Moran M.J., Shapiro H.N. Fundamentals of Engineering Thermodynamics

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13.113 Using the result of Problem 13.111(b), determine the

total specific flow exergy at locations 1, 2, and 3 and the rate

of exergy destruction, each in Btu/min, for the device of

Problem 12.102. Let the environment be a mixture of dry air

and water vapor at 95

8F, 1 atm with y

5 0.022, y

5 0.978.

Also, let c

5 0.24 Btu/lb ? 8R and c

5 0.44 Btu/lb ? 8R

13.114 Figure P13.114 provides data for a cooling tower at steady

state. Using results from Problem 13.111 as needed, determine

the rate of exergy destruction, in Btu/min. Let the environment

be a mixture of dry air and water vapor at 90

8F, 1 atm with

5 0.024, y

5 0.976. Also, let c

5 0.24 Btu/lb ? 8R and

5 0.44 Btu/lb ? 8R.

Liquid water, T

= 140°F

= 123 lb/min

··

Liquid water, T

= 123.7°F

= m

Makeup water

= 90°F

= 1.28 lb/min

Atmospheric air

= 95°F,

= 0.0138,

= 400 lb/min

Moist air

= 105°F

= 0.017

Fig. P13.114

c DESIGN & OPEN-ENDED PROBLEMS: EXPLORING ENGINEERING PRACTICE

13.1D Middle school science students may wonder how

gasoline powers their family’s car. Prepare a 30-minute

presentation suitable for an eighth-grade science class to

explain the basic operation of a spark-ignition engine while

touching on relevant chemical reactions and emission

concerns. Include instructional aids and a group activity to

enhance your presentation.

13.2D Complete two of the following and report your findings

in a memorandum, including sample calculations, discussion

of the modeling used, and references to the technical

literature, as appropriate.

(a) Assuming electricity is provided by a coal-fired vapor

power plant, determine how much coal, in kg, is saved over

the lifetime of a compact fluorescent light bulb in comparison

to the coal required to power a comparable incandescent

light bulb over the same interval.

(b) When crude oil is brought to earth’s surface, it is often

accompanied by natural gas, which is commonly flared,

releasing carbon dioxide to the atmosphere. What are other

options for managing such natural gas than flaring it? Rank-

order them in terms of economic feasibility.

that produces hydrogen from water using electricity from the

grid. The hydrogen is said to be an emissions-free fuel for

vehicles, emergency back-up power units for buildings, and

other applications. Investigate how the hydrogen energy

station works and critically evaluate the claims that are

made by the manufacturer.

(d) From the engineering literature, obtain the standard

chemical exergy, in kJ/kg, for gold and lead. Also, for each

determine the current market price, in $/kg. Investigate the

difference between exergetic value and economic value, and

on this basis rationalize the chemical exergy and market-

price values determined for these commodities.

13.3D Municipal solid waste (MSW), often called garbage or

trash, consists of the combined solid waste produced by

homes and workplaces. In the United States, a portion of the

annual MSW accumulation is burned to generate steam for

producing electricity, and heating water and buildings, while

several times as much MSW is buried in sanitary landfills.

Investigate these two MSW disposal approaches. For each

approach, prepare a list of at least three advantages and

three disadvantages, together with brief discussions of each

advantage and disadvantage. Report your findings in a

PowerPoint presentation suitable for a community planning

group.

13.4D Mounting evidence suggests that more frequent severe

weather events, melting of glaciers and polar icecaps, and

other observed changes in nature are linked to emissions

from fossil fuel combustion. Researchers have developed

detailed computer simulations to predict global weather

patterns and investigate the effects of CO

and other gaseous

emissions on the weather. Investigate the underlying modeling

assumptions and characteristics of computer simulations used

to predict global weather patterns. Write a report, including

at least three references, of your findings.

13.5D Federal and state governmental legislation places

restrictions on harmful emissions associated with combustion

of fossil fuels in power plants and vehicles. For stationary

power plants and for vehicles in your geographic area,

identify the restricted emissions, their allowed levels, their

associated hazards, and means employed to achieve desired

emission levels. Investigate and discuss political, social,

economic, and technological issues that influenced adoption

of these regulations, and project whether any future

adjustments to them might occur. Summarize your findings

in a report citing at least three references.

13.6D Many utilities are converting power plants from coal to

alternative fuels due to environmental considerations. Conduct

a case study of a power plant in your geographic region that

has converted or is planning to convert from coal to an

alternative fuel. Provide a schematic of the coal-based system

and the alternative fuel-based system and describe pertinent

features of each. Investigate the advantages of using the new

fuel, physical plant changes with associated costs required to

accommodate the new fuel, and the impact of the fuel change

on system performance and operational cost. Summarize your

findings in a PowerPoint presentation suitable for your class.

Design & Open-Ended Problems: Exploring Engineering Practice 843

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844 Chapter 13

Reacting Mixtures and Combustion

13.7D A proposed simple gas turbine will produce power at a

rate of 500 kW by burning fuel with 200% theoretical air in

the combustor. Air temperature and pressure at the

compressor inlet are 298 K and 100 kPa, respectively. Fuel

enters the combustor at 298 K, while products of combustion

consisting of CO

, H

O, O

, and N

exit the combustor with

no significant change in pressure. Metallurgical considerations

require the turbine inlet temperature to be no greater than

1500 K. Products of combustion exit the turbine at 100 kPa.

The compressor has an isentropic efficiency of 85%, while

the turbine isentropic efficiency is 90%. Three fuels are

being considered: methane (CH

), ethylene (C

), and

ethane (C

). On the basis of minimal fuel use, recommend

a fuel, turbine inlet temperature, and compressor pressure

ratio for the gas turbine. Summarize your findings in a

report, supported by well-documented sample calculations

and a full discussion of the thermodynamic modeling used.

13.8D Fuel cell systems installed as part of a distributed

generation network require auxiliary components to support

the fuel cell stack. Depending on the type of fuel cell, these

components may provide fuel reforming, humidification, and

appropriate flows of fuel and oxidizer. Auxiliary components

also accommodate heat transfer, power conditioning, and

electrical connection. Additionally, the fuel cell system requires

various controls. Identify and research a fuel cell system for

combined heat and power integrated with a building in your

geographic region. Describe each component in the fuel cell

system and create a schematic of the system to include fuel

cell stack, its auxiliary components, and its integration with

the building to provide electricity and heating. Contact the

building supervisor to identify any installation, operational,

and/or maintenance issues. Estimate total costs (components,

installation, and annual fuel and operating costs) for the fuel

cell system and compare to the prior system’s cost, assuming

the same annual electricity and heating requirements. Summarize

your findings in a PowerPoint presentation.

13.9D The chemical exergies of common hydrocarbons C

can

be represented in terms of their respective lower heating value,

V, by an expression of the form

1LHV2

5 c

1 c

a22 c

where c

, c

, and c

are constants. Evaluate the constants to

obtain an expression applicable to the gaseous hydrocarbons

of Table A-26 (Model II).

13.10D Coal gasification is a means for using coal to produce

various energy products, usually including electricity. In

particular, coal gasification is a key feature of Integrated

Gasification Combined Cycle (IGCC) Plants (Sec. 9.10).

Evaluate the exergetic efficiencies of the gasifier and

synthetic gas (syngas) cleanup units within such an IGCC

Plant. Use operating data obtained from the engineering

literature for the plant together with standard chemical

exergies. In a report, give the exergetic efficiency values you

obtain, supported by well-documented sample calculations, a

full discussion of the thermodynamic modeling used, and at

least three references.

13.11D Coal mining operations in certain regions of the United

States have created vast amounts of waste coal known as

culm. Some power plants have been built near culm banks to

generate electricity from this waste resource. Using a gravimetric

analysis for culm of 43% carbon, 36% ash, 15% moisture, 3%

oxygen, and trace amounts of hydrogen, nitrogen, and sulfur,

evaluate its standard chemical exergy relative to the exergy

reference of Table A-26, Model II. Also determine the standard

chemical exergy of anthracite coal. Investigate the advantages

and disadvantages of using culm instead of coal in power

plants. Write a report summarizing your findings, including a

comparison of the chemical exergy values obtained, sample

calculations, and at least three references.

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846

ENGINEERING CONTEXT

The objective of the present chapter is to consider the concept of equi-

librium in greater depth than has been done thus far. In the first part of the chapter, we develop the funda-

mental concepts used to study chemical and phase equilibrium. In the second part of the chapter, the study

of reacting systems initiated in Chap. 13 is continued with a discussion of chemical equilibrium in a single

phase. Particular emphasis is placed on the case of reacting ideal gas mixtures. The third part of the chap-

ter concerns phase equilibrium. The equilibrium of multicomponent, multiphase, nonreacting systems is

considered and the phase rule is introduced.

In Sec. 14.1, equilibrium criteria are introduced.

Philip and Karen Smith/Getty Images, Inc.

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1Chemical and

Phase Equilibrium

When you complete your study of this chapter, you will be able to...

demonstrate understanding of key concepts related to chemical and phase equilibrium,

including criteria for equilibrium, the equilibrium constant, and the Gibbs phase rule.

apply the equilibrium constant relationship, Eq. 14.35, to relate pressure, temperature, and

equilibrium constant for ideal gas mixtures involving individual and multiple reactions.

use chemical equilibrium concepts with the energy balance.

determine the equilibrium flame temperature.

apply the Gibbs phase rule, Eq. 14.68.

LEARNING OUTCOMES

847

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848 Chapter 14

Chemical and Phase Equilibrium

Equilibrium Fundamentals

In this part of the chapter, fundamental concepts are developed that are useful in the

study of chemical and phase equilibrium. Among these are equilibrium criteria and

the chemical potential concept.

14.1 Introducing Equilibrium Criteria

A system is said to be in thermodynamic equilibrium if, when it is isolated from its

surroundings, there would be no macroscopically observable changes. An important

requirement for equilibrium is that the temperature be uniform throughout the sys-

tem or each part of the system in thermal contact. If this condition were not met,

spontaneous heat transfer from one location to another could occur when the system

was isolated. There must also be no unbalanced forces between parts of the system.

These conditions ensure that the system is in thermal and mechanical equilibrium,

but there is still the possibility that complete equilibrium does not exist. A process

might occur involving a chemical reaction, a transfer of mass between phases, or both.

The objective of this section is to introduce criteria that can be applied to decide

whether a system in a particular state is in equilibrium. These criteria are developed

using the conservation of energy principle and the second law of thermodynamics as

discussed next.

Consider the case of a simple compressible system of fixed mass for which tem-

perature and pressure are uniform with position throughout. In the absence of overall

system motion and ignoring the influence of gravity, the energy balance in differential

form (Eq. 2.36) is

dU 5 dQ 2 dW

If volume change is the only work mode and pressure is uniform with position

throughout the system, dW 5 p dV. Introducing this in the energy balance and solv-

ing for dQ gives

dQ 5 dU 1 p dV

Since temperature is uniform with position throughout the system, the entropy bal-

ance in differential form (Eq. 6.25) is

dS 5

1 ds

Eliminating dQ between the last two equations

T dS 2 dU 2 p dV 5 T ds (14.1)

Entropy is produced in all actual processes and conserved only in the absence of

irreversibilities. Hence, Eq. 14.1 provides a constraint on the direction of processes. The

only processes allowed are those for which ds $ 0. Thus

T dS 2 dU 2 p dV $ 0 (14.2)

Equation 14.2 can be used to study equilibrium under various conditions.

a process taking place in an insulated, constant-volume ves-

sel, where dU 5 0 and dV 5 0, must be such that

dS4

U, V

$ 0 (14.3)

thermodynamic equilibrium

Fixed U, V

max

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14.1 Introducing Equilibrium Criteria 849

Equation 14.3 suggests that changes of state of a closed system at constant internal

energy and volume can occur only in the direction of increasing entropy. The expres-

sion also implies that entropy approaches a maximum as a state of equilibrium is

approached. This is a special case of the increase of entropy principle introduced in

Sec. 6.8.1. b b b b b

An important case for the study of chemical and phase equilibria is one in which

temperature and pressure are fixed. For this, it is convenient to employ the Gibbs

function

in extensive form

G 5 H 2 TS 5 U 1 pV 2 TS

Forming the differential

dG 5 dU 1 p dV 1 V dp 2 T dS 2 S dT

or on rearrangement

dG 2 V dp 1 S dT 521T dS 2 dU 2 p dV2

Except for the minus sign, the right side of this equation is the same as the expres-

sion appearing in Eq. 14.2. Accordingly, Eq. 14.2 can be written as

dG 2 V dp 1 S dT # 0 (14.4)

where the inequality reverses direction because of the minus sign noted above.

It can be concluded from Eq. 14.4 that any process taking place at a specified

temperature and pressure (dT 5 0 and dp 5 0) must be such that

dG4

T, p

# 0 (14.5)

This inequality indicates that the Gibbs function of a system at fixed T and p

decreases during an irreversible process. Each step of such a process results in a

decrease in the Gibbs function of the system and brings the system closer to

equilibrium. The equilibrium state is the one having the minimum value of the

Gibbs function. Therefore, when

dG4

T, p

5 0 (14.6)

we have equilibrium. In subsequent discussions, we refer to Eq. 14.6 as the equilibrium

criterion

Equation 14.6 provides a relationship among the properties of a system when it is at

an equilibrium state. The manner in which the equilibrium state is reached is unimportant,

however, for once an equilibrium state is obtained, a system exists at a particular T and

p and no further spontaneous changes can take place. When applying Eq. 14.6, therefore,

we may specify the temperature T and pressure p, but it is unnecessary to require addi-

tionally that the system actually achieves equilibrium at fixed T and fixed p.

14.1.1

Chemical Potential and Equilibrium

In the present discussion, the Gibbs function is considered further as a prerequisite

for application of the equilibrium criterion dG]

T, p

5 0 introduced above. Let us begin

by noting that any extensive property of a single-phase, single-component system is

a function of two independent intensive properties and the size of the system. Select-

ing temperature and pressure as the independent properties and the number of

moles n as the measure of size, the Gibbs function can be expressed in the form

G 5 G(T, p, n). For a single-phase, multicomponent system, G may then be considered

a function of temperature, pressure, and the number of moles of each component

present, written G 5 G(T, p, n

, n

, . . . , n

Gibbs function

equilibrium criterion

Fixed T, p

min

multicomponent system

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850 Chapter 14

Chemical and Phase Equilibrium

If each mole number is multiplied by a, the size of the system changes by the same

factor and so does the value of every extensive property. Thus, for the Gibbs function

we may write

aG1T, p, n

, n

, . . . , n

25 G1T, p, an

, an

, . . . , an

Differentiating with respect to a while holding temperature, pressure, and the mole

numbers fixed and using the chain rule on the right side gives

G 5

01an

. . .

01an

This equation holds for all values of a. In particular, it holds for a 5 1. Setting a 5 1,

the following expression results:

G 5

i51

T, p, n

(14.7)

where the subscript n

denotes that all n’s except n

are held fixed during differentiation.

The partial derivatives appearing in Eq. 14.7 have such importance for our study

of chemical and phase equilibrium that they are given a special name and symbol.

The chemical potential of component i, symbolized by m

, is defined as

5 a

T, p, n

(14.8)

The chemical potential is an intensive property. With Eq. 14.8, Eq. 14.7 becomes

G 5

i51

(14.9)

The equilibrium criterion given by Eq. 14.6 can be written in terms of chemical

potentials, providing an expression of fundamental importance for subsequent discus-

sions of equilibrium. Forming the differential of G1T, p, n

, . . . , n

2 while holding tem-

perature and pressure fixed results in

dG4

T, p

i51

T, p, n

The partial derivatives are recognized from Eq. 14.8 as the chemical potentials, so

dG4

T, p

i51

(14.10)

With Eq. 14.10, the equilibrium criterion dG]

T, p

5 0 can be placed in the form

i51

5 0

(14.11)

Like Eq. 14.6, from which it is obtained, this equation provides a relationship among

properties of a system when the system is at an equilibrium state where the tempera-

ture is T and the pressure is p. Like Eq. 14.6, this equation applies to a particular state,

and the manner in which that state is attained is not important.

14.1.2

Evaluating Chemical Potentials

Means for evaluating the chemical potentials for two cases of interest are introduced

in this section: a single phase of a pure substance and an ideal gas mixture.

TAKE NOTE...

Equations 14.8 and 14.9

correspond to Eqs. 11.107

and 11.108, respectively.

chemical potential

TAKE NOTE...

Equations 14.10 and

14.11 are special forms of

Eqs. 11.112 and 11.113,

respectively.

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SINGLE PHASE OF A PURE SUBSTANCE. An elementary case considered

later in this chapter is that of equilibrium between two phases involving a pure sub-

stance. For a single phase of a pure substance, Eq. 14.9 becomes simply

G 5 nm

m 5

5 g

(14.12)

That is, the chemical potential is just the Gibbs function per mole.

IDEAL GAS MIXTURE. An important case for the study of chemical equilibrium

is that of an ideal gas mixture. The enthalpy and entropy of an ideal gas mixture are

given by

H 5

i51

1T2





and



S 5

i51

1T, p

where p

5 y

p is the partial pressure of component i. Accordingly, the Gibbs function

takes the form

G 5 H 2 TS 5

i51

1T 22 T

i51

1T, p

1T 22 T s

1T, p



1ideal gas2

(14.13)

Introducing the molar Gibbs function of component i

1T, p

25 h

1T 22 T s

1T, p



1ideal gas2 (14.14)

Equation 14.13 can be expressed as

G 5

i51

1T, p



1ideal gas2

(14.15)

Comparing Eq. 14.15 to Eq. 14.9 suggests that

5 g

1T, p



1ideal gas2 (14.16)

That is, the chemical potential of component i in an ideal gas mixture is equal to its

Gibbs function per mole of i, evaluated at the mixture temperature and the partial

pressure of i in the mixture. Equation 14.16 can be obtained formally by taking the

partial derivative of Eq. 14.15 with respect to n

, holding temperature, pressure, and

the remaining n’s constant, and then applying the definition of chemical potential,

Eq. 14.8.

The chemical potential of component i in an ideal gas mixture can be expressed

in an alternative form that is somewhat more convenient for subsequent applications.

Using Eq. 13.23, Eq. 14.14 becomes

5 h

1T22 T s

1T, p

5 h

1T22 T as8

1T22 R ln

ref

5 h

1T22 T s 8

1T21 RT ln

ref

TAKE NOTE...

Expressions for the inter-

nal energy, enthalpy, and

entropy of ideal gas mix-

tures are summarized in

Table 13.1.

14.1 Introducing Equilibrium Criteria 851

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852 Chapter 14

Chemical and Phase Equilibrium

where p

ref

is 1 atm and y

is the mole fraction of component i in a mixture at tem-

perature T and pressure p. The last equation can be written compactly as

5 g8

1 RT ln

ref





1ideal gas2

(14.17)

where g 8

is the Gibbs function of component i evaluated at temperature T and a

pressure of 1 atm. Additional details concerning the chemical potential concept are

provided in Sec. 11.9. Equation 14.17 is the same as Eq. 11.144 developed there.

BIOCONNECTIONS The human body, as well as all living things, exist in a

type of equilibrium that is considered in biology to be a dynamic equilibrium called

homeostasis. The term refers to the ability of the body to regulate its internal state,

such as body temperature, within specific limits necessary for life despite the conditions

of the external environment, or the level of exertion (see accompanying figure). Within the

body, feedback mechanisms regulate numerous variables that must be kept under control.

If this dynamic equilibrium cannot be maintained because of the severe influence of the

surroundings, living things can become sick and die.

The numerous control mechanisms in the body are varied, but they share some common

elements. Generally, they involve a feedback loop that includes a way to sense a fluctuation

of a variable from its desired condition and generate a corrective action to bring the variable

back into the desired range. Feedback loops exist within the body at the levels of cells,

body fluid systems (such as the circulatory system), tissues, and organs. These feedback

loops also interact to maintain homeostasis much as a home thermostat regulates the

heating and cooling system in a house. The mechanisms are highly stable as long as the

external conditions are not too severe.

An example of the complexity of homeostasis is the way in which the body maintains

the levels of glucose in the blood within desired limits. Glucose is an essential “fuel” for

the processes within cells. Glucose absorbed by the digestive system from food, or stored

as glycogen in the liver, is distributed by the blood stream to the cells. The body senses

the level of glucose in the blood and various glands produce hormones to stimulate the

conversion of glucose from stored glycogen, if needed to supplement food intake, or to

inhibit the release of glucose, as necessary to maintain the desired levels. Several different

hormones and numerous organs within the body are involved. The result is a balance that

is highly stable within the limits needed by the body for homeostatic equilibrium.

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