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: [email protected]
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
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