to avoid confusion with M for the Mach number. In many pieces of rocket literature is called molecular weight.
Example 3-1. An ideal rocket chamber is to operate at sea level using propellants whose combustion products have a specific heat ratio k of 1.30. Determine the required chamber pressure and nozzle area ratio between throat and exit if the nozzle exit Mach number is 2.40. The nozzle inlet Mach number may be considered to be negligibly small.
SOLUTION. For optimum expansion the nozzle exit pressure should be equal to the atmospheric pressure which has the value 0.1013 MPa. If the chamber velocity is small, the chamber pressure is equal to the total or stagnation pressure, which is, from Eq. 3-13, p0=p[\+i2(k-\)M2]
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