Book cover for Thermodynamics: An Engineering Approach

Thermodynamics: An Engineering Approach

Yunus A. Cengel, Michael A. Boles

ISBN #9781259822674

9th Edition

2,694 Questions

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Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter covers the core principles of compressible flow, emphasizing the importance of understanding stagnation properties, speed of sound, and Mach number. It provides detailed insights into isentropic flow relations and advanced nozzle designs, and explains the behavior of shock waves and expansion fans under various conditions. Additionally, the chapter introduces Rayleigh flow for accounting heat transfer in ducts, and culminates with practical problems that bridge theory and real-world applications in high-speed aerothermal systems.

Learning Objectives

1

Describe the fundamentals of compressible flow including stagnation properties, speed of sound, and Mach number.

2

Explain isentropic flow relations and their application in the design of converging and converging–diverging nozzles.

3

Analyze shock phenomena such as normal shocks, oblique shocks, and Prandtl–Meyer expansion waves.

4

Apply the principles of Rayleigh flow to evaluate heat transfer effects in constant-area ducts.

5

Utilize compressible flow concepts in the design and analysis of high-speed aerothermal and propulsion systems.

Key Concepts

CONCEPT

DEFINITION

Compressible Flow

The study of fluid flows in which density changes are significant, typically at high speeds.

Stagnation Properties

Properties of the fluid (e.g., temperature, pressure, density) when it is brought to rest isentropically.

Speed of Sound

The rate at which pressure disturbances propagate through a fluid; dependent on the medium's properties.

Mach Number

A dimensionless quantity representing the ratio of the flow speed to the local speed of sound.

Isentropic Flow Relations

Mathematical relationships that describe how flow properties change under constant entropy conditions.

Converging Nozzles

Nozzles that decrease in cross-sectional area, typically accelerating a subsonic flow to its throat.

Converging–Diverging Nozzles

Nozzles designed with a converging section to accelerate the flow to choked conditions and a diverging section for supersonic expansion.

Normal Shocks

Shock waves that occur perpendicular to the flow direction, causing abrupt changes in flow properties.

Oblique Shocks

Shock waves that are inclined relative to the flow direction, resulting in less severe property changes compared to normal shocks.

Prandtl–Meyer Expansion Waves

Expansion fans occurring in supersonic flows that gradually turn and accelerate the flow while decreasing pressure.

Rayleigh Flow

Flow in ducts where heat transfer affects the fluid properties while the area remains constant.

Example Problems

Example 1

A high-speed aircraft is cruising in still air. How does the temperature of air at the nose of the aircraft differ from the temperature of air at some distance from the aircraft?

Example 2

What is dynamic temperature?

Example 3

In air-conditioning applications, the temperature of air is measured by inserting a probe into the flow stream. Thus, the probe actually measures the stagnation temperature. Does this cause any significant error?

Example 4

How and why is the stagnation enthalpy $h_{0}$ defined? How does it differ from ordinary (static) enthalpy?

Example 5

Air flows through a device such that the stagnation pressure is $0.4 \mathrm{MPa}$, the stagnation temperature is $400^{\circ} \mathrm{C},$ and the velocity is $520 \mathrm{~m} / \mathrm{s}$. Determine the static pressure and temperature of the air at this state.

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Step-by-Step Explanations

QUESTION

How do you calculate the stagnation temperature given the static temperature (T) and Mach number (M) for an ideal gas?

STEP-BY-STEP ANSWER:

Step 1: Recall the stagnation temperature formula: T0 = T * (1 + ((gamma - 1) / 2) * M^2), where gamma is the specific heat ratio.
Step 2: Identify the known values for static temperature (T), Mach number (M) and the specific heat ratio (gamma, typically 1.4 for air).
Step 3: Substitute the values into the formula.
Step 4: Calculate the multiplier (1 + ((gamma - 1) / 2) * M^2) and multiply by T to obtain T0.
Final Answer: The stagnation temperature T0 is computed as T * (1 + ((gamma - 1) / 2) * M^2).

Stagnation Properties

QUESTION

How can you determine the exit Mach number in a converging nozzle given the throat conditions and isentropic relations?

STEP-BY-STEP ANSWER:

Step 1: Recognize that the flow is isentropic, meaning that entropy remains constant.
Step 2: Use the area-Mach number relation for isentropic flow, which relates the nozzle cross-sectional area to the Mach number.
Step 3: Identify the throat flow conditions (typically where Mach = 1) and the exit area.
Step 4: Solve the area-Mach number relation equation for the exit Mach number using the given exit area.
Step 5: Verify that the computed Mach number is consistent with subsonic or supersonic flow, as appropriate for a converging nozzle.
Final Answer: The exit Mach number is obtained by solving the area-Mach number relation under isentropic conditions, ensuring continuity with throat conditions.

Isentropic Flow Relations in Nozzle Design

QUESTION

What steps are involved in calculating the downstream properties of a normal shock wave given the upstream Mach number?

STEP-BY-STEP ANSWER:

Step 1: Begin by noting the upstream Mach number (M1) and the ratio of specific heats (gamma).
Step 2: Use the normal shock relations to compute the downstream Mach number (M2).
Step 3: Calculate the stagnation pressure and temperature ratios across the shock.
Step 4: Apply the conservation equations of mass, momentum, and energy for a normal shock.
Step 5: Compile the results to determine properties such as pressure, temperature, and density downstream of the shock.
Final Answer: Downstream flow properties are derived from the upstream Mach number using normal shock relations in conjunction with the conservation laws.

Normal Shocks

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Common Mistakes

  • Confusing stagnation properties with static properties, leading to incorrect temperature or pressure calculations.
  • Overlooking the importance of the specific heat ratio (gamma) in both isentropic and shock relations.
  • Assuming that flow remains subsonic in all nozzle sections, ignoring the possibility of choked flow in converging nozzles.
  • Misapplying shock relations by neglecting the conservation laws during abrupt property changes in normal shocks.
  • Failing to consider heat transfer effects in Rayleigh flow, which can lead to significant deviations in predicted fluid properties.