Thermal radiation and the second law

Andrés Agudelo* 1,2

Cristóbal Cortés 2

1 Group of Efficient Management of Energy – GIMEL –, Facultad de Ingeniería, Universidad

de Antioquia, Calle 67 No. 53 – 108, Medellín, Colombia.

2 Centre of Research for Energy and Resources Consumption – CIRCE –, Universidad de

Zaragoza, María de Luna 3, 50018, Zaragoza , Spain

Received

Abstract

The purpose of this paper is to collect and interrelate the fundamental concepts about second

law analysis of thermal radiation. This heat transfer mode plays a leading role in solar energy

utilization and in high-temperature devices, representing a significant contribution to

irreversibility that is frequently omitted in engineering analysis. Entropy and exergy of

thermal radiation are reviewed first. Radiative transfer processes are reviewed next, including

exchange between surfaces, the presence of a participative medium, and the analysis of

combined heat transfer modes. Emphasis is put on grey body radiation when treating with

non-black body radiation, due to its relevance in engineering applications. The mathematical

formulation of second law analysis of thermal radiation is complex, which limits its use in

conventional heat transfer analysis. For this reason, numerical approaches reported to date

deal with quite simple cases, leaving an open promising field of research.

Keywords: Second law, thermal radiation, entropy, exergy

* Corresponding autor (Andrés Agudelo). Address: CIRCE, Departamento de Ingeniería Mecánica, Centro Politécnico Superior, Universidad de Zaragoza. María de Luna 3, 50018, Zaragoza, Spain. Tel. (+34) 976762582. Fax (+34) 976732078. E-mail: andres.agudelo@unizar.es

2

Nomenclature

A = Surface area [m2]

b = Exergy flux [W/m2]

'b = Directional exergy radiation intensity [W/m2·sr]

'b = Spectral directional exergy radiation intensity [W/m2·sr·μm]

c = Speed of light in vacuum (2.988 x 108 m/s)

C = Concentration factor

f = Dilution factor

F = Function used to calculate the fluxes of grey radiation

h = Planck’s constant (6.626 x 10-34 J·s)

i = Energy flux [W/m2]

'i = Spectral directional radiation intensity [W/m2·sr·μm]

k = Boltzmann’s constant (1.381 x 10-23 J/K)

a ,K = Spectral isotropic absorption coefficient [m-1]

e,K = Spectral isotropic extinction coefficient [m-1]

s ,K = Spectral isotropic scattering coefficient [m-1]

l = Entropy flux [W/K·m2]

'l = Directional entropy radiation intensity [W/K·m2·sr]

'l = Spectral directional entropy radiation intensity [W/K·m2·sr·μm]

L = Rate of entropy emission [W/K]

m = Variable used to calculate the fluxes of grey radiation

m = Mass flow rate [kg/s]

n = Coefficient to compare the entropy fluxes of thermal radiation and conduction

p = State of polarisation

Q = Heat transfer [J]

Q = Heat transfer rate [W]

Q = Spectral energy transfer between matter and radiation

s = Specific entropy [J/kg·K]. Also direction of radiative transfer

S = Entropy [J/K]

S = Entropy transfer rate [W/K]

T = Temperature [K]

'T = Monochromatic radiation temperature [K]

U = Energy (internal) [J]

V = Volume [m3]

W = Specific power output [W/m2]

x = Dimensionless variable used to calculate radiative intensities. Also an integration variable

0' 'x ,x = Spectral dimensionless variables used to calculate radiative intensities

X = Function used to calculate the fluxes of grey radiation

Greek symbols

δ = Inexact differential

ε = Emissivity

= Scattering phase function

= Maximum conversion efficiency of thermal radiation

' = Spectral quality factor

λ = Wavelength [μm]

ν = Frequency [s-1]

σ = Stefan-Boltzmann constant (5.670 x 10-8 W/m2K)

4

' = Ratio of emitted to combined spectral energy intensity

= Solid angle [sr]

' = Generic scattering direction

Subscripts

0 = Environment

b = Boundary

BR = Black body radiation

c = Solar converter

cc = Conduction and convection

CV = Control volume

d = Exergy destruction

e = Exit or outlet of a flow

ea = Emission and absorption

emit = Emission

gen = Entropy generation

GR = Grey body radiation

HC = Heat conduction

i = Inlet of a flow

j = Generic heat transfer surface

M = Material medium

NBR = Non-black body radiation

ph = Photovoltaic

R = Radiation field

s = Scattering. Also sun

th = Thermal

TR = Thermal radiation

λ = Spectral property

Superscripts

= Incoming flux to surface of higher temperature

+ = Outgoing flux from surface of higher temperature

A = Surface of a solid wall

V = Volume in the radiation field

W = Radiative exergy flux obtained by Wright et al.

Abbreviations

BR = Black body radiation

GR = Grey body radiation

HC = Heat conduction

NBR = Non-black body radiation

RETE = Radiation entropy transfer equation

SLA = Second law analysis

TR = Thermal radiation

1. Introduction

The second law of thermodynamics is a well-known tool for the evaluation and optimisation

of energy systems, and has been used as a fundamental criterion in the search for improved

energy efficiency and rational use of natural resources [1-5]. Thermal radiation (TR) is

characteristic of any material system at temperatures above the absolute zero and becomes an

important form of heat transfer in devices that operate at high-temperatures [6]. Radiation is

the dominant form of heat transfer in applications such as furnaces, boilers, and other

combustion systems [7], and represents an important source of entropy generation,

6

contributing significantly to inefficiency [8]. Despite this fact, irreversibilities associated with

radiation phenomena are often excluded in conventional exergy analysis and there are few

quantitative approaches for its practical calculation, most of which are quite simple cases

reported recently [8-14].

The study of second law implications in radiation phenomena dates back to the end of

nineteenth century with the work of Kirchhoff and Wien [15, 16]. The entropy formula for

monochromatic black body radiation (BR) intensity was first obtained by Planck [17].

Alternative ways to obtain Planck’s formula were presented by Lewis [18], Rosen [19], and

Ore [15]. The major application of radiation entropy after Planck has been for the case of BR,

especially for solar radiation energy [20-31], although some studies treat the entropy of

radiation with an arbitrary spectrum [8, 10, 20, 32-34]. The exergy formula for TR was first

derived for BR by Petela in the context of solar engineering [16, 27], followed by some other

expressions by other researchers [16, 23, 25, 27]. The expression of Petela is the most widely

used formula to give the upper theoretical limit for work extraction from BR [20, 27, 31, 35,

36]. The exergy of radiation with an arbitrary spectrum was first determined by Karlsson [37],

whose result was questioned by Wright et al. [33] based on a set of assumptions not proved

yet. These authors presented an alternative expression to calculate the exergy of non-black

body radiation (NBR). Recently, Candau [36] confirmed the results of Karlsson, and Liu and

Chu [12] have used it to obtain the exergy transfer equation. The most advanced

developments in the field of second law analysis (SLA) of TR consist of the study of radiative

transfer processes, covering radiative exchange between surfaces [38], the presence of

participating media, and the irreversibility of radiative transfer when several modes of heat

transfer are present [8, 11-13, 39-42].

There are several relevant papers related to second law aspects of thermal radiation that have

been published in the Energy journal in recent years. We have the work of Edgerton [23] that

defines thermodynamic properties of BR such as internal energy, pressure, entropy, and

available energy. This author also relates entropy flux with energy flux for diffuse BR. Later,

Suzuki [43] calculated the exergy of solar radiation as that of conduction-convection heat

transfer when performing a second law analysis of solar collectors. In the late eighties and

early nineties Bădescu presented a series of papers analysing the efficiency of solar energy

conversion. In the first paper [44], the effect of solar radiation concentration on maximum

efficiency of converters is analysed, finding that efficiency decreases rapidly as the

concentration factor is reduced. This gives a very low efficiency for flat-plate collectors. The

next paper [45] compares several formulas for the efficiency of solar collectors, taking as

reference the one given by Castañs et al., [46], which uses the concept of effective

temperature. The formula of Petela gave the best description of converter’s efficiency. The

third paper of Bădescu [47] deals with the maximum efficiency of converters, taking into

account the diffuse character of solar radiation at the Earth’s surface. It was found that

selective converters have higher efficiency that non-selective ones. Again, the formula of

Petela gave the best approach to the efficiency of both types of converters when an

appropriate effective temperature for the sky is used.

More recently, Mahmud and Fraser [48] presented an analysis of entropy generation in a

vertical porous channel with combined convection and radiation heat transfer by means of

simulation. They follow the approach of Arpaci [49-51] by assuming that the optical thin

hypothesis for fluids is of general validity, treating radiative transfer as a diffusion process to

calculate its contribution to entropy generation. Szargut [52] presents an exergy balance of the

Earth, in which he uses the formula of Petela to calculate the exergy of solar radiation.

Hermann [53] uses the same approach, but additionally he uses the approximation of Wright

et al., [33] to calculate the exergy of solar radiation at the surface of the Earth. Finally, Lior et

al., [54] present an application of exergy analysis to transport phenomena in which they

8

calculate the entropy transfer associated with radiation transfer the same as if it was heat

conduction.

This work presents a review of publications about SLA of TR. The starting point is the

entropy formula and its application to BR and NBR, followed by the exergy of radiation.

Finally, the application of these concepts to radiative transfer is tackled by introducing the

fluxes of entropy and exergy in the radiative exchange between surfaces, and the entropy and

exergy radiative transfer equations, including some applications of entropy generation in

enclosures and combined transfer modes. All of the notions relating the SLA of TR are spread

through the technical literature over the past century. The aim of this paper is to gather these

concepts together, from fundamentals to state-of-the-art theory and applications.

Although the theoretical foundations of SLA of TR are known, its application to practical

engineering cases is difficult to perform due to the complex nature of the equations. In order

to obtain numerical results for situations of practical interest it is first necessary to write the

radiative transfer equations in order to make it compatible with conventional heat transfer

calculation methods. Due to these limitations there is a lack of practical quantitative cases

available. There is an extensive field of application for the theory presented here that must be

investigated to achieve improved energy efficiency in solar applications and in high-

temperature energy systems.

2. Entropy of thermal radiation

Planck established that every monochromatic ray or pencil of radiation has its definite

entropy, which depends on its energy and frequency, and which is propagated with it. This

concept gave origin to the idea of entropy radiation, which, as in the case of energy, is

measured “by the amount of entropy which passes in unit time through a unit area in a

definite direction” [17]. Every ray passing through a medium in a point has also its own

temperature (monochromatic temperature). As a consequence, in a point in a medium there is

an infinite number of temperatures, independent of the temperature of the medium itself. In

stable thermodynamic equilibrium there is one only temperature, which is common to the

medium and to all the rays of different frequencies that cross it in every direction [17].

In this section Planck’s entropy formula is presented and its application to BR and NBR is

discussed. Finally, a comparison is made between the entropy flux of TR and heat conduction

(HC), showing that the former is greater, especially for NBR.

2.1 The entropy formula

The formula for spectral or monochromatic directional entropy intensity of equilibrium

radiation was obtained first by Planck in the beginning of the twentieth century, based on

statistical and quantum physics [17]:

4 2 5 2 5 2 5 2 5 21 1

' ' ' '' i i i ipkc W

l ln lnphc phc phc phc K m sr m

(1)

where λ is wavelength, p is the state of polarisation (p = 2 for unpolarised radiation, and p = 1

for plane-polarised radiation), k is Boltzmann’s constant, c is the speed of light in vacuum, h

is Planck’s constant, and 'i is the spectral directional intensity of radiation.

As originally deduced, Eq. (1 is valid for equilibrium radiation only, for which the spectral

intensity of radiation will be the black body distribution obtained by Planck [17]:

2

5 2

1

1'

,BR hc kT

phc Wi

e m sr m

(2)

where T is the temperature.

The entropy formula presented in Eq. (1 was obtained later by Lewis [18], using a statistical

basis and some developments made by Einstein in regard to the distribution of BR presented

by Planck (Eq. (2). Planck also obtained the following relationship, which is a thermodynamic

definition of the temperature of radiation [55]:

10

1'

' '

l

i T

(3)

where 'T is the temperature of monochromatic radiation. For BR, 'T T (independent of

frequency) [22]. Using this result, Planck obtained the following functional relationship

between spectral entropy and energy intensity [55]:

23

'' i

l f

(4)

where f denotes a certain function of a single argument and ν is frequency. Equation (4 is a

generalised form of Wien’s displacement law. Starting from this expression and from the

ordinary statistical definition of entropy, Rosen [19] obtained the explicit form of this

function, arriving at Eq. (1 for the case plane-polarised radiation.

An alternative method for obtaining Eq. (1 is to use the definition given in Eq. (3: the value of

1/T is obtained from Eq. (2 and the result is integrated [15, 17].

The monochromatic or spectral directional radiation temperature is the temperature of a black

body that would emit the same spectral directional intensity, 'i , at the wavelength λ [16, 22,

38]. The expression for the temperature of monochromatic radiation can be obtained from Eq.

(2:

2 5

1

1

'

'

hcT

k phcln

i

(5)

For isotropic radiation, the temperature at a point is uniquely defined [8]. This temperature is

applicable to non-equilibrium or NBR by using the measured or estimated non-equilibrium

radiation intensity, 'i [22, 37, 38, 56], and vice versa:

2

5

1

1'

'

hc kT

phci

e

(6)

2.2 Black body radiation

The black body and BR serve as a reference for comparing radiative properties of real

surfaces and NBR [6]. The spectral directional entropy intensity for BR can be obtained by

replacing the BR intensity (Eq. (2) into Eq. (1:

4

1 1 1 11 1

1 1 1 1'

,BR x x x x

pkcl ln ln

e e e e

(7)

With

hcx

kT (8)

The distribution of Eq. (7 is shown in Figure 1.

The shape of the curves in this figure is very similar to that of radiation intensity, exhibiting a

strong effect of temperature. In order to investigate the relationship between temperature and

wavelength corresponding to the maximum of entropy radiation intensity we need to

differentiate Eq. (7 with respect to λ and to equalise the resulting expression to zero. In so

doing, we obtain:

5

2 24 2 2

4 1 1 1 11 1

1 1 1 1

1 11

1 10

1 1

',BR

x x x x

x xx x

x x

l pkcln ln

e e e e

hce ln hce lnpkc e e

kT e kT e

(9)

The numerical solution of Eq. (9 is:

13002 195767

4 791267'

,BR,max

hcl T . m K

k . (10)

12

This is the Wien’s displacement law for BR entropy intensity. As shown in Figure 2, the locus

of peak entropy values observed in Figure 1 behaves in a similar way to that of energy. The

constant of 3002.195767 μm·K is about 3.6% higher than its energy counterpart. As a result,

for a given temperature, the peak BR entropy intensity will always be reached at a higher

wavelength than the corresponding peak value of radiation intensity. Also, for a given

wavelength, the maximum entropy intensity will be obtained at a higher temperature than the

peak radiation intensity.

The directional entropy radiation intensity is found by integrating Eq. (7 over the wavelength

spectrum [17, 22, 23]:

32

0

2

3' 'BR ,BR

p Wl l d T

K m sr

(11)

where σ is the Stefan-Boltzmann constant.

Equation (11 can also be obtained by considering radiation enclosed in a given volume

(system) as an ideal photon or Bose gas and performing a conventional thermodynamic

analysis [16, 23, 57-59]. For BR it is found that both Gibbs free energy and chemical potential

are equal to zero [16, 24, 57, 59, 60]. Integration of Eq. (11 over solid angle (hemisphere)

gives the total entropy flux for BR [10]:

32

2

2

3'

BR BR

p Wl l d T

K m

(12)

where is the solid angle.

This entropy flux can be written as [23]:

1 4 3 42

3BR BR

pl i (13)

where the BR energy flux for unpolarised radiation is [6]:

42BR

Wi T

m

(14)

The net rate of entropy emission from a black surface of area A can be calculated as:

BR BR

A

WL l dA

K (15)

2.3 Non-black body radiation

In most engineering systems radiative transfer has a NBR spectrum. The definition of entropy

made by Planck (Eq. (1) is valid for monochromatic radiation, and can be applied to NBR by

substituting the corresponding measured or estimated spectral directional radiation intensity

on it [22, 39]. NBR is a superposition of monochromatic rays that do not interact with each

other. An approximation for the entropy flux of unpolarised NBR is that of a black body with

the same energy flux [10], which after Eq. (13 can be written as:

1 4 3 44

3NBR NBRl i (16)

This approximation overestimates the entropy flux, since the equality applies only to diffuse

BR. The error of this expression depends on the energy flux of the NBR. The case of grey

body radiation is analysed next.

2.3.1 Grey body radiation

Diffuse grey body radiation (GR) is the simplest case of NBR. GR is very interesting because

the emission of many real surfaces can be approximated as diffuse-grey. The spectral

radiation intensity of diffuse GR can be expressed as:

' ',GR ,BRi i (17)

where ε is the emissivity of the material.

14

After replacing the result of Eq. (17 into Eq. (1, spectral directional entropy intensity of

diffuse radiation from a grey body can be written as [13]:

41 1

1 1 1 1'

,GR x x x x

pkcl ln ln

e e e e

(18)

This distribution is shown in Figure 3 for several emissivity values at a fixed temperature of

2000 K. It can be seen in this figure how entropy intensity decreases as emissivity is reduced.

Figure 3 shows that entropy radiation for NBR is lower than that of BR at the same

temperature, but not proportionally, i.e. ' ',BR ,GRl l is not a constant value. It must be recalled

that the curves in Figure 3 correspond to different energy radiation intensities. The reduction

of entropy intensity caused by emissivity is due to a corresponding reduction in energy

intensity. It is known that for the same energy flux or the same temperature, NBR entropy

flux is lower than that of BR [34, 61], although at a given temperature, the entropy-to-energy

ratio of NBR is higher than that of BR [22].

Directional entropy radiation intensity of diffuse GR is found by integrating Eq. (18 over the

spectrum:

43 2

2 30

1 11 1 1 1

'GR x x x x

pkl T x ln ln dx

c h e e e e

(19)

The definite integral in Eq. (19 is only function of emissivity:

2

0

1 11 1 1 1x x x x

F x ln ln dxe e e e

(20)

The Stefan-Boltzmann constant is defined as [62]:

5 4

2 3

2

15

k

c h

(21)

Using this result and the definition of Eq. (20, Eq. (19 can be written as:

34 4

45 2 45

4 3 4' 'GR BR

pl F T F l

(22)

There is no closed solution for the integral of Eq. (20. Stephens and Obrien presented an

infinite series solution [22] which is not suitable for practical engineering calculations.

Landsberg and Tonge [20, 63] proposed an approximate limiting solution for low emissivities

(ε < 0.1):

44

45F X

(23)

With

0 9652 0 2777 0 0511X . . ln . (24)

An approximation for this function has recently been proposed by Pons [64], which is valid

for a wider emissivity range (0.05 ≤ ε ≤ 0.8):

0 973 0 275 0 0273X . . ln . (25)

The absolute error of this approximation is below 5 x 10-4. An alternative approximation to

the integral of Eq. (20 has been obtained by Wright et al. [10, 22]:

44

45F m ln

(26)

Using numerical integration, the authors found that the best fit of this approximation

corresponds to expressing m as a linear function of emissivity:

2 311 0 175m . . (27)

With this result, Eq. (26 yields a maximum error of 0.16% in the range 0.05 ≤ ε ≤ 1.

Using Eqs. (22, (26 and (27, directional entropy radiation intensity of diffuse GR can be

written as:

16

34

45 21 2 311 0 175

4 3'GR

pl . . ln T

(28)

The entropy flux for diffuse GR can be obtained by integrating the above equation over the

solid angle:

34

2 451 2 311 0 175

3 4GR

pl . . ln T

(29)

Using the approximation of Eqs. (16 and (17, the entropy flux of unpolarised diffuse GR is

[10]:

3 4 34

3GRl T (30)

This expression has an error below 1% for ε > 0.5, but for lower emissivity values it can

result in important deviations, as can be seen in Figure 4. As expected from Eqs. (29 and (30,

the error is independent of temperature.

2.4 Entropy flux of thermal radiation in engineering

The entropy balance for a generic control volume (CV) in conventional thermodynamic

analysis is stated as [65]:

jCVi i e e gen

j i ej

QdSm s m s S

dt T

(31)

where jQ is the heat transfer rate at CV boundary with uniform temperature Tj, m is mass

flow rate, s is specific entropy, and genS is the entropy generation rate inside the CV.

The first term in the right hand side of the above equation assumes that the entropy flux of

thermal radiation is equal to that of HC or convection, which is not true. From Eqs. (12 and

(14) it can be seen that for unpolarised BR the entropy flux is 4/3 times the energy flux

divided by the emission temperature, and for NBR it can be even greater [34]. Therefore, the

entropy balance equation should be modified to take into account the real entropy flux of

thermal radiation:

gene

eei

iij A

jTRj

jccCV SsmsmdAlT

Q

dt

dS

j

,

, (32)

where jccQ , is the net heat transfer rate by conduction and convection, and lTR,j is the entropy

flux from CV boundary surface Aj.

In order to compare the entropy flux of thermal radiation with that of conduction, a

dimensionless coefficient is defined as the ratio of thermal radiation entropy flux and that

corresponding to HC at a surface [22, 34]:

TR TR

HC TR

l ln

l i T (33)

As said before, the value of n for BR is 4/3. Wright [34] makes a detailed analysis of this

coefficient for GR. From Eqs. (12 and (29, the entropy flux of GR can be written as a function

of that of BR:

4

451 2 311 0 175

4GR BRl . . ln l

(34)

By recalling that Eq. (17 holds also for the energy flux, i, the coefficient n for unpolarised

diffuse GR can be written as:

4

4 451 2 311 0 175

3 4GRn . . ln

(35)

This result is presented in Figure 5, where it can be observed that nGR shows asymptotic

behaviour as the emissivity approaches zero, and the lower limit is that of BR (4/3).

18

It is evident from this figure that the entropy flux of free1 NBR is higher than that of HC, and

that it is incorrect to calculate the entropy flux of TR in a similar way to that of HC. For non-

polarised diffuse GR the error is greater as the emissivity of the surface decreases.

In several papers Arpaci proposes a general expression for the local volumetric rate of entropy

generation for combined transport processes including the contribution of thermal radiation

[49-51, 66]. This author assumes that radiative transfer can be treated as a diffusion process

for any optical thickness in order to calculate entropy generation. Nevertheless, radiation is a

long-range phenomenon, so the local radiative heat flux (and entropy generation) is not

determined by the local temperature gradient but by the temperature distribution of the entire

enclosure [11]. In order to prove that it is incorrect to calculate the entropy generation of

radiative transfer as a diffusion process, Liu and Chu [11] used a one-dimensional problem

consisting of an absorbing, emitting and isotropic scattering grey medium bounded by two

black body walls. By using as a reference the formulas derived by Caldas and Semião [8],

these authors found that the diffusion approximation is valid only for extremely high optical

thickness, giving incoherent or incorrect results for small optical thickness.

The different nature of the entropy flux associated with radiative transfer also makes

necessary a modification of the Clausius inequality in order for it to be applicable to all heat

transfer modes [67]:

0

TR

b

cc ST

Q (36)

where STR is the net TR entropy exchange, and subscript b refers to the boundary of the

system.

1 Free radiation: only emission from the surface is considered.

19

3. Exergy of thermal radiation

Exergy is the maximum amount of work that can be obtained from a system until it reaches

total equilibrium with its environment [2]. It is a useful concept for the evaluation of energy

systems because it allows a rational inventory of system inefficiencies, contributing to

improved operation and design [1, 3]. Radiation heat transfer plays an important role in high

temperature engineering equipment and SLA of the radiative processes is a powerful tool for

system improvement.

The exergy of TR was first derived for solar energy (BR) and most of the work on this subject

is related to solar engineering applications [16, 27]. Although the formula for the exergy of

NBR is known, it has not been applied in quantitative engineering analysis.

3.1 Black body radiation

In 1964 Petela determined the exergy flux of unpolarised BR at temperature T [27]:

4 3 40 0 2

4 1

3 3BR

Wb T T T T

m

(37)

where T0 is the temperature of the environment, being the only relevant property of the

environment required to calculate the exergy of TR [35].

Gordon [68] obtained the above result by considering BR as a finite heat capacity (hot)

reservoir (photon gas) and applying a standard thermodynamic analysis to calculate the

maximum work output.

Since the beginning, exergy of TR was expressed in terms of an efficiency defined as the ratio

of exergy to energy [16, 27]:

4

0 04 11

3 3TR

TRTR

T Tb

i T T

(38)

This is the maximum conversion efficiency for undiluted (extraterrestrial) solar radiation

(BR). This formula is valid for any T > T0, and for T < T0 provided that

20

(T/T0) > (1/41/3) 0.63 [31]. Below this value the calculated efficiency will be greater than 1.

Equation (38 has been widely used and discussed through the years. Some researchers have

proposed modifications to it on the basis of the impossibility of obtaining such a high

efficiency [16, 23, 25, 69]. Edgerton [23] proposed the use of the Carnot factor and

considered that radiation pressure was also available for doing work:

01

41

3BR,

T

T

(39)

Jetter [25] recognised that the exergy flux of BR is given by Eq. (37, but arrived at a different

expression for the maximum conversion efficiency of thermal radiation equal to the Carnot

factor:

02 1BR,

T

T (40)

The result of Jetter was also obtained by De Vos and Pauwels [69], and Lewins [57] using

different approaches. Spanner, and Gribik and Osterle also proposed an alternative formula

for the maximum conversion efficiency [16, 27, 59, 70]:

03

41

3BR,

T

T (41)

Bejan [16] proved that the formulas of Eqs. (38, (40 and (41 are really valid for particular

conversion processes and that there was no conflict among them. The most widely used

formula to calculate the exergy of BR is Eq. (37 [20, 21, 27-29, 31, 35, 36, 58, 59, 71, 72],

derived from Petela’s formula (Eq. (38), although the difference in the results of the previous

equations is less than 2% in practical solar engineering applications [73].

Isvoranu and Bădescu [74] have presented recently an assessment of the exergy of TR, giving

emphasis to BR (diluted and undiluted). Following the conventional approach, these authors

developed several expressions for the conversion efficiency by considering different

21

processes and irreversibilities, confirming the result of Bejan [16] that the different formulas

are applicable to particular processes.

In the field of solar energy conversion, de Vos [60] defines the conversion efficiencies for

thermal and photovoltaic converters using energy-related quantities:

4,s

thc Tf

W

4,

sphc TCf

W

(42)

where W is the power produced per unit of surface area, Ts is the temperature of the sun, f is

the dilution factor (f = 2.16 10-5 for the Earth), and C is the concentration factor (C ≥ 1).

It is highlighted that the efficiency of photovoltaic conversion in significantly lower than the

efficiency of thermal conversion. This difference is expected to be reduced when considering

exergy quantities, since photovoltaic converters produce electric power directly.

3.2 Non-black body radiation

Exergy of NBR was studied by Karlsson [37], who obtained the expression for the spectral

directional exergy radiation intensity. Wright et al. [33] questioned this result and proposed a

formula for calculating the exergy flux of GR. Candau [36] confirmed Karlsson’s definition

of the exergy of NBR obtained two decades before.

Karlsson [37] considered the environment as BR of temperature T0 and analysed the radiative

exchange of an unpolarised monochromatic beam impinging perpendicularly on a black

surface at T0. The temperature of the beam is given by Eq. (5. If the beam-surface system

reaches equilibrium without doing work, all of its exergy (beam’s exergy) will be transformed

into lost work. Using the energy and entropy balances of the system as well as the Gouy-

Stodola theorem, he obtained the definition of spectral directional exergy radiation intensity

for unpolarised radiation:

22

0 0 0 2' ' ' ' '

,BR ,BR

Wb i i T T l l T

m sr m

(43)

Using the definitions of Eqs. (1, (2, (6 and (7, the above expression can be written as [37]:

0

00 04

2 11

1 1

' '

' '

' x x,' ' '

, ' x x

Tkc e eb T x ln

T e e

(44)

where the x variables are defined according to Eq. (8:

00

'',

hcx

kT and '

'

hcx

kT

(45)

with

0 0' ',T T T (46)

Following the traditional approach, Karlsson defined a quality factor as the ratio of spectral

directional exergy and energy [37]:

00 0

1 11

1

' '

'

x x' ''

' ' ' x

b x e eln

i x x e

(47)

This factor is always positive for 0' '

,T T (T T0) and equal to zero for 0' '

,T T (T = T0).

Karlsson obtained the same result of Petela (Eq. (38) by defining a mean BR quality factor

[37]:

4

0 0 0

0

4 11

3 3

'

BR'

b c dT T

T Ti c d

(48)

The integrand in the numerator is the spectral exergy density and that of the denominator is

the spectral energy density.

23

Candau [36] used an absorber-Carnot engine system to derive the monochromatic or spectral

directional exergy radiation intensity, obtaining the same result as Karlsson (Eq. (43).

3.2.1 Grey body radiation

The dimensionless monochromatic temperature of diffuse GR is shown in Figure 6, which is

calculated from Eqs. (5 and (17. It is evident from this figure that monochromatic temperature

of diffuse GR is lower for decreasing emissivity at any surface temperature. As wavelength

increases the temperature decreases from its BR value, although the effect is stronger for

higher temperatures. This behaviour was first observed by Zhang and Basu [38].

The spectral quality factor for diffuse GR is shown in Figure 7, using a value of 298.15 K for

T0. It is observed in this figure that the spectral quality factor for diffuse GR decreases with

wavelength and that for a given emissivity it is higher for higher temperatures. An important

feature of this figure is that the quality factor increases as emissivity is augmented for all

wavelengths and its maximum value corresponds to BR, which means that the exergy-to-

energy ratio is higher for BR than for GR. This behaviour suggests that NBR has a lower

exergy-to-energy ratio than BR.

Directional exergy radiation intensity and exergy flux are defined in the same way as its

entropy counterparts (Eqs. (11 and (12). The directional exergy radiation intensity for diffuse

GR is found by substituting Eqs. (2, (7, (17 and (18 into Eq. (43 and performing the

integration. After using Eqs. (11 and (28, one obtains:

4

4 0 04

4 45 11 2 311 0 175

3 4 3'GR

T Tb T . . ln

T T

(49)

The exergy flux for this simple case of diffuse radiation will be:

'GR GRb b (50)

24

Wright et al. [33] affirm that Eq. (43 gives a level of work output that cannot be theoretically

achieved because part of the entropy production calculated is inherent and cannot be avoided

since it appears that it is not possible for NBR to exist in equilibrium with matter, even with

its own emitting material. These authors obtain an alternative expression for the exergy flux

of diffuse or isotropic NBR based on the analysis of an enclosed NBR system in a two step

process: 1) spontaneous transformation of NBR into BR with the same energy but higher

entropy upon contact with absorbing material (irreversible process); 2) reversible conversion

of the BR as the system equilibrates with the environment (BR at T0). The resulting exergy

flux for diffuse radiation is:

1 4 3 4 40 0

4

3 3WNBR NBR NBRb i T i T

(51)

This result applied to GR yields [33]:

4

4 3 4 0 04 1

3 3WGR

T Tb T

T T

(52)

Using these results the authors concluded that NBR has a lower energy content or exergy-to-

energy ratio than that of BR, and that the exergy of NBR can be a very small fraction of its

energy for low emissivities [33]. These findings are the same as those mentioned in the

discussion of Figure 7.

In order to compare the two expressions for diffuse GR the exergy-to-energy ratio or

maximum conversion efficiency is calculated for both Eqs. (50 and (52 using the flux form of

Eq. (17 (see Figure 8).

For BR (ε = 1) both results coincide and are equivalent to that of Eq. (38. The curves in

Figure 8 have a lower temperature limit below which they are not valid. This limit increases

with emissivity reduction in such a way that for low emissivities the ratio is valid only for

temperatures slightly higher than T0. As expected, the exergy of BR is zero for the

25

temperature of the environment. The curves obtained using Karlsson’s formula are always

higher than that corresponding to the approach of Wright et al., since the former author

considers the total lost work as exergy, while the latter don’t include the portion

corresponding to the transformation of GR into BR. The minimum value for this ratio is

always zero when using the formula of Wright et al., while for the formula of Karlsson it is

greater than zero for ε < 1 and increases as emissivity is reduced.

4. Radiative transfer processes

Radiation heat transfer is of primary interest for the design and evaluation of systems where

the useful effect is obtained by means of the interaction of a high-temperature energy source

with energy carriers at lower temperatures. Radiative transfer processes in engineering

practice cover a wide number of applications, many of which include direct exchange

between surfaces with the participation of a medium (gas). With the use of SLA radiative

transfer can be enhanced in order to increase energy conversion efficiency. In this section we

present the fundamentals of SLA in radiative exchange between surfaces as well as the

radiative transfer equations for entropy and exergy.

4.1 Radiative exchange between surfaces

Emission and absorption processes are inherently irreversible [17] and sometimes difficult to

grasp because entropy transfer with radiation from (to) a surface does not have a general form

since it depends strongly on the characteristics of the incident, emitted and reflected fluxes

[67].

4.1.1 Entropy flux and entropy generation

The net radiative energy flux from a surface is found by subtracting the incoming flux from

the outgoing flux (radiosity), which contains both reflected and emitted fluxes. The net

entropy flux cannot be evaluated in the same way since spectral entropy intensity depends on

the net energy flux travelling in one direction (Eq. (1). This flux can be the superposition of

26

several fluxes, and in general depends on the characteristics of incoming radiation, on the

temperature and on radiative properties of the surface material [22]. As a consequence, the

entropy flux cannot be based on the superposition of individual entropy fluxes, each

calculated for a different energy flux: the outgoing entropy flux from a surface must be

calculated considering the combined emitted and reflected energy spectrums [22, 33]. Zhang

and Basu [38] stress that both the entropy intensity and the monochromatic temperature must

be defined based on the combined intensity, which is the net intensity in every point of space

in a given direction and spectral interval.

In order to calculate the entropy transfer between surfaces, Zhang and Basu [38] assume that

the ratio of the entropy intensity of an individual ray to that of the combined radiation is equal

to the ratio of the respective individual (energy) intensity to the combined intensity. This way

the absorbed, emitted, and reflected entropy can be calculated. The ratio of emitted to

combined spectral (energy) intensity is defined for outgoing radiation as:

' ' ',BRi i (53)

This allows the calculation of emitted entropy radiation intensity:

' ' ' ',emitl l i (54)

This expression confirms that the energy emitted by a surface depends only on its temperature

and emissivity, while the emitted entropy also depends on the incoming radiation [38]. The

emitted entropy flux can be obtained by integration of the above equation over spectrum and

solid angle.

Using these ideas, Zhang and Basu [38] analysed the radiative transfer between two large

parallel plates at temperatures T1 and T2 (T1 > T2), separated by vacuum, with diffuse-grey

surfaces of emissivities ε1 and ε2. The combined intensities for incoming and outgoing

27

radiation (respect to plate 1), denoted with superscripts – and +, respectively, are obtained by

means of the ray tracing method [38]:

1 1 1 2 2

1 2

1

1 1 1

' ',BR , ,BR,' i i

i

(55)

1 2 1 2 2

1 2

1

1 1 1

' ',BR , ,BR,' i i

i

(56)

Entropy fluxes are calculated as:

0

'l l d

(57)

0

'l l d

(58)

where the entropy intensities are calculated by replacing the intensities of Eqs. (55 and (56

into Eq. (1.

Liu and Chu [13] performed a spectral entropy balance in an opaque surface, obtaining the

spectral entropy generation rate at the solid boundary:

'A 'gen,

M

idS dA d l d

T

(59)

where TM is the local temperature of the surface medium.

The authors used this result to study the entropy generation in a square enclosure filled with a

semitransparent gas, including entropy generation by absorption and emission, and by

scattering.

4.1.2 Exergy flux and exergy destruction

Liu and Chu [12] determined the net local rate of exergy gain of the wall medium (M) due to

absorption of radiation:

28

01A 'M ,

M

Tdb dA d i d

T

(60)

Also, using the result of Eq. (59 they found the spectral radiative exergy destruction at a wall

surface to be consistent with the Gouy-Stodola theorem:

0A Ad , gen,db T dS (61)

where subscript d refers to exergy destruction.

To date, there is no knowledge of quantitative application of these equations or any other

approach for the exergy analysis of radiative transfer between surfaces.

4.2 Radiative entropy transfer in participative media

4.2.1 Radiative entropy transfer equation

Planck was the first to investigate the irreversible interaction of radiation and matter [17, 39],

but Wildt [41] was the first to formulate the transfer equation for radiant entropy in terms of

the source function, in the context of stellar atmospheres in equilibrium. This work was

continued by Oxenius [39], who treated the irreversible interaction of a radiation field with

matter by means of a microscopic model of a self-excited isothermal gas in steady state,

composed of atoms with only two discrete energy levels. Oxenius also formulated the

radiative entropy transfer equation (RETE) in terms of the source function and found that the

energy exchange of a ray with a material system is always an irreversible process and that the

local entropy generation, integrated over the entire spectrum is always positive. Another

important result of this work is that “the local entropy production of an anisotropic radiation

field is greater than that of the corresponding isotropic radiation field”. Sen [42] started from

the model of Oxenius and added a background continuum. This author affirms that the

divergence of radiant entropy is the exact expression for the rate of entropy production per

unit volume. The main findings of his work are that the addition of a continuum causes an

29

overall increase in the entropy production, and that the model proposed, as well as that of

Oxenius, can give negative values of spectral entropy production for some situations. This is

attributed to the fact that the thermodynamic description of the system may not be complete

[42].

Kröll [40] studied the interaction of a non-polarised radiation field and a plasma. The

radiation field was described as an ideal Bose gas and the plasma as an ideal gas. The

interactions considered were absorption, emission and scattering of radiation. This author

wrote the spectral entropy generation as a bilinear form in generalised forces and fluxes,

analogous to that of entropy generation in classical irreversible thermodynamics, by

introducing generalised temperatures.

The RETE is based on the radiative transfer equation, which can be written as [75]:

4

's ,' ' ' ' ' '

e , a , ,BR

KdiK i K i i , d

ds

(62)

where s is the direction of exchange, is the phase function, and a ,K , s ,K and e,K are

the spectral absorption, scattering and extinction coefficients, respectively.

The RETE can be obtained from Eq. (62 by means of the relation of Eq. (3 [8]:

4

4 '

' ''' ',BR s , ' '

e , a ,' ' '

ii Kdl iK K , d

ds T T T

(63)

4.2.2 Radiative entropy generation

Entropy generation in a volume dV is determined as the sum of the entropy increments of the

radiation field and matter [8, 13]:

V V Vgen, R, M ,dS dS dS (64)

where the subscripts M and R refer to the medium and to the radiation field, respectively.

30

The local net increment of entropy in the radiation field in a volume dV can be obtained by

integrating Eq. (63 over a 4π solid angle [8, 13]:

4 44

' ''',BR s ,V ' '

R , e , a ,' ' '

ii KidS dVd K K , d d

T T T

(65)

The local net increment of entropy in the matter filling the volume dV is [8, 13]:

4

a ,V ' 'M , ,BR M

M M

K dVdQdS i i T d

T T

(66)

where Q is the spectral energy exchange between the medium and the radiation field by

absorption and emission.

Entropy generation may be separated into two contributions: absorption-emission and

scattering. The mechanisms of emission and absorption are analysed together since they have

no separated existence [8]. The part associated with emission and absorption is [8, 13]:

4

1 1ea ' 'gen, a , ,BR M '

M

dS K dVd i T i dT T

(67)

The part associated with scattering is [8, 13]:

4 4

1 1

4s ' ' ' ' 'gen, s , '

dS K i , d i d dVdT

(68)

Caldas and Semião [8] used the model described here to calculate the entropy generation in a

rectangular enclosure bounded by grey walls and filled with a hot grey anisotropic medium.

The major findings of their work are that emission and absorption are the main sources of

irreversibility, with a much smaller contribution of scattering, and that entropy change in the

radiation field is much larger than that of matter. Also important is that entropy generation is

higher at zones presenting strong gradients of properties and higher interaction between

matter and radiation.

31

Liu and Chu [13] extended the analysis of Caldas and Semião to include entropy generation at

solid surfaces. They calculated the spectral entropy generation rates due to radiation at solid

boundaries, to emission and absorption, and to scattering:

4

'A 'gen,

MA

iS l d dA

T

(69)

4

1 1ea ' 'gen, a , ,BR '

MV

S K i i d dVT T

(70)

4 4

1 1

4s ' ' ' ' 'gen, s , '

V

S K i , d i d dVT

(71)

These authors used the above model to calculate radiative entropy generation in an enclosure

bounded by diffuse grey isothermal surfaces. As Caldas and Semião, they found that the main

contributions to entropy generation are emission and absorption processes. They also found

that with the increase of wall emissivity, spectral entropy generation rate at solid walls

decreased while that corresponding to the medium increased. Numerical results were

satisfactorily validated with the thermodynamic results for each case.

4.3 Radiative exergy transfer in participative media

4.3.1 Radiative exergy transfer equation

Liu and Chu [12] started from the spectral directional radiative exergy intensity obtained by

Karlsson [37] and Candau [36] (Eq. (43) to obtain:

0

' ' 'db di dlT

ds ds ds (72)

The radiative exergy transfer equation can be obtained by substituting Eqs. (62 and (63 into

the above expression, as indicated by Liu and Chu [12]:

32

0 0

4

11 1

4

'' ' ' ' ' ' '

a , ,BR M s ,' '

db T TK i i T K i i , d

ds T T

(73)

The first term on the right hand side of the above equation represents the contribution of

emission and absorption processes to radiative exergy intensity. The second term represents

the contribution of scattering.

4.3.2 Radiative exergy destruction

Liu and Chu [12] obtained the expression for spectral radiative exergy destruction in a volume

dV due to the interaction of radiation and matter:

0

4

0

4 4

1 1

1 1

4

V ' 'd , a , ,BR M '

M

' ' ' ' 's , '

db K T dVd i T i dT T

K T dVd i , d i dT

(74)

Using Eqs. (67 and (68, the spectral radiative exergy destruction in volume dV can be written

in a way consistent with the Gouy-Stodola theorem:

0V ea sd , gen, gen,db T dS dS (75)

To date, there are no quantitative applications of these equations to radiative transfer

processes.

4.4 Combined heat transfer modes

The most recent applications of SLA of TR involve the entropy generation in combined heat

transfer processes. Caldas and Semião [14] present an analysis of the interaction of turbulence

and radiation on radiative entropy generation. Turbulence affects radiative transfer through

fluctuations of temperature that influence the radiation field, and of species concentration and

pressure that also affect the absorption coefficient. These authors use an averaging procedure

for properties and apply the model to the same case as in reference [8]. They found that

interaction between turbulence and radiation changes significantly the magnitude of radiative

33

entropy generation and heat transfer, although its pattern is not significantly modified. The

effect is directly proportional to turbulence intensity.

Ben Nejma et al. [9] perform a numerical calculation of entropy generation due to combined

gas radiation and forced convection in participating media for a laminar flow of a non-grey

gas in the entrance region of a duct formed by two parallel plates. The radiative SLA follows

from the results of references [8] and [13]. These researchers found that more entropy

generation is obtained when the temperature difference between the gas and boundaries

(diffuse grey walls) is increased. By comparing heating and cooling modes they observed a

much higher entropy generation for heating conditions. They also found that the greater the

separation between plates, the higher the entropy generation of the system. The effect of

reducing gas pressure and diminishing emissivity of the walls was a decrease in radiative

entropy generation.

5. Conclusions

The study of radiation using the second law of thermodynamics dates back to the end of

nineteenth century and its theoretical foundations are based on the work of Planck at the

beginning of the twentieth century. The original results of Planck regarding entropy and

temperature of monochromatic radiation have been confirmed and extended in the last

decades by many researchers in several fields of knowledge. Nowadays, such theoretical

developments are being combined with practical heat transfer calculation methods in order to

obtain quantitative results for engineering applications. This has permitted to gain

thermodynamic insight into more and more complex problems, proving the usefulness of

second law analysis.

This work presents the fundamental second law aspects of thermal radiation in three major

thematic blocks. The first part treats the concept of thermal radiation entropy and its historical

development and application to black body and non-black body radiation, as well as the

34

calculation of entropy fluxes in engineering analysis. This is followed by a section on

radiative exergy, its definition and expressions for black body and non-black body radiation.

In the development of the concepts related to non-black body radiation especial emphasis is

put on diffuse grey radiation because of its interest to practical engineering situations. The

final part is dedicated to entropy and exergy analyses of radiative transfer processes,

considering the exchange between surfaces and the presence of a participative medium, and

presenting some brief comments on the application to combined heat transfer modes.

The theoretical basis for second law analysis of radiative transfer processes is well established

but its complex nature hinders the application to practical engineering systems. As a

consequence there are little numerical results reported, which limits the understanding of the

true potential of this refined tool.

According to the author’s knowledge this is the first time that all these concepts relating

second law analysis of thermal radiation are put together and interrelated in order to give a

general view of the subject.

Acknowledgements

Andrés Agudelo wants to acknowledge the support of the Programme Alan, the European

Union Programme of High Level Scholarships for Latin America, under scholarship No.

E07D403613CO.

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41

Figure captions

Figure 1. Spectral directional entropy radiation intensity for BR

Figure 2. Graphical representation of Eq. 10

Figure 3. Spectral directional entropy radiation intensity for GR

Figure 4. Error of using Eq. 30 to calculate the entropy flux of GR

Figure 5. Dimensionless coefficient for comparison of TR and HC entropy flux for GR

Figure 6. Monochromatic temperature of diffuse GR

Figure 7. Spectral quality factor of diffuse GR

Figure 8. Exergy-to-energy ratio of diffuse GR