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Chapter 4: Problem 181

\( \mathrm{A}\) small solid-fuel rocket motor is fired on a test stand. The combustion chamber is circular, with \(100 \mathrm{mm}\) diameter. Fuel, of density \(1660 \mathrm{kg} / \mathrm{m}^{3},\) burns uniformly at the rate of \(12.7 \mathrm{mm} / \mathrm{s}\). Measurements show that the exhaust gases leave the rocket at ambient pressure, at a speed of \(2750 \mathrm{m} / \mathrm{s}\). The absolute pressure and temperature in the combustion chamber are \(7.0 \mathrm{MPa}\) and \(3610 \mathrm{K},\) respectively. Treat the combustion products as an ideal gas with molecular mass of 25.8 . Evaluate the rate of change of mass and of linear momentum within the rocket motor. Express the rate of change of linear momentum within the motor as a percentage of the motor thrust.

Short Answer

Expert verified
The rate of change of mass and of linear momentum within the rocket motor can be calculated using the given data and formulas pertaining to ideal gas behavior, thrust, and momentum. The rate of change of linear momentum can also be expressed as a percentage of the motor thrust using basic percentage calculations. The detailed calculations involve use of mathematical formulas and the data provided in the exercise.

Step by step solution

01

Compute the rate of change of mass

To compute the rate of change of mass, use the relation of rate of change with respect to area, density, and velocity. The area of the combustion chamber is given by the formula \( A = \pi \left(\frac{D}{2}\right)^2 \) where \( D = 100 \, \text{mm} \). So we have \( A = \pi \left(\frac{100 \times 10^{-3}}{2}\right)^2 \, \text{m}^{2} \). The rate of burning fuel is \( 12.7 \, \text{mm/s} = 12.7 \times 10^{-3} \, \text{m/s} \) and the fuel density is \( 1660 \, \text{kg/m}^{3} \). The rate of change of mass (\( \dot{m} \)) can be calculated using the formula \( \dot{m} = \rho \, v \, A \)
02

Calculate the rate of change of linear momentum

The rate of change of linear momentum within the rocket can be determined by multiplying the mass flow rate calculated previously with the exhaust gas velocity. The velocity given in the exercise is 2750 m/s. So the rate of change of linear momentum \( \dot{p} \) can be calculated as \( \dot{p} = \dot{m} \, v \)
03

Evaluate the motor thrust

Thrust for this rocket motor can be calculated using the equation of thrust for an ideal gas which is \( T = \dot{m} \, v + (P_c - P_0) \, A_e \), where \( P_c \) is the absolute pressure in the combustion chamber, \( P_0 \) is the ambient pressure and \( A_e \) is the exit area. Given that exhaust gases leave the rocket at ambient pressure, \( P_c = P_0 \), hence \( T = \dot{m} \, v \)
04

Express rate of change of momentum as percentage of thrust

This can be computed by dividing the rate of linear momentum by the thrust and then multiplying the result by 100 to convert it to a percentage. The formula for this computation is \( \text{Percentage} = \left(\frac{\dot{p}}{T}\right) \times 100 \)

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Most popular questions from this chapter

A rocket sled accelerates from rest on a level track with negligible air and rolling resistances. The initial mass of the sled is \(M_{0}=600 \mathrm{kg}\). The rocket initially contains \(150 \mathrm{kg}\) of fuel. The rocket motor burns fuel at constant rate \(m=15 \mathrm{kg} / \mathrm{s}\). Exhaust gases leave the rocket nozzle uniformly and axially at \(V_{c}=2900 \mathrm{m} / \mathrm{s}\) relative to the nozzle, and the pressure is atmospheric. Find the maximum speed reached by the rocket sled. Calculate the maximum acceleration of the sled during the run.

\(\mathrm{A}\) -home-made" solid propellant rocket has an initial mass of \(20 \mathrm{lbm} ; 15 \mathrm{lbm}\) of this is fuel. The rocket is directed vertically upward from rest, burns fuel at a constant rate of \(0.5 \mathrm{lbm} / \mathrm{s},\) and ejects exhaust gas at a speed of \(6500 \mathrm{ft} / \mathrm{s}\) relative to the rocket. Assume that the pressure at the exit is atmospheric and that air resistance may be neglected. Calculate the rocket speed after 20 s and the distance traveled by the rocket in 20 s. Plot the rocket speed and the distance traveled as functions of time.

\( \mathrm{A}\) jet of oil \((\mathrm{SG}-0.8)\) strikes a curved blade that turns the fluid through angle \(\theta-180^{\circ}\). The jet area is \(1200 \mathrm{mm}^{2}\) and its speed relative to the stationary norzle is \(20 \mathrm{m} / \mathrm{s}\). The blade moves toward the nozzle at \(10 \mathrm{m} / \mathrm{s}\). Determine the force that must be applied to maintain the blade speed constant.

Water is discharged at a flow rate of \(0.3 \mathrm{m}^{3} / \mathrm{s}\) from a narrow slot in a 200 -mm-diameter pipe. The resulting horizontal twodimensional jet is \(1 \mathrm{m}\) long and \(20 \mathrm{mm}\) thick, but of nonuniform velocity, the velocity at location (2) is twice that at location (1). The pressure at the inlet section is \(50 \mathrm{kPa}\) (gage). Calculate (a) the velocity in the pipe and at locations (1) and (2) and (b) the forces required at the coupling to hold the spray pipe in place. Neglect the mass of the pipe and the water it contains.

\( \mathrm{A}\) rocket sled has mass of \(5000 \mathrm{kg}\). including \(1000 \mathrm{kg}\) of fuel. The motion resistance in the track on which the sled rides and that of the air total \(k U,\) where \(k\) is \(50 \mathrm{N} \cdot \mathrm{s} / \mathrm{m}\) and \(U\) is the speed of the sled in \(\mathrm{m} / \mathrm{s}\). The exit speed of the exhaust gas relative to the rocket is \(1750 \mathrm{m} / \mathrm{s}\), and the exit pressure is atmospheric. The rocket burns fuel at the rate of \(50 \mathrm{kg} / \mathrm{s}\). (a) Plot the sled speed as a function of time. (b) Find the maximum speed. (c) What percentage increase in maximum speed would be obtained by reducing \(k\) by 10 percent?
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