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An approach to view factor calculation for radiation transfer simulation in 2D axisymmetric geometries
Nuclear Instruments and Methods in Physics Research A 415 (1998) 127—132
An approach to view factor calculation for radiation transfer simulation in 2D axisymmetric geometries E.G. Vasina*, V.M. Chekshin VNIEF, Russian Federal Nuclear Center, Arzamas-167, 607190 Sarov, Nizhni Novgorod Region, Russia
Abstract An approach to view factor calculation for 2D axisymmetric geometry is proposed with analytical determination of integration range based on building projection of obscuring element on the plane. ( 1998 Elsevier Science B.V. All rights reserved.
One of the methods of radiation transfer calculation is the view factor method [1—5]. Reliable and efficient codes for view factor computation are required for successful implementation of this method. A new approach to the analytical determination of the integration range, which is the most cumbersome part of view factor calculation, is proposed. The approach is implemented in the code for calculation of diffusion view factors in 2D axially symmetric geometry. View factors characterize mutual influence of arbitrarily located surface elements with account of possible shadowing. For isotropic sources view factor is defined by the formula (see, e.g., Ref. [1]])
PP
a " ij
si sj
cos a cos a ds ds i j i j, nr2 ij
* Corresponding author. Fax: #7 83130 54565; e-mail: veg—[email protected].
where S , S are i and j surface elements areas; a , i j i a are the angles between surface normals at certain j points and straight line connecting these points; r is the distance between the points (see, Fig. 1). ij Integration is made over mutually visible surface segments. In 2D axisymmetric geometry integration over angle of rotation around the axis of symmetry is substituted for one of surface elements S or S with multiplication by 2p. So, to determine i j integration range, it is necessary to find surface S segment visible from surface S points. Since the j i determination of integration range is the most cumbersome in view factor calculation, special attention should be paid to development of efficient algorithm for it. Consider regions bounded by surfaces of rotation with straight generating line (cylinders, cones, disks). Exclusion of spheres and toruses from consideration is, in our opinion, reasonable limitation, simplifying the problem, yet providing wide applicability of the developed approach and code. Consider two conic elements i and j in the simplest case of the absence of obscuring elements (Fig. 2).
0168-9002/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 8 ) 0 0 5 0 6 - 3
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E.G. Vasina, V.M. Chekshin/Nucl. Instr. and Meth. in Phys. Res. A 415 (1998) 127—132
Fig. 1. View factor definition.
Fig. 2. Visible segment of conic element.
If none of the pair i, j is disk, the segment of element i visible from element j is determined by an interval over angle of revolution of cone j generatrix such that every point of cone j surface from this interval is visible from cone i generatrix points. In this case visible segment is the same for all points of cone j and cone i generatrices. Cases when one of the elements i, j is a disk, are illustrated in Fig. 3. In these cases visible segments are determined for pairs of rings — elements i, j sub-
divisions along x-axis. The finer the subdivisions, the more accurate integration range determination. In the absence of obscuring elements integration range for a pair of rings n , n is easily found from i j simple geometric considerations. Let us consider a single obscuring element with generatrix LM (see Fig. 4). Here we describe an approach to the determination of integration range for a pair of rings n and n of elements i and j, i i respectively. Built projection of obscuring element from upper n ring point A onto the ring n plane is i j seen (see shaded area in Fig. 4). Obscuring element projection can be of three types: smaller circle contained in the larger one; two intersecting circles; two distant circles (see Fig. 5) (shadowed areas are cross-hatched). Note that, due to axial symmetry, centers of both projected circles lie on the y-axis of n plane. j We now write out the formulas to describe the shadow geometry. Let (x , y , 0) be coordinates of observation point i i A (Fig. 4); x"x — equation of n plane for projection; j j (x , 0, 0) — coordinates of the first projected circle 1 center O ; 1 (x , y , 0) — coordinates of the first projected 1 1 circle upper point ¸; (x , 0, 0) — coordinates of the second projected 2 circle center O ; 2 (x , y , 0) — coordinates of the second projected 2 2 circle upper point M. Coordinates of O@ , projection of O on n plane 1 1 j from point A are (x , y@ , 0), where j 1 y@ "y [x !x ]/[x !x ]. 1 i 1 j 1 i
Fig. 3. (a) Integration range for a pair disk — cone depends on observation point position on the disk. (b) Integration range for a pair cone — disk is determined by straight line on the disk.
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E.G. Vasina, V.M. Chekshin/Nucl. Instr. and Meth. in Phys. Res. A 415 (1998) 127—132
Fig. 4. Determination of integration range for a pair of rings n and n in the presence of one obscuring element. i i
The projected circle radius is easily calculated: r "y [x !x ]/[x !x ]. 1 1 j i 1 i Similarly, for the second circle, we have: O@ coor2 dinates: (x , y@ , 0), where y@ "y (x !x )/(x !x ); j 2 2 i 2 j 2 i r "y (x !x )/(x !x ). 2 2 j i 2 i With projected circles coordinates and radii known, shadow shape is easily found out. In the last two cases (Fig. 5, middle and right) it is also necessary to build a tangent line to the two circles (one tangent line equation is sufficient due to symmetry against y axis). Tangent line hits two tangency points in plane x"x , its equation is: j y!y z!z t1 " t1 . y !y z !z t2 t1 t2 t1 Tangency points coordinates are easily found:
by bold lines in Fig. 4) does not fall into the shaded area. Different cases of n circle — shadow geometry are j possible. For example, n circle is distant from the j shadow, or contained in both projected circles (no shadowing) or circle n lies completely within the j shadowed area (zero visibility) or circle n is parj tially shadowed (see Fig. 4). Determine intersections if n circle with both j projected circles and tangent line (if any). Equation of circle n : j y2#z2"y2; j equation of the first projected circle (y!y@ )2#z2"r2; 1 1
r !r r !r 2 1 , z "r 1! 2 1 ; y "y@ !r 2 1 1 y@ !y@ t1 1 t1 y@ !y@ 1 1 2 2
equation of the second projected circle
r !r r !r 2 1 , z "r 1! 2 1 , y "y@ !r 2 2 2 y@ !y@ t2 2 t2 y@ !y@ 1 1 2 2 where r , r are projected circles radii, y@ , y@ are 1 2 1 2 y-coordinates of the circles centers. Determine shadowing of ring n by one obscurj ing element. Visible segment of the circle n (shown j
The equation of tangent line is given above. If intersection of n circle with one of the projecj ted circles (for example, first) takes place, intersection point C coordinates are 1 y@ 2#y2!r2 1, j y " 1 z "Jy2!y2 ; c1 c1 j c1 2y@ 1
S A S A
B B
(y!y@ )2#z2"r2. 2 2
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E.G. Vasina, V.M. Chekshin/Nucl. Instr. and Meth. in Phys. Res. A 415 (1998) 127—132
Fig. 5. Shape of shadow from one obscuring element on n plane. (left) Smaller circle contained in the larger one. (middle) Two j intersecting circles#tangent lines. (right) Two distant circles#tangent lines.
and angle u value in this point is u"arccos (y /y ). c1 j Similar formulas are obtained for the second circle. Note, that while determining intersection of n circle with tangent line, only line segment bej tween two tangency points should be considered. Coordinates of intersection points of n circle and j straight tangent line are found from solution of the system of equations: y2#z2"y2; j y!y z!z t1 " t1 . y !y z !z t2 t1 t2 t1 The y-coordinates of the intersection points are y "!cd$Jy2(1#c2)!d2, j c1,2 where z !z t1 , d"z !cy . c" t2 t1 1 y !y t2 t1 The corresponding angle u values are:
Fig. 6. One arc of n circle out of five is visible in this case: (/ , j c1 / ). c2
A B
y c1,2 . y j So, a set of angles corresponding to intersection points of the circle n with two projected circles and j tangent line is obtained. / "arccos c1,2
E.G. Vasina, V.M. Chekshin/Nucl. Instr. and Meth. in Phys. Res. A 415 (1998) 127—132
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Fig. 7. Test problem geometry.
Fig. 8. View factor relative error as a function of the element number.
To determine the integration range it is necessary to find out which of the arcs lie beyond the shaded area. Whether the whole arc is visible or not is determined by one arbitrary point of the arc, the position of which relative to the shaded area is checked. Angular interval between 0 and p is considered due to the problem symmetry. Fig. 6 exemplifies the situation: only one arc (/ , / ) is c1 c2 visible out of five: (0, / ), (/ , / ), (/ , / ), (/ , 1 1 c1 c1 c2 c2 / ), (/ , p). 2 2 To obtain shadowing from several obscuring elements, visible segments are found accounting for each obscuring element independently, and then segments obscured by none of them are determined.
In order to minimize the computation time, preliminary selection of potentially obscuring elements is done. Generatrix LM is checked for hitting the triangle formed by point A and upper and lower n ring points (see Fig. 4). In terms of projections it j means nonzero intersection of line segment limited by minimum and maximum y-coordinates of obscuring element projection with line segment (!y , j y ) corresponding to the n circle. j j Let us describe one of the computations carried out to test the accuracy and efficiency of the proposed approach. Computation was carried out with double precision for 87 surface elements; element was subdivided over rotation angle maximum by 100,
II. TARGETS
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E.G. Vasina, V.M. Chekshin/Nucl. Instr. and Meth. in Phys. Res. A 415 (1998) 127—132
along the axis of symmetry by 9; absolute and relative errors of integration were 0.00001 and 0.001, respectively (See Figs. 7 and 8). Computation time was 120 s on IBM PC-486 (66 MHz).
References [1] R. Siegel, J.R. Howell, Thermal Radiation Heat Transfer, Mir, Moscow, 1975, p. 840.
[2] N.N. Chentsov, G.V. Dumkina, N.S. Shoidina, Calculation of radiation view factors in axisymmetric geometry, EPhJ 34 (2) (1978) 306. [3] Yu.A. Dement’ev, R.F. Mashinin, V.F. Mironova, N.N. Chentsov, VANT, Serias Techniques and Codes for Numerical Solution of Mathematical Physics Problems 1 (12) (1983) 32. [4] E. Garelis, T.E. Rudy, R.B. Hickman, J. Quant. Spectrosc. and Radiat. Transfer 34 (5) (1985) 417. [5] Ian Ashdown, Radiosity. A Programmer’s Perspective. Wiley, New York, 1994, 500 c.
An approach to view factor calculation for radiation transfer simulation in 2D axisymmetric geometries E.G. Vasina*, V.M. Chekshin VNIEF, Russian Federal Nuclear Center, Arzamas-167, 607190 Sarov, Nizhni Novgorod Region, Russia
Abstract An approach to view factor calculation for 2D axisymmetric geometry is proposed with analytical determination of integration range based on building projection of obscuring element on the plane. ( 1998 Elsevier Science B.V. All rights reserved.
One of the methods of radiation transfer calculation is the view factor method [1—5]. Reliable and efficient codes for view factor computation are required for successful implementation of this method. A new approach to the analytical determination of the integration range, which is the most cumbersome part of view factor calculation, is proposed. The approach is implemented in the code for calculation of diffusion view factors in 2D axially symmetric geometry. View factors characterize mutual influence of arbitrarily located surface elements with account of possible shadowing. For isotropic sources view factor is defined by the formula (see, e.g., Ref. [1]])
PP
a " ij
si sj
cos a cos a ds ds i j i j, nr2 ij
* Corresponding author. Fax: #7 83130 54565; e-mail: veg—[email protected].
where S , S are i and j surface elements areas; a , i j i a are the angles between surface normals at certain j points and straight line connecting these points; r is the distance between the points (see, Fig. 1). ij Integration is made over mutually visible surface segments. In 2D axisymmetric geometry integration over angle of rotation around the axis of symmetry is substituted for one of surface elements S or S with multiplication by 2p. So, to determine i j integration range, it is necessary to find surface S segment visible from surface S points. Since the j i determination of integration range is the most cumbersome in view factor calculation, special attention should be paid to development of efficient algorithm for it. Consider regions bounded by surfaces of rotation with straight generating line (cylinders, cones, disks). Exclusion of spheres and toruses from consideration is, in our opinion, reasonable limitation, simplifying the problem, yet providing wide applicability of the developed approach and code. Consider two conic elements i and j in the simplest case of the absence of obscuring elements (Fig. 2).
0168-9002/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 9 8 ) 0 0 5 0 6 - 3
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E.G. Vasina, V.M. Chekshin/Nucl. Instr. and Meth. in Phys. Res. A 415 (1998) 127—132
Fig. 1. View factor definition.
Fig. 2. Visible segment of conic element.
If none of the pair i, j is disk, the segment of element i visible from element j is determined by an interval over angle of revolution of cone j generatrix such that every point of cone j surface from this interval is visible from cone i generatrix points. In this case visible segment is the same for all points of cone j and cone i generatrices. Cases when one of the elements i, j is a disk, are illustrated in Fig. 3. In these cases visible segments are determined for pairs of rings — elements i, j sub-
divisions along x-axis. The finer the subdivisions, the more accurate integration range determination. In the absence of obscuring elements integration range for a pair of rings n , n is easily found from i j simple geometric considerations. Let us consider a single obscuring element with generatrix LM (see Fig. 4). Here we describe an approach to the determination of integration range for a pair of rings n and n of elements i and j, i i respectively. Built projection of obscuring element from upper n ring point A onto the ring n plane is i j seen (see shaded area in Fig. 4). Obscuring element projection can be of three types: smaller circle contained in the larger one; two intersecting circles; two distant circles (see Fig. 5) (shadowed areas are cross-hatched). Note that, due to axial symmetry, centers of both projected circles lie on the y-axis of n plane. j We now write out the formulas to describe the shadow geometry. Let (x , y , 0) be coordinates of observation point i i A (Fig. 4); x"x — equation of n plane for projection; j j (x , 0, 0) — coordinates of the first projected circle 1 center O ; 1 (x , y , 0) — coordinates of the first projected 1 1 circle upper point ¸; (x , 0, 0) — coordinates of the second projected 2 circle center O ; 2 (x , y , 0) — coordinates of the second projected 2 2 circle upper point M. Coordinates of O@ , projection of O on n plane 1 1 j from point A are (x , y@ , 0), where j 1 y@ "y [x !x ]/[x !x ]. 1 i 1 j 1 i
Fig. 3. (a) Integration range for a pair disk — cone depends on observation point position on the disk. (b) Integration range for a pair cone — disk is determined by straight line on the disk.
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E.G. Vasina, V.M. Chekshin/Nucl. Instr. and Meth. in Phys. Res. A 415 (1998) 127—132
Fig. 4. Determination of integration range for a pair of rings n and n in the presence of one obscuring element. i i
The projected circle radius is easily calculated: r "y [x !x ]/[x !x ]. 1 1 j i 1 i Similarly, for the second circle, we have: O@ coor2 dinates: (x , y@ , 0), where y@ "y (x !x )/(x !x ); j 2 2 i 2 j 2 i r "y (x !x )/(x !x ). 2 2 j i 2 i With projected circles coordinates and radii known, shadow shape is easily found out. In the last two cases (Fig. 5, middle and right) it is also necessary to build a tangent line to the two circles (one tangent line equation is sufficient due to symmetry against y axis). Tangent line hits two tangency points in plane x"x , its equation is: j y!y z!z t1 " t1 . y !y z !z t2 t1 t2 t1 Tangency points coordinates are easily found:
by bold lines in Fig. 4) does not fall into the shaded area. Different cases of n circle — shadow geometry are j possible. For example, n circle is distant from the j shadow, or contained in both projected circles (no shadowing) or circle n lies completely within the j shadowed area (zero visibility) or circle n is parj tially shadowed (see Fig. 4). Determine intersections if n circle with both j projected circles and tangent line (if any). Equation of circle n : j y2#z2"y2; j equation of the first projected circle (y!y@ )2#z2"r2; 1 1
r !r r !r 2 1 , z "r 1! 2 1 ; y "y@ !r 2 1 1 y@ !y@ t1 1 t1 y@ !y@ 1 1 2 2
equation of the second projected circle
r !r r !r 2 1 , z "r 1! 2 1 , y "y@ !r 2 2 2 y@ !y@ t2 2 t2 y@ !y@ 1 1 2 2 where r , r are projected circles radii, y@ , y@ are 1 2 1 2 y-coordinates of the circles centers. Determine shadowing of ring n by one obscurj ing element. Visible segment of the circle n (shown j
The equation of tangent line is given above. If intersection of n circle with one of the projecj ted circles (for example, first) takes place, intersection point C coordinates are 1 y@ 2#y2!r2 1, j y " 1 z "Jy2!y2 ; c1 c1 j c1 2y@ 1
S A S A
B B
(y!y@ )2#z2"r2. 2 2
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E.G. Vasina, V.M. Chekshin/Nucl. Instr. and Meth. in Phys. Res. A 415 (1998) 127—132
Fig. 5. Shape of shadow from one obscuring element on n plane. (left) Smaller circle contained in the larger one. (middle) Two j intersecting circles#tangent lines. (right) Two distant circles#tangent lines.
and angle u value in this point is u"arccos (y /y ). c1 j Similar formulas are obtained for the second circle. Note, that while determining intersection of n circle with tangent line, only line segment bej tween two tangency points should be considered. Coordinates of intersection points of n circle and j straight tangent line are found from solution of the system of equations: y2#z2"y2; j y!y z!z t1 " t1 . y !y z !z t2 t1 t2 t1 The y-coordinates of the intersection points are y "!cd$Jy2(1#c2)!d2, j c1,2 where z !z t1 , d"z !cy . c" t2 t1 1 y !y t2 t1 The corresponding angle u values are:
Fig. 6. One arc of n circle out of five is visible in this case: (/ , j c1 / ). c2
A B
y c1,2 . y j So, a set of angles corresponding to intersection points of the circle n with two projected circles and j tangent line is obtained. / "arccos c1,2
E.G. Vasina, V.M. Chekshin/Nucl. Instr. and Meth. in Phys. Res. A 415 (1998) 127—132
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Fig. 7. Test problem geometry.
Fig. 8. View factor relative error as a function of the element number.
To determine the integration range it is necessary to find out which of the arcs lie beyond the shaded area. Whether the whole arc is visible or not is determined by one arbitrary point of the arc, the position of which relative to the shaded area is checked. Angular interval between 0 and p is considered due to the problem symmetry. Fig. 6 exemplifies the situation: only one arc (/ , / ) is c1 c2 visible out of five: (0, / ), (/ , / ), (/ , / ), (/ , 1 1 c1 c1 c2 c2 / ), (/ , p). 2 2 To obtain shadowing from several obscuring elements, visible segments are found accounting for each obscuring element independently, and then segments obscured by none of them are determined.
In order to minimize the computation time, preliminary selection of potentially obscuring elements is done. Generatrix LM is checked for hitting the triangle formed by point A and upper and lower n ring points (see Fig. 4). In terms of projections it j means nonzero intersection of line segment limited by minimum and maximum y-coordinates of obscuring element projection with line segment (!y , j y ) corresponding to the n circle. j j Let us describe one of the computations carried out to test the accuracy and efficiency of the proposed approach. Computation was carried out with double precision for 87 surface elements; element was subdivided over rotation angle maximum by 100,
II. TARGETS
132
E.G. Vasina, V.M. Chekshin/Nucl. Instr. and Meth. in Phys. Res. A 415 (1998) 127—132
along the axis of symmetry by 9; absolute and relative errors of integration were 0.00001 and 0.001, respectively (See Figs. 7 and 8). Computation time was 120 s on IBM PC-486 (66 MHz).
References [1] R. Siegel, J.R. Howell, Thermal Radiation Heat Transfer, Mir, Moscow, 1975, p. 840.
[2] N.N. Chentsov, G.V. Dumkina, N.S. Shoidina, Calculation of radiation view factors in axisymmetric geometry, EPhJ 34 (2) (1978) 306. [3] Yu.A. Dement’ev, R.F. Mashinin, V.F. Mironova, N.N. Chentsov, VANT, Serias Techniques and Codes for Numerical Solution of Mathematical Physics Problems 1 (12) (1983) 32. [4] E. Garelis, T.E. Rudy, R.B. Hickman, J. Quant. Spectrosc. and Radiat. Transfer 34 (5) (1985) 417. [5] Ian Ashdown, Radiosity. A Programmer’s Perspective. Wiley, New York, 1994, 500 c.