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International Journal of Thermal Sciences 49 (2010) 1848e1855

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International Journal of Thermal Sciences

journal homepage: www.elsevier .com/locate/ i j ts

Determination of loss coefficients for high-temperature flow devices: Anentropy-based approach

D. Makhanlall, L.H. Liu*, H.C. ZhangSchool 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

Article history:Received 18 January 2010Received in revised form26 March 2010Accepted 20 April 2010Available online 21 May 2010

Keywords:Jet flowEntropy productionExergy lossLoss coefficientRadiation

* Corresponding author. Tel.: þ86 451 86402237.E-mail address: lhliu@hit.edu.cn (L.H. Liu).

1290-0729/$ e see front matter � 2010 Elsevier Masdoi:10.1016/j.ijthermalsci.2010.04.015

a b s t r a c t

Thermodynamic analysis of a high-temperature confined turbulent gas-jet is presented in this paper. Thenumerical model is two dimensional, steady, and includes the effect of gravity in the governing equa-tions. Computations are carried out with a commercial CFD code and the local exergy losses are deter-mined as post processed quantities. The analysis takes into account the second law effects of viscousdissipation, heat conduction and convection, and radiative heat transfer. The study is extended byconducting a parametric investigation to determine the effects of Reynolds number, inlet fluidtemperature, optical thickness, and Planck number on the exergy loss coefficient, which is defined as thetotal exergy destroyed per unit mechanical energy input. The results show that exergy loss troughradiation entropy production is higher than that due to heat conduction and convection when the inletgas temperature is high. It has also been found that in contrast to the conventional head loss coefficient,the exergy loss coefficient increases with inlet gas temperature, optical thickness, and Planck number.

� 2010 Elsevier Masson SAS. All rights reserved.

1. Introduction

In fluids engineering devices, even though energy is conserved,the quality of the energy always decreases. High-quality energyrefers to energy from which a great amount of useful work can beextracted, and exergy is used to refer to the work potential of thatenergy. Bejan [1,2], Sahin [3,4], Datta [5], and Herwig and Kock [6]conducted extensive research to show that heat transfer, fluidfriction and other thermodynamic irreversibilies lead to theproduction of entropy, which destroys useful energy (exergy) ina system. The local entropy generation rate is a key to improvingenergy processes and applications (Bejan [2]).

At present, it is common practice to use global loss parameterssuch as head loss and pressure recovery coefficients to characterizethe performance of a fluid engineering system. However, thisapproach typically does not provide any information on the specificlocations and sources of flow losses within a fluid system. An alter-native formulation that allows the entropy production in a fluidelement to be converted to a local flow loss parameter was recentlypresented by Naterer and Adeyinka [8e10]. It can be shown that theloss term due to viscous dissipation in the mechanical energy equa-tion can be related to entropy production by.

son SAS. All rights reserved.

hl ¼1_m

ZV

T _S000p;fdV (1)

Here, hl is the head loss in Bernoulli’s equation, _S000p;f is the local

entropy production rate due to viscous dissipation, T is the localtemperature, and _m is the mass flow rate. This relation confirmsthat flow loss is a local volumetric phenomenon involving the rateof entropy production. In a recent study by Zhang et al. [11], it wasshown that Eqn. (1) is a useful tool for developing loss correlationsfor micro-flow devices. Another unique advantage of Eqn. (1) is thatit allows designers to use local loss mapping to detect locations ofhigh entropy production, thereby allowing local design changes ofgeometrical or other parameters to improve system efficiency(Naterer and Camberos [7]).

Though eqn. (1) is a valuable local alternative to conventionalglobal loss characterization, it has limitations when analyzing high-temperature engineering devices. In many high-temperaturesystems, such as solar collectors, boilers and furnaces, inefficienciesassociated with radiative heat transfer becomes an important issue.Unfortunately, even though numerous studies have examined thematter, there still remain uncertainties how exergy loss throughentropy generation in radiating situations should be handled. Forthis reason, the effects of thermal radiation are often excludedwhen conducting second law analysis, even for high-temperaturesystems. In their analysis, Arpaci [12,13], and Arpaci and Esmaeeli

Nomenclature

A Wall surface area, m2

Ce Exergy loss coefficientCf Head loss coefficientc0 Speed of light in vacuum, m s�1

cp Specific heat at constant pressure, J Kg�1 K�1

di Inlet length of the channel, mdo Outlet length of the channel, mg Gravitational acceleration, 9.81 m s�2

h Planck’s constant, J sH Height of the channel, mIl Spectral radiative intensity, W m�3 sr�1

Ib,l Blackbody spectral radiative intensity, W m�3 sr�1

k Turbulent kinetic energy, m2 s�2

kb Boltzmann’s constant, J K�1

keff Effective thermal conductivity, W m�1 K�1

km Thermal conductivity, W m�1 K�1

L Length of the channel, mLl Spectral radiative entropy intensity,

W m�2 mm�1 sr�1 K�1

nw Unit outward normal vector of boundary wall

Pl Planck number,km

4sHT30

r Spatial position vector

Red Reynolds number,ruRdm

s Direction vector_S000p;c Local entropy production due to heat conduction and

convection, W m�3 K�1

_S000p;f Local entropy production due to viscous dissipation,

W m�3 K�1

_S000p;r Local radiation entropy production in the medium,

W m�3 K�1

_S00Ap;r Local radiation entropy production at the wall,

W m�2 K�1

T Temperature, KT0 Reference temperature, K

Tl Spectral radiation temperature, KV Volume, m3

uin Inlet velocity of gas, m s�1

ux;uy Velocity components, m s�1

x; y Coordinates, m_X000d;f Local exergy loss through viscous dissipation, J/m3 s_X000d;c Local exergy loss through heat conduction and

convection, J/m3 s_X000d;r Local exergy loss through radiative processes in the

gas, J/m3 s_X00Ad;r Local exergy loss through radiation processes at the

wall, J/m2 s_Xd Total exergy loss, J/s_Xd;f Total exergy loss through viscous dissipation, J/s_Xd;c Total exergy loss through heat conduction and

convection, J/s_Xd;r Total exergy loss through radiation processes in the

gas, J/s_XAd;r Total exergy loss through radiation processes at the

wall, J/s

Greek Symbolsa Inverse effective Prandtl number3 Turbulent dissipation rate, m2 s�3

3w Emissivityka Absorption coefficient, m�1

ks Scattering coefficient, m�1

l Wavelength, mm Viscosity, kg m�1 s�1

meff Effective viscosity, kg m�1 s�1

mt Turbulent or eddy viscosity, kg m�1 s�1

r Density, kg m�3

s StefaneBoltzmann constant, W m�2 K�4

F Scattering phase functionu Single scattering albedoU Solid angle, sr

D. Makhanlall et al. / International Journal of Thermal Sciences 49 (2010) 1848e1855 1849

[14] used the following diffusion approximation to calculate localradiation entropy production.

_S000

p;r ¼ �qr$VTT2 (2)

Here, qr is the radiative heat flux. It can be shown that thisconduction-type formula is only applicable in problems concerningvery large optical thickness (Liu and Chu [15]).

In the present study, we analyze a high-temperature gas-jetdevice based on the second law. The gas is modeled as a semi-transparent medium that absorbs, emits and scatters radiativeenergy. The radiative entropy production is calculated from theequations derived by Caldas and Semiao [16] and Liu and Chu [17].Using the commercial CFD code FLUENT 6.1.22, the distributions ofvelocity and temperature are determined. Exergy losses associatedwith viscous dissipation, heat conduction and convection, andradiative heat transfer are calculated by post processing usinga FORTRAN 77 code. In addition, a parametric investigation iscarried out to determine the effects of Reynolds number, inlet fluidtemperature, optical thickness, and Planck number on the entropy-based head loss and the exergy loss coefficient.

2. Mathematical formulae and simulation method

The geometrical configuration of the high-temperature jetflow device under consideration is shown in Fig. 1. The heightand length of the system are H ¼ 1.0 m and L ¼ 0.4 m, respec-tively. The length of the inlet and outlet spouts aredi ¼ 0.08 m, and do ¼ 0.2 m, respectively. The effect of gravity istaken under consideration. The gravitational acceleration isg ¼ 9.81 m/s2. The walls of the model are considered to bethermodynamically black, opaque and fixed at a temperature of300 K. The inlet and outlet are assumed blackbody with the sametemperature as the local fluid medium.

A gas-jet enters the chamber with temperature Th,in ¼ 1000 Kand Reynolds number Re ¼ 73 733 (turbulent jet flow). Theconfined jet is assumed to be steady, two-dimensionalsymmetric, and incompressible. The simulation is started withthe following values assigned to the molecular viscosity, thermalconductivity, specific heat, density, absorption coefficient andscattering coefficient, respectively: mm ¼ 1.087 � 10�5 kg/m s,km ¼ 0.0332 W/m K, cp ¼ 2250 J/kg K, r ¼ 0.6679 kg/m3,ka ¼ 0.62, and ks ¼ 0.02.

Fig. 1. Schematic of the confined gas-jet (H ¼ 1.0 m; L ¼ 0.4 m; di ¼ 0.08 m; do ¼ 0.2 m;g ¼ 9.81 m/s2).

D. Makhanlall et al. / International Journal of Thermal Sciences 49 (2010) 1848e18551850

2.1. Solution procedure

The calculation of the two-dimensional statistically stationaryturbulent flow field is based on the solution of the RANS equations,closed with the k� 3 model derived from the renormalized groupmethod (RNG) of Yakhot and Orszag [18]. When the RNG k� 3

model is employ, an effective viscosity and an effective thermalconductivity are use in the solution of the RANS equations. They aredefined respectively as follows

meff ¼ mm þ mt; mt ¼ rCmk2

3(3)

keff ¼ acpmeff (4)

Here, mt is the turbulent viscosity, k is the turbulent kineticenergy, 3 is the turbulent dissipation rate, Cm ¼ 0.0845 is a simula-tion constant, and a is the inverse effective Prandtl number. Theinverse effective Prandtl number is computed using the followingformula derived analytically by the RNG theory:

���� a� 1:3929a0 � 1:3929

����0:6321���� aþ 2:3929a0 þ 2:3929

����0:3679 ¼ mmmeff

(5)

where

a0 ¼ kmmmcp

(6)

More details on the turbulence model are available in [18,19].In many high-temperature systems, thermal radiation becomes

an important mode of heat transfer and the analysis of suchsystems requires considerable more computational effort due tothe integro-differential nature of gas radiation. The radiativeintensity distribution in an absorbing, emitting and scatteringmedium can be obtained from the radiative transfer equationderived by Modest [20]:

dIlds

¼ �ðkaþ ksÞIlðr;sÞþ kaIblðrÞþks4p

Z4p

Ilðr;s0ÞFðs;s0ÞdU0 (7)

with,

Ilðrw; sÞ ¼ 3wIblðrwÞ þ1� 3w

p

Znw$s<0

Ilðrw; s0Þðnw$s0ÞdU0 (8)

as the boundary condition. In the above equations, Il(r,s) is thespectral radiative intensity, Ib l(r) is the blackbody spectral intensity,Fðs0; sÞ is themedium scattering phase function, l is thewavelength,U is the solid angle, r is a unit position vector, s is a unit directionvector, 3w is thewall emissivity, andnw is the unit surface normal. Tosolve Eqns. (7) and (8)weused the discrete ordinatemethod (DOM).DOM is one of the oldest and still themost extensively usedmethodby physicists and engineers to solve the radiative transfer equation.

2.2. Calculation of local entropy production rates

In non-reacting high-temperature jet flow configurations,entropy production results from fluid friction and the heat transferprocess. The entropy production due to friction, and heat conduc-tion and convection can be determined from the equations given byBejan [1,2]. Similar to the method used by Yapici et al. [21], the localvolumetric entropy production due to viscous dissipation, and heatconduction and convection are calculated respectively using theeffective viscosity and effective thermal conductivity:

_S000p;f ¼ meff

T

"�vuxvy

þ vuyvx

�2þ2�vuxvx

�2þ2�vuyvy

�2#

(9)

_S000p;c ¼ keff

T2

"�vTvx

�2

þ�vTvy

�2#

(10)

A radiation beam not only carries energy but also entropy andexergy. Therefore, in the second law analysis of high-temperatureengineering devices, thermodynamic irreversibilities associatedwith radiative heat transfer must also be evaluated. The spectralradiative entropy intensity carried by incoherent and unpolarizedradiation is defined by Planck [22] as.

Llðr; sÞ ¼ 2kbc0l�4

( 1þ Ilðr; sÞ

2hc20l�5

!ln

1þ Ilðr; sÞ

2hc20l�5

!

�

Ilðr; sÞ2hc20l

�5

!ln

Ilðr; sÞ2hc20l

�5

!)ð11Þ

where, c0 is the speed of light, h is Planck’s constant, and kb isBoltzmann’s constant.

Using Eqn. (11), an expression for local radiative entropyproduction rate in an absorbing, emitting and scattering mediumcan be derived (Caldas and Semiao [16]):

Fig. 2. Flow structure of the turbulent gas-jet (1 ¼ core zone, 2 ¼ mixing zone;3 ¼ near-wall zone).

D. Makhanlall et al. / International Journal of Thermal Sciences 49 (2010) 1848e1855 1851

_S000

p;r ¼ZN0

Z4p

24� ðka þ ksÞ Ilðr; sÞTlðr; sÞ

þ kaIb;lðrÞTlðr; sÞ

þ ks4p

Z4p

Ilðr; s0ÞTlðr; sÞ

Fðs0; sÞdU035dUdl

þZN0

Z4p

ka

hIlðr; sÞ � Ib;lðrÞ

iTðrÞ dUdl

(12)

Here,

Tlðr; sÞ ¼ hc0lkb

1

lnh2hc20l

�5=Ilðr; sÞ þ 1

i (13)

is the spectral radiation temperature.In addition, radiation absorption and emission processes at solid

walls also lead to entropy production. The local radiative entropyproduction rate per unit surface area at opaquewalls can be definedas (Liu and Chu [17], Chu [23]):

_S00Ap;r ¼

ZN0

Z4p

�Ilðrw; sÞTðrwÞ � Llðrw; sÞ

�ðnw$sÞdUdl (14)

2.3. Calculation of exergy destruction rateand exergy loss coefficient

Exergy is defined as the work potential that can be extractedfrom an energy source. Local exergy can be more readily inter-preted physically than entropy production because it containsthe same dimensional units as energy. The exergy destroyed ina process is the extent to which the operation of an actual systemdeparts from the theoretical limit of the ideal system. Thisdeparture is proportional to the entropy production, and isknown as the GuoyeStodola theorem cited by Bejan [2]. It can beshown that the relation between radiative exergy loss andentropy production is also consistent with the GuoyeStodolatheorem (Liu and Chu [24]).

In the high-temperature gas-jet system, exergy is destroyedthrough entropy production associated with viscous dissipation,heat conduction and convection, and radiative heat transfer (Liuand Chu [25]). All of these irreversible processes dissipate usefulmechanical energy into less useful internal energy. The local exergyloss due to viscous dissipation, heat conduction and convection,radiative heat transfer in the medium, and radiative heat transfer atthe wall can be calculated respectively as follows.

_X000

d;f ¼ T0 _S000p;f (15)

_X000

d;c ¼ T0 _S000p;c (16)

_X000

d;r ¼ T0 _S000

p;r (17)

_X00Ad;r ¼ T0 _S

00Ad;r (18)

where, T0¼ 300 K is the temperature of the reference environment.The exergy of the reference environment is zero, and the exergy ofa system is zero when it is in equilibrium with the referenceenvironment (Rosen and Dincer [26]).

The total exergy loss of the gas-jet can be calculated by addition:

_Xd ¼ _Xd;f þ _Xd;c þ _Xd;r þ _XAd;r (19)

Here,

_Xd;f ¼ZV

T0 _S000p;fdV (20)

_Xd;c ¼ZV

T0 _S000p;cdV (21)

_Xd;r ¼ZV

T0 _S000p;rdV (22)

_XAd;r ¼

ZA

T0 _S00Ap;rdA (23)

D. Makhanlall et al. / International Journal of Thermal Sciences 49 (2010) 1848e18551852

The exergy loss coefficient Ce of the confined gas-jet can bedefined as.

Ce ¼_Xd

0:5 _mu2in(24)

3. Results and discussion

3.1. Grid independence studies

For turbulent flow simulations, an extra parameter defined as.

y*hrC1=4

m k1=2p ypmm

(25)

has to be checked ([19]). Here, kp and yp refer to the turbulencekineticenergy at, and the distance from the wall to the first grid point,respectively. The performance of the widely used k� 3 turbulencemodel in FLUENT worsens if the first grid point happens to be in thebuffer layer. Unfortunately, y* is a post processed quantity, andseveral iterations are required to arrive at the right grid size. Evenwhen the computational geometry remains the same, an increase ofvelocity leads to higher shear stresses and a higher y*.

In this study, a non-uniform grid system consisting of 12 000cells is used with cell clustering near the walls. A two-sided cellclustering factor is used to adjust the distance between the solidwall and the closest node. Employing the standard wall function,the numerical results were considered to be grid independentwhen 11.25 � y* � 200, and the change in total entropy productionwas in the order of �1%.

3.2. Distribution of velocity, temperature, head lossand exergy destruction rates

Fig. 2 shows the typical flow structure of the high-temperatureturbulent gas-jet. The flow is symmetrical about the centerline of

Fig. 3. Distribution of (a) velocity (

the channel. Three flow regions can be identified: a core zone,a mixing zone, and a near-wall region. The core zone is a high-temperature high-velocity region. The local temperature andvelocity in the jet core are almost at the same values as the gas inlettemperature and velocity. The jet core disappears along the axisrapidly. The mixing zone exemplifies the region with strongturbulent intensities. Here, turbulent eddies produce largetemperature and velocity gradients in the flow field. The gas-jetinlet turbulent intensity can be estimated from the followingempirical correlation ([19]):

Ii ¼ 0:16ðRedÞ�1=8 (26)

The flow at the near-wall region is a low-velocity flow againstgravity.

In thermo-fluid systems, the local volumetric rate of entropyproduction is closely related to the velocity and temperaturedistribution. Regions with large temperature gradients will typi-cally have high entropy production rates due to heat conductionand convection (see Eqn. (10)). Entropy production due to viscousdissipation is related to the velocity distribution. Exergy loss islinked with entropy production trough the GuoyeStodola theoremand will be large in locations with high entropy production rates.The contour plots of velocity and temperature are shown in Fig. 3aand b, respectively. Fig. 4a shows the head loss distribution, whilethe local exergy loss in the gas due to viscous dissipation, heatconduction and convection, and radiation are shown in Fig. 4bed,respectively.

At the jet core, where the velocity and temperature gradientsare relatively small, exergy loss trough entropy production associ-ated with viscous dissipation, and heat conduction and convectioncan be neglected. However, the strong turbulent perturbationowing to the gas entrainment leads to large local velocity andtemperature gradients at the beginning section of the mixing zone.This result in large exergy destruction rates at the inlet in themixing layer (see Fig. 4b and c). The mixing layer expands along the

m/s) and (b) temperature (K).

Fig. 4. Distribution of (a) head loss (J/kg m3) and exergy destruction rate in the gas (W/m3) due to (b) viscous dissipation, (c) heat conduction and convection, and (d) thermalradiation.

D. Makhanlall et al. / International Journal of Thermal Sciences 49 (2010) 1848e1855 1853

direction of the flow, and the velocity and temperature gradientgradually decreases, both in the flow direction and from the centerto the edge of the mixing layer. Therefore, the local exergy loss (andentropy production) due to viscous dissipation and heat conductionand convection in the mixing zone reduces in the direction of thegas flow, and also from the center to the edge of the mixing layer. Atthe near-wall region, the temperature gradient increases in thedirection of the wall. At the isothermal walls there exists a narrowzone with a large temperature gradient. FLUENT code predicts thethickness of this narrow layer to be approximately 0.004 m. Thishigh-temperature gradient layer causes large exergy destructionrates at the wall (see Fig. 4c).

Fig. 4d shows the distribution of the local radiation exergy lossin the gas. In contrast to entropy production due to heat conductionand convection, the radiation entropy production is not dependenton the local temperature gradient. Thermal radiation is a long rangephenomenon. Therefore, the local radiation entropy production isdependent on the temperature distribution of the entire system.Fig. 4d shows that the radiative exergy destruction rate is largest inthe jet core and at the wall.

For the gas-jet, the head loss is calculated using Eqn. (1) and is68.10 J/kg. The total exergy loss calculated from Eqn. (19) is about35 kW. Of the total exergy destroyed, it is predicted that about 60%is due to wall radiation effects, 18% is due to radiative absorption,emission, and scattering in the gas, and 22% is due to heatconduction and convection. The total exergy loss due to viscousdissipation can be neglected.

Table 1Effects of inlet gas temperature.

T (K) Cf Ce hl (J/kg) _Xd (W) _Xd;f =_Xd

(%)

_Xd;c=_Xd

(%)

_Xd;r=_Xd

(%)

_XAd;r=

_Xd(%)

500 0.63 31 70.57 2804 1.20 64.37 15.81 18.621000 0.61 380 68.10 34 191 0.05 22.05 17.53 60.372000 0.61 4382 68.10 394 408 0.01 4.13 10.46 85.40

3.3. Parametric investigation

In the parametric study carried out here, the effect of the inletgas temperature, Reynolds number, optical thickness, and Plancknumber on the total exergy loss of the gas-jet is determined. Inaddition, we also look at the exergy loss coefficient (Ce) and thehead loss coefficient (Cf). Using Eqn. (1), the head loss coefficient isdefined as.

Cf ¼ hl0:5u2in

¼ 10:5 _mu2in

ZV

T _S000

p;fdV (27)

3.3.1. Effects of inlet gas-jet temperatureThe effect of the gas inlet temperature on the total exergy

destruction rate is shown in Table 1. When only the inlet gastemperature is increased, the total useful energy destroyed troughentropy production becomes larger. The numerical results showthat this is mainly through the increase of radiative exergy loss atthe wall. Heat conduction and convection appears to be the maincause of exergy loss when the gas inlet temperature is reduced to500 K. Since the mechanical energy input remains constant, theexergy loss coefficient also increases strongly with increasing gasinlet temperature. However, with the increase of the inlet gastemperature from 500 K to 2000 K, the total head loss reduces from70.57 J/kg to 60.10 J/kg, while the head loss coefficient remainspractically constant at 0.6.

3.3.2. Effects of Reynolds numberTable 2 shows how the jet flow device reacts when only the inlet

Reynolds number is changed. With the increase of Reynoldsnumber from 49 155 to 122 889, the total rate of exergy destructionincreases from 31 956 to 38 103. However, the reduction of theexergy loss coefficient means that the mechanical energy input isbetter utilized as Reynolds number becomes larger. Though thehead loss increases with Reynolds number, the head loss coefficientremains almost constant at 0.6. Thus, from a second law point of

Table 2Effects of Reynolds number.

Red Cf Ce hl (J/kg) _Xd (W) _Xd;f =_Xd

(%)

_Xd;c=_Xd

(%)

_Xd;r=_Xd

(%)

_XAd;r=

_Xd(%)

49 155 0.64 1197 31.75 31 956 0.02 18.64 18.44 62.9073 733 0.61 380 68.10 34 191 0.05 22.05 17.53 60.3798 311 0.59 169 118.28 36 191 0.11 24.96 16.79 58.14122 889 0.59 91 183.75 38 103 0.20 27.59 16.13 56.08

0.003 0.006 0.009 0.012 0.015 0.01810

20

30

40

50

60

70

Xd,r

/Xd

XA

d,r/X

d

a

Rel

ativ

e ex

ergy

loss

(%

)

Pl

Xd,c

/Xd

40

50

60

70

XA

d,r/X

dve

exe

rgy

loss

(%

)

b

D. Makhanlall et al. / International Journal of Thermal Sciences 49 (2010) 1848e18551854

view, the system operates better at a higher inlet Reynolds number.The numerical results also show that the percentage of the totalexergy destroyed through entropy production associated with heatconduction and convection increases with Reynolds number.

3.3.3. Effects of optical thicknessOptical thickness (or optical distance) s is an important

parameter in problems involving radiation absorption and emis-sion. In this analysis, the optical thickness is based on the channelheight H. Table 3 shows the effects of s on the system performance.For the high-temperature fluid, approximately 80% of the totalexergy is destroyed trough radiation entropy production whens ¼ 1.24. The increase of the exergy loss coefficient with opticalthickness means that the efficiency of the system reduces as sbecomes larger. However, this is not seenwhen observing the headloss coefficient. The head loss and the head loss coefficient remainsalmost constant when the optical thickness is increase froms ¼ 1.24 to s ¼ 1.24. Hence, the conventional loss parameter is lesssuitable for analyzing high-temperature systems.

0.003 0.006 0.009 0.012 0.015 0.018

10

20

30X

d,c/X

d

Rel

ati

Pl

Xd,r

/Xd

Fig. 5. (a) Effect of Planck number on the relative exergy loss (in %) due to heat transferin the case Th,in ¼ 750 K. (b) Effect of Planck number on the relative exergy loss (in %)due to heat transfer in the case Th,in ¼ 1000 K.

240

280

320

360

400

440

Ce

Th,in = 1000 K

Th,in = 750 K

3.4. Effects of Planck number

The numerical results obtained in this study shows that exergyis destroyed primarily through the production of entropy associ-ated with heat transfer. To measure the relative importance of heatconduction and convection to radiative heat transfer, we used thePlanck number. The gas-jet Planck number can be defined as.

Pl ¼ km4sHT30

(28)

Here s ¼ 5.6696 � 10�8 W/m2 K4 is the StefaneBoltzmannconstant.

The analysis is carried out for two different gas inlet tempera-tures. The results are presented in Fig. 5a and b. It is apparent thatwith increasing Planck number, the percentage of exergy destroyedtrough entropy production associated with heat conduction andconvection increases. When the gas inlet temperature drops to750 K, conduction and convection heat transfer quickly becomesthe main cause of exergy loss with increasing Planck number. Fig. 6shows the effect of Planck number on the exergy loss coefficient.The loss coefficient increases almost linearly with Planck number.At higher inlet gas temperature, the loss coefficient of the gas-jet islarger, but its rate of increase with Planck number remains almostthe same.

Table 3Effects of optical thickness.

s Cl Ce hl(W) _Xd (W) _Xd;f =_Xd

(%)

_Xd;c=_Xd

(%)

_Xd;r=_Xd

(%)

_XAd;r=

_Xd(%)

0.10 0.61 234 68.17 21 069 0.08 34.32 9.78 55.820.62 0.61 380 68.10 34 191 0.05 22.05 17.53 60.371.24 0.60 486 68.04 43 738 0.04 17.92 17.58 64.46

120

160

200

Pl0.003 0.006 0.009 0.012 0.015 0.018

Fig. 6. Effect of Planck number on the exergy loss coefficient at different inlet gastemperatures.

D. Makhanlall et al. / International Journal of Thermal Sciences 49 (2010) 1848e1855 1855

4. Conclusions

In this paper, a high-temperature confined turbulent gas-jet hasbeen analyzed based on the second law of thermodynamics. Thelocal exergy destroyed through entropy production associated withviscous dissipation, heat conduction and convection, and radiativeheat transfer has been mapped, and the effects of inlet gastemperature, Reynolds number, optical thickness, and Plancknumber on the exergy loss coefficient was studied. The resultsshow that:

(1) radiation entropy production is the main cause of exergydestruction when the gas inlet temperature is high.

(2) The exergy loss coefficient becomes larger when the inlet gastemperature, optical thickness and Planck number increases.

(3) The conventional head loss coefficient is not suitable foranalyzing high-temperature turbulent flow systems.

Acknowledgements

The supports of this work by the National Natural ScienceFoundation of China (No.: 50836002) and the Changjiang Scholarsand Innovative Research Team in University (IRT0914) are grate-fully acknowledged.

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