# Optimum Height and Tilt Angle of the Solar Receiver for a 30 kWe Solar Tower Power Plant for the Electricity Production in the Sahelian Zone

Authors: Kory Faye; Ababacar Thiam; Mactar Faye
Journal: International Journal of Photoenergy
 (2021)

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## Abstract

This work investigated the prediction of the optimum height and tilt angle of the solar receiver of a 30 kWe solar tower power plant for the electricity production in the Sahelian zone. Initially, the solar field is sized to determine the total reflecting surface area of the mirrors and the number of heliostats. A PS10-like radially staggered heliostat field is used to design the heliostat layout in the field using a Matlab code. The concentrated solar flux at the input of the receiver was determined using Soltrace software by the Monte Carlo ray tracing (MCRT) method. The sizing results show that the total reflecting surface area is 350 m2 for an optical efficiency of 76.4% and a reference DNI of 600 W/m2. The solar field layout indicates 175 heliostats of 2 m2 surface area and 1.5 m height each. The simulation results show that the optimum height and tilt angle of the solar receiver are 26 m and 65°, respectively.


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## Body

## 1. Introduction

The increasing use of renewable energy sources to reduce the use of fossil fuels is an essential path for the transition to a sustainable energy future with low carbon emissions [1]. Solar energy is considered the most viable renewable energy source in areas with high solar potential [2]. It can be converted into electricity in two different ways: photovoltaic (PV) and concentrating solar power (CSP) technologies [3]. The manufacture of photovoltaic technology is expensive and hardly exists in Africa. Thus, CSP technology is considered as one of the promising ways for future sustainable electricity production [4]. Among the CSP technologies, solar tower power plants (STPP) present the best performance due to their capacity to provide high temperatures and therefore high efficiency [5]. The STPP technology can be also hybridized with other energy sources and have the ability to store heat [6].In many African countries, in particular the Sahelian zone has significant potential for the installation of STPP due to strong direct solar irradiation (DNI) of between 1400 and 2000 kWh/m2/year [7]. In addition, the electrification rate in the sub-Saharan zone is around 32% [8] on average with a strong variation from one country to another. This rate is much lower in rural areas (16%) where around 70% of the population lives [8]. The use of STPP technology can decrease this low electrification rate.The solar field represents the key subsystem of the STPP because it contributes around 50% [9] of the total cost of the plant and causes 40% [10] of the overall energy losses. So the performance of a solar STPP depends strongly on the solar field efficiency [11]. Indeed, the solar field efficiency depends mainly on the heliostat layout in the field, the tracking system, and the control systems [11]. The optimization of heliostat layout can reduce energy losses and therefore maximize the concentrated solar flux at the input of the receiver.Many researchers have been interested in the different categories of the heliostat layout of the solar field to find the optimal layout. Wei et al. [12] have experimented and optimized four models of heliostat field layout including north-south cornfield, north-south stagger, radial cornfield, and radial stagger. They conclude that the north-south cornfield layout is the optimal decision for the 1 MWe STPP. Zhou and Zhao [13] have presented a new method of heliostat field layout design for two typical models: cornfield and radial stagger. The results showed an optical efficiency of 82.5% and a ground coverage of 33.6% for the radial stagger against an optical efficiency of 86.3% and a coverage of 38.3% for the cornfield. Lipps and Vant-Hull [14] have presented four categories of heliostat field layout: radial cornfield, radial stagger, north-south cornfield, and north-south stagger for a 100 MWe central receiver system. The results showed that radial staggers are better than cornfield because they reduce the field area and increase the concentrated solar flux.In addition, several parameters affecting the heliostat layout have been developed and optimized for maximizing concentrated solar flux. Siala and Elayeb [15] have presented a graphical method for the radial stagger layout. The heliostats are divided into groups and rings while respecting the principle of no-blocking, the safety distance (ds), and the area density criteria. This method is also represented by mathematical equations thus facilitating its computer implementation. Collado [16] has developed a radial stagger layout for surrounded heliostat field which only uses two parameters for optimization, i.e., a no-blocking factor and safety distance. He confirmed that this method needs some improvement to find the optimal mean radial (ΔR) and azimuthal spacing of the heliostat field. Wei et al. [17] have proposed a new method for designing the heliostat field layout for a STPP. The heliostat boundary is constrained by the receiver aperture (acceptance angle (δr) and opening radius (Rout)) and the efficiency factor. They showed that the design time is reduced when calculating the concentrated solar flux. Zhang et al. [18] have defined a new factor called efficiency of available land. This factor is a function of atmospheric attenuation losses (ηatt) and spilling losses (ηspil). It was used to position in an optimum way the heliostats of a 1 MW STPP. Its performance shows an annual optical efficiency (ηopt) of 71.36%. Collado and Guallar [19] have presented the details of the optimization of a large STPP like Noor III (150 MWe, 7400 heliostats) performed with the campo code [20]. They demonstrated that this code is able to optimize the heliostat field with only two parameters: the minimum (ΔRmin) and maximum (ΔRmax) radial spacing. This article [21] analyzes the site slope effect on the optical efficiency (ηopt) of the solar field. Latitude (λ) is treated as a control variable since it modifies all the field characteristics solar. This study reveals that choosing the optimal site improves optical efficiency by up to 1.65%.The above research works are concerned with different types of heliostat layouts in the solar field and the parameters involved in the design of any heliostat layout. To our knowledge, no study has yet been carried out to assess the influence of the height and tilt angle of the solar receiver on the concentrated solar flux. The objective of the present study is to determine the optimum height and tilt angle of the solar receiver of a 30 kWe STPP intended for the production of electricity in the Sahelian zone.To achieve this objective, the solar field is firstly sized to determine the total reflective area of the heliostats and the number of heliostats. Then, a calculation code is developed in Matlab software to determine the heliostat layout and the positions and orientations of each heliostat. Finally, we have varied the height and tilt angle of the solar receiver to determine the peak flux.


## 2. Materials and Methods

### 2.1. Solar Field Sizing

The sizing of the solar field is done by determining the total reflecting area (AT) of the heliostats. The method described in Figure 1 is suitable [4]. It consists of determining the total reflective area of the heliostats from the electrical power (Ep) of the STPP. This area can be determined by the following expression [4]:(1)AT=Epηopt·ηrec·ηconv·DNI,where ηopt is the optical efficiency of the heliostat, ηrec is the solar receiver efficiency, ηconv is the conversion yield, and DNI is the direct normal irradiation in the local area.Figure 1
Sizing principle of a solar tower power plant.The optical efficiency (ηopt) of the heliostats reflects the quantity of energy reaching the receiver following the reflection of solar rays by the heliostats [12]. It determines the set of field losses such as atmospheric attenuation (ηaat), cosine (ηcos), shading and blocking (ηsh&bloc), reflecting (ηref), and spilling losses (ηspil) [22].(2)ηopt=ηaat·ηcos·ηsh&bloc·ηref·ηspil.The cosine losses represent the cosine of the angle between the incident solar ray and the vector normal to the reflecting surface of the heliostat. Therefore, they depend on both the position of the sun and the heliostat [23]. Atmospheric attenuation losses depend on weather conditions and the distance between the heliostats and the solar receiver [12]. Shading and blocking losses represent the phenomenon whereby concentrated solar rays are blocked by adjacent heliostats during scattering [24]. Spilling losses occur when the receiver cannot intercept all of the solar rays reflected by the heliostats.The receiver efficiency (ηrec) is defined as the net thermal power absorbed compared to the total power intercepted at the input of the receiver. Therefore, in this definition is included the absorption losses (ηab) and the conduction (ηcond), convection (ηconv), and radiation (ηrad) losses (see equation (3)) [4]. The conversion yield (ηcon) of the power unit is composed of the electric efficiency (ηturb) of the micro gas turbine and the electrical generator efficiency (ηG) (see equation (4)) [4].(3)ηrec=ηab·ηcond·ηconv·ηrad,(4)ηcon=ηturb·ηG.


### 2.2. Heliostat Layout Design

Among the various heliostat layout designs mentioned above, we have adapted the radial stagger layout. This method minimizes the shading and blocking losses (ηsh&bloc) and represents the most widespread layout [15]. For this, the limit of the solar field shown in Figure 2 is a function of the tower’s height (hT) and the geometry of the solar receiver (acceptance angle (θa), inlet diameter (Rin), and tilt angle (βr)). The acceptance angle is defined as the maximum angle according to which any ray reflected from the solar field with an incident angle (θ) less than this can be received by the receiver [25]. The opening of the receiver is circular, and consequently, the limits of its projection on the solar field are described by an ellipse of semiminor axis (a1) and semimajor axis (a2) given by the following equations [26, 27]:(5)a1=hT·tanθa·1+tan2θa1−tan2βr·tan2θa,(6)a2=hT·tanθa1−tan2βr·tan2θa1+tan2βr·1+tan2βrtan4θa.Figure 2
Projection limit on the field from the opening of the solar receiver.In the radial stagger layout, the heliostats are divided into groups, themselves distributed in numbered rings starting from the tower. The even rings are called the essential rings, and the odd rings are called the staggered rings. This method takes into account the tilt angle of the solar field (βL). In this method, the trajectory of a heliostat is represented by a circle as shown in Figure 3(a). Its characteristic diameter (DM) given by equation (7) is equal to the diagonal (D) of the heliostat plus the interheliostat safety distance (ds) [15].(7)DM=D+ds.Figure 3
Parameters defining the design of the (a) heliostat and the (b) no-blocking principle.
(a)(b)Using the dimensions of a heliostat, the characteristic diameter is written(8)DM=1+f2+ds·Lh,where f is the ratio between the width and the length of the heliostat and Lh is the length of the heliostat.


#### 2.2.1. Azimuthal Spacing

The azimuthal spacing in the first ring of each group representing the minimum spacing is equal to DM [15]. In the other rings, the azimuthal spacing is chosen while taking care of the principle of no-blocking. For any group j, the azimuth angle (γj) can be expressed by the following relation [15]:(9)γj=DM2Ri,j,where Ri,j represents the radius in a ring i and a group j.The heliostats that have the same azimuth angle can be classified in the same group. The angular direction (Ψm) is an angle between the north axis and the distribution axes. It can be given by the following equation [15].(10)Ψm=±n·γj,where n=0,2,4,⋯, for the essential rings, n=1,3,5,⋯, for the staggered rings, + for the northeast half field, and − for the northwest half field.


#### 2.2.2. Radial Spacing

The radius of each ring is determined by the nature of the ring. The radius of the first ring is generally given as a function of the receiver’s height (hT). The values proposed by Collado and Turegano [28] and Falcone [29] are generally between hT/2 and hT. We choose Rmin such that(11)Rmin=hT2+1.For the second ring in any groupj, the radius can be expressed on a solar field inclined at an angle (βL) by [13](12)Rmin+1=Rmin+DM·cos30·cosβL.So the minimum radial spacing (ΔRmin) between the heliostats can be given by(13)ΔRmin=DM·cos30·cosβL.The maximum radial spacing (ΔRmax) should be determined by equation (17) according to the principle of no-blocking between adjacent heliostats as shown in Figure 3(b). The detailed calculations of this principle are given by equations (14), (15), and (16) [13].(14)zm=Rmin·tanβL+Hh,(15)d=Rmin2+hT−zm2,(16)γ=arcsinDM2·d+arcsinRmind−βL,(17)ΔRmax=DM·cosβLcosγ.Finally, the radius of the other rings can be determined by the following relation [13]:(18)Rmin+n=Rmin+ΔRmin+RcoefΔRmax−ΔRmin,where Rcoef is the optimized coefficient of the radius.


### 2.3. Positions and Orientations of the Heliostats

The first step to find the position and the orientation of the heliostats in the solar field is to determine the position of the sun. The sun’s position is characterized by the altitude (α) and the azimuth angle (A). Figure 4(a) shows the angles defining the apparent position of the sun. The sun’s altitude is calculated as follows [30, 31]:(19)α=sin−1sinλ·sinδ+cosδ·cosλ·cosω,where λ is the local latitude, δ is the solar declination angle, and ω is the hour angle. The solar declination angle (δ) depends on the day number (N) of the year and can be determined by [30](20)δ=23,45sin·360365·284+N.Figure 4
Positions of the (a) sun and the (b) heliostat.
(a)(b)The hour angle is measured from solar noon and is given by [30](21)ω=15·12−TSV,where TSV is the solar time. Then, the solar azimuth angle (A) representing the angle between the projection of the sun’s direction and the north direction is given by [30](22)A=arcsincosδ·sinwcosα.The sun vector (S⟶), as illustrated in Figure 4(b), can be defined by equation (23) where i⟶, j⟶, and k⟶ represent, respectively, the unit vectors of the x, y, and z axes.(23)S⟶=Sxi⟶+Syj⟶+Szk⟶.By projection in the frameO,i,j,k, the three components (Sx,Sy,Sz) of the sun vector are determined as follows [32]:(24)S=Sx=cosα·sinA,Sy=cosα·cosA,Sz=sinA.A heliostat is located in the solar field by the coordinates of its center (see Figure4(b)) which are determined once the angular direction (Ψm) of the heliostat and the radius (R) of the ring to which it belongs are known. Thus, the coordinates (Hx,Hy,Hz) of each heliostat can be determined by the following system of equations [13]:(25)H=Hx=R·sinΨm,Hy=R·cosΨm,Hz=Hh.The cosine losses (ηcos) which represents the greatest loss in the solar field is due to the angle between the incident solar ray and the normal vector at the heliostat’s area. Therefore, it depends on both the position of the heliostat and the sun. To determine the normal vector (n⟶) at the heliostat’s area, the vector (R⟶) located between the solar receiver and the heliostat must be defined first [33].(26)R⟶=T⟶−C⟶T⟶−C⟶,where T⟶ and C⟶ represent, respectively, the position vectors of the sun and the heliostat. Thus, the normal vector (n⟶) at the heliostat’s area is given by [23](27)n⟶=R⟶+S⟶2·1+S⟶.R⟶.Then, the target vector (N⟶) which determines the heliostat’s position and orientation is given by(28)N⟶=H⟶+n⟶.


### 2.4. Tilt Angle of the Receiver

The method described in Figure5 is used to determine the optimum tilt angle of the solar receiver. This method consists of calculating the coordinates of the receiver, i.e., Aimpoint (Xaimpoint, Yaimpoint, and Zaimpoint) for different tilt angles of the solar receiver. The system of equations given by the following relation calculates the coordinates of the receiver for each tilt angle.(29)Aimpoint=Xap=0,Yap=Lr·cosδr,Zap=Ztarget−ΔZ,where Ztarget is the height of the tower, Lr is the length of the solar receiver, and ΔZ=Lr·sinδr.Figure 5
Method for determining the tilt angle of the solar receiver.


### 2.5. Operating Parameters

We estimated the different efficiency factors that determine optical efficiency (ηopt). We have also chosen the various factors that participate in the sizing of the solar field such as the solar receiver efficiency (ηrec), the turbine efficiency (ηturb), and the electric generator efficiency (ηG). All of its operating parameters are provided in Table 1.Table 1
Various estimated efficiency factors.
CharacteristicsParametersValuesRef.Efficiency factorsηaat0.95[32]ηcos0.90[4]ηsh&.bloc0.95[32]ηref0.97[34]ηspil0.97[4]Conversion yieldηturb0.26[35]ηG0.90[36]Receiverηrec0.80[37]


## 2.1. Solar Field Sizing

The sizing of the solar field is done by determining the total reflecting area (AT) of the heliostats. The method described in Figure 1 is suitable [4]. It consists of determining the total reflective area of the heliostats from the electrical power (Ep) of the STPP. This area can be determined by the following expression [4]:(1)AT=Epηopt·ηrec·ηconv·DNI,where ηopt is the optical efficiency of the heliostat, ηrec is the solar receiver efficiency, ηconv is the conversion yield, and DNI is the direct normal irradiation in the local area.Figure 1
Sizing principle of a solar tower power plant.The optical efficiency (ηopt) of the heliostats reflects the quantity of energy reaching the receiver following the reflection of solar rays by the heliostats [12]. It determines the set of field losses such as atmospheric attenuation (ηaat), cosine (ηcos), shading and blocking (ηsh&bloc), reflecting (ηref), and spilling losses (ηspil) [22].(2)ηopt=ηaat·ηcos·ηsh&bloc·ηref·ηspil.The cosine losses represent the cosine of the angle between the incident solar ray and the vector normal to the reflecting surface of the heliostat. Therefore, they depend on both the position of the sun and the heliostat [23]. Atmospheric attenuation losses depend on weather conditions and the distance between the heliostats and the solar receiver [12]. Shading and blocking losses represent the phenomenon whereby concentrated solar rays are blocked by adjacent heliostats during scattering [24]. Spilling losses occur when the receiver cannot intercept all of the solar rays reflected by the heliostats.The receiver efficiency (ηrec) is defined as the net thermal power absorbed compared to the total power intercepted at the input of the receiver. Therefore, in this definition is included the absorption losses (ηab) and the conduction (ηcond), convection (ηconv), and radiation (ηrad) losses (see equation (3)) [4]. The conversion yield (ηcon) of the power unit is composed of the electric efficiency (ηturb) of the micro gas turbine and the electrical generator efficiency (ηG) (see equation (4)) [4].(3)ηrec=ηab·ηcond·ηconv·ηrad,(4)ηcon=ηturb·ηG.


## 2.2. Heliostat Layout Design

Among the various heliostat layout designs mentioned above, we have adapted the radial stagger layout. This method minimizes the shading and blocking losses (ηsh&bloc) and represents the most widespread layout [15]. For this, the limit of the solar field shown in Figure 2 is a function of the tower’s height (hT) and the geometry of the solar receiver (acceptance angle (θa), inlet diameter (Rin), and tilt angle (βr)). The acceptance angle is defined as the maximum angle according to which any ray reflected from the solar field with an incident angle (θ) less than this can be received by the receiver [25]. The opening of the receiver is circular, and consequently, the limits of its projection on the solar field are described by an ellipse of semiminor axis (a1) and semimajor axis (a2) given by the following equations [26, 27]:(5)a1=hT·tanθa·1+tan2θa1−tan2βr·tan2θa,(6)a2=hT·tanθa1−tan2βr·tan2θa1+tan2βr·1+tan2βrtan4θa.Figure 2
Projection limit on the field from the opening of the solar receiver.In the radial stagger layout, the heliostats are divided into groups, themselves distributed in numbered rings starting from the tower. The even rings are called the essential rings, and the odd rings are called the staggered rings. This method takes into account the tilt angle of the solar field (βL). In this method, the trajectory of a heliostat is represented by a circle as shown in Figure 3(a). Its characteristic diameter (DM) given by equation (7) is equal to the diagonal (D) of the heliostat plus the interheliostat safety distance (ds) [15].(7)DM=D+ds.Figure 3
Parameters defining the design of the (a) heliostat and the (b) no-blocking principle.
(a)(b)Using the dimensions of a heliostat, the characteristic diameter is written(8)DM=1+f2+ds·Lh,where f is the ratio between the width and the length of the heliostat and Lh is the length of the heliostat.


### 2.2.1. Azimuthal Spacing

The azimuthal spacing in the first ring of each group representing the minimum spacing is equal to DM [15]. In the other rings, the azimuthal spacing is chosen while taking care of the principle of no-blocking. For any group j, the azimuth angle (γj) can be expressed by the following relation [15]:(9)γj=DM2Ri,j,where Ri,j represents the radius in a ring i and a group j.The heliostats that have the same azimuth angle can be classified in the same group. The angular direction (Ψm) is an angle between the north axis and the distribution axes. It can be given by the following equation [15].(10)Ψm=±n·γj,where n=0,2,4,⋯, for the essential rings, n=1,3,5,⋯, for the staggered rings, + for the northeast half field, and − for the northwest half field.


### 2.2.2. Radial Spacing

The radius of each ring is determined by the nature of the ring. The radius of the first ring is generally given as a function of the receiver’s height (hT). The values proposed by Collado and Turegano [28] and Falcone [29] are generally between hT/2 and hT. We choose Rmin such that(11)Rmin=hT2+1.For the second ring in any groupj, the radius can be expressed on a solar field inclined at an angle (βL) by [13](12)Rmin+1=Rmin+DM·cos30·cosβL.So the minimum radial spacing (ΔRmin) between the heliostats can be given by(13)ΔRmin=DM·cos30·cosβL.The maximum radial spacing (ΔRmax) should be determined by equation (17) according to the principle of no-blocking between adjacent heliostats as shown in Figure 3(b). The detailed calculations of this principle are given by equations (14), (15), and (16) [13].(14)zm=Rmin·tanβL+Hh,(15)d=Rmin2+hT−zm2,(16)γ=arcsinDM2·d+arcsinRmind−βL,(17)ΔRmax=DM·cosβLcosγ.Finally, the radius of the other rings can be determined by the following relation [13]:(18)Rmin+n=Rmin+ΔRmin+RcoefΔRmax−ΔRmin,where Rcoef is the optimized coefficient of the radius.


## 2.2.1. Azimuthal Spacing

The azimuthal spacing in the first ring of each group representing the minimum spacing is equal to DM [15]. In the other rings, the azimuthal spacing is chosen while taking care of the principle of no-blocking. For any group j, the azimuth angle (γj) can be expressed by the following relation [15]:(9)γj=DM2Ri,j,where Ri,j represents the radius in a ring i and a group j.The heliostats that have the same azimuth angle can be classified in the same group. The angular direction (Ψm) is an angle between the north axis and the distribution axes. It can be given by the following equation [15].(10)Ψm=±n·γj,where n=0,2,4,⋯, for the essential rings, n=1,3,5,⋯, for the staggered rings, + for the northeast half field, and − for the northwest half field.


## 2.2.2. Radial Spacing

The radius of each ring is determined by the nature of the ring. The radius of the first ring is generally given as a function of the receiver’s height (hT). The values proposed by Collado and Turegano [28] and Falcone [29] are generally between hT/2 and hT. We choose Rmin such that(11)Rmin=hT2+1.For the second ring in any groupj, the radius can be expressed on a solar field inclined at an angle (βL) by [13](12)Rmin+1=Rmin+DM·cos30·cosβL.So the minimum radial spacing (ΔRmin) between the heliostats can be given by(13)ΔRmin=DM·cos30·cosβL.The maximum radial spacing (ΔRmax) should be determined by equation (17) according to the principle of no-blocking between adjacent heliostats as shown in Figure 3(b). The detailed calculations of this principle are given by equations (14), (15), and (16) [13].(14)zm=Rmin·tanβL+Hh,(15)d=Rmin2+hT−zm2,(16)γ=arcsinDM2·d+arcsinRmind−βL,(17)ΔRmax=DM·cosβLcosγ.Finally, the radius of the other rings can be determined by the following relation [13]:(18)Rmin+n=Rmin+ΔRmin+RcoefΔRmax−ΔRmin,where Rcoef is the optimized coefficient of the radius.


## 2.3. Positions and Orientations of the Heliostats

The first step to find the position and the orientation of the heliostats in the solar field is to determine the position of the sun. The sun’s position is characterized by the altitude (α) and the azimuth angle (A). Figure 4(a) shows the angles defining the apparent position of the sun. The sun’s altitude is calculated as follows [30, 31]:(19)α=sin−1sinλ·sinδ+cosδ·cosλ·cosω,where λ is the local latitude, δ is the solar declination angle, and ω is the hour angle. The solar declination angle (δ) depends on the day number (N) of the year and can be determined by [30](20)δ=23,45sin·360365·284+N.Figure 4
Positions of the (a) sun and the (b) heliostat.
(a)(b)The hour angle is measured from solar noon and is given by [30](21)ω=15·12−TSV,where TSV is the solar time. Then, the solar azimuth angle (A) representing the angle between the projection of the sun’s direction and the north direction is given by [30](22)A=arcsincosδ·sinwcosα.The sun vector (S⟶), as illustrated in Figure 4(b), can be defined by equation (23) where i⟶, j⟶, and k⟶ represent, respectively, the unit vectors of the x, y, and z axes.(23)S⟶=Sxi⟶+Syj⟶+Szk⟶.By projection in the frameO,i,j,k, the three components (Sx,Sy,Sz) of the sun vector are determined as follows [32]:(24)S=Sx=cosα·sinA,Sy=cosα·cosA,Sz=sinA.A heliostat is located in the solar field by the coordinates of its center (see Figure4(b)) which are determined once the angular direction (Ψm) of the heliostat and the radius (R) of the ring to which it belongs are known. Thus, the coordinates (Hx,Hy,Hz) of each heliostat can be determined by the following system of equations [13]:(25)H=Hx=R·sinΨm,Hy=R·cosΨm,Hz=Hh.The cosine losses (ηcos) which represents the greatest loss in the solar field is due to the angle between the incident solar ray and the normal vector at the heliostat’s area. Therefore, it depends on both the position of the heliostat and the sun. To determine the normal vector (n⟶) at the heliostat’s area, the vector (R⟶) located between the solar receiver and the heliostat must be defined first [33].(26)R⟶=T⟶−C⟶T⟶−C⟶,where T⟶ and C⟶ represent, respectively, the position vectors of the sun and the heliostat. Thus, the normal vector (n⟶) at the heliostat’s area is given by [23](27)n⟶=R⟶+S⟶2·1+S⟶.R⟶.Then, the target vector (N⟶) which determines the heliostat’s position and orientation is given by(28)N⟶=H⟶+n⟶.


## 2.4. Tilt Angle of the Receiver

The method described in Figure5 is used to determine the optimum tilt angle of the solar receiver. This method consists of calculating the coordinates of the receiver, i.e., Aimpoint (Xaimpoint, Yaimpoint, and Zaimpoint) for different tilt angles of the solar receiver. The system of equations given by the following relation calculates the coordinates of the receiver for each tilt angle.(29)Aimpoint=Xap=0,Yap=Lr·cosδr,Zap=Ztarget−ΔZ,where Ztarget is the height of the tower, Lr is the length of the solar receiver, and ΔZ=Lr·sinδr.Figure 5
Method for determining the tilt angle of the solar receiver.


## 2.5. Operating Parameters

We estimated the different efficiency factors that determine optical efficiency (ηopt). We have also chosen the various factors that participate in the sizing of the solar field such as the solar receiver efficiency (ηrec), the turbine efficiency (ηturb), and the electric generator efficiency (ηG). All of its operating parameters are provided in Table 1.Table 1
Various estimated efficiency factors.
CharacteristicsParametersValuesRef.Efficiency factorsηaat0.95[32]ηcos0.90[4]ηsh&.bloc0.95[32]ηref0.97[34]ηspil0.97[4]Conversion yieldηturb0.26[35]ηG0.90[36]Receiverηrec0.80[37]


## 3. Results and Discussion

### 3.1. DNI Estimate

In this study, we used direct normal irradiation (DNI) data measured in Senegal. This data is available for one year, which is acceptable for the concentrating solar power (CSP) technologies. From these data, we chose those of March 21, which corresponds to the spring equinox where the sun passes at the zenith (i.e., vertical instead). The profile of the DNI for the day of March 21 is shown in Figure6. For the sizing of the 30 kWe solar tower power plant, we chose a reference DNI of 600 W/m2. This choice made it possible to obtain a solar operating range of 6 h.Figure 6
Daily profiles of DNI of March 21 in Senegal.


### 3.2. Result of the Sizing

For the sizing of the solar field, we chose an indirect pressurized air volumetric solar receiver with an efficiency (ηrec) of 80% [37], a Capstone turbine (C30) with an electrical efficiency (ηturb) of 26% [35], and an electric generator (ηG) yield of 90% [36]. The results showed that the optical efficiency (ηopt) of the solar field is 76.4% and the total reflecting area (AT) of the heliostats is 350 m2.The size of the heliostats in solar tower power plants is relatively large (121 m2 at PS10 and PS20) [38]. This large size of the heliostats increases mechanical stress, wind resistance, and production costs. Therefore, in this study, we chose small heliostats that could easily be handled for upkeep and maintenance. The chosen mini heliostats have a reflecting area of 2 m2 and a height of 1.5 m. The surface shape of the heliostat mirror is square. So the solar field is composed by 175 heliostats. The site for the installation of the solar tower power plant is located in the department of Podor (Saint-Louis, Senegal) where the north latitude is about 16° 40′ and the west longitude is about 14° 57′.The solar field is a horizontal plane and is tilted 0° with respect to the solar receiver. The main characteristics of the solar field subsystems are provided in Table 2.Table 2
Characteristics of solar field subsystems.
SubsystemsParametersValuesUnitsHeliostatsFormSquare[-]Number175[-]Height1.50[m]Area2[m2]Reflectivity0.90[-]Transmittivity0[-]TowerOrientationNorth[-]ReceiverTypeVolumetric[-]Acceptance angle25[°]Transmittivity0.90[-]Reflectivity0[-]Length1[m]Opening radius0.70[m]Opening area1.54[m2]


### 3.3. Heliostat Layout Results

The radial stagger layout of the solar field shown in Figure7 is designed for the 30 kWe STPP. The heliostats are placed north of the tower in the solar field. For an interheliostat security factor (ds) of 0.3, the characteristic diameter (DM) of the heliostats is 2.42 m. The minimum (ΔRmin) and maximum (ΔRmax) radial spacing between the heliostats are 2.1 m and 2.45 m, respectively, for the optimized coefficient of the radius (Rcoef) of 0.6. The coordinates of each heliostat, the azimuth angle (γj), and the radius (R) of each ring are calculated. Figure 7 shows the heliostat layout in the solar field for the heights (hT) of 20 m, 22 m, 24 m, 26 m, 28 m, and 29 m. The characteristics of each heliostat layout are given in Table 3.Figure 7
Layouts of the solar field for different heights of the tower.
(a)
Height of the solartower=20m(b)
Height of the solartower=22m(c)
Height of the solartower=24m(d)
Height of the solartower=26m(e)
Height of the solartower=28m(f)
Height of the solartower=29mTable 3
Characteristics of each heliostat layout design.
Heights of the tower20 m22 m24 m26 m28 m29 mSemiminor axisa1 (m)15.1415.9617.4118.8620.3121.04Semimajor axisa2 (m)18.7218.9720.6922.4224.1525.01Field area (m2)890.5951.31132.11328.61540.91652.9Number of rings171615141413Essential rings988776Staggered rings887777Radius of the first ring (m)111213141515.5Radius of the last ring (m)45.6844.5643.4442.3343.3241.72As can be clearly seen in Table3, the semiminor (a1) axis and the semimajor (a2) axis increase with the height of the tower, which leads to an increase in the solar field area. However, the number of rings decreases as the height of the tower increases. This shows that the more of field area increases, the number of heliostats in a ring is high, and therefore, the number of rings decreases. The radius of the first ring depends on the tower height and that of the last ring depends not only on the tower height but also on the number of rings.


### 3.4. Simulation Results

After calculating the parameters of each heliostat and their design over the field using Matlab software, an Excel file is generated indicating all the positions and orientations of the heliostats. This card is incorporated in Soltrace to indicate the positions of the heliostats, the tower, and the solar receiver as illustrated in Figure8(a). Modelling and simulation of the solar field under Soltrace reproduce the trajectory of the solar rays and establish the concentrated flux map at the entrance of the receiver. Soltrace software takes into account the reflectivity (ρref) of the mirrors and adjustment errors such as slope error (σslope) and specularity error (σsp). Assuming that the reflectivity of the mirror equals to 0.90, the transmittivity equals to 0, the slope error equals to 0.95 mrad, and the specularity error equals to 0.3 mrad. The solar receiver transmittivity is 0.90, the reflectivity is 0, the slope error is 0.9 mrad, and the secularity error is 0.2 mrad. For simulation, we launched 1,000,000 solar rays to visualize the energy flow at the input of the solar receiver. Figure 8(b) shows the visualization in the Soltrace environment of the intersections between the solar rays and the elements of the solar field (heliostats, solar receiver, and tower).Figure 8
Design of (a) heliostat layout and (b) visualization of rays in Soltrace.The solar flux distribution over the opening of the receiver is shown in Figure9(a). As can be seen in this figure, the spilling losses are almost zero for an input radius (Rin) of the receiver of 0.7 m. With a direct normal irradiation (DNI) of 850 W/m2, the maximum concentrated solar flux at the input of the receiver is approximately 120 kW/m2 (i.e., 184.8 kW). Figure 9(b) shows the three-dimensional (3D) flux map which clearly shows the maximum solar flux.Figure 9
(a) Distribution of the solar flux at the input of the receiver and (b) the 3D flux map.
(a)(b)We studied the influence of the tower’s height (hT) on the concentrated solar flux at the input of the solar receiver located at the top of the tower. For this, we varied the tower’s height from 20 to 29 m to generate the maximum solar flux (peak flux) at the input of the solar receiver. The simulation results are shown in Figure 10. As we can clearly see from this figure, the solar flux starts to increase from 131.25 kW/m2 for a height of 20 m up to about 134.40 kW/m2 for a height of 26 m where it begins to decrease. In this interval (20 to 26 m), the solar receiver increasingly intercepts all of the solar rays reflected by the heliostats. But after 26 m, the spilling losses (ηspil) seem to appear because the solar receiver is no longer able to intercept all the concentrated solar rays. These results made it possible to choose the height of the tower of 26 m because it allows obtaining a maximum solar flux at the input of the receiver which is approximately 134.4 kW/m2.Figure 10
Maximum solar flux depending on the height of the tower.The influence of the tilt angle (βr) of the solar receiver on the concentrated solar flux is also studied in this work. The solar receiver’s height (hT) is 26 m, since it is the optimum value calculated previously. We varied the tilt angle from 0 to 90° to determine the maximum solar flux (peak flux). Figure 11 shows the simulation results obtained. The tendency is similar for the solar receiver’s height. We note an increase in the solar flux up to values between approximately 60 and 65°. After 65°, the decrease of the peak flux is pronounced. This decrease is caused by the spilling losses (ηspil) at the opening of the solar receiver. In addition, due to the fact that the cosine losses (ηcos) and atmospheric attenuation losses (ηatt) increase, these also lead to the decrease of the concentrated solar flux. The tilt angle is therefore an important parameter of the solar receiver which must be well chosen when it is desired to obtain the maximum solar flux. With these results, it can be concluded that the optimum tilt angle is approximately 65° which corresponds to the maximum solar flux equals to 93 kW/m2 (143.22 kW).Figure 11
Maximum solar flux depending on the tilt angle of the receiver.


## 3.1. DNI Estimate

In this study, we used direct normal irradiation (DNI) data measured in Senegal. This data is available for one year, which is acceptable for the concentrating solar power (CSP) technologies. From these data, we chose those of March 21, which corresponds to the spring equinox where the sun passes at the zenith (i.e., vertical instead). The profile of the DNI for the day of March 21 is shown in Figure6. For the sizing of the 30 kWe solar tower power plant, we chose a reference DNI of 600 W/m2. This choice made it possible to obtain a solar operating range of 6 h.Figure 6
Daily profiles of DNI of March 21 in Senegal.


## 3.2. Result of the Sizing

For the sizing of the solar field, we chose an indirect pressurized air volumetric solar receiver with an efficiency (ηrec) of 80% [37], a Capstone turbine (C30) with an electrical efficiency (ηturb) of 26% [35], and an electric generator (ηG) yield of 90% [36]. The results showed that the optical efficiency (ηopt) of the solar field is 76.4% and the total reflecting area (AT) of the heliostats is 350 m2.The size of the heliostats in solar tower power plants is relatively large (121 m2 at PS10 and PS20) [38]. This large size of the heliostats increases mechanical stress, wind resistance, and production costs. Therefore, in this study, we chose small heliostats that could easily be handled for upkeep and maintenance. The chosen mini heliostats have a reflecting area of 2 m2 and a height of 1.5 m. The surface shape of the heliostat mirror is square. So the solar field is composed by 175 heliostats. The site for the installation of the solar tower power plant is located in the department of Podor (Saint-Louis, Senegal) where the north latitude is about 16° 40′ and the west longitude is about 14° 57′.The solar field is a horizontal plane and is tilted 0° with respect to the solar receiver. The main characteristics of the solar field subsystems are provided in Table 2.Table 2
Characteristics of solar field subsystems.
SubsystemsParametersValuesUnitsHeliostatsFormSquare[-]Number175[-]Height1.50[m]Area2[m2]Reflectivity0.90[-]Transmittivity0[-]TowerOrientationNorth[-]ReceiverTypeVolumetric[-]Acceptance angle25[°]Transmittivity0.90[-]Reflectivity0[-]Length1[m]Opening radius0.70[m]Opening area1.54[m2]


## 3.3. Heliostat Layout Results

The radial stagger layout of the solar field shown in Figure7 is designed for the 30 kWe STPP. The heliostats are placed north of the tower in the solar field. For an interheliostat security factor (ds) of 0.3, the characteristic diameter (DM) of the heliostats is 2.42 m. The minimum (ΔRmin) and maximum (ΔRmax) radial spacing between the heliostats are 2.1 m and 2.45 m, respectively, for the optimized coefficient of the radius (Rcoef) of 0.6. The coordinates of each heliostat, the azimuth angle (γj), and the radius (R) of each ring are calculated. Figure 7 shows the heliostat layout in the solar field for the heights (hT) of 20 m, 22 m, 24 m, 26 m, 28 m, and 29 m. The characteristics of each heliostat layout are given in Table 3.Figure 7
Layouts of the solar field for different heights of the tower.
(a)
Height of the solartower=20m(b)
Height of the solartower=22m(c)
Height of the solartower=24m(d)
Height of the solartower=26m(e)
Height of the solartower=28m(f)
Height of the solartower=29mTable 3
Characteristics of each heliostat layout design.
Heights of the tower20 m22 m24 m26 m28 m29 mSemiminor axisa1 (m)15.1415.9617.4118.8620.3121.04Semimajor axisa2 (m)18.7218.9720.6922.4224.1525.01Field area (m2)890.5951.31132.11328.61540.91652.9Number of rings171615141413Essential rings988776Staggered rings887777Radius of the first ring (m)111213141515.5Radius of the last ring (m)45.6844.5643.4442.3343.3241.72As can be clearly seen in Table3, the semiminor (a1) axis and the semimajor (a2) axis increase with the height of the tower, which leads to an increase in the solar field area. However, the number of rings decreases as the height of the tower increases. This shows that the more of field area increases, the number of heliostats in a ring is high, and therefore, the number of rings decreases. The radius of the first ring depends on the tower height and that of the last ring depends not only on the tower height but also on the number of rings.


## 3.4. Simulation Results

After calculating the parameters of each heliostat and their design over the field using Matlab software, an Excel file is generated indicating all the positions and orientations of the heliostats. This card is incorporated in Soltrace to indicate the positions of the heliostats, the tower, and the solar receiver as illustrated in Figure8(a). Modelling and simulation of the solar field under Soltrace reproduce the trajectory of the solar rays and establish the concentrated flux map at the entrance of the receiver. Soltrace software takes into account the reflectivity (ρref) of the mirrors and adjustment errors such as slope error (σslope) and specularity error (σsp). Assuming that the reflectivity of the mirror equals to 0.90, the transmittivity equals to 0, the slope error equals to 0.95 mrad, and the specularity error equals to 0.3 mrad. The solar receiver transmittivity is 0.90, the reflectivity is 0, the slope error is 0.9 mrad, and the secularity error is 0.2 mrad. For simulation, we launched 1,000,000 solar rays to visualize the energy flow at the input of the solar receiver. Figure 8(b) shows the visualization in the Soltrace environment of the intersections between the solar rays and the elements of the solar field (heliostats, solar receiver, and tower).Figure 8
Design of (a) heliostat layout and (b) visualization of rays in Soltrace.The solar flux distribution over the opening of the receiver is shown in Figure9(a). As can be seen in this figure, the spilling losses are almost zero for an input radius (Rin) of the receiver of 0.7 m. With a direct normal irradiation (DNI) of 850 W/m2, the maximum concentrated solar flux at the input of the receiver is approximately 120 kW/m2 (i.e., 184.8 kW). Figure 9(b) shows the three-dimensional (3D) flux map which clearly shows the maximum solar flux.Figure 9
(a) Distribution of the solar flux at the input of the receiver and (b) the 3D flux map.
(a)(b)We studied the influence of the tower’s height (hT) on the concentrated solar flux at the input of the solar receiver located at the top of the tower. For this, we varied the tower’s height from 20 to 29 m to generate the maximum solar flux (peak flux) at the input of the solar receiver. The simulation results are shown in Figure 10. As we can clearly see from this figure, the solar flux starts to increase from 131.25 kW/m2 for a height of 20 m up to about 134.40 kW/m2 for a height of 26 m where it begins to decrease. In this interval (20 to 26 m), the solar receiver increasingly intercepts all of the solar rays reflected by the heliostats. But after 26 m, the spilling losses (ηspil) seem to appear because the solar receiver is no longer able to intercept all the concentrated solar rays. These results made it possible to choose the height of the tower of 26 m because it allows obtaining a maximum solar flux at the input of the receiver which is approximately 134.4 kW/m2.Figure 10
Maximum solar flux depending on the height of the tower.The influence of the tilt angle (βr) of the solar receiver on the concentrated solar flux is also studied in this work. The solar receiver’s height (hT) is 26 m, since it is the optimum value calculated previously. We varied the tilt angle from 0 to 90° to determine the maximum solar flux (peak flux). Figure 11 shows the simulation results obtained. The tendency is similar for the solar receiver’s height. We note an increase in the solar flux up to values between approximately 60 and 65°. After 65°, the decrease of the peak flux is pronounced. This decrease is caused by the spilling losses (ηspil) at the opening of the solar receiver. In addition, due to the fact that the cosine losses (ηcos) and atmospheric attenuation losses (ηatt) increase, these also lead to the decrease of the concentrated solar flux. The tilt angle is therefore an important parameter of the solar receiver which must be well chosen when it is desired to obtain the maximum solar flux. With these results, it can be concluded that the optimum tilt angle is approximately 65° which corresponds to the maximum solar flux equals to 93 kW/m2 (143.22 kW).Figure 11
Maximum solar flux depending on the tilt angle of the receiver.


## 4. Conclusion

The solar tower power plant offers interesting potential for the production of electricity in the Sahelian zone where the solar potential is high and the electrification rate remains low. The objective of this study was to determine the optimum height and tilt angle of the solar receiver of a 30 kWe STPP intended for the production of electricity in the Sahelian zone. The sizing of the solar field had given the following results: an optical efficiency of 76.4% for a reference DNI of 600 W/m2, a total reflective area of the heliostats of 350 m2, and a number of heliostats of 175 of 2 m2 area and 1.5 m height. The simulation results showed that the optimum height and tilt angle of the solar receiver were 26 m and 65°, respectively.These results will serve as a basis for the sizing of the compound parabolic concentrator (CPC) and the modelling of heat transfers within the solar receiver.



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*Source: 1961134-2021-07-22.xml*