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Chapter 12: Problem 62

At a particular location in an insulated duct, the Mach number of the airflow is 0.2 the pressure is \(500 \mathrm{kPa}\), and the temperature is \(127^{\circ} \mathrm{C}\). At a downstream location where the Mach number is equal to \(0.6,\) estimate the pressure, temperature, and velocity of the air.

Short Answer

Expert verified
Pressure: 353.5 kPa, Temperature: 368.71 K, Velocity: 223.2 m/s.

Step by step solution

01

Convert Initial Temperature to Kelvin

The initial temperature given is in Celsius, so we need to convert it to Kelvin as it's the standard unit for gas equations.The formula is: \[ T(K) = T(掳C) + 273.15 \]Substitute the given value: \[ T = 127 + 273.15 = 400.15 \, K \]
02

Apply the Isentropic Flow Relations

In an insulated duct, isentropic relations are applicable. Use the relation between pressures and Mach numbers: \[ \frac{P_2}{P_1} = \left( \frac{T_2}{T_1} \right)^{\frac{\gamma}{\gamma-1}} \] where \( \gamma = 1.4 \) for air. Also, \[ \frac{T_2}{T_1} = \frac{1 + \frac{\gamma-1}{2} M_1^2}{1 + \frac{\gamma-1}{2} M_2^2} \]Substitute \( M_1 = 0.2 \), \( M_2 = 0.6 \), and the initial temperature value:\[ \frac{T_2}{400.15} = \frac{1 + \frac{1.4-1}{2} \times 0.2^2}{1 + \frac{1.4-1}{2} \times 0.6^2} \]
03

Calculate the Temperature at Downstream Location

From the equation: \[ \frac{T_2}{400.15} = \frac{1 + 0.2}{1 + 0.108} \]\[ \frac{T_2}{400.15} = \frac{1.02}{1.108} \]\[ T_2 = 400.15 \times \frac{1.02}{1.108} \approx 368.71 \, K \]
04

Calculate the Pressure at Downstream Location

Use the temperature ratio from step 3 in the pressure relation:\[ \frac{P_2}{500} = \left( \frac{368.71}{400.15} \right)^{\frac{1.4}{0.4}} \]\[ \frac{P_2}{500} = (0.9217)^{3.5} \approx 0.707 \]So, \( P_2 \approx 0.707 \times 500 \approx 353.5 \, \text{kPa} \).
05

Calculate the Velocity of the Air

The velocity of the air, given the Mach number, can be found using:\[ V = M \times a \]where \( a = \sqrt{\gamma R T} \), and \( R = 287 \ m^2/s^2/K \) for air. First calculate the speed of sound \( a_2 \) at \( T_2 \):\[ a_2 = \sqrt{1.4 \times 287 \times 368.71} \approx 372 \ m/s \]So, the velocity \( V_2 \) is:\[ V_2 = 0.6 \times 372 \approx 223.2 \ m/s \]

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Key Concepts

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

Mach Number
The Mach number is a dimensionless quantity that represents the ratio of the speed of an object moving through a fluid to the speed of sound in that fluid. It provides a convenient way to describe the flow velocity compared to the sound waves in the medium. A Mach number less than 1 indicates subsonic flow, whereas a Mach number greater than 1 denotes supersonic flow.
For example, in the given exercise, the Mach number transitions from 0.2 to 0.6. Both of these values are subsonic, meaning the airflow is slower than the speed of sound at both the initial and downstream locations. Understanding the Mach number is crucial because it affects how pressure, temperature, and flow characteristics change in different regions of a duct or channel.
It's important to note that in compressible flow, such as air in a duct, flow properties change significantly with Mach number, influencing both engineering designs and theoretical calculations.
Pressure and Temperature Relations
Pressure and temperature are closely related in isentropic flow, which is a type of flow where entropy remains constant. In the context of the problem, as the Mach number increases from 0.2 to 0.6, the flow properties change due to isentropic relations.
Isentropic flow allows us to relate pressures and temperatures with Mach numbers using specific equations. For temperature: \[ \frac{T_2}{T_1} = \frac{1 + \frac{\gamma-1}{2} M_1^2}{1 + \frac{\gamma-1}{2} M_2^2} \]This formula illustrates how temperatures at different Mach numbers can be compared using the ratio.
For pressure, the relation is:\[ \frac{P_2}{P_1} = \left( \frac{T_2}{T_1} \right)^{\frac{\gamma}{\gamma-1}} \]The change in temperature prompts a corresponding change in pressure. As derived in the problem, these relationships precisely calculate the different flow properties at varying Mach numbers, aiding engineers in designing efficient systems.
Speed of Sound
The speed of sound is a critical component in calculating flow properties in a duct. It is determined by the medium's specific properties, such as temperature and the type of gas. The speed of sound in a perfect gas can be calculated by the formula:\[ a = \sqrt{\gamma R T} \]where \( \gamma \) is the specific heat ratio, \( R \) is the specific gas constant, and \( T \) is the absolute temperature.
In the exercise, this formula helps compute the speed of sound at a downstream location, given a new temperature. With the speed of sound known, airflow velocity is easily determined using the Mach number:\[ V = M \times a \]Understanding the speed of sound is crucial because it enables precise calculation of flow velocities and assessments of how different conditions, such as temperature, impact airflow in engineering applications. This, in turn, influences the design and safety of various systems that rely on ducted airflows, such as HVAC systems and aircraft engines.

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Most popular questions from this chapter

Show that the pressure coefficient, \(C_{\mathrm{p}},\) for the isentropic flow of an ideal gas can be expressed in terms of \(k\) and Ma such that $$C_{\mathrm{p}}=\frac{p_{0}-p}{\frac{1}{2} \rho V^{2}}=\frac{2}{k \mathrm{Ma}^{2}}\left\\{\left[1+\frac{k-1}{2} \mathrm{Ma}^{2}\right]^{\frac{k}{k-1}}-1\right\\}$$ where \(p_{0}\) is the stagnation pressure, \(p\) is the static pressure, \(V\) is the free-stream speed of the fluid, \(k\) is the specific heat ratio, and Ma is the free-stream Mach number.

A CD nozzle with a throat diameter of \(40 \mathrm{~mm}\) connects air in a source reservoir to a discharge reservoir, where the diameter of the nozzle exit is \(90 \mathrm{~mm}\). When the pressure and temperature in the source reservoir are \(101 \mathrm{kPa}\) and \(15^{\circ} \mathrm{C}\), respectively, a normal shock occurs at the nozzle exit. Determine the back pressure on the nozzle and the mass flow rate through the nozzle under this condition.

If the magnitude of the pressure fluctuations generated by a source triple from \(6 \mathrm{~Pa}\) to \(18 \mathrm{~Pa}\), what is the change in decibel level?

A supersonic aircraft is cruising at an elevation of \(10 \mathrm{~km}\) in a standard atmosphere, and the (stagnation) pressure measured by a Pitot tube mounted on the aircraft is \(100 \mathrm{kPa}\). Estimate the air speed and Mach number at which the aircraft is traveling.

Air flows through an insulated pipe section that is \(0.50 \mathrm{~m}\) long and has a diameter of \(50 \mathrm{~mm}\). At the entrance to the pipe, the Mach number is \(0.6,\) the pressure is \(150 \mathrm{kPa}\), and the temperature is \(27^{\circ} \mathrm{C}\). If the average friction factor in the pipe is \(0.020,\) determine the Mach number, temperature, and pressure at the exit of the pipe section.
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