1. Introduction To Solar Panel Technology

2. Surface Manifold Hypothesis

Figure 1. Shows Thin Fabric printable solar cells which can be folded in any shape

Figure 1. Shows Thin Fabric printable solar cells which can be folded in any shape

Figure 2. Dimension of flat PV Panel before performing surface manifold

Figure 2. Dimension of flat PV Panel before performing surface manifold

2.1. Curving Surface (along x-axis)

In this type of surface manifold, one side of the plate i.e. $x-axis$ is curved using either wavy (2) or zigzag (3) curve see figure 3. However, $y-axis$ is left straight (uncurved). $\lambda$ is the wave length, $\delta$ is the wave amplitude, this is an extension of zigzag and wavy line in article [11] but this is in 3D surface.

$$f\left(x\right)=\frac{\delta }{4}\cos \left(\frac{2\pi x}{\lambda }\right)$$

(2)

$$f\left(x\right)=\frac{\delta }{2\pi }{\cos }^{-1}\left(\cos \left(\frac{2\pi x}{\lambda }\right)\right)$$

(3)

To find the surface area of the above (2) and (3), see (4).

$$S=\left(y_2-y_1\right){\int }_{x_1}^{x_2}\sqrt{1+{\left(\frac{df\left(x\right)}{dx}\right)}^2} dx$$

(4)

Figure 3. Folded surface along x-axis (a) is Zigzag surface from (3) and (b) is wavy surface from (2).

Figure 3. Folded surface along x-axis (a) is Zigzag surface from (3) and (b) is wavy surface from (2).

2.2. Curving Surface (along x-axis, and y-axis)

Another approach is to extend the above curve surface from the x-axis only to both x-axis and y-axis. See equations (5) and (6).

$$f\left(x,y\right)=\frac{\delta }{4}\left(\sin \left(\frac{2\pi }{\lambda }x\right)+\sin \left(\frac{2\pi }{\lambda }y\right)\right)$$

(5)

$$f\left(x,y\right)=\frac{\delta }{2\pi }\left({\sin }^{-1}\left(\sin \left(\frac{2\pi }{\lambda }x\right)\right)+{\cos }^{-1}\left(\cos \left(\frac{2\pi }{\lambda }y\right)\right)\right)$$

(6)

Figure 4. (a) Wavy Surface using equation (5) and (b) Zigzag Surface using equation (6)

Figure 4. (a) Wavy Surface using equation (5) and (b) Zigzag Surface using equation (6)

Furthermore, we can use the ripple concept to generate a circular wave (7), a circular zigzag (8), or a rectangular wavy surface like in (9) to produce another surface manifold. $p$ is a constant value for modifying the surface in equation (9).

$$f\left(x,y\right)=\frac{\delta }{4}\left[\sin \left(\frac{2\pi }{\lambda }\left[x^2+y^2\right]\right)+\sin \left(\frac{2\pi }{\lambda }\left[x^2+y^2\right]\right)\right]$$

(7)

$$f\left(x,y\right)=\frac{\delta }{2\pi }\left[{\sin }^{-1}\left(\sin \left(\frac{2\pi }{\lambda }\left[x^2+y^2\right]\right)\right)+{\sin }^{-1}\left(\sin \left(\frac{2\pi }{\lambda }\left[x^2+y^2\right]\right)\right)\right]$$

(8)

$$f\left(x,y\right)=\frac{\delta }{4}\left[\cos \left(\frac{2\pi }{\lambda }[x^2+p]\right)+\cos \left(\frac{2\pi }{\lambda }{[y}^2+p]\right)\right]$$

(9)

Figure 5. (a) Circular Wavy Ripple Plate from Equation (7); (b) Circular Zigzag Ripple Plate from Equation (8); and (c) Rectangular Wavy Ripple from Equation (9).

Figure 5. (a) Circular Wavy Ripple Plate from Equation (7); (b) Circular Zigzag Ripple Plate from Equation (8); and (c) Rectangular Wavy Ripple from Equation (9).

The surface area of the above surface manifold produced by curving along the x and y- axes can be obtained by using the below equation (10). This solution can be obtained numerically using MATLAB or any other numerical solver.

$$S={\int }_{x_1}^{x_2}{\int }_{y_1}^{y_2}\sqrt{1+{\left(\frac{d}{dx}f\left(x,y\right)\right)}^2+{\left(\frac{d}{dy}f\left(x,y\right)\right)}^2}dxdy$$

(10)

Since we have defined our equation using sin(x) and cos(x), to determine which of these produces longer length either sin(x) or cos(x) function, let’s find the length of curves produced by the two functions using equation (11). Given that $x_1=0, x_2=3$ so for $f\left(x\right)=\mathrm{s}\mathrm{i}\mathrm{n}\left(x\right)$ and $f\left(x\right)=\mathrm{c}\mathrm{o}\mathrm{s}\left(x\right)$.

$$l={\int }_{x_0}^{x_2}\sqrt{1+{\left(\frac{d}{dx}f\left(x\right)\right)}^2}dx$$

(11)

Solving the equation (11), the result shows that the length for sin(x) is 3.6203 and for cos(x) is 3.6781.

The following conclusion is made: cos(x) provides more curve surface than its counterpart sin(x). So, throughout this text, cos(x) can be used instead of sin(x) for an optimal surface manifold where those functions are applied.

2.3. Protruding Shapes on the Panel Surface

a) Cone Shape:

Given that the surface area of a cone is given as $S=\pi r\sqrt{r^2+h^2}$, and the supposed total number of cones that can fit inside a given plate surface is $n$, therefore the total surface area of the panel plus the protruding cones is given as: -

$$S=\left(\left(x_2-x_1\right)\left(y_2-y_1\right)-n\pi r\left[r-\sqrt{r^2+h^2} \right]\right)$$

(12)

b) Pyramidal shape:

In this way, the surface of the panel can be protruded with many pyramidal shapes, as shown in Figure 6 below.

$$S=2a\sqrt{{\left(\frac{a}{2}\right)}^2+h^2}$$

(13)

Figure 6. Square base Pyramidal Shape

Figure 6. Square base Pyramidal Shape

For comparison of the pyramidal surface area with cone shape and hemisphere shape, the pyramidal with a square base whose side length “a” is obtained from circle base of the cone and hemisphere whose radius is $r$, so side of the square base of the pyramid is $a=r\sqrt{\pi }$. This ensure that all have same base area occupied.

Total surface area of the panel’s top part surface with $n$ given number of pyramidal shapes is.

$$S=\left(\left(x_2-x_1\right)\left(y_2-y_1\right)-na\left[a-2\sqrt{{\left(\frac{a}{2}\right)}^2+h^2} \right]\right)$$

(14)

c) Hemisphere:

Curve Surface area of a hemisphere without base area is given as $2\pi r^2$. Therefore, the surface area of the panel with $n$ given number of hemispheres protruding out of its surface is given as.

$$S=\left(x_2-x_1\right)\left(y_2-y_1\right)+n\pi r^2$$

(15)

3. Geometry Surface Area Analysis.

Table 1. Shows Comparison of some selected shape surface area having same base area.

Figure 7. shows, from (a) to (g), different surface manifolds for increasing the surface area of the given Plate designed using ANSYS Space-Claim and Design Modular

Figure 7. shows, from (a) to (g), different surface manifolds for increasing the surface area of the given Plate designed using ANSYS Space-Claim and Design Modular

Table 2. shows the sample curve and protruding surface of a panel and their surface area, plus the percentage increase in surface area from Figure 7.

4. Discussion

5. Conclusion

6. Future Extension

References

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