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Chapter 11: Problem 49

The gas entering a rocket nozzle has a stagnation pressure of \(1500 \mathrm{kPa}\) and a stagnation temperature of \(3000^{\circ} \mathrm{C}\). The rocket is traveling in the still Standard Atmosphere at \(30,000 \mathrm{m}\). Find the throat and exit area for a flow rate of \(10 \mathrm{kg} / \mathrm{s}\). Assume \(k=1.35, R=\) \(287.0 \mathrm{N} \cdot \mathrm{m} / \mathrm{kg} \cdot \mathrm{K} .\) The gas is perfectly, expanded to the ambient pressure.

Short Answer

Expert verified
First, convert the stagnation temperature to Kelvin. Then, use the flow rate equation to calculate the throat area, and find the exit pressure from standard tables. Finally, plug all these values into the equation to calculate the exit area.

Step by step solution

01

Conversion of Units

First, convert the given temperature from degree Celsius to Kelvin using the formula \(T(K) = T(°C) + 273.15\). Thus, the stagnation temperature, \(T_0\) will be \(3000°C + 273.15 = 3273.15 K\).
02

Calculate Throat Area

Apply the following formula to find the throat area, \(A^*\), which is given by, \[A^* = \frac{m \cdot R \cdot T_0}{P_0 \sqrt{k}}\] where \(m = 10 kg/s\) (mass flow rate), \(P_0 = 1500 kPa\) (stagnation pressure), \(R = 287 J/kg-K\) (universal gas constant), \(T_0\) is the stagnation temperature and \(k = 1.35\). Calculate \(A^*\) using these values.
03

Find the Exit Pressure

The exit pressure is the pressure of the atmosphere at 30,000m altitude. This can be found in standard tables, let's denote it as \(P_e\)
04

Calculate Exit Area

Substitute the values into the following equation to find the exit area - \[ A_e = \frac{2}{k+1} \left( \frac{P_0}{P_e} \right) ^{\frac{1}{k-1}} A^* \] where \(P_e\) is the exit pressure and \(A^*\) is the throat area found in Step 2.

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Most popular questions from this chapter

The static pressure to stagnation pressure ratio at a point in a gas flow field is measured with a Pitot-static probe as being equal to \(0.6 .\) The stagnation temperature of the gas is \(20^{\circ} \mathrm{C}\). Determine the flow speed in \(\mathrm{m} / \mathrm{s}\) and the Mach number if the gas is air. What error would be associated with assuming that the flow is incompressible?

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