**2. Outline of the problem**

The reciprocity principle enunciated by Lambert in 1760 and expressed in Eqn. (1), yields the following well-known integral equation (2) that acts as the theoretical basis for form factor calculation between two surfaces.

$$d \; \mathcal{O}\_{1\cdot 2} = \{E\_{b1} \cdot E\_{b2}\} \cos \theta\_1 ^\ast \cos \theta\_2 ^\ast \frac{^d A\_1 ^\ast dA\_2}{^\ast r^{\ast \cdot \cdot ^2}}\tag{1}$$

$$\mathcal{O}\_{1\cdot 2} = (E\_{b1} - E\_{b2}) \mathbf{f}\_{A\_1^\* A\_2} \cos \theta\_1 \mathbf{^\*} \cos \theta\_2 \mathbf{^\*} \frac{^d A\_1 \mathbf{^\*} ^d A\_2}{^{\ast \ast} r^{\ast}} \tag{2}$$

Where the terms are depicted in Figure 1,

Ebi= radiant power emitted by the corresponding surface 1 or 2

Ai = area of surface, dAi = differential of area

r = distance radiovector

θi =angle between radiovector at differential element *i* and the normal to the surface

The previous expression states that radiant interchange for every given form depends on its shape and its relative position in the three-dimensional space (Figure 1). From the times of Lambert to our days, researchers and scientists in the fields of geometric optics and radiative transfer have sought to provide solutions to the canonical equation (2) for a variety of forms [1]. This is no minor feat, since the said equation leads in most cases to a quadruple integration and the fourth degree primitive of even simple mathematical expressions often implies lengthy calculations.

Given the fact that this equation depends on geometric parameters, it is reasonable to think that there should be an easier way to approach the problem rather than dealing directly with the integral; also, with the aid of computer simulation, mathematical solutions of complex functions can be approached in a simple and friendly way. Curvilinear forms present some characteristics that make them suitable for a different treatment in terms of radiative transfer.

## **3. Form factors derived from the sphere**

Starting from simple forms several form factors can be calculated without hardly any calculus; later, this logic can be applied to more complex configurations. Let us consider first the simplest

**Figure 1.** The reciprocity principle and equation for arbitrary surfaces A1 and A2

frequently occur in the technical domains but were not feasible for exact calculation during a number of years. The research is duly accomplished by presenting the equations needed to evaluate interreflections in curvilinear geometries. Thus, heat transfer simulation is enhanced by such results leading to create innovative software which has been expanded in turn by the

The reciprocity principle enunciated by Lambert in 1760 and expressed in Eqn. (1), yields the following well-known integral equation (2) that acts as the theoretical basis for form factor

*cosθ*1\**cosθ*2\*

*d A*1\**d A*<sup>2</sup>

*d A*1\**d A*<sup>2</sup>

*<sup>π</sup>*\**<sup>r</sup>* <sup>2</sup> (1)

*<sup>π</sup>*\**<sup>r</sup>* <sup>2</sup> (2)

*d*∅1-2 =(*Eb*<sup>1</sup> - *Eb*2)*cosθ*1\**cosθ*2\*

*A*1 *∫ A*2

=angle between radiovector at differential element *i* and the normal to the surface

The previous expression states that radiant interchange for every given form depends on its shape and its relative position in the three-dimensional space (Figure 1). From the times of Lambert to our days, researchers and scientists in the fields of geometric optics and radiative transfer have sought to provide solutions to the canonical equation (2) for a variety of forms [1]. This is no minor feat, since the said equation leads in most cases to a quadruple integration and the fourth degree primitive of even simple mathematical expressions often implies lengthy

Given the fact that this equation depends on geometric parameters, it is reasonable to think that there should be an easier way to approach the problem rather than dealing directly with the integral; also, with the aid of computer simulation, mathematical solutions of complex functions can be approached in a simple and friendly way. Curvilinear forms present some characteristics that make them suitable for a different treatment in terms of radiative transfer.

Starting from simple forms several form factors can be calculated without hardly any calculus; later, this logic can be applied to more complex configurations. Let us consider first the simplest

∅1-2 =(*Eb*<sup>1</sup> - *Eb*2)*∫*

Ebi= radiant power emitted by the corresponding surface 1 or 2

= differential of area

authors.

4 Solar Radiation Applications

Ai

θi

calculations.

**2. Outline of the problem**

calculation between two surfaces.

= area of surface, dAi

r = distance radiovector

Where the terms are depicted in Figure 1,

**3. Form factors derived from the sphere**

form, a sphere that irradiates energy from its inner surface; the irradiated energy is entirely received by itself; so that, being the sphere surface 1, the only factor that has to be considered is:

$$\mathbf{F}\_{11} = \mathbf{1} \tag{3}$$

Bearing this in mind, in a similar surface, for instance a hemisphere, the form factor is accordingly F11= ½. The configuration factor of a differential area to a disk of radius *r* under the center of the disk at precisely the distance *r*, provides a hint in that it is also ½ [2]. For a point of the hemisphere the factor required is ½.

Stimulated by this result, volumes composed of only two surfaces, one being planar and the other spherical, were analyzed. The first case was the spherical cap which is a generalization of the hemisphere.

**Figure 2.** A spherical cap of height h and radius of the base a

Extending the reciprocity principle to a spherical cap (Fig. 2) of radius R (surface 1), and its entire base (surface 2) the factor was obtained from the relation *A*<sup>1</sup> · *F*<sup>12</sup> = *A*<sup>2</sup> · *F*21; since F21=1, and there is no F22 for planar surfaces, *F*<sup>12</sup> <sup>=</sup> *<sup>A</sup>*<sup>2</sup> *A*1 , in this particular case:

$$F\_{12} = \frac{a^2}{a^2 + h^2} \tag{4}$$

$$F\_{11} = \frac{h^{-2}}{a^2 + h^{-2}} = \frac{h}{2^\*R} \tag{5}$$

Two important laws are inferred from here, which have been defined as Cabeza-Lainez laws:

Cabeza-Lainez first law:

If a volume is encircled by two surfaces preseting one of them positive of thempositive curvature, and the second being planar, the exchange factor from the curved surfaceto the other equals the inverse ratio of areas of the aforementioned figures. The notion of positive curvature of the element is introduced to foresee stagnation of radiant flux.

Cabeza-Lainez second law:

Within a spherical surface the form factor of any given area over itself is precisely the fraction between that area and the sphere

The second law requires of more deduction as follows

Given that a spherical cap represents an Yth fraction of the total area of the sphere of radius R, and recalling from trigonometry that,

$$\left(h^{\;2} + a^{\;2}\right) = 2 \cdot R \cdot h \tag{6}$$

Thus,

$$\text{If } Y \cdot \{h^2 + a^2\} = 4 \cdot R^2 \text{ (5); } Y = 2 \cdot \frac{R}{h} \text{ (6); } h = 2 \cdot \frac{R}{Y} \tag{7}$$

Consequently,

**Figure 2.** A spherical cap of height h and radius of the base a

and there is no F22 for planar surfaces, *F*<sup>12</sup> <sup>=</sup> *<sup>A</sup>*<sup>2</sup>

Cabeza-Lainez first law:

6 Solar Radiation Applications

Cabeza-Lainez second law:

Thus,

between that area and the sphere

and recalling from trigonometry that,

The second law requires of more deduction as follows

Extending the reciprocity principle to a spherical cap (Fig. 2) of radius R (surface 1), and its entire base (surface 2) the factor was obtained from the relation *A*<sup>1</sup> · *F*<sup>12</sup> = *A*<sup>2</sup> · *F*21; since F21=1,

*A*1

*<sup>a</sup>* <sup>2</sup> <sup>+</sup> *<sup>h</sup>* <sup>2</sup> <sup>=</sup> *<sup>h</sup>*

Two important laws are inferred from here, which have been defined as Cabeza-Lainez laws:

If a volume is encircled by two surfaces preseting one of them positive of thempositive curvature, and the second being planar, the exchange factor from the curved surfaceto the other equals the inverse ratio of areas of the aforementioned figures. The notion of positive

Within a spherical surface the form factor of any given area over itself is precisely the fraction

Given that a spherical cap represents an Yth fraction of the total area of the sphere of radius R,

*<sup>F</sup>*<sup>12</sup> <sup>=</sup> *<sup>a</sup>* <sup>2</sup>

*<sup>F</sup>*<sup>11</sup> <sup>=</sup> *<sup>h</sup>* <sup>2</sup>

curvature of the element is introduced to foresee stagnation of radiant flux.

, in this particular case:

*<sup>a</sup>* <sup>2</sup> <sup>+</sup> *<sup>h</sup>* <sup>2</sup> (4)

(*h* <sup>2</sup> + *a* 2) =2 · *R* · *h* (6)

2\**<sup>R</sup>* (5)

$$F\_{11} = \frac{h}{2 \cdot R} = \frac{h^{-2} \cdot Y}{4 \cdot R^{-2}} = \frac{1}{Y} \tag{8}$$

Cabeza-Lainez second law:

The configuration factor of an Yth part of the sphere over itself is precisely the inverse of Y.

Thus, the assumption for the hemisphere is confirmed; in the quarter of sphere F11 has to be 1/4 and successively for every portion of the given sphere.

This law will hold true even if we are not dealing with spherical caps but for any fragment of the surface. Taking a critical look at the canonical equation (1) adapted to the sphere, it is logical to establish a relationship between r, *cosθ* and the radius R (Figure 3).

**Figure 3.** Differential surfaces in the sphere of centre C and luminance L used to find the radiative exchange

Substituting, these terms in the canonical equation (1):

$$
\mathcal{L}\mathcal{D}\_{1\cdot 2} = \frac{E\_{b1}}{4\cdot\pi\cdot R^{-2}} \mathbf{f}\_{A\_1} \mathbf{f}\_{A\_2} \, dA\_1 \cdot dA\_2 \tag{9}
$$

4*πR* <sup>2</sup> is the total area of the sphere. Thus, the radiative flux transfer is dependent on the size of the surfaces but not on their position in the sphere and for given areas it is also a constant. Trying to obtain *F*<sup>11</sup> <sup>=</sup> *<sup>∅</sup>*<sup>11</sup> *Eb*1.*A*<sup>1</sup> from equation (7) gives the expression:

$$F\_{11} = \frac{A\_1}{4 \cdot \pi \cdot R^{-2}} = \frac{1}{Y} \tag{10}$$

This means that spherical surfaces present these unique properties (Eqs. 3 and 8) which are crucial for our discussion crucial for our discussion.

Now Cabeza-Lainez laws can be applied to more complex volumes that involve portions of the sphere. Considering a sector of the sphere comprised between to semicircles forming an internal angle *x* from 0 to 180 degrees:

**Figure 4.** Denomination of surfaces in a sector of the sphere, 1 and 2 are planar semicircles, 3 is curved.

As has been discussed, the Y portion of the sphere is, in this case 1 *<sup>Y</sup>* <sup>=</sup> *<sup>x</sup>* 360 and thus,

$$F\_{33} = \frac{x}{360} \tag{11}$$

Accordingly,

$$F\_{31} = F\_{32} = \frac{1}{2} \cdot \left(1 - \frac{x}{360}\right) \tag{12}$$

And introducing the areas of the semicircles, *<sup>π</sup><sup>R</sup>* <sup>2</sup> 2

$$F\_{13} = F\_{23} = \frac{\chi}{90} \cdot \left(1 - \frac{\chi}{360}\right) \tag{13}$$

Following the discussion, these pair of semicircles can form any angle x between 0 and 360 degrees (Fig. 5). So that, the following equation, which has not been found expressed previ‐ ously in the literature, is proposed in order to obtain the energy balance between the half disks, where x represents the value of their internal angle (Figure 5).

**Figure 5.** Two semicircles of the same radius R with a common edge forming an angle X

Now Cabeza-Lainez laws can be applied to more complex volumes that involve portions of the sphere. Considering a sector of the sphere comprised between to semicircles forming an

**Figure 4.** Denomination of surfaces in a sector of the sphere, 1 and 2 are planar semicircles, 3 is curved.

*<sup>F</sup>*<sup>33</sup> <sup>=</sup> *<sup>x</sup>*

<sup>2</sup> ·(1 - *<sup>x</sup>*

<sup>90</sup> ·(1 - *<sup>x</sup>*

Following the discussion, these pair of semicircles can form any angle x between 0 and 360 degrees (Fig. 5). So that, the following equation, which has not been found expressed previ‐ ously in the literature, is proposed in order to obtain the energy balance between the half disks,

2

*<sup>F</sup>*<sup>31</sup> <sup>=</sup> *<sup>F</sup>*<sup>32</sup> <sup>=</sup> <sup>1</sup>

*<sup>F</sup>*<sup>13</sup> <sup>=</sup> *<sup>F</sup>*<sup>23</sup> <sup>=</sup> *<sup>x</sup>*

where x represents the value of their internal angle (Figure 5).

*<sup>Y</sup>* <sup>=</sup> *<sup>x</sup>*

<sup>360</sup> (11)

<sup>360</sup> ) (12)

<sup>360</sup> ) (13)

360 and thus,

As has been discussed, the Y portion of the sphere is, in this case 1

And introducing the areas of the semicircles, *<sup>π</sup><sup>R</sup>* <sup>2</sup>

Accordingly,

internal angle *x* from 0 to 180 degrees:

8 Solar Radiation Applications

$$F\_{12} = 1 - \frac{\text{x}}{90} + \frac{\text{x}^2}{32400} \tag{14}$$

**Figure 6.** Radiative exchanges between two semicircles with a common edge and forming an internal angle *x*

The latter expression (Eq. 14) is a good indicator of the factor between two inclined and equal surfaces with a common edge. If they are not too dissimilar from the semicircle, a factor that is usually lengthy and cumbersome to calculate can be devised easily.

Let us now return to the first principle, the expression *<sup>h</sup>* <sup>2</sup>*<sup>R</sup>* (Eq. 5), applied to the spherical cap. Form factors between the contained surfaces are as follows:

$$\,\_{11}F\_{11} = \frac{h}{2 \cdot R} = \frac{h^{-2}}{h^{-2} + a^2} \tag{15}$$

$$F\_{12} = \frac{a^2}{h^2 + a^2} \tag{16}$$

$$F\_{21} = 1\tag{17}$$

If we introduce at this point the dimensionless parameter β, we can simplify equation 16 as,

$$
\beta^2 = \frac{h^{-2}}{a^2} \tag{18}
$$

$$F\_{12} = \frac{1}{\beta^2 + 1} \tag{19}$$

Since this principle is more general than the second one, we can extend it to non-spherical surfaces.

#### **4. Application to common surfaces**

#### **4.1. Prolate semispheroid**

Surface 1 is the spheroid and surface 2 is the circular disk that works as a base to the former, h>a.

Firstly the dimensionless parameter m is introduced:

$$
\Delta m = \sqrt{1 - \frac{a^2}{h^2}}\tag{20}
$$

By virtue of the first principle,

$$\,\_{1}F\_{12} = \frac{a^\*m}{a^\*m + h^\* \text{arcsin } (m)}\tag{21}$$

#### Radiative Heat Transfer for Curvilinear Surfaces http://dx.doi.org/10.5772/59797 11

**Figure 7.** Prolate spheroid

The latter expression (Eq. 14) is a good indicator of the factor between two inclined and equal surfaces with a common edge. If they are not too dissimilar from the semicircle, a factor that

<sup>2</sup>*<sup>R</sup>* (Eq. 5), applied to the spherical cap.

*<sup>h</sup>* <sup>2</sup> <sup>+</sup> *<sup>a</sup>* <sup>2</sup> (15)

*<sup>h</sup>* <sup>2</sup> <sup>+</sup> *<sup>a</sup>* <sup>2</sup> (16)

*F*<sup>21</sup> =1 (17)

*<sup>a</sup>* <sup>2</sup> (18)

*<sup>β</sup>* <sup>2</sup> <sup>+</sup> <sup>1</sup> (19)

*<sup>h</sup>* <sup>2</sup> (20)

*<sup>a</sup>*\**<sup>m</sup>* <sup>+</sup> *<sup>h</sup>* \*arcsin (*m*) (21)

is usually lengthy and cumbersome to calculate can be devised easily.

*<sup>F</sup>*<sup>11</sup> <sup>=</sup> *<sup>h</sup>*

<sup>2</sup> · *<sup>R</sup>* <sup>=</sup> *<sup>h</sup>* <sup>2</sup>

If we introduce at this point the dimensionless parameter β, we can simplify equation 16 as,

Since this principle is more general than the second one, we can extend it to non-spherical

Surface 1 is the spheroid and surface 2 is the circular disk that works as a base to the former,

*<sup>m</sup>*<sup>=</sup> <sup>1</sup> - *<sup>a</sup>* <sup>2</sup>

*<sup>F</sup>*<sup>12</sup> <sup>=</sup> *<sup>a</sup>*\**<sup>m</sup>*

*<sup>β</sup>* <sup>2</sup> <sup>=</sup> *<sup>h</sup>* <sup>2</sup>

*<sup>F</sup>*<sup>12</sup> <sup>=</sup> <sup>1</sup>

*<sup>F</sup>*<sup>12</sup> <sup>=</sup> *<sup>a</sup>* <sup>2</sup>

Let us now return to the first principle, the expression *<sup>h</sup>*

surfaces.

10 Solar Radiation Applications

h>a.

**4. Application to common surfaces**

Firstly the dimensionless parameter m is introduced:

**4.1. Prolate semispheroid**

By virtue of the first principle,

Form factors between the contained surfaces are as follows:

$$F\_{21} = 1\tag{22}$$

$$F\_{11} = \frac{h^\* \text{arcsin (m)}}{a^\* m + h^\* \text{arcsin (m)}} \tag{23}$$

And making,

$$\left\|\beta\right\|^2 = \frac{\hbar^{\frac{2}{2}}}{a^2};\ m = \sqrt{1 - \frac{1}{\beta^{\frac{2}{2}}}}\tag{24}$$

$$F\_{12} = \frac{\sqrt{1 \cdot \frac{1}{\rho^2}}}{\sqrt{1 \cdot \frac{1}{\rho^2}} + \rho^\* \text{arcsin}\left[\sqrt{1 \cdot \frac{1}{\rho^2}}\right]} \tag{25}$$

#### **4.2. Oblate semiespheroid**

Surface 1 is the spheroid and surface 2 is the circular disk that works as a base to the former, h<a

Denote the parameter m1,

$$m\_1 = \sqrt{\frac{a^2}{h^2} - 1} \tag{26}$$

**Figure 8.** Oblate spheroid

$$\;\_1F\_{12} = \frac{a^\*m\_1}{a^\*m\_1 + h^\*\text{arcsinh}\,\text{(}m\_1\text{)}} \;\_1F\_{21} = 1\tag{27}$$

By the first principle and,

$$F\_{11} = \frac{h^\* \text{arcsinh}\left(m\_1\right)}{a^\* m\_1 + h^\* \text{arcsinh}\left(m\_1\right)}\tag{28}$$

With the same procedure as before to make the expression dimensionless

$$m\_1 = \sqrt{\frac{1}{\beta^2} - 1} \tag{29}$$

$$\,\_2F\_{12} = \frac{m\_1}{m\_1 + \beta^\* \text{arcsinh}\left(m\_1\right)}\tag{30}$$

#### **4.3. Paraboloid of revolution**

Surface 1 is the paraboloid and surface 2 is the circular disk that works as a base to the former

$${}^{1}F\_{12} = \frac{{}^{6}a^{\*}{}^{h}{}^{2}}{[{}^{2} + {}^{4}{}^{4}{}^{h}{}^{2}]^{3} {}^{3} {}^{2} {}^{3}}} ; F\_{21} = 1 \tag{31}$$

$$F\_{11} = 1 - \frac{6^\* a^\* h^{-2}}{\prod (a^2 + 4^\* h^{-2})^{3/2} \cdot a^{3}}\tag{32}$$

$$\beta = \frac{h}{a} \quad ; F\_{12} = \frac{6^\* \beta^{\*2}}{\prod (1 + 4^\* \beta^{\*2})^{3/2} \cdot 1} \tag{33}$$

#### **4.4. Right cone**

1 is the surface of the cone and 2 is the circular base

$$F\_{12} = \frac{a}{\sqrt{a^2 + h^2}} \; ; \; F\_{21} = 1 \tag{34}$$

**Figure 9.** Paraboloid of revolution

*<sup>F</sup>*<sup>12</sup> <sup>=</sup> *<sup>a</sup>*\**m*<sup>1</sup>

With the same procedure as before to make the expression dimensionless

*<sup>F</sup>*<sup>11</sup> <sup>=</sup> *<sup>h</sup>* \*arcsinh (*m*1)

*<sup>m</sup>*<sup>1</sup> <sup>=</sup> <sup>1</sup>

*<sup>F</sup>*<sup>12</sup> <sup>=</sup> *<sup>m</sup>*<sup>1</sup>

*<sup>F</sup>*<sup>12</sup> <sup>=</sup> 6\**a*\**<sup>h</sup>* <sup>2</sup>

*<sup>F</sup>*<sup>12</sup> <sup>=</sup> *<sup>a</sup>*

*<sup>β</sup>* <sup>=</sup> *<sup>h</sup>*

1 is the surface of the cone and 2 is the circular base

*<sup>F</sup>*<sup>11</sup> =1 - 6\**a*\**<sup>h</sup>* <sup>2</sup>

*<sup>a</sup>* ; *<sup>F</sup>* <sup>12</sup> <sup>=</sup> 6\**<sup>β</sup>* <sup>2</sup>

Surface 1 is the paraboloid and surface 2 is the circular disk that works as a base to the former

By the first principle and,

**Figure 8.** Oblate spheroid

12 Solar Radiation Applications

**4.3. Paraboloid of revolution**

**4.4. Right cone**

*<sup>a</sup>*\**m*<sup>1</sup> <sup>+</sup> *<sup>h</sup>* \*arcsinh (*m*1) ; *<sup>F</sup>* <sup>21</sup> =1 (27)

*<sup>a</sup>*\**m*<sup>1</sup> <sup>+</sup> *<sup>h</sup>* \*arcsinh (*m*1) (28)

*<sup>β</sup>* <sup>2</sup> - <sup>1</sup> (29)

*<sup>m</sup>*<sup>1</sup> <sup>+</sup> *<sup>β</sup>*\*arcsinh (*m*1) (30)

(*<sup>a</sup>* <sup>2</sup> <sup>+</sup> 4\**<sup>h</sup>* 2)3/2 - *<sup>a</sup>* <sup>3</sup> ; *<sup>F</sup>*<sup>21</sup> =1 (31)

(*<sup>a</sup>* <sup>2</sup> <sup>+</sup> 4\**<sup>h</sup>* 2)3/2 - *<sup>a</sup>* <sup>3</sup> (32)

(1 <sup>+</sup> 4\**<sup>β</sup>* 2)3/2 - <sup>1</sup> (33)

*<sup>a</sup>* <sup>2</sup> <sup>+</sup> *<sup>h</sup>* <sup>2</sup> ; *<sup>F</sup>* <sup>21</sup> =1 (34)

$$F\_{11} = 1 - \frac{a}{\sqrt{a^2 + h^2}}\tag{35}$$

Introducing the parameter *β*,

$$F\_{12} = \frac{1}{\sqrt{1 \cdot \beta^{-2}}} \tag{36}$$

It is possible to compare the performance in terms of F12, of all the figures found up to now, where the cone shows better performance followed by the paraboloid.

**Figure 10.** Cone

**Figure 11.** Comparison of form factors for different shapes

#### **4.5. Ellipsoid**

In this case, 1 is the surface of the ellipsoid and 2 is the elliptic base; y is a parameter equal to 1.6. The example shows that the first principle is not limited to surfaces of revolution.

$$\;\_1F\_{12} = \frac{\;\_ {a^\*b^\*}\sqrt[3]{3}}{2^\*\sqrt[3]{a^{\*\*}b^\*\;+a^\*b^\*h^{-\*}+a^\*b^\*h^{\*\*}}} ; \;\_1F\_{21} = 1\tag{37}$$

$$F\_{11} = 1 - \frac{a^\*b^\*\overline{\sqrt{3}}}{2^\*\overline{\sqrt{a^\*b^\*b^\* + a^\*b^\*h^{\*\frac{\mathfrak{z}}{3}} + \dotsb^\*b^{\*\mathfrak{z}}h^{\*\mathfrak{z}}}} \tag{38}$$

**Figure 12.** Ellipsoid

As the area of the ellipsoid is not exact, we can expect errors on the range of 1% depending on the values of a, b and h.

This principle can be also used in other surfaces, for example, for two complementary caps within the sphere of radius r,

**Figure 13.** Sphere divided in two caps of diverse heights

**Figure 11.** Comparison of form factors for different shapes

In this case, 1 is the surface of the ellipsoid and 2 is the elliptic base; y is a parameter equal to

As the area of the ellipsoid is not exact, we can expect errors on the range of 1% depending on

This principle can be also used in other surfaces, for example, for two complementary caps

2\* *<sup>a</sup> <sup>y</sup>*\**<sup>b</sup> <sup>y</sup>* <sup>+</sup> *<sup>a</sup> <sup>y</sup>*\**<sup>h</sup> <sup>y</sup>* <sup>+</sup> <sup>+</sup> *<sup>b</sup> <sup>y</sup>*\**<sup>h</sup> <sup>y</sup> <sup>y</sup>* ; *<sup>F</sup>* <sup>21</sup> =1 (37)

2\* *<sup>a</sup> <sup>y</sup>*\**<sup>b</sup> <sup>y</sup>* <sup>+</sup> *<sup>a</sup> <sup>y</sup>*\**<sup>h</sup> <sup>y</sup>* <sup>+</sup> <sup>+</sup> *<sup>b</sup> <sup>y</sup>*\**<sup>h</sup> <sup>y</sup> <sup>y</sup>* (38)

1.6. The example shows that the first principle is not limited to surfaces of revolution.

*<sup>F</sup>*<sup>12</sup> <sup>=</sup> *<sup>a</sup>*\**b*\* <sup>3</sup>*<sup>y</sup>*

*<sup>F</sup>*<sup>11</sup> =1 - *<sup>a</sup>*\**b*\* <sup>3</sup>*<sup>y</sup>*

**4.5. Ellipsoid**

14 Solar Radiation Applications

**Figure 12.** Ellipsoid

the values of a, b and h.

within the sphere of radius r,

As an immediate consequence of Cabeza-Lainez laws, *r* being the radius of the inner circle and *h* the respective heights of the caps,

$$F\_{11} = F\_{21} = \frac{h\_1^{-2}}{h\_1^{-2} + r^2} = \frac{r^2}{h\_2^{-2} + r^2} = \frac{h\_1 " h\_2}{h\_2^{-2} + r^2} = \frac{h\_1^{-2} + r^2}{(h\_1 + h\_1)^2} = \frac{h\_1}{(h\_1 + h\_2)} = \frac{h\_1}{2^r R} \tag{39}$$

$$F\_{22} = F\_{12} = \frac{h\_2^{\*2}}{h\_2^{\*2} + r^2} = \frac{r^{\*2}}{h\_1^{\*2} + r^{\*2}} = \frac{h\_1 " h\_2}{h\_1^{\*2} + r^{\*2}} = \frac{h\_2}{\left(h\_1 + h\_2\right)}\tag{40}$$

If now the caps within the same sphere are of any size and arbitrary position,

In this case by virtue of Cabeza-Lainez Law,

$$\,^1F\_{11} = \frac{\left. \,^1\mathbf{r}\_1^2 \right.}{\left. \,^2\mathbf{r}\_1 \right.} ; \,^2F\_{22} = \frac{\left. \,^2\mathbf{r}\_2 \right.}{\left. \,^2\mathbf{r}\_2 \right.} \tag{41}$$

And now we need to apply the canonical equation 9 again, substituting the respective areas of the caps; *A*<sup>1</sup> =2.*π*.*R*.*h* <sup>1</sup>. ; *A*<sup>2</sup> =2.*π*.*R*.*h* <sup>2</sup>

$$\bigcirc\_{1\cdot2} \bigcirc\_{1\cdot2} = \frac{E\_{b1}}{4^{\ast}\pi^{\ast}R^{\ast2}} \bigcirc\_{A\_1} \bigcirc\_{A\_2} dA\_1 \mathfrak{\*} dA\_2 \tag{42}$$

$$F\_{12} = \frac{h\_1 \text{\*} h\_2}{h\_1 \text{\*} + a^2}; \; F\_{21} = \frac{h\_1 \text{\*} h\_2}{h\_2 \text{\*} + a\_2^2} \tag{43}$$

**Figure 14.** Two caps of arbitrary size

In the special situation that the caps are parallel, which equates a truncated cone, the flux would be *Eb*1.π.*h* 1.*h* <sup>2</sup> and the fraction of energy from disk 1 to disk 2 (or their surrounding caps), equates *h*1*\*h*<sup>2</sup> / *<sup>a</sup>* <sup>2</sup> or *h*1*\*h*<sup>2</sup> / *a*<sup>2</sup> 2. In the case that the bases are of equal radius a, h1=h2=h. If the perpendicular distance between the disks, called 2b, is known (Figure 15), the height of the cap would be,

$$h = \sqrt{a^2 + b^2} - b \tag{44}$$

Thus, the fraction is obtained as,

$$F\_{12} = F\_{21} = \frac{a^2 + 2^\*b^2 - 2^\*b^\*\sqrt{a^2 + b^2}}{a^2} \tag{45}$$

By virtue of equation 45 it is feasible to address radiative transfer in several figures composed of three surfaces and limited by parallel disks like truncate paraboloids, caps and especially cylinders. Appropriate equations can be easily formed in which only two values need to be found. To the circles in the extremes of the cylinder a spherical cap could be connected (fig. 16) and the radiative transfer would not be altered significantly since we have previously described the performance of caps limited by circles. In the particular case that the cap is a hemisphere, the factor already determined ought to be multiplied by 0.5 and subsequently for different curvatures, bearing in mind that the unity is the circle and null would imply a "theoretical" whole sphere 1

<sup>1</sup> Note that values under 0.5 can also be found for this relationship in a sort of globular cap with an area bigger than the hemisphere.

**Figure 15.** Surfaces defined by a cylindrical volume used to find the radiative transfer

In the special situation that the caps are parallel, which equates a truncated cone, the flux would be *Eb*1.π.*h* 1.*h* <sup>2</sup> and the fraction of energy from disk 1 to disk 2 (or their surrounding caps),

perpendicular distance between the disks, called 2b, is known (Figure 15), the height of the

*<sup>F</sup>*<sup>12</sup> <sup>=</sup> *<sup>F</sup>*<sup>21</sup> <sup>=</sup> *<sup>a</sup>* <sup>2</sup> <sup>+</sup> 2\**<sup>b</sup>* <sup>2</sup> - 2\**b*\* *<sup>a</sup>* <sup>2</sup> <sup>+</sup> *<sup>b</sup>* <sup>2</sup>

By virtue of equation 45 it is feasible to address radiative transfer in several figures composed of three surfaces and limited by parallel disks like truncate paraboloids, caps and especially cylinders. Appropriate equations can be easily formed in which only two values need to be found. To the circles in the extremes of the cylinder a spherical cap could be connected (fig. 16) and the radiative transfer would not be altered significantly since we have previously described the performance of caps limited by circles. In the particular case that the cap is a hemisphere, the factor already determined ought to be multiplied by 0.5 and subsequently for different curvatures, bearing in mind that the unity is the circle and null would imply a

1 Note that values under 0.5 can also be found for this relationship in a sort of globular cap with an area bigger than the

2. In the case that the bases are of equal radius a, h1=h2=h. If the

*h* = *a* <sup>2</sup> + *b* <sup>2</sup> - *b* (44)

*<sup>a</sup>* <sup>2</sup> (45)

equates *h*1*\*h*<sup>2</sup> / *<sup>a</sup>* <sup>2</sup>

16 Solar Radiation Applications

**Figure 14.** Two caps of arbitrary size

cap would be,

or *h*1*\*h*<sup>2</sup> / *a*<sup>2</sup>

Thus, the fraction is obtained as,

"theoretical" whole sphere 1

hemisphere.

The space of figure 16 has been used throughout the history of buildings in cathedrals, opera houses, museums and assembly halls. If both extremes are curved, such shape is still found at bunkers, water tanks and pressure vessels of power reactors.

**Figure 16.** Volume composed of a cylinder and a spherical cap used to find the radiative transfer among those surfaces

#### **4.6. Two opposed spherical caps with a common axis**

In order to calculate the radiative exchanges in this relatively complex figure, we need to determine beforehand the following nine geometric parameters that depend on the geometric variables shown in Figure 17.

$$z = \frac{r\_1^2 - r\_2^2}{4^4 b};\ R = \sqrt{(z+b)^2 + r\_2^2} \tag{46}$$

$$l = \sqrt{(r\_1 - r\_2)^2 + 4^\*b^2} \tag{47}$$

$$Q = \text{R}^{\cdot 2} \text{ - } z^2 + b^2 \text{ - } \text{2}^\* \text{R}^\* b \tag{48}$$

$$\mathbf{Q\_1 = r\_1^2 \cdot Q \; :\; Q\_2 = r\_2^2 \cdot Q} \tag{49}$$

$$D\_1 = h\_1^2 + r\_1^2\tag{50}$$

$$D\_2 = h\_2^{\;2} + r\_2^{\;2} \tag{51}$$

$$D\_3 = l^\*(r\_1 + r\_2) \tag{52}$$

**Figure 17.** Volume composed by spherical cap, truncated cone and hemispheroid.

And then we would obtain the corresponding nine form factors involved,

$$\{F\_{11} = \frac{h\_1}{D\_1}^2; \; F\_{12} = \frac{Q}{D\_1}; \; F\_{13} = \frac{Q\_1}{D\_1}\tag{53}$$

$$F\_{22} = \frac{h\_2}{D\_2};\ F\_{21} = \frac{Q}{D\_2};\ F\_{23} = \frac{Q\_2}{D\_2} \tag{54}$$

$$F\_{31} = \frac{Q\_1}{D\_3};\; F\_{32} = \frac{Q\_2}{D\_3};\;\; F\_{33} = \mathbf{1} - \frac{Q\_1 Q\_2}{D\_3} \tag{55}$$

In this simple way the problem is completely solved

#### **4.7. Straight cone**

**4.6. Two opposed spherical caps with a common axis**

*<sup>z</sup>* <sup>=</sup> *<sup>r</sup>*<sup>1</sup> <sup>2</sup> - *<sup>r</sup>*<sup>2</sup> 2

*Q*<sup>1</sup> =*r*<sup>1</sup>

**Figure 17.** Volume composed by spherical cap, truncated cone and hemispheroid.

variables shown in Figure 17.

18 Solar Radiation Applications

In order to calculate the radiative exchanges in this relatively complex figure, we need to determine beforehand the following nine geometric parameters that depend on the geometric

4\**<sup>b</sup>* ; *<sup>R</sup>* <sup>=</sup> (*<sup>z</sup>* <sup>+</sup> *<sup>b</sup>*)<sup>2</sup> <sup>+</sup> *<sup>r</sup>*<sup>2</sup>

<sup>2</sup> - *<sup>Q</sup>* ;*Q*<sup>2</sup> <sup>=</sup>*r*<sup>2</sup>

<sup>2</sup> <sup>+</sup> *<sup>r</sup>* <sup>1</sup>

<sup>2</sup> <sup>+</sup> *<sup>r</sup>*<sup>2</sup>

*D*<sup>1</sup> =*h*<sup>1</sup>

*D*<sup>2</sup> =*h*<sup>2</sup>

2 (46)

<sup>2</sup> - *Q* (49)

<sup>2</sup> (50)

<sup>2</sup> (51)

*D*<sup>3</sup> =*l*\*(*r*<sup>1</sup> + *r*<sup>2</sup> ) (52)

*<sup>l</sup>* <sup>=</sup> (*r*<sup>1</sup> - *<sup>r</sup>*2)<sup>2</sup> <sup>+</sup> 4\**b*<sup>2</sup> (47)

*Q* =*R* <sup>2</sup> - *z* <sup>2</sup> + *b* <sup>2</sup> - 2\**R*\**b* (48)

This is a limit case of the previous discussion.

**Figure 18.** Right cone with a circular base

As the former also includes the cone, by making *r*<sup>0</sup> =0 and *h*<sup>1</sup> =*h*<sup>2</sup> =0, *<sup>Q</sup>*<sup>2</sup> =0, *<sup>z</sup>* <sup>=</sup> *<sup>r</sup>*<sup>1</sup> 2 <sup>4</sup>*<sup>b</sup>* , *R* = *z* + *b*, *Q* =0, *Q*<sup>1</sup> =0, *Q*<sup>2</sup> =0

$$l = \sqrt{r\_1^2 + 4^\*b^2} \tag{56}$$

If *D*<sup>1</sup> =*r*<sup>1</sup> 2 , *D*<sup>2</sup> =0 then

$$D\_3 = \sqrt{r\_1^2 + 4^\*b^2} \tag{57}$$

Only three factors remain,

$$F\_{11} = 1\tag{58}$$

$$F\_{31} = \frac{r\_1}{\sqrt{r\_1^2 + 4^\*b^2}}\tag{59}$$

$$F\_{33} = 1 - \frac{r\_1}{\sqrt{r\_1^2 + 4^\*b^2}}\tag{60}$$

F31 is obviously the ratio of areas of the cone to its base which proves that the equation is true, by virtue of Cabeza-Lainez Law.

#### **4.8. Paraboloid, truncated cone and spheroid**

If for instance, the upper extreme of the volume is a paraboloid and the lower surface is an oblate ellipsoid (Figure 19), we can still maintain the same factors with the following simple adaptations,

**Figure 19.** Volume composed by a paraboloid, a truncated cone and a spheroid.

$$F\_{22} = 1 - \frac{6^4 r\_2 "{h\_2}^2}{\left[ {r\_2}^2 + {4^4 h\_2}^2 \right]^{3/2} \cdot {r\_2}^3} \tag{61}$$

as in the paraboloid alone

*D*<sup>3</sup> = *r*<sup>1</sup>

*<sup>F</sup>*<sup>31</sup> <sup>=</sup> *<sup>r</sup>*<sup>1</sup> *r*1

*<sup>F</sup>*<sup>33</sup> =1 - *<sup>r</sup>*<sup>1</sup> *r*1

F31 is obviously the ratio of areas of the cone to its base which proves that the equation is true,

If for instance, the upper extreme of the volume is a paraboloid and the lower surface is an oblate ellipsoid (Figure 19), we can still maintain the same factors with the following simple

Only three factors remain,

20 Solar Radiation Applications

by virtue of Cabeza-Lainez Law.

adaptations,

**4.8. Paraboloid, truncated cone and spheroid**

**Figure 19.** Volume composed by a paraboloid, a truncated cone and a spheroid.

*<sup>F</sup>*<sup>22</sup> =1 - 6\**r*2\**<sup>h</sup>* <sup>2</sup>

(*r*2 <sup>2</sup> <sup>+</sup> 4\**<sup>h</sup>* <sup>2</sup> 2

2)3/2 - *<sup>r</sup>*<sup>2</sup>

<sup>3</sup> (61)

<sup>2</sup> + 4\**b*<sup>2</sup> (57)

*F*<sup>11</sup> =1 (58)

<sup>2</sup> <sup>+</sup> 4\**b*<sup>2</sup> (59)

<sup>2</sup> <sup>+</sup> 4\**b*<sup>2</sup> (60)

$$F\_{21} = \frac{6^4 h\_2^{2\*} Q}{r\_2^4 \left[ \left( r\_2^2 + 4^\* h\_2^{\*} \right)^{3/2} - r\_2^3 \right]} \tag{62}$$

$$F\_{2,3} = \frac{6^4 h\_2^{2\star} \text{(\$r\_2^2\$ - Q)}}{r\_2 \text{[\$\left(\$r\_2^2 + 4^\star h\_2^{2\star}\right)^{3/2} - a\_2^3\$]}} \tag{63}$$

$$F\_{11} = \frac{h\_1 \text{\*} \text{arcsinh } (m\_1)}{r \text{\*} m\_1 + h\_1 \text{\*} \text{arcsinh } (m\_1)} \tag{64}$$

as it were in the oblate elipsoid alone

$$\text{for } \mathbf{i} \text{ is now} = \sqrt{\frac{r\_1^2}{h\_1^2} - 1} \tag{65}$$

$$F\_{12} = \frac{m\_1 \text{\*} Q}{r\_1 \text{\*} (r^\* m\_1 + h\_1 \text{\*} \text{arcsinh } (m\_1))}\tag{66}$$

$$F\_{13} = \frac{m\_1 \text{\*} \{r\_1^2 \text{ - } Q\}}{r\_1 \text{\*} \{r\_1 \text{\*} m\_1 + h\_1 \text{\*} \text{arcsinh}\,\{m\_1\}\}} \tag{67}$$

F31, F32 and F33 have the same values as before as these correspond to the truncated cone and bear only nominal relation with the surfaces of the extremes,

$$F\_{31} = \frac{Q\_1}{D\_3} \tag{68}$$

$$F\_{32} = \frac{Q\_2}{D\_3} \tag{69}$$

$$F\_{33} = 1 - \frac{Q\_1 Q\_2}{D\_3} \tag{70}$$

Similar results will be obtained when the truncate is a paraboloid instead of a cone as it is the case in rocket nozzles.

#### **4.9. Summary of the findings**

All the aforementioned form factors have been obtained by logical deduction. In order to provide researchers and designers with all this factors in a compact format, the following table is presented, which comprises all the volume configurations presented in this chapter.

F21 is always the unit as shown by first law


**Table 1.** Resume of form factors for curved surfaces.

#### **5. Interreflections amongst surfaces in a closed volume**

Until this point the discussion has dealt with primary transmission of energy but, in a closed space, if the surfaces have some degree of reflectivity a significant part of the flux would be re-irradiated and the concepts of emitters and receivers entwine.

Under such circumstance, the global balance of radiant power can be found through expression 71,

$$E\_{tot} = E\_{dir} + E\_{ref} \tag{71}$$

Edir is defined as the direct power received while Eref stands for the reflected energy. The two quantities added yield the global balance of radiant energy Etot. If the problem entails several surfaces, expression 71 is expanded for an array of equations. To resolve it, we define before‐ hand the matrices Fd and Fr, whose elements are described as follows in a three-dimensional fashion, (see figure 16):

$$F\_d = \begin{vmatrix} F\_{11} \ast \rho\_1 & F\_{12} \ast \rho\_2 & F\_{13} \ast \rho\_3 \\ F\_{21} \ast \rho\_1 & F\_{22} \ast \rho\_2 & F\_{23} \ast \rho\_3 \\ F\_{31} \ast \rho\_1 & F\_{32} \ast \rho\_2 & F\_{33} \ast \rho\_3 \end{vmatrix} \tag{72}$$

#### Radiative Heat Transfer for Curvilinear Surfaces http://dx.doi.org/10.5772/59797 23

$$F\_r = \begin{pmatrix} 1 & \text{-}F\_{12}\text{\*}\rho\_2 & \text{-}F\_{13}\text{\*}\rho\_3\\ \text{-}F\_{21}\text{\*}\rho\_1 & 1 & \text{-}F\_{23}\text{\*}\rho\_3\\ \text{-}F\_{31}\text{\*}\rho\_1 & \text{-}F\_{32}\text{\*}\rho\_2 & 1 \end{pmatrix} \tag{73}$$

Each term in equations 72 and 73 is presented in the form Fij (F11, F12...). This stands for the configuration factors already found, from one of the surfaces *i* to another adjacent surface *j*. The term ρ<sup>i</sup> is defined as the reflective quotient which corresponds to a given surface *i*.

A detailed explanation for the phenomenon is given in [3]. Formerly, as volumes considered were limited by planes, all the elements in the diagonal of matrix F*<sup>d</sup>* were equal to zero and we could not deal with the problem while, for curved surfaces, the values of the diagonal are different from null and need to be calculated with the expressions hereby presented.

Once the value of these matrices is obtained, it is easy to establish the following relationship between direct and reflected radiation:

$$\left[F\_r \, ^\ast E\_{ref} = F\_d \, ^\ast E\_{dir}\right] \tag{74}$$

$$F\_{rd} = F\_r^{-1\*} F\_d \tag{75}$$

$$E\_{ref} = F\_{rd} \, ^\ast E\_{dir} \tag{76}$$

As the value of reflected radiation is known, the problem is solved. However, we have to bear in mind that the number of surfaces should be augmented depending on the dimensions of the case study. The procedure for interreflection can be considered iterative depending on the accuracy that is required for a particular problem [3].

The simplest case of repeated reflections appears in the sphere and is wont to be employed in lieu of the former calculations with matrices.From expression 9 and successive, it was deducted that energy impinging on a point of the sphere from an emitter contained in the same surface equates the quotient between the area of the emitting surface and the total area 4 πR2, and it can be expressed under the form W/A.

After a relevant number of reflections, the total power distributed over the sphere is defined by:

$$E\_{ref} = E^\* \frac{\mathcal{W}}{A} \* \left(\rho + \rho^{-2} + \dots \rho^n\right) \tag{77}$$

As,

**SURF. Area of the revolution surface** F1<sup>→</sup><sup>1</sup>

*h* <sup>2</sup>

*<sup>h</sup>* <sup>2</sup> - <sup>1</sup>

*h\**arcsin (*m*) *a\*m* + *h\**arcsin (*m*)

*h\**arcsinh (*m*1) *a\*m*<sup>1</sup> + *h\**arcsinh (*m*1)

(*<sup>a</sup>* <sup>2</sup> <sup>+</sup> <sup>4</sup>*\*h* 2)3/2 - *<sup>a</sup>* <sup>3</sup>

*a* <sup>2</sup> + *h* <sup>2</sup>

<sup>1</sup> - (*ab* <sup>3</sup> *<sup>y</sup>* ) / <sup>2</sup> *<sup>a</sup> yb <sup>y</sup>* <sup>+</sup> *<sup>a</sup> yh <sup>y</sup>* <sup>+</sup> <sup>+</sup> *<sup>b</sup> yh <sup>y</sup> <sup>y</sup>*

Until this point the discussion has dealt with primary transmission of energy but, in a closed space, if the surfaces have some degree of reflectivity a significant part of the flux would be

Under such circumstance, the global balance of radiant power can be found through expression

Edir is defined as the direct power received while Eref stands for the reflected energy. The two quantities added yield the global balance of radiant energy Etot. If the problem entails several surfaces, expression 71 is expanded for an array of equations. To resolve it, we define before‐ hand the matrices Fd and Fr, whose elements are described as follows in a three-dimensional

> *F*11\**ρ*<sup>1</sup> *F*12\**ρ*<sup>2</sup> *F*13\**ρ*<sup>3</sup> *F*21\**ρ*<sup>1</sup> *F*22\**ρ*<sup>2</sup> *F*23\**ρ*<sup>3</sup> *F*31\**ρ*<sup>1</sup> *F*32\**ρ*<sup>2</sup> *F*33\**ρ*<sup>3</sup>

*Etot* =*Edir* + *Eref* (71)

) (72)

*<sup>h</sup>* <sup>2</sup> <sup>1</sup> - <sup>6</sup>*\*a\*h* <sup>2</sup>

<sup>2</sup>*π<sup>a</sup>* <sup>2</sup> *arcsen*(*m*) · (*<sup>h</sup>* <sup>+</sup> *am*) *a* · *m*

*<sup>m</sup>*<sup>=</sup> <sup>1</sup> - *<sup>a</sup>* <sup>2</sup>

<sup>2</sup>*π<sup>a</sup>* <sup>2</sup> *arcsenh* (*m*) · (*<sup>h</sup>* <sup>+</sup> *am*) *a* · *m*

> *<sup>π</sup> <sup>a</sup>* <sup>2</sup> <sup>+</sup> <sup>4</sup>*<sup>h</sup>* <sup>2</sup> <sup>3</sup> - *<sup>a</sup>* <sup>3</sup> 6*a* <sup>3</sup>

circular base *<sup>π</sup><sup>a</sup> <sup>a</sup>* <sup>2</sup> <sup>+</sup> *<sup>h</sup>* <sup>2</sup> <sup>1</sup> - *<sup>a</sup>*

<sup>4</sup>*π*( *<sup>a</sup> yb <sup>y</sup>* <sup>+</sup> *<sup>a</sup> yh <sup>y</sup>* <sup>+</sup> *<sup>b</sup> yh*

**Table 1.** Resume of form factors for curved surfaces.

*y* =<sup>8</sup> 5

<sup>3</sup> ) 1 *y*

**5. Interreflections amongst surfaces in a closed volume**

re-irradiated and the concepts of emitters and receivers entwine.

*Fd* =(

*<sup>m</sup>*<sup>=</sup> *<sup>a</sup>* <sup>2</sup>

Prolate semispheroid with circular base

22 Solar Radiation Applications

Oblate semispheroid with elliptic base

Revolution paraboloid with circular base

Revolution Ellipsoid

71,

fashion, (see figure 16):

Straight cone with

F1<sup>→</sup><sup>2</sup>

*a\*m a\*m* + *h\**arcsin (*m*)

*a\*m*<sup>1</sup> *a\*m*<sup>1</sup> + *h\**arcsinh (*m*1)

6*\*a\*h* <sup>2</sup> (*<sup>a</sup>* <sup>2</sup> <sup>+</sup> <sup>4</sup>*\*h* 2)3/2 - *<sup>a</sup>* <sup>3</sup>

> *a a* <sup>2</sup> + *h* <sup>2</sup>

(*ab* 3 *<sup>y</sup>* ) / 2 *<sup>a</sup> yb <sup>y</sup>* <sup>+</sup> *<sup>a</sup> yh <sup>y</sup>* <sup>+</sup> <sup>+</sup> *<sup>b</sup> yh <sup>y</sup> <sup>y</sup>*

$$\lim\_{n \to \infty} \left( \frac{\rho^{\frac{n+1}{n+1}} - 1}{\rho - 1} - 1 \right) = \frac{\rho}{1 - \rho} \tag{78}$$

$$E\_{ref} = E^\* \frac{\mathcal{W}}{A} \* \left(\frac{\rho}{1 \cdot \rho}\right) \tag{79}$$

In the precedent discussion ρ includes the mean of all reflection quotients ρ<sup>i</sup> inside the sphere, while E represents the direct power exiting from the source. Such expression would be technically applicable to all kinds of surfaces, but its accuracy dwindles when the actual volume is not akin to a sphere. If such is the case, equation 79 would be less acceptable.

Since the reflectivity of the internal surfaces can be changed on demand, the way to treat glazed elements or voids is to assign them a high absorption coefficient to ensure that those elements play a limited role in the global energy balance.

## **6. Conclusions**

An ever-increasing number of configuration factors for curved geometries, has been deducted. The authors have extracted the former in total conformity with the procedures of optical mechanics and thus the new factors can be termed as exact in contrast with other random or discretized methods.

This represents an indubitable advance of knowledge for radiative heat transfer that is already being implemented in computer models. However, the details of the simulation procedures are not discussed in this chapter in the credence that other scientists will arrive with perfect ease to the required algorithms.

Thus, this new form factors have been programmed in computer algorithms, creating a powerful tool that is able to enrich the repertoire of forms and spaces suitable for simulation. This procedure will benefit energy-conscious engineering and architecture, as has been demonstrated by the authors in previous publications [7, 8,9,10] Indeed, the prototypes based on the science of heat transfer are sure to progress in their accuracy and sophistication. Radiative devices and fixtures can be conceived departing from the findings exposed previ‐ ously on a more scientific basis and this will be beneficial to expand the innumerable boons of solar radiation.

Contemplating the ruins of the colossal statues of Ramses in Egypt, Shelley once wrote:

*My name is Ozymandias, King of Kings, Look on my works ye Mighty And Despair*

## **Acknowledgements**

Jose Cabeza would like to thank his family in Japan and Spain for failing to understand his work.

#### **Author details**

*Eref* =*E*\*

In the precedent discussion ρ includes the mean of all reflection quotients ρ<sup>i</sup>

play a limited role in the global energy balance.

**6. Conclusions**

24 Solar Radiation Applications

discretized methods.

solar radiation.

work.

**Acknowledgements**

ease to the required algorithms.

*W <sup>A</sup>* \*( *<sup>ρ</sup>*

while E represents the direct power exiting from the source. Such expression would be technically applicable to all kinds of surfaces, but its accuracy dwindles when the actual volume is not akin to a sphere. If such is the case, equation 79 would be less acceptable.

Since the reflectivity of the internal surfaces can be changed on demand, the way to treat glazed elements or voids is to assign them a high absorption coefficient to ensure that those elements

An ever-increasing number of configuration factors for curved geometries, has been deducted. The authors have extracted the former in total conformity with the procedures of optical mechanics and thus the new factors can be termed as exact in contrast with other random or

This represents an indubitable advance of knowledge for radiative heat transfer that is already being implemented in computer models. However, the details of the simulation procedures are not discussed in this chapter in the credence that other scientists will arrive with perfect

Thus, this new form factors have been programmed in computer algorithms, creating a powerful tool that is able to enrich the repertoire of forms and spaces suitable for simulation. This procedure will benefit energy-conscious engineering and architecture, as has been demonstrated by the authors in previous publications [7, 8,9,10] Indeed, the prototypes based on the science of heat transfer are sure to progress in their accuracy and sophistication. Radiative devices and fixtures can be conceived departing from the findings exposed previ‐ ously on a more scientific basis and this will be beneficial to expand the innumerable boons of

Contemplating the ruins of the colossal statues of Ramses in Egypt, Shelley once wrote:

Jose Cabeza would like to thank his family in Japan and Spain for failing to understand his

*My name is Ozymandias, King of Kings, Look on my works ye Mighty And Despair*

<sup>1</sup> - *<sup>ρ</sup>* ) (79)

inside the sphere,

Jose Maria Cabeza Lainez1,2\*, Jesus Alberto Pulido Arcas3 , Manuel-Viggo Castilla4 , Carlos Rubio Bellido4 and Juan Manuel Bonilla Martínez5


#### **References**

