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Chapter 11: Problem 12

A convergent-divergent nozzle originally designed to give an exit Mach number of \(1.8\) with air is used with argon \((\gamma=5 / 3)\). What is the ratio of the entry stagnation pressure to the exit pressure when a normal shock is formed just inside the nozzle exit?

Short Answer

Expert verified
The pressure ratio \(\frac{P_0}{P_e}\) is 7.2.

Step by step solution

01

Understand the given parameters

The exit Mach number, originally designed for air: \(M_e = 1.8\)Specific heat ratio of argon: \(\gamma_{\text{argon}} = \frac{5}{3} = 1.67\)
02

Determine the Mach number after the shock

For a normal shock, the relationship between the Mach numbers before and after the shock is given by:\[M{'_2} = \sqrt{\frac{(\gamma - 1)M^2 + 2}{2\gamma M^2 - (\gamma - 1)}}\]Substituting the values, \(M_2 = \sqrt{\frac{(1.67 - 1)1.8^2 + 2}{2\cdot1.67\cdot1.8^2 - (1.67 - 1)}} = \sqrt{\frac{0.67\cdot1.8^2 + 2}{2\cdot1.67\cdot1.8^2 - 0.67}} = \sqrt{\frac{2.1708}{9.024}} \approx 0.494\)
03

Use the normal shock relations to find the pressure ratio

The pressure ratio across a normal shock is given by:\[\frac{P_2}{P_1} = 1 + \frac{2\gamma}{\gamma + 1} (M_1^2 - 1)\]Substituting the given values,\(\frac{P_2}{P_1} = 1 + \frac{2\cdot1.67}{1.67 + 1} (1.8^2 - 1) = 1 + \frac{3.34}{2.67} (3.24 - 1) = 1 + 1.25 \cdot 2.24 \approx 4.8\)
04

Calculate stagnation pressure to exit pressure ratio

The stagnation-to-static pressure ratio behind the shock is influenced by the isentropic relations. Therefore,\(\frac{P_0}{P_2} = \left(1 + \frac{\gamma - 1}{2} M_2^2 \right)^{\frac{\gamma}{\gamma - 1}}\), and\(\frac{P_0}{P_e} = \left(1 + \frac{1.67 - 1}{2} \times 0.494^2 \right)^{\frac{1.67}{0.67}} = \left(1 + 0.333 \times 0.244 \right)^{2.49} = \left(1 + 0.08 \right)^{2.49} \approx 1.200^{2.49} \approx 1.5\)Finally, combining both results, \(\frac{P_0}{P_e} = 4.8 \times 1.5 = 7.2\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Convergent-Divergent Nozzle
A convergent-divergent nozzle is an essential tool in fluid mechanics for controlling the velocity and pressure of the flow of gases. This nozzle has two distinct sections:
  • Convergent section, where the cross-sectional area decreases
  • Divergent section, where the cross-sectional area increases
The gas accelerates in the convergent section until the speed of sound (Mach 1) is reached at the throat, the narrowest part of the nozzle. If the nozzle is designed correctly, the gas will continue to accelerate to supersonic speeds in the divergent section. This type of nozzle is commonly used in applications such as jet engines and rockets to efficiently accelerate the gas to high velocities.
Normal Shock Waves
Normal shock waves occur in the divergent section of a convergent-divergent nozzle when the flow shifts from supersonic to subsonic speeds. This sudden deceleration leads to a rise in pressure and temperature, accompanied by a decrease in velocity and Mach number. For a normal shock, the properties of the flow before and after the shock are related through various equations. These include the Mach number, pressure, and temperature ratios. Understanding normal shock waves is crucial for predicting and analyzing the behavior of high-speed gas flows in nozzles.
Mach Number
The Mach number is a dimensionless quantity representing the ratio of the speed of a flow to the speed of sound in that medium. It is given by the formula: \[ M = \frac{v}{a} \] where \(v\) is the flow velocity and \(a\) is the speed of sound. The Mach number determines whether the flow is subsonic (\(M < 1\)), sonic (\(M = 1\)), or supersonic (\(M > 1\)). For example, in the provided exercise, the nozzle is designed to have an exit Mach number of 1.8, which indicates supersonic flow. Variable Mach numbers must be considered for proper calculations in fluid mechanics, especially when dealing with shock waves and isentropic processes.
Pressure Ratios
Pressure ratios are crucial when analyzing the changes in flow properties across different sections of a nozzle or through shock waves. In subsonic flows, pressure decreases with an increase in velocity. However, in supersonic flows, pressure increases with an increase in velocity. Across a normal shock, the pressure ratio \( \frac{P_2}{P_1}\) can be given by: \[ \frac{P_2}{P_1} = 1 + \frac{2\gamma \left(M_1^2 - 1\right)}{\gamma + 1} \] where \(\gamma\) is the specific heat ratio (1.67 for argon). This ratio helps in determining the resultant pressures after a shock, critical for designing efficient nozzles.
Isentropic Relations
Isentropic relations describe the behavior of the flow when it is both adiabatic (no heat transfer) and reversible. These principles are used to analyze the flow before and after the shock in a nozzle. For example, the relation for pressure and Mach number is given by: \[ \frac{P_0}{P} = \left(1 + \frac{\gamma - 1}{2}M^2\right)^{\frac{\gamma}{\gamma - 1}} \] Here, \(P_0\) is the stagnation pressure (total pressure when flow is brought to rest), \(P\) is the static pressure, and \(M\) is the Mach number. These relations help estimate the changes in flow properties for designed conditions in the nozzle.

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Most popular questions from this chapter

Determine the mass flow rate of air under adiabatic conditions through a \(50 \mathrm{~mm}\) diameter pipe \(85 \mathrm{~m}\) long from a large reservoir in which the pressure and temperature are \(300 \mathrm{kPa}\) and \(15^{\circ} \mathrm{C}\). The pressure at the outlet of the pipe is \(120 \mathrm{kPa}\). Assume a negligible pressure drop at the pipe inlet and a mean value of \(0.006\) for \(f\). To what value would the inlet pressure need to be raised to increase the mass flow rate by \(50 \%\) ?

A Pitot-static tube in a wind-tunnel gives a static pressure reading of \(40.7 \mathrm{kPa}\) and a stagnation pressure \(98.0 \mathrm{kPa}\). The stagnation temperature is \(90{ }^{\circ} \mathrm{C}\). Calculate the air velocity upstream of the Pitot-static tube.

A supersonic air stream at \(35 \mathrm{kPa}\) is deflected by a wedgeshaped two-dimensional obstacle of total angle \(20^{\circ}\) mounted with its axis parallel to the oncoming flow. Shock waves are observed coming from the apex at \(40^{\circ}\) to the original flow direction. What is the initial Mach number and the pressure rise through the shocks? If the stagnation temperature is \(30^{\circ} \mathrm{C}\) what is the velocity immediately downstream of the shock waves?

A normal shock wave forms in front of a two-dimensional blunt-nosed obstacle in a supersonic air stream. The pressure at the nose of the obstacle is three times the static pressure upstream of the shock wave. Determine the upstream Mach number, the density ratio across the shock and the velocity immediately after the shock if the upstream static temperature is \(10^{\circ} \mathrm{C}\). If the air were subsequently expanded isentropically to its original pressure what would its temperature then be?

An aircraft flies at \(8000 \mathrm{~m}\) altitude where the atmospheric pressure and temperature are respectively \(35.5 \mathrm{kPa}\) and \(-37^{\circ} \mathrm{C}\). An air-speed indicator (similar to a Pitot-static tube) reads \(740 \mathrm{~km} \cdot \mathrm{h}^{-1}\), but the instrument has been calibrated for variable-density flow at sea-level conditions (101.3 kPa and \(15^{\circ} \mathrm{C}\) ). Calculate the true air speed and the stagnation temperature.
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