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Chapter 4: Problem 108

Gases leaving the propulsion nozale of a rocket are modeled as flowing radially outward from a point upstream from the nozzle throat. Assume the speed of the exit flow, \(V_{e}\) has constant magnitude. Develop an expression for the axial thrust, \(T_{a}\) developed by flow leaving the nozzle exit plane. Compare your result to the one-dimensional approximation, \(T=m v_{e^{-}}\) Evaluate the percent error for \(\alpha=15^{\circ} .\) Plot the percent error versus a for \(0

Short Answer

Expert verified
For \(\alpha=15^{\circ}\), the percent error calculated between the axial thrust and the one-dimensional approximation of thrust is \(|1-\cos(15^{\circ})|\times 100\%\).

Step by step solution

01

Develop the expression for the axial thrust

The axial thrust, \(T_{a}\), can be calculated using the formula \(T_{a}=m V_{e} \cos(\alpha)\) where m is the mass flow rate of the gases, \(V_{e}\) is the exit flow velocity, and \(\alpha\) is the angle of flow with respect to the axial direction.
02

Compare the derived expression to the one-dimensional approximation

The one-dimensional approximation of thrust is given by the formula \(T=m v_{e^{-}}\). Here, since the speed of the exit flow, \(V_{e}\) is given as constant, \(V_{e}=v_{e^{-}}\). Hence, we can see that the only difference between the two expressions is the \(\cos(\alpha)\) term. The one-dimensional approximation assumes the gas flow is purely axial, i.e., \(\alpha=0\), for which \(\cos(0)=1\). Therefore, the two expressions would be equal at \(\alpha=0\).
03

Evaluate the percent error at \(\alpha=15^{\circ}\)

The percent error can be evaluated using the formula: \(\text{Percent error} = \frac{|T_{a} - T|}{T} \times 100\% = \frac{|m V_{e} \cos(\alpha) - m v_{e^-}|}{m v_{e^-}} \times 100\%\). Substituting \(\alpha=15^{\circ}\) and considering that \(V_{e}=v_{e^-}\), we obtain an error of \(|1-\cos(15^{\circ})|\times 100\%\).
04

Plot the percent error for the range \(0

For each angle \(\alpha\) in the range \(0<a<22.5^{\circ}\), evaluate the percent error using the formula derived in Step 3. The resulting graph will have percent error on the y-axis and \(\alpha\) on the x-axis.

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Most popular questions from this chapter

\( \mathrm{A}\) 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.

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