4. Calculation of the optical-energetical characteristics of the LSF: the general approach

In the helio-engineering practice, different techniques for evaluation of energetical characteristics of mirror concentrating systems (MCS) are employed, depending on the purpose, demanded accuracy, etc [23–29]. However, if a big MCS as a solar furnace in Odeillo (France) or Large Solar Furnace near Tashkent (Uzbekistan) is concerned, there are specific features which must be taken into account.

An important feature of a big-scale MCS is the use of multi-heliostat system to create illumination field on its focal zone. Thanks to this feature, there is a

The reflecting surface of the concentrator is a rectangular-stepped cutting of a paraboloid of revolution with a focal distance of 18 m. The height of the midsection of the concentrator is 42 m, and the width is 54 m. The total area of the midsection

rhomb-shaped facets with sides of 447 mm and different apex angles are installed

imately 2 ha of ground area. Technical characteristics of the installation are also

Some advantages of the LSF should be noted as compared with other solar furnaces, resulting from the specific nature of the optical circuit of the furnace and latest engineering developments used in the furnace's operation: the heliostat automated control system (HACS), computer vision system (CVS), photometers, radiometers, melting furnaces of different designs, modern meteorological station in the territory of the furnace, high-accuracy methods for arranging the furnace's

Some projects have been implemented in recent years at the entity "Sun" of the Academy of Sciences of the Republic of Uzbekistan, where the LSF is located; the aim of these projects is to improve its operational characteristics and extend its

One of the most significant finished projects is the commissioning of the modern meteorological station on the territory of the "Sun" entity in 2013 with the financial support of the Asian Development Bank. This station is equipped with highfrequency pyranometers and pyrheliometers to measure solar radiation. The station measures global horizontal radiation and diffuse horizontal and beam normal solar radiation [20]. As is known, beam normal solar radiation is important for the LSF, and depending on its level, it is possible to implement certain high-temperature processes. With the commissioning of the meteorological station, it is possible to define more correctly and promptly the energy characteristics of the furnace using the calibration coefficients. Certain types of operations, such as, preventive maintenance, adjustment and cleaning of the reflecting elements, regulation of the solar sensors of the heliostats, different high-temperature processes requiring certain process modes in the focal zone of the furnace, etc., are performed at the LSF depending on the climatic conditions and solar radiation level. With the intensive operation of the LSF, planning of the work on it is important, and the selection of the type of operations is primarily related to the radiation environment of the location. For example, according to long-term observations, many high-

temperature processes are carried out when the beam solar radiation has a value of 600 W/m2 or larger. At present, various types of operations are planned in advance at the LSF according to the time of day and season based on statistical processing of

Efficient operation of the LSF significantly depends on the technical parameters of the HACS for tracking the Sun due to the specific nature of its optical circuit. At present, the capabilities of the existing HACS developed from outmoded element bases do not meet modern requirements, although it functions. This is why a new modern HACS has been developed at the LSF in recent years; this will make it

reflecting mirrors, various tests and optical benches, etc [14–21].

In total, there are 22,790 facets; the total area of the reflecting mirrors (facets) is

The general pattern of the device is shown in Figure 2a and the general view of

; the heliostat field, process tower, and concentrator occupy approx-

. The concentrator consists of 214 blocks. 50

of the reflecting surface is 1906 m<sup>2</sup>

described in the works [9, 10, 12, 13].

the installation were shown in Figure 2b.

3. Operating experience of the LSF

about 5200 m<sup>2</sup>

functional capabilities.

meteorological data.

114

on each block. The thickness of mirrors is 5 mm.

A Guide to Small-Scale Energy Harvesting Techniques

possibility of a flexible control upon flux density distribution by individual control of each heliostat operation regime. This is a very important point for many technological processes and special investigations.

system connected with the particular heliostat. In this case the integrating bound-

Thousand kW High-Temperature Solar Furnace in Parkent (Uzbekistan) – Energetical…

Let us determine the Jacobian of this transformation. The origin of the new system O is placed at the heliostat's center; axes OX1 and OY1 are directed along the height and width correspondingly and the OZ1 axis along the heliostat's normal. To determine the Jacobian transformation, we apply geometrical definitions. Consider elementary area dS on the surface of the paraboloidal concentrator. For the normal

!

A projection of the elementary surface on dS along the direction of the vector K

In its turn a projection of this area onto the heliostat surface is determined from

! <sup>¼</sup> dS N ! <sup>g</sup> K � � !

Nmz

N ! <sup>m</sup> K � � !

ð Þ 0; 0; 1 we obtain obvious formula

� � � � � �

. Substituting explicit expressions

Nmz

N ! <sup>m</sup> K � � !

dSK <sup>¼</sup> dS N ! mK !

> N ! <sup>g</sup> K

On the other hand, the projection of the elementary area onto the midship

dSm <sup>¼</sup> dx dy <sup>¼</sup> dS

N ! <sup>g</sup> K � � !

We note that the above presented vector expressions are written taking into

<sup>m</sup> and K !

� �ðKx sin hn <sup>þ</sup> Ky cos hn cos An <sup>þ</sup> Kz cos hn sin An p

D ¼ j j cos hn cos An

From these relations we obtain the following expression for the Jacobian

� � � � � �

� � � � ¼

! <sup>g</sup> , N !

!

<sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>y</sup><sup>2</sup> <sup>þ</sup> <sup>p</sup><sup>2</sup> <sup>p</sup> ; �<sup>y</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>y</sup><sup>2</sup> <sup>þ</sup> <sup>p</sup><sup>2</sup> <sup>p</sup> ; <sup>p</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>y</sup><sup>2</sup> <sup>þ</sup> <sup>p</sup><sup>2</sup> <sup>p</sup>

!

(2)

� � � � � �

aries will coincide with the heliostat borders.

DOI: http://dx.doi.org/10.5772/intechopen.83411

<sup>m</sup> <sup>¼</sup> �<sup>x</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

where p is the focal parameter of the paraboloid.

dSg <sup>¼</sup> dx<sup>1</sup> dy<sup>1</sup> <sup>¼</sup> dSK

<sup>D</sup> <sup>¼</sup> dSm dSg � � � �

for these vectors into the last formula, finally we get

section of the concentrator dSm has the form

account unit normal of the vectors N

xKx þ yKy þ pKz

For the particular case, when K

<sup>m</sup> at this point, we have

N !

at this point will be equal to

the expression

transformation:

D ¼

117

� � � � � �

N !

Exposure distribution of the whole system is the sum of individual exposure distributions created by particular heliostats. Because of this fact, it is enough to consider the problem of illumination distribution created by a particular heliostat for arbitrary moment and given direction of beams reflected from the heliostat.

We will use the Cartesian coordinate system with axes: X (direction to Zenith), Y (to West) and Z (to South); the center of this system is in the focus of the furnace. Let the direction of beams reflected from the heliostat be characterized by a unit

vector K ! Kx;Ky;Kz � �. Then the unit vector of the heliostat's normal is determined by the expression

$$
\overrightarrow{N}\_{\xi} = \frac{\overrightarrow{\mathcal{S}} + \overrightarrow{K}}{\sqrt{2\left(1 + \overrightarrow{\mathcal{S}}\overrightarrow{K}\right)}}
$$

where S ! is the coordinate of the Sun's unit vector

$$\begin{aligned} S\_x &= \sin h\_t = \cos \rho \cos \delta \cos kt + \sin \rho \sin \delta \\ S\_{\mathcal{Y}} &= \cos h\_t \sin A\_{\mathcal{t}} = \cos \delta \sin kt \\ S\_x &= \cos h\_t \cos A\_{\mathcal{t}} = \sin \rho \cos \delta \cos kt - \cos \rho \sin \delta \end{aligned}$$

where δ is the declination of the Sun, k = 15 deg./h, φ is the geographical latitude, and t is time.

The angle coordinates of the heliostat An and hn are determined by expressions

$$N\_{\rm gx} = \sin h\_{\rm n}, \quad N\_{\rm gy} = \cos h\_{\rm n} \cos A\_{\rm n}, \quad N\_{\rm gz} = \cos h\_{\rm n} \sin A\_{\rm n}$$

It is well known that the energy density at arbitrary point A(xa, ya, za) of the receiver is determined by integration over the last reflecting surface according to the formula [23]:

$$E = \int\_{\boldsymbol{\alpha}} B d\_{\boldsymbol{\alpha}\_x} = \int\_{\boldsymbol{\alpha}} B \frac{d\mathbf{S} \left(\overrightarrow{\mathbf{N}}\_m \, \overrightarrow{\mathbf{M}} \overrightarrow{\mathbf{A}}\right)}{\boldsymbol{\Lambda} \boldsymbol{\Lambda}^4} \left(\overrightarrow{\mathbf{M}} \overrightarrow{\mathbf{A}} \, \overrightarrow{\mathbf{N}}\_A\right) \tag{1}$$

where N ! <sup>m</sup>, normal to the surface at point M; dS, elementary area around point M; N ! <sup>A</sup>, normal of the receiving surface; B, energetical brightness in a reflected beam.

A generalized approach for determining B is presented in [23].

Integration boundaries in this formula are determined by the coordinate of the heliostat center position, direction of the reflected beams K ! , and asimuthal and local angles А<sup>n</sup> and hn, which determine the heliostat's orientation.

As it is known in large-scale MCS, usually each heliostat illuminates the particular target area on the concentrator at the given moment [13]. It is easy to see that in the case of rectangle heliostat, the projection of its borders onto the concentrator midship section (a projection of concentrator surface into XY plane) will be different ellipses. Determination of the integration boundaries is a rather difficult problem due to complicated shapes. That is why it is desirable to move to the coordinate Thousand kW High-Temperature Solar Furnace in Parkent (Uzbekistan) – Energetical… DOI: http://dx.doi.org/10.5772/intechopen.83411

system connected with the particular heliostat. In this case the integrating boundaries will coincide with the heliostat borders.

Let us determine the Jacobian of this transformation. The origin of the new system O is placed at the heliostat's center; axes OX1 and OY1 are directed along the height and width correspondingly and the OZ1 axis along the heliostat's normal. To determine the Jacobian transformation, we apply geometrical definitions. Consider elementary area dS on the surface of the paraboloidal concentrator. For the normal

N ! <sup>m</sup> at this point, we have

possibility of a flexible control upon flux density distribution by individual control of each heliostat operation regime. This is a very important point for many techno-

Exposure distribution of the whole system is the sum of individual exposure distributions created by particular heliostats. Because of this fact, it is enough to consider the problem of illumination distribution created by a particular heliostat for arbitrary moment and given direction of beams reflected from the heliostat. We will use the Cartesian coordinate system with axes: X (direction to Zenith), Y (to West) and Z (to South); the center of this system is in the focus of the furnace. Let the direction of beams reflected from the heliostat be characterized by a unit

> Ng !

is the coordinate of the Sun's unit vector

Sy ¼ coshs sinAs ¼ cos δ sinkt

<sup>¼</sup> <sup>S</sup> ! þ K ! ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1þ S ! K � � ! r

Sx ¼ sinhs ¼ cos φ cos δ coskt þ sin φ sin δ

Sz ¼ coshs cosAs ¼ sin φ cos δ coskt � cos φ sin δ

where δ is the declination of the Sun, k = 15 deg./h, φ is the geographical latitude,

The angle coordinates of the heliostat An and hn are determined by expressions

Ngx ¼ sin hn, Ngy ¼ cos hn cos An, Ngz ¼ cos hn sin An

It is well known that the energy density at arbitrary point A(xa, ya, za) of the receiver is determined by integration over the last reflecting surface according to

dS N!

<sup>A</sup>, normal of the receiving surface; B, energetical brightness in a reflected

Integration boundaries in this formula are determined by the coordinate of the

As it is known in large-scale MCS, usually each heliostat illuminates the particular target area on the concentrator at the given moment [13]. It is easy to see that in the case of rectangle heliostat, the projection of its borders onto the concentrator midship section (a projection of concentrator surface into XY plane) will be different ellipses. Determination of the integration boundaries is a rather difficult problem due to complicated shapes. That is why it is desirable to move to the coordinate

<sup>m</sup> MA � � �!

<sup>m</sup>, normal to the surface at point M; dS, elementary area around point

MA<sup>4</sup> MA

�! �NA ! Þ

!

, and asimuthal and local

(1)

�

� �. Then the unit vector of the heliostat's normal is determined

logical processes and special investigations.

A Guide to Small-Scale Energy Harvesting Techniques

vector K !

by the expression

where S !

and t is time.

the formula [23]:

where N !

M; N !

beam.

116

E ¼ ð

ω

Bd<sup>ω</sup><sup>z</sup> ¼

heliostat center position, direction of the reflected beams K

angles А<sup>n</sup> and hn, which determine the heliostat's orientation.

ð

ω B

A generalized approach for determining B is presented in [23].

Kx;Ky;Kz

$$\vec{N}\_m = \left(\frac{-\chi}{\sqrt{\mathbf{x}^2 + \mathbf{y}^2 + \mathbf{p}^2}}, \frac{-\chi}{\sqrt{\mathbf{x}^2 + \mathbf{y}^2 + \mathbf{p}^2}}, \frac{p}{\sqrt{\mathbf{x}^2 + \mathbf{y}^2 + \mathbf{p}^2}}\right)$$

where p is the focal parameter of the paraboloid.

A projection of the elementary surface on dS along the direction of the vector K ! at this point will be equal to

$$d\mathcal{S}\_K = \frac{d\mathcal{S}}{\vec{N}\_m \vec{K}}$$

In its turn a projection of this area onto the heliostat surface is determined from the expression

$$d\mathbf{S}\_{\mathbf{g}} = d\mathbf{x}\_1 d\mathbf{y}\_1 = \frac{d\mathbf{S}\_K}{\vec{N}\_{\mathbf{g}} \ \vec{K}} = \frac{d\mathbf{S}}{\left(\vec{N}\_{\mathbf{g}} \ \vec{K}\right) \left(\vec{N}\_m \ \vec{K}\right)}$$

On the other hand, the projection of the elementary area onto the midship section of the concentrator dSm has the form

$$d\mathbb{S}\_{\mathfrak{m}} = d\mathfrak{x} \, dy = \frac{d\mathbb{S}}{N\_{\mathfrak{m}\mathfrak{x}}}.$$

From these relations we obtain the following expression for the Jacobian transformation:

$$D = \left| \frac{d\mathbb{S}\_m}{d\mathbb{S}\_\emptyset} \right| = \left| \frac{\left(\overrightarrow{N}\_\emptyset \,\overrightarrow{K}\right)\left(\overrightarrow{N}\_m \,\overrightarrow{K}\right)}{N\_{mx}} \right| \tag{2}$$

We note that the above presented vector expressions are written taking into account unit normal of the vectors N ! <sup>g</sup> , N ! <sup>m</sup> and K ! . Substituting explicit expressions for these vectors into the last formula, finally we get

$$D = \left| \frac{\left( \varkappa K\_x + y K\_y + pK\_x \right) (K\_x \sin \ \ h\_n + K\_y \cos \ \ h\_n \cos \ \ A\_n + K\_x \cos \ \ h\_n \sin \ A\_n)}{p} \right|$$

For the particular case, when K ! ð Þ 0; 0; 1 we obtain obvious formula

$$D = |\cos h\_n \cos A\_n|$$

Below we apply the developed approach to calculation of energetical brightness at given point of the receiver taking into account errors of reflecting surfaces, shadowing and blocking effects, etc.

The real normal is given with reference to the vector N

¼ L3x; L3y; L3<sup>z</sup>

Y1

!

The direction of unit vectors may be chosen in different ways, depending on the character of deviation of the real vector from the ideal one. Let the unit vectors lay

<sup>Þ</sup> are directing unit vectors �

! <sup>m</sup>, i.e.,

Thousand kW High-Temperature Solar Furnace in Parkent (Uzbekistan) – Energetical…

� � <sup>¼</sup> Nm

on the meridional and sagittal sections. In this case the unit vectors L<sup>1</sup>

determined via the vector product of previously determined vectors:

!

can be determined using L<sup>2</sup>

<sup>L</sup>3xL3<sup>z</sup>; <sup>L</sup>3zL3<sup>y</sup>; � <sup>L</sup><sup>2</sup>

q

<sup>μ</sup> <sup>þ</sup> tg <sup>2</sup>

!

angle errors of the surface, is unambiguously determined by two angles μ and ν, which are actually its projections onto two mutually orthogonal planes, for example, meridional and sagittal planes [23]. It is easy to see that the angle γ is determined via

<sup>γ</sup> <sup>¼</sup> tg <sup>2</sup>

<sup>γ</sup> ffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Nxr ¼ cos γ tgμ, Nyr ¼ cos γ tgν, Nzr ¼ cos γ

Error density distribution of BSF reflecting surfaces for angles μ and ν can be determinate by experiments. Here we supposed that these errors are to be distributed according to the Gaussian function. For a set of random values μ and ν, individually having Gaussian distribution with zero mean, a two-dimensional probability distribution in the absence of correlations has the form [31].

> exp � <sup>1</sup> 2

μ2 σ2 μ þ ν2 σ2 ν

" # !

1 2πσμσν

Now the following expression may be written for components of the real

tg <sup>2</sup>

Or if smallness of angles μ and ν is taken into account

Wð Þ¼ μ; ν

h i � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 <sup>3</sup><sup>x</sup> <sup>þ</sup> <sup>L</sup><sup>2</sup> 3y

auxiliary Cartesian reference frame X<sup>1</sup>

DOI: http://dx.doi.org/10.5772/intechopen.83411

and Z axis, directed along the vector N

L2 !

> L1 ! ¼ L<sup>2</sup> ! �L<sup>3</sup> ! ¼

The angle between the ideal direction N

μ and ν, according to the following relation:

Now, the unit vector L<sup>1</sup>

normal:

119

L3 !

where L<sup>1</sup> ! ; L<sup>2</sup> ! ; L<sup>3</sup> !

L2<sup>x</sup>; l2<sup>y</sup>; L2<sup>z</sup> � � <sup>¼</sup> <sup>L</sup><sup>3</sup>

!

!

Nmx; Nmy; Nmz � �,

�ð Þ¼ <sup>0</sup>; <sup>0</sup>; �<sup>1</sup> �L3<sup>y</sup>; <sup>L</sup>3<sup>x</sup>; <sup>0</sup> � �

q

!

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 <sup>3</sup><sup>x</sup> <sup>þ</sup> <sup>L</sup><sup>2</sup> 3y

<sup>m</sup> and real normal γ, which characterizes

ν (3)

<sup>μ</sup><sup>2</sup> <sup>þ</sup> <sup>ν</sup><sup>2</sup> <sup>p</sup> (4)

and L<sup>3</sup> ! :

<sup>3</sup><sup>x</sup> <sup>þ</sup> <sup>L</sup><sup>2</sup> 3y

Z1 is introduced with the origin at point M

<sup>m</sup>. For this purpose an

!

and L<sup>2</sup> !

can be

(5)

At first the relation between the new and old variables should be determined. Let us consider point G with coordinates (x1, y1, 0) on the heliostat's reference frame. Transformation of this point into the other reference frame FXYZ is performed by the following expression:

$$
\begin{pmatrix} \varkappa\_f \\ \varkappa\_f \\ \varkappa\_f \end{pmatrix} = T\_1(A\_n) \, T\_2(-h\_n) \begin{pmatrix} \varkappa\_1 \\ \varkappa\_1 \\ \mathbf{0} \end{pmatrix} + \begin{pmatrix} \varkappa\_0 \\ \varkappa\_0 \\ \varkappa\_0 \end{pmatrix}
$$

where T<sup>1</sup> and T<sup>2</sup> are anticlockwise rotation matrixes around X and Y axes, respectively [30], and (x0, y0, z0) are the coordinates of the heliostat's center in the FXYZ frame. Now we have to find the intersection point of the vector K ! with the surface of the paraboloid, i.e., solve the following set of equations

$$\frac{x - x\_f}{K\_x} = \frac{y - y\_f}{K\_y} = \frac{z - z\_f}{K\_x} \quad \left\{ \begin{array}{c} \hline \\ K\_x \end{array} \right\} $$
 
$$x^2 + y^2 - p^2 + 2pz = 0$$

from which we obtain

$$\mathbf{x} = \mathbf{x}\_f + t\mathbf{K}\_x, \ \mathbf{y} = \mathbf{y}\_f + t\mathbf{K}\_y, \ \mathbf{z} = \mathbf{z}\_f + t\mathbf{K}\_{z\mathbf{v}}$$

where

$$t = \frac{-b + \sqrt{b^2 - ac}}{a}, a = K\_x^2 + K\_y^2, b = \varkappa\_f K\_x + \jmath\_f K\_y + pK\_{x^\*}c = \varkappa\_f^2 + \jmath\_f^2 - 2p\_{z\_f}$$

Exposure B at the given target point A is determined by the inverse ray tracing method when a ray is directed from this point to the current point M on the concentrator, then a normal to the reflecting surface at this point is determined, and the ray reflected in the heliostat's direction is found. After that the crossing point of this ray with the given heliostat's reflective surface and the corresponding normal at this point is determined. Finally, the direction of the last reflected ray is found, and depending on the fact, whether this ray hits solar disk (determine the angle φ) or not (B = 0), the energetical brightness is determined. Distribution of the brightness over the solar disk is taken into consideration according to the Jose formula [23]:

$$B(\rho) = 1.23 \frac{E\_0}{\pi \rho\_0} \cdot \frac{1 + 1.5641 \sqrt{1 - \frac{\sin^2(\rho)}{\sin^2(\rho\_0)}}}{2.5641},$$

where φ<sup>0</sup> is the apparent angular size of the solar disk (16 min) and E<sup>0</sup> is the solar radiation.

During the ray tracing procedure, shadowing and blocking conditions of the given heliostat by other heliostats, technological tower, and other installations are taken into account.

Thousand kW High-Temperature Solar Furnace in Parkent (Uzbekistan) – Energetical… DOI: http://dx.doi.org/10.5772/intechopen.83411

The real normal is given with reference to the vector N ! <sup>m</sup>. For this purpose an auxiliary Cartesian reference frame X<sup>1</sup> Y1 Z1 is introduced with the origin at point M and Z axis, directed along the vector N ! <sup>m</sup>, i.e.,

$$
\overrightarrow{L\_3} = \left(L\_{3x}, L\_{3y}, L\_{3x}\right) = \overrightarrow{N\_m}\left(N\_{mx}, N\_{my}, N\_{mx}\right),
$$

$$
\text{where } \left(\overrightarrow{L\_1}, \overrightarrow{L\_2}, \overrightarrow{L\_3}\right) \text{ are directly unit vectors.}
$$

The direction of unit vectors may be chosen in different ways, depending on the character of deviation of the real vector from the ideal one. Let the unit vectors lay on the meridional and sagittal sections. In this case the unit vectors L<sup>1</sup> ! and L<sup>2</sup> ! can be determined via the vector product of previously determined vectors:

$$
\overrightarrow{L\_2} \text{ (} L\_{2\text{x}}, l\_{2\text{y}}, L\_{2\text{z}}\text{) } = \overrightarrow{L\_3} \times (\mathbf{0}, \mathbf{0}, -\mathbf{1}) = \frac{\left(-L\_{3\text{y}}, L\_{3\text{x}}, \mathbf{0}\right)}{\sqrt{L\_{3\text{x}}^2 + L\_{3\text{y}}^2}}
$$

Now, the unit vector L<sup>1</sup> ! can be determined using L<sup>2</sup> ! and L<sup>3</sup> ! :

$$\overrightarrow{L\_1} = \overrightarrow{L\_2} \times \overrightarrow{L\_3} = \frac{\left[L\_{3\text{x}}L\_{3\text{x}}, L\_{3\text{x}}L\_{3\text{y}}, -\left(L\_{3\text{x}}^2 + L\_{3\text{y}}^2\right)\right]}{\sqrt{L\_{3\text{x}}^2 + L\_{3\text{y}}^2}}$$

The angle between the ideal direction N ! <sup>m</sup> and real normal γ, which characterizes angle errors of the surface, is unambiguously determined by two angles μ and ν, which are actually its projections onto two mutually orthogonal planes, for example, meridional and sagittal planes [23]. It is easy to see that the angle γ is determined via μ and ν, according to the following relation:

$$\text{tg}^2 \chi = \text{tg}^2 \mu + \text{tg}^2 \nu \tag{3}$$

Or if smallness of angles μ and ν is taken into account

$$
\gamma \cong \sqrt{\mu^2 + \nu^2} \tag{4}
$$

Now the following expression may be written for components of the real normal:

$$N\_{\pi r} = \cos \gamma \text{ tg}\mu, \quad N\_{yr} = \cos \gamma \text{ tg}\nu, \quad N\_{\pi r} = \cos \gamma$$

Error density distribution of BSF reflecting surfaces for angles μ and ν can be determinate by experiments. Here we supposed that these errors are to be distributed according to the Gaussian function. For a set of random values μ and ν, individually having Gaussian distribution with zero mean, a two-dimensional probability distribution in the absence of correlations has the form [31].

$$\mathcal{W}(\mu, \nu) = \frac{1}{2\pi\sigma\_{\mu}\sigma\_{\nu}} \exp\left[-\frac{1}{2} \left(\frac{\mu^2}{\sigma\_{\mu}^2} + \frac{\nu^2}{\sigma\_{\nu}^2}\right)\right] \tag{5}$$

Below we apply the developed approach to calculation of energetical brightness

At first the relation between the new and old variables should be determined. Let us consider point G with coordinates (x1, y1, 0) on the heliostat's reference frame. Transformation of this point into the other reference frame FXYZ is

> x1 y1 0

1

CA <sup>þ</sup>

9 >=

>;

<sup>y</sup> , b <sup>¼</sup> xfKx <sup>þ</sup> yfKy <sup>þ</sup> pKz, c <sup>¼</sup> <sup>x</sup><sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � sin <sup>2</sup>ð Þ <sup>φ</sup> sin <sup>2</sup> φ<sup>0</sup> ð Þ

<sup>2</sup>:<sup>5641</sup> ,

q

x0 y0 z0

1

CA

!

<sup>f</sup> <sup>þ</sup> <sup>y</sup><sup>2</sup>

<sup>f</sup> � 2pzf

with the

0

B@

0

B@

<sup>¼</sup> <sup>z</sup> � zf Kz

at given point of the receiver taking into account errors of reflecting surfaces,

CA <sup>¼</sup> <sup>T</sup>1ð Þ An <sup>T</sup>2ð Þ �hn

FXYZ frame. Now we have to find the intersection point of the vector K

surface of the paraboloid, i.e., solve the following set of equations

x � xf Kx

where T<sup>1</sup> and T<sup>2</sup> are anticlockwise rotation matrixes around X and Y axes, respectively [30], and (x0, y0, z0) are the coordinates of the heliostat's center in the

> <sup>¼</sup> <sup>y</sup> � yf Ky

<sup>x</sup><sup>2</sup> <sup>þ</sup> <sup>y</sup><sup>2</sup> � <sup>p</sup><sup>2</sup> <sup>þ</sup> <sup>2</sup>pz <sup>¼</sup> <sup>0</sup>

x ¼ xf þ tKx, y ¼ yf þ tKy, z ¼ zf þ tKz,

Exposure B at the given target point A is determined by the inverse ray tracing

1 þ 1:5641

where φ<sup>0</sup> is the apparent angular size of the solar disk (16 min) and E<sup>0</sup> is the

During the ray tracing procedure, shadowing and blocking conditions of the given heliostat by other heliostats, technological tower, and other installations are

method when a ray is directed from this point to the current point M on the concentrator, then a normal to the reflecting surface at this point is determined, and the ray reflected in the heliostat's direction is found. After that the crossing point of this ray with the given heliostat's reflective surface and the corresponding normal at this point is determined. Finally, the direction of the last reflected ray is found, and depending on the fact, whether this ray hits solar disk (determine the angle φ) or not (B = 0), the energetical brightness is determined. Distribution of the brightness over the solar disk is taken into consideration according to the Jose

shadowing and blocking effects, etc.

performed by the following expression:

xf yf zf

A Guide to Small-Scale Energy Harvesting Techniques

1

0

B@

from which we obtain

p a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>b</sup><sup>2</sup> � a c

, a <sup>¼</sup> <sup>K</sup><sup>2</sup>

Bð Þ¼ φ 1:23

E0 πφ<sup>0</sup> �

<sup>x</sup> <sup>þ</sup> <sup>K</sup><sup>2</sup>

where

formula [23]:

solar radiation.

118

taken into account.

<sup>t</sup> <sup>¼</sup> �<sup>b</sup> <sup>þ</sup>

where σ<sup>2</sup> <sup>μ</sup>, σ<sup>2</sup> <sup>ν</sup> are dispersions of corresponding random values.

Generation of random numbers with a given distribution function is the special issue. To obtain pseudorandom numbers with Gaussian distribution function, we used (after thorough testing) the following expression:

$$\xi = a + \sigma \left[ \sin \left( 2\pi \eta\_1 \right) \sqrt{-2 \ln \left( \eta\_2 \right)} \right]$$

where η<sup>1</sup> and η<sup>2</sup> are random numbers uniformly distributed over the interval [0:1]. Note that high-level programming languages have such built-in functions.

Note that in the case of Rayleigh distribution, i.e., when σμ ¼ σν ¼ σ, an explicit expression for the distribution density (Eq. (5)) may be derived. Omitting simple transformations we obtain the following result:

$$\mathcal{W}(\boldsymbol{\gamma}) = \frac{\boldsymbol{\gamma}}{\sigma^2} \, \exp\left(-\frac{\boldsymbol{\gamma}^2}{2\sigma^2}\right), \quad \boldsymbol{\gamma} > \mathbf{0} \tag{6}$$

characteristics of the furnace, it is necessary to consider many factors, which are in a certain temporary state. This is why, when specifically referring to the energy state of the furnace, it is necessary to provide these factors with the corresponding information. Despite this, some characteristic features of the energy parameters of the LSF can be refined, and it is necessary to analyze a large amount of information on the energy density distribution in the focal zone from certain heliostats, shelves, and groups of heliostats for the various system conditions in order to determine

Thousand kW High-Temperature Solar Furnace in Parkent (Uzbekistan) – Energetical…

DOI: http://dx.doi.org/10.5772/intechopen.83411

The aim of this paragraph is a detailed analysis of the LSF energy characteristics based on numerical calculations. The peculiarities of the methods for calculating the LSF energy characteristics and their implementation for specific problems are given in [25, 28, 29]. Some characteristic features of the LSF energy characteristics are

Each heliostat illuminates a certain area of the concentrator in the normal operation mode of the device. A scaled circuit of the concentrator midsection with the block circuits (solid line) and relevant heliostat zones (dotted line) is shown in Figure 3. The numbers of the heliostats are given in the left angle of their zone. The upper contour line of the building roof adjacent to the process tower, which insignificantly blocks the light flux from the heliostats, is also shown in Figure 3. As is clear from Figure 3, the heliostats 55 and 62 are most inefficient (less than 50% of

The LSF energy spot is formed from the energy contributions (irradiance/energy density) of certain heliostats. The energy contributions of the heliostats depend on the place of their location, reflection coefficient, mirror inaccuracy, adjustment

The authors developed a program to calculate the energy characteristics taking into account the real influencing factors in order to study the peculiarities of the LSF energy characteristics [25]. The program uses Monte Carlo method to calculate

Midsection of concentrator of blocks of facets and corresponding zones of heliostats.

given in these works but with no detailed theoretical or design analysis.

them.

the heliostat area is used).

state, etc.

Figure 3.

121

as this takes place

$$
\sigma\_{\gamma} = \sqrt{D\gamma} = \sigma\sqrt{2}
$$

So, the distribution for deviations of the angle γ does not obey the Gaussian law. Experimental results obtained for mirrors of the BSF also indicate the non-Gaussian character of error distributions.

Now the vector N ! <sup>r</sup> must be transformed to the basic reference frame:

$$\begin{aligned} N\_{\text{xx}} &= L\_{\text{1x}} N\_{\text{x}r} + L\_{\text{2x}} N\_{\text{yr}} + L\_{\text{3x}} N\_{\text{x}r} & N\_{\text{my}} &= L\_{\text{1y}} N\_{\text{x}r} + L\_{\text{2y}} N\_{\text{yr}} + L\_{\text{3y}} N\_{\text{x}r} \\ N\_{\text{mx}} &= L\_{\text{1x}} N\_{\text{x}r} + L\_{\text{2x}} N\_{\text{yr}} + L\_{\text{3x}} N\_{\text{x}r} \end{aligned}$$

Real normals for other reflecting parts may be determined in the same way. In the case when total errors have distribution law distinct from Gaussian, direction of the real normal can be determined using the error density distribution function [31].

It should be emphasized that in view of the character of the integrand function, chosen approach to solution of the problem, specific features of the algorithm, and possibilities of modern computers, it is desirable to calculate the integral (Eq.(1)) by the statistical method (Monte Carlo procedure) [31]. Moreover, in the problem under consideration, simultaneous determination of exposure values is possible in all given target points (not only at one point) using one set of random numbers is possible. In this way calculation steps are significantly reduced.

Corresponding software based on the above described technique has been developed in Delphi programming language. Using these programs case calculations of the BSF energetical characteristics are performed.
