Modern Engineering Thermodynamics

CHAPTER 11
More Thermodynamic Relations
CONTENTS
11.1 Kynning (Introduction) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
11.2 Two New Properties: Helmholtz and Gibbs Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
11.3 Gibbs Phase Equilibrium Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
11.4 Maxwell Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
11.5 The Clapeyron Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
11.6 Determining u, h, and s from p, v, and T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
11.7 Constructing Tables and Charts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
11.7.1 Saturation tables ............................................................379
11.7.2 Superheated vapor tables . ....................................................380
11.8 Thermodynamic Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
11.9 Gas Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
11.10 Compressibility Factor and Generalized Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
11.11 Is Steam Ever an Ideal Gas? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 398
11.1 KYNNING (INTRODUCTION)
We have a problem. It turns out that no meters, gauges, or instruments of any kind can be used to directly
measure the internal energy or the enthalpy or the entropy of a system. How, then, do you get numerical values
for thermodynamic properties that are not directly measurable? In Chapter 3, we discuss this subject briefly and
find that numerical values of properties that are not directly measurable (e.g., u, h, ands) cansometimesbe
calculated from the numerical values of properties that are measurable (e.g., p, v, and T).
For elementary materials, such as incompressible solids (or liquids) and ideal gases, we have relatively simple
equations of state that provide the necessary relations. For example, the specific internal energy, specific enthalpy,
and specific entropy of an incompressible material are related to its temperature, pressure, and specific volume by
ðu 2 − u 1 Þ incomp = cðT 2 − T 1 Þ, ðh 2 − h 1 Þ incomp = cðT 2 − T 1 Þ + vðp 2 − p 1 Þ, and ðs 2 − s 1 Þ incomp = c lnðT 2 /T 1 Þ
where c is the specific heat of the material. And, in the case of an ideal gas, these properties are related by
ðu 2 − u 1 Þ ideal gas
= c v ðT 2 − T 1 Þ, ðh 2 − h 1 Þ ideal gas
= c p ðT 2 − T 1 Þ, and ðs 2 − s 1 Þ ideal gas
= c p lnðT 2 /T 1 Þ − R lnðp 2 /p 1 Þ
Complex materials require more sophisticated equations of state plus a knowledge of various mathematical
property interrelationships to be able to evaluate their unmeasurable thermodynamic properties. In this chapter,
we build on the equations introduced in Chapter 3 to formulate new property relations that can be used to
compute numerical values for u, h, and s for complex real materials.
Modern Engineering Thermodynamics. DOI: 10.1016/B978-0-12-374996-3.00011-7
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