30079_41.pdf

CHAPTER 41

THERMODYNAMICS FUNDAMENTALS

Adrian Bejan

Department of Mechanical Engineering and Materials Science

Duke University

Durham, North Carolina

41,1 INTRODUCTION

1 331

41.5 RELATIONS AMONG

THERMODYNAMIC

PROPERTIES

13 39

41.2 THE FIRST LAW OF

THERMODYNAMICS FOR

CLOSED SYSTEMS 13 33

41.6 IDEAL GASES

13 41

41.7 INCOMPRESSIBLE

SUBSTANCES

41.3 THE SECOND LAW OF

THERMODYNAMICS FOR

CLOSED SYSTEMS 13 35

13 44

41.8 TWO-PHASE STATES 13 44

41.4 THE LAWS OF

THERMODYNAMICS FOR

OPEN SYSTEMS

41.9 ANALYSIS OF ENGINEERING

SYSTEM COMPONENTS 13 47

13 38

41.1 INTRODUCTION

Thermodynamics has historically grown out of man's determination—as Sadi Carnot put it—to cap-

ture "the motive power of fire." Relative to mechanical engineering, thermodynamics describes the

relationship between mechanical work and other forms of energy. There are two facets of contem-

porary thermodynamics that must be stressed in a review such as this. The first is the equivalence of

work and heat as two possible forms of energy exchange. This facet is encapsulated in the first law

of thermodynamics. The second aspect is the irreversibility of all processes (changes) that occur in

nature. As summarized by the second law of thermodynamics, irreversibility or entropy generation

is what prevents us from extracting the most possible work from various sources; it is also what

prevents us from doing the most with the work that is already at our disposal. The objective of this

chapter is to review the first and second laws of thermodynamics and their implications in mechanical

engineering, particularly with respect to such issues as energy conversion and conservation. The

analytical aspects (the formulas) of engineering thermodynamics are reviewed primarily in terms of

the behavior of a pure substance, as would be the case of the working fluid in a heat engine or in a

refrigeration machine. In the next chapter we review in greater detail the newer field of entropy

generation minimization (thermodynamic optimization).

SYMBOLS AND UNITS

c specific heat of incompressible substance, J/(kg • K)

cp specific heat at constant pressure, J/(kg • K)

CT constant temperature coefficient, m3/kg

cv specific heat at constant volume, J/(kg • K)

COP coefficient of performance

E energy, J

/ specific Helmholtz free energy (u - TV), J/kg

F force vector, N

Mechanical Engineers' Handbook, 2nd ed., Edited by Myer Kutz.

g gravitational acceleration, m/sec2

g specific Gibbs free energy (h — TV), J/kg

h specific enthalpy (u + Pu), J/kg

K isothermal compressibility, m2/N

m mass of closed system, kg

m mass flow rate, kg/sec

mt mass of component in a mixture, kg

M mass inventory of control volume, kg

M molar mass, g/mol or kg/kmol

n number of moles, mol

N0 Avogadro's constant

P pressure

8Q infinitesimal heat interaction, J

Q heat transfer rate, W

r position vector, m

R ideal gas constant, J/(kg • K)

s specific entropy, J/(kg • K)

S entropy, J/K

Sgen entropy generation, J/K

5gen entropy generation rate, W/K

T absolute temperature, K

u specific internal energy, J/kg

U internal energy, J

v specific volume m3/kg

v specific volume of incompressible substance, m3/kg

V volume, m3

V velocity, m/sec

8W infinitesimal work interaction, J

Wi^ rate of lost available work, W

Wsh rate of shaft (shear) work transfer, W

x linear coordinate, m

x quality of liquid and vapor mixture

Z vertical coordinate, m

j8 coefficient of thermal expansion, 1/K

y ratio of specific heats, cp/cv

j] "efficiency" ratio

Tjj first-law efficiency

iTn second-law efficiency

6 relative temperature, °C

SUBSCRIPTS

()in inlet port

()out outlet port

()rev reversible path

()H high-temperature reservoir

()L low-temperature reservoir

()max maximum

()T turbine

()c compressor

()N nozzle

()D diffuser

()j initial state

()2 final state

()0 reference state

()f saturated liquid state (/ = "fluid")

()g saturated vapor state (g = "gas")

()5 saturated solid state (s = "solid")

()* moderately compressed liquid state

()+ slightly superheated vapor state

Definitions

THERMODYNAMIC SYSTEM is the region or the collection of matter in space selected for analysis.

ENVIRONMENT is the thermodynamic system external to the system of interest, that is, external to

the region selected for analysis or for discussion.

BOUNDARY is the real or imaginary surface delineating the system of interest. The boundary separates

the system from its environment. The boundary is an unambiguously defined surface. The bound-

ary has zero thickness.

CLOSED SYSTEM is the thermodynamic system whose boundary is not penetrated (crossed) by the

flow of mass.

OPEN SYSTEM, or flow system, is the thermodynamic system whose boundary is permeable to mass

flow. Open systems have their own nomenclature, so that the thermodynamic system is usually

referred to as the control volume, the boundary of the open system is the control surface, and the

particular regions of the boundary that are crossed by mass flows are the inlet or outlet ports.

STATE is the condition (the being) of a thermodynamic system at a particular point in time, as

described by an ensemble of quantities called thermodynamic properties (e.g., pressure, volume,

temperature, energy, enthalpy, entropy). Thermodynamic properties are only those quantities that

depend solely on the instantaneous state of the system. Thermodynamic properties do not depend

on the "history" of the system between two different states. Quantities that depend on the system

evolution (path) between states are not thermodynamic properties (examples of nonproperties are

the work, heat, and mass transfer interactions; the entropy transfer interactions; the entropy gen-

eration; and the lost available work—see also the definition of process).

EXTENSIVE PROPERTIES are properties whose values depend on the size of the system (e.g., mass,

volume, energy, enthalpy, entropy).

INTENSIVE PROPERTIES are properties whose values do not depend on the system size (e.g., pressure,

temperature). The collection of all intensive properties (or the properties of an infinitesimally

small element of the system, including the per-unit-mass properties, such as specific energy and

specific entropy) constitutes the intensive state.

PHASE is the collection of all system elements that have the same intensive state (e.g., the liquid

droplets dispersed in a liquid-vapor mixture have the same intensive state, that is, the same

pressure, temperature, specific volume, specific entropy, etc.).

PROCESS is the change of state from one initial state to a final state. In addition to the end states,

knowledge of the process implies knowledge of the interactions experienced by the system while

in communication with its environment (e.g., work transfer, heat transfer, mass transfer, and

entropy transfer). To know the process also means to know the path (the history, or the succession

of states) followed by the system from the initial to the final state.

CYCLE is a special process in which the final state coincides with the initial state.

41.2 THE FIRST LAW OF THERMODYNAMICS FOR CLOSED SYSTEMS

The first law of thermodynamics is a statement that brings together three concepts in thermodynamics:

work transfer, heat transfer, and energy change. Of these concepts, only energy change or, simply,

energy, is, in general, a thermodynamic property. Before stating the first law and before writing down

the equation that accounts for this statement, it is necessary to review1 the concepts of work transfer,

heat transfer, and energy change.

Consider the force Fx experienced by a certain system at a point on its boundary. Relative to this

system, the infinitesimal work transfer interaction between system and environment is

8W = -Fxdx

where_the boundary displacement dx is defined as positive in the direction of the force Fx. When the

force F and the displacement of its point of application dr are not collinear, the general definition

of infinitesimal work transfer is

8W = -F - dr

The work transfer interaction is considered positive when the system does work on its environment—

in other words, when F and dr point in opposite directions. This sign convention has its origin in

heat engine engineering, since the purpose of heat engines as thermodynamic systems is to deliver

work while receiving heat.

In order for a system to experience work transfer, two things must occur: (1) a force must be

present on the boundary, and (2) the point of application of this force (hence, the boundary) must

move. The mere presence of forces on the boundary, without the displacement or the deformation of

the boundary, does not mean work transfer. Likewise, the mere presence of boundary displacement

without a force opposing or driving this motion does not mean work transfer. For example, in the

free expansion of a gas into an evacuated space, the gas system does not experience work transfer

because throughout the expansion the pressure at the imaginary system-environment interface is zero.

If a closed system can interact with its environment only via work transfer (i.e., in the absence

of heat transfer 8Q discussed later), then it is observed that the work transfer during a change of

state from state 1 to state 2 is the same for all processes linking states 1 and 2,

if2 \

- 8W\ =E2-El

V1 / 8Q=0

In this special case the work transfer interaction (W1_2)5G=0 is a property of the system, since its

value depends solely on the end states. This thermodynamic property is the energy change of the

system, E2 — Ev The statement that preceded the last equation is the first law of thermodynamics

for closed systems that do not experience heat transfer.

Heat transfer is, like work transfer, an energy interaction that can take place between a system

and its environment. The distinction between 8Q and dW is made by the second law of thermody-

namics discussed in the next section: Heat transfer is the energy interaction accompanied by entropy

transfer, whereas work transfer is the energy interaction taking place in the absence of entropy

transfer. The transfer of heat is driven by the temperature difference established between the system

and its environment.2 The system temperature is measured by placing the system in thermal com-

munication with a test system called thermometer. The result of this measurement is the relative

temperature 9 expressed in degrees Celsius, 0(°C), or Fahrenheit, 0(°F); these alternative temperature

readings are related through the conversion formulas

0(°Q - 5/9[0(°F) - 32]

0(°F) = 5/90(°C) + 32

1°F - 5/9°C

The boundary that prevents the transfer of heat, regardless of the magnitude of the

system-environment temperature difference, is termed adiabatic. Conversely, the boundary that is

the locus of heat transfer even in the limit of vanishingly small system-environment temperature

difference is termed diathermal.

It is observed that a closed system undergoing a change of state 1 —> 2 in the absence of work

transfer experiences a heat-transfer interaction whose magnitude depends solely on the end states:

(!28Q) =E2-E1

\Jl /8W=0

In the special case of zero work transfer, the heat-transfer interaction is a thermodynamic property

of the system, which is by definition equal to the energy change experienced by the system in going

from state 1 to state 2. The last equation is the first law of thermodynamics for closed systems

incapable of experiencing work transfer. Note that, unlike work transfer, the heat transfer is considered

positive when it increases the energy of the system.

Most thermodynamic systems do not manifest the purely mechanical (8Q = 0) or purely thermal

(8W = 0) behavior discussed to this point. Most systems manifest a coupled mechanical and thermal

behavior. The preceding first-law statements can be used to show that the first law of thermodynamics

for a process executed by a closed system experiencing both work transfer and heat transfer is

J* 8Q - £ 8W = E2- El

heat work energy

transfer transfer change

energy interactions (property)

(nonproperties)

The first law means that the net heat transfer into the system equals the work done by the system

on the environment, plus the increase in the energy of the system. The first law of thermodynamics

for a cycle or for an integral number of cycles executed by a closed system is

<j> 5Q = <j> 8W = 0

Note that the net change in the thermodynamic property energy is zero during a cycle or an integral

number of cycles.

The energy change term E2 — El appearing on the right-hand side of the first law can be replaced

by a more general notation that distinguishes between macroscopically identifiable forms of energy

storage (kinetic, gravitational) and energy stored internally,

E2 - E, = U2 - U, + ^ - ^ + mgZ2 - mgZ,

energy internal kinetic

gravitational

change energy energy

energy

change change

change

If the closed system expands or contracts quasi-statically (i.e., slowly enough, in mechanical equi-

librium internally and with the environment) so that at every point in time the pressure P is uniform

throughout the system, then the work transfer term can be calculated as being equal to the work done

by all the boundary pressure forces as they move with their respective points of application,

f 8W= I PdV

Ji Ji

The work-transfer integral can be evaluated provided the path of the quasi-static process, P(V), is

known; this is another illustration that the work transfer is path-dependent (i.e., not a thermodynamic

property).

41.3 THE SECOND LAW OF THERMODYNAMICS FOR CLOSED SYSTEMS

A temperature reservoir is a thermodynamic system that experiences only heat-transfer interactions

and whose temperature remains constant during such interactions. Consider first a closed system

executing a cycle or an integral number of cycles while in thermal communication with no more than

one temperature reservoir. To state the second law for this case is to observe that the net work

transfer during each cycle cannot be positive,

<P 8W < 0

In other words, a closed system cannot deliver work during one cycle, while in communication with

one temperature reservoir or with no temperature reservoir at all. Examples of such cyclic operation

are the vibration of a spring-mass system, or the bouncing of a ball on the pavement: in order for

these systems to return to their respective initial heights, that is, in order for them to execute cycles,

the environment (e.g., humans) must perform work on them. The limiting case of frictionless cyclic

operation is termed reversible, because in this limit the system returns to its initial state without

intervention (work transfer) from the environment. Therefore, the distinction between reversible and

irreversible cycles executed by closed systems in communication with no more than one temperature

reservoir is

<j> SW = 0 (reversible)

<j> 8W < 0 (irreversible)

To summarize, the first and second laws for closed systems operating cyclically in contact with no

more than one temperature reservoir are (Fig. 41.1)

j> 8W = j> 8Q < 0

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