Performance of method of lines solution of discrete ordinates method in the freeboard of a bubbling fluidized bed combustor

Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 503 – 516

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Performance of method of lines solution of discrete ordinates method in the freeboard of a bubbling +uidized bed combustor Nevin Sel.cuk ∗ , Aykan Batu, Isil Ayranci Department of Chemical Engineering, Middle East Technical University, 06531 Ankara, Turkey

Abstract Predictive accuracy of the method of lines (MOL) solution of discrete ordinates method (DOM) is assessed by applying it to the prediction of incident radiative +uxes on the freeboard walls of a bubbling +uidized bed combustor, and comparing its predictions with those of the zone method and measurements. Freeboard is treated as a three-dimensional rectangular enclosure containing gray, absorbing, emitting and isotropically scattering medium. Radiative properties of the medium are calculated by using Leckner’s correlations for gas and Mie theory for particles. Data for application and validation are generated from METU 0:3 MWt atmospheric bubbling +uidized bed combustor test rig burning lignite in its own ash. Comparisons reveal that MOL solution of DOM provides accurate and computationally e@cient solutions for wall +uxes in the freeboard of +uidized bed combustors containing particle laden combustion gases. Parametric studies are also carried out to analyze the sensitivity of predicted heat +ux proAles to the presence of particles, particle load and anisotropic scattering. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Method of lines; Discrete ordinates method; Bubbling +uidized bed combustors; Radiative heat +ux measurements

1. Introduction Radiative heat transfer in an absorbing, emitting, scattering medium is an important aspect in many practical engineering problems such as the modeling of energy transport in combustion chambers and furnaces. In +uidized bed combustion, the e@ciency depends upon the heat recovered in the freeboard region, where the dominant component of heat transfer is radiation to which the major contributor is the emittance from combustion oC-gases and +y-ash particles. Therefore, modeling

∗

Corresponding author. Tel.: +90-312-210-2603; fax: +90-312-210-1264. E-mail address: [email protected] (N. Sel.cuk).

0022-4073/02/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 2 - 4 0 7 3 ( 0 1 ) 0 0 2 2 5 - 4

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Nomenclature Amc b; n B D32 Fc fN (Dp ) g Lm M Mt n NX NY NZ p(Dp ) Qa ; Qs uo Vp (Dp ) W (Dp ) wm x ; ;  

mass speciAc cross section, m2 =kg Rosin–Rammler constants, dimensionless particle load, kg=m3 Sauter mean diameter, m carryover +ow rate, kg=h number distribution of particles, dimensionless asymmetry factor, dimensionless mean beam length, m total number of discrete ordinates, dimensionless total weight of the particles, kg unit surface normal pointing away from the surface, dimensionless number of grids in x direction, dimensionless number of grids in y direction, dimensionless number of grids in z direction, dimensionless diCerential weight distribution, dimensionless absorption and scattering e@ciencies, dimensionless superAcial velocity, m=s volume of particle of diameter Dp ; m3 cumulative weight fraction, dimensionless quadrature weight for ordinate m, dimensionless particle size parameter, dimensionless direction cosines in x; y; z directions, dimensionless direction vector of radiant intensity, dimensionless

Subscripts and superscripts av b bottom inc m m
p top w

size averaged property blackbody bottom surface of the freeboard incident discrete direction incoming ordinate direction particle top surface of the freeboard side surface of the freeboard

of radiative heat transfer in such systems necessitates an accurate knowledge of radiative properties of the particle laden combustion gases. Calculation of the radiative properties, on the other hand, requires information on temperatures, gas composition as well as particle concentration, composition, shape and size distribution.

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Di@culties encountered in obtaining such data from industrial scale units lead to the use of small scale test rigs equipped for measurements. A pioneering study on measurement of emissivity and transmissivity of particle laden combustion gases in freeboard of a simulated atmospheric bubbling +uidized bed combustor (ABFBC), where silica sand is +uidized in propane–air combustion products has been performed by Lindsay et al. [1]. It was reported that dominant component of heat transfer is radiation from particle laden combustion gases. The investigation was extended to prediction of the eCect of scattering on the emissivity of the particle hold up in the freeboard of a 1 MWt ABFBC burning coal with and without limestone addition in a bed of sand particles by Filla et al. [2]. Predictions of particle cloud emissivity have shown signiAcant eCect of scattering on the radiative properties of particle laden +ue gases. An extensive experimental study in an ABFBC providing data required both for the calculation of radiative properties and for the assessment of the accuracy of the radiative heat exchange models is due to Kozan and Sel.cuk [3]. Data on +ow rates, concentrations and temperatures together with incident radiative heat +uxes in the freeboard of a 0:3 MWt ABFBC burning lignite in its own ash were measured during the steady state operation of the rig. Radiative exchange in the freeboard of the combustor was modeled by using a simple engineering approach, i.e. a well-stirred enclosure model in conjunction with radiosity-irradiation method (RIM) and radiative properties of particle laden combustion gases estimated from measured data. SigniAcant discrepancies between the predicted and measured incident radiative +uxes were observed. This was considered to be due to the single zone treatment of the freeboard in which a constant uniform incident heat +ux was used to represent the relatively steep variation of measured proAle along the freeboard wall. The objective of the present study is to apply a recently emerging solution technique, method of lines (MOL) solution of discrete ordinates method (DOM) to the prediction of radiative +uxes on the freeboard wall of the test rig. MOL solution of DOM (MOL of DOM) is based on implementation of false transients approach to discrete ordinates representation of radiative transfer equation (RTE). It is a promising method when RTE is to be solved in conjunction with the time-dependent conservation equations for mass, momentum, energy and species. In this article, MOL of DOM, which has been shown to be an accurate method upon its assessment on idealized test problems with participating media [4 – 6], is subjected to a more challenging problem of prediction of radiative +uxes in the freeboard of the 0:3 MWt ABFBC test rig. Steady state operating data on +ow rates, medium and wall temperature distributions, gas composition as well as particle concentration, composition, shape and size distribution are used for the calculation of radiative properties of the particle laden combustion gases. Predictive accuracy of the method is assessed by comparing its predictions with incident +uxes measured on the freeboard walls. The predictive accuracy and computational e@ciency of the method are also evaluated by comparison with the zone method (ZM) of analysis. The sensitivity of the heat +uxes to the presence of particles, particle load and anisotropic scattering are also examined. 2. Description of the test rig The main body of the test rig is a modular combustor formed by Ave modules of 1 m height with internal cross-section of 0:45 m × 0:45 m. The Arst and the Afth modules refer to bed and cooler, respectively, and the three in between are the freeboard modules. There are two cooling surfaces

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Table 1 Operating conditions for the experiment

SuperAcial velocity, uo (m=s) Carryover +ow rate, Fc (kg=h) Particle density, p (kg= m3 ) Sauter mean diameter, D32 (m) Particle load, B (kg= m3 )

3.1 25 536.7 35.6 0.01

Mass speciAc cross-section, Amc (m2 = kg) Average H2 O concentration (%) Average CO2 concentration (%) ◦ Average medium temperature ( C) Mean beam length, Lm (m)

78.6 10 10 890 0.38

of 0:35 m2 and 4:3 m2 in the bed and cooler modules, respectively. At various heights along the combustor, there are 14 ports for thermocouples and 10 ports for gas sampling probes into which radiation probe is also inserted. Inner walls of the combustor are lined with alumina based refractory bricks of 6 cm thickness. In order to measure the concentrations of O2 , CO, CO2 ; SO2 and NO=NOx along the combustor at steady state, combustion gas is sampled from the combustor and passed through gas conditioning system where the sample is Altered, dried and cooled to be fed to the analyzers. The process values such as +ow rates and temperatures of each stream, gas composition and temperature along the combustor are logged to a PC by means of a data acquisition and control system, Bailey INFI 90. Further details for the test rig can be found elsewhere [3,7–9]. Radiative heat +uxes incident on the refractory side-walls of the freeboard were measured by a Medtherm 48P-20-22K heat +ux transducer during the steady state operation of the test rig. Details of the transducer are available elsewhere [3,8]. The radiometer eliminates the eCects of convection and measures only the incident radiative heat +ux. The radiometer probe was inserted into the gas sampling ports at Ave diCerent heights along freeboard +ush with the inner surface of the refractory side-wall. The radiometer output for incident radiative heat +ux was read from a voltmeter and the readings were then converted to heat +uxes using the certiAed calibration of the transducer. The steady state operating conditions required for the radiative property estimation are presented in Table 1. Temperature measurements were carried out on a discrete grid of points along the freeboard at steady state operation. In order to facilitate the use of these measurements as input data in the calculation of radiative exchange, the experimental data were represented by high order polynomials given in Fig. 1. For radiative property estimation of particle laden combustion gases, particles collected from the cyclone downstream of the freeboard were subjected to particle size distribution analysis by laser light scattering technique. Measured distribution is expressed by the Rosin–Rammler function and displayed in Fig. 2. 3. Approximation of the freeboard as a three-dimensional (3-D) radiation problem In order to apply the radiation models to the freeboard, it is required to provide temperatures and radiative properties of the surfaces and the medium. The freeboard section of the combustor is treated as a 3-D rectangular enclosure containing gray absorbing, emitting and isotropically scattering medium bounded by diCuse, gray=black walls. The cooler boundary at the top, which consists of gas lanes and cooler tubes, is represented by an equivalent gray surface of eCective emissivity and temperature related to area weighted average emissivity and emissive power of the components, respectively. Details of the treatment of tube-row=gas-lane combination can be found elsewhere

N. Selc)uk et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 503 – 516 1.0

Cumulative Weight Fraction, W (Dp)

1000

COOLER

800

600

BED

Temperature (oC)

507

400

200

gas temperature wall temperature

experimental Rosin-Rammler fit

0.8

0.6

0.4

0.2

0.0

0 0

1

2

3

4

5

Height, z (m)

Fig. 1. Temperature proAles along freeboard. Polynomials for temperature proAles: Tg (z) = −5:2962z 4 + 43:956z 3 − 146:52z 2 + 240:79z + 748:6 (◦ C); Tw (z) = −11:164z 3 + 54:123z 2 − 109:95z + 938:8 (◦ C).

0

500

1000

1500

2000

Particle Size, Dp (µm)

Fig. 2. Carryover particle size distribution Rosin–Rammler distribution function: W = exp(−bDpn ), b = 2:90 × 10−3 , n = 1:21, R2 = 0:998.

Table 2 Radiative properties of the medium and the surfaces

Gas absorption coe@cient, #g (1=m) Absorption coe@cient of particle cloud, #p (1=m) Scattering coe@cient of particle cloud, $s (1=m) Extinction coe@cient of the particles, %p = #p + $s (1=m) Absorption coe@cient of the medium, # = #p + #g (1=m) Extinction coe@cient of the medium, % = # + $s (1=m) Scattering albedo of the medium, ! = $s =% Emissivity of top surface, 'top Emissivity of side surfaces, 'w Emissivity of bottom surface, 'bottom ◦ Temperature of top surface ( C), Ttop ◦ Temperature of bottom surface ( C), Tbottom

0.43 0.16 0.45 0.61 0.59 1.04 0.43 0.87 0.33 1.00 549 873

[3,8]. The resulting temperature and eCective emissivity of the equivalent gray surface are reported in Table 2. The boundary with the bed section at the bottom is represented as a black surface due to Hohlraum eCect [8]. The physical system and the treatment of the freeboard is schematically illustrated in Fig. 3. Final step in the preparation of the input data is the estimation of radiative properties of the particle laden combustion gases consisting of CO2 ; H2 O and +y-ash particles bounded by the freeboard walls. This mixture is treated as an absorbing, emitting and isotropically scattering gray medium. Its radiative properties are assumed to be uniform and constant throughout the freeboard. This assump-

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Cooler

508

Freeboard

Fly-ash particles and combustion gases

Equivalent gray surface

Top surface εtop, Ttop

Absorbing, emitting, isotropically scattering medium

Medium Tg(z), κ ,σs

Gray side surfaces

Side surfaces εw, Tw(z)

3.35m Refractory lined side walls

z Imaginary black surface

Bed

0.9 m

0.45 m

y

x

Bottom surface εbottom, T bottom

0.45 m

The physical system: Freeboard of the AFBC and its boundaries

Treatment of the freeboard as a 3-D rectangular enclosure

Solution domain, boundary conditions and 5x5x35 grid used in MOL solution of DOM

Fig. 3. Treatment of freeboard as a 3-D enclosure and solution domain for MOL of DOM.

tion is based on uniform CO2 and H2 O concentrations measured along the freeboard and the fact that particle concentration and size distribution can be represented by the material sampled from the cyclone [9]. The radiative properties of the participating combustion gases are estimated by using Leckner’s correlations [10], which require the partial pressures of carbon dioxide and water vapor, the gas temperature and mean beam length, Lm . The mean beam length in the present study is based on the freeboard region. Calculation of the gas emissivity 'g through Leckner’s correlations leads to gas absorption coe@cient expressed by #g = −(1=Lm ) ln(1 − 'g ):

(1)

Radiative properties of the cloud of +y-ash particles depend on the composition, size distribution and particle loading. The ash content of the +y-ash particles determined by chemical analysis was 98% indicating that the +y-ash can be treated as pure ash in the radiative property estimation. The spectral dependence of complex index of refraction is neglected and a representative value of m = 1:5 − 0:02i is used as given in [11]. Independent scattering is assumed to take place in the freeboard of the test rig as the particle volume fraction is in the order of 10−5 . This assumption is

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conArmed by the scattering regime map of Tien and Drolen [12]. Assuming also that the particles are spherical, e@ciency factors of a single particle can be obtained from Mie theory [13] for a given particle size parameter (x = )Dp =*) and complex index of refraction. The size parameter is determined by using a representative wavelength (3 m) suggested for combustion systems in [14]. Utilizing the assumption of independent scattering, e@ciency factors for the particle cloud can be determined by taking average over the e@ciencies of the single particles. In the case of a homogeneous polydispersion and single wavelength, size averaged e@ciency factors characterize the radiative properties of the particle cloud. In order to And the size averaged e@ciency factors, the cumulative weight distribution of the particles is expressed as a Rosin–Rammler distribution function (Fig. 2) n

W (Dp ) = e(−b·Dp ) ;

(2)

where b and n are constants and W (Dp ) is the cumulative weight distribution function. Once the constants b and n are determined, the cumulative weight distribution is related to diCerential weight distribution p(Dp ) and number distribution fN (Dp ) with the following relations: p(Dp ) = − fN (Dp ) =

d W (Dp ); dDp p(Dp ) ; p Vp (Dp )=Mt

(3) (4)

where Mt is the total weight of the particles, p is the density of the particles and Vp (Dp ) is the volume of a spherical particle of diameter Dp . The size averaged absorption e@ciency of the particle cloud Qa; av is given by [15]
∞ Qa (Dp )()Dp2 =4)fN (Dp ) dDp ; (5) Qa; av = 0
∞ ()Dp2 =4)fN (Dp ) dDp 0 where Qa (Dp ) is the absorption e@ciency factor obtained from Mie theory. Similar relation holds for the size averaged scattering e@ciency factor Qs; av . The absorption and scattering coe@cients of the system of particles are then calculated from [15] #p = Qa; av BAmc ;

(6)

$s = Qs; av BAmc ;

(7)

Fc ; uo Ac

(8)

where B=

Amc =

3=2 : p D32

(9)

Amc denotes the mass speciAc cross-section, B is the particle load, uo represents the superAcial velocity, Fc is the carryover +ow rate of particles collected from the cyclone with density p ; D32

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refers to the Sauter mean diameter of the size distribution of particles and Ac is the cross-section of the combustor. Radiative properties of the particle laden combustion gases and radiative properties and temperatures of the bounding surfaces are summarized in Table 2. These data together with medium and side-wall temperature proAles given in Fig. 1 provide the input data supplied to the radiation models.

4. Application of the radiation models to the freeboard 4.1. Method of lines solution of discrete ordinates method In this method, false-transients approach is implemented to the discrete ordinates representation of the RTE, and resulting equations are solved by MOL which is an e@cient numerical technique for solution of PDEs. DOM is based on representation of the continuous angular domain by a discrete set of ordinates with appropriate angular weights, spanning the total solid angle of 4) sr [16]. The discrete ordinates representation of RTE for the 3-D rectangular enclosure containing gray, absorbing, emitting and scattering medium takes the following form: m

M @I m @I m @I m $s   + m + m = −#I m + #Ib − $s I m + -(m ; m )wm I m @x @y @z 4) m =1

(10)

where, I m [≡I (x; y; z; m ; m ; m )] is the total radiation intensity at position (x; y; z) at discrete direction m ; m denotes the discrete direction (m = 1; 2; : : : ; M ); M is the total number of discrete directions, m ; m , and m are the direction cosines of m with x; y and z axes, respectively, wm is the angular quadrature weight associated with the incoming direction m ; # and $s are the absorption and scattering coe@cients respectively. Ib (≡ $T 4 =)) is the black-body radiation intensity at the position (x; y; z) and -(m ; m ) represents the phase function for scattering between m and m discrete directions which is equal to unity for isotropic scattering. The boundary conditions at the two opposite, diCuse, gray surfaces with normal vectors parallel to the x-axis can be written as: at x = 0;

I m = ' w Ib ; w +

(1 − 'w )   wm |m |I m ; )   ¡0

m ¿ 0;

(11)

(1 − 'w )   wm |m |I m ; )   ¿0

m ¡ 0;

(12)

m

at x = L;

I m = ' w Ib ; w +

m

where I m is the intensity of radiation leaving the surface, 'w denotes the surface emissivity, Ib; w is the total black-body radiation intensity at the temperature of the surface. Similar expressions hold for boundaries in other coordinate directions. In order to solve for the radiative intensity distribution, the rectangular enclosure is subdivided into NX ×NY ×NZ control volumes of equal size and angular domain is discretized into M ordinates.

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511

Writing Eq. (10) for each ordinate at each control volume results in NX × NY × NZ × M coupled PDEs to be solved simultaneously. The classical numerical solution procedure employed to solve discrete ordinates equations is based on conversion of the main equation (Eq. (10)) into an algebraic Anite-diCerence form by employing control-volume approach and application of an interpolation scheme to relate face intensities of the control volume to cell-center value. The solution is obtained by recursive evaluation of radiative intensities by sweeping the enclosure in accord with the direction of physical propagation of the radiation beam, recalculation of boundary conditions and repetition of the sweeps until a prescribed convergence is achieved between two iterations [16]. In this paper, discrete ordinates equations are solved by MOL approach instead of the classical ordinate sweeping technique. The solution of discrete ordinates equations with MOL is carried out by adoption of the false-transients approach which involves incorporation of a pseudo-time derivative of intensity into the discrete ordinate equation for each direction [4]. Application of the false-transients approach to Eq. (10) and rearranging yields M @I m @I m @I m @I m $s   wm  I m : = −m − m − m − #I m + #Ib − $s I m + @t @x @y @z 4) m =1

(13)

The PDE system is transformed into an ODE initial-value problem by using the method of lines approach [17]. The transformation is carried out by representation of the position derivatives with approximate diCerence equations. Starting from an initial condition for radiation intensities in all discrete directions, the resulting ODE system is integrated until steady state by using a powerful ODE solver. The ODE solver takes the burden of time discretization and chooses the time steps in a way that maintains the accuracy and stability of the evolving solution. Any initial condition can be chosen to start the integration, as its eCect on the steady state solution decays to insigniAcance. As a result, evolution of radiative intensity with time at each node and ordinate is obtained. The steady state intensity values give the solution to Eq. (10) because the artiAcial time derivative vanishes at steady state. Once the radiation intensity distribution in the enclosure is evaluated by using the RTE together with its boundary conditions, incident radiative +ux on the walls, which is the quantity of interest for comparison with measurements can be readily calculated using:  

qw wm |n · m |I m : (14) ; inc = n·m ¡0

In the present work, for the implementation of the DOM, the SN angular quadrature scheme proposed by Carlson and Lathrop [18] is selected. The choice is based on an assessment study carried out by Sel.cuk and Kayakol [19]. For the diCerence relations of spatial derivatives, three-point upwind diCerencing scheme DSS014, assessed previously for accuracy [4 – 6] is employed. The utilized ODE solver is the RKF45 (Runge–Kutta–Fehlberg integration) subroutine previously found to be as accurate but less CPU intensive than Livermore solver for ordinary diCerential equations (LSODE) [6]. Numerical accuracy and computational economy of the MOL of DOM with respect to spatial and angular discretizations were investigated by comparing its steady state predictions with those obtained by the reference case carried out with Anest spatial and angular discretizations. Comparison is illustrated in condensed form in Table 3. As can be seen from the table, errors decrease with number of grids at the expense of computational time and CPU times increase threefold whereas

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Table 3 Grid reAnement study for MOL of DOM

Number of control volumes 1×1×7 3 × 3 × 23 5 × 5 × 35 7 × 7 × 53

S4

S6

% Errora

CPU time ratiob

% Errora

CPU time ratiob

6.895 2.680 1.103 0.382

0.0008 0.014 0.056 0.18

6.759 2.649 1.102 0.379

0.0025 0.034 0.13 0.42

a % Error: Average percent relative error for predicted incident heat +uxes at grid points with respect to the predictions of the reference case with S6 and 9 × 9 × 67. b CPU time ratio: CPU time=CPU time of the reference case.

the errors do not improve signiAcantly with the angular discretization. Hence, from the viewpoints of accuracy and computational economy, the use of S4 and 5 × 5 × 35 control volumes was found su@cient for the problem under consideration. 4.2. Zone method One of the most accurate procedures available for mathematical modeling of radiation Aelds within enclosures is the ZM, described in detail by Hottel and SaroAm [20]. The method is readily applicable to multi-dimensional radiative transfer in absorbing, emitting and isotropically scattering media. In the present study, the freeboard is subdivided into 10 volume zones along the height leading to 12 surface zones including the top and bottom surfaces. The incident radiative heat +ux equations for surface and volume zones given by Noble [21] are solved simultaneously by Gauss–Jordan elimination method to obtain incident radiation on surface and volume zones. Multiple integrals deAning the direct exchange areas between any pair of zones are obtained by direct numerical integration [22]. 5. Validation of the models against experimental data Fig. 4 illustrates comparison between the incident radiative heat +uxes predicted by the ZM and MOL of DOM with the measurements. As can be seen from the Agure, the incident +ux decreases from the bed surface toward the cooler and that the predictions of both methods are in good agreement with each other and with the measurements. For comparative testing purposes, point values of the predicted +uxes were compared with the measurements at discrete points. Table 4 shows the percentage relative errors in +uxes predicted by both methods. As can be seen from the table, percentage errors are of the same order of magnitude. SigniAcant discrepancy between the predictions and the measurement at the uppermost port is considered to be due to the fact that the top surface of the enclosure is approximated by an equivalent gray surface consisting of cold tube-row=hot gas-lane combination for modeling purposes, whereas the radiometer probe is aCected mostly by the cooling tubes as the port for the measurement is located nearly adjacent to a cooler tube.

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513

100

COOLER

80

BED

Incident Radiative Heat Flux (kW/m2)

120

60

40 MOL solution of DOM Zone Method Measurements

20

0 0

1

2

3

4

5

Height, z (m)

Fig. 4. Incident radiative heat +uxes on freeboard wall.

Table 4 Incident radiative heat +uxes on freeboard wall

Height (m)

1.23 1.83 2.91 3.44 4.19

Experimental (kW= m2 )

108.9 96.4 90.2 71.5 28.0

Predictions (kW= m2 )

Relative error (%)

ZM

MOL of DOM

ZM

MOL of DOM

103.0 102.8 94.0 81.8 51.4

99.0 98.8 90.1 78.0 47.3

−5:4

−9:1

6.6 4.2 14.4 83.6

2.4

−0:1

9.1 69.0

CPU times for MOL of DOM and ZM using IBM Risc=6000 Model 590 were compared. MOL of DOM is found to require 230 s which is an order of magnitude smaller than that required by ZM including the direct exchange area calculation. Sensitivity of incident heat +ux to the presence of particles was analyzed by comparing the predictions of both methods with and without particles (Fig. 5). As can be seen from the Agure, eCect of particles on predicted heat +uxes is not considerable. This may be due to the relatively low particle loads typically encountered in the freeboards of ABFBCs. A parametric study is also performed by using MOL of DOM to analyze the eCects of particle load and anisotropic scattering on the incident wall heat +uxes. The real case analyzed previously with isotropic scattering assumption is taken as basis and three diCerent cases are generated by increasing the particle load and=or by incorporating strong anisotropy into the problem. The particle load is increased 1000-fold to an order of magnitude typical of circulating FBCs [15]. The eCect of anisotropy is analyzed by utilizing linear anisotropy assumption, for which the discrete ordinates

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COOLER

80

60

40 MOL of DOM - gas only ZM - gas only MOL of DOM - gas + particle ZM - gas + particle

20

0

100

80

COOLER

100

BED

Incident Radiative Heat Flux (kW/m2)

120

BED

Incident Radiative Heat Flux (kW/m2)

120

60

40 base case anisotropic high particle load high particle load + anisotropic

20

0

0

1

2

3

4

5

0

1

Height, z (m)

Fig. 5. Sensitivity of radiative heat +ux to the presence of particles. Gas only: %p = 0, gas + particle: %p = 0:61.

2

3

4

5

Height, z (m)

Fig. 6. ECect of high particle load and anisotropy on incident radiative heat +ux. Base case: B=0:01 kg=m3 , 3g=0, anisotropic: B = 0:01 kg=m3 , 3g = 0:9, high particle load: B = 10 kg=m3 , 3g = 0, high particle load + anisotropic: B = 10 kg=m3 , 3g = 0:9.

representation of scattering phase function takes the following form: -(m ; m ) = 1 + 3g(m m + m m + m m );

(15)

where g is the asymmetry factor. Strong, forward scattering is assumed (3g = 0:9) in accordance with the scattering nature of +y-ash particles. Fig. 6 shows the incident radiative +ux proAles on the side walls of the freeboard obtained for the three cases together with the base case. It can be seen that 1000-fold increase in particle load increases the +uxes both for isotropic and anisotropic scattering treatments. On the other hand, incorporation of anisotropic instead of isotropic scattering to the solution has negligible eCect on predicted radiative +uxes for both particle loads concerned. 6. Conclusions Radiative heat +uxes incident on the side walls of freeboard of an atmospheric bubbling +uidized bed combustor have been predicted by using the MOL of DOM and the ZM. The freeboard containing particle laden combustion gases is treated as a 3-D enclosure with absorbing, emitting and isotropically scattering medium. Radiative properties of the medium are calculated by using Leckner’s correlations for gas and Mie’s theory for particles. The input data required for the application and validation of the radiation models are generated from METU 0:3 MWt ABFBC test rig operating under steady state conditions. On the basis of comparisons between predictions and measurements, the following conclusions have been reached. • Both the MOL of DOM and the ZM reproduce the measured heat +uxes reasonably well. • MOL of DOM is as accurate as the ZM and much faster for lightly scattering media.

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Conclusions drawn from the parametric studies carried out for the eCect of particle load and anisotropy on the predicted radiative +uxes are as follows: • Presence of particles in the participating medium does not aCect the magnitude of predicted incident +uxes signiAcantly due to low particle load in the freeboards of bubbling +uidized bed combustors. • Increasing the particle load to the order of magnitude typically encountered in circulating FBCs leads to signiAcant rise in incident radiative +uxes at the wall. • ECect of anisotropy on incident radiative heat +uxes on the side-walls is negligible even in the presence of high particle loads. The data produced for the radiative analysis in the freeboard section of the combustor may serve in the future as a test problem for the assessment of the predictive accuracy of various multi-dimensional radiation models in participating media. References [1] Lindsay JJ, Morton W, Newey DC. Radiative heat transfer in the freeboard region of a +uidised bed. Proceedings of the Fifth Engineering Foundation Conference on Fluidization, Engineering Foundation, New York, NY, 1986. p. 385 –92. [2] Filla M, Scalabrin A, Tonfoni C. Scattering of thermal radiation in the freeboard of a 1 MWt +uidized bed combustor with coal and limestone feeding. Twenty-Sixth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA, 1996. p. 3295 –300. [3] Kozan M, Sel.cuk N. Investigation of radiative heat transfer in freeboard of a 0:3 MWt AFBC test rig. Combust Sci Tech 2000;153:113–26. [4] YTucel A. Solution of the discrete ordinate equations for radiatively participating medium by the method of lines. In: Vichnevetsky R, Knight D, Richter G, editors. Advances in Computer Methods for Partial DiCerential Equations VII. New Brunswick, NJ: IMACS, 1992. p. 838–44. [5] Sel.cuk N, Kirba.s G, Tarhan T. Evaluation of method of lines solution of discrete ordinates method and Anite volume method in a planar medium. CHMT 99, Proceedings of the International Conference on Computational Heat and Mass Transfer, GazimaVgusa, Northern Cyprus, 1999. p. 358– 64. [6] Sel.cuk N, Kirba.s G. The method of lines solution of discrete ordinates method for radiative heat transfer in enclosures. Numer Heat Transfer Part B 2000;37:379–92. [7] Sel.cuk N, DeVgirmenci E, GTogV ebakan Y. Assessment of catalyst deactivation model for sulfur retention in +uidized bed combustors. Combust Sci Tech 2000;153:95–111. [8] Kozan M. Investigation of radiative heat transfer in freeboard of a 0:3 MWt AFBC test rig. M.Sc. thesis, METU Chemical Engineering Department, Ankara, 1999. [9] DeVgirmenci E. Dynamic simulation of +uidized bed combustors. Ph.D. thesis, METU Chemical Engineering Department, Ankara, 2000. [10] Leckner B. Spectral and total emissivity of water vapor and carbon dioxide. Combust Flame 1978;19:33–48. [11] Viskanta R, Ungan A, MengTuc. MP. Predictions of radiative properties of pulverized coal and +y-ash polydispersions. ASME Paper, 81-HT-24, 1981. [12] Tien CL, Drolen BL. Thermal radiation in particulate media with dependent and independent scattering. Ann Rev Numer Fluid Mech Heat Transfer 1987;1:1–32. [13] Deirmendjian D. Electromagnetic scattering of spherical polydispersions. USA: Elsevier, 1969. [14] Modest MF. Radiative heat transfer. New York, NY: McGraw-Hill, 1993. [15] Neubronner M, Vortmeyer D. Thermal radiation of +y ashes—dependence on size distribution and chemical composition. In: Hewitt GF, editor. Proceedings of the Tenth (International) Heat Transfer Conference, Institution of Chemical Engineers, Warwickshire, UK, 1994. p. 117–22.

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N. Selc)uk et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 73 (2002) 503 – 516

[16] Fiveland WA. Three dimensional radiative heat-transfer solutions by the discrete-ordinates method. J Thermophys 1988;2:309–16. [17] Schiesser WE. The numerical method of lines in integration of partial diCerential equations. New York: Academic Press, 1991. [18] Carlson BG, Lathrop KD. Transport theory—the method of discrete ordinates. In: Greenspan H, Kelber CN, Okrent D, editors. Computing methods in reactor physics. New York: Gordon and Breach, 1968. [19] Sel.cuk N, Kayakol N. Evaluation of angular quadrature and spatial diCerencing schemes for discrete ordinates method in rectangular furnaces. ASME HTD-Vol 325, vol. 3. 1996. p. 151–8. [20] Hottel HC, SaroAm AF. Radiative transfer. New York: McGraw-Hill, 1967. [21] Noble JJ. The zone method: explicit matrix relations for total exchange areas. Int J Heat Mass Transfer 1975;18:261–9. [22] Rhine JM, Tucker RJ. Modelling of gas-Ared furnaces and boilers. British gas plc. Great Britain: McGraw-Hill, 1991.