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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59,300 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter centers on the energy analysis of closed systems with a focus on polytropic processes. It emphasizes that the process is characterized by the equation PV^n = constant, leading to a derivation of work done as (P2V2 - P1V1)/(1-n) for n ? 1 and a logarithmic expression for n = 1. Understanding the role of the polytropic exponent is crucial for accurately analyzing energy balances in systems such as piston–cylinder devices, compressors, turbines, and even biological systems.

Learning Objectives

1

Understand the concept of a polytropic process and its mathematical representation using PV^n = constant.

2

Derive and calculate the work done during a polytropic process, distinguishing between cases when n ? 1 and when n = 1.

3

Analyze the influence of the polytropic exponent (n) on energy transformations in closed systems.

4

Apply energy balance concepts to practical systems such as piston–cylinder devices, compressors, and turbines.

5

Compare thermodynamic processes, including recognizing isothermal processes as a special case of the polytropic process.

Key Concepts

CONCEPT

DEFINITION

Polytropic Process

A thermodynamic process that follows the equation PV^n = constant, which can represent a variety of processes depending on the value of n.

Polytropic Exponent (n)

A parameter that characterizes the specific nature of the polytropic process; it determines how pressure and volume change and influences the work done during the process.

Work Done (W)

The energy transferred during a process, calculated by integrating pressure with respect to volume. For a polytropic process, W = (P2V2 - P1V1)/(1-n) when n ≠ 1, and a logarithmic form is used for n = 1.

Energy Analysis

The assessment of energy transformations within a system, which in closed systems includes internal energy, enthalpy changes, and the work done during various thermodynamic processes.

Example Problems

Example 1

Is the boundary work associated with constant-volume systems always zero?

Example 2

On a $P$ - $U$ diagram, what does the area under the process curve represent?

Example 3

An ideal gas at a given state expands to a fixed final volume first at constant pressure and then at constant temperature. For which case is the work done greater?

Example 4

Calculate the total work, in $\mathrm{kJ},$ for process $1-3$ shown in Fig. P4-4 when the system consists of $2 \mathrm{~kg}$ of nitrogen.

Example 5

Calculate the total work, in Btu, produced by the process of Fig. P4-5E.
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Step-by-Step Explanations

QUESTION

How do you determine the work done during a polytropic process in a closed system?

STEP-BY-STEP ANSWER:

Step 1: Begin with the polytropic process definition, PV^n = constant, where n is the polytropic exponent.
Step 2: Express work as the integral of pressure with respect to volume: W = ∫[V1 to V2] P dV.
Step 3: Substitute the pressure from the process equation into the integral, i.e., P = constant/V^n.
Step 4: Integrate the expression. When n ≠ 1, the integration yields: W = (P2V2 - P1V1)/(1-n).
Step 5: Recognize that for n = 1 (an isothermal process), the integration results in a logarithmic form: W = P1V1 ln(V2/V1).
Final Answer: The work done during a polytropic process is (P2V2 - P1V1)/(1-n) for n ≠ 1, and a logarithmic expression for n = 1.

Calculating Work in a Polytropic Process

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Common Mistakes

  • Misinterpreting the polytropic exponent (n) and its role in defining the process.
  • Incorrectly applying the work formula without distinguishing between the cases for n ? 1 and n = 1.
  • Omitting proper integration limits and failing to properly set up the integral for work done.
  • Assuming all thermodynamic processes are similar without recognizing the unique aspects of polytropic processes.