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Prediction of spectral radiative transfer in a condensed cylindrical medium using discrete ordinates method
J. Quant. Spectrosc. Rodiat. Transfer Vol. 58, No. 3, pp. 329-345, 1997 0 1997 Elsevier Science Ltd. All rights reserved
Pergamon
Printed in Great Britain 0022-4073/97 1617.00+ 0.00
PII: !SOO22=4073@7)00037-X
PREDICTION OF SPECTRAL RADIATIVE TRANSFER IN A CONDENSED CYLINDRICAL MEDIUM USING DISCRETE ORDINATES METHOD KONG HOON LEE? and RAYMOND
VISKANTAI
School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907-1288, U.S.A. (Received
18 November
1996)
Abstract-Spectral radiative transfer is analyzed in a condensed cylindrical medium using discrete ordinates method. In order to appropriately consider the directional variations of the reflectivities at the specularly bounding interfaces, two types of quadrature sets are examined and the S-8 LSH quadrature set is used. The BK7 optical quality glass is considered as an example of the semitransparent medium, and the spectral radiative fluxes are predicted. The results obtained show that for I, 2 2.8 pm the BK7 glass can be treated as an optically thick medium, and the radiative fluxes differ little for the glass with the dimensions larger than those being considered in the present study. For 0.4 < 1 < 2.8 pm, the characteristics of the spectral absorption coefficients of the glass become important, and the radiative fluxes depend on the opacity of the glass. The predicted spectral radiative fluxes differ from those based on the opaque solutions, and this indicates that the spectral radiative transfer must be considered carefully in the semitransparent medium. 0 1997 Elsevier Science Ltd
1. INTRODUCTION
Accurate prediction of temperature distributions in glass and other semitransparent materials at high temperature are essential during various fabricating operations. For example, the annealing of glass plates requires prediction of the temperature distribution from the transient combined conduction and radiation heat transfer analysis. During the process of annealing of optical quality glass which is utilized for optical components in imaging systems, the temperature gradients in the glass must be controlled very carefully by imposing a constant cooling rate of the order of 0.01 “C/h, because a true geometric and color image of an object is generated by using the property of refraction, and even small stresses due to the temperature gradients in the annealing range cause permanent strains and inhomogeneities of the refractive index.’ This can be detrimental to optical glasses of high quality. Moreover, the stress-strain relation in glass plate during annealing or tempering and danger of fracture is also strongly related to the temperature gradients.2 However, the local temperature and the local temperature gradient cannot be predicted accurately using the concept of the effective (apparent) thermal conductivity, and rigorous treatment of radiative transfer is needed.3.4 When semitransparent materials are manufactured or processed at high temperature, both volumetric absorption and emission must be accounted for when predicting the temperature distribution in the absorbing-emitting substance. Typically, such materials are homogeneous and scattering of radiation can be neglected in comparison to absorption, and this makes it easier to carry out the analysis of radiative heat transfer.5 However, since semitransparent materials have the spectral absorption and refraction characteristics which depend strongly on wavelength and specularly reflecting boundaries, it is difficult to solve the radiative transfer equation. In general, glass is semitransparent to thermal radiation in approximately the spectral range of 0 < I c 5 ,um, and is effectively opaque beyond 5 pm. This selectivity has to be properly accounted for in predicting radiative transfer inside the glass and radiation exchange between the surfaces and the surroundings.4 tPermanently at Advanced Institute of Machinery and Design, Seoul National University, Seoul 151-742, Korea. $‘Towhom all correspondence should be addressed. 329
330
Kong Hoon Lee and Raymond Viskanta
The purpose of the present study is to analyze radiative heat transfer in a cylindrical semitransparent medium as the first step to predicting the transient combined conductive and radiative heat transfer in a cylindrical disk of glass during annealing. For this purpose, the discrete ordinates method (DOM, S-N method) is used to solve the radiative transfer equation (RTE). Discrete ordinates method is widely accepted, and a comprehensive discussion is available.6 The numerical analysis was performed for an axisymmetric cylindrical medium and the model is validated by comparing to an exact analytical solution.4 2. ANALYSIS
2.1. Radiative transfer equation (RTE) Consider the radiative energy transport within a cylindrical semitransparent medium shown in Fig. 1. The medium is surrounded by isothermal black walls, and the space between the medium and the surrounding wall is either a vacuum (n;,’ = 1) or is occupied by a transparent entity having the refractive index of unity. In addition, the following assumptions are made in analyzing radiative transfer in the system: (1) the radiation field is axisymmetric and two dimensional; (2) the medium emits and absorbs but does not scatter thermal radiation; (3) the medium is in local thermodynamic equilibrium for which Planck’s and Kirchhoffs laws are valid; (4) the spatial dimensions of the medium are much larger than the wavelength of radiation for the semitransparent band, i.e., the coherence effects are negligible; (5) the refractive index of the medium at a given wavelength is uniform and does not depend on temperature in the considered range, therefore, the refraction of radiation within the medium is absent; (6) the interface between the medium and the bounding entity is optically smooth; and (7) the surrounding walls are black gray diffuse emitters of radiation. Under the above assumptions, the radiative transfer equation (RTE) for a spectrally absorbing-emitting medium can be written as6 WVZAr,Q)
= kiLni Z&Yl - ZAr,W
(1)
where Zj.(r, R) is the spectral radiative intensity, which is a function of position, direction and wavelength: Z,,,(T, nJ is the spectral intensity of blackbody radiation given by Planck’s function. Integration of Eq. (1) over all directions (Q = 4~) yields the spectral radiant energy equation V.F;. = 4mc;.nfZbi(7’) - K;.G;.
(2)
where the spectral radiative flux vector Fj. and the spectral irradiance Gj, are defined as F;. = sR=4n
Z@,fi)QdQ
Fig. 1. Schematic of physical situation and coordinate system.
(3)
Prediction of spectral radiative transfer in a condensed cylindrical medium
331
and
s
(4)
h.(r,WQ
G;. =
R= 4n
respectively. It is the integrated over the entire spectrum Eq. (1) that is used in the thermal energy equation to determine the temperature distribution in problems involving simultaneous conduction and radiationP The radiative transfer equation, Eq. (l), for the spectrally emitting and absorbing axisymmetric medium along a specified discrete direction Qm.‘(tP, $9 can be expressed as
In the above equation, 4 is the azimuthal angle for the direction (m, f) relative to the local radial direction, and pm.‘,q”‘.’and t;“‘,’are direction cosines and defined by p = sinfkos4,
q = sintkin~,
t = cost7 .
(6)
2.2. Boundary conditions The interface conditions between the medium and the bounding entity constitute the boundary conditions for Eq. (1). The interaction of radiation at the smooth interface between two dielectric media is governed by Snell’s and Fresnel’s laws?’ n,sintl = n;.‘sinO’.
(7)
When the radiation is transmitted from a denser into an optically less dense medium, the angle of refraction approaches 90” as an angle of incidence increases toward the critical angle (e,), which is defined by 8, = sin - ‘(ni’/ni)
.
03)
For an angle of incidence larger than the critical angle, the outgoing radiation from the medium is totally reflected inward. Similarly, the incoming radiation from the surrounding wall is concentrated into the range of directions associated with the angle of refraction smaller than 8,. The relation of the incident and reflected radiation at the smooth interface is represented by Fresnel’s equation. The directional reflectivity of the interface between two dielectric media of different reflective indices is obtained by combining Fresnel’s equation for reflection and Snell’s law for refraction:’
p;.(e)= i
I(h.iosfl-n;Pj-O Z n;,’ k0se + n,P
l(n;.cose-P):
+ Z
n;.cose
1: +
:zIl
(9)
P
Finally, the boundary condition for Eq. (1) at an optically smooth bounding surface is written as L.(r,Q) = [l -
p;.(ell(nj.ln;,‘)2~~,(~;,,) + p;.(Wj.(r,W for 42-n> 0
(10)
8 = cos - ‘@In), R’ = 2(Ch)n
(11)
where
JQSRT
58.3LB
- R
332
Kong Hoon Lee and Raymond Viskanta
The boundary conditions for the spectral intensity in the specified discrete direction, GF’, are obtained from the symmetry condition and Eq. (10) Zil”’= ZY’
at r=O
ZY’ = p?jZY’ + (1 - p;$)(nj./n;.‘)2Zbi.( TUT)at r = R Zy’ = p:V:‘,’ f (1 - p~~)(nj./nj.‘)2Z,,(~“~)at 2 = 0
Z?’ = p?:Z?,’ + (1 - p~:)(nj./n;.‘)2Z,,(~“,) at z = L
(12)
The discrete reflectivities, p”‘, in the above equations are defined to preserve the reflected radiative flux in the normal direction to the interface.
In the above equation, 8 is the angle between the direction of propagation Q and the inward normal II to an interface as defined by Eq. (11). 2.3. Discrete ordinates approximation Closed form analytical solutions of the RTE, Eq. (5), are not possible because the boundary conditions are complicated functions of direction; therefore, the discrete ordinates method (DOM) is used to obtain a numerical solution of the problem. The discrete ordinates equations in axisymmetric cylindrical geometry are complicated by the presence of the angular derivative term, a(~Z,)/&$. Carlson and Lathrop* have proposed a direct differencing technique for calculating the angular derivatives at the quadrature points. Using this technique, the second term on the left-hand-side of Eq. (5) can be written as a~ -1 a(qn’Jz,“‘)
(14)
r
The directions I+ l/2 define the boundaries of the angular range of ordinates, and the second term of numerator on the right-hand-side of Eq. (14) represents transport out and into the angular range, respectively. Since the angular ranges of the ordinates are not constant, the geometrical coefficients, a”‘,‘+I”, are also not constant and need to be determined. Since the values of a depend only on the differencing scheme and are independent of radiative intensity, a may be determined by examining a constant intensity field,’ that is, Zj.= ?:I,;. = constant, and the following formula is obtained,
This expression is used as a recursion formula for a’“,‘+‘I2using the fact that am.li2= 0 at 4”’ = 0 and ylm.li2 = 0. Thus, the spatially discretized radiative transfer equation in terms of the radiative intensities on the control surface can be obtained by integrating over the ring-shaped control volume, /Am$4”z~;- A,z~:) + 5”+t,z::
- &.I$
(16)
+ (A, - A,9)(amJ+iZ!$+ f _ am,‘-iZ!$ i)/,m./
where A. = 2w,(z, - z,.), A,v= 21cr,&, - z,.), A, = 71(ri - r.f) = A,,
(17)
Vp= ~(2~ - z,.)(r,Z- rt) .
(18)
Prediction of spectral radiative transfer in a condensed cylindrical medium
333
E-__-__E Ar
Fig. 2. Control volume
In order to reduce the number of the unknowns in Eq. (16), a differencing scheme needs to be introduced. For example, the radiation intensity, I;‘$, at the center of the control volume can be related to the intensities at the control surfaces (I~~;(,Z;‘;‘,l~~ and I;:;: shown in Fig. 2)
= (1 - fb)ZlliP f + fJ$+
f for
$“.’ > 0
and
o)‘.’> 0
(19)
The value of the weighting factor must be restricted within the range 0 < f I 1. Takingf = l/2 yields the diamond or central differencing scheme proposed by Carlson and Lathrop,’ and taking f = 1 yields the step or upwind differencing scheme. In implementing the discrete ordinates approximation for the cylindrical geometry, it has been a common practice to assume a uniform distribution of the weighting factor over the calculation domain, regardless of the control volume size.6 When the uniform weighting factors are used, the numerical predictions often exhibit physically unrealistic negative intensities, and the solution procedure has to be repeated until the negative values are removed, while the value of weighting factor is gradually decreased from its initial value of 1.0.9 In the present study, an exponential-type scheme which was proposed by Song and Viskanta’” in order to obtain positive radiative intensities is used together with the central and upwind schemes. In the exponential-type scheme” the weighting factors are calculated as follows
+exp(
-
w)]-’
(20)
f,=[l+exp(
-
%)I-’
(21)
f,=[l
fo
I +exp
=
[
(
K;.
~,l~“.‘pp’
-
(a”#.‘+f + c(“‘.‘- :)(A, _ A,)
11 -I
’
(22)
If the absorption coefficients are constant, the weighting factors given by Eqs (20)-(22) depend only on geometry and need to be evaluated only once.
334
Kong Hoon Lee and Raymond Viskqnta
The final form of both directionally and spatially discretized algebraic equation for positive ,u”‘.’ and (“I.’directions are obtained from Eq. (16) and (19) (23) where a = Ml
- f,) +
4fillfr
(24)
AfM; = Aelf; = A,l.f;
(25)
c = (A, - A,J[rP’+ i(1 - f4) + urn.‘-if,]/(cP’f~)
(26)
b = [A,(1 - f.-) +
d=
(27)
K;,&.
Eq. (23) is applicable to the case where both .P’ and cm.’are positive. For the other combinations of @“.’and t”‘*‘,the discretized equations can be obtained by integrating over appropriate control volumes. Once the radiation field has been determined, the radiative fluxes in the radial and axial directions are written in terms of discrete variables as follows
E.,,=
F;..;=
s s
N
2m
z,(r,z,fi)pdSZ N 1 c ,‘,%P’ m=,,=, n =4n N
2m
Z;.(r,z,Q)tdQ 2: 1 ~Z;m.‘~mGY’.‘.
R=4n
fPt=I,=l
(29)
2.4. Selection of discrete ordinate quadrature sets The accuracy of the discrete ordinates method depends on the choice of the quadrature set. Although the choice is arbitrary, a completely symmetric quadrature is preferred in order to preserve geometric invariance of the solution. The quadrature set used in the present study is based on the moment matching technique of Lathrop and Carlson” and Fiveland.‘* If the zeroth moment and sequential even moments (0,2,4..) are satisfied, level symmetric even (LSE) quadrature sets are obtained and if the zeroth and sequential moments (0,1,2,3..) are satisfied, level symmetric odd (LSO) quadrature sets are obtained.” LSE quadrature sets are not desirable because they do not satisfy the first moment, which defines the radiative flux. Fiveland’* obtained level symmetric higher-order (LSH) quadrature sets which satisfy the selected moments. The S-4 LSH quadrature sets are the same as the LSO quadrature, and the S-8 LSH quadrature sets satisfy the zeroth, the first and sequential even moments (0,1,2,4,6). Since the scattering of radiation is negligible in comparison to the absorption, in the present study the quadrature that satisfy the higher order of moments than the first moment is not needed. However, the directional dependence of specular boundaries is affected by the choice of the quadrature set to be used, because a weight represents the part of area on a unit sphere for each ordinate direction, and the average value of reflectivity within the range of a weight, Eq. (13), varies with the type of the quadrature sets. Thus, two kinds of quadrature sets (i.e., LSO and LSH) are examined. The quadrature sets of S-4 and S-8 are shown in Table 1. The direction cosines and weights given in Table 1 cover the solid angle of 471steradians. The weights, UP’, are positive and nonoverlapping and sum up to the area of a unit sphere, because they represent areas on the unit sphere. As the angular distribution of the radiative intensity is symmetric with respect to the r-z plane, the weights are twice of those used in the general three-dimensional problem, and then only half the number of directions[N(N + 2)/2] needs to be considered. 2.5. Radiative properties
of optical quality glass
As an example for an analysis of spectral radiative transfer in optical quality glass, BK7 glass is used. The glass is of optical quality which is sodium rich, and has excellent transmission
335
Predictionof spectralradiative transferin a condensed cylindrical medium Table I. Discreteordinatesquadraturesetsfor S-N approximation
I
m s-4
S-8 LSO"
( - 5) I
( - PI
&l.l
p
v’
P’ ( + PC) 2
kO.295876
0.295876
50.908248
n/3
2
4 3
kO.908248 kO.295876
0.295876 0.908248
& 0.295876 + 0.295876
sl3 nl3
I
2
5 0.142255
0.142255
kO.979554
0.342472
2
1 2
4 3
kO.577350 + 0.142255
0.142255 0.577350
+ 0.804009 + 0.804009
0.198457 0.198457
3
I 2 3
6 5 4
+ 0.804009 + 0.577350 k 0.142255
0.142255 0.577350 0.804009
kO.577350 kO.577350 kO.577350
0.198457 0.923436 0.198457
I 2 3 4
8 7 6 5
kO.979554 + 0.804009 + 0.577350 +0.142255
0.142255 0.577350 0.804009 0.979554
+ 0.142255 kO.142255 + 0.142255 kO.142255
0.342472 0.198457 0.198457 0.342472
I
I
2
kO.158489
0.158489
,0.974558
0.302780
2
I 2
4 3
* 0.577350 +_0.158489
0.158489 0.577350
+ 0.800967 + 0.800967
0.280735 0.280735
3
I 2 3
6 5 4
+ 0.800967 kO.577350 + 0.158489
0.158489 0.577350 0.800967
kO.577350 + 0.577350 kO.577350
0.280735 0.548844 0.280735
1 2 3 4
8 7 6 5
kO.974558 f 0.800967 kO.577350 kO.158489
0.158489 0.577350 0.800967 0.974558
kO.158489 k 0.158489 + 0.158489 kO.158489
0.302780 0.280735 0.280735 0.302780
4
S-8 LSH'*
( + 5)
4
5
5
characteristics in the visible and near infrared, making it a popular glass for optical systems. The properties of BK7 glass, refractive index and absorption coefficient, given by the manufacturer? are used in the present study. The refractive index of BK7 glass with Sellmeier dispersion formula is given by (30) 1.6
1.5
-.
-.
-.
-.
. . \
Absorption coefficient Refractiveindex
1
2
Wavelength
3
4
-
5
1.2
(pm)
Fig. 3. Spectralabsorptioncoefficient and refractive index of BK7 glass. $SchottGlaswerke, Mainz, Germany
336
Kong Hoon Lee and Raymond Viskanta
where the unit of wavelength, A, is pm and B, = l.O3961212,C, = 6.00069867 x 10e3, B2 = 2.31792344 x 10-4, C, = 2.00197144 x 10-2, B, = l.O1046945,C, = 1.03560653 x lo*. As shown in Fig. 3, the refractive index decreases with wavelength. The absorption coefficient, K;.,is relatively low in the visible and near infrared region. BK7 optical glass has good transmissivity in this spectral region. The emissivity in the opaque spectral region (1 > 5 pm) is 6 = 0.9. 2.6. Method of solution The solution procedure involves solving the radiative transfer equation in each of the ordinate directions, using a set of N(N + 2)/2 coupled discretized equations. The calculation starts at the grids adjacent the boundary surfaces along the specified direction with the boundary conditions, Eq. (12). However, before starting the solution, an initial estimation of the azimuthal component of intensity needs to be obtained by solving Eq. (23) in the special &directions where tl = 0 and the values of the weights and geometric coefficients, a, are assigned to be zero. The radiative intensity at the center of a control volume can be explicitly calculated from the downstream surface values using Eq. (23), and the intensities at the downstream surfaces can be obtained by extrapolation using Eq. (19). The evaluation of the radiative intensity proceeds in the direction of increasing (or decreasing) r and z, when the direction cosines p and l are positive (or negative), respectively. For a given t (or superscript m), the evaluation proceeds in the order of increasing ~1(or superscript I). Since the distributions of the radiative intensity in different directions are coupled to each other through the specular boundary conditions, the solution procedure has to be iterative. The updating of radiative intensity field is repeated until the radiative intensity at every node and in every direction does not change within the accuracy of five significant digits. The number of iterations required to obtain the converged solution increases as the opacity of the medium decreases. The grid spacing used is not uniform, but is more compact near the symmetry axis and the bounding interfaces, This was done in order to resolve the steep spatial change in the physical variables expected in these regions. 3. RESULTS
AND DISCUSSION
3.1. Validation In order to validate the solution method, one-dimensional radiative transfer was considered in a semitransparent plate. It was assumed that the reflectivity at the curved surface (r = R) was unity for the entire range of incident angles to obtain one-dimensional problem from the two-dimensional formulation. The temperatures in the medium and the surrounding walls were assumed to be uniform and cold (7& = 0), respectively. In addition, the refractive index was assumed to be a constant (n = 1.46), and the optical thickness r,( = KL) was used for the comparison in each spectral band. The results are compared in Figs 4 and 5 together with the integral (exact) solution obtained under the same conditions.4 For the large optical thickness (7, = 100) the radiative flux is maximum near the surface, and it means that the radiative cooling occurs mainly in the very thin layer adjacent to the bounding surface. In the case of a opaque medium (TV + co), only the surface exchange needs to be accounted for. As the optical thickness decreases, the radiative cooling becomes more significant throughout the larger fraction of layer thickness, and the maximum value of the gradient of the radiative flux decreases. For zL = 1, the radiative flux increases almost linearly toward the surface from the center plane, and the radiative cooling is nearly uniform across the semitransparent medium. Figure 4 shows the radiative flux distribution for different quadrature sets. A 31 x 61 grid and an exponential-type differencing scheme are used. For TV= 1, the S-4 approximation overpredicts the radiative flux across the plate. The radiative flux calculated using the S-4 approximation is larger than that obtained from the integral solution by 11% at z = L. This overprediction results from the evaluation of the average reflectivity, Eq. (13), because it is not appropriately averaged over the angular range of the ordinates. The directional reflectivity varies strongly near the critical
Prediction of spectral radiative transfer in a condensed cylindrical medium
!
-
,s * P.,0.8 LL _ X 3 G=
331
Integral solution 0
S-8 LSH quadrature
0
S-8 LSO quadrature
A
S-4
ti
r” 0.6 -
z ‘3 m e !I * 0.4
-
8 P 0 ._ 2 fj
._ n
0.2 -
Fig. 4. Comparison of the radiative flux for the different quadrature sets.
angle 8,, and it is, therefore, very important to appropriately average the steep variation over the angular range of the ordinates in order to correctly treat the specular boundary conditions. To obtain the more accurate results with discrete ordinates method, a higher order of quadrature set is needed. In the present study, two kinds of the S-8 quadrature sets were tested as shown in Fig. 4. The radiative flux obtained with the S-8 LSO quadrature set is in closer agreement with the integral
-
Integral solution Cl
Exponential scheme
0
Central scheme Upwind scheme
A
Fig. 5. Comparison of the radiative fluxes for the different differencing schemes.
338
Kong Hoon Lee and Raymond Viskanta Table 2. Dimensionless radiative fluxes (F-/r&) for the different grid systems at z/L = 1 when rL = 1 Grid system (N_ x N,) 21 x 41
31 x 61
41 x 81
S-8 LSH quadrature Exponential scheme Central scheme Upwind scheme
0.63473 0.63792 0.62609
S-8 LSO quadrature S-4 quadrature
0.6503 1 0.71364
Integral solution
0.63451 0.63445 0.63754 0.63741 0.62829 ’ 0.62903 Exponential scheme 0.65014 0.71036
0.65007 0.71368
0.63664
solution than with the S-4 quadrature set, and the radiative flux with the S-8 LSH quadrature set shows fairly good agreement with the integral solution through the entire range of optical thicknesses. However, this agreement can vary with the refractive index. Even though a comparison is not presented in this paper, if the refractive index is larger (for example, n = 1.5), the S-8 LSO quadrature set yields improved results than the LSH quadrature set, because the critical angle and the angular distribution of reflectivity, Eq. (9) are also changed. For the smaller value (n = 1.42), the S-8 LSH quadrature set still yields accurate results. Finally, over the range of the refractive index shown in Fig. 3, the S-8 LSH quadrature set is the preferred one for the present study. Figure 5 shows the radiative fluxes obtained with three different differencing schemes using the 31 x 61 grid and the S-8 LSH quadrature set. The results obtained with the exponential-type and central differencing scheme are in good agreement with the integral solution, but the upwind scheme underpredicts the radiative flux. The uniform weighting factors were used in the central and upwind scheme only for the purpose of comparison even though they could yield unreasonable negative intensity. When the upwind and exponential-type schemes were used, the negative intensity was not predicted. However, for the central scheme, the calculations converged slowly, and much more computing time and a larger number of iterations were required than for other schemes due to the tendency of the solution to oscillate which was caused by the negative intensity. The number of iterations required for convergence were about a hundred times as many as those for the other schemes. Sensitivity tests for the various grid systems were also carried out. The grid spacing used is not uniform and is more compact near the symmetry axis and the bounding interfaces to resolve the steep spatial change of the radiative flux in these regions. Some of the results for 21 x 41, 31 x 61, and 41 x 81 grid systems are included in Table 2. For the purpose of comparison, only the results for rL = 1 are given in Table 2. However, the results for the other optical thicknesses also have the similar trends. As shown in Table 2, the radiative flux obtained with the S-8 LSH quadrature sets and the exponential-type differencing scheme is in good agreement with the integral solution. They are nearly independent on the grid systems. In the present study, the finer (31 x 61) grid than the coarser (21 x 41) grid is used to resolve the steep spatial changes of the radiation flux near the bounding surfaces, even though the 21 x 41 grid also yields acceptable results. The sensitivity tests have established that the S-8 LSH quadrature set, the exponential-type differencing scheme, and the 31 x 61 grid system are adequate for the present study. 3.2. Spectral radiative transfer in the optical quality glass The results for spectral radiative transfer in BK7 optical quality glass are given in Figs 6 - 11 and Table 3, and they are obtained using the absorption coefficient and the refractive index shown in Fig. 3. The optical quality glass disk considered has the radius R = 100 mm and the thickness L = 50 mm. The results are obtained in the range of 300 K < T < 1100 K and zU:.,= 300 K. In order to obtain the results, the S-8 LSH quadrature set, the 31 x 61 grid, and the exponential-type differencing scheme were used in the discrete ordinates method. The effect of the number of bands on the spectral radiative flux has been examined. Each band is uniformly spaced and four to one hundred bands are considered in the temperature range of 300 K < T I 1100 K. The results obtained using 24 bands are in good agreement with those
Prediction
of spectral
radiative
transfer
in a condensed
cylindrical
medium
339
12-
IO-
8-
6-
4-
2.
O0
1
2
3
Wavelength Fig. 6. Effect of the number
4
5
(pm)
of bands on the spectral radiative flux in the axial direction and T= 1000 K, 7;,, = 300 K.
at z = L. r = 0.5R
obtained using 100 bands with a discrepancy of less than 1O/i. Thus, 24 bands were used to obtain all the results reported in the remainder of the paper. The effect of the number of bands on the radiative fluxes for less than 24 bands is shown in Fig. 6 at a uniform temperature of 1000 K with the surrounding wall temperature of 300 K. The radiative flux obtained using 16 bands is close 20 T=llOOK -F ---_
)i.r F ?..z
16.
19
L
1OOOK
\
1
I 4 I
Wavelength Fig.
7. Spectral
radiative
(pm)
flux distribution with temperature at 7;., = 300 K and 2 = 0.5L. r = R and F,., at 2 = L, I = 0.5R).
N,,,, = 24 (F,., at
340
Kong Hoon Lee and Raymond Viskanta
1
4
3
2
5
Wavelength (pm) Fig. 8. Hemispherical
spectral radiative fluxes and spectral blackbody temperature at rUr = 300 K and Nbnd = 24.
intensity distribution
with
to that obtained using 24 bands. Eight bands also yield results for the flux that may be acceptable for engineering calculations. The effect of the number of bands on the spectral radiative flux, F+, at a bounding surfaces (z = 50 mm) is quantitatively compared in Table 3. The spectral radiative flux shows little difference 50 -
40
T=llOOK
--me
T=lOOOK
.-.-.-
T=gOOK
.......... T=BOOK
g
30
2 S E Q 20
10
~ 0
30
35
40
45
1
z(mm)
Fig. 9. Radiative flux distribution with temperature in the axial direction at r = OSR. ru., = 300 K, and Nbnd = 24.
341
Prediction of spectral radiative transfer in a condensed cylindrical medium 50
40 -
-
T=llOOK
----
T=lOOOK
.-.-.-
T= 9OOK
. . . . . . . T= 6OOK
0
20
40
60
60
100
r(mm) Fig. IO. Radiative flux distribution with temperature in the radial direction N band= 24.
at I = OSL. 7;,, = 300 K. and
with the number of bands for the larger wavelengths (A = 3.125 pm and 4.375 pm), and this agreement is also revealed in Fig. 6. For the shorter wavelengths, the spectral radiative flux differs from that obtained with 24 bands and this shows that the smaller number of bands, in particular N band= 4, cannot appropriately resolve the spectral radiative transfer. 601
70 -
50 -
$3 40.
25 LLZ30
-
20 -
10.
0.
T W) Fig. 11. Comparison of radiative fluxes in the semitransparent (2 > 5 pm) at the bounding
surfaces,
(0 < I I 5 pm) and opaque 7;,, = 300 K. and Nhrnd = 24.
spectral
range
342
Kong Hoon Lee and Raymond Viskanta Table 3. Spectral radiative fluxes (FJ with the number of bands at a bounding surface (z = 50 mm) IkWlm4 Number of bands, Nbnd 8 16
4
24
T=lOOOK 0.625 1.875 3.125 4.375
0.12668 3.36762
0.00410 3.58861
11.27508
I 1.27476
8.34236
8.34086 T=900
0.625
0.03087
1.875
1.52394
3.125 4.375
6.68475 5.68694
0.00100 1.71159 6.68493 5.68633
0.625 1.875 3.125 4.375
0.00540 0.57876 3.50219 3.54359
0.00017 0.68501 3.50254 3.54352
0.00052 2.57866 11.51153 8.31997
0.00016 2.20319 11.55442 8.31603
K 0.00008 1.15062 6.86096 5.69635
o.OOOQ2 0.95992 6.89363 5.69818
0.00001 0.42195 3.60509 3.56602
O.OOOOO 0.34098 3.62453 3.57018
T=8OOK
Figure 7 shows the spectral radiative flux distributions in the radial and axial directions with temperature as a parameter. The spectral blackbody emitted flux, E,,(T) = n?i’Z,, is also included in Fig. 8 for the purpose of comparison. For rZ< 1 pm, the medium is nearly transparent to the radiation in the temperature range considered. Even though the absorption coefficient increases for 1 < 1 pm (as shown in Fig. 3), the radiative transfer for 1 < 1 pm is negligible, because the blackbody intensity in this spectral range is too small. The spectral blackbody intensity at 1100 K has maximum in the range of L > 2.5 pm, and the maximum of blackbody intensity shift towards the longer wavelength as the medium temperature is decreased. Thus, at T = 1100 K, about 71% of the energy of blackbody radiation is emitted in the range of 0 < 1 I 5 pm, but at T = 400 K, about 93% of the energy of black radiation is emitted at wavelengths greater than 5 pm to which optical quality glass is opaque. Therefore, the spectral absorption characteristics of the glass becomes more important as the medium temperature increases. As the glass is nearly transparent in the visible spectrum range, radiative transfer plays an important role in the infrared spectral range (1 I 1 I 5 pm). The radiative flux distributions in the r-direction, F,,,, and in the z-direction, Fj,.~,are identical for 1 2 2.8 pm, and differ from each other only in the spectral range of 0.4 < 1 < 2.8 pm. For 1 5 0.4 pm, the distributions are also identical, but they are not shown in Fig. 7, because the absolute values of the radiative fluxes are very small in this spectral range. The reason for the differences in the 0.4 < 1 < 2.8 pm spectral range is that the opacity of the medium in the radial direction is larger than that in the axial direction. The diameter of the glass disk is four times greater as is its thickness. For 1 2 2.8 pm, the absorption coefficient of the BK7 glass is very large, and thus the opacity also becomes large, regardless of the dimension of the medium in that direction. Garden” has calculated the spectral and total emittance as the function of the optical thickness for window glass plate from one-dimensional analysis. For the window glass, as the optical thickness (rc,L) increases the spectral emittance of the glass also increases, and for the large optical thickness, the spectral emittance approaches a constant value. In the present study, since the optical quality glass has the spectral absorption coefficient similar to that of the window glass, similar trends are shown in Fig. 7. For 1 2 2.8 pm, since the radiative fluxes in the two directions are identical. It is clear from the exact solution of RTE, Eq. (l), that the radiative fluxes in the two directions are identical when the opacities in the two directions are large. When the temperature of the medium is constant and the radiative properties do not depend on the temperature, the exact solution of RTE, Eq. (1) in the direction a, can be written as
Zi(r,i2)
= Z,,,.(Q)e-'As +
$Zbj.(T)(l- e-“3
(31)
Prediction of spectral radiative transfer in a condensed cylindrical medium
343
where S is the distance that the radiation has propagated from the bounding interface along the direction 0. For convenience we defined I+ as the intensity for C&n > 0 and I; for 4% < 0.When the spectral absorption coefficient or the distance between two boundary surfaces is large enough, the intensities at a boundary surface are as follows (32) I; = Z,,,, for k,S-+O.
(33)
Thus, the intensity, Z+ , is the same as the spectral blackbody intensity at the specified temperature and wavelength when the distance between the interfaces are large enough. The spectral radiative fluxes for R.n > 0, F> , is compared to the spectral blackbody emitted flux with temperature in Fig. 8. As expected from Eq. (32), the spectral radiative fluxes. F,!r and F::, are identical with the spectral blackbody emitted flux for I, 2 2.8 pm in the entire temperature range considered. Substituting the boundary condition, Eq. (lo), into Eq. (33), the intensity, I;, for C&n < 0 can be written as
1j.T= [1 - p;.(~)l(~;./~;.‘)ZZ~,(7;,,) + pi(8)$Z,,(T) for
rc,S+O
(34)
where pj(e) is the reflectivity by Eq. (9), and 0 is already defined by Eq. (11). Thus, I,- is also constant if the temperature of the surrounding wall is constant. Therefore, when the spectral absorption coefficients are large enough for 1 2 2.8 pm, the spectral radiative fluxes in the axial and radial directions are identical if the dimensions of the medium in the two directions are sufficiently large. Optical glass disks with other aspect ratios and dimensions {i.e.. (a) L/D = 1.4 [Z. = 50 mm, D = 200 mm], (b) L/D = 1 [t = 50 mm, D= 50mm] and (c) L/D = I [L = 200 mm, D = 200 mm]) have been considered. The results show that the spectral radiative fluxes, F,,, and F,,:, are the same for 1 2 2.8 pm, and FTr and F:, are equal to Z&,(T) since the spectral optical dimensions (i.e., K,L and rc,D) are in the optically thick range ( % 1). For cases (b) and (c), the spectral radiative flux in the radial direction F,,, and the radiative flux in the axial direction F,,: are almost equal to each other for 1 < 2.8 pm, because the optical dimensions in the two directions are equal. The results are similar in trends with those revealed in Fig. 7 and are, therefore, not shown for the sake of brevity. 3.3. Total radiative fluxes The effect of temperature on the total radiative fluxes in the axial and radial directions are shown in Figs 9 and 10, respectively. The radiative fluxes shown in Figs 9 and 10 were obtained by integrating the spectral radiative fluxes defined by Eq. (28) and (29) over the semitransparent spectral range (0 I i I 5 pm)
s 5Jm
F I.Yrn=
F,,,dE, for
0
i = r or z .
(35)
The radiative fluxes in the two directions increase as the glass temperature increases, because the blackbody emission increases with temperature. The variation of the radiative flux in the radial direction is larger than that in the axial direction near the respective bounding interface, and the radiative fluxes at the bounding surfaces are also larger. These trends are due to the spectral variation of the radiative flux with the opacity of the glass for 0.4 < 3, < 2.8 pm as shown in Figs 7 and 8. Moreover, the radiative exchange with the surroundings, in general, is confined to the region near the bounding surfaces as the optical thickness of the medium increases. In Figs 4 and 5, it is shown very clearly that the radiative flux varies strongly near the surfaces when the optical thickness is large and is nearly linear when the optical thickness is relatively small. Similarly, the radiative flux in the axial z-direction varies more smoothly than that in the radial direction because the thickness of the disk is smaller than the diameter, and thus the opacity is also smaller.
344
Kong Hoon Lee and Raymond Viskanta
A comparison of the radiative fluxes in the semitransparent (0 < J I 5 pm) and opaque (A> 5 pm) spectral region at the two bounding surfaces is given in Fig. 11. The radiative flux F,,, is defined by Eq. (35) and &,, is defined in the opaque spectral range
s cc
L,p =
Fj..idn for
i = r or
Z.
(36)
5w
For T I 1000 K, the radiative flux in the opaque spectral range is larger than that in the semitransparent range, but for T 2 1100 K, the opposite is true. This is due to the spectral blackbody emission with wavelength in the medium as mentioned before. The fraction of the blackbody emission to contribute to the radiative transfer within the semitransparent spectral range increases from 1.33 to 71.2% as the temperature of the medium increases from 300 to 1100 K. At temperatures less than 400 K (since more than 90% of the blackbody emission is emitted in the opaque spectral range to which the glass is relatively opaque) radiative transfer in the semitransparent spectral range is small. When T = 800K, about 50% of the blackbody emission is emitted in the opaque spectral range, and the radiative flux in the opaque range is about twice as large as that in the semitransparent range. For T 2 1100 K, more than 70% of the blackbody emission is emitted in the semitransparent spectral range and the radiative flux becomes larger than that in the opaque range. The radiative flux in the radial direction is still larger than that in the axial direction due to the greater opacity of the medium in that direction. In addition, the difference between the radiative fluxes in the two directions gradually increase as the temperature increases. This increase in the difference between the radiative fluxes with the temperature is due to the spectral radiative transfer in the intermediate spectral range as shown in Figs 7 and 8. 4. CONCLUSIONS
Radiative transfer in an axisymmetric disk of optical quality glass is considered. The radiative transfer equation is solved numerically using the S-N approximation of the discrete ordinates method. The numerical procedure used is validated by comparing the predicted fluxes with those based on an integral method of solution for one-dimensional radiative transfer in a semitransparent plate. Based on the numerical results obtained, the following conclusions can be drawn: 1. The S-8 level symmetric higher order (LSH) quadrature yields improved predictions of radiative transfer than the S-8 level symmetric odd (LSO) quadrature. 2. The exponential-type differencing scheme yields improved convergence and predictions over the central and upwind schemes. 3. For ,l 2 2.8 pm the optical dimensions in the axial and radial directions are sufficiently large and, therefore, the spectral radiative fluxes in the axial (F,,.) and radial (F,,,) directions are the same. 4. Near the bounding interfaces the total radiative fluxes increase very sharply, exhibiting a “boundary layer” type behavior where most of the radiant heat loss (i.e., cooling) of the system with largest temperature gradients would be expected to occur. 5. The total radiative fluxes at the two bounding surfaces increase as the temperature of the glass increases. For T 2 1100 K, the total radiative fluxes in the semitransparent spectral range (0 c 1s 5 pm) at the two bounding surfaces, I;;,,,, are larger than those in the opaque spectral range (1 > 5 pm), &._,, because more than 70% of the spectral blackbody emission is emitted in the semitransparent spectral range. REFERENCES 1. Bach, H. and Neuroth, N., eds., The Properties of Optical Glass. Springer, Berlin, 1995. 2. Gardon, R., J. Am. Ceramic Sot., 1961, 44, 305. Field, R. E. and Viskanta, R., J. Am. Ceramic Sot., 1990, 73, 2047. :: Viskanta, R. and Anderson, E. E., in Advances in Heat Transfer, eds. T. F. Irvine and J. P. Hartnett. Academic Press, New York, 1975, Vol. 11, p; 318. 5. Bergman, T. L. and Viskanta, R., in Radiative Transfer-Z, ed. M.P. Mengiic. Begell House, New York, 1996, p. 13. 6. Modest, M. F., Radiative Heat Transfer. McGraw-Hill, New York, 1993.
Prediction of spectral radiative transfer in a condensed cylindrical medium
34s
7. Siegel, R. and Howell, J. R., Thermal Radialion Heat Transfer, 3rd edn. McGraw-Hill, New York, 1992. 8. Carlson, B. G. and Lathrop, K. D., in Computing Methods in Reaction Physics, eds. H. Greenspan, C. N. Kelber and D. Okrent. Gordon & Breach, New York, 1968, p. 171. 9. Jamaluddin, A. S. and Smith, D. J., Combustion Sci. Tech., 1988, 62, 173. IO. Song, M. and Viskanta, R., in Proceedings of National Heat Transfer Conference, ASME HTD. ASME, New York, 1996, Vol. 325, p. 55. I I. Lathrop, K. D. and Carlson, B. G., Technical Reporf ,!,A-3186. Los Alamos Scientific Laboratory, Los Alamos, New Mexico, 1965. 12. Fiveland, W. A., in Fundamentals of Radiation Heat Transfer. ASME HTD, 1991, Vol. 160, p. 89. 13. Gardon, R., f. Am. Ceramic Sot., 1956, 39, 278.
Pergamon
Printed in Great Britain 0022-4073/97 1617.00+ 0.00
PII: !SOO22=4073@7)00037-X
PREDICTION OF SPECTRAL RADIATIVE TRANSFER IN A CONDENSED CYLINDRICAL MEDIUM USING DISCRETE ORDINATES METHOD KONG HOON LEE? and RAYMOND
VISKANTAI
School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907-1288, U.S.A. (Received
18 November
1996)
Abstract-Spectral radiative transfer is analyzed in a condensed cylindrical medium using discrete ordinates method. In order to appropriately consider the directional variations of the reflectivities at the specularly bounding interfaces, two types of quadrature sets are examined and the S-8 LSH quadrature set is used. The BK7 optical quality glass is considered as an example of the semitransparent medium, and the spectral radiative fluxes are predicted. The results obtained show that for I, 2 2.8 pm the BK7 glass can be treated as an optically thick medium, and the radiative fluxes differ little for the glass with the dimensions larger than those being considered in the present study. For 0.4 < 1 < 2.8 pm, the characteristics of the spectral absorption coefficients of the glass become important, and the radiative fluxes depend on the opacity of the glass. The predicted spectral radiative fluxes differ from those based on the opaque solutions, and this indicates that the spectral radiative transfer must be considered carefully in the semitransparent medium. 0 1997 Elsevier Science Ltd
1. INTRODUCTION
Accurate prediction of temperature distributions in glass and other semitransparent materials at high temperature are essential during various fabricating operations. For example, the annealing of glass plates requires prediction of the temperature distribution from the transient combined conduction and radiation heat transfer analysis. During the process of annealing of optical quality glass which is utilized for optical components in imaging systems, the temperature gradients in the glass must be controlled very carefully by imposing a constant cooling rate of the order of 0.01 “C/h, because a true geometric and color image of an object is generated by using the property of refraction, and even small stresses due to the temperature gradients in the annealing range cause permanent strains and inhomogeneities of the refractive index.’ This can be detrimental to optical glasses of high quality. Moreover, the stress-strain relation in glass plate during annealing or tempering and danger of fracture is also strongly related to the temperature gradients.2 However, the local temperature and the local temperature gradient cannot be predicted accurately using the concept of the effective (apparent) thermal conductivity, and rigorous treatment of radiative transfer is needed.3.4 When semitransparent materials are manufactured or processed at high temperature, both volumetric absorption and emission must be accounted for when predicting the temperature distribution in the absorbing-emitting substance. Typically, such materials are homogeneous and scattering of radiation can be neglected in comparison to absorption, and this makes it easier to carry out the analysis of radiative heat transfer.5 However, since semitransparent materials have the spectral absorption and refraction characteristics which depend strongly on wavelength and specularly reflecting boundaries, it is difficult to solve the radiative transfer equation. In general, glass is semitransparent to thermal radiation in approximately the spectral range of 0 < I c 5 ,um, and is effectively opaque beyond 5 pm. This selectivity has to be properly accounted for in predicting radiative transfer inside the glass and radiation exchange between the surfaces and the surroundings.4 tPermanently at Advanced Institute of Machinery and Design, Seoul National University, Seoul 151-742, Korea. $‘Towhom all correspondence should be addressed. 329
330
Kong Hoon Lee and Raymond Viskanta
The purpose of the present study is to analyze radiative heat transfer in a cylindrical semitransparent medium as the first step to predicting the transient combined conductive and radiative heat transfer in a cylindrical disk of glass during annealing. For this purpose, the discrete ordinates method (DOM, S-N method) is used to solve the radiative transfer equation (RTE). Discrete ordinates method is widely accepted, and a comprehensive discussion is available.6 The numerical analysis was performed for an axisymmetric cylindrical medium and the model is validated by comparing to an exact analytical solution.4 2. ANALYSIS
2.1. Radiative transfer equation (RTE) Consider the radiative energy transport within a cylindrical semitransparent medium shown in Fig. 1. The medium is surrounded by isothermal black walls, and the space between the medium and the surrounding wall is either a vacuum (n;,’ = 1) or is occupied by a transparent entity having the refractive index of unity. In addition, the following assumptions are made in analyzing radiative transfer in the system: (1) the radiation field is axisymmetric and two dimensional; (2) the medium emits and absorbs but does not scatter thermal radiation; (3) the medium is in local thermodynamic equilibrium for which Planck’s and Kirchhoffs laws are valid; (4) the spatial dimensions of the medium are much larger than the wavelength of radiation for the semitransparent band, i.e., the coherence effects are negligible; (5) the refractive index of the medium at a given wavelength is uniform and does not depend on temperature in the considered range, therefore, the refraction of radiation within the medium is absent; (6) the interface between the medium and the bounding entity is optically smooth; and (7) the surrounding walls are black gray diffuse emitters of radiation. Under the above assumptions, the radiative transfer equation (RTE) for a spectrally absorbing-emitting medium can be written as6 WVZAr,Q)
= kiLni Z&Yl - ZAr,W
(1)
where Zj.(r, R) is the spectral radiative intensity, which is a function of position, direction and wavelength: Z,,,(T, nJ is the spectral intensity of blackbody radiation given by Planck’s function. Integration of Eq. (1) over all directions (Q = 4~) yields the spectral radiant energy equation V.F;. = 4mc;.nfZbi(7’) - K;.G;.
(2)
where the spectral radiative flux vector Fj. and the spectral irradiance Gj, are defined as F;. = sR=4n
Z@,fi)QdQ
Fig. 1. Schematic of physical situation and coordinate system.
(3)
Prediction of spectral radiative transfer in a condensed cylindrical medium
331
and
s
(4)
h.(r,WQ
G;. =
R= 4n
respectively. It is the integrated over the entire spectrum Eq. (1) that is used in the thermal energy equation to determine the temperature distribution in problems involving simultaneous conduction and radiationP The radiative transfer equation, Eq. (l), for the spectrally emitting and absorbing axisymmetric medium along a specified discrete direction Qm.‘(tP, $9 can be expressed as
In the above equation, 4 is the azimuthal angle for the direction (m, f) relative to the local radial direction, and pm.‘,q”‘.’and t;“‘,’are direction cosines and defined by p = sinfkos4,
q = sintkin~,
t = cost7 .
(6)
2.2. Boundary conditions The interface conditions between the medium and the bounding entity constitute the boundary conditions for Eq. (1). The interaction of radiation at the smooth interface between two dielectric media is governed by Snell’s and Fresnel’s laws?’ n,sintl = n;.‘sinO’.
(7)
When the radiation is transmitted from a denser into an optically less dense medium, the angle of refraction approaches 90” as an angle of incidence increases toward the critical angle (e,), which is defined by 8, = sin - ‘(ni’/ni)
.
03)
For an angle of incidence larger than the critical angle, the outgoing radiation from the medium is totally reflected inward. Similarly, the incoming radiation from the surrounding wall is concentrated into the range of directions associated with the angle of refraction smaller than 8,. The relation of the incident and reflected radiation at the smooth interface is represented by Fresnel’s equation. The directional reflectivity of the interface between two dielectric media of different reflective indices is obtained by combining Fresnel’s equation for reflection and Snell’s law for refraction:’
p;.(e)= i
I(h.iosfl-n;Pj-O Z n;,’ k0se + n,P
l(n;.cose-P):
+ Z
n;.cose
1: +
:zIl
(9)
P
Finally, the boundary condition for Eq. (1) at an optically smooth bounding surface is written as L.(r,Q) = [l -
p;.(ell(nj.ln;,‘)2~~,(~;,,) + p;.(Wj.(r,W for 42-n> 0
(10)
8 = cos - ‘@In), R’ = 2(Ch)n
(11)
where
JQSRT
58.3LB
- R
332
Kong Hoon Lee and Raymond Viskanta
The boundary conditions for the spectral intensity in the specified discrete direction, GF’, are obtained from the symmetry condition and Eq. (10) Zil”’= ZY’
at r=O
ZY’ = p?jZY’ + (1 - p;$)(nj./n;.‘)2Zbi.( TUT)at r = R Zy’ = p:V:‘,’ f (1 - p~~)(nj./nj.‘)2Z,,(~“~)at 2 = 0
Z?’ = p?:Z?,’ + (1 - p~:)(nj./n;.‘)2Z,,(~“,) at z = L
(12)
The discrete reflectivities, p”‘, in the above equations are defined to preserve the reflected radiative flux in the normal direction to the interface.
In the above equation, 8 is the angle between the direction of propagation Q and the inward normal II to an interface as defined by Eq. (11). 2.3. Discrete ordinates approximation Closed form analytical solutions of the RTE, Eq. (5), are not possible because the boundary conditions are complicated functions of direction; therefore, the discrete ordinates method (DOM) is used to obtain a numerical solution of the problem. The discrete ordinates equations in axisymmetric cylindrical geometry are complicated by the presence of the angular derivative term, a(~Z,)/&$. Carlson and Lathrop* have proposed a direct differencing technique for calculating the angular derivatives at the quadrature points. Using this technique, the second term on the left-hand-side of Eq. (5) can be written as a~ -1 a(qn’Jz,“‘)
(14)
r
The directions I+ l/2 define the boundaries of the angular range of ordinates, and the second term of numerator on the right-hand-side of Eq. (14) represents transport out and into the angular range, respectively. Since the angular ranges of the ordinates are not constant, the geometrical coefficients, a”‘,‘+I”, are also not constant and need to be determined. Since the values of a depend only on the differencing scheme and are independent of radiative intensity, a may be determined by examining a constant intensity field,’ that is, Zj.= ?:I,;. = constant, and the following formula is obtained,
This expression is used as a recursion formula for a’“,‘+‘I2using the fact that am.li2= 0 at 4”’ = 0 and ylm.li2 = 0. Thus, the spatially discretized radiative transfer equation in terms of the radiative intensities on the control surface can be obtained by integrating over the ring-shaped control volume, /Am$4”z~;- A,z~:) + 5”+t,z::
- &.I$
(16)
+ (A, - A,9)(amJ+iZ!$+ f _ am,‘-iZ!$ i)/,m./
where A. = 2w,(z, - z,.), A,v= 21cr,&, - z,.), A, = 71(ri - r.f) = A,,
(17)
Vp= ~(2~ - z,.)(r,Z- rt) .
(18)
Prediction of spectral radiative transfer in a condensed cylindrical medium
333
E-__-__E Ar
Fig. 2. Control volume
In order to reduce the number of the unknowns in Eq. (16), a differencing scheme needs to be introduced. For example, the radiation intensity, I;‘$, at the center of the control volume can be related to the intensities at the control surfaces (I~~;(,Z;‘;‘,l~~ and I;:;: shown in Fig. 2)
= (1 - fb)ZlliP f + fJ$+
f for
$“.’ > 0
and
o)‘.’> 0
(19)
The value of the weighting factor must be restricted within the range 0 < f I 1. Takingf = l/2 yields the diamond or central differencing scheme proposed by Carlson and Lathrop,’ and taking f = 1 yields the step or upwind differencing scheme. In implementing the discrete ordinates approximation for the cylindrical geometry, it has been a common practice to assume a uniform distribution of the weighting factor over the calculation domain, regardless of the control volume size.6 When the uniform weighting factors are used, the numerical predictions often exhibit physically unrealistic negative intensities, and the solution procedure has to be repeated until the negative values are removed, while the value of weighting factor is gradually decreased from its initial value of 1.0.9 In the present study, an exponential-type scheme which was proposed by Song and Viskanta’” in order to obtain positive radiative intensities is used together with the central and upwind schemes. In the exponential-type scheme” the weighting factors are calculated as follows
+exp(
-
w)]-’
(20)
f,=[l+exp(
-
%)I-’
(21)
f,=[l
fo
I +exp
=
[
(
K;.
~,l~“.‘pp’
-
(a”#.‘+f + c(“‘.‘- :)(A, _ A,)
11 -I
’
(22)
If the absorption coefficients are constant, the weighting factors given by Eqs (20)-(22) depend only on geometry and need to be evaluated only once.
334
Kong Hoon Lee and Raymond Viskqnta
The final form of both directionally and spatially discretized algebraic equation for positive ,u”‘.’ and (“I.’directions are obtained from Eq. (16) and (19) (23) where a = Ml
- f,) +
4fillfr
(24)
AfM; = Aelf; = A,l.f;
(25)
c = (A, - A,J[rP’+ i(1 - f4) + urn.‘-if,]/(cP’f~)
(26)
b = [A,(1 - f.-) +
d=
(27)
K;,&.
Eq. (23) is applicable to the case where both .P’ and cm.’are positive. For the other combinations of @“.’and t”‘*‘,the discretized equations can be obtained by integrating over appropriate control volumes. Once the radiation field has been determined, the radiative fluxes in the radial and axial directions are written in terms of discrete variables as follows
E.,,=
F;..;=
s s
N
2m
z,(r,z,fi)pdSZ N 1 c ,‘,%P’ m=,,=, n =4n N
2m
Z;.(r,z,Q)tdQ 2: 1 ~Z;m.‘~mGY’.‘.
R=4n
fPt=I,=l
(29)
2.4. Selection of discrete ordinate quadrature sets The accuracy of the discrete ordinates method depends on the choice of the quadrature set. Although the choice is arbitrary, a completely symmetric quadrature is preferred in order to preserve geometric invariance of the solution. The quadrature set used in the present study is based on the moment matching technique of Lathrop and Carlson” and Fiveland.‘* If the zeroth moment and sequential even moments (0,2,4..) are satisfied, level symmetric even (LSE) quadrature sets are obtained and if the zeroth and sequential moments (0,1,2,3..) are satisfied, level symmetric odd (LSO) quadrature sets are obtained.” LSE quadrature sets are not desirable because they do not satisfy the first moment, which defines the radiative flux. Fiveland’* obtained level symmetric higher-order (LSH) quadrature sets which satisfy the selected moments. The S-4 LSH quadrature sets are the same as the LSO quadrature, and the S-8 LSH quadrature sets satisfy the zeroth, the first and sequential even moments (0,1,2,4,6). Since the scattering of radiation is negligible in comparison to the absorption, in the present study the quadrature that satisfy the higher order of moments than the first moment is not needed. However, the directional dependence of specular boundaries is affected by the choice of the quadrature set to be used, because a weight represents the part of area on a unit sphere for each ordinate direction, and the average value of reflectivity within the range of a weight, Eq. (13), varies with the type of the quadrature sets. Thus, two kinds of quadrature sets (i.e., LSO and LSH) are examined. The quadrature sets of S-4 and S-8 are shown in Table 1. The direction cosines and weights given in Table 1 cover the solid angle of 471steradians. The weights, UP’, are positive and nonoverlapping and sum up to the area of a unit sphere, because they represent areas on the unit sphere. As the angular distribution of the radiative intensity is symmetric with respect to the r-z plane, the weights are twice of those used in the general three-dimensional problem, and then only half the number of directions[N(N + 2)/2] needs to be considered. 2.5. Radiative properties
of optical quality glass
As an example for an analysis of spectral radiative transfer in optical quality glass, BK7 glass is used. The glass is of optical quality which is sodium rich, and has excellent transmission
335
Predictionof spectralradiative transferin a condensed cylindrical medium Table I. Discreteordinatesquadraturesetsfor S-N approximation
I
m s-4
S-8 LSO"
( - 5) I
( - PI
&l.l
p
v’
P’ ( + PC) 2
kO.295876
0.295876
50.908248
n/3
2
4 3
kO.908248 kO.295876
0.295876 0.908248
& 0.295876 + 0.295876
sl3 nl3
I
2
5 0.142255
0.142255
kO.979554
0.342472
2
1 2
4 3
kO.577350 + 0.142255
0.142255 0.577350
+ 0.804009 + 0.804009
0.198457 0.198457
3
I 2 3
6 5 4
+ 0.804009 + 0.577350 k 0.142255
0.142255 0.577350 0.804009
kO.577350 kO.577350 kO.577350
0.198457 0.923436 0.198457
I 2 3 4
8 7 6 5
kO.979554 + 0.804009 + 0.577350 +0.142255
0.142255 0.577350 0.804009 0.979554
+ 0.142255 kO.142255 + 0.142255 kO.142255
0.342472 0.198457 0.198457 0.342472
I
I
2
kO.158489
0.158489
,0.974558
0.302780
2
I 2
4 3
* 0.577350 +_0.158489
0.158489 0.577350
+ 0.800967 + 0.800967
0.280735 0.280735
3
I 2 3
6 5 4
+ 0.800967 kO.577350 + 0.158489
0.158489 0.577350 0.800967
kO.577350 + 0.577350 kO.577350
0.280735 0.548844 0.280735
1 2 3 4
8 7 6 5
kO.974558 f 0.800967 kO.577350 kO.158489
0.158489 0.577350 0.800967 0.974558
kO.158489 k 0.158489 + 0.158489 kO.158489
0.302780 0.280735 0.280735 0.302780
4
S-8 LSH'*
( + 5)
4
5
5
characteristics in the visible and near infrared, making it a popular glass for optical systems. The properties of BK7 glass, refractive index and absorption coefficient, given by the manufacturer? are used in the present study. The refractive index of BK7 glass with Sellmeier dispersion formula is given by (30) 1.6
1.5
-.
-.
-.
-.
. . \
Absorption coefficient Refractiveindex
1
2
Wavelength
3
4
-
5
1.2
(pm)
Fig. 3. Spectralabsorptioncoefficient and refractive index of BK7 glass. $SchottGlaswerke, Mainz, Germany
336
Kong Hoon Lee and Raymond Viskanta
where the unit of wavelength, A, is pm and B, = l.O3961212,C, = 6.00069867 x 10e3, B2 = 2.31792344 x 10-4, C, = 2.00197144 x 10-2, B, = l.O1046945,C, = 1.03560653 x lo*. As shown in Fig. 3, the refractive index decreases with wavelength. The absorption coefficient, K;.,is relatively low in the visible and near infrared region. BK7 optical glass has good transmissivity in this spectral region. The emissivity in the opaque spectral region (1 > 5 pm) is 6 = 0.9. 2.6. Method of solution The solution procedure involves solving the radiative transfer equation in each of the ordinate directions, using a set of N(N + 2)/2 coupled discretized equations. The calculation starts at the grids adjacent the boundary surfaces along the specified direction with the boundary conditions, Eq. (12). However, before starting the solution, an initial estimation of the azimuthal component of intensity needs to be obtained by solving Eq. (23) in the special &directions where tl = 0 and the values of the weights and geometric coefficients, a, are assigned to be zero. The radiative intensity at the center of a control volume can be explicitly calculated from the downstream surface values using Eq. (23), and the intensities at the downstream surfaces can be obtained by extrapolation using Eq. (19). The evaluation of the radiative intensity proceeds in the direction of increasing (or decreasing) r and z, when the direction cosines p and l are positive (or negative), respectively. For a given t (or superscript m), the evaluation proceeds in the order of increasing ~1(or superscript I). Since the distributions of the radiative intensity in different directions are coupled to each other through the specular boundary conditions, the solution procedure has to be iterative. The updating of radiative intensity field is repeated until the radiative intensity at every node and in every direction does not change within the accuracy of five significant digits. The number of iterations required to obtain the converged solution increases as the opacity of the medium decreases. The grid spacing used is not uniform, but is more compact near the symmetry axis and the bounding interfaces, This was done in order to resolve the steep spatial change in the physical variables expected in these regions. 3. RESULTS
AND DISCUSSION
3.1. Validation In order to validate the solution method, one-dimensional radiative transfer was considered in a semitransparent plate. It was assumed that the reflectivity at the curved surface (r = R) was unity for the entire range of incident angles to obtain one-dimensional problem from the two-dimensional formulation. The temperatures in the medium and the surrounding walls were assumed to be uniform and cold (7& = 0), respectively. In addition, the refractive index was assumed to be a constant (n = 1.46), and the optical thickness r,( = KL) was used for the comparison in each spectral band. The results are compared in Figs 4 and 5 together with the integral (exact) solution obtained under the same conditions.4 For the large optical thickness (7, = 100) the radiative flux is maximum near the surface, and it means that the radiative cooling occurs mainly in the very thin layer adjacent to the bounding surface. In the case of a opaque medium (TV + co), only the surface exchange needs to be accounted for. As the optical thickness decreases, the radiative cooling becomes more significant throughout the larger fraction of layer thickness, and the maximum value of the gradient of the radiative flux decreases. For zL = 1, the radiative flux increases almost linearly toward the surface from the center plane, and the radiative cooling is nearly uniform across the semitransparent medium. Figure 4 shows the radiative flux distribution for different quadrature sets. A 31 x 61 grid and an exponential-type differencing scheme are used. For TV= 1, the S-4 approximation overpredicts the radiative flux across the plate. The radiative flux calculated using the S-4 approximation is larger than that obtained from the integral solution by 11% at z = L. This overprediction results from the evaluation of the average reflectivity, Eq. (13), because it is not appropriately averaged over the angular range of the ordinates. The directional reflectivity varies strongly near the critical
Prediction of spectral radiative transfer in a condensed cylindrical medium
!
-
,s * P.,0.8 LL _ X 3 G=
331
Integral solution 0
S-8 LSH quadrature
0
S-8 LSO quadrature
A
S-4
ti
r” 0.6 -
z ‘3 m e !I * 0.4
-
8 P 0 ._ 2 fj
._ n
0.2 -
Fig. 4. Comparison of the radiative flux for the different quadrature sets.
angle 8,, and it is, therefore, very important to appropriately average the steep variation over the angular range of the ordinates in order to correctly treat the specular boundary conditions. To obtain the more accurate results with discrete ordinates method, a higher order of quadrature set is needed. In the present study, two kinds of the S-8 quadrature sets were tested as shown in Fig. 4. The radiative flux obtained with the S-8 LSO quadrature set is in closer agreement with the integral
-
Integral solution Cl
Exponential scheme
0
Central scheme Upwind scheme
A
Fig. 5. Comparison of the radiative fluxes for the different differencing schemes.
338
Kong Hoon Lee and Raymond Viskanta Table 2. Dimensionless radiative fluxes (F-/r&) for the different grid systems at z/L = 1 when rL = 1 Grid system (N_ x N,) 21 x 41
31 x 61
41 x 81
S-8 LSH quadrature Exponential scheme Central scheme Upwind scheme
0.63473 0.63792 0.62609
S-8 LSO quadrature S-4 quadrature
0.6503 1 0.71364
Integral solution
0.63451 0.63445 0.63754 0.63741 0.62829 ’ 0.62903 Exponential scheme 0.65014 0.71036
0.65007 0.71368
0.63664
solution than with the S-4 quadrature set, and the radiative flux with the S-8 LSH quadrature set shows fairly good agreement with the integral solution through the entire range of optical thicknesses. However, this agreement can vary with the refractive index. Even though a comparison is not presented in this paper, if the refractive index is larger (for example, n = 1.5), the S-8 LSO quadrature set yields improved results than the LSH quadrature set, because the critical angle and the angular distribution of reflectivity, Eq. (9) are also changed. For the smaller value (n = 1.42), the S-8 LSH quadrature set still yields accurate results. Finally, over the range of the refractive index shown in Fig. 3, the S-8 LSH quadrature set is the preferred one for the present study. Figure 5 shows the radiative fluxes obtained with three different differencing schemes using the 31 x 61 grid and the S-8 LSH quadrature set. The results obtained with the exponential-type and central differencing scheme are in good agreement with the integral solution, but the upwind scheme underpredicts the radiative flux. The uniform weighting factors were used in the central and upwind scheme only for the purpose of comparison even though they could yield unreasonable negative intensity. When the upwind and exponential-type schemes were used, the negative intensity was not predicted. However, for the central scheme, the calculations converged slowly, and much more computing time and a larger number of iterations were required than for other schemes due to the tendency of the solution to oscillate which was caused by the negative intensity. The number of iterations required for convergence were about a hundred times as many as those for the other schemes. Sensitivity tests for the various grid systems were also carried out. The grid spacing used is not uniform and is more compact near the symmetry axis and the bounding interfaces to resolve the steep spatial change of the radiative flux in these regions. Some of the results for 21 x 41, 31 x 61, and 41 x 81 grid systems are included in Table 2. For the purpose of comparison, only the results for rL = 1 are given in Table 2. However, the results for the other optical thicknesses also have the similar trends. As shown in Table 2, the radiative flux obtained with the S-8 LSH quadrature sets and the exponential-type differencing scheme is in good agreement with the integral solution. They are nearly independent on the grid systems. In the present study, the finer (31 x 61) grid than the coarser (21 x 41) grid is used to resolve the steep spatial changes of the radiation flux near the bounding surfaces, even though the 21 x 41 grid also yields acceptable results. The sensitivity tests have established that the S-8 LSH quadrature set, the exponential-type differencing scheme, and the 31 x 61 grid system are adequate for the present study. 3.2. Spectral radiative transfer in the optical quality glass The results for spectral radiative transfer in BK7 optical quality glass are given in Figs 6 - 11 and Table 3, and they are obtained using the absorption coefficient and the refractive index shown in Fig. 3. The optical quality glass disk considered has the radius R = 100 mm and the thickness L = 50 mm. The results are obtained in the range of 300 K < T < 1100 K and zU:.,= 300 K. In order to obtain the results, the S-8 LSH quadrature set, the 31 x 61 grid, and the exponential-type differencing scheme were used in the discrete ordinates method. The effect of the number of bands on the spectral radiative flux has been examined. Each band is uniformly spaced and four to one hundred bands are considered in the temperature range of 300 K < T I 1100 K. The results obtained using 24 bands are in good agreement with those
Prediction
of spectral
radiative
transfer
in a condensed
cylindrical
medium
339
12-
IO-
8-
6-
4-
2.
O0
1
2
3
Wavelength Fig. 6. Effect of the number
4
5
(pm)
of bands on the spectral radiative flux in the axial direction and T= 1000 K, 7;,, = 300 K.
at z = L. r = 0.5R
obtained using 100 bands with a discrepancy of less than 1O/i. Thus, 24 bands were used to obtain all the results reported in the remainder of the paper. The effect of the number of bands on the radiative fluxes for less than 24 bands is shown in Fig. 6 at a uniform temperature of 1000 K with the surrounding wall temperature of 300 K. The radiative flux obtained using 16 bands is close 20 T=llOOK -F ---_
)i.r F ?..z
16.
19
L
1OOOK
\
1
I 4 I
Wavelength Fig.
7. Spectral
radiative
(pm)
flux distribution with temperature at 7;., = 300 K and 2 = 0.5L. r = R and F,., at 2 = L, I = 0.5R).
N,,,, = 24 (F,., at
340
Kong Hoon Lee and Raymond Viskanta
1
4
3
2
5
Wavelength (pm) Fig. 8. Hemispherical
spectral radiative fluxes and spectral blackbody temperature at rUr = 300 K and Nbnd = 24.
intensity distribution
with
to that obtained using 24 bands. Eight bands also yield results for the flux that may be acceptable for engineering calculations. The effect of the number of bands on the spectral radiative flux, F+, at a bounding surfaces (z = 50 mm) is quantitatively compared in Table 3. The spectral radiative flux shows little difference 50 -
40
T=llOOK
--me
T=lOOOK
.-.-.-
T=gOOK
.......... T=BOOK
g
30
2 S E Q 20
10
~ 0
30
35
40
45
1
z(mm)
Fig. 9. Radiative flux distribution with temperature in the axial direction at r = OSR. ru., = 300 K, and Nbnd = 24.
341
Prediction of spectral radiative transfer in a condensed cylindrical medium 50
40 -
-
T=llOOK
----
T=lOOOK
.-.-.-
T= 9OOK
. . . . . . . T= 6OOK
0
20
40
60
60
100
r(mm) Fig. IO. Radiative flux distribution with temperature in the radial direction N band= 24.
at I = OSL. 7;,, = 300 K. and
with the number of bands for the larger wavelengths (A = 3.125 pm and 4.375 pm), and this agreement is also revealed in Fig. 6. For the shorter wavelengths, the spectral radiative flux differs from that obtained with 24 bands and this shows that the smaller number of bands, in particular N band= 4, cannot appropriately resolve the spectral radiative transfer. 601
70 -
50 -
$3 40.
25 LLZ30
-
20 -
10.
0.
T W) Fig. 11. Comparison of radiative fluxes in the semitransparent (2 > 5 pm) at the bounding
surfaces,
(0 < I I 5 pm) and opaque 7;,, = 300 K. and Nhrnd = 24.
spectral
range
342
Kong Hoon Lee and Raymond Viskanta Table 3. Spectral radiative fluxes (FJ with the number of bands at a bounding surface (z = 50 mm) IkWlm4 Number of bands, Nbnd 8 16
4
24
T=lOOOK 0.625 1.875 3.125 4.375
0.12668 3.36762
0.00410 3.58861
11.27508
I 1.27476
8.34236
8.34086 T=900
0.625
0.03087
1.875
1.52394
3.125 4.375
6.68475 5.68694
0.00100 1.71159 6.68493 5.68633
0.625 1.875 3.125 4.375
0.00540 0.57876 3.50219 3.54359
0.00017 0.68501 3.50254 3.54352
0.00052 2.57866 11.51153 8.31997
0.00016 2.20319 11.55442 8.31603
K 0.00008 1.15062 6.86096 5.69635
o.OOOQ2 0.95992 6.89363 5.69818
0.00001 0.42195 3.60509 3.56602
O.OOOOO 0.34098 3.62453 3.57018
T=8OOK
Figure 7 shows the spectral radiative flux distributions in the radial and axial directions with temperature as a parameter. The spectral blackbody emitted flux, E,,(T) = n?i’Z,, is also included in Fig. 8 for the purpose of comparison. For rZ< 1 pm, the medium is nearly transparent to the radiation in the temperature range considered. Even though the absorption coefficient increases for 1 < 1 pm (as shown in Fig. 3), the radiative transfer for 1 < 1 pm is negligible, because the blackbody intensity in this spectral range is too small. The spectral blackbody intensity at 1100 K has maximum in the range of L > 2.5 pm, and the maximum of blackbody intensity shift towards the longer wavelength as the medium temperature is decreased. Thus, at T = 1100 K, about 71% of the energy of blackbody radiation is emitted in the range of 0 < 1 I 5 pm, but at T = 400 K, about 93% of the energy of black radiation is emitted at wavelengths greater than 5 pm to which optical quality glass is opaque. Therefore, the spectral absorption characteristics of the glass becomes more important as the medium temperature increases. As the glass is nearly transparent in the visible spectrum range, radiative transfer plays an important role in the infrared spectral range (1 I 1 I 5 pm). The radiative flux distributions in the r-direction, F,,,, and in the z-direction, Fj,.~,are identical for 1 2 2.8 pm, and differ from each other only in the spectral range of 0.4 < 1 < 2.8 pm. For 1 5 0.4 pm, the distributions are also identical, but they are not shown in Fig. 7, because the absolute values of the radiative fluxes are very small in this spectral range. The reason for the differences in the 0.4 < 1 < 2.8 pm spectral range is that the opacity of the medium in the radial direction is larger than that in the axial direction. The diameter of the glass disk is four times greater as is its thickness. For 1 2 2.8 pm, the absorption coefficient of the BK7 glass is very large, and thus the opacity also becomes large, regardless of the dimension of the medium in that direction. Garden” has calculated the spectral and total emittance as the function of the optical thickness for window glass plate from one-dimensional analysis. For the window glass, as the optical thickness (rc,L) increases the spectral emittance of the glass also increases, and for the large optical thickness, the spectral emittance approaches a constant value. In the present study, since the optical quality glass has the spectral absorption coefficient similar to that of the window glass, similar trends are shown in Fig. 7. For 1 2 2.8 pm, since the radiative fluxes in the two directions are identical. It is clear from the exact solution of RTE, Eq. (l), that the radiative fluxes in the two directions are identical when the opacities in the two directions are large. When the temperature of the medium is constant and the radiative properties do not depend on the temperature, the exact solution of RTE, Eq. (1) in the direction a, can be written as
Zi(r,i2)
= Z,,,.(Q)e-'As +
$Zbj.(T)(l- e-“3
(31)
Prediction of spectral radiative transfer in a condensed cylindrical medium
343
where S is the distance that the radiation has propagated from the bounding interface along the direction 0. For convenience we defined I+ as the intensity for C&n > 0 and I; for 4% < 0.When the spectral absorption coefficient or the distance between two boundary surfaces is large enough, the intensities at a boundary surface are as follows (32) I; = Z,,,, for k,S-+O.
(33)
Thus, the intensity, Z+ , is the same as the spectral blackbody intensity at the specified temperature and wavelength when the distance between the interfaces are large enough. The spectral radiative fluxes for R.n > 0, F> , is compared to the spectral blackbody emitted flux with temperature in Fig. 8. As expected from Eq. (32), the spectral radiative fluxes. F,!r and F::, are identical with the spectral blackbody emitted flux for I, 2 2.8 pm in the entire temperature range considered. Substituting the boundary condition, Eq. (lo), into Eq. (33), the intensity, I;, for C&n < 0 can be written as
1j.T= [1 - p;.(~)l(~;./~;.‘)ZZ~,(7;,,) + pi(8)$Z,,(T) for
rc,S+O
(34)
where pj(e) is the reflectivity by Eq. (9), and 0 is already defined by Eq. (11). Thus, I,- is also constant if the temperature of the surrounding wall is constant. Therefore, when the spectral absorption coefficients are large enough for 1 2 2.8 pm, the spectral radiative fluxes in the axial and radial directions are identical if the dimensions of the medium in the two directions are sufficiently large. Optical glass disks with other aspect ratios and dimensions {i.e.. (a) L/D = 1.4 [Z. = 50 mm, D = 200 mm], (b) L/D = 1 [t = 50 mm, D= 50mm] and (c) L/D = I [L = 200 mm, D = 200 mm]) have been considered. The results show that the spectral radiative fluxes, F,,, and F,,:, are the same for 1 2 2.8 pm, and FTr and F:, are equal to Z&,(T) since the spectral optical dimensions (i.e., K,L and rc,D) are in the optically thick range ( % 1). For cases (b) and (c), the spectral radiative flux in the radial direction F,,, and the radiative flux in the axial direction F,,: are almost equal to each other for 1 < 2.8 pm, because the optical dimensions in the two directions are equal. The results are similar in trends with those revealed in Fig. 7 and are, therefore, not shown for the sake of brevity. 3.3. Total radiative fluxes The effect of temperature on the total radiative fluxes in the axial and radial directions are shown in Figs 9 and 10, respectively. The radiative fluxes shown in Figs 9 and 10 were obtained by integrating the spectral radiative fluxes defined by Eq. (28) and (29) over the semitransparent spectral range (0 I i I 5 pm)
s 5Jm
F I.Yrn=
F,,,dE, for
0
i = r or z .
(35)
The radiative fluxes in the two directions increase as the glass temperature increases, because the blackbody emission increases with temperature. The variation of the radiative flux in the radial direction is larger than that in the axial direction near the respective bounding interface, and the radiative fluxes at the bounding surfaces are also larger. These trends are due to the spectral variation of the radiative flux with the opacity of the glass for 0.4 < 3, < 2.8 pm as shown in Figs 7 and 8. Moreover, the radiative exchange with the surroundings, in general, is confined to the region near the bounding surfaces as the optical thickness of the medium increases. In Figs 4 and 5, it is shown very clearly that the radiative flux varies strongly near the surfaces when the optical thickness is large and is nearly linear when the optical thickness is relatively small. Similarly, the radiative flux in the axial z-direction varies more smoothly than that in the radial direction because the thickness of the disk is smaller than the diameter, and thus the opacity is also smaller.
344
Kong Hoon Lee and Raymond Viskanta
A comparison of the radiative fluxes in the semitransparent (0 < J I 5 pm) and opaque (A> 5 pm) spectral region at the two bounding surfaces is given in Fig. 11. The radiative flux F,,, is defined by Eq. (35) and &,, is defined in the opaque spectral range
s cc
L,p =
Fj..idn for
i = r or
Z.
(36)
5w
For T I 1000 K, the radiative flux in the opaque spectral range is larger than that in the semitransparent range, but for T 2 1100 K, the opposite is true. This is due to the spectral blackbody emission with wavelength in the medium as mentioned before. The fraction of the blackbody emission to contribute to the radiative transfer within the semitransparent spectral range increases from 1.33 to 71.2% as the temperature of the medium increases from 300 to 1100 K. At temperatures less than 400 K (since more than 90% of the blackbody emission is emitted in the opaque spectral range to which the glass is relatively opaque) radiative transfer in the semitransparent spectral range is small. When T = 800K, about 50% of the blackbody emission is emitted in the opaque spectral range, and the radiative flux in the opaque range is about twice as large as that in the semitransparent range. For T 2 1100 K, more than 70% of the blackbody emission is emitted in the semitransparent spectral range and the radiative flux becomes larger than that in the opaque range. The radiative flux in the radial direction is still larger than that in the axial direction due to the greater opacity of the medium in that direction. In addition, the difference between the radiative fluxes in the two directions gradually increase as the temperature increases. This increase in the difference between the radiative fluxes with the temperature is due to the spectral radiative transfer in the intermediate spectral range as shown in Figs 7 and 8. 4. CONCLUSIONS
Radiative transfer in an axisymmetric disk of optical quality glass is considered. The radiative transfer equation is solved numerically using the S-N approximation of the discrete ordinates method. The numerical procedure used is validated by comparing the predicted fluxes with those based on an integral method of solution for one-dimensional radiative transfer in a semitransparent plate. Based on the numerical results obtained, the following conclusions can be drawn: 1. The S-8 level symmetric higher order (LSH) quadrature yields improved predictions of radiative transfer than the S-8 level symmetric odd (LSO) quadrature. 2. The exponential-type differencing scheme yields improved convergence and predictions over the central and upwind schemes. 3. For ,l 2 2.8 pm the optical dimensions in the axial and radial directions are sufficiently large and, therefore, the spectral radiative fluxes in the axial (F,,.) and radial (F,,,) directions are the same. 4. Near the bounding interfaces the total radiative fluxes increase very sharply, exhibiting a “boundary layer” type behavior where most of the radiant heat loss (i.e., cooling) of the system with largest temperature gradients would be expected to occur. 5. The total radiative fluxes at the two bounding surfaces increase as the temperature of the glass increases. For T 2 1100 K, the total radiative fluxes in the semitransparent spectral range (0 c 1s 5 pm) at the two bounding surfaces, I;;,,,, are larger than those in the opaque spectral range (1 > 5 pm), &._,, because more than 70% of the spectral blackbody emission is emitted in the semitransparent spectral range. REFERENCES 1. Bach, H. and Neuroth, N., eds., The Properties of Optical Glass. Springer, Berlin, 1995. 2. Gardon, R., J. Am. Ceramic Sot., 1961, 44, 305. Field, R. E. and Viskanta, R., J. Am. Ceramic Sot., 1990, 73, 2047. :: Viskanta, R. and Anderson, E. E., in Advances in Heat Transfer, eds. T. F. Irvine and J. P. Hartnett. Academic Press, New York, 1975, Vol. 11, p; 318. 5. Bergman, T. L. and Viskanta, R., in Radiative Transfer-Z, ed. M.P. Mengiic. Begell House, New York, 1996, p. 13. 6. Modest, M. F., Radiative Heat Transfer. McGraw-Hill, New York, 1993.
Prediction of spectral radiative transfer in a condensed cylindrical medium
34s
7. Siegel, R. and Howell, J. R., Thermal Radialion Heat Transfer, 3rd edn. McGraw-Hill, New York, 1992. 8. Carlson, B. G. and Lathrop, K. D., in Computing Methods in Reaction Physics, eds. H. Greenspan, C. N. Kelber and D. Okrent. Gordon & Breach, New York, 1968, p. 171. 9. Jamaluddin, A. S. and Smith, D. J., Combustion Sci. Tech., 1988, 62, 173. IO. Song, M. and Viskanta, R., in Proceedings of National Heat Transfer Conference, ASME HTD. ASME, New York, 1996, Vol. 325, p. 55. I I. Lathrop, K. D. and Carlson, B. G., Technical Reporf ,!,A-3186. Los Alamos Scientific Laboratory, Los Alamos, New Mexico, 1965. 12. Fiveland, W. A., in Fundamentals of Radiation Heat Transfer. ASME HTD, 1991, Vol. 160, p. 89. 13. Gardon, R., f. Am. Ceramic Sot., 1956, 39, 278.