Energy 35 (2010) 5151e5160
Contents lists available at ScienceDirect
Energy
journal homepage: www.elsevier.com/locate/energy
SLA (Second-law analysis) of transient radiative transfer processes
D. Makhanlall, L.H. Liu*, H.C. Zhang
School of Energy Science and Engineering, Harbin Institute of Technology, 92 West Dazhi Street, Harbin 150001, People’s Republic of China
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 14 April 2010
Received in revised form
8 July 2010
Accepted 30 July 2010
Available online 15 September 2010
This paper concerns a SLA (second-law analysis) of transient radiative heat transfer in an absorbing,
emitting and scattering medium. Based on Planck’s definition of radiative entropy, transient radiative
entropy transfer equation and local radiative entropy generation in semitransparent media with uniform
refractive index are derived. Transient radiative exergy transfer equation and local radiative exergy
destruction are also derived based on Candau’s definition of radiative exergy. The analytical results are
consistent with the GouyeStodola theorem of classical thermodynamics. As an application concerning
transient radiative transfer, exergy destruction of diffuse pulse radiation in a semitransparent slab is
studied. The transient radiative transfer equation is solved using the discontinuous finite element based
discrete ordinates equation. Transient radiative exergy destruction is calculated by a post-processing
procedure.
Ó 2010 Elsevier Ltd. All rights reserved.
Keywords:
Second-law analysis
Transient radiative entropy generation
Transient radiative exergy destruction
Diffuse pulse radiation
Transient radiative transfer equation
1. Introduction
SLA (Second-law analysis) is a very effective method to
analyze the process of energy transfer. Since SLA tracks the loss
of work potential through entropy generation, it provides new
insights that cannot be obtained from energy analysis alone.
Bejan [1e3] investigated extensively entropy generation and
presented systematically concept and optimization method for
entropy generation minimization. Recently, the principle of
maximum irreversible entropy has also been linked to the Bejan
theory [4e6].
Thermal radiation is an important factor in thermodynamic
analysis of high-temperature systems. Correct evaluation of
radiation entropy generation is important in determining the
second-law performance of these systems. Planck [7] was the
first to investigate the interaction of light and matter with
respect to its irreversibility. Recently, Caldas and Semiao [8]
presented a numerical simulation method for entropy generation analysis of steady-state radiative transfer in participating
media with uniform refractive index. This method is completely
compatible with standard radiative transfer calculations such as
DOM (discrete ordinates method). Liu et al. [9] extended this
method to analyze steady-state radiative entropy generation in
enclosures filled with semitransparent media. Radiative entropy
* Corresponding author. Tel.: þ86 451 86402237.
E-mail address: lhliu@hit.edu.cn (L.H. Liu).
0360-5442/$ e see front matter Ó 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.energy.2010.07.051
generation in semitransparent graded index medium was also
studied recently by Liu [10]. A radiative entropy transfer equation for semitransparent graded index medium was derived and
verified. The need for re-stating fundamental equations to cope
with the unique character of thermal radiation was noted by
Wright [11]. Wright re-formulated Clausius expression to allow
general applicability to all cycles with processes involving any
form of heat transfer. The general exergy balance equation for
thermodynamic systems was also re-stated by Wright et al.
[12,13] so that it correctly applies to radiative heat transfer. The
analysis of entropy generation in the complex problems of
combined non-grey gas radiation and forced convection, and
combined non-grey gas radiation and free convection was conducted by Nejma et al. [14,15]. All the main concepts relating to
SLA of thermal radiation developed so far were very recently
reviewed for the first time, by Agudelo and Cortez [16]. An in
depth study on the concepts of radiative entropy generation and
radiative exergy destruction and their application in the analysis
of various high-temperature systems was recently conducted by
Chu et al. [17].
Though most of the research on radiative transfer concerns
cases for which the transient term in the radiative transfer equation can be neglected, there has been growing interest in investigating TRT (transient radiative transfer) in semitransparent
media recently. This effort has been motivated principally due to
the development and application of short-pulse lasers in many
emerging applications. Many numerical methods for solving the
TRT equation have been developed. Guo et al. [18] simulated
5152
D. Makhanlall et al. / Energy 35 (2010) 5151e5160
Nomenclature
A
Atot
l
AAl I
AAl II
Aae
l
Asl
aAl
aae
l
asl
c
eAM;l
eARf ;l
eVM;l
eVR;l
eVRf ;l
Gl
h
Il
Ib,l
kb
D
Ll
Surface area
Transient spectral radiative exergy destruction of the
system
Transient spectral radiative exergy destruction at plate I
Transient spectral radiative exergy destruction at plate II
Transient spectral radiative exergy destruction of the
system due to absorption-emission in medium
Transient spectral radiative exergy destruction of the
system due to scattering in medium
Transient local spectral radiative exergy destruction at
wall matter
Transient local spectral radiative exergy destruction
due to absorption-emission in medium
Transient local spectral radiative exergy destruction
due to scattering in medium
Speed of light
Transient local net increment of spectral exergy in the
wall matter
Transient local net increment of spectral radiative
exergy flow in the radiative field at the wall
Transient local net increment of spectral radiative
exergy in the medium
Transient local net increment of spectral radiative
exergy in the radiative field of the medium
Transient local net increment of spectral radiative
exergy flow in the radiative field of the medium
Transient spectral incident radiation
Planck’s constant
Spectral radiative intensity
Spectral blackbody intensity
Boltzmann’s constant
Distance between plate I and plate II
Spectral radiative entropy intensity
three-dimensional TRT in scattering-absorbing media by the MC
(Monte Carlo) method. Pilon et al. [19] developed a modified
method of characteristics to solve the TRT. Wu et al. [20] and Tan
and Hsu [21] used a time-dependent IE (integral equation)
formulation to develop a modeling method of the TRT. Very
recently, the DOM and the FVM (finite volume method) were also
extended to solve the TRT [22e25]. However, due to hyperbolic
wave characteristics of the TRT equation, the propagation speed of
light within medium is predicted with some inaccuracies with the
DOM and the FVM. Recently, Li et al. [26e28] introduced a DFEM
(discontinuous finite element method) to solve steady-state radiative heat transfer in semitransparent medium. In DFEM the
continuity at inter-element boundaries is relaxed so that field
variable is considered discontinuous across the element boundaries. Liu et al. [29,30] extended the DFEM to solve the TRT
problems in semitransparent medium. Numerical examples illustrated good performance of the DFEM in solving TRT problems.
In this work, the authors extend the SLA of steady-state radiative
transfer in participating media carried out by Liu et al. [31]. Here,
thermodynamic analysis of the TRT in semitransparent absorbing,
emitting and scattering media with uniform refractive index is
considered. As in the case of steady-state radiative transfer, the
analytical results for entropy generation and exergy destruction are
verified by checking for consistency with classical thermodynamic
theorem. An application concerning diffuse pulse radiation transfer
in an absorbing, non-emitting, and isotropically scattering media is
also studied.
Lbl
M
nw
r
0
s,s
SAM;l
SAG;l
SARf ;l
SVM;l
SVG;l
SVR;l
SVRf ;l
t
t*
T
T0
Tl
V
wm
Spectral blackbody radiative entropy intensity
Number of discrete directions
Unit outward normal vector of boundary wall
Spatial position vector
Direction vectors
Transient local net increment of spectral radiative
entropy in the radiative field at the wall
Transient local spectral radiative entropy generation at
the wall
Transient local net increment of spectral radiative
entropy flow in the radiative field at boundary
Transient local net increment of spectral radiative
entropy in the medium
Transient local spectral radiative entropy generation in
the medium
Transient local net increment of spectral radiative
entropy in the radiative field of the medium
Transient local net increment of spectral radiative
entropy flow in the radiative field of the medium
Time
Dimensionless time
temperature
Reference temperature
Spectral radiation temperature
Volume
Weight of direction m
Greek Symbols
Spectral absorption coefficient
Spectral scattering coefficient
Wavelength
Scattering phase function
Single scattering albedo
Spectral radiative exergy intensity
Solid angles
ka,l
ks,l
l
Fl
u
Jl
0
U,U
2. Entropy generation in TRT processes
The following analysis concerns incoherent and unpolarized
radiation. The TRT equation in semitransparent medium can be
written as [32]
1 DIl ðr; s; tÞ
1 vIl ðr; s; tÞ
¼
þ s,VIl ðr; s; tÞ
c
Dt
c
vt
¼ ðkal þ ksl ÞIl ðr; s; tÞ þ kal Ibl ðTðr; tÞÞ
Z
0
k
þ sl
Il ðr; s0 ; tÞFl ðs; s0 ; tÞdU
4p
ð1Þ
4p
where, kal and ksl are the spectral absorption and the spectral
scattering coefficients, respectively; Il is the spectral radiative
intensity, Ibl is the spectral blackbody intensity, Fl is the scattering
phase function, c is the speed of light in the medium, U is the solid
angle, r is a position vector, and s is a unit direction vector.
The spectral radiation temperature corresponding to any spectral radiative intensity Il(r,s,t) is defined as
"
#1
5
hc
2hc2 l
Tl ðr; s; tÞ ¼
ln
þ1
lkb Il ðr; s; tÞ
(2)
where, h is Planck’s constant, kb is Boltzmann’s constant, and l is
wavelength.
D. Makhanlall et al. / Energy 35 (2010) 5151e5160
5153
2.1. The transient radiative entropy transfer equation
A radiation beam not only carries energy, but also entropy and
exergy. The spectral radiative entropy intensity carried by a radiation beam with intensity Il is defined by Planck [4] as
Il ðr; sÞ
Il ðr; sÞ
4
Ll ½Il ðr; sÞ ¼ 2kb cl
þ
1
ln
þ
1
2 5
2 5
2hc
l
2hc l
Il ðr; sÞ
Il ðr; sÞ
ln
5
5
2hc2 l
2hc2 l
ð3Þ
In a transient radiation field, the intensities changes with time.
Accordingly,
Il ðr; s; tÞ
I ðr; s; tÞ
4
Ll ½Il ðr; s; tÞ ¼ 2kb cl
þ 1 ln l
þ1
5
5
2
2
2hc
l
2hc l
Il ðr; s; tÞ
Il ðr; s; tÞ
ln
5
5
2hc2 l
2hc2 l
ð4Þ
The derivative of Eq. (4) with respect to radiative intensity Il equals
the inverse of spectral radiation temperature:
"
#
5
dLl ½Il ðr; s; tÞ
k l
2hc2 l
1
¼ b ln
þ1 ¼
hc
Il ðr; s; tÞ
dIl ðr; s; tÞ
Tl ðr; s; tÞ
(5)
Fig. 1. Transient radiative energy and entropy transfer through a differential volume.
Eq. (5) is always positive, proving that in a time-varying radiation
field the entropy intensity always increases with increasing radiative intensity. The transfer process of radiative entropy is similar to
that of radiation energy (see Fig. 1). Using Eqs. (1) and (5), the
transient spectral radiative entropy transfer equation can be written
as follows:
1 DLl ðr; s; tÞ
1 vLl ðr; s; tÞ
¼
þ s,VLl ðr; s; tÞ
c
Dt
c
vt
I ðr; s; tÞ
I ðTðr; tÞÞ
¼ ðkal þ ksl Þ l
þ kal bl
Tl ðr; s; tÞ
Tl ðr; s; tÞ
Z
0
ksl
Il ðr; s0 ; tÞ
0
Fl ðs; s ; tÞdU
þ
p
4
Tl ðr; s; tÞ
and the transient local net increment of spectral radiative entropy
flow in the radiative field can be written as:
dSVRf ;l ðr; tÞ ¼ dV
Z
dA
ð6Þ
In a pure radiation problem, entropy (exergy) variations in
radiative field and matter are considered. As can be seen in Fig. 1,
after radiation propagates through a differential volume dV, the
transient local net increment of spectral radiative entropy in the
radiative field can be written as
4p
4p
Z
ðnA ,sÞLl ðr; s; tÞdUdA
(8)
4p
where dA is surface area of the differential volume dV, and nA is an
outward normal unit vector of surface dA.
Through radiation absorption and emission, matter gets or
releases heat and entropy. By definition, the transient local net
increment of entropy flow in the matter under consideration as
spectral radiation goes through a differential volume dV is given as
2.2. Transient radiative entropy generation in semitransparent
medium
dSVR;l ðr; tÞ ¼ dV
s,VLl ðr; s; tÞdU
¼
4p
Z
Z
dSVM;l ðr; tÞ ¼ k dV
dQl ðr; tÞ
¼ al
Tðr; tÞ
Tðr; tÞ
Z
½Il ðr; s; tÞ Ibl ðTðr; tÞÞdU
4p
(9)
1 vLl ðr; s; tÞ
dU
c
vt
(7)
By combination of Eqs. (7)e(9), the transient local spectral radiative
entropy generation in a differential volume dV can be written as:
dSVG;l ðr; tÞ ¼ dSVR;l ðr; tÞ þ dSVRf ;l ðr; tÞ þ dSVM;l ðr; tÞ
2
3
Z
Z
Z
0
k dV
Il ðr; s; tÞ
Ibl ðTðr; tÞÞ ksl
Il ðr; s0 ; tÞ
0
4
Fl ðs; s ; tÞdU 5 þ al
½Il ðr; s; tÞ Ibl ðTðr; tÞÞdU
þ kal
þ
¼ dV
ðkal þ ksl Þ
4p
Tl ðr; s; tÞ
Tl ðr; s; tÞ
Tl ðr; s; tÞ
Tðr; tÞ
4p
4p
4p
(10)
5154
D. Makhanlall et al. / Energy 35 (2010) 5151e5160
Similar to that in steady-state, the transient local spectral radiative
entropy generation given by Eq. (10) can be divided into two parts.
One is due to absorption and emission, and the other is due to
scattering. Accordingly,
dSae
ðr; tÞ ¼ kal dV
G;l
Z
½Ibl ðTðr; tÞÞ Il ðr; s; tÞ
4p
where, T(rw,t) is the wall temperature. By combination of Eqs. (14)
and (15), the transient spectral radiative entropy generation at
opaque walls can be written as
1
1
dU
Tl ðr; s; tÞ Tðr; tÞ
9
82
3
Z
=
< Z I ðTðr; tÞÞ
p
I
ðr;
s;
tÞ
ð4
I
ðr;
tÞ
G
ðr;
tÞÞ
l
l
bl
bl
¼ kal dV 4
dU dU 5 ;
:
Tl ðr; s; tÞ
Tl ðr; s; tÞ
Tðr; tÞ
4p
dSsG;l ðr; s; tÞ
¼ ksl dV
8
< Z
:
4p
3 9
2
Z
=
0
1
0
0
41
Il ðr; s ; tÞFl ðs; s ; tÞdU Il ðr; s; tÞ5dU
;
Tl ðr; s; tÞ 4p
Gl ðr; tÞ ¼
(12)
4p
where,
Z
(11)
4p
Il ðr; s; tÞdU
(13)
4p
dSAG;l ðrw ; tÞ ¼ dSARf ;l ðrw ; tÞ þ dSAM;l ðrw ; tÞ
Z Il ðrw ; s; tÞ
¼ dA
Ll ðrw ; s; tÞ ðnw ,sÞdU
Tðrw ; tÞ
(16)
4p
is the transient spectral incident radiation.
2.3. Transient radiative entropy generation in opaque solid surfaces
3. Exergy destruction in TRT processes
Fig. 2 illustrates the transient radiative entropy transfer process
on a differential surface area dA. The transient local net increment
of spectral radiative entropy flow in the radiative field at the opaque wall surface can be written as follows
3.1. The transient radiative exergy transfer equation
dSARf ;l ðrw ; tÞ ¼ dA
Z
Ll ðrw ; s; tÞðnw ,sÞdU
Jl ðr; sÞ ¼ Il ðr; sÞ Ibl ðT0 Þ T0 fLl ½Il ðr; sÞ Lbl ½Ibl ðT0 Þg
(14)
4p
and the transient local net increment of spectral radiative entropy
flow in the wall matter under consideration after radiation
processes at a differential surface dA is given by
dQ ðrw ;tÞ
dA
dSAM;l ðrw ;tÞ ¼ l
¼
Tðrw ;tÞ
Tðrw ;tÞ
Z
The spectral radiative exergy intensity is defined by Candau [33]
as
(17)
where, T0 is the temperature of the reference environment. The
definition of the environment in conventional exergy analysis
completely suffices for thermal radiation exergy analysis [12]. The
transient form of Eq. (17) is readily obtained:
Jl ðr; s; tÞ ¼ Il ðr; s; tÞ Ibl ðT0 Þ T0 fLl ½Il ðr; s; tÞ Lbl ½Ibl ðT0 Þg
(18)
Il ðrw ;s;tÞðnw ,sÞdU
4p
Fig. 2. Transient radiative entropy transfer at a unit surface area.
(15)
The similarities of exergy transfer with energy and entropy
transfer can be observed from Fig. 3. Taking the substantial (total)
derivative of Eq. (18), and using Eqs. (1) and (6), the transient
spectral radiative exergy transfer equation can be written as
follows:
Fig. 3. Transient radiative exergy transfer through a differential volume.
D. Makhanlall et al. / Energy 35 (2010) 5151e5160
5155
1 DJl ðr; s; tÞ
1 DIl ðr; s; tÞ T0 DLl ðr; s; tÞ
¼
c
c
Dt
c
Dt
Dt
3
2
Z
0
T0
T0
1
0
0
4
¼ kal 1 Il ðr; s ; tÞFl ðs; s ; tÞdU 5
½I ðr; s; tÞ Ibl ðTðr; tÞÞ ksl 1 Il ðr; s; tÞ Tl ðr; s; tÞ l
Tl ðr; s; tÞ
4p
(19)
4p
3.2. Transient radiative exergy destruction in semitransparent
media
As can be seen in Fig. 3, after radiation propagates through
a differential volume dV, the transient local net increment of
spectral radiative exergy in the radiative field can be written as:
T
deVM;l ðr;tÞ ¼ 1 0 dQl ðr;tÞ
Tðr;tÞ
Z
T
½Il ðr;s;tÞIbl ðTðr;tÞÞdU
¼ kal dV 1 0
Tðr;tÞ
(22)
4p
By combination of Eqs. (20)e(22), transient local spectral radiative
exergy destruction in volume can be written as:
daVl ðr;tÞ ¼ deVR;l ðr;tÞdeVRf ;l ðr;tÞdeVM;l ðr;tÞ
Z ¼ kal T0 dV
4p
3
2
Z
Z
0
1
1
1
1
0
0
4
½I ðr;s;tÞIbl ðTðr;tÞÞdU þ ksl T0 dV
Il ðr;s ;tÞFl ðs;s ;tÞdU Il ðr;s;tÞ5dU
Tðr;tÞ Tl ðr;s;tÞ l
Tl ðr;s;tÞ 4p
4p
4p
(23)
Z
In Eq. (23), the following relation has been used
1 vJl ðr; s; tÞ
dU
c
vt
deVR;l ðr; tÞ ¼ dV
4p
(20)
Z
and the transient local net increment of spectral radiative exergy
flow in the radiative field can be written as:
Z
s,VJl ðr; s; tÞdU
deVRf ;l ðr; tÞ ¼ dV
Z
4p
Z
ðnA ,sÞJl ðr; s; tÞdUdA
¼
dA
(21)
4p
2
4Il ðr; s; tÞ 1
4p
Z
3
0
Il ðr; s ; tÞFl ðs; s ; tÞdU 5dU ¼ 0
0
0
(24)
4p
As in the steady-state case [31], the radiative exergy destruction
given by Eq. (23) can be divided into two parts. One is due to
absorption and emission, and the other is due to scattering.
Accordingly,
4p
daae
l ðr; tÞ ¼ kal T0 dV
Z
½Ibl ðTðr; tÞÞ Il ðr; s; tÞ
4p
1
1
dU
Tl ðr; s; tÞ Tðr; tÞ
9
82
3
Z
=
< Z I ðTðr; tÞÞ
p
ðr;
ð4
ðTðr;
ðr;
I
I
s;
tÞ
tÞÞ
G
tÞÞ
l
l
l
l
b
b
¼ kal T0 dV 4
dU dU5 ;
:
Tl ðr; s; tÞ
Tl ðr; s; tÞ
Tðr; tÞ
4p
dasl ðr; s; tÞ ¼ ksl T0 dV
8
< Z
:
4p
(25)
4p
2
3 9
Z
=
0
1
1
4
Il ðr; s0 ; tÞFl ðs; s0 ; tÞdU Il ðr; s; tÞ5dU
;
Tl ðr; s; tÞ 4p
(26)
4p
Through radiation absorption and emission, matter gets or releases
heat and exergy. The transient local net increment of exergy flow in
the matter under consideration as radiation goes through a differential volume is given by
3.3. Transient radiative exergy destruction in opaque solid surfaces
Fig. 4 illustrates the transient radiative exergy transfer process
on a differential surface area dA. The transient local net increment
5156
D. Makhanlall et al. / Energy 35 (2010) 5151e5160
h
i
s
daVl ðr; tÞ ¼ T0 dSae
G;l ðr; tÞ þ dSG;l ðr; tÞ
(31)
daAl ðrw ; tÞ ¼ T0 dSAG;l ðrw ; tÞ
(32)
4. Diffuse pulse radiation transfer in semitransparent media:
exergy destruction analysis
Fig. 4. Transient radiative exergy transfer at a unit surface area.
of spectral radiative exergy flow in the radiative field at the wall
surface can be written as
Z
deARf ;l ðrw ;tÞ ¼ dA
Jl ðrw ;s;tÞðnw ,sÞdU
4p
Z
¼ dA
fIl ðrw ;s;tÞT0 Ll ðrw ;s;tÞgðnw ,sÞdU
(27)
4p
In Eq. (27), the following relation has been used:
Z
Ibl ðT0 Þðnw ,sÞdU 4p
Z
T0 Lbl ½Ibl ðT0 Þðnw ,sÞdU
4p
Z
¼ fIbl ðT0 Þ T0 Lbl ½Ibl ðT0 Þg
ðnw ,sÞdU ¼ 0
(28)
4p
The transient local net increment of spectral radiative exergy flow
in the wall matter under consideration after radiation processes at
a differential surface dA is given by
T0
dQ_ l ðrw ; tÞ
1
Tðrw ; tÞ
Z
T0
¼ dA 1 Il ðrw ; s; tÞðnw ,sÞdU
Tðrw ; tÞ
deAM;l ðrw ; tÞ ¼
(29)
4p
By combination of Eqs. (27) and (29), transient spectral radiative
exergy destruction at opaque walls can be written as
We consider a TRT problem in an absorbing, non-emitting, and
isotropically scattering medium bounded by two parallel black
plates (see Fig. 5). The thickness of the medium is D ¼ 1.0 m, and the
plates are initially at 0 K. The medium has a uniform temperature
Tg ¼ 300 K. The optical thickness and the scattering albedo of the
medium are s ¼ ðkal þ ksl ÞD ¼ 1:0 and u ¼ ksl =ðkal þ ksl Þ ¼ 0:5
respectively. At time t ¼ 0 s, the temperature of plate I (x/D ¼ 0) is
suddenly raised to Th ¼ 500 K. The problem is equivalent to
a thermodynamic system in which heat is transported from a hightemperature heat source to a low-temperature heat sink by thermal
radiation. One-dimensional heat transfer is assumed. The temperature gradient of the medium during the heat transfer process is
assumed to be zero (i.e., vT/vx ¼ 0). The temperature of the reference environment is T0 ¼ 300 K.
4.1. Numerical simulation schemes
The TRT equation is solved using the DFEM based discrete
ordinates equation. The DOM used in this work refer to Ref. [32].
The DOM consists of dividing the entire solid angle into M different
directions with corresponding unit direction vectors Um and
weights wm. A Fortran 77 computer code based on the preceding
calculation procedure was written. Grid refinement studies were
also performed for the physical model to ensure that the essential
physics are independent of grid size. In the following analysis, the
slab was divided uniformly into 200 elements, and the direction
cosine was divided uniformly into 140 discrete directions. Details
on the procedure for solving TRT with the DFEM is discussed in
[29,30] and thus not repeated here. A textbook approach on the
DFEM can be found in [34].
Once the intensity distribution is determined as a function of
time and location, exergy calculations are carried out by postprocessing procedures. For the numerical scheme used, transient
local spectral radiative exergy destruction in the volume and in the
walls are calculated from the following approximation equations:
(
aae
l ðr; tÞ ¼ kal T0
daAl ðrw ; tÞ ¼ deARf ;l ðrw ; tÞ deAM;l ðrw ; tÞ
Z Il ðrw ; s; tÞ
¼ T0 dA
Ll ðrw ; s; tÞ ðnw ,sÞdU
Tðrw ; tÞ
(30)
M
X
wm
I ðTðr; tÞÞ
T m ðr; tÞ bl
m¼1 l
4pIbl ðTðr; tÞÞ Gl ðr; tÞ
Tðr; tÞ
Ilm ðr; tÞ
)
4p
3.4. Consistency with GouyeStodola theorem
The exergy destroyed in a process is the extent to which the
operation of an actual system departs from the theoretical limit of
the ideal system. This departure is proportional to the entropy
generation, and is known as the GouyeStodola theorem [1]. As in
the stead-state case, the analytical results derived here show that
radiative entropy generation and exergy destruction are linked
through the GouyeStodola theorem of classical thermodynamics.
Accordingly,
Fig. 5. Physical geometry of slab.
ð33Þ
D. Makhanlall et al. / Energy 35 (2010) 5151e5160
"
#
0
M
M I m ðr; tÞ
0
X
0
1 X
m ;m m m
l
F
al ðr; tÞ ¼ kal T0
w w
T m ðr; tÞ
4p m ¼ 1
m0 l
)
0
M I m ðr; tÞ
X
m
l
w
m
0 Tl ðr; tÞ
5157
(
s
ð34Þ
m
aAl ðr; tÞ ¼ T0
M I m ðr ; tÞ
X
w
l
m¼1
Tðrw ; tÞ
m
m
Lm
l ðrw ; tÞ ðnw ,s Þw
(35)
where,
Z
Il ðr; s; tÞdUz
Gl ðr; tÞ ¼
4p
M
X
m¼1
Ilm ðr; tÞwm
(36)
4.2. Results and discussion
Fig. 7. Transient distributions of spectral radiative heat flux.
4.2.1. The spectral incident radiation and spectral radiative heat flux
The spectral direction-integrated intensity (spectral incident
radiation) Gl is determined from Eq. (36). At different dimensionless times, Gl is calculated at different locations throughout
the problem domain. Fig. 6 shows the results for five different
dimensionless times. As in Ref. [29], the dimensionless time is
defined as t * ¼ ðkal þ ksl Þct: By defining the dimensionless time
step as Dt * ¼ ðkal þ ksl ÞcDt ¼ 0:0001; the wave front of the
pulse radiation arrives, theoretically, at the locations x/D ¼ 0.3,
0.6 and 0.9 at the times t* ¼ 0.3, 0.6 and 0.9, respectively. At
times t* 1.0, the wave front strikes plate II at x/D ¼ 1.0. The
radiation wavelength is l ¼ 6.0 mm, which is approximately the
wavelength of the maximum blackbody emissive power at
a temperature of 500 K.
At each time step, the radiative heat flux equation (Eq. (37)) was
also solved. Fig. 7 shows the spectral radiative heat flux Ql as
a function of location at different dimensionless times. As shown in
Figs. 6 and 7, the DFEM predicted the correct propagation speed
within the medium.
Z
Ql ðr; s; tÞ ¼
4p
Il ðr; s; tÞ n,sdUz
M
X
m¼1
Ilm ðr; tÞmm wm
Fig. 6. Transient distributions of spectral incident radiation.
(37)
4.2.2. The transient local spectral exergy destruction rate in the
medium
At each time step, the transient local spectral radiative exergy
destruction in the gas medium due to absorption and scattering are
computed respectively using Eq. (33) and Eq. (34) accordingly. Fig. 8
shows the distribution of local spectral exergy destruction due to
radiative absorption in the gas at different dimensionless times. As
can be seen, exergy destruction in the gas due to absorption is
correctly predicted to be most significant in the vicinity of plate I,
where the incident radiation is found to be largest at all times.
The local distribution of transient spectral radiative exergy
destruction in the medium due to scattering is shown in Fig. 9 at
different dimensionless times. It can be seen that the rate of exergy
destruction due to scattering is also largest in the medium at plate I.
The maximum rate of exergy destruction due to scattering reduces
with time, while the rate of exergy destruction due to heat
absorption increases with time everywhere in the slab until the
system reaches steady-state condition.
4.2.3. The transient total spectral exergy destruction rate
The transient total spectral exergy destruction rate of the entire
system can be determined as follows
Fig. 8. Transient distributions of local spectral radiative exergy destruction due to
absorption in medium.
5158
D. Makhanlall et al. / Energy 35 (2010) 5151e5160
Fig. 9. Transient distributions of local spectral radiative exergy destruction due to
scattering in medium.
tot
ae
AI
s
AII
Al ðtÞ ¼ Al ðtÞ þ Al ðtÞ þ Al ðtÞ þ Al ðtÞ
(38)
where,
Al ðtÞ ¼ kal T0
ae
Z ( X
M
wm
m¼1 l
V
I ðr; tÞ
T m ðr; tÞ bl
Ilm ðr; tÞ
)
4pIbl ðr; tÞ Gl ðr; tÞ
dV
Tðr; tÞ
Al ðtÞ ¼ kal T0
s
ð39Þ
Z (
"
#
0
M
M I m ðr; tÞ
0
X
0
1 X
m ;m m m
l
F
w w
m
4p m ¼ 1
0 Tl ðr; tÞ
V
)
0
M I m ðr; tÞ
X
m
l
dV
w
m
0 Tl ðr; tÞ
m
ð40Þ
m
AAl I ðtÞ ¼ T0
Z
AI
AAl II ðtÞ ¼ T0
M I m ðr ; tÞ
X
w
l
m¼1
Tw ðrw ; tÞ
Z X
M I m ðr ;tÞ
w
l
AII
m¼1
Tw ðrw ;tÞ
m
m
Lm
l ðrw ; tÞ ðnw ,s Þw dAI
ðr
;tÞ
ðnw ,sm Þwm dAII
Lm
w
l
(41)
Fig. 10. Spectral radiative exergy destruction rates as a function of dimensionless time.
law, ðlTÞmax ¼ 2898 mm K (Modest [32]), is l ¼ 5.796 mm. As can
be seen, the total spectral radiative exergy destruction rate of the
system is larger for wavelengths with higher spectral blackbody
emissive power. The numerical results show that for wavelengths
l ¼ 3.0 mm, l ¼ 4.0 mm, l ¼ 6.0 mm and l ¼ 8.0 mm, the total spectral
radiative exergy destruction rate at steady-state is approximately
35 W/mm, 95 W/mm, 140 W/mm and 110 W/mm, respectively.
4.2.5. Effect of scattering albedo
Figs. 12e14 show the variation of spectral radiative exergy
destruction rates as a function of dimensionless time for different
values of slab scattering albedo. The medium optical thickness is
s ¼ 1.0,and the radiation wavelength is l ¼ 6.0 mm. As can be seen in
Fig. 12, with increase of the scattering albedo, transient spectral
radiative exergy destruction reduces due to radiative absorption,
but increases due to radiative scattering. This is corresponding to
the analytical results (see Eqs. (11) and (12)). For higher values of
scattering albedo, the scattering coefficient is larger, but the
absorption coefficient is smaller. Fig. 13 shows that with an increase
of the scattering albedo, transient spectral radiative exergy
destruction reduces at plate I, but increases at plate II. The transient
spectral radiative exergy destruction rate of the entire system,
(42)
Fig. 10 shows the transient spectral radiative exergy destruction
rates as a function of dimensionless time. As the pulse radiation
proceeds towards plate II, the total spectral radiative exergy
destruction rate of the system increases rapidly. For t* < 1.0, the
wave front has not arrive at x/D ¼ 1.0. Therefore, no exergy is
destroyed at plate II for t* 1.0. The system reaches steady-state
condition at approximately t* ¼ 3.0. The total spectral radiative
exergy destruction rate of the system at steady-state is about
140 W/mmfor wavelength l ¼ 6.0 mm.
4.2.4. Effect of wavelength
Fig. 11 shows the distribution of the transient total spectral
radiative exergy destruction rate of the system for different wavelengths. The wavelength l ¼ 6.0 mm is approximately the wavelength of the maximum blackbody emissive power of plate I. Since
plate I has a temperature of 500 K, the wavelength of the maximum
blackbody emissive power calculated from Wien’s displacement
Fig. 11. Transient total spectral radiative exergy destruction rate for different
wavelengths.
D. Makhanlall et al. / Energy 35 (2010) 5151e5160
Fig. 12. Effect of scattering albedo on transient spectral radiative exergy destruction in
the medium.
5159
Fig. 15. Effect of optical thickness on transient spectral radiative exergy destruction in
the medium.
shown in Fig. 14, also reduces for larger scattering albedo. This is
due to the large reduction in spectral radiative exergy destruction
associated with absorption in the gas and radiative transfer at plat I.
Fig. 13. Effect of scattering albedo on transient spectral radiative exergy destruction in
wall.
Fig. 14. Effect of scattering albedo on transient spectral radiative exergy destruction of
entire system.
4.2.6. Effect of optical thickness
Figs. 15e17 show the variation of spectral radiative exergy
destruction rates as a function of dimensionless time for different
values of slab optical thickness. The medium scattering albedo is
u ¼ 0.5, and the radiation wavelength is l ¼ 6.0 mm. The numerical
results in Figs. 15 and 16 show that with increase of optical thickness, the transient spectral radiative exergy destruction rate
increases due to medium absorption, but decreases due to radiation
at solid boundaries. This is because the absorption coefficient and
the scattering coefficient are larger when optical thickness
increases at constant scattering albedo. Fig. 17 shows the variation
of the transient total spectral radiative exergy destruction rate of
the system. The initial rate of total exergy destruction is largest for
the case u ¼ 2.0. However, at steady-state, the increased spectral
radiative exergy destruction due to medium absorption is almost
balanced by the reduced spectral radiative exergy destruction at
Fig. 16. Effect of optical thickness on transient spectral radiative exergy destruction in
wall.
5160
D. Makhanlall et al. / Energy 35 (2010) 5151e5160
Fig. 17. Effect of optical thickness on transient spectral radiative exergy destruction of
entire system.
the boundaries. Hence, the steady-state exergy destruction rate of
the system is little affected with the change of optical thickness.
5. Conclusions
With the recent development of short-pulse lasers, TRT in
participating medium has become an important process in a variety
of emerging engineering and biomedical applications. Hence, SLA
of TRT processes represents a valuable topic for thermodynamic
optimization purposes. In this work, transient radiative entropy
and transient radiative exergy transfer equations have been
derived. To verify these transfer equations, the formulae of local
entropy generation and local exergy destruction for TRT processes
in semitransparent medium are developed. The analytical results
are consistent with the GouyeStodola theorem in classical thermodynamics. Numerical solution, based on the DOM and the DFEM,
is demonstrated for the special case of diffuse-pulse radiation. The
analysis presented here can be used to study the performance of
high-temperature systems in which TRT is an important mode of
heat transfer.
Acknowledgements
The supports of this work by the National Natural Science
Foundation of China (No.: 50836002) and the Changjiang Scholars
and Innovative Research Team in University (IRT0914) are gratefully acknowledged.
References
[1]
[2]
[3]
[4]
Bejan A. Entropy generation through heat and fluid flow. Wiley; 1982.
Bejan A. Entropy generation minimization. CRC Press; 1996.
Bejan A. Advanced engineering thermodynamics. Wiley; 1997.
Lucia U. Statistical approach of the irreversible entropy variation. Physica A
2008;387:3454e60.
[5] Lucia U. Irreversibility entropy variation and the problem of the trend to
equilibrium. Physica A 2007;376:289e92.
[6] Lucia U. Probability, ergodicity, irreversibility and dynamical systems.
Proceedings of the Royal Society A 2008;464:1089e104.
[7] Planck M. The theory of heat radiation. Dover; 1959.
[8] Caldas M, Semiao V. Entropy generation through radiative transfer in
participating media: analysis and numerical computation. Journal of Quantitative Spectroscopy and Radiative Transfer 2005;96:423e37.
[9] Liu LH, Chu SX. Verification of numerical simulation method for entropy
generation of radiative heat transfer in semitransparent medium. Journal of
Quantitative Spectroscopy and Radiative Transfer 2007;103:43e56.
[10] Liu LH. Radiative entropy transfer equation and numerical computation of
radiative entropy generation in semitransparent graded index medium.
Chinese Journal of Computational Physics 2009;26:267e74.
[11] Wright SE. The Clausius inequality corrected for heat transfer involving
radiation. International Journal of Engineering Science 2007;45:1007e16.
[12] Wright SE, Rosen MA, Scott DS, Haddow JB. The exergy flux of radiative heat
transfer for the special case of blackbody radiation. Exergy, an International
Journal 2002;2:24e33.
[13] Wright SE, Rosen MA, Scott DS, Haddow JB. The exergy flux of radiative heat
transfer with an arbitrary spectrum. Exergy, an International Journal
2002;2:69e77.
[14] Nejma FB, Mazgar AN, Abdallah N, Charrada K. Entropy generation through
combined non-grey gas radiation and forced convection between two parallel
plates. Energy 2008;33:1169e78.
[15] Mazgar AN, Nejma FB, Charrada K. Entropy generation through combined
non-grey gas radiation and natural convection in vertical pipe. Progress in
Computational Fluid Dynamics 2009;9:495e506.
[16] Agudelo A, Cortés C. Thermal radiation and the second law. Energy 2010;35:
679e91.
[17] Chu SX. Thermodynamic analysis of incoherent radiative transfer process.
Ph.D. dissertation. Harbin, P.R. China: Harbin Institute of Technology; 2009.
[18] Guo ZX, Aber J, Garetz BA, Kumar S. Monte Carlo simulation and experiments
of pulsed radiative transfer. Journal of Quantitative Spectroscopy & Radiative
Transfer 2002;73:159e68.
[19] Pilon L, Katika K. Modified method of characteristics for transient radiative
transfer. Journal of Quantitative Spectroscopy & Radiative Transfer 2006;
98:220e37.
[20] Wu CY, Wu SH. Integral equation formulation for transient radiative transfer
in an anisotropically scattering medium. International Journal of Heat and
Mass Transfer 2002;43:2009e20.
[21] Tan ZM, Hsu PF. An integral formulation of transient radiative transfer. Journal
of Heat Transfer 2001;123:466e75.
[22] Guo Z, Kumar S. Three-dimensional discrete ordinates method in transient
radiative transfer. Journal of Thermophysics and Heat Transfer 2002;
16:289e96.
[23] Chai JC. One-dimensional transient radiation heat transfer modeling using
a finite-volume method. Numerical Heat Transfer, Part B 2003;44:
187e208.
[24] Chai JC. Transient radiative transfer in irregular two-dimensional geometries.
Journal of Quantitative Spectroscopy & Radiative Transfer 2004;84:281e94.
[25] Chai JC, Hsu PF, Lam YC. Three-dimensional transient radiative transfer
modeling using the finite-volume method. Journal of Quantitative Spectroscopy & Radiative Transfer 2004;86:299e313.
[26] Li BQ, Cui X. A discontinuous finite element formulation for internal radiation
problems. Numerical Heat Transfer, Part B 2004;46:223e42.
[27] Li BQ, Cui X. A discontinuous finite element formulation for multidimensional
radiative transfer in absorbing, emitting, and scattering media. Numerical
Heat Transfer, Part B 2004;46:399e428.
[28] Li BQ, Cui X. Discontinuous finite element solution of 2-D radiative transfer
with and without axisymmetry. Journal of Quantitative Spectroscopy &
Radiative Transfer 2005;96:383e407.
[29] Liu LH, Liu LJ. Discontinuous finite element approach for transient radiative
transfer equation. Journal of Heat Transfer 2007;126:1069e74.
[30] Liu LH, Hsu PF. Analysis of transient radiative transfer in semitransparent
graded index medium. Journal of Quantitative Spectroscopy & Radiative
Transfer 2007;105:357e76.
[31] Liu LH, Chu SX. Radiative exergy transfer equation. Journal of Thermophysics
and Heat Transfer 2007;21:819e22.
[32] Modest MF. Radiative heat transfer. Academic Press; 2003.
[33] Candau Y. On the exergy of radiation. Solar Energy 2003;75:241e7.
[34] Li BQ. Discontinuous finite elements in fluid dynamics and heat transfer.
Springer; 2006.