# Innovative Tool to Determine Radiative Heat Transfer Inside Spherical Segments
## 4 Discussion  
To continue, we will discuss in this section, the applicability of the results derived from the new form factor equations for several usual configurations. In the first place, we will study the expressions deduced in Equations (24)-(28) for a new case of two tangent circles which form an arbitrary angle from 0 to \(\pi\) (Figure 6). If the surfaces \(\mathrm{A}_{1}\) and \(\mathrm{A}_{2}\) are equal, the postulate reduces to  
\[\mathrm{F}_{12}=\frac{2\mathrm{A}_{1}+\mathrm{A}_{3}(\mathrm{F}_{33}-1)}{2 \mathrm{A}_{1}}=\frac{1}{2}\Bigg{(}\frac{2\mathrm{A}_{1}-\mathrm{A}_{3}}{ \mathrm{A}_{1}}+\frac{\mathrm{A}_{3}^{2}}{\mathrm{A}_{\mathrm{s}}\mathrm{A}_{ 1}}\Bigg{)}=1+\frac{\mathrm{A}_{3}}{2\mathrm{A}_{1}}\Bigg{(}\frac{\mathrm{A}_{3 }}{\mathrm{A}_{\mathrm{s}}}-1\Bigg{)} \tag{29}\]  
\[\mathrm{F}_{13}=\frac{\mathrm{A}_{3}}{2\mathrm{A}_{1}}\Bigg{(}1-\frac{\mathrm{A }_{3}}{\mathrm{A}_{\mathrm{s}}}\Bigg{)} \tag{30}\]  
Should the angle between the said circles stand at \(\pi/2\) (perpendicular), by common geometry rules we can substitute the respective values of the magnitudes in Equation (25) to obtain \(\mathrm{F}_{12}\). The area of the two equal circles A\({}_{1}\) = A\({}_{2}\) is \(\pi\)R\({}^{2}/2\), the former Equation (29) can then be cleared to  
\[\mathrm{F}_{12}=1+\frac{\mathrm{A}_{3}(\mathrm{F}_{33}-1)}{\pi\mathrm{R}^{2}}=1 +\frac{\mathrm{A}_{3}^{2}}{\mathrm{A}_{5}\pi\mathrm{R}^{2}}-\frac{\mathrm{A}_{ 3}}{\pi\mathrm{R}^{2}} \tag{31}\]  
A\({}_{3}\) is obtained as the total area of the sphere, \(4\pi\)R\({}^{2}\), minus two spherical segments whose lateral area equates to \(\pi\)(a\({}^{2}\) + h\({}^{2}\)), where a represents half of the base of the segment and h is the perpendicular distance from the base to the top of the segment. Therefore, by simple trigonometry, h is R(1\(-\surd 2/2\)) and a equates to R\(\surd 2/2\). \[\mathrm{a}^{2}=\frac{\mathrm{R}^{2}}{2};\,\mathrm{h}^{2}=\mathrm{R}^{2}\bigg{(} \frac{3}{2}-\surd 2\bigg{)}=\frac{\mathrm{R}^{2}}{2}\bigg{(}3-2\surd 2\bigg{)} \tag{32}\]  
\[\mathrm{a}^{2}+\mathrm{h}^{2}=\frac{\mathrm{R}^{2}}{2}\bigg{(}4-2\surd 2\bigg{)}\]  
\[\mathrm{A}_{3}=4\pi\mathrm{R}^{2}-2\pi\frac{\mathrm{R}^{2}}{2}\bigg{(}1+3-2 \surd 2\bigg{)}=4\pi\mathrm{R}^{2}-2\pi\mathrm{R}^{2}\Big{(}2-\surd 2\bigg{)}=\pi\mathrm{R}^{2}2\sqrt{2} \tag{33}\]  
Squaring this area in accordance, we obtain,  
\[\mathrm{A}_{3}^{2}=8\pi^{2}\mathrm{R}^{4} \tag{34}\]  
Substituting into Equation (29),  
\[\mathrm{F}_{12}=1+\frac{8\pi^{2}\mathrm{R}^{4}}{4\pi\mathrm{R}^{2}\pi\mathrm{ R}^{2}}-\frac{\pi\mathrm{R}^{2}\Big{(}2\surd 2\Big{)}}{\pi\mathrm{R}^{2}} \tag{35}\]  
Thus, the sought factor \(\mathrm{F}_{12}\) is obtained as  
\[\mathrm{F}_{12}=1+2-2\surd 2=3-2\surd 2=0.1715 \tag{36}\]  
For a general case of any subtended angle \(\alpha=2\theta\), the area of the circles as compared to the sphere yields  
\[\mathrm{A}_{1}=\mathrm{A}_{2}=\pi\mathrm{R}^{2}\cos^{2}\theta \tag{37}\]  
To compare with the above perpendicular case, remember that \(\cos\theta\) was there equal to \(\cos(\pi/4)=\surd 2/2\), and \(\mathrm{A}_{1}=\mathrm{A}_{2}=\pi\mathrm{R}^{2}/2\). In this more encompassing case, the spherical caps resulting from the limit sections, present the parameters a = R/2 cos\(\theta\) and h = R(1 - sin\(\theta\)). \[\mathrm{F}_{13}=\frac{\mathrm{A}_{3}}{2\mathrm{A}_{1}}\Bigg{(}1-\frac{\mathrm{A }_{3}}{\mathrm{A}_{\mathrm{s}}}\Bigg{)} \tag{30}\]0 Form factor calculation between two circles with a tangent point forming an arbitrary angle \(\alpha\).