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

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 them.

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

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 heliostat area is used).

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 state, etc.

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

Figure 3. Midsection of concentrator of blocks of facets and corresponding zones of heliostats.

where σ<sup>2</sup>

<sup>μ</sup>, σ<sup>2</sup>

as this takes place

Now the vector N

function [31].

120

character of error distributions.

!

Nmz ¼ L1zNxr þ L2zNyr þ L3zNzr

<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

<sup>ξ</sup> <sup>¼</sup> <sup>a</sup> <sup>þ</sup> <sup>σ</sup> sin 2πη<sup>1</sup> ð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

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

<sup>σ</sup><sup>2</sup> exp � <sup>γ</sup><sup>2</sup>

σγ <sup>¼</sup> ffiffiffiffiffiffi

h i p

2σ<sup>2</sup> � �

<sup>D</sup><sup>γ</sup> <sup>p</sup> <sup>¼</sup> <sup>σ</sup> ffiffi

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

Nmx ¼ L1xNxr þ L2xNyr þ L3xNzr, Nmy ¼ L1yNxr þ L2yNyr þ L3yNzr,

Real normals for other reflecting parts may be determined in the same way.

direction of the real normal can be determined using the error density distribution

Corresponding software based on the above described technique has been developed in Delphi programming language. Using these programs case calculations

Analysis of the characteristics of the processing modes of material synthesis and heat treatment, i.e., isochronic maps of the energy density distribution in the focal zone of the LSF, shows that they are very different. When referring to the energy

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

In the case when total errors have distribution law distinct from Gaussian,

possible. In this way calculation steps are significantly reduced.

5. Characteristic features of the energy modes of the LSF

of the BSF energetical characteristics are performed.

5.1 Preliminary notes and some assumptions

<sup>r</sup> must be transformed to the basic reference frame:

2 p

�2ln η<sup>2</sup> ð Þ

, γ . 0 (6)

used (after thorough testing) the following expression:

A Guide to Small-Scale Energy Harvesting Techniques

transformations we obtain the following result:

Wð Þ¼ γ

γ

the integral. Comparative analysis of the calculation results shows that the calculation accuracy is 2–4%. The intensity of the solar beam is determined by backtracking of the beam path. The program interface is shown in Figure 4, along with the layout of the heliostats and numbers of the heliostats of certain shelves.

γ varies according to a Gaussian law with dispersion σ, and φ varies uniformly

This program was used to carry out some sequences of the numerical calculation

The scope of calculations is quite large, which is why some peculiarities of the LSF energy characteristics are revealed from analysis of data, which are given

The mirror inaccuracies of the heliostats σ<sup>g</sup> are measured on the horizontal axis. Some individual lines correspond to various inaccuracies of the concentrator mirrors σc. It is easy to see that the change in focal irradiance is expectedly more sensitive to mirror inaccuracies of the concentrator than the heliostats. However, beginning with 2.5–3.0 angular minutes, the influence of the mirror inaccuracies of the heliostats and concentrator on the character of the irradiance distribution does not significantly differ and is approximately similar. In practice, we have the mirror inaccuracies within 3–10 angular minutes. This is why, for decreasing (by an order of magnitude) calculation variants, it is assumed that the mirror inaccuracies of the heliostats and concentrator are similar (σ<sup>c</sup> = σ<sup>g</sup> = σ). As mentioned above, it is considered that these mirror inaccuracies obey a normal law of errors. Under such an assumption, the dependence of the focal irradiance of the LSF on the mirror inaccuracy will be a function of one variable, which can be represented as the

, Rsc = Rsg = 0.6).

Figure 5a shows the dependence of the focal irradiance EF on the mirror

inaccuracy of the heliostats σ<sup>g</sup> and concentrator σ<sup>c</sup> (E<sup>0</sup> = 700 W/m<sup>2</sup>

EF <sup>¼</sup> <sup>43578</sup>E0RscRsg

1 � 0:034σ þ 0:028σ<sup>2</sup>

where σ is the root-mean-square mirror inaccuracy per minute. The relative error of this formula is no more than 1.2%. A diagram of this dependence is shown

Let us consider the size of the focal spot in different cases. Usually, the value where the irradiance is 10% of the focal irradiance is selected as the size of the focal spot in solar concentrators. In the case of the LSF, for descriptive reasons and convenience of analysis, various levels of irradiance within 3–11 W/cm<sup>2</sup> were used as the contour (isometric) line of the focal spot. The contour lines of the focal spot

A certain idea of the character of the irradiance distribution for various mirror

inaccuracies can be obtained from this, taking into account the value EF from Figure 5b. It can be seen that the shape of the focal spot is not round, but a bit prolate along the horizontal axis. This is evidently related to the shape of the midsection of the concentrator. It can be also seen from Figure 5b that beginning from the second or third minute of the mirror inaccuracy, the character of the energy distribution (extension of the spot contour line) varies gradually and

W cm<sup>2</sup> 

, Rsc = Rsg = 0.6) are shown in

to determine the irradiance distribution (energy density) of certain heliostats, shelves of heliostats, groups of heliostats, and the entire LSF with different inaccuracies of the reflecting surfaces of the heliostats and concentrator (1–10 angular

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

5.2 Analysis of the results of numerical calculations

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

5.2.1 Analysis of the energy characteristics of the whole system

within 0–2π.

minutes).

below.

following empirical formula:

for various mirror inaccuracies (E<sup>0</sup> = 700 W/m<sup>2</sup>

relatively uniformly as a function of σ.

in Figure 5b.

Figure 6.

123

The input parameters of the program are the beam solar radiation E<sup>0</sup> [W/cm2 ]; Rsc and Rsg are the reflection coefficients of the mirrors of the concentrators and heliostats, the root-mean-square mirror inaccuracy of the heliostats σ<sup>g</sup> and concentrator σc, the relative boundaries of integration of the heliostats, the number of points in the focal plane, the number of playouts in the Monte Carlo method, etc. The energy density distribution was determined for a flat receiver located in the focal plane of the concentrator in all calculations.

In the solar energy problems, the reflecting properties of the real mirror are characterized by the root-mean-square angular inaccuracy σ defined as a rootmean-square value of deviations of the real normal from the ideal (γ). It is assumed in many calculations that the meridional (μ) and sagittal (ν) components of the angle of deviation from the normal (γ <sup>2</sup> ≈ μ <sup>2</sup> + ν 2 ) obey a Gaussian law for integral paraboloid concentrators [23]. The sagittal component of deviation v is often neglected. However, in the case of the LSF, due to the large sizes of the device, the curvature of some concentrator facets (with dimensions of 45 � 45 cm in various sections), especially the peripheral facets, is very similar and small; i.e., the facets are almost isotropic reflectors. The reflecting surfaces of the heliostats also consist of flat reflectors. For this reason, it is assumed in calculations that the probability distribution of the deviation of the mirror normal γ obeys a Gaussian law with dispersion σ. In other words, if the direction of the real normal relative to the ideal is expressed by the formula

$$l\_{\mathbf{x}} = \sin \chi \cos \rho, \quad l\_{\mathbf{y}} = \sin \chi \sin \rho, \quad l\_{\mathbf{z}} = \cos \chi,$$

Figure 4. Program interface.

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

γ varies according to a Gaussian law with dispersion σ, and φ varies uniformly within 0–2π.

This program was used to carry out some sequences of the numerical calculation to determine the irradiance distribution (energy density) of certain heliostats, shelves of heliostats, groups of heliostats, and the entire LSF with different inaccuracies of the reflecting surfaces of the heliostats and concentrator (1–10 angular minutes).

## 5.2 Analysis of the results of numerical calculations

### 5.2.1 Analysis of the energy characteristics of the whole system

The scope of calculations is quite large, which is why some peculiarities of the LSF energy characteristics are revealed from analysis of data, which are given below.

Figure 5a shows the dependence of the focal irradiance EF on the mirror inaccuracy of the heliostats σ<sup>g</sup> and concentrator σ<sup>c</sup> (E<sup>0</sup> = 700 W/m<sup>2</sup> , Rsc = Rsg = 0.6).

The mirror inaccuracies of the heliostats σ<sup>g</sup> are measured on the horizontal axis. Some individual lines correspond to various inaccuracies of the concentrator mirrors σc. It is easy to see that the change in focal irradiance is expectedly more sensitive to mirror inaccuracies of the concentrator than the heliostats. However, beginning with 2.5–3.0 angular minutes, the influence of the mirror inaccuracies of the heliostats and concentrator on the character of the irradiance distribution does not significantly differ and is approximately similar. In practice, we have the mirror inaccuracies within 3–10 angular minutes. This is why, for decreasing (by an order of magnitude) calculation variants, it is assumed that the mirror inaccuracies of the heliostats and concentrator are similar (σ<sup>c</sup> = σ<sup>g</sup> = σ). As mentioned above, it is considered that these mirror inaccuracies obey a normal law of errors. Under such an assumption, the dependence of the focal irradiance of the LSF on the mirror inaccuracy will be a function of one variable, which can be represented as the following empirical formula:

$$E\_F = \frac{43578 E\_0 R\_\kappa R\_\text{sg}}{1 - 0.034\sigma + 0.028\sigma^2} \left[\frac{W}{cm^2}\right]^2$$

where σ is the root-mean-square mirror inaccuracy per minute. The relative error of this formula is no more than 1.2%. A diagram of this dependence is shown in Figure 5b.

Let us consider the size of the focal spot in different cases. Usually, the value where the irradiance is 10% of the focal irradiance is selected as the size of the focal spot in solar concentrators. In the case of the LSF, for descriptive reasons and convenience of analysis, various levels of irradiance within 3–11 W/cm<sup>2</sup> were used as the contour (isometric) line of the focal spot. The contour lines of the focal spot for various mirror inaccuracies (E<sup>0</sup> = 700 W/m<sup>2</sup> , Rsc = Rsg = 0.6) are shown in Figure 6.

A certain idea of the character of the irradiance distribution for various mirror inaccuracies can be obtained from this, taking into account the value EF from Figure 5b. It can be seen that the shape of the focal spot is not round, but a bit prolate along the horizontal axis. This is evidently related to the shape of the midsection of the concentrator. It can be also seen from Figure 5b that beginning from the second or third minute of the mirror inaccuracy, the character of the energy distribution (extension of the spot contour line) varies gradually and relatively uniformly as a function of σ.

the integral. Comparative analysis of the calculation results shows that the calcula-

backtracking of the beam path. The program interface is shown in Figure 4, along with the layout of the heliostats and numbers of the heliostats of certain shelves. The input parameters of the program are the beam solar radiation E<sup>0</sup> [W/cm2

Rsc and Rsg are the reflection coefficients of the mirrors of the concentrators and heliostats, the root-mean-square mirror inaccuracy of the heliostats σ<sup>g</sup> and concentrator σc, the relative boundaries of integration of the heliostats, the number of points in the focal plane, the number of playouts in the Monte Carlo method, etc. The energy density distribution was determined for a flat receiver located in the

In the solar energy problems, the reflecting properties of the real mirror are characterized by the root-mean-square angular inaccuracy σ defined as a rootmean-square value of deviations of the real normal from the ideal (γ). It is assumed in many calculations that the meridional (μ) and sagittal (ν) components of the

<sup>2</sup> ≈ μ <sup>2</sup> + ν 2

lx ¼ sin γ cos φ, ly ¼ sin γ sin φ, lz ¼ cos γ,

paraboloid concentrators [23]. The sagittal component of deviation v is often neglected. However, in the case of the LSF, due to the large sizes of the device, the curvature of some concentrator facets (with dimensions of 45 � 45 cm in various sections), especially the peripheral facets, is very similar and small; i.e., the facets are almost isotropic reflectors. The reflecting surfaces of the heliostats also consist of flat reflectors. For this reason, it is assumed in calculations that the probability distribution of the deviation of the mirror normal γ obeys a Gaussian law with dispersion σ. In other words, if the direction of the real normal relative to the ideal is

];

) obey a Gaussian law for integral

tion accuracy is 2–4%. The intensity of the solar beam is determined by

focal plane of the concentrator in all calculations.

A Guide to Small-Scale Energy Harvesting Techniques

angle of deviation from the normal (γ

expressed by the formula

Figure 4. Program interface.

122

Figure 7 shows the dependence (in the section) of the irradiance distribution in the focal zone of the LSF for the specified parameters with various inaccuracies of

Figure 8 shows the energy density distribution in the focal zone of the LSF in the

form of isometric lines with equal irradiances and its three-dimensional

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

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

the reflecting mirrors.

Contour lines of focal spot.

Figure 6.

Figure 7.

125

Irradiance distribution in the section of focal zone of LSF.

Figure 5.

(a) Dependence of focal irradiance EF on the mirror inaccuracy of the heliostats σ<sup>g</sup> and concentrator σ<sup>c</sup> and (b) dependence of focal irradiance EF on the mirror inaccuracy (case σ<sup>g</sup> = σc).

Thus, as follows from the given analysis and without violating the generality of the results, the characteristic parameters of the LSP can be calculated, and its energy characteristics can be analyzed on their basis. On the basis of this data, the energy characteristics of the furnace can also be estimated for its other parameters. The following values were used as the characteristic parameters of the LSF: E<sup>0</sup> = 700 W/m2 is the beam solar radiation, and Rsc = Rsg = 0.6 is the reflection coefficient of the mirrors of the concentrator and heliostats; the mirror inaccuracies of the concentrator and heliostats are σ =7(σ = σ<sup>c</sup> = σg) angular minutes.

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

Figure 6. Contour lines of focal spot.

Figure 7 shows the dependence (in the section) of the irradiance distribution in the focal zone of the LSF for the specified parameters with various inaccuracies of the reflecting mirrors.

Figure 8 shows the energy density distribution in the focal zone of the LSF in the form of isometric lines with equal irradiances and its three-dimensional

Figure 7. Irradiance distribution in the section of focal zone of LSF.

Thus, as follows from the given analysis and without violating the generality of

the results, the characteristic parameters of the LSP can be calculated, and its energy characteristics can be analyzed on their basis. On the basis of this data, the energy characteristics of the furnace can also be estimated for its other parameters.

(a) Dependence of focal irradiance EF on the mirror inaccuracy of the heliostats σ<sup>g</sup> and concentrator σ<sup>c</sup> and

(b) dependence of focal irradiance EF on the mirror inaccuracy (case σ<sup>g</sup> = σc).

A Guide to Small-Scale Energy Harvesting Techniques

Figure 5.

124

The following values were used as the characteristic parameters of the LSF: E<sup>0</sup> = 700 W/m2 is the beam solar radiation, and Rsc = Rsg = 0.6 is the reflection coefficient of the mirrors of the concentrator and heliostats; the mirror inaccuracies

of the concentrator and heliostats are σ =7(σ = σ<sup>c</sup> = σg) angular minutes.

representation specifying the irradiance levels. The maximum energy density at the

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

It should be noted that the furnace power in the selected parameters has a value

Calculations are carried out to select the empirical formulas for the best approximation of the two-dimensional calculated data. The selection of the different types of functions shows that for this purpose, a two-dimensional Gaussian function

> x � xc wx � �<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>i</sup>¼<sup>1</sup> Ei � E xi; yi � � � � <sup>2</sup> N

2

The LSF conditions with various inaccuracies of the reflecting mirrors are considered, and the parameters of the empirical formulas and standard root-meansquare deviations of irradiance are determined for each case according to the fol-

calculations, the amount of light energy per unit time (power) arriving at the focal zone of the LSF is determined in all calculations, and it was compared to the Wo value. It turned out that the difference between these values does not exceed 2–3%.

5.2.2 Representation of the calculated data by empirical formulas

(with six parameters) gives the best fit, which is given by

E xð Þ¼ ; <sup>y</sup> <sup>E</sup><sup>0</sup> <sup>þ</sup> <sup>A</sup> exp � <sup>1</sup>

E<sup>σ</sup> ¼

racies of the heliostats and concentrator.

∑<sup>N</sup>

where i is number of the current points, Ei is the calculated data, and N is point number, in our case, 2601. Table 1 gives the determined parameters of the two-dimensional Gaussian functions and Eσ values for the various mirror inaccu-

It is clear from Table 1 that with an increase in mirror inaccuracies, the empirical formulas describe the calculated data better. Despite this, when using these formulas, it is necessary to make certain in each case that it is possible to use them. It should be noted that in all cases, the total power of the focal spot determined using the empirical formulas almost coincides with the power value determined by the calculated data. It also should be mentioned that in all cases, the sum of

σ, мин E<sup>0</sup> A xc wx yc wy E<sup>σ</sup> σ<sup>с</sup> = σ<sup>g</sup> = 2 4.71 1164.28 0.00006 0.074 0.00014 0.079 12.25 σ<sup>с</sup> = σ<sup>g</sup> = 4 7.02 840.87 0.000095 0.084 �0.00022 0.09 8.21 σ<sup>с</sup> = σ<sup>g</sup> = 7 9.24 474.82 �0.00031 0.106 0.00015 0.115 7.34 σ<sup>с</sup> = σ<sup>g</sup> = 10 10.53 290.26 0.0004 0.129 0.00058 0.139 6.10 σ<sup>с</sup> = σ<sup>g</sup> = 12 11.03 217.65 0.00056 0.143 0.00031 0.155 5.35 σ<sup>с</sup> = 7, σ<sup>g</sup> = 5 8.79 572.7 0.00054 0.098 0.0004 0.106 7.97 σ<sup>с</sup> = 7, σ<sup>g</sup> = 4 8.58 624.0 0.00039 0.094 0.000024 0.102 8.21 σ<sup>с</sup> = 5, σ<sup>g</sup> = 3 7.24 841.64 0.00031 0.084 �0.00012 0.09 8.42

s

. The asymmetry of the focal spot can be also seen in

•1906m2 = 480.3кW. To check the correctness of the

� 1 2

y � yc wy � �<sup>2</sup> ( ) (7)

center of the spot is 506 W/cm<sup>2</sup>

of W0 = 0.6•0.6•700 W/m2

lowing formula:

Table 1.

127

Parameters of empirical formulas.

the quantitative measures in Figure 8.

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

#### Figure 8.

(a) Energy density distribution from total system (isometric lines) and (b) energy density distribution from total system (three-dimensional).

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

representation specifying the irradiance levels. The maximum energy density at the center of the spot is 506 W/cm<sup>2</sup> . The asymmetry of the focal spot can be also seen in the quantitative measures in Figure 8.

It should be noted that the furnace power in the selected parameters has a value of W0 = 0.6•0.6•700 W/m2 •1906m2 = 480.3кW. To check the correctness of the calculations, the amount of light energy per unit time (power) arriving at the focal zone of the LSF is determined in all calculations, and it was compared to the Wo value. It turned out that the difference between these values does not exceed 2–3%.

#### 5.2.2 Representation of the calculated data by empirical formulas

Calculations are carried out to select the empirical formulas for the best approximation of the two-dimensional calculated data. The selection of the different types of functions shows that for this purpose, a two-dimensional Gaussian function (with six parameters) gives the best fit, which is given by

$$E(\mathbf{x}, \mathbf{y}) = E\_0 + A \exp\left\{ -\frac{1}{2} \left( \frac{\mathbf{x} - \mathbf{x}\_c}{w\_\mathbf{x}} \right)^2 - \frac{1}{2} \left( \frac{\mathbf{y} - \mathbf{y}\_c}{w\_\mathbf{y}} \right)^2 \right\} \tag{7}$$

The LSF conditions with various inaccuracies of the reflecting mirrors are considered, and the parameters of the empirical formulas and standard root-meansquare deviations of irradiance are determined for each case according to the following formula:

$$E\_{\sigma} = \sqrt{\frac{\sum\_{i=1}^{N} \left(E\_i - E\left(\mathbf{x}\_i, \mathbf{y}\_i\right)\right)^2}{N}}$$

where i is number of the current points, Ei is the calculated data, and N is point number, in our case, 2601. Table 1 gives the determined parameters of the two-dimensional Gaussian functions and Eσ values for the various mirror inaccuracies of the heliostats and concentrator.

It is clear from Table 1 that with an increase in mirror inaccuracies, the empirical formulas describe the calculated data better. Despite this, when using these formulas, it is necessary to make certain in each case that it is possible to use them. It should be noted that in all cases, the total power of the focal spot determined using the empirical formulas almost coincides with the power value determined by the calculated data. It also should be mentioned that in all cases, the sum of


#### Table 1. Parameters of empirical formulas.

Figure 8.

126

system (three-dimensional).

A Guide to Small-Scale Energy Harvesting Techniques

(a) Energy density distribution from total system (isometric lines) and (b) energy density distribution from total

(Ei � E(xi,yi)) over all points is approximately zero. The empirical formulas were determined using the OriginPro8 graphics package (www.originLab.com).

#### 5.2.3 Analysis of the power characteristics of the focal spot and average concentration

The value of the spot power and average radiation concentration in certain zones of the focal spot is important for some problems. The spot powers W in different square zones with the center at the focus of the concentrator (d is zone diameter) and average radiation concentrations Eaver in these zones are determined (Figure 9).

## 5.2.4 Analysis of the energy characteristics of certain LSF shelves

The energy characteristics of certain LSF shelves are also studied.

The three-dimensional irradiance distributions of certain shelves of heliostats are given for comparative analysis in Figure 10. The irradiance levels are specified in the belt lines. It is clear from Figure 10 that the contribution of shelf 1 is the smallest from the viewpoint of the maximum irradiance, and the spot is more prolate along the X axis (vertically). The irradiance distributions from shelves 2, 3, and 8 have an approximately round symmetry. It is also seen that the focal spot of shelves 4, 5, 6, and 7 is more prolate along the Y axis (horizontally). The focal irradiance is maximum for shelf 4. The vertical section of the irradiance distribution from certain shelves is shown in Figure 11. The maximum values of irradiance by the heliostat shelves are as follows: 19, 41, 69, 93, 80, 80, 72, and 52 W/cm2 , and their sum is 506 W/cm<sup>2</sup> . The quantitative characteristics of the irradiance distribution of certain shelves can be determined from the diagram.

Note that inclusion or exclusion of some individual shelves of heliostats in the tracking mode allows a certain set of possible irradiance distributions to be obtained. It is easy to see that the number of variants of inclusion/exclusion of the heliostat shelves Np is 256. Actually,

$$N\_p = \sum\_{k=0}^{8} C\_8^k = \sum\_{k=0}^{8} \frac{8!}{(8-k)!k!} = (1+8+28+56+70+56+28+8+1) = 256$$

Figure 12 shows the dependence of the maximum irradiance (points in the diagram) of the distribution on the number N of the variant of inclusion/exclusion of the heliostat shelves. For example, the entry 01111111 means that all shelves except for first are engaged. Transfer from the number of the variant N to such a type of the entry is as follows: N is converted to the binary representation, and the

Energy density distribution from certain shelves of LSF heliostats.

Three-dimensional energy density distribution from certain shelves of LSF heliostats.

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

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

Figure 10.

Figure 11.

129

Figure 9. Power and average concentration in different zones of focal spot.

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

Figure 10. Three-dimensional energy density distribution from certain shelves of LSF heliostats.

Figure 11. Energy density distribution from certain shelves of LSF heliostats.

Figure 12 shows the dependence of the maximum irradiance (points in the diagram) of the distribution on the number N of the variant of inclusion/exclusion of the heliostat shelves. For example, the entry 01111111 means that all shelves except for first are engaged. Transfer from the number of the variant N to such a type of the entry is as follows: N is converted to the binary representation, and the

(Ei � E(xi,yi)) over all points is approximately zero. The empirical formulas were determined using the OriginPro8 graphics package (www.originLab.com).

5.2.3 Analysis of the power characteristics of the focal spot and average concentration

5.2.4 Analysis of the energy characteristics of certain LSF shelves

A Guide to Small-Scale Energy Harvesting Techniques

tion of certain shelves can be determined from the diagram.

8! ð Þ 8 � k !k!

Power and average concentration in different zones of focal spot.

their sum is 506 W/cm<sup>2</sup>

Np ¼ ∑ 8 k¼0 Ck <sup>8</sup> ¼ ∑ 8 k¼0

Figure 9.

128

heliostat shelves Np is 256. Actually,

The energy characteristics of certain LSF shelves are also studied.

The value of the spot power and average radiation concentration in certain zones of the focal spot is important for some problems. The spot powers W in different square zones with the center at the focus of the concentrator (d is zone diameter) and average radiation concentrations Eaver in these zones are determined (Figure 9).

The three-dimensional irradiance distributions of certain shelves of heliostats are given for comparative analysis in Figure 10. The irradiance levels are specified in the belt lines. It is clear from Figure 10 that the contribution of shelf 1 is the smallest from the viewpoint of the maximum irradiance, and the spot is more prolate along the X axis (vertically). The irradiance distributions from shelves 2, 3, and 8 have an approximately round symmetry. It is also seen that the focal spot of shelves 4, 5, 6, and 7 is more prolate along the Y axis (horizontally). The focal irradiance is maximum for shelf 4. The vertical section of the irradiance distribution from certain shelves is shown in Figure 11. The maximum values of irradiance by the heliostat shelves are as follows: 19, 41, 69, 93, 80, 80, 72, and 52 W/cm2

Note that inclusion or exclusion of some individual shelves of heliostats in the

tracking mode allows a certain set of possible irradiance distributions to be obtained. It is easy to see that the number of variants of inclusion/exclusion of the

. The quantitative characteristics of the irradiance distribu-

¼ ð1 þ 8 þ 28 þ 56 þ 70 þ 56 þ 28 þ 8 þ 1Þ ¼ 256

, and

Based on the analysis of illumination of the concentrator by the heliostats, four groups of heliostats are distinguished. The heliostats with a close distance from the optical axis of the concentrator are combined in each group. The numbers of

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

Group 2: 16, 17, 18, 24, 25, 27, 28, 33, 36, 41, 44, 49, 50, 51, 52, 58, 59, and 17

Group 3: 8, 9, 10, 15, 19, 23, 29, 32, 37, 40, 45, 48, 53, 57, 60, and 15 heliostats

Group 4: 1, 2, 3, 4, 5, 6, 7, 11, 12, 13, 14, 20, 21, 22, 30, 31, 38, 39, 46, 47, 54, 55,

The new number of the heliostats is given in brackets. In Figure 3, heliostats belonging to one group are designated, respectively, by rectangles, triangles, circles, and rhombs. As the analysis of the numerical calculations has shown, the shape of the focal spot of certain groups of heliostats is almost symmetric, except for group 4, where a slight asymmetry is observed. This is why the energy density distributions from certain groups in the horizontal section (Y axis) are shown in Figure 13.

In the general case, to determine the necessary possible focal irradiance (or the required irradiance), it suffices to construct a diagram of the dependence of the cumulative sum of the focal irradiance on the number of the heliostat, i.e.,

This diagram is similar to that in Figure 12 in the character of construction; only the new numbers of heliostats are measured on the horizontal axis. A diagram of

The authors performed a comparative analysis of the energy density distributions from certain heliostats. The focal irradiances of certain heliostats can be determined from Table 2, knowing the total focal irradiance of the LSF, 506 W/m2

The isometric line corresponding to a value of 1 W/cm<sup>2</sup> is selected from the

.

EFð Þi , n ¼ 1 � 62

Group 1: 26, 34, 35, 42, 43, and 5 heliostats (1–5, rectangle).

56, 61, 62, and 25 heliostats (38–62, rhombs).

y nð Þ¼ ∑ i¼n i¼1

5.2.6 Analysis of the focal spot of the individual heliostats

Energy density distribution from certain groups of heliostats.

function y(n) for such a new group of heliostats is shown in Figure 14.

heliostats in these groups are as follows:

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

heliostats (6–22, circles).

(23–37, triangles).

Figure 13.

131

Figure 12. Variants of inclusion/exclusion of shelves of LSF heliostats.

obtained number is read in reverse order. For example, 25410 = 111111102, and the entry (combination) has the form 01111111. In the figure, the line Example corresponds to the required maximum density 325 W/cm<sup>2</sup> , and such energy density is achieved using four variants: A, B, C, and D.

#### 5.2.5 Heliostat rearrangement and its energy characteristics

Let us consider the energy characteristics of certain heliostats of the LSE. The focal irradiances and character of irradiance distribution of the heliostats significantly differ from each other. Analysis of the calculation results showed that the relative focal irradiance (normalized to the focal irradiance of the LSF) of the heliostats varies insignificantly with various mirror inaccuracies. In Table 2, this data is presented for 33 heliostats, taking symmetry into account.

Note that the maximum contribution of the heliostat to the focal irradiance is higher when the center of the heliostat is closer to the optical axis of the concentrator. This is why the heliostats can be grouped according to this principle. It can be expected that the shape of the focal spot of certain groups will be more symmetric.


#### Table 2.

The relative focal irradiance of the heliostats.

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

Based on the analysis of illumination of the concentrator by the heliostats, four groups of heliostats are distinguished. The heliostats with a close distance from the optical axis of the concentrator are combined in each group. The numbers of heliostats in these groups are as follows:

Group 1: 26, 34, 35, 42, 43, and 5 heliostats (1–5, rectangle). Group 2: 16, 17, 18, 24, 25, 27, 28, 33, 36, 41, 44, 49, 50, 51, 52, 58, 59, and 17 heliostats (6–22, circles). Group 3: 8, 9, 10, 15, 19, 23, 29, 32, 37, 40, 45, 48, 53, 57, 60, and 15 heliostats (23–37, triangles). Group 4: 1, 2, 3, 4, 5, 6, 7, 11, 12, 13, 14, 20, 21, 22, 30, 31, 38, 39, 46, 47, 54, 55, 56, 61, 62, and 25 heliostats (38–62, rhombs).

The new number of the heliostats is given in brackets. In Figure 3, heliostats belonging to one group are designated, respectively, by rectangles, triangles, circles, and rhombs. As the analysis of the numerical calculations has shown, the shape of the focal spot of certain groups of heliostats is almost symmetric, except for group 4, where a slight asymmetry is observed. This is why the energy density distributions from certain groups in the horizontal section (Y axis) are shown in Figure 13.

In the general case, to determine the necessary possible focal irradiance (or the required irradiance), it suffices to construct a diagram of the dependence of the cumulative sum of the focal irradiance on the number of the heliostat, i.e.,

$$y(n) = \sum\_{i=1}^{i=n} E\_F(i), \quad n = 1 - 62$$

This diagram is similar to that in Figure 12 in the character of construction; only the new numbers of heliostats are measured on the horizontal axis. A diagram of function y(n) for such a new group of heliostats is shown in Figure 14.

#### 5.2.6 Analysis of the focal spot of the individual heliostats

The authors performed a comparative analysis of the energy density distributions from certain heliostats. The focal irradiances of certain heliostats can be determined from Table 2, knowing the total focal irradiance of the LSF, 506 W/m2 . The isometric line corresponding to a value of 1 W/cm<sup>2</sup> is selected from the

Figure 13. Energy density distribution from certain groups of heliostats.

obtained number is read in reverse order. For example, 25410 = 111111102, and the entry (combination) has the form 01111111. In the figure, the line Example corre-

Let us consider the energy characteristics of certain heliostats of the LSE. The focal irradiances and character of irradiance distribution of the heliostats significantly differ from each other. Analysis of the calculation results showed that the relative focal irradiance (normalized to the focal irradiance of the LSF) of the heliostats varies insignificantly with various mirror inaccuracies. In Table 2, this

Note that the maximum contribution of the heliostat to the focal irradiance is higher when the center of the heliostat is closer to the optical axis of the concentrator. This is why the heliostats can be grouped according to this principle. It can be expected that the shape of the focal spot of certain groups will be more symmetric.

No. % No. % No. % No. % No. % 1 0.67 2 0.56 3 1.05 6 0.45 7 1.38 8 1.12 9 2.06 13 0.44 14 0.79 15 2.18 16 1.74 17 3.21 22 0.79 23 1.04 24 2.96 25 2.41 26 4.34 31 0.93 32 1.2 33 3.45 34 2.25 39 1.02 40 0.97 41 3.39 42 2.17 47 0.8 48 1.04 49 2.96 50 2.34 55 0.25

, and such energy density is

sponds to the required maximum density 325 W/cm<sup>2</sup>

5.2.5 Heliostat rearrangement and its energy characteristics

56 1.31 57 1.24 58 2.55

The relative focal irradiance of the heliostats.

data is presented for 33 heliostats, taking symmetry into account.

achieved using four variants: A, B, C, and D.

Variants of inclusion/exclusion of shelves of LSF heliostats.

A Guide to Small-Scale Energy Harvesting Techniques

Figure 12.

Table 2.

130

#### Figure 14. Increasing energy density from certain groups of heliostats.

two-dimensional diagram of the irradiance distribution from a certain heliostat represented as isometric lines of a similar irradiance. It is clear that this isometric line corresponds to the contour line of the focal spot of the considered heliostat. The contour lines of the spots of certain heliostats (33 heliostats) are shown in Figure 15a and b. For descriptive reasons, so as not to overload the diagram, the heliostats are conditionally divided into two groups, central (left) and extreme (right). The contour line of the focal spot of the selected heliostat can be accurately traced from the diagram.

solar radiation, a clear sunny day is selected according to data from the Parkent meteorological station: 18 June 2013. It can be seen from Figure 16 that within the time interval of 10:00–16:00, i.e., within 6 h (13:00 is approximately midday), the character of the energy density distribution does not change significantly. It should be noted that a similar analysis should be carried out for other periods of time.

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

Let us note, lastly, that the performed theoretical and design analyses of the LSF energy characteristics and their revealed features, general patterns, and presented detailed data allow to correctly determine the configuration parameters of the LSF for various required process modes in the focal zones of the furnace. The long-term experience in application of the LSF for synthesis and thermal processing of various high-temperature materials has shown that the accuracy in implementing certain process modes significantly affects the final characteristics of the product. This is why the obtained results have important practical value for efficient operation of

the LSF of the Academy of Sciences of the Republic of Uzbekistan.

Institute of Materials Science of the Academy of Sciences of the Republic of

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

6. Conclusion

Figure 16.

Variation in energy density within 1 day.

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

Author details

Akbarov Rasul

133

Uzbekistan, Tashkent, Uzbekistan

\*Address all correspondence to: aryu12@mail.ru

provided the original work is properly cited.

The character of the energy density distribution can be estimated if the maximum irradiance and geometry of the focal spot of the heliostat are known.

#### 5.2.7 Analysis of the changes of the energy distribution within a day

One of the most important problems is the change in the focal energy density distribution within a day, occurring due to the daily variation in beam solar radiation. As an example, Figure 16 shows a diagram characterizing the change in the focal energy density distribution along the horizontal axis of the LSF. For the beam

Figure 15. Contour lines and orientation of focal spots of certain heliostats of LSF (Figures in contours - numbers of heliostats)

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

#### Figure 16.

two-dimensional diagram of the irradiance distribution from a certain heliostat represented as isometric lines of a similar irradiance. It is clear that this isometric line corresponds to the contour line of the focal spot of the considered heliostat. The contour lines of the spots of certain heliostats (33 heliostats) are shown in Figure 15a and b. For descriptive reasons, so as not to overload the diagram, the heliostats are conditionally divided into two groups, central (left) and extreme (right). The contour line of the focal spot of the selected heliostat can be accurately

The character of the energy density distribution can be estimated if the maximum irradiance and geometry of the focal spot of the heliostat are known.

One of the most important problems is the change in the focal energy density distribution within a day, occurring due to the daily variation in beam solar radiation. As an example, Figure 16 shows a diagram characterizing the change in the focal energy density distribution along the horizontal axis of the LSF. For the beam

Contour lines and orientation of focal spots of certain heliostats of LSF (Figures in contours - numbers of heliostats)

5.2.7 Analysis of the changes of the energy distribution within a day

traced from the diagram.

Increasing energy density from certain groups of heliostats.

A Guide to Small-Scale Energy Harvesting Techniques

Figure 14.

Figure 15.

132

Variation in energy density within 1 day.

solar radiation, a clear sunny day is selected according to data from the Parkent meteorological station: 18 June 2013. It can be seen from Figure 16 that within the time interval of 10:00–16:00, i.e., within 6 h (13:00 is approximately midday), the character of the energy density distribution does not change significantly. It should be noted that a similar analysis should be carried out for other periods of time.

## 6. Conclusion

Let us note, lastly, that the performed theoretical and design analyses of the LSF energy characteristics and their revealed features, general patterns, and presented detailed data allow to correctly determine the configuration parameters of the LSF for various required process modes in the focal zones of the furnace. The long-term experience in application of the LSF for synthesis and thermal processing of various high-temperature materials has shown that the accuracy in implementing certain process modes significantly affects the final characteristics of the product. This is why the obtained results have important practical value for efficient operation of the LSF of the Academy of Sciences of the Republic of Uzbekistan.
