Abstract

A general formula for calculating radiative fluxes from point sources of radiation incident on spherical objects was derived using some fundamental laws of classical radiometry. This formula was derived in the Cartesian coordinate system, 0xyz, where the coordinates, x, y, and z, determine the position of the spherical object with respect to the point source. The obtained solution was dependent on the radius of the object, and on the function describing the intensity of the radiation. A specific solution for calculating fluxes of isotropic radiations was presented and selected calculations were illustrated graphically.

© 2004 Optical Society of America

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References

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  1. G. Brescia, R. Moreira, L. Braby and E. Castell-Perez, �??Monte Carlo simulation and dose distribution of low energy electron irradiation of an apple,�?? J. Food Engin. 60, 31-39 (2003).
    [CrossRef]
  2. C. Sasse, K. Muinonen, J. Piironen, and G. Dröse, �??Albedo measurements on single particles,�?? J. Quant. Spectrosc. Radiat. Transfer 55, 673-681 (1996).
    [CrossRef]
  3. R. Sommer, A. Cabaj, T. Sandu, and M. Lhotsky, �??Measurments of UV radiation using suspension of microorganisms,�?? J. Photochem. Photobiol. B 53, 1-6 (1999).
    [CrossRef]
  4. J.F. Diehl, �??Food irradiation-past, present and future,�?? Radiat. Phys. Chem. 63, 211-215 (2002).
    [CrossRef]
  5. R. W. Durante, �??Food processors requirements met by radiation processing,�?? Radiat. Phys. Chem. 63, 289-294 (2002).
    [CrossRef]
  6. G. W. Gould, �??Potential of irradiation as a component of mild combination preservation procedures,�?? Radiat. Phys. Chem. 48, 366 (1996).
    [CrossRef]
  7. C. M. Bruhn, �??Consumer acceptance of irradiated food: theory and reality,�?? Radiat. Phys. Chem. 52, 129-133 (1998).
    [CrossRef]
  8. M. Strojnik and G Paez, �??Radiometry,�?? in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds., (Marcel Dekker, New York, 2001), pp. 649-699.
  9. J. Tallarida, Pocket Book of Integrals and Mathematical Formulas, (CRC, Boca Raton, 1999), Chpt. 4.
    [CrossRef]
  10. S. Wolfram, Mathematica-A System for Doing Mathematics by Computer (Addison-Wesley, Reading, Mass. 1993), pp. 44-186.

Handbook of Optical Engineering (1)

M. Strojnik and G Paez, �??Radiometry,�?? in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds., (Marcel Dekker, New York, 2001), pp. 649-699.

J. Food Engin. (1)

G. Brescia, R. Moreira, L. Braby and E. Castell-Perez, �??Monte Carlo simulation and dose distribution of low energy electron irradiation of an apple,�?? J. Food Engin. 60, 31-39 (2003).
[CrossRef]

J. Photochem. Photobiol. B (1)

R. Sommer, A. Cabaj, T. Sandu, and M. Lhotsky, �??Measurments of UV radiation using suspension of microorganisms,�?? J. Photochem. Photobiol. B 53, 1-6 (1999).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

C. Sasse, K. Muinonen, J. Piironen, and G. Dröse, �??Albedo measurements on single particles,�?? J. Quant. Spectrosc. Radiat. Transfer 55, 673-681 (1996).
[CrossRef]

Radiat. Phys. Chem. (4)

J.F. Diehl, �??Food irradiation-past, present and future,�?? Radiat. Phys. Chem. 63, 211-215 (2002).
[CrossRef]

R. W. Durante, �??Food processors requirements met by radiation processing,�?? Radiat. Phys. Chem. 63, 289-294 (2002).
[CrossRef]

G. W. Gould, �??Potential of irradiation as a component of mild combination preservation procedures,�?? Radiat. Phys. Chem. 48, 366 (1996).
[CrossRef]

C. M. Bruhn, �??Consumer acceptance of irradiated food: theory and reality,�?? Radiat. Phys. Chem. 52, 129-133 (1998).
[CrossRef]

Other (2)

J. Tallarida, Pocket Book of Integrals and Mathematical Formulas, (CRC, Boca Raton, 1999), Chpt. 4.
[CrossRef]

S. Wolfram, Mathematica-A System for Doing Mathematics by Computer (Addison-Wesley, Reading, Mass. 1993), pp. 44-186.

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Figures (4)

Fig. 1.
Fig. 1.

An intersection of a spherical object with the center 01 at a distance r from the origin 0 of the main coordinate system, 0xyz, erected at point source P. The additional coordinate system, 01 x 1 y 1, is erected at the center 01 of the spherical object. The aperture, A, at the distance, d, from the source P limits the width, ρ, of the radiation cross-section in the plane z=constant at zd.

Fig. 2.
Fig. 2.

Geometrical illustration of the transformation described by Eqs. (5)-(6) and definition of parameters used in Eq. (7) to determine the ellipse made by the external contour of the shadow of the spherical object with the center 01 on the 0′xy plane.

Fig. 3.
Fig. 3.

Definition of geometrical parameters used for obtaining Eqs. (9)(11).

Fig. 4.
Fig. 4.

The relative value of the total radiative flux, ΦP (x,y,z)/ΦP(x=0,y=0,z), as a dependency on x and y at z=100, R=1, and ρ=100 (a), and as a dependency on y and z at x=0, R=1, and ρ=100 (b). All geometrical variables (x, y, z, R, and ρ) are expressed in relative units.

Equations (21)

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d Φ P ( x , y , z ) = I ( x , y , z ) d ω P ( x , y , z ) ,
d ω P ( x , y , z ) = d S ( x , y , z ) cos θ r 2 = z d x d y ( x 2 + y 2 + z 2 ) 3 2 ,
d Φ P ( x , y , z ) = I ( x , y , z ) z d x d y ( x 2 + y 2 + z 2 ) 3 2 .
d Φ P ( x , y , z ; x 1 , y 1 ) = I ( x , y , z ; x 1 , y 1 ) z d x 1 d y 1 [ ( x x 1 ) 2 + ( y y 1 ) 2 + z 2 ] 3 2 .
x 1 = x 11 cos φ y 11 sin φ ,
y 1 = x 11 sin φ y 11 cos φ ,
( x 11 c 2 a 1 ) 2 a 2 + y 11 2 b 2 = 1 ,
b = R a ,
a = ( a 1 + a 2 ) / 2 = R z r 2 R 2 ,
a 1 = r 2 R z r 2 R 2 + R r 2 z 2 ,
a 2 = r 2 R z r 2 R 2 R r 2 z 2 ,
J = cos φ sin φ sin φ cos φ = 1 ,
d Φ P ( x , y , z ; x 11 , y 11 ) = I ( x , y , z ; x 11 , y 11 ) z d x 11 d y 11 ( x 11 2 2 r xy x 11 + y 11 2 + r 2 ) 3 2 ,
Φ P ( x , y , z ) = { b 2 a ( 2 a 2 b 2 ) a d x 11 y 11 y 11 I ( x , y , z ; x 11 , y 11 ) z d y 11 ( y 11 2 + x 11 2 2 r xy + r 2 ) 3 2 , if r xy ρ 2 a + b 2 a , b 2 a ρ r xy d x 11 y 11 y 11 I ( x , y , z ; x 11 , y 11 ) z d y 11 ( y 11 2 + x 11 2 2 r xy x 11 + r 2 ) 3 2 , if r xy > ρ 2 a + b 2 a , 0 , if r xy > ρ + b 2 a ,
y 11 = b a 2 [ x 11 ( a 2 b 2 ) / a ] 2 / a ,
y 11 = b a 2 [ x 11 ( a 2 b 2 ) / a ] 2 / a ,
4 π r 2 I ( x , y , z ) = 4 π z 2 I ( x = 0 , y = 0 , z ) = 4 π z 2 I z ,
I ( x , y , z ; x 11 , y 11 ) = I z z 2 y 11 2 + x 11 2 2 r xy x 11 + r 2
Φ P ( x , y , z ) = { I z z 3 b 2 a ( 2 a 2 b 2 ) a d x 11 y 11 y 11 d y 11 ( y 11 2 + x 11 2 2 r xy x 11 + r 2 ) 5 2 , if r xy ρ 2 a + b 2 a , I z z 3 b 2 a ρ r xy d x 11 y 11 y 11 d y 11 ( y 11 2 + x 11 2 2 r xy x 11 + r 2 ) 5 2 , if r xy > ρ 2 a + b 2 a , 0 , if r xy > ρ + b 2 a .
Φ P ( x , y , z ) = { I z z 3 b 2 a ( 2 a 2 b 2 ) a f ( x , y , z ; x 11 ) d x 11 , if r xy ρ 2 a + b 2 a , I z z 3 b 2 a ρ r xy f ( x , y , z ; x 11 ) d x 11 , if r xy ρ 2 a + b 2 a , 0 , if r xy > ρ + b 2 a ,
f ( x , y , z : x 11 ) = 2 b 3 b 2 ( 2 a 2 b 2 ) + 2 a ( a 2 b 2 ) x 11 a 2 x 11 2 [ ( r x y x 11 ) 2 + z 2 ] 2 × 4 a 2 b 4 2 b 6 + 3 a 4 r 2 + a [ 4 b 2 ( a 2 b 2 ) 6 a 3 r x y ] x 11 + a 2 ( 3 a 2 2 b 2 ) x 11 2 { b 4 ( 2 a 2 b 2 ) + r 2 a 4 + 2 a [ b 2 ( a 2 b 2 ) a 3 r x y ] x 11 + a 2 ( a 2 b 2 ) x 11 2 } 3 / 2 .

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