## Problems

(1) Consider the nozzle represented in Fig. 4.35. An axisymmetric bell nozzle with zero divergence angle and the dimensions (in mm) indicated in the figure is assumed. There is zero ambient pressure at the nozzle exit.

The calculation can be based on the approximation of a steady, one-dimensional, isentropic flow of a thermally and calorically perfect gas with an average ratio of 1.30 of the specific heats and a gas constant of 330 m2/(s2 K). The stagnation temperature can be taken equal to the flame

Figure 4.35 Axisymmetric bell nozzle. (Dimensions are in mm.)

temperature of3200 K in the combustion chamber. The stagnation pressure, which can be taken equal to the chamber pressure, is 800 N/cm2.

(a) Find the Mach number at the nozzle exit.

(b) Find the pressure, density, temperature, and velocity at the nozzle throat.

(c) Find the mass flow rate, the theoretical thrust (including the "pressure thrust"), and the specific impulse.

(d) Graph as functions of the axial coordinate the Mach number, the temperature, and the pressure.

(2) Consider a solid-propellant motor with the grain cross section indicated in Fig. 4.36. This cross section can be assumed to be uniform over the length of 1500 mm of the grain. The density of the propellant is 1700 kg/m3. The nozzle throat area is 4000 mm2, the nozzle exit area 0.300 m2. The motor operates in vacuum. The flame temperature in the gas passage perforation is 3300 K and can be taken equal to the stagnation temperature of the flow.

A perfect gas is assumed with ratio of the specific heats of 1.25 and a gas constant of 280 m2 / (s2 K).

(a) Given a constant burn rate of 3.5 mm/s of the propellant, show graphically the inner contour of the propellant at 0,20, 40, 60, and 80 seconds after ignition and find the mass flow rate at these times.

(b) Find the stagnation pressure (= pressure in the center perforation) at these times.

(c) Find the thrust (including the "pressure thrust") at these times.

(d) From the burn rate of 3.5 mm/s at ignition and a burn rate pressure exponent of 0.6, find improved values for the burn rates at 20, 40, 60, and 80 seconds by using the results obtained in (b). With these improved numbers, recalculate the thrust at these times.

(3) For the rocket motor defined in Problem 1 and using the results obtained there, find

(a) The ambient pressure at which there will be a normal shock at the nozzle exit plane.

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