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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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