Spherical object in radiation field from a point source

Stanislaw Tryka

doi:10.1364/OPEX.12.000512

1. Introduction

Many natural objects and industrial products of spherical shape are irradiated in nature by non-ionizing and ionizing radiation from a single point source or from many point sources. For example, non-ionizing and ionizing radiation from distant stars continuously irradiates planets, planetoids and moons, while a such radiation from non-natural point sources may be used for irradiation of fruits, vegetables and seeds, various microorganisms as well as molecules and atoms when they may be considered as spherical bodies [1–3]. In addition, a variety of spherical products may be irradiated by artificial point sources of radiation [4–7].

Treatment of some foods by non-ionizing and ionizing radiation have shown very promising results in destroying harmful pathogens and extending shelf life of some food products without any detectable physical and chemical changes. Such methods of irradiation have been successfully tested in laboratories and should find wide application in the future [1, 3–7]. Therefore, to study the effect of this radiation on irradiated objects quantitatively accurate analytical methods for estimating the radiative fluxes are necessary.

The main goal of this research was to derive an analytical formula for calculating radiative fluxes emitted by various point source models into a space subtended by a spherical object. As an example, selected results for an isotropic point source in various configurations with respect to the spherical object were calculated and illustrated graphically by three-dimensional surface plots.

2. Basic equations

An infinitesimal element of the radiative flux, dΦ_P(x, y, z), emitted by a point source, P, located at the point P(0, 0, 0), in the direction determined by the Cartesian coordinates x, y and z is defined by the expression [8]

d Φ_{P} (x, y, z) = I (x, y, z) d ω_{P} (x, y, z),

where I(x, y, z) is the intensity of the radiation (radiant intensity) emitted by the source P and dω_P(x, y, z) is the infinitesimal element of the solid angle subtended by the surface element dS(x, y, z) at the point P(0, 0, 0). The infinitesimal solid angle is defined as [8]

d ω_{P} (x, y, z) = \frac{d S (x, y, z) \cos θ}{r^{2}} = \frac{z d x d y}{{(x^{2} + y^{2} + z^{2})}^{3 ⁄ 2}},

where r ²=x ²+y ²+z ², and θ is the angle between r and the normal to the surface element dS(x, y, z). Substituting Eq. (2) into (1) we get

d Φ_{P} (x, y, z) = I (x, y, z) \frac{z d x d y}{{(x^{2} + y^{2} + z^{2})}^{3 ⁄ 2}} .

For the center 01 of the spherical object, at the distance z from the origin 0 of the coordinate system 0xyz, we can introduce an additional coordinate system 0₁ x ₁ y ₁ z as shown in Fig. 1. In these coordinates, Eq. (3) becomes

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 z≥d.

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d Φ_{P} (x, y, z; x_{1}, y_{1}) = I (x, y, z; x_{1}, y_{1}) \frac{z d x_{1} d y_{1}}{{[{(x - x_{1})}^{2} + {(y - y_{1})}^{2} + z^{2}]}^{3 ⁄ 2}} .

It is easy to see, that after integration of Eq. (4) with respect to the variables x ₁ and y ₁ it is possible to obtain expressions for calculating radiative flux as function of the x, y, and z coordinates determining the center 0₁ of the spherical object.

Transforming coordinates x ₁ and x ₂ as

x_{1} = x_{11} \cos φ - y_{11} \sin φ,

y_{1} = x_{11} \sin φ - y_{11} \cos φ,

we can obtain an equation of the ellipse for an external contour of a shadow made by the spherical object on the 0₁ x ₁₁ y ₁₁ plane (see Fig. 2) given by [9]

\frac{{(x_{11} - c^{2} a^{- 1})}^{2}}{a^{2}} + \frac{y_{11}^{2}}{b^{2}} = 1,

where a and b denote half-lengths of the major and minor ellipse axes, respectively, and c ²=a ²-b 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.

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The geometrical relationships in Fig. 2 lead us to the expression for the half-length of the minor ellipse axis

b = \sqrt{R a},

and from dependencies between some variables shown in Fig. 3 we can obtain the half-length of the major ellipse axis

a = (a_{1} + a_{2}) / 2 = \frac{R z}{\sqrt{r^{2} - R^{2}}},

where

a_{1} = \frac{r^{2} R}{z \sqrt{r^{2} - R^{2}} + R \sqrt{r^{2} - z^{2}}},

a_{2} = \frac{r^{2} R}{z \sqrt{r^{2} - R^{2}} - R \sqrt{r^{2} - z^{2}}},

and R is the radius of the spherical object.

Fig. 3. Definition of geometrical parameters used for obtaining Eqs. (9)–(11).

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The Jacobian, J, of the transformation described by Eqs. (5)–(6) is given by

J = ∣ \begin{matrix} \cos φ & - \sin φ \\ \sin φ & \cos φ \end{matrix} ∣ = 1,

and does not vanish for any angle φ. Therefore, Eq. (4) can be rewritten as

d Φ_{P} (x, y, z; x_{11}, y_{11}) = I (x, y, z; x_{11}, y_{11}) \frac{z d x_{11} d y_{11}}{{(x_{11}^{2} - {2 r}_{xy} x_{11} + y_{11}^{2} + r^{2})}^{3 ⁄ 2}},

where r_xy=(x ²+y ²)^1/2.

It should be noted here that Eq. (13) determines the infinitesimal radiative flux incident on a spherical object when the center 0₁ of this object lies at the distance r>R from the point source P.

3. The total flux of radiation incident from a point source on a spherical object

The total flux, Φ_P(x,y,z), of radiation (the total radiative flux) incident from a point source on a spherical object will be given as a double definite integral of dΦ_P(x,y,z;x ₁₁,y ₁₁) with respect to x ₁₁ and y ₁₁ with the limits for the variable y ₁₁ given as the function of x ₁₁ derived from Eq. (7). The limits for the variable x ₁₁ are determined by the condition ρ-r_xy-2a+b ²/a=0 and some geometrical dependencies between the parameters of the ellipse drawn in Fig. 2. Thus, the formula for Φ_P(x,y,z) obtains the form

Φ_{P} (x, y, z) = {\begin{matrix} \int_{- b^{2} ⁄ a}^{(2 a^{2} - b^{2}) ⁄ a} d x_{11} \int_{y_{11}^{'}}^{y_{11}^{'}} \frac{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} \leq ρ - 2 a + b^{2} ⁄ a, \\ \int_{- b^{2} ⁄ a}^{ρ - r_{xy}} d x_{11} \int_{y_{11}^{'}}^{y_{11}^{'}} \frac{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, \end{matrix}

where

y_{11}^{'} = - b \sqrt{a^{2} - {[x_{11} - (a^{2} - b^{2}) / a]}^{2}} / a,

and

y_{11}^{″} = b \sqrt{a^{2} - {[x_{11} - (a^{2} - b^{2}) / a]}^{2}} / a,

It can easily be seen that integrals in Eq. (14) may be further calculated if the radiant intensity function, I(x, y, z, x ₁₁, y ₁₁), is known. For some intensity functions, the inner integrals in Eq. (14) may be evaluated analytically. Then the obtained formula may be simply programmed and used directly in practice.

4. Radiative flux from an isotropic point source incident on a spherical object

In this section, we will present a particular solution to Eq. (14) obtained for radiation emitted by an isotropic point source. The intensity of isotropic radiation from a point source fulfils the relation

4 π r^{2} I (x, y, z) = 4 π z^{2} I (x = 0, y = 0, z) = 4 π z^{2} I_{z},

where I(x=0, y=0, z)=I_z=constant, and denotes the intensity of radiation emitted in the direction of the z-axis.

Applying Eq. (15) to the space subtended by the spherical object, we get

I (x, y, z; x_{11}, y_{11}) = \frac{I_{z} z^{2}}{y_{11}^{2} + x_{11}^{2} - 2 r_{xy} x_{11} + r^{2}}

and Eq. (14) becomes

Φ_{P} (x, y, z) = {\begin{matrix} I_{z} z^{3} \int_{- b^{2} ⁄ a}^{(2 a^{2} - b^{2}) ⁄ a} d x_{11} \int_{y_{11}^{″}}^{y_{11}^{″}} \frac{d y_{11}}{{(y_{11}^{2} + x_{11}^{2} - 2 r_{xy} x_{11} + r^{2})}^{5 ⁄ 2}}, if r_{xy} \leq ρ - 2 a + b^{2} ⁄ a, \\ I_{z} z^{3} \int_{- b^{2} ⁄ a}^{ρ - r_{xy}} d x_{11} \int_{y_{11}^{″}}^{y_{11}^{″}} \frac{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 . \end{matrix}

Equation (17) can be integrated analytically with respect to y ₁₁ yielding

Φ_{P} (x, y, z) = {\begin{matrix} I_{z} z^{3} \int_{- b^{2} ⁄ a}^{(2 a^{2} - b^{2}) ⁄ a} f (x, y, z; x_{11}) {d x}_{11}, if r_{xy} \leq ρ - 2 a + b^{2} ⁄ a, \\ I_{z} z^{3} \int_{- b^{2} ⁄ a}^{ρ - r_{xy}} f (x, y, z; x_{11}) d x_{11}, if r_{xy} \leq ρ - 2 a + b^{2} ⁄ a, \\ 0, if r_{xy} > ρ + b^{2} ⁄ a, \end{matrix}

where

\begin{array}{l} f (x, y, z : x_{11}) = \frac{2 b}{3} \frac{\sqrt{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}} \\ \times \frac{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}} . \end{array}

Evaluating the integrals in Eq. (18) with respect to x ₁₁, we do not obtain any closed form expressions. However, an analytical solution to Eq. (18) exists and may be given by superposition of some elementary functions with elliptic integrals of all three kinds. This solution due to its very long and complicated form is not presented here. For our purposes, Eq. (18) was programmed for numerical simulation of some sample results.

5. Selected results of computer simulation

Selected results of numerical simulation from Eq. (18) obtained using Mathematica 2.2.3 software [10] are presented in Fig. 4(a) and (b). The data illustrate the total radiative flux, Φ_P(x,y,z), normalized with respect to the total flux Φ_P(x=0,y=0,z) incident on the object at the point 0′(0,0,z), as a dependency on x and y at given z, R, and ρ, Fig. 4(a), and as the function of y and z at given x, R, and ρ, Fig. 4(b). Figure 4(a) shows that the flux Φ_P(x,y,z)/Φ_P(x=0,y=0,z) depends on x and on y, analogously. However, Fig. 4(b) illustrates that the flux Φ_P(x,y,z)/Φ_P(x=0,y=0,z) is more sharply dependent on the variation of z than on the variation of y.

Fig. 4. The relative value of the total radiative flux, Φ_P(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.

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6. Conclusions

We have shown that the total radiative flux, Φ_P(x,y,z), incident on spherical objects from any arbitrary point source models may be evaluated from Eq. (14) determined by two double integral expressions dependent on the radiant intensity function I(x, y, z, x ₁₁, y ₁₁). However, no general analytical solution to Eq. (14) can be given until the function I(x, y, z, x ₁₁, y ₁₁) is known. Usually, even then the radiant intensity function is known it is still difficult to obtain an analytical equation in closed form for Φ_P(x,y,z) calculation. Therefore, in most cases, numerical procedures for double integral evaluation must be used to estimate the total flux Φ_P(x,y,z) of radiation.

For some radiant intensity functions I(x, y, z, x ₁₁, y ₁₁) the inner integrals in Eq. (14) may be expressed in closed form. In these situations it is easy to calculate the flux Φ_P(x,y,z) by applying one of many simple numerical procedures for single integral estimation. Sometimes the outer integrals in Eq. (14) may also be expressed simply. In most cases, however, the expressions obtained may be very complicated, so that the numerical procedures of single integral evaluation may be more practical than extended closed analytical solutions.

Equation (14) was then applied to evaluate the radiative flux emitted into the space subtended by the spherical object from an isotropic point source model. For this source model, the solution to Eq. (14) is represented by Eq. (18) described by two single integral expressions. These single integral expressions are calculated as a complex superposition of elementary functions with elliptic integrals of all three kinds. However, this complex solution was not presented in the paper and the flux Φ_P(x,y,z) was calculated numerically.

Selected results of numerical simulations were presented in Figs. 4(a) and (b). The three-dimensional relationships of the radiative flux on the spherical object position with respect to the x, y, and z coordinates indicate that the flux is more sensitive to the variation of z than to x or y. These results indicate that the radiative flux changes considerably when the spherical objects moves in the radiation field from a point source. Such variation must be taken into account when systems for spherical object irradiation are designed. The calculation of the radiative flux may also be helpful in estimating the dose of radiation in the processes of spherical food product irradiation if these products are moving in the radiation field from a single point source or from several point sources.

References and links

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Spherical object in radiation field from a point source

Abstract

1. Introduction

2. Basic equations

3. The total flux of radiation incident from a point source on a spherical object

4. Radiative flux from an isotropic point source incident on a spherical object

5. Selected results of computer simulation

6. Conclusions

References and links

Cited By

Figures (4)

Equations (21)

Optics Express