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Determination of view factors by contour integral technique
Annals of Nuclear Energy 36 (2009) 1681–1688
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Determination of view factors by contour integral technique Santosh B. Bopche *, Arunkumar Sridharan 1 Department of Mechanical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 18 March 2009 Received in revised form 9 September 2009 Accepted 16 September 2009 Available online 14 October 2009
This paper presents the application of contour integral technique to derive the diffuse radiation view factor expressions (analytical) for elements of nuclear reactor fuel bundle. The cases considered are: (i) view factor between two cylindrical rods of equal diameter and finite length, (ii) view factor between two cylindrical rods with interference by another rod and (iii) view factor between a cylindrical rod and a non-concentric cylindrical enclosure. The contour integral method is significantly more accurate than the area-integration method. The view factor results based on the analytical expressions derived for these finite length geometries are compared with that of the exact expressions available in the literature for infinite length. It is observed that the use of infinite length approximations in finite length cases can lead to significant errors. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction The analysis of the energy exchange by radiation between two surfaces depends largely on the knowledge of view factors, accurate evaluation of which is very important for many applications. View factor between two components depends only on the geometry and its orientation/relative position. The geometric view factor between two isothermal, black, diffusely emitting and reflecting surfaces is defined as,
F AiAj ¼
1 Ai
Z Z Ai
cos bi cos bj
pR2
Aj
dAi dAj
ð1Þ
where bi and bj are the angles between the normal and the line joining two infinitesimal areas dAi and dAj of the respective surfaces. R is the distance between two areas. Ai is the total radiating area of surface i. To simplify the analytical evaluation of Eq. (1) Sparrow (1963) transforms the double area integral to double contour integral by using Stokes’ theorem. Appling Stokes’ theorem twice Eq. (1) reduces to Eq. (2).
F Ai Aj
1 ¼ 2pAi
(Z Z Ci
ln R dxi dxj þ ln R dyi dyj þ ln R dzi dzj
)
Cj
ð2Þ where Ci and Cj represent the contours bounding the view areas of surfaces Ai and Aj. dx, dy and dz are the elemental lengths and R is the distance between elements on the contours of respective sur* Corresponding author. Tel.: +91 9892 518668; fax: +91 22 2572 6875. E-mail addresses: [email protected], [email protected] (S.B. Bopche), [email protected] (A. Sridharan). 1 Tel.: +91 22 2576 7580; fax: +91 22 2572 6875. 0306-4549/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2009.09.007
faces. In the integration care has been taken to evaluate the integral over those parts of Aj which are directly visible from Ai or its infinitesimal element dAi, as reported by Sparrow (1963). Sparrow and Jonsson (1963) present results of the view factors for diffuse interchange of radiant energy between cylindrical tubes which are oriented parallel to one another, as shown in Fig. 1. It has application in the analysis of fin-tube radiators, boiler tubes, etc. Fig. 1a portrays the angle factor between elemental areas dAi and dAj on the surfaces of the tubes. Each element is of length dz which subtends an arc of 180 on the circumference. Fig. 1b portrays the angle factor between the elemental area dAi and the finite area Aj. Fig. 1c and d are analogous to Fig. 1a and b, respectively, except that the participating areas are confined to either portion (i.e. front or back) of the tubes. Applying ‘Contour Integral technique,’ the authors carried out analysis for obtaining an integral expression. The expression thus obtained is solved numerically to get the view factor values between elemental ring areas, dAi and dAj, z distance apart and also between elemental areas dAi and Aj, the finite area of other tube for z = 0 to z ! 1. The same procedure is followed for the cases shown in Fig. 1c and d. It is reported that, the local angle factor decreases with increase in longitudinal distance (z/S) between two elements. The increase in tube-to-tube separation distance may either increase or decrease the view factor, depending upon the size of tubes (e.g. S/r), where, 2S is the spacing between two tubes and r is the outer radius. Expressions of local view factor FdAiAj, z?1 for both the cases are also presented thereof. Hahne and Bassiouni (1980) have used the contour integral representation of the angle factor for radiant interchange between the two inner circumferential halves of cylindrical tubing at different temperatures, one half as ‘absorber’ and the other as ‘cover’ for evacuated tube type solar collector application. Both areas are
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Nomenclature a, r radius of tube/rod/pipe (m) A finite area of the surface (m2) contours of respective surfaces Ci, Cj d diameter of cylindrical rod (m) e eccentricity (m) dA infinitesimal areas of the respective surfaces i and j (m2) Fx, Fy, Fz view factor components view factor between two geometries of infinite length Finfinite Finterference view Factor between two rods in case of interference view factor value using cross string method Fcs Fci, Fij view factor value using present contour integration expressions L length of the geometry (m) n number of divisions of the contour/line/surface area p pitch/centre to centre distance of two cylindrical rods (m) radius of cylindrical surfaces, i and j (m) ri, rj R, S distance between two infinitesimal/finite elements of surfaces i and j (m) radius of cylindrical enclosure (pressure tube of nuclear R1 reactor), j (m)
assumed to be at different but uniform temperatures. The view factor results of exact analytical expression derived for finite length cylindrical tube are compared with infinite length value i.e. 2=p for ðr=LÞ 6 0:01 . Juul (1982) has derived a simple double-integral expression for the diffuse radiation view factor between two parallel cylinders of finite lengths. A new integration scheme is developed by which the dz
Aj
dAj z
z dAi dAi
r
2S r (a) dF dAi-dAj
r
2S r (b) dF dAi-Aj
dz
Aj
z
dAj
z dAi dAi
r
2S (c) dF
r dAi-dAj
r
2S (d) dF
r dAi-Aj
Fig. 1. Surface elements of parallel oriented cylinders with/without partition (Sparrow and Jonsson, 1963).
R2 x z X, Y, Z
eccentricity or distance between center of rod and center of a cylindrical enclosure (m) interference distance (m) distance between two elements or longitudinal dimension of the geometry (m) coordinates of the system with respective origin o. (alphabets A to Z are constants, defined at the end of each obtained view factor formulation)
Greek symbols included angle at the centre of that geometry or direction cosine angle (radian) b included angle at the centre of that geometry or direction cosine angle, or angle between the normal to a point and a line joining the two elements on surface i and j (radian) d included angle at the centre of that geometry (radian) n constant used in Eq. (10)
a
Subscripts i, j surfaces acting as a source, sink for radiation
quadruple integral for the view factor is reduced to a simple double integral, the subsequent integration of which is carried out numerically. It is reported that ‘no closed-form solution appears possible except for the limiting case of infinitely long cylinders’ for which an analytical expression for the view factor, Finfinite is derived by applying the cross string method. The author computed the view factor between two cylinders of equal finite length by numerical integration, accuracy of which is limited compare to the analytical solution. Shapiro (1985) has reported that the numerical integration of double contour integral requires significantly less computer time as compared to the double area integral equations. In addition to that it (the line/contour integration method) is significantly more accurate than the area-integration method. The surfaces between which view factors were calculated are plane quadrilaterals. Contour integral method requires a subdivision of the contour of the quadrilateral into n divisions while Area integral method requires a subdivision of the surface area. Dividing each of the four lines of quadrilateral into n divisions results in a total of 4n nodes around the contour and n2 nodes for the surface area. Operations counts reported for the area-integration method as (114n4 + 86n2) and for contour integral method as (464n2 + 24n). Timing studies show the Contour integral method is faster than the area-integration method. Ambirajan and Venkateshan (1993) have suggested a method to improve the accuracy of the numerical results using contour double integral formula (CDIF), both for non intersecting surfaces and for intersecting surfaces. Calculated view factor values are compared well with values computed using view factor algebra. It is concluded that evaluating the view factor using CDIF with the trapezoidal rule coupled with Romberg extrapolation yields accurate results. The method is also capable of giving very accurate values of the view factor for areas bounded by curved contours. Rao and Sastri (1996) present a computationally efficient and simpler method of evaluating view factors between two plane surfaces. It is concluded that the Guassian quadrature method applied to the double contour integral equations with nonlinear transformation to map the boundary is found to be the most accurate, computationally faster and very general. Carlson and Garcia (1984) evaluate the view factor between the wall and end (internal and external base) of a cylinder using contour integral expressions.
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Byrd (1993) presents an expression for view factor derived for the case of parallel rectangular planes that are not directly opposed. Chung and Kim (1982) have reported that finite element method together with Guassian quadrature integration is a capable tool for irregular geometries. The view factors values between two parallel planes and two intersecting planes at angle (30°, 60°, 90°, 120° and 150°) are evaluated using finite element techniques, and compared with analytical and contour integration solution. From the results, it is observed that a close approximation to analytical and contour integral solution can be achieved by the finite element method. Analytical formulae for view factor between two cylindrical geometries are available in the standard texts (Sparrow and Cess, 1977; Modest, 1993; Siegel and Howell, 1972; Howell, 1982) for infinite length. The assumption of infinite length geometry for view factor evaluation, is valid for the case of minimum spacing between the two, e.g. for lower p/d ratios in case of cylinders. It may invite considerable deviation from the realistic results in case of higher spacing/(p/d) ratios. The contour integral technique developed by E.M. Sparrow was applied and accordingly the problem was formulated for the view factor between ‘two cylinders of same diameters with and without interference’ and ‘cylindrical rod and cylindrical enclosure’, of finite lengths. The objective of the present study is to obtain an analytical– numerical solution for ‘view factor between the finite length fuel rods in a fuel rod bundle’. Each bundle is enclosed in a cylindrical tube called ‘pressure tube’. The pressure tube provides the sink to decay heat removal between the fuel rods and moderator in case of severe accidents in a reactor. Hence, evaluation of view factor between rods and pressure tube is critical. It necessitates mainly the view factor between two cylindrical rods with and without interference of another rod and between a cylindrical rod and a cylindrical pressure tube surface. Contour integral technique simplifies the analytical evaluation of view factor. To obtain a closed-form solution, assumption of infinite length geometries, which may deviate the results from realistic one, is not mandatory.
Z
F Ai Aj ¼
Z Z
1 ln R dxi dxj þ 2pAi Cj Ci ! Z Z þ ln R dzi dzj Cj
Z Z Cj
ln R dyi dyj ð3Þ
Ci
F Ai Aj ¼ F x þ F y þ F z
ð4Þ
where Fx and Fy can be termed as view factor components obtained considering changes only in the X and Y-coordinates along the contours, respectively. Similarly, Fz is the view factor value obtained considering the contours perpendicular to X–Y plane, which are p (rod pitch) distance apart. Because of complexities of the terms involved in the integration, the values of Fx and Fy were obtained numerically. Analytical formulation was possible for Fz for all the view factor cases considered in this paper. The required double integrals can be solved using standard tables (Jeffrey, 1995). 2.1. Problem formulation As shown in Fig. 3, dAi and dAj are two points on the contours, of the two cylinders. These can be specified as:
z=L X
−π 2
dAi
R
(j)
dAj z=0
−π 2 P Fig. 2. Contour representations of two cylindrical rods.
Y
∫
+π 2
L 0
+ αi
∫ ‘i'
∫
0 L
L o
+π 2
‘j' +α j
dAi
− αi
dAj
−π 2
−α j
∫
p
0
X
−π 2
L
Fig. 3. View surface and included angle representations in X–Y plane.
xi ¼ r i cos ai ; R¼
Ci
+π 2
(i)
2. View factor between two cylindrical rods of same diameter and equal length The view factor expression between two cylindrical rods of same diameter and finite length is discussed herewith taking into consideration only the view areas between two cylindrical rods. The schematic of the arrangement is shown in Fig. 2. The contour integral Eq. (2) can be rewritten as Eqs. (3) and (4),
+π 2
Y
xj ¼ p rj cos aj ;
yi ¼ ri sin ai ;
yj ¼ rj sin aj
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðr i cos ai p þ r j cos aj Þ2 þ ðri sin ai rj sin aj Þ2 þ ðzi zj Þ2
where ai and aj are the included angles at the center of respective cylindrical surfaces viewing each other. R is the distance between two infinitesimal elements. Substituting, the above coordinates and putting in the limits of integration;
Fx ¼
1 2 p Ai
Z Z Ci
ln R dxi dxj Cj
It results in, (p=2 p=2 p 2 2 ðr=LÞ X X þ cos aj þ sin ai sin aj Fx ¼ ln cos ai 2 4p r p=2 p=2
p=2 X p=2 p 2 X sin ai sin aj dai daj þ þ cos aj ln cos ai r p=2 p=2 ) 2 # 2 L þ sin ai sin aj þ sin ai sin aj dai daj ð5Þ r
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Similarly, ‘Fy’ is obtained as, ( p=2 p=2 p 2 2 ðr=LÞ X X þ cos Fy ¼ ln cos a a þ sin a sin a i j i j 4p2 r p=2 p=2 p=2 X p=2 2 p X cos ai cos aj dai daj þ ln cos ai þ cos aj r p=2 p=2 ) 2 # 2 L þ sin ai sin aj þ cos ai cos aj dai daj ð6Þ r
0.12
0.1
Fi-j
0.08
0.06
These Eqs. (5) and (6) were evaluated numerically. Analytical expression was possible for,
Fz ¼
1 2pAi
Cj
0.04
ln R dzi dzj ;
Z Z
L=Infinity, Eq. (8) L/d=1 E+08 L/d=1000 L/d=875 L/d=750 L/d=625 L/d=500 L/d=375 L/d=250 L/d=125 L/d=62.5 L/d=25 L/d=12.5
0.02
Ci
The expression for Fz, simplified for two parallel cylindrical rods, is as given below.
! ! ! 1
X2 1 þ X2 1 þ W2 þ l ln þ WZ ln Fz ¼
XY ln 4p2 1 þ X2 1 þ W2 W2
L L
þ 4Y tan1 ð7Þ 4Z tan1 p a
0 0
2
4
6
8
10
12
14
16
p/d Fig. 4. View factor between two cylinders of same radius and finite length.
F Infinite ¼
"rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #
p2 p 1 d 1 þ sin d p d p 1
ð8Þ
where
a¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 þ 4r 2 ;
p X¼ ; L
p Y¼ ; r
a W¼ ; L
a Z¼ ; r
l¼
L r
2.2. Results and discussion The results of Eqs. (5)–(7) obtained, for finite length cylinders, were compared with those available in standard texts (Modest, 1993; Siegel and Howell, 1972; Howell, 1982) for view factor between two cylinders of same diameter and infinite length, as given in Eq. (8). Where, p is the pitch/center to center distance between two cylinders and d as diameter of cylinders.
The values of view factor between two cylindrical rods for various pitch–diameter ratios and for some finite lengths are as shown in Table 1. In case of finite length cylindrical rods if infinite length–view factor values are used it may cause major deviation from realistic results. Fig. 4 shows that, view factor between two cylindrical rods decreases with increase in pitch–diameter ratios. Higher p/d ratio demands more number of rods to enclose/encircle the ith rod, which reduces rod to rod view factor. It is also due to the fact that, view factor is inversely proportional to the square of the distance between their surface elements. It is also seen that, view factor increases with increase in its length to diameter ratio and approaches to that of infinite length rods, at finite length–diameter
Table 1 View factor between two cylindrical rods for finite length. L/d p/d
12.5
25
62.5
125
250
500
750
1000
1E+08
Infinite L
1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15
0.07185 0.05261 0.0404 0.03212 0.02621 0.02183 0.01848 0.01586 0.01376 0.01205 0.01064 0.00946 0.00841 0.00763 0.00686 0.00627 0.0057 0.00524 0.0048 0.00445 0.0041 0.00381 0.00354 0.00331 0.00308 0.00289 0.00271 0.00255
0.08131 0.06108 0.04806 0.03911 0.03263 0.02775 0.02397 0.02096 0.01851 0.0165 0.01481 0.01338 0.01208 0.01109 0.01012 0.00936 0.00861 0.00802 0.00742 0.00694 0.00647 0.00607 0.00568 0.00536 0.00504 0.00476 0.00449 0.00426
0.088867 0.068035 0.054523 0.045154 0.038318 0.033132 0.029073 0.025817 0.023152 0.020933 0.019058 0.017456 0.015986 0.014865 0.013746 0.012866 0.011989 0.011281 0.010576 0.009996 0.009419 0.008936 0.008456 0.008049 0.007643 0.007296 0.00695 0.00665
0.09206 0.07103 0.05735 0.04784 0.04089 0.0356 0.03145 0.02811 0.02537 0.02308 0.02114 0.01948 0.01804 0.01678 0.01567 0.01469 0.01381 0.01302 0.01231 0.01166 0.01107 0.01053 0.01004 0.00958 0.00916 0.00877 0.00841 0.00807
0.09394 0.0728 0.05904 0.04947 0.04245 0.03711 0.03291 0.02953 0.02675 0.02443 0.02246 0.02077 0.0193 0.01802 0.01689 0.01588 0.01498 0.01417 0.01343 0.01277 0.01216 0.0116 0.01109 0.01062 0.01019 0.00978 0.0094 0.00905
0.095016 0.073833 0.060032 0.050421 0.043377 0.038008 0.033787 0.030385 0.027587 0.025247 0.023261 0.021555 0.020074 0.018777 0.017632 0.016614 0.015702 0.014881 0.014138 0.013463 0.012847 0.012282 0.011762 0.011283 0.010839 0.010427 0.010043 0.009686
0.09542 0.07421 0.0604 0.05078 0.04372 0.03835 0.03412 0.03071 0.0279 0.02556 0.02357 0.02186 0.02037 0.01907 0.01792 0.0169 0.01598 0.01516 0.01441 0.01373 0.01311 0.01255 0.01202 0.01154 0.0111 0.01068 0.0103 0.00994
0.095626 0.074418 0.060596 0.050967 0.043909 0.038527 0.034294 0.030882 0.028074 0.025725 0.023731 0.022018 0.020531 0.019227 0.018076 0.017051 0.016134 0.015308 0.014561 0.013881 0.01326 0.012691 0.012167 0.011683 0.011236 0.01082 0.010433 0.010072
0.096376 0.075143 0.061301 0.051655 0.044581 0.039186 0.034942 0.031519 0.028702 0.026344 0.024342 0.022622 0.021127 0.019817 0.01866 0.01763 0.016707 0.015876 0.015123 0.014439 0.013813 0.01324 0.012712 0.012225 0.011773 0.011354 0.010963 0.010599
0.110696 0.081376 0.064555 0.05356 0.04579 0.04000 0.035516 0.031938 0.029018 0.026588 0.024534 0.022775 0.021252 0.01992 0.018746 0.017702 0.016769 0.015929 0.015169 0.014479 0.013848 0.013271 0.012739 0.012249 0.011795 0.011373 0.010981 0.010614
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ratio of the order of 1E+08. It is attributed to the increased values of local view factor between the respective ith surface elements and the finite jth surface.
Case 1: For (0 < x/r < 1)
xi ¼ p r cos a1 ;
xj ¼ r cos a1 ; yi ¼ r sin a1 ; 1 r x yj ¼ r sin a1 a1 ¼ sin ; r qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 A ¼ ðp 2r cos a1 Þ2 & B ¼ ðp r cos a1 Þ2 þ ðr sin a1 rÞ
3. View factor between two cylindrical rods with interference The situation of two cylindrical rods with interference from third rod occurs in nuclear reactors, where a bundle of rods is closely packed. Interference or partial obstruction of the view between two geometries significantly alters their view factor value and hence the radiative heat exchange.
Case 2: For (2 > x/r > 1)
xi ¼ p r cos a2 ;
xj ¼ r cos a2 ; yi ¼ r sin a2 ; 1 x r ; yj ¼ r sin a2 ; a2 ¼ sin rqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 A ¼ ðp 2r cos a2 Þ2 & B ¼ ðp r cos a2 Þ2 þ ðr sin a2 þ rÞ
3.1. Approach to the problem
Case 3: For (x/r = 1)
As discussed earlier, during evaluation of view factor between two circular cylinders i and j, semi-cylindrical surface areas of both are considered as the maximum view areas. The surface areas influenced by the interference of third cylindrical rod k, are shown by dark contours, as shown in Fig. 5. Let, x is the interference distance (due to third rod), r is the radius of the rods & x/r represents the ‘Interference ratio’, (0 < x/ r 6 2) For the case with (x/r < 1), the contour-coordinates are specified as follows:
xi ¼ p r cos a1 ;
a1 ¼ sin1
xj ¼ r cos a1 ;
a1 ¼ a2 ¼ 0; A ¼ ðp 2rÞ & B ¼ 3.2. Results and discussion
The total view factor values (i.e. sum of Fx, Fy and Fz) obtained, for different values of interference, x (mm) are presented in Table 2. It is seen that with increase in interference, view factor values reduce for the range of p/d considered. It is also observed that with increase in pitch–diameter ratios (p/d) and interference ratios (x/r), view factor between two interfered geometries decreases and the variation is less in case of higher pitch–diameter ratios. The contour integral expressions can be used for interference ratio, ‘x/r’ ranges from 0 to 2. These are checked for half interference (x/r = 1), for cases with more than 50% interference and also for the limiting case of complete interference i.e. (1 < x/r < 2). Table 4 shows these results for a constant pitch to diameter ratio (p/d = 3.4033) and length to diameter ratio (L/d = 84). The view factor values obtained using above analysis are compared with the algebraic expression derived using cross string method for the present case. According to cross string method, view factor between two infinitely long cylinders in case of interference, is obtained as,
rx ; r
yi ¼ r sin a1 ; yj ¼ r sin a1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼ ðp r cos a1i r cos a1i Þ2 þ ðr sin a1i r sin a1j Þ2 þ ðzi zj Þ2 The integral expressions for Fx and Fy as discussed in previous case were solved numerically. The analytical expression derived, solving for Fz, is as given below;
! ! 1
X2 1 þ X2 F zinterference ¼ þ 2l ln
2XY ln 8p2 1 þ X2 1 þ W2 !
1 þ W2 1 L 1 L þ 8Y tan 8Z tan þ 2WZ ln p a W2 !! 2 2 ðX þ 1ÞðV þ 1Þ þ 2TU lnðT 2 þ 1Þ þ l ln ðT 2 þ 1Þ2 VS lnðV 2 þ 1Þ XY ln X 2 þ 1 þ 2XY lnðXÞ
1 þ 2SV lnðVÞ 4TU lnðTÞ þ 4 ðYÞ tan1 X
1 1
þ ðSÞ tan1 ð2UÞ tan1 ð9Þ
V T
F cs ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 þ 4r 2 ;
a¼
Z ¼ a=r
V ¼ A=L;
X ¼ p=L;
S ¼ A=r;
Y ¼ p=r;
T ¼ B=L;
1 4p r
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p 2r ðp þ Lp Þ 2 p2 4r 2 þ 4r cos1 p 2 ð10Þ
where
(rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) p 2 2 Lp ¼ 2 r sin n þ ðx r þ r cos nÞ þ rn & 2 0 0 11
p B 2ðr xÞ r B CC þ cos1 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAA n ¼ @tan1 p 2 2 2 ðr xÞ þ ðp=2Þ
where
l ¼ L=r;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðp r Þ2 þ r2
W ¼ a=L;
U ¼ B=r
-Y
‘j’
−α2
α1
r
∫
L 0
∫
0
∫
L
0 L
− α
α1 = α 2 = 0 ; x r = 1
∫
0 L
∫
x
α1 = π 2
‘k’ p
α1
0
‘i’ 2
X
L
∫
L o
Fig. 5. View factor between two rods in case of interference.
α1 = π 2
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Table 2 View factor results for various interference ratios (x/r), at L/d = 84.03. L/d = 84.03 p/d
x/r = 0.168
x/r = 0.252
x/r = 0.336
x/r = 0.504
x/r = 0.672
x/r = 0.840
1.51 2.06 2.68 3.40 4.20 5.08 6.05 7.10 8.23 9.45 10.75 12.14 13.61 15.16 16.80 18.53 20.33 22.22 24.20 26.26 28.40 30.63 32.94 35.33 37.81
0.088517 0.066655 0.051197 0.040121 0.032019 0.025964 0.021344 0.017755 0.01492 0.012648 0.010805 0.009292 0.008039 0.006992 0.00611 0.005362 0.004723 0.004176 0.003703 0.003294 0.002938 0.002627 0.002354 0.002115 0.001904
0.087171 0.065717 0.050494 0.039569 0.031571 0.025591 0.021028 0.017482 0.014681 0.012438 0.010618 0.009125 0.007889 0.006856 0.005986 0.005249 0.00462 0.004081 0.003616 0.003214 0.002864 0.002559 0.002291 0.002057 0.00185
0.0853 0.064448 0.049565 0.038853 0.031 0.025123 0.020636 0.017149 0.014394 0.012188 0.010398 0.008931 0.007715 0.0067 0.005846 0.005122 0.004505 0.003976 0.003521 0.003126 0.002784 0.002485 0.002224 0.001994 0.001792
0.079934 0.060893 0.047012 0.036921 0.029482 0.023897 0.019625 0.0163 0.013673 0.011567 0.009859 0.008458 0.007299 0.006331 0.005517 0.004827 0.00424 0.003737 0.003304 0.00293 0.002605 0.002322 0.002075 0.001858 0.001667
0.07246 0.055997 0.043539 0.034323 0.027465 0.022286 0.018311 0.01521 0.012755 0.010785 0.009187 0.007876 0.00679 0.005884 0.005122 0.004477 0.003928 0.003457 0.003053 0.002704 0.002401 0.002137 0.001907 0.001705 0.001528
0.063123 0.049848 0.039186 0.031082 0.024962 0.020299 0.0167 0.013881 0.011643 0.009845 0.008384 0.007184 0.00619 0.00536 0.004662 0.004071 0.003568 0.003138 0.002768 0.002449 0.002172 0.001931 0.001721 0.001537 0.001376
Table 3 Comparison with infinite length interference rod view factor results. p/d
3.4033
x/r
L/d
Fzinterference
Fcs
0.1680
80.03 168.06 252.10 420.16 840.33 1E+08
0.04012068 0.0423317 0.04319233 0.04395824 0.044608 0.045436384
0.046446
0.3361
80.03 168.06 252.10 420.16 840.33 1E+08
0.03885308 0.04111883 0.04200251 0.04278988 0.04345866 0.044313024
0.0444483
0.5042
80.03 168.06 252.10 420.16 840.33 1E+08
0.0369208 0.0391882 0.040074 0.040864 0.0415357 0.042395239
0.041111
ever, with increase in length of rods the view factor values approach the values obtained by cross string method. Cross string method is used as a tool just to check whether values obtained by contour integral formula are realistic or not. 4. View factor between a finitely long cylinder and a nonconcentric enclosure
The view factor formula derived using contour integral technique for interference rod case, agrees well with cross string method, as shown in Table 3. It is seen that with increase in interference ratio, view factor between two cylinders i and j decreases. HowTable 4 View factor values for (2 > x/r > 1) and (x/r = 1) obtained from contour integral expressions. x/r
Fzinterference
p/d = 3.4
1 1.176
0.02744 0.02285
L/d = 84
1.345 1.513 1.681 1.849 2
0.01802 0.01286 0.00755 0.00244 1.3E16 (zero)
View factor evaluation between a cylinder of finite length and non-concentric cylindrical enclosure (an arc) for radiation heat transfer is a common case in nuclear reactors (between hot fuel rod and a pressure tube/sink surface) during loss of coolant situation, in heat exchangers (between hot pipes and the outer cylindrical shell) and in solar collectors, between a coolant tube and an evacuated cylindrical cover etc. The view factor for this case is presented in the current section. The problem is formulated as shown in Fig. 6. The following nomenclature is used for the same.a is an included angle at the center of a cylindrical rod by its view surface, ðp=2Þ 6 a 6 ðp=2Þ.a1 & a2 are the included angles at the center of sink surface/enclosure by its extremity. The contour integral expressions yield better results for the ranges of angles, as given below.
a1;min ¼ sin1 ðr=R1 Þ and a1;max ¼ cos1 ½ðR2 þ rÞ=R1 a2;min ¼ cos1 ½ðR2 rÞ=R1 and a2;max ¼ p sin1 ðr=R1 Þ b and d are the included angles by additional view areas of a rod (i.e. beyond the range of angle a),
0
1
r B C b ¼ cos1 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 2 2 ðR1 cos a1 R2 Þ þ ðR1 sin a1 Þ
R1 sin a1 ½0 6 b 6 ðp=2Þ tan1 R1 cos a1 R2 d ¼ cos1 ðr=R1 Þ p þ a2
0 6 d 6 cos1 ðr=R1 Þ p þ a2;max
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S.B. Bopche, A. Sridharan / Annals of Nuclear Energy 36 (2009) 1681–1688
where
+Y
∫
α2
Aj
0 L
R2 R1
∫
L
+α −α
0
α1 δ
β
+X
∫
L
∫
Ai
0
-X
0
L
Fig. 6. View factor between cylinder and a non-concentric cylindrical enclosure.
R1 is radius of cylindrical enclosure/sink/pressure tube, R2 is the offset distance between rod and sink surface centers (e) and r is the radius of cylindrical rod. Thus, the total required view factor values for Fx and Fy, were obtained numerically using FORTRAN compiler and the analytical expression obtained for view factor component, Fz is given in Eq. (11).
Fz ¼
! !! ! 2 2 1
ðL=CÞ þ 1 ðL=JÞ þ 1 E2 þ ln þ l ln ln ðL=EÞ2 þ 1
2 2 2
rL 8p ðL=EÞ þ 1 ðL=HÞ þ 1 2
C rL
!
2
ln ðL=CÞ þ 1 þ 2l ln
2
CJ H þ rL EH
!
2
ln ðL=HÞ þ 1
!
J2 C L E L 2 ln ðL=JÞ þ 1 þ 4 tan1 tan1 rL r C r E
2
þ ln
ðL=qÞ þ 1 ðL=pÞ2 þ 1
4.1. Results and discussion The view factor results are compared with standard view factor formula for infinitely long lengths of rod and a cylindrical enclosure available in view factor catalog (Howell, 1982) as given by Eq. (12) for Fig. 7.
F infinite;12 ¼
!
2
J H L 1
ðL=mÞ þ 1 1 L 1
tan tan þ þ
l ln 2 r J r H 8p2 ðL=nÞ þ 1 !!
l ¼ L=r qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C ¼ ðR2 þ r R1 cos a1 Þ2 þ ðR1 sin a1 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E ¼ ðR2 r R1 cos a1 Þ2 þ ðR1 sin a1 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H ¼ ðR2 þ r R1 cos a2 Þ2 þ ðR1 sin a2 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J ¼ ðR2 r R1 cos a2 Þ2 þ ðR1 sin a2 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m ¼ ðR2 þ r cos b R1 cos a1 Þ2 þ ðr sin b R1 sin a1 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ¼ ðR2 þ r R1 cos a1 Þ2 þ ðR1 sin a1 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ¼ ðR2 þ r cos b R1 cos a2 Þ2 þ ðr sin b R1 sin a2 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ¼ ðR2 þ r R1 cos a2 Þ2 þ ðR1 sin a2 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼ ðR2 r cos d R1 cos a1 Þ2 þ ðr sin d R1 sin a1 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S ¼ ðR2 r R1 cos a1 Þ2 þ ðR1 sin a1 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T ¼ ðR2 r cos d R1 cos a2 Þ2 þ ðr sin d R1 sin a2 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U ¼ ðR2 r R1 cos a2 Þ2 þ ðR1 sin a2 Þ2
n2 2 þ ln ðL=nÞ þ 1 rL
1 a2 a1 1þE a2 tan þ tan1 2p 1E 2 2
1 þ E a 1 tan ; E ¼ e=R1 tan1 1E 2
ð12Þ
The total view factor values obtained, for the above discussed cylindrical geometries of finite length, were compared with that of Eq. (12), as shown in Fig. 8. It was observed that with increase
2 2 m mq p 2 2 ln ðL=mÞ þ 1 þ 2l ln ln ðL=pÞ þ 1 þ rL rL np
A2
2
q m L n L 2 ln ðL=qÞ þ 1 þ 4 tan1 tan1 rL r m r n
!
2 q L p L
1
ðL=RÞ þ 1 tan1 tan1 þ l ln þ
2 r q r p 8p2 ðL=SÞ þ 1
þ ln
!! 2 ðL=U Þ þ 1 2
ðL=TÞ þ 1
α2
!
þ
R1
S2 R2 2 2 lnððL=SÞ þ 1Þ ð Þ lnððL=RÞ þ 1Þ rL rL
! !
RU T2 U2 2 2 ln ðL=TÞ þ 1 lnððL=UÞ þ 1Þ þ þ 2l ln rL rL ST
α1
R L S L U L tan1 tan1 þ tan1 þ4 r R r S r U
T L
tan1
r T
A1
e ð11Þ Fig. 7. Infinitely long cylinder and non-concentric cylindrical enclosure.
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S.B. Bopche, A. Sridharan / Annals of Nuclear Energy 36 (2009) 1681–1688
α1=20°,α2=100° Infinite 'L' L/d=1E+08 F i -j
L/d=420.17 L/d=252 L/d=168.07 L/d=84.03 L/d=42.02 L/d=8.40 SR=[R1/(R2+r)] Fig. 8. View factor between cylindrical rod and a sink surface (an arc).
Computational time t, execution time in seconds
Contour Integral Technique
5. Conclusions To evaluate radiative heat exchange between the various three dimensional geometries, accurate values of view factor can be obtained by using the contour integral technique. In the present paper, exact expressions for diffused radiation view factor between two cylindrical rods/pipes, of equal diameter and same finite length, between two cylindrical pipes/rods in case of (partial or complete) interference and between cylindrical source and sink surface have been presented. These expressions can be applied for the surfaces maintained at different and uniform temperatures. The reciprocity relation can be used to evaluate view factor between surface j and surface i. Care has to be taken while evaluating radiative heat transfer area of the respective surface. The main findings of this work are: In addition to spacing between two radiative surfaces, length of the geometry also influences the view factor results significantly. Use of infinite length formulae for finite length situations can lead to significant errors. A contour integral technique developed by E.M. Sparrow is proved to be the simplest, effective and accurate tool for view factor evaluation between three dimensional geometries.
References
n, number of segments of contours Fig. 9. Computational time for contour integral method.
in length of the geometries, view factor between them increases and approach to the infinite length–view factors at L/d = 1E+08 that lie within ±5% deviation. For increasing spacing ratios it shows a decreasing trend for almost all L/d ratios. The cases discussed in this paper are applicable for radiative heat exchange in a nuclear fuel bundle during loss of coolant situation. The computational time, t on an average for the above discussed finite length geometries are plotted with respect to the number of divisions, n of the contours. It is shown in Fig. 9. Computation is done using a FORTRAN code with LINUX as an operating system.
Ambirajan, A., Venkateshan, S.P., 1993. Accurate determination of diffuse view factors between planer surfaces. Int. J. Heat Mass Transfer 36 (8), 2203–2208. Byrd, L.W., 1993. View factor algebra for two arbitrarily sized non-opposing parallel rectangles. ASME J. Heat Transfer 115, 517–518. Carlson, R.W., Garcia, J., 1984. View factor for radiant heat transfer between the wall and end of a cylinder. Ann. Nuclear Energy 11 (4), 187–196. Chung, T.J., Kim, J.Y., 1982. Radiation view factors by finite elements. ASME J. Heat Transfer 104, 792–795. Hahne, E., Bassiouni, M.K., 1980. The angle factor for radiant interchange within a constant radius cylindrical enclosure. Lett. Heat Mass Transfer 7, 303–309. Howell, J.R., 1982. A Catalogue of Radiation Configuration Factors. McGraw Hill, New York. Jeffrey, Alan, 1995. Handbook of Mathematical Formulas and Integrals. Academic Press. Juul, N.H., 1982. View factors in radiation between two parallel oriented cylinders of finite lengths. ASME J. Heat Transfer 104, 384–388. Modest, M.F., 1993. Radiative Heat Transfer. Academic Press. Rao, V.R., Sastri, V.M.K., 1996. Efficient evaluation of diffuse view factors for radiation. Int. J. Heat Mass Transfer 39 (6), 1281–1286. Shapiro, A.B., 1985. Computer implementation, accuracy, and timing of radiation view factor algorithms. ASME J. Heat Transfer 107, 730–732. Siegel, R., Howell, J.R., 1972. Thermal Radiation Heat Transfer. McGraw Hill, New York. Sparrow, E.M., 1963. A new and simpler formulation for radiative angle actors. ASME J. Heat Transfer 81, 81–87. Sparrow, E.M., Cess, R.D., 1977. Radiation Heat Transfer. McGraw Hill. Sparrow, E.M., Jonsson, V.K., 1963. Angle factors for radiant interchange between parallel-oriented tubes. ASME J. Heat Transfer 85, 382–384.
Contents lists available at ScienceDirect
Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
Determination of view factors by contour integral technique Santosh B. Bopche *, Arunkumar Sridharan 1 Department of Mechanical Engineering, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 18 March 2009 Received in revised form 9 September 2009 Accepted 16 September 2009 Available online 14 October 2009
This paper presents the application of contour integral technique to derive the diffuse radiation view factor expressions (analytical) for elements of nuclear reactor fuel bundle. The cases considered are: (i) view factor between two cylindrical rods of equal diameter and finite length, (ii) view factor between two cylindrical rods with interference by another rod and (iii) view factor between a cylindrical rod and a non-concentric cylindrical enclosure. The contour integral method is significantly more accurate than the area-integration method. The view factor results based on the analytical expressions derived for these finite length geometries are compared with that of the exact expressions available in the literature for infinite length. It is observed that the use of infinite length approximations in finite length cases can lead to significant errors. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction The analysis of the energy exchange by radiation between two surfaces depends largely on the knowledge of view factors, accurate evaluation of which is very important for many applications. View factor between two components depends only on the geometry and its orientation/relative position. The geometric view factor between two isothermal, black, diffusely emitting and reflecting surfaces is defined as,
F AiAj ¼
1 Ai
Z Z Ai
cos bi cos bj
pR2
Aj
dAi dAj
ð1Þ
where bi and bj are the angles between the normal and the line joining two infinitesimal areas dAi and dAj of the respective surfaces. R is the distance between two areas. Ai is the total radiating area of surface i. To simplify the analytical evaluation of Eq. (1) Sparrow (1963) transforms the double area integral to double contour integral by using Stokes’ theorem. Appling Stokes’ theorem twice Eq. (1) reduces to Eq. (2).
F Ai Aj
1 ¼ 2pAi
(Z Z Ci
ln R dxi dxj þ ln R dyi dyj þ ln R dzi dzj
)
Cj
ð2Þ where Ci and Cj represent the contours bounding the view areas of surfaces Ai and Aj. dx, dy and dz are the elemental lengths and R is the distance between elements on the contours of respective sur* Corresponding author. Tel.: +91 9892 518668; fax: +91 22 2572 6875. E-mail addresses: [email protected], [email protected] (S.B. Bopche), [email protected] (A. Sridharan). 1 Tel.: +91 22 2576 7580; fax: +91 22 2572 6875. 0306-4549/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2009.09.007
faces. In the integration care has been taken to evaluate the integral over those parts of Aj which are directly visible from Ai or its infinitesimal element dAi, as reported by Sparrow (1963). Sparrow and Jonsson (1963) present results of the view factors for diffuse interchange of radiant energy between cylindrical tubes which are oriented parallel to one another, as shown in Fig. 1. It has application in the analysis of fin-tube radiators, boiler tubes, etc. Fig. 1a portrays the angle factor between elemental areas dAi and dAj on the surfaces of the tubes. Each element is of length dz which subtends an arc of 180 on the circumference. Fig. 1b portrays the angle factor between the elemental area dAi and the finite area Aj. Fig. 1c and d are analogous to Fig. 1a and b, respectively, except that the participating areas are confined to either portion (i.e. front or back) of the tubes. Applying ‘Contour Integral technique,’ the authors carried out analysis for obtaining an integral expression. The expression thus obtained is solved numerically to get the view factor values between elemental ring areas, dAi and dAj, z distance apart and also between elemental areas dAi and Aj, the finite area of other tube for z = 0 to z ! 1. The same procedure is followed for the cases shown in Fig. 1c and d. It is reported that, the local angle factor decreases with increase in longitudinal distance (z/S) between two elements. The increase in tube-to-tube separation distance may either increase or decrease the view factor, depending upon the size of tubes (e.g. S/r), where, 2S is the spacing between two tubes and r is the outer radius. Expressions of local view factor FdAiAj, z?1 for both the cases are also presented thereof. Hahne and Bassiouni (1980) have used the contour integral representation of the angle factor for radiant interchange between the two inner circumferential halves of cylindrical tubing at different temperatures, one half as ‘absorber’ and the other as ‘cover’ for evacuated tube type solar collector application. Both areas are
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Nomenclature a, r radius of tube/rod/pipe (m) A finite area of the surface (m2) contours of respective surfaces Ci, Cj d diameter of cylindrical rod (m) e eccentricity (m) dA infinitesimal areas of the respective surfaces i and j (m2) Fx, Fy, Fz view factor components view factor between two geometries of infinite length Finfinite Finterference view Factor between two rods in case of interference view factor value using cross string method Fcs Fci, Fij view factor value using present contour integration expressions L length of the geometry (m) n number of divisions of the contour/line/surface area p pitch/centre to centre distance of two cylindrical rods (m) radius of cylindrical surfaces, i and j (m) ri, rj R, S distance between two infinitesimal/finite elements of surfaces i and j (m) radius of cylindrical enclosure (pressure tube of nuclear R1 reactor), j (m)
assumed to be at different but uniform temperatures. The view factor results of exact analytical expression derived for finite length cylindrical tube are compared with infinite length value i.e. 2=p for ðr=LÞ 6 0:01 . Juul (1982) has derived a simple double-integral expression for the diffuse radiation view factor between two parallel cylinders of finite lengths. A new integration scheme is developed by which the dz
Aj
dAj z
z dAi dAi
r
2S r (a) dF dAi-dAj
r
2S r (b) dF dAi-Aj
dz
Aj
z
dAj
z dAi dAi
r
2S (c) dF
r dAi-dAj
r
2S (d) dF
r dAi-Aj
Fig. 1. Surface elements of parallel oriented cylinders with/without partition (Sparrow and Jonsson, 1963).
R2 x z X, Y, Z
eccentricity or distance between center of rod and center of a cylindrical enclosure (m) interference distance (m) distance between two elements or longitudinal dimension of the geometry (m) coordinates of the system with respective origin o. (alphabets A to Z are constants, defined at the end of each obtained view factor formulation)
Greek symbols included angle at the centre of that geometry or direction cosine angle (radian) b included angle at the centre of that geometry or direction cosine angle, or angle between the normal to a point and a line joining the two elements on surface i and j (radian) d included angle at the centre of that geometry (radian) n constant used in Eq. (10)
a
Subscripts i, j surfaces acting as a source, sink for radiation
quadruple integral for the view factor is reduced to a simple double integral, the subsequent integration of which is carried out numerically. It is reported that ‘no closed-form solution appears possible except for the limiting case of infinitely long cylinders’ for which an analytical expression for the view factor, Finfinite is derived by applying the cross string method. The author computed the view factor between two cylinders of equal finite length by numerical integration, accuracy of which is limited compare to the analytical solution. Shapiro (1985) has reported that the numerical integration of double contour integral requires significantly less computer time as compared to the double area integral equations. In addition to that it (the line/contour integration method) is significantly more accurate than the area-integration method. The surfaces between which view factors were calculated are plane quadrilaterals. Contour integral method requires a subdivision of the contour of the quadrilateral into n divisions while Area integral method requires a subdivision of the surface area. Dividing each of the four lines of quadrilateral into n divisions results in a total of 4n nodes around the contour and n2 nodes for the surface area. Operations counts reported for the area-integration method as (114n4 + 86n2) and for contour integral method as (464n2 + 24n). Timing studies show the Contour integral method is faster than the area-integration method. Ambirajan and Venkateshan (1993) have suggested a method to improve the accuracy of the numerical results using contour double integral formula (CDIF), both for non intersecting surfaces and for intersecting surfaces. Calculated view factor values are compared well with values computed using view factor algebra. It is concluded that evaluating the view factor using CDIF with the trapezoidal rule coupled with Romberg extrapolation yields accurate results. The method is also capable of giving very accurate values of the view factor for areas bounded by curved contours. Rao and Sastri (1996) present a computationally efficient and simpler method of evaluating view factors between two plane surfaces. It is concluded that the Guassian quadrature method applied to the double contour integral equations with nonlinear transformation to map the boundary is found to be the most accurate, computationally faster and very general. Carlson and Garcia (1984) evaluate the view factor between the wall and end (internal and external base) of a cylinder using contour integral expressions.
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Byrd (1993) presents an expression for view factor derived for the case of parallel rectangular planes that are not directly opposed. Chung and Kim (1982) have reported that finite element method together with Guassian quadrature integration is a capable tool for irregular geometries. The view factors values between two parallel planes and two intersecting planes at angle (30°, 60°, 90°, 120° and 150°) are evaluated using finite element techniques, and compared with analytical and contour integration solution. From the results, it is observed that a close approximation to analytical and contour integral solution can be achieved by the finite element method. Analytical formulae for view factor between two cylindrical geometries are available in the standard texts (Sparrow and Cess, 1977; Modest, 1993; Siegel and Howell, 1972; Howell, 1982) for infinite length. The assumption of infinite length geometry for view factor evaluation, is valid for the case of minimum spacing between the two, e.g. for lower p/d ratios in case of cylinders. It may invite considerable deviation from the realistic results in case of higher spacing/(p/d) ratios. The contour integral technique developed by E.M. Sparrow was applied and accordingly the problem was formulated for the view factor between ‘two cylinders of same diameters with and without interference’ and ‘cylindrical rod and cylindrical enclosure’, of finite lengths. The objective of the present study is to obtain an analytical– numerical solution for ‘view factor between the finite length fuel rods in a fuel rod bundle’. Each bundle is enclosed in a cylindrical tube called ‘pressure tube’. The pressure tube provides the sink to decay heat removal between the fuel rods and moderator in case of severe accidents in a reactor. Hence, evaluation of view factor between rods and pressure tube is critical. It necessitates mainly the view factor between two cylindrical rods with and without interference of another rod and between a cylindrical rod and a cylindrical pressure tube surface. Contour integral technique simplifies the analytical evaluation of view factor. To obtain a closed-form solution, assumption of infinite length geometries, which may deviate the results from realistic one, is not mandatory.
Z
F Ai Aj ¼
Z Z
1 ln R dxi dxj þ 2pAi Cj Ci ! Z Z þ ln R dzi dzj Cj
Z Z Cj
ln R dyi dyj ð3Þ
Ci
F Ai Aj ¼ F x þ F y þ F z
ð4Þ
where Fx and Fy can be termed as view factor components obtained considering changes only in the X and Y-coordinates along the contours, respectively. Similarly, Fz is the view factor value obtained considering the contours perpendicular to X–Y plane, which are p (rod pitch) distance apart. Because of complexities of the terms involved in the integration, the values of Fx and Fy were obtained numerically. Analytical formulation was possible for Fz for all the view factor cases considered in this paper. The required double integrals can be solved using standard tables (Jeffrey, 1995). 2.1. Problem formulation As shown in Fig. 3, dAi and dAj are two points on the contours, of the two cylinders. These can be specified as:
z=L X
−π 2
dAi
R
(j)
dAj z=0
−π 2 P Fig. 2. Contour representations of two cylindrical rods.
Y
∫
+π 2
L 0
+ αi
∫ ‘i'
∫
0 L
L o
+π 2
‘j' +α j
dAi
− αi
dAj
−π 2
−α j
∫
p
0
X
−π 2
L
Fig. 3. View surface and included angle representations in X–Y plane.
xi ¼ r i cos ai ; R¼
Ci
+π 2
(i)
2. View factor between two cylindrical rods of same diameter and equal length The view factor expression between two cylindrical rods of same diameter and finite length is discussed herewith taking into consideration only the view areas between two cylindrical rods. The schematic of the arrangement is shown in Fig. 2. The contour integral Eq. (2) can be rewritten as Eqs. (3) and (4),
+π 2
Y
xj ¼ p rj cos aj ;
yi ¼ ri sin ai ;
yj ¼ rj sin aj
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðr i cos ai p þ r j cos aj Þ2 þ ðri sin ai rj sin aj Þ2 þ ðzi zj Þ2
where ai and aj are the included angles at the center of respective cylindrical surfaces viewing each other. R is the distance between two infinitesimal elements. Substituting, the above coordinates and putting in the limits of integration;
Fx ¼
1 2 p Ai
Z Z Ci
ln R dxi dxj Cj
It results in, (p=2 p=2 p 2 2 ðr=LÞ X X þ cos aj þ sin ai sin aj Fx ¼ ln cos ai 2 4p r p=2 p=2
p=2 X p=2 p 2 X sin ai sin aj dai daj þ þ cos aj ln cos ai r p=2 p=2 ) 2 # 2 L þ sin ai sin aj þ sin ai sin aj dai daj ð5Þ r
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Similarly, ‘Fy’ is obtained as, ( p=2 p=2 p 2 2 ðr=LÞ X X þ cos Fy ¼ ln cos a a þ sin a sin a i j i j 4p2 r p=2 p=2 p=2 X p=2 2 p X cos ai cos aj dai daj þ ln cos ai þ cos aj r p=2 p=2 ) 2 # 2 L þ sin ai sin aj þ cos ai cos aj dai daj ð6Þ r
0.12
0.1
Fi-j
0.08
0.06
These Eqs. (5) and (6) were evaluated numerically. Analytical expression was possible for,
Fz ¼
1 2pAi
Cj
0.04
ln R dzi dzj ;
Z Z
L=Infinity, Eq. (8) L/d=1 E+08 L/d=1000 L/d=875 L/d=750 L/d=625 L/d=500 L/d=375 L/d=250 L/d=125 L/d=62.5 L/d=25 L/d=12.5
0.02
Ci
The expression for Fz, simplified for two parallel cylindrical rods, is as given below.
! ! ! 1
X2 1 þ X2 1 þ W2 þ l ln þ WZ ln Fz ¼
XY ln 4p2 1 þ X2 1 þ W2 W2
L L
þ 4Y tan1 ð7Þ 4Z tan1 p a
0 0
2
4
6
8
10
12
14
16
p/d Fig. 4. View factor between two cylinders of same radius and finite length.
F Infinite ¼
"rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #
p2 p 1 d 1 þ sin d p d p 1
ð8Þ
where
a¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 þ 4r 2 ;
p X¼ ; L
p Y¼ ; r
a W¼ ; L
a Z¼ ; r
l¼
L r
2.2. Results and discussion The results of Eqs. (5)–(7) obtained, for finite length cylinders, were compared with those available in standard texts (Modest, 1993; Siegel and Howell, 1972; Howell, 1982) for view factor between two cylinders of same diameter and infinite length, as given in Eq. (8). Where, p is the pitch/center to center distance between two cylinders and d as diameter of cylinders.
The values of view factor between two cylindrical rods for various pitch–diameter ratios and for some finite lengths are as shown in Table 1. In case of finite length cylindrical rods if infinite length–view factor values are used it may cause major deviation from realistic results. Fig. 4 shows that, view factor between two cylindrical rods decreases with increase in pitch–diameter ratios. Higher p/d ratio demands more number of rods to enclose/encircle the ith rod, which reduces rod to rod view factor. It is also due to the fact that, view factor is inversely proportional to the square of the distance between their surface elements. It is also seen that, view factor increases with increase in its length to diameter ratio and approaches to that of infinite length rods, at finite length–diameter
Table 1 View factor between two cylindrical rods for finite length. L/d p/d
12.5
25
62.5
125
250
500
750
1000
1E+08
Infinite L
1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15
0.07185 0.05261 0.0404 0.03212 0.02621 0.02183 0.01848 0.01586 0.01376 0.01205 0.01064 0.00946 0.00841 0.00763 0.00686 0.00627 0.0057 0.00524 0.0048 0.00445 0.0041 0.00381 0.00354 0.00331 0.00308 0.00289 0.00271 0.00255
0.08131 0.06108 0.04806 0.03911 0.03263 0.02775 0.02397 0.02096 0.01851 0.0165 0.01481 0.01338 0.01208 0.01109 0.01012 0.00936 0.00861 0.00802 0.00742 0.00694 0.00647 0.00607 0.00568 0.00536 0.00504 0.00476 0.00449 0.00426
0.088867 0.068035 0.054523 0.045154 0.038318 0.033132 0.029073 0.025817 0.023152 0.020933 0.019058 0.017456 0.015986 0.014865 0.013746 0.012866 0.011989 0.011281 0.010576 0.009996 0.009419 0.008936 0.008456 0.008049 0.007643 0.007296 0.00695 0.00665
0.09206 0.07103 0.05735 0.04784 0.04089 0.0356 0.03145 0.02811 0.02537 0.02308 0.02114 0.01948 0.01804 0.01678 0.01567 0.01469 0.01381 0.01302 0.01231 0.01166 0.01107 0.01053 0.01004 0.00958 0.00916 0.00877 0.00841 0.00807
0.09394 0.0728 0.05904 0.04947 0.04245 0.03711 0.03291 0.02953 0.02675 0.02443 0.02246 0.02077 0.0193 0.01802 0.01689 0.01588 0.01498 0.01417 0.01343 0.01277 0.01216 0.0116 0.01109 0.01062 0.01019 0.00978 0.0094 0.00905
0.095016 0.073833 0.060032 0.050421 0.043377 0.038008 0.033787 0.030385 0.027587 0.025247 0.023261 0.021555 0.020074 0.018777 0.017632 0.016614 0.015702 0.014881 0.014138 0.013463 0.012847 0.012282 0.011762 0.011283 0.010839 0.010427 0.010043 0.009686
0.09542 0.07421 0.0604 0.05078 0.04372 0.03835 0.03412 0.03071 0.0279 0.02556 0.02357 0.02186 0.02037 0.01907 0.01792 0.0169 0.01598 0.01516 0.01441 0.01373 0.01311 0.01255 0.01202 0.01154 0.0111 0.01068 0.0103 0.00994
0.095626 0.074418 0.060596 0.050967 0.043909 0.038527 0.034294 0.030882 0.028074 0.025725 0.023731 0.022018 0.020531 0.019227 0.018076 0.017051 0.016134 0.015308 0.014561 0.013881 0.01326 0.012691 0.012167 0.011683 0.011236 0.01082 0.010433 0.010072
0.096376 0.075143 0.061301 0.051655 0.044581 0.039186 0.034942 0.031519 0.028702 0.026344 0.024342 0.022622 0.021127 0.019817 0.01866 0.01763 0.016707 0.015876 0.015123 0.014439 0.013813 0.01324 0.012712 0.012225 0.011773 0.011354 0.010963 0.010599
0.110696 0.081376 0.064555 0.05356 0.04579 0.04000 0.035516 0.031938 0.029018 0.026588 0.024534 0.022775 0.021252 0.01992 0.018746 0.017702 0.016769 0.015929 0.015169 0.014479 0.013848 0.013271 0.012739 0.012249 0.011795 0.011373 0.010981 0.010614
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ratio of the order of 1E+08. It is attributed to the increased values of local view factor between the respective ith surface elements and the finite jth surface.
Case 1: For (0 < x/r < 1)
xi ¼ p r cos a1 ;
xj ¼ r cos a1 ; yi ¼ r sin a1 ; 1 r x yj ¼ r sin a1 a1 ¼ sin ; r qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 A ¼ ðp 2r cos a1 Þ2 & B ¼ ðp r cos a1 Þ2 þ ðr sin a1 rÞ
3. View factor between two cylindrical rods with interference The situation of two cylindrical rods with interference from third rod occurs in nuclear reactors, where a bundle of rods is closely packed. Interference or partial obstruction of the view between two geometries significantly alters their view factor value and hence the radiative heat exchange.
Case 2: For (2 > x/r > 1)
xi ¼ p r cos a2 ;
xj ¼ r cos a2 ; yi ¼ r sin a2 ; 1 x r ; yj ¼ r sin a2 ; a2 ¼ sin rqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 A ¼ ðp 2r cos a2 Þ2 & B ¼ ðp r cos a2 Þ2 þ ðr sin a2 þ rÞ
3.1. Approach to the problem
Case 3: For (x/r = 1)
As discussed earlier, during evaluation of view factor between two circular cylinders i and j, semi-cylindrical surface areas of both are considered as the maximum view areas. The surface areas influenced by the interference of third cylindrical rod k, are shown by dark contours, as shown in Fig. 5. Let, x is the interference distance (due to third rod), r is the radius of the rods & x/r represents the ‘Interference ratio’, (0 < x/ r 6 2) For the case with (x/r < 1), the contour-coordinates are specified as follows:
xi ¼ p r cos a1 ;
a1 ¼ sin1
xj ¼ r cos a1 ;
a1 ¼ a2 ¼ 0; A ¼ ðp 2rÞ & B ¼ 3.2. Results and discussion
The total view factor values (i.e. sum of Fx, Fy and Fz) obtained, for different values of interference, x (mm) are presented in Table 2. It is seen that with increase in interference, view factor values reduce for the range of p/d considered. It is also observed that with increase in pitch–diameter ratios (p/d) and interference ratios (x/r), view factor between two interfered geometries decreases and the variation is less in case of higher pitch–diameter ratios. The contour integral expressions can be used for interference ratio, ‘x/r’ ranges from 0 to 2. These are checked for half interference (x/r = 1), for cases with more than 50% interference and also for the limiting case of complete interference i.e. (1 < x/r < 2). Table 4 shows these results for a constant pitch to diameter ratio (p/d = 3.4033) and length to diameter ratio (L/d = 84). The view factor values obtained using above analysis are compared with the algebraic expression derived using cross string method for the present case. According to cross string method, view factor between two infinitely long cylinders in case of interference, is obtained as,
rx ; r
yi ¼ r sin a1 ; yj ¼ r sin a1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼ ðp r cos a1i r cos a1i Þ2 þ ðr sin a1i r sin a1j Þ2 þ ðzi zj Þ2 The integral expressions for Fx and Fy as discussed in previous case were solved numerically. The analytical expression derived, solving for Fz, is as given below;
! ! 1
X2 1 þ X2 F zinterference ¼ þ 2l ln
2XY ln 8p2 1 þ X2 1 þ W2 !
1 þ W2 1 L 1 L þ 8Y tan 8Z tan þ 2WZ ln p a W2 !! 2 2 ðX þ 1ÞðV þ 1Þ þ 2TU lnðT 2 þ 1Þ þ l ln ðT 2 þ 1Þ2 VS lnðV 2 þ 1Þ XY ln X 2 þ 1 þ 2XY lnðXÞ
1 þ 2SV lnðVÞ 4TU lnðTÞ þ 4 ðYÞ tan1 X
1 1
þ ðSÞ tan1 ð2UÞ tan1 ð9Þ
V T
F cs ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 þ 4r 2 ;
a¼
Z ¼ a=r
V ¼ A=L;
X ¼ p=L;
S ¼ A=r;
Y ¼ p=r;
T ¼ B=L;
1 4p r
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p 2r ðp þ Lp Þ 2 p2 4r 2 þ 4r cos1 p 2 ð10Þ
where
(rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) p 2 2 Lp ¼ 2 r sin n þ ðx r þ r cos nÞ þ rn & 2 0 0 11
p B 2ðr xÞ r B CC þ cos1 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAA n ¼ @tan1 p 2 2 2 ðr xÞ þ ðp=2Þ
where
l ¼ L=r;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðp r Þ2 þ r2
W ¼ a=L;
U ¼ B=r
-Y
‘j’
−α2
α1
r
∫
L 0
∫
0
∫
L
0 L
− α
α1 = α 2 = 0 ; x r = 1
∫
0 L
∫
x
α1 = π 2
‘k’ p
α1
0
‘i’ 2
X
L
∫
L o
Fig. 5. View factor between two rods in case of interference.
α1 = π 2
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Table 2 View factor results for various interference ratios (x/r), at L/d = 84.03. L/d = 84.03 p/d
x/r = 0.168
x/r = 0.252
x/r = 0.336
x/r = 0.504
x/r = 0.672
x/r = 0.840
1.51 2.06 2.68 3.40 4.20 5.08 6.05 7.10 8.23 9.45 10.75 12.14 13.61 15.16 16.80 18.53 20.33 22.22 24.20 26.26 28.40 30.63 32.94 35.33 37.81
0.088517 0.066655 0.051197 0.040121 0.032019 0.025964 0.021344 0.017755 0.01492 0.012648 0.010805 0.009292 0.008039 0.006992 0.00611 0.005362 0.004723 0.004176 0.003703 0.003294 0.002938 0.002627 0.002354 0.002115 0.001904
0.087171 0.065717 0.050494 0.039569 0.031571 0.025591 0.021028 0.017482 0.014681 0.012438 0.010618 0.009125 0.007889 0.006856 0.005986 0.005249 0.00462 0.004081 0.003616 0.003214 0.002864 0.002559 0.002291 0.002057 0.00185
0.0853 0.064448 0.049565 0.038853 0.031 0.025123 0.020636 0.017149 0.014394 0.012188 0.010398 0.008931 0.007715 0.0067 0.005846 0.005122 0.004505 0.003976 0.003521 0.003126 0.002784 0.002485 0.002224 0.001994 0.001792
0.079934 0.060893 0.047012 0.036921 0.029482 0.023897 0.019625 0.0163 0.013673 0.011567 0.009859 0.008458 0.007299 0.006331 0.005517 0.004827 0.00424 0.003737 0.003304 0.00293 0.002605 0.002322 0.002075 0.001858 0.001667
0.07246 0.055997 0.043539 0.034323 0.027465 0.022286 0.018311 0.01521 0.012755 0.010785 0.009187 0.007876 0.00679 0.005884 0.005122 0.004477 0.003928 0.003457 0.003053 0.002704 0.002401 0.002137 0.001907 0.001705 0.001528
0.063123 0.049848 0.039186 0.031082 0.024962 0.020299 0.0167 0.013881 0.011643 0.009845 0.008384 0.007184 0.00619 0.00536 0.004662 0.004071 0.003568 0.003138 0.002768 0.002449 0.002172 0.001931 0.001721 0.001537 0.001376
Table 3 Comparison with infinite length interference rod view factor results. p/d
3.4033
x/r
L/d
Fzinterference
Fcs
0.1680
80.03 168.06 252.10 420.16 840.33 1E+08
0.04012068 0.0423317 0.04319233 0.04395824 0.044608 0.045436384
0.046446
0.3361
80.03 168.06 252.10 420.16 840.33 1E+08
0.03885308 0.04111883 0.04200251 0.04278988 0.04345866 0.044313024
0.0444483
0.5042
80.03 168.06 252.10 420.16 840.33 1E+08
0.0369208 0.0391882 0.040074 0.040864 0.0415357 0.042395239
0.041111
ever, with increase in length of rods the view factor values approach the values obtained by cross string method. Cross string method is used as a tool just to check whether values obtained by contour integral formula are realistic or not. 4. View factor between a finitely long cylinder and a nonconcentric enclosure
The view factor formula derived using contour integral technique for interference rod case, agrees well with cross string method, as shown in Table 3. It is seen that with increase in interference ratio, view factor between two cylinders i and j decreases. HowTable 4 View factor values for (2 > x/r > 1) and (x/r = 1) obtained from contour integral expressions. x/r
Fzinterference
p/d = 3.4
1 1.176
0.02744 0.02285
L/d = 84
1.345 1.513 1.681 1.849 2
0.01802 0.01286 0.00755 0.00244 1.3E16 (zero)
View factor evaluation between a cylinder of finite length and non-concentric cylindrical enclosure (an arc) for radiation heat transfer is a common case in nuclear reactors (between hot fuel rod and a pressure tube/sink surface) during loss of coolant situation, in heat exchangers (between hot pipes and the outer cylindrical shell) and in solar collectors, between a coolant tube and an evacuated cylindrical cover etc. The view factor for this case is presented in the current section. The problem is formulated as shown in Fig. 6. The following nomenclature is used for the same.a is an included angle at the center of a cylindrical rod by its view surface, ðp=2Þ 6 a 6 ðp=2Þ.a1 & a2 are the included angles at the center of sink surface/enclosure by its extremity. The contour integral expressions yield better results for the ranges of angles, as given below.
a1;min ¼ sin1 ðr=R1 Þ and a1;max ¼ cos1 ½ðR2 þ rÞ=R1 a2;min ¼ cos1 ½ðR2 rÞ=R1 and a2;max ¼ p sin1 ðr=R1 Þ b and d are the included angles by additional view areas of a rod (i.e. beyond the range of angle a),
0
1
r B C b ¼ cos1 @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 2 2 ðR1 cos a1 R2 Þ þ ðR1 sin a1 Þ
R1 sin a1 ½0 6 b 6 ðp=2Þ tan1 R1 cos a1 R2 d ¼ cos1 ðr=R1 Þ p þ a2
0 6 d 6 cos1 ðr=R1 Þ p þ a2;max
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S.B. Bopche, A. Sridharan / Annals of Nuclear Energy 36 (2009) 1681–1688
where
+Y
∫
α2
Aj
0 L
R2 R1
∫
L
+α −α
0
α1 δ
β
+X
∫
L
∫
Ai
0
-X
0
L
Fig. 6. View factor between cylinder and a non-concentric cylindrical enclosure.
R1 is radius of cylindrical enclosure/sink/pressure tube, R2 is the offset distance between rod and sink surface centers (e) and r is the radius of cylindrical rod. Thus, the total required view factor values for Fx and Fy, were obtained numerically using FORTRAN compiler and the analytical expression obtained for view factor component, Fz is given in Eq. (11).
Fz ¼
! !! ! 2 2 1
ðL=CÞ þ 1 ðL=JÞ þ 1 E2 þ ln þ l ln ln ðL=EÞ2 þ 1
2 2 2
rL 8p ðL=EÞ þ 1 ðL=HÞ þ 1 2
C rL
!
2
ln ðL=CÞ þ 1 þ 2l ln
2
CJ H þ rL EH
!
2
ln ðL=HÞ þ 1
!
J2 C L E L 2 ln ðL=JÞ þ 1 þ 4 tan1 tan1 rL r C r E
2
þ ln
ðL=qÞ þ 1 ðL=pÞ2 þ 1
4.1. Results and discussion The view factor results are compared with standard view factor formula for infinitely long lengths of rod and a cylindrical enclosure available in view factor catalog (Howell, 1982) as given by Eq. (12) for Fig. 7.
F infinite;12 ¼
!
2
J H L 1
ðL=mÞ þ 1 1 L 1
tan tan þ þ
l ln 2 r J r H 8p2 ðL=nÞ þ 1 !!
l ¼ L=r qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C ¼ ðR2 þ r R1 cos a1 Þ2 þ ðR1 sin a1 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E ¼ ðR2 r R1 cos a1 Þ2 þ ðR1 sin a1 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H ¼ ðR2 þ r R1 cos a2 Þ2 þ ðR1 sin a2 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J ¼ ðR2 r R1 cos a2 Þ2 þ ðR1 sin a2 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m ¼ ðR2 þ r cos b R1 cos a1 Þ2 þ ðr sin b R1 sin a1 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ¼ ðR2 þ r R1 cos a1 Þ2 þ ðR1 sin a1 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ¼ ðR2 þ r cos b R1 cos a2 Þ2 þ ðr sin b R1 sin a2 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ¼ ðR2 þ r R1 cos a2 Þ2 þ ðR1 sin a2 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R ¼ ðR2 r cos d R1 cos a1 Þ2 þ ðr sin d R1 sin a1 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S ¼ ðR2 r R1 cos a1 Þ2 þ ðR1 sin a1 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T ¼ ðR2 r cos d R1 cos a2 Þ2 þ ðr sin d R1 sin a2 Þ2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U ¼ ðR2 r R1 cos a2 Þ2 þ ðR1 sin a2 Þ2
n2 2 þ ln ðL=nÞ þ 1 rL
1 a2 a1 1þE a2 tan þ tan1 2p 1E 2 2
1 þ E a 1 tan ; E ¼ e=R1 tan1 1E 2
ð12Þ
The total view factor values obtained, for the above discussed cylindrical geometries of finite length, were compared with that of Eq. (12), as shown in Fig. 8. It was observed that with increase
2 2 m mq p 2 2 ln ðL=mÞ þ 1 þ 2l ln ln ðL=pÞ þ 1 þ rL rL np
A2
2
q m L n L 2 ln ðL=qÞ þ 1 þ 4 tan1 tan1 rL r m r n
!
2 q L p L
1
ðL=RÞ þ 1 tan1 tan1 þ l ln þ
2 r q r p 8p2 ðL=SÞ þ 1
þ ln
!! 2 ðL=U Þ þ 1 2
ðL=TÞ þ 1
α2
!
þ
R1
S2 R2 2 2 lnððL=SÞ þ 1Þ ð Þ lnððL=RÞ þ 1Þ rL rL
! !
RU T2 U2 2 2 ln ðL=TÞ þ 1 lnððL=UÞ þ 1Þ þ þ 2l ln rL rL ST
α1
R L S L U L tan1 tan1 þ tan1 þ4 r R r S r U
T L
tan1
r T
A1
e ð11Þ Fig. 7. Infinitely long cylinder and non-concentric cylindrical enclosure.
1688
S.B. Bopche, A. Sridharan / Annals of Nuclear Energy 36 (2009) 1681–1688
α1=20°,α2=100° Infinite 'L' L/d=1E+08 F i -j
L/d=420.17 L/d=252 L/d=168.07 L/d=84.03 L/d=42.02 L/d=8.40 SR=[R1/(R2+r)] Fig. 8. View factor between cylindrical rod and a sink surface (an arc).
Computational time t, execution time in seconds
Contour Integral Technique
5. Conclusions To evaluate radiative heat exchange between the various three dimensional geometries, accurate values of view factor can be obtained by using the contour integral technique. In the present paper, exact expressions for diffused radiation view factor between two cylindrical rods/pipes, of equal diameter and same finite length, between two cylindrical pipes/rods in case of (partial or complete) interference and between cylindrical source and sink surface have been presented. These expressions can be applied for the surfaces maintained at different and uniform temperatures. The reciprocity relation can be used to evaluate view factor between surface j and surface i. Care has to be taken while evaluating radiative heat transfer area of the respective surface. The main findings of this work are: In addition to spacing between two radiative surfaces, length of the geometry also influences the view factor results significantly. Use of infinite length formulae for finite length situations can lead to significant errors. A contour integral technique developed by E.M. Sparrow is proved to be the simplest, effective and accurate tool for view factor evaluation between three dimensional geometries.
References
n, number of segments of contours Fig. 9. Computational time for contour integral method.
in length of the geometries, view factor between them increases and approach to the infinite length–view factors at L/d = 1E+08 that lie within ±5% deviation. For increasing spacing ratios it shows a decreasing trend for almost all L/d ratios. The cases discussed in this paper are applicable for radiative heat exchange in a nuclear fuel bundle during loss of coolant situation. The computational time, t on an average for the above discussed finite length geometries are plotted with respect to the number of divisions, n of the contours. It is shown in Fig. 9. Computation is done using a FORTRAN code with LINUX as an operating system.
Ambirajan, A., Venkateshan, S.P., 1993. Accurate determination of diffuse view factors between planer surfaces. Int. J. Heat Mass Transfer 36 (8), 2203–2208. Byrd, L.W., 1993. View factor algebra for two arbitrarily sized non-opposing parallel rectangles. ASME J. Heat Transfer 115, 517–518. Carlson, R.W., Garcia, J., 1984. View factor for radiant heat transfer between the wall and end of a cylinder. Ann. Nuclear Energy 11 (4), 187–196. Chung, T.J., Kim, J.Y., 1982. Radiation view factors by finite elements. ASME J. Heat Transfer 104, 792–795. Hahne, E., Bassiouni, M.K., 1980. The angle factor for radiant interchange within a constant radius cylindrical enclosure. Lett. Heat Mass Transfer 7, 303–309. Howell, J.R., 1982. A Catalogue of Radiation Configuration Factors. McGraw Hill, New York. Jeffrey, Alan, 1995. Handbook of Mathematical Formulas and Integrals. Academic Press. Juul, N.H., 1982. View factors in radiation between two parallel oriented cylinders of finite lengths. ASME J. Heat Transfer 104, 384–388. Modest, M.F., 1993. Radiative Heat Transfer. Academic Press. Rao, V.R., Sastri, V.M.K., 1996. Efficient evaluation of diffuse view factors for radiation. Int. J. Heat Mass Transfer 39 (6), 1281–1286. Shapiro, A.B., 1985. Computer implementation, accuracy, and timing of radiation view factor algorithms. ASME J. Heat Transfer 107, 730–732. Siegel, R., Howell, J.R., 1972. Thermal Radiation Heat Transfer. McGraw Hill, New York. Sparrow, E.M., 1963. A new and simpler formulation for radiative angle actors. ASME J. Heat Transfer 81, 81–87. Sparrow, E.M., Cess, R.D., 1977. Radiation Heat Transfer. McGraw Hill. Sparrow, E.M., Jonsson, V.K., 1963. Angle factors for radiant interchange between parallel-oriented tubes. ASME J. Heat Transfer 85, 382–384.