30079_42.pdf

CHAPTER 42

EXERGY ANALYSIS AND ENTROPY

GENERATION MINIMIZATION

Adrian Bejan

Department of Mechanical Engineering and Materials Science

Duke University

Durham, North Carolina

42.1 INTRODUCTION

13 51

42.6 HEAT TRANSFER

13 59

42.2 PHYSICAL EXERGY 13 53

42.7 STORAGE SYSTEMS 13 61

42.3 CHEMICAL EXERGY 13 55

42.8 SOLAR ENERGY

CONVERSION

13 62

42.4 ENTROPY GENERATION

MINIMIZATION

13 57

42.9 POWER PLANTS

13 62

42.5 CRYOGENICS

13 58

42.1 INTRODUCTION

In this chapter, we review two important methods that account for much of the newer work in

engineering thermodynamics and thermal design and optimization. The method of exergy analysis

rests on thermodynamics alone. The first law, the second law, and the environment are used simul-

taneously in order to determine (i) the theoretical operating conditions of the system in the reversible

limit and (ii) the entropy generated (or exergy destroyed) by the actual system, that is, the departure

from the reversible limit. The focus is on analysis. Applied to the system components individually,

exergy analysis shows us quantitatively how much each component contributes to the overall irre-

versibility of the system.1"3

Entropy generation minimization (EGM) is a method of modeling and optimization. The entropy

generated by the system is first developed as a function of the physical characteristics of the system

(dimensions, materials, shapes, constraints). An important preliminary step is the construction of a

system model that incorporates not only the traditional building blocks of engineering thermodynam-

ics (systems, laws, cycles, processes, interactions), but also the fundamental principles of fluid me-

chanics, heat transfer, mass transfer and other transport phenomena. This combination makes the

model "realistic" by accounting for the inherent irreversibility of the actual device. Finally, the

minimum entropy generation design (Sgen min) is determined for the model, and the approach of any

other design (5gen) to the limit of realistic ideality represented by Sgenmin is monitored in terms of the

entropy generation number Ns = Sgen/Sgenmin > 1.

To calculate 5gen and minimize it, the analyst does not need to rely on the concept of exergy. The

EGM method represents an important step beyond thermodynamics. It is a new method4 that combines

thermodynamics, heat transfer, and fluid mechanics into a powerful technique for modeling and

optimizing real systems and processes. The use of the EGM method has expanded greatly during the

last two decades.5

SYMBOLS AND UNITS

specific nonflow availability, J/kg

nonflow availability, J

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

A area, m2

b specific flow availability, J/kg

B flow availability, J

B duty parameter for plate and cylinder

Bs duty parameter for sphere

BQ duty parameter for tube

Be dimensionless group, 5g'en Ar/(5g'en Ar + S'^>AP)

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

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

C heat leak thermal conductance, W/K

C* time constraint constant, sec/kg

D diameter, m

e specific energy, J/kg

E energy, J

ech specific flow chemical exergy, J/kmol

et specific total flow exergy, J/kmol

ex specific flow exergy, J/kg

~ex specific flow exergy, J/kmol

EQ exergy transfer via heat transfer, J

Ew exergy transfer rate, W

Ex flow exergy, J

EGM the method of entropy generation minimization

/ friction factor

FD drag force, N

g gravitational acceleration, m/sec2

G mass velocity, kg/(sec • m2)

h specific enthalpy, J/kg

h heat transfer coefficient, W/(m2K)

h° total specific enthalpy, J/kg

H° total enthalpy, J

k thermal conductivity, W/(m K)

L length, m

m mass, kg

m mass flow rate, kg/sec

M mass, kg

N mole number, kmol

N molal flow rate, kmol/sec

Ns entropy generation number, Sgen/Sgenmin

Nu Nusselt number

Ntu number of heat transfer units

P pressure, N/m2

Pr Prandtl number

q' heat transfer rate per unit length, W/m

Q heat transfer, J

Q heat transfer rate, W

r dimensionless insulation resistance

R ratio of thermal conductances

ReD Reynolds number

s specific entropy, J/(kg • K)

S entropy, J/K

Sgen entropy generation, J/K

5gen entropy generation rate, W/K

Sgen entropy generation rate per unit length, W/(m • K)

5g'en entropy generation rate per unit volume, W/(m3 K)

t time, sec

tc time constraint, sec

T temperature, K

U overall heat transfer coefficient, W/(m2 K)

f/oo free stream velocity, m/sec

v specific volume, m3/kg

V volume, m3

V velocity, m/sec

W power, W

x longitudinal coordinate, m

z elevation, m

AP pressure drop, N/m2

A7 temperature difference, K

77 first law efficiency

Tjn second law efficiency

8 dimensionless time

fji viscosity, kg/(sec • m)

fjf chemical potentials at the restricted dead state, J/kmol

/t0l chemical potentials at the dead state, J/kmol

v kinematic viscosity, m2/sec

£ specific nonflow exergy, J/kg

H nonflow exergy, J

Hch nonflow chemical exergy, J

Hr nonflow total exergy, J

p density, kg/m3

Subscripts

()B base

()c collector

()c Carnot

( )H high

( )L low

()m melting

()max maximum

()min minimum

()opt optimal

()p pump

()rev reversible

(), turbine

()0 environment

()00 free stream

42.2 PHYSICAL EXERGY

Figure 42.1 shows the general features of an open thermodynamic system that can interact thermally

(g0) and mechanically (P0 dV/dt) with the atmospheric temperature and pressure reservoir (ro, P0).

The system may have any number of inlet and outlet ports, even though only two such ports are

illustrated. At a certain point in time, the system may be in communication with any number of

additional temperature reservoirs (7\, . . . , Tn), experiencing the instantaneous heat transfer interac-

tions, Qi, . . . , Qn- The work transfer rate W represents all the possible modes of work transfer,

specifically, the work done on the atmosphere (P0 dVldf) and the remaining (useful, deliverable)

portions such as P dV/dt, shaft work, shear work, electrical work, and magnetic work. The useful

part is known as available work (or simply exergy) or, on a unit time basis,

£,= *-P0f

Fig. 42.1 Open system in thermal and mechanical communication with the ambient.

Reprinted by permission.)

The first law and the second law of thermodynamics can be combined to show that the available

work transfer rate from the system of Fig. 42.1 is given by the Ew equation:1"3

Ew = ~ (E - roS + P0V) + i (l - jj &

Accumulation Exergy transfer

of nonflow exergy via heat transfer

+ £ m(h° - T0s) _ ^ m(h° - T0s) _ T *

out

^O^gen

Intake of Release of Destruction

flow exergy via flow exergy via of exergy

mass flow mass flow

where £", V, and S are the instantaneous energy, volume, and entropy of the system, and h° is shorthand

for the specific enthalpy plus the kinetic and potential energies of each stream, h° = h + l/iV2 + gz.

The first four terms on the right-hand side of the Ew equation represent the energy rate delivered as

useful power (to an external user) in the limit of reversible operation (Ew>rev, Sgen = 0). It is worth

noting that the Ew equation is a restatement of the Gouy-Stodola theorem (see Section 41.4), or the

proportionality between the rate of exergy (work) destruction and the rate of entropy generation

^W,rev ~ ^W ~ -*0^gen

A special exergy nomenclature has been devised for the terms formed on the right side of the

Ew equation. The exergy content associated with a heat transfer interaction (Qt, Tt) and the environ-

ment (T0) is the exergy of heat transfer,

^ = a(i-|)

This means that the heat transfer with the environment (Q0, T0) carries zero exergy relative to the

environment T0.

Associated with the system extensive properties (E, S, V) and the two specified intensive properties

of the environment (ro, P0) is a new extensive property: the thermomechanical or physical nonflow

availability,

A = E - T0S + P0V

a = e - T0s + P0v

Let A0 represent the nonflow availability when the system is at the restricted dead state (T0, P0), that

is, in thermal and mechanical equilibrium with the environment, A0 = EQ - T^Q + P0V0. The

difference between the nonflow availability of the system in a given state and its nonflow availability

in the restricted dead state is the thermomechanical or physical nonflow exergy,

~=A-A0 = E-E0-T0(S-S0) + P0(V - Vo)

£ = a-a0 = e-e0- T0(s - s0) + P0(v - v0)

The nonflow exergy represents the most work that would become available if the system were to

reach its restricted dead state reversibly, while communicating thermally only with the environment.

In other words, the nonflow exergy represents the exergy content of a given closed system relative

to the environment.

Associated with each of the streams entering or exiting an open system is the thermomechanical

or physical flow availability,

B = H° - T0S

b = h° - T0s

At the restricted dead state, the nonflow availability of the stream is B0 = H°Q - TQS0. The difference

B - B0 is known as the thermomechanical or physical flow exergy of the stream,

Ex = B - B0 = H° - HI - T0(S - So)

ex = b - b0 = h° - hi - T0(s - s0)

Physically, the flow exergy represents the available work content of the stream relative to the restricted

dead state (T0, P0). This work could be extracted in principle from a system that operates reversibly

in thermal communication only with the environment (ro), while receiving the given stream (m, h°,

s) and discharging the same stream at the environmental pressure and temperature (m, h°Q, s0).

In summary, the Ew equation can be rewritten more simply as

EW = -~ + 2 EQi + 5>^ - S mex - roSgen

ai /=l in out

Examples of how these exergy concepts are used in the course of analyzing component by component

the performance of complex systems can be found in Refs. 1-3. Figure 42.2 shows one such example.1

The upper part of the drawing shows the traditional description of the four components of a simple

Rankine cycle. The lower part shows the exergy streams that enter and exit each component, with

the important feature that the heater, the turbine and the cooler destroy significant portions (shaded,

fading away) of the entering exergy streams. The numerical application of the Ew equation to each

component tells the analyst the exact widths of the exergy streams to be drawn in Fig. 42.2. In

graphical or numerical terms, the "exergy wheel" diagram1 shows not only how much exergy is being

destroyed but also where. It tells the designer how to rank order the components as candidates for

optimization according to the method of entropy generation minimization (Sections 42.4-42.9).

To complement the traditional (first law) energy conversion efficiency, TJ = (Wt — Wp)/QH in Fig.

42.2, exergy analysis recommends as figure of merit the second law efficiency,

Wt ~ Wp

T7ii - EQn

where Wt - Wp is the net power output (i.e., Ew earlier in this section). The second law efficiency

can have values between 0 and 1, where 1 corresponds to the reversible limit. Because of this limit,

i7n describes very well the fundamental difference between the method of exergy analysis and the

method of entropy generation minimization (EGM), because in EGM the system always operates

irreversibly. The question in EGM is how to change the system such that its Sgen value (always finite)

approaches the minimum Sgen allowed by the system constraints.

42.3 CHEMICAL EXERGY

Consider now a nonflow system that can experience heat, work, and mass transfer in communication

with the environment. The environment is represented by T0, P0, and the n chemical potentials jm0i

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