# A New Principle for Building Simulation of Radiative Heat Transfer in the Presence of Spherical Surfaces

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{3}, the so-called form factor F

_{33}, in other words, the fraction of energy emitted by such sphere’s fragment over itself, is defined as the ratio between the area of the said fragment and the whole sphere (Equation (1)). Hence, it yields ½ for a hemisphere, ¼ for a quarter of sphere and successively. Due to its simplicity, it constituted a paramount finding [13,14,19,20].

_{3}divided by the total area of the sphere, A

_{3}/A

_{sphere}[13], as in Figure 1.

_{1}and E

_{2}represent the amount of energy (in W/m

^{2}) emitted in Lambertian mode by each surface 1 and 2 [21].

**θ**and

_{1}**θ**, stand for the inclination angles formed (in radians) by the perpendiculars to the unit area elements of areas

_{2}**dA**and

_{1}**dA**,

_{2}**r**

_{12}being the random radius vector (in meters) that connects both elementary surfaces

**dA**and

_{1}**dA**[21].

_{2}_{1}− A

_{2}) that enclose within them a fragment of the same sphere (A

_{3}), the form factor from surface 1 to surface 2 can be defined as:

_{33}is the factor of a sphere’s fragment over itself that was defined previously in Cabeza-Lainez’s first principle as the ratio between the area of the said fragment and the whole sphere (Equation (1)), which resulted in a value of ½ for a hemisphere, ¼ for a quarter of sphere and successively. It is written A

_{3}/A

_{s}. Where A

_{s}stands for the total area of the sphere or the constant 4πR

^{2}, and where R is the radius of the said sphere [13].

_{1}and A

_{2}are, as described, the respective areas of surfaces 1, 2 (segments of a circle), and A

_{3}is the comprised fragment of a sphere (Figure 3 and Figure 4). Determination of areas 1 and 2 offers no particular problem, but finding the surface of a fragment of a sphere might at times require the employ of spherical trigonometry and the internal angles involved, for instance, through Girard’s theorem, to be able to substitute in the formula of the third property.

_{1}and A

_{2}, if they pass or not through the center of the sphere or even if they possess any tangent point. That is, both planar surfaces can even be independent circular sections of the sphere. The form factors between the sphere and any of the planar surfaces cannot be obtained at present in any other way or procedure, hence the novelty presented. Projective or geometric methods were detained at the inaccessibility of finding the projection of a conical section generated by a sphere on the view cone [25]; finite element and other numerical procedures encountered the obstacle of non-discretizable surfaces when curves were involved [26].

## 3. Methodology of Proof and Results

_{11}+ F

_{12}+ F

_{13}+ … F

_{1N}= 1

_{33}, which has been previously defined in Cabeza-Lainez’s first principle of form factors (Equation (1)).

_{33}is, therefore, a constant for a given sub-surface pertaining to the total sphere [22].

_{31}+ F

_{32}+ F

_{33}= 1;

_{12}+ F

_{13}= 1;

_{21}+ F

_{23}= 1;

_{12}= 1 − F

_{13};

_{23}= 1 − F

_{21};

_{13}= 1 − F

_{12};

_{1}F

_{12}= A

_{2}F

_{21}

_{31}= 1 − F

_{32}− F

_{33};

_{3}, we obtain

_{12}, we can immediately find F

_{21}by using reciprocity theorem (8)

_{23}, F

_{13}, F

_{32}(9) and F

_{31}(10 and 11), and the exact exchanges between the sphere and the planar limiting surfaces are found.

#### 3.1. Perpendicular Circles within the Sphere

^{2}/2 and A

_{1}= A

_{2}, such equation turns out to be:

_{3}is found by subtraction of the area of the two identical caps of the sphere; the area of one segment is π(a

^{2}+ h

^{2}), h is the height and a the radius of the base circumference of the cap. Theretofore, a = R√2/2 and h = R(1 − √2/2), where R is the radius of the sphere.

_{12}is obtained as

_{12}would be, as it is known, 1/5 = 0.2, a very similar quantity which attests to the validity of our calculations. Knowing the F

_{12}and the F

_{33}form factors of the problem, the radiant exchanges between the circles and the enclosing sphere, namely, F

_{23}, F

_{13}, F

_{32}and F

_{31,}are obtained from Equations (5)–(7). This implies a complete novelty even for such simple and usual arrangement of surfaces.

_{12}to evaluate the radiative exchanges that occur in Figure 8 and Figure 10, such exchanges, are unfeasible to calculate in an exact manner even for a perpendicular angle not to mention any other inclinations of the semicircles [28,29]. Therefore, the fourth principle can be considered as a pivotal advance, which is now subsumed in the third property, as we will show below. As it happens, the fourth property was discovered before the third property, but being less general, the author has chosen to alter their correlative order.

#### 3.2. Calculations for an Eighth of Sphere

_{3}= πR

^{2}/2.

_{1}= A

_{2}= πR

^{2}/2 and subsequently both areas are equal to A

_{3}, to find the above quantity (Equation (26)).

#### 3.3. Estimations for a Fifth of Sphere

#### 3.4. The Case of a Fourth of Sphere

#### 3.5. Calculations for a Third of Sphere

#### 3.6. Null Base Case, Half of a Sphere

#### 3.7. Cases Obtained by Numerical Calculus

## 4. Complementary Considerations for Inter-Reflections

**E**is the fraction of direct energy and

_{dir}**E**is the reflected fraction. If these quantities are added, we obtain the global amount of radiation

_{ref}**E**.

_{tot}**F**and

_{d}**F**. For the volume enclosed by the three surfaces (see Figure 5), such matrices would have the ensuing form [32,33,34,35]:

_{r}**F**as previously found, yielding the energy transfer between the respective surfaces involved. Here, ρ

_{ij}_{i}represents the ratio of reflection (direct or otherwise) assigned to a particular element i [2,14].

_{d}(Equation (36)), the third element of the diagonal is not null as the

**F**factor (with double i) for the same sub-indexes element has definite values for non-planar surfaces as the sphere, unlike the exchange that takes place in a cuboid [1].

_{ii}_{12}, and then we have the form factor of this circle to the corresponding spherical cap ${F}_{12}^{1}$, the final factor between the more remote circle and the said cap F would be the product of the two factors, that is [33],

## 5. Discussion

#### 5.1. Example of Application

**F**= 0.085805

_{1+2, 3+4}**F**= 0.228031

_{2 3}**F**= 0.04815

_{1+2, 3}**F**= 0.303317

_{2, 3+4}**A**= 36 × 14 × 0.085805 + 14 × 16 × 0.228031 − 36 × 14 × 0.04815 − 14 × 6 × 0.303317

_{1}F_{14}= 14 × 30 ×

**F**= 12.31908

_{14}**F**= 0.029331

_{14}**F**= 0.125705 is the desired value to find the exchange of energy.

_{41}**F**= 0.125705, the value of illuminance on surface 4 would be 3733.53 lux, the lower surface of the quarter of sphere as we have just discussed in the fourth principle (Section 3.4), receives exactly ¼ of the former value, that is: 933.38 lux. Accordingly, the floor plan, by simple operations not presented here, would receive an average of 350 lux (See Figure 20).

_{41}#### 5.2. Recollection on the Findings

- The total exchanges of radiant energy between two planar surfaces that cut a sphere in any way and the comprising fragment of the same sphere. The said planar surfaces can intersect, be tangent or stay completely independent of one another. The exchange case for arbitrary intersections and of these with the fragment of the sphere is an absolute novelty which cannot be obtained by any other procedure. The remaining non-intersecting layout is perfectly encompassed under a modern and inclusive formulation, and the exchanges between the disks and the inner sphere are completely innovative.
- The possible inter-reflections of the compound despite the existence of a curved surface. This was not previously solved in the references since reflections were always limited to cuboidal spaces.
- The introduction of the commutative property and its subsequent algebra in form factors on a purely surface-to-surface basis for volumes that integrate the circle in their composition.

## 6. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

**Nomenclature of canonical equation,**

**E**and

_{1}**E**are the amounts of energy (in W/m

_{2}^{2}) emitted in Lambertian fashion by surfaces

**1**and

**2**.

**θ**and

_{1}**θ**, stand for the inclination angles (in radians) formed by the perpendiculars to the unit area elements of area

_{2}**dA**and

_{1}**dA**.

_{2}**r**

_{12}is the arbitrary vector (in m) that connects the surfaces

**dA**and

_{1}**dA**.

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**Figure 1.**Hemisphere with fragment of sphere A

_{3}in red tones, which is used to determine F

_{33}, or the factor of the fraction over itself.

**Figure 2.**Radiative exchanges due to manifold surface sources A

_{1}and A

_{2}, separated by an arbitrary distance

**r**.

**Figure 3.**Profile view of surfaces A

_{1}, A

_{2}(in orange) and A

_{3}(in cyan) forming part of a complete sphere of radius R.

**Figure 4.**Imaginary perspective view of A

_{1}a circular segment with radius r

_{1}and another A

_{2}of radius r

_{2}, in concordance with A

_{3}(spherical fragment).

**Figure 5.**The three surfaces involved in radiative exchange for the third property. Two planar sections (usually circular) and a spherical fragment that works as an enclosure of both sections.

**Figure 9.**Graphic representation of the Cabeza-Lainez fourth principle (Equation (24)) of form factors between semicircles, marking its value for angle = Pi/2 =1.57 that is 0.25 or 1/4.

**Figure 13.**Average calculation by Matlab of the configuration factor from a semicircle over another perpendicular of equal radius as discussed previously. View in perspective.

**Figure 14.**Graph of the numerical calculation performed with Matlab of the average between perpendicular semicircles. View in perspective.

**Figure 15.**Graphical representation of the calculations for the configuration factor in a point-by-point distribution between a semicircle and a perpendicular circular segment.

**Figure 16.**Detail of numerical determination by Matlab of the average factor of Figure 15 in planar view to appreciate more clearly the shape of the circular segment.

**Figure 19.**Simplified graph of the layout of the Scenic Triclinium in Tivoli (Rome), excerpted to study the radiation exchange between the reflecting pool and the adjoining domed structure.

**Figure 20.**Illuminance distribution under the dome of the Triclinium due only to the thermal (arched) aperture in the summer’s solstice.

**Figure 21.**Ilja Doganoff. Railway Sheds of conoidal shape and semicircular clerestories in Bulgaria (1958). Source: Author.

**Figure 22.**Interior spherical dome of the church of St. Louis (1730). Seville. Spain. Source: Author.

**Figure 23.**The Latina Market (1960) with spherical vaults and openings. Madrid. Spain. Source: Author.

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**MDPI and ACS Style**

Cabeza-Lainez, J.
A New Principle for Building Simulation of Radiative Heat Transfer in the Presence of Spherical Surfaces. *Buildings* **2023**, *13*, 1447.
https://doi.org/10.3390/buildings13061447

**AMA Style**

Cabeza-Lainez J.
A New Principle for Building Simulation of Radiative Heat Transfer in the Presence of Spherical Surfaces. *Buildings*. 2023; 13(6):1447.
https://doi.org/10.3390/buildings13061447

**Chicago/Turabian Style**

Cabeza-Lainez, Joseph.
2023. "A New Principle for Building Simulation of Radiative Heat Transfer in the Presence of Spherical Surfaces" *Buildings* 13, no. 6: 1447.
https://doi.org/10.3390/buildings13061447