# Shaping High Efficiency, High Temperature Cavity Tubular Solar Central Receivers

^{1}

^{2}

^{*}

## Abstract

**:**

^{2}and 25 m

^{2}for high, and low convection losses, respectively.

## 1. Introduction

#### 1.1. Context

^{2}[1]. Tubular receivers are operated in all the commercial solar thermal power plants even if other options as porous receivers have been developed at pilot scale [1]. Molten salt working temperature results in heat-to-electricity efficiency of approximately 42%. Getting higher efficiencies (~50% and more) is possible at 700–750 °C with supercritical carbon dioxide (sCO

_{2}) cycles [2] and at approximately 850 °C with combined cycles [3]. Such high operating temperatures result in great challenges on the solar receiver design, construction materials and heat transfer fluids. Central receiver designs are, either external or cavity types. At high temperatures, external receiver must tolerate high incident solar flux densities (high concentration, ~1000 suns), in order to maintain acceptable radiation losses [4]. Consequently, high wall-to-fluid heat transfer coefficients are compulsory in external tubular solar receivers to maintain the wall temperature within its allowable working condition as it is the case with liquid sodium [5]. The other option is using direct HTF heating design as falling particles solar receivers [1].

_{th}). Researches on cavity receivers have been mainly focused on small-size cavities integrated with dish solar concentrators [7,8,9]. Numerical three-dimensional (3D) study of the combined natural convection and radiation heat losses of downward facing cavity receivers of different shapes was presented in [7], in the temperature range 250–650 °C. The results indicated that convection losses decrease strongly with the inclination angle of the cavity for all the receiver shapes. A correlation was proposed to predict the convective losses in a small range of Rayleigh number (Ra = 2 × 10

^{8}–6 × 10

^{8}). Le Roux et al. [8] have studied tubular receiver in the power range of 1–100 kW to be used in a small-scale solar thermal Brayton cycle using a micro-turbine. The numerical study integrated the optical efficiency of a dish concentrator (tracking error) and the thermal efficiency of the receiver. A 3D numerical simulation of cylindrical cavity receivers, with a 45° (π/4 radians), inclination was proposed in [9]. It was shown that the receiver efficiency varied sharply with the aperture size and that the ratio of radiation to convection losses was ranging from approximately two to four at low direct normal irradiance (DNI) (i.e., low temperature), and high DNI, respectively. Grange et al. [10] modeled a medium-scale cavity receiver to power a Brayton cycle. Receiver efficiency of solar power tower was studied for example in [11,12,13,14]. Rodriguez-Sanchez et al. [11] reported an operation efficiency of molten salt external receiver of the Solar Two demonstration power plant. The results showed that receiver efficiency was around 76% for full load and 69% for half load. These data are lower than previous estimations, 87%, and 80% respectively, reported when the external tube temperature was assumed independent of the incident power. Kim et al. [12] developed a simplified model of heat losses of SPT receivers using correlations derived from numerical simulations. External and cavity with a 9 m

^{2}-absorber have been considered in the temperature range 600–900 °C. For low wind velocity, calculated efficiency of Solar Two external receiver was 88% [12]. Generally, for cavity receivers at 900 °C the ratio of radiation to convection losses ranged between 2 and 7 except with head-on high wind condition (10 m/s) that resulted in a ratio approximately equal to one [12]. A cavity-type molten salt solar receiver model was developed in [13] in combination with a solar field optical model. The receiver model included a thermo-hydraulic approach of molten salt flow in 12 m-long tubes. Qiu et al. [14] have proposed a similar approach, including a strategy for a direct steam generation cavity solar receiver tested at Dahan pilot-plant (China). They demonstrated that the cavity effect could improve the optical efficiency throughout the whole year. The effect of directional variation of optical properties in cavity solar receiver was discussed in [15] using Monte Carlo simulations. They concluded that the higher the diffuse ratio, the higher the efficiency and the lower the influence of incident radiation pattern.

_{b}(T

_{b}= (T

_{c}+ T

_{a})/2, where T

_{c}and T

_{a}are the cavity and ambient air temperature respectively). Nusselt number versus Rayleigh number correlations have been deduced for different regimes and cavity orientations. The comparison with experimental data resulted in a convective heat transfer coefficient of 7.2 and 9.7 W/m

^{2}K for the inactive, and the absorber tube surfaces, respectively. Samanes et al. [18] compared different correlations for convective losses prediction and selected the Clausing’s approach. A numerical analysis of convective losses of inclined cavity was proposed in [19]. They concluded that in no wind condition the Clausing model and the simulation results match very well (for both horizontal and inclined cavities). For inclined cavity with wind, an increase of the convective losses was predicted due to a shrinking of the stagnant zone, which is in contrast to the predictions of the Clausing model. Table A1, presented in Appendix A, lists the previously cited studies and their main findings.

#### 1.2. Background

^{2}s is a safe limit for 50 mm intern diameter (ID) tubes and approximately 50 μm particles. The particles can experience a different regime along the tube height in particular bubbling and slugging [29]. Axisymmetric slugging must be avoided due to the associated reduction of the particle-to-wall heat transfer coefficient. Fortunately, slugging is strongly reduced with increases in temperature. Nevertheless, this issue results in a limited height of the tubes. General particle flow in each tube in a multi-tube solar receiver [30] is not affected by the presence of adjacent tubes. In particular, any difference in particle mass flow rate was observed. Concerning the maximum power of a single receiver, it is clearly related to the maximum acceptable height of the tubes, this issue is discussed in [31].

#### 1.3. Objective

- The acceptable length of the absorber tubes limits the maximum power of the solar receiver. Based on a previous study [29] and further unpublished experimental results, the length of the tubes is fixed at 7 m. Accounting for this limited length of the tubes, the nominal power of a single solar receiver is approximately 50 MW [31].
- The wall-to-fluidized bed heat transfer coefficient is chosen in agreement with previous experimental data [26].
- The tubes are vertical. Vertical position of the absorber tubes is mandatory because tilting will result in a strong channeling and particle segregation. Indeed, this would cause a very non-homogeneous bed, with a dense bed (without bubbles) near the solar-irradiated part of the tube and all the bubbles in the opposite region [34]. This will cause a dramatic decrease of wall-to-bed heat transfer.
- The solar radiation acceptable flux density is another constraint related to the working temperature limit of the tube wall. This value is discussed in the next section.

## 2. Receiver’s Geometry and Preliminary Considerations

#### 2.1. General Description and Material Properties

^{3}and its specific heat variation with temperature is given by Equation (1). An apparent specific heat is calculated, based on the average temperature of the particle in the tube (${T}_{ave}\text{}=$ 650 °C), which gives ${C}_{p,olivine}\text{}=$ 1.3 kJ/kgK:

#### 2.2. Geometry Parametrization

#### 2.2.1. The Absorber

#### 2.2.2. The Aperture

#### 2.3. Tubes Number Calculation

_{2}cycle [44]. Alternatively, the extracted power can be formulated based on the targeted efficiency ${\eta}_{rec}$ and the incident solar radiation power ${P}_{rec}$ as follow: ϕabc = η

_{rec}P

_{rec}.

^{2}s, with ${D}_{t}$ the internal tube diameter (50 mm), an extracted power of 50 MW

_{th}, the targeted efficiency of 85% and equalizing the two formulations, one obtains a total tube number of 360. This value is kept constant in the whole analysis. The chosen particle mass flux, 250 kg/m

^{2}s, is a tradeoff between very high values that can result in chocking and small values that will not satisfy the constraint on power extraction and that are difficult to stabilize. Moreover, we have tested this mass flux with a cold mockup. The pressure loss in the absorber does not change with the particles mass flow rate, because the driving force is the pressure inside the dispenser, as explained in the Section 1.2 (Background). The pressure loss depends on the mean particle volume fraction, which is approximately 30%.

#### 2.4. Incident Concentrated Solar Flux

^{2}K based on experimental data [26]. In practice, this coefficient is based on the mean logarithmic temperature difference in the tube and an exchange surface ${A}_{t}=\frac{\pi}{2}{D}_{t}{H}_{t}$, corresponding to the irradiated part of the exchange surface of the tube. This mean logarithmic temperature difference involves the tube wall temperatures at the inlet and outlet of the particles. For the sake of simplicity, no temperature variation along the tube wall is considered, which simplifies the formulation (Equation (7)):

^{2}. Given the high temperatures involved and the cavity effect, which will increase the net flux received by the absorber, the incident concentrated solar flux density is set to 400 kW/m

^{2}in this study. The latter corresponds to a wall temperature of approximately 950 °C.

^{2}which corresponds to roughly 27% of the incident solar radiation. Such order of magnitude unquestionably legitimates the choice of a cavity type receiver and the need for a geometrical optimization of the cavity to lower the radiation losses.

## 3. Numerical Modelling

#### 3.1. Radiative Balance

#### 3.2. Losses and Efficiency

#### 3.2.1. Absorbed Power

#### 3.2.2. Radiative Losses

#### 3.2.3. Convective Losses

^{−5}of the receiver power. Such order of magnitude justifies neglecting this term in the model:

_{air}in Equations (13) and (14) is not the same for active and passive surfaces according to Clausing (see Introduction). However, as explained in the Introduction, the estimation of convective losses in cavity solar receivers is a challenging subject and thus, the two convective exchange coefficients are assumed to be identical to focus on the scope of this study, which is to assess the influence of the receiver geometry on its efficiency. It is set to 10 W/m

^{2}K for the whole study; this value overestimates the convective losses. The effect of this assumption on the simulation results is discussed in Section 5.

#### 3.2.4. Receiver Power and Efficiency

## 4. Results

- The incident flux on the absorber tubes is 400 kW/m
^{2}and the active and passive surfaces wall temperatures are homogenous and equal to 950 °C (see preliminary calculations). - The ambient air temperature is set at 15 °C while the air temperature inside the cavity is set to 500 °C.
- The total number of tubes N
_{t}is set at 360 but is rounded to have a ratio N_{t}/M as an integer.

#### 4.1. Influence of the Absorber Geometry

_{ape}= 7 m, the aperture length L

_{ape}= 4 m and height H

_{ape}= 5 m, for a zero inclination (i.e., an aperture parallel to the absorber).

_{th}.

#### 4.2. Influence of the Aperture’s Distance

^{2}, an arc angle $\theta =$ π or π/2 and a number of panels $M=$ 5. Correspondingly, the angle $\beta $ between the absorber and the vertical passive surfaces varies between 9 and 68° (i.e., between 0.16 and 1.19 rad), and the angle ${\gamma}_{h}$ with the top passive surface varies between 27 and 79° (i.e., between 0.47 and 1.38 rad). Figure 6 depicts the changes in the losses and efficiency.

#### 4.3. Influence of the Aperture Inclination

^{2}unchanged) strongly decreases the view factor between the absorber and the aperture, which strongly decreases the radiative losses, while it slightly increases the dimensions of the passive surfaces, which corresponds to the small increase in convective losses.

#### 4.4. Influence of the Aperture Dimensions

_{ape}= 7 m and an aperture inclination angle $\alpha =$ 30°. Figure 8 shows the evolution of the different losses and efficiency. The angles $\beta $,${\gamma}_{h}$ and ${\gamma}_{b}$ range from 43 to 54° (i.e., from 0.75 to 0.94 rad), from 54 to 90° (i.e., from 0.94 to 1.57 rad), and from 85 to 89° (i.e., from 1.48 to 1.55 rad) respectively.

^{2}.

^{2}and 25 m

^{2}aperture areas, the corresponding concentration ratios are approximately 2500 and 2000 respectively. These values are realistic but are attainable only with high quality (optical and mechanical) heliostats.

## 5. Discussion

^{2}K applied on all the internal surface area of the receiver, including active and passive surfaces at 950 °C. This coefficient value is the maximum given by Clausing [16,17]. Other considerations lead to the conclusion that convective losses have probably been overestimated. First, the inclination of the aperture (Figure 7a–c) results in the formation of a stagnant zone in the upper part of the receiver. Consequently, convective losses are very small in this region. Second, the literature review presented in Section 1.2 (Background) indicates that at high temperature the ratio of radiation to convection losses is approximately 4, whereas in our simulation it is approximately 2. Consequently, additional calculation have been performed accounting for an overall convective heat transfer coefficient of 5 W/m

^{2}K (or reducing the heat exchange surface by 2 with 10 W/m

^{2}K). Figure 10 provides the results for ${h}_{air}=$ 10 and 5 W/m

^{2}K in a 3D representation, showing the values of the parameters leading to a receiver efficiency equal or larger than 85%. Aperture areas of less 10 m

^{2}have not been considered in the simulation since smaller values are not realistic.

^{2}K, there is no configuration leading to a receiver efficiency equal or larger than 85% for aperture greater than 20 m

^{2}. For this aperture area, the receiver thermal efficiency reaches 85% (with $\theta =$ 2π/3 rad, ${d}_{ape}=$ 9 m, ${H}_{ape}=$ 5 m and ${L}_{ape}=$ 4 m) and the radiative and convective losses are 9.3% and 5.7% (a ratio of 1.6), respectively. Reducing the convective losses results in a significant change of this limit. For ${h}_{air}=$ 5 W/m

^{2}K, an efficiency of 85.5% is reached with an aperture area of 25 m

^{2}(with $\theta =$ 2π/3, ${d}_{ape}=$ 9 m, ${H}_{ape}=$ 5 m and ${L}_{ape}=$ 5 m). The respective radiative and convective losses are 11.6% and 2.9% (a ratio of 4). Since the radiative losses decrease with the reduction of the aperture size (i.e., the increase of the concentration ratio), the ratio of radiative to convection losses also decreases with the concentration ratio.

^{2}and ${h}_{air}=$ 5 W/m

^{2}K and the minimum is 84% at 1000 °C for 25 m

^{2}and ${h}_{air}=$ 10 W/m

^{2}K. These data indicate that increasing the wall-to-fluidized bed heat transfer coefficient, which results in a decrease of the absorber wall temperature, has a positive effect on the receiver efficiency. In any case, during operation, the receiver temperature is maintained approximately constant by varying the solid mass flow rate inside the tubes [31].

## 6. Conclusions

_{th}cavity tubular solar receiver, using particles as HTF within chosen design constraints, selected based on previous studies [27,28,29,30,31,32,33,34]. The absorber is composed of $M=$ 5 panels to house 360 tubes (7 m long) in an arc circle of an angle $\theta $. According to previous design data, the wall temperature is 950 °C for particles outlet temperature of 750 °C. An either vertical or inclined aperture is used, to accommodate the main direction of the reflected solar beam by the heliostats. Various configurations are defined to reach the targeted receiver’s thermal efficiency of at least 85%. The dominant parameters that govern the receiver efficiency are the aperture area and the distance between the aperture and the absorber. In this context, the assumption on convective losses appears to be a key factor that affects the acceptable aperture surface area. For a distance between the aperture and the absorber of 9 m (and with $\theta =$ 2π/3), the efficiency threshold of 85% is reached for aperture surface areas equal or less than 20 m

^{2}for high convection losses and it increases to 25 m

^{2}for low convection losses. This result is consistent with the data published in [31] for a non-optimized geometry. In the two cases of convection losses, decreasing the distance between the aperture and the absorber decreases the aperture area that allows reaching the targeted efficiency.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

${A}_{t}$ | Irradiated area of a tube (m^{2}) |

$c$ | Chord of the circle arc (m) |

${C}_{p}$ | Specific heat (J/kgK) |

${d}_{ape}$ | Aperture—absorber distance (m) |

${d}_{\alpha}$ | Shift of the aperture coordinates in the y axis due to its inclination (m) |

${D}_{t}$ | Internal diameter of a tube (m) |

${e}_{t}$ | Thickness of a tube (m) |

$f$ | Arrow of the arc circle (m) |

${F}_{ij}$ | View factor between the i and j surfaces (-) |

${G}_{p}$ | Mass flux of particles (kg/m^{2}s) |

${h}_{t,part}$ | Heat transfer coefficient between wall tubes and fluidized particles (W/m^{2}K) |

${h}_{air}$ | Convective heat transfer coefficient with the air (W/m^{2}K) |

${H}_{ape}$ | Aperture height (m) |

${H}_{\alpha}$ | Shift of the aperture coordinates in the z axis due to its inclination (m) |

${J}_{i}$ | Radiosity of the i surface (W/m^{2}) |

${L}_{ape}$ | Aperture length (m) |

$\dot{{m}_{p}}$ | Mass flow rate of particles per tube (kg/s) |

$M$ | Number of tubes panels in the absorber |

$N$ | Number of tubes in each panel |

${N}_{t}$ | Total number of tubes in the absorber |

${P}_{rec}$ | Receiver power input (W) |

$r$ | Radius of curvature of the arc circle (m) |

${r}^{IR}$ | Reflectivity in the infrared spectral band (-) |

${r}^{sol}$ | Reflectivity in the solar spectral band (-) |

${S}_{ape}$ | Aperture area (m^{2}) |

${S}_{i}$ | Area of the i surface (m^{2}) |

${S}_{tube}$ | Internal section of a tube (m^{2}) |

${T}_{cav}$ | Temperature of the air inside the cavity (K) |

${T}_{ext}$ | Temperature of the air outside the cavity (K) |

${T}_{wall}$ | Wall temperature (inside the cavity) (K) |

${T}_{wall}^{ext}$ | Wall temperature outside the cavity (K) |

${T}_{part}^{in}$ | Inlet temperature of the particles in a tube (K) |

${T}_{part}^{out}$ | Outlet temperature of the particles in a tube (K) |

$\alpha $ | Inclination angle of the aperture relative to the vertical plane (rad) |

${\alpha}^{IR}$ | Absorptivity in the infrared spectral band (-) |

${\alpha}^{sol}$ | Absorptivity in the solar spectral band (-) |

$\beta $ | Angle between the vertical absorber plane and the passive vertical surfaces (rad) |

${\gamma}_{b}$ | Angle between the absorber plane and the top passive surface (rad) |

${\gamma}_{h}$ | Angle between the absorber plane and the bottom passive surface (rad) |

$\mathsf{\Delta}{T}_{lm,part}$ | Mean logarithmic temperature of a tube (K) |

${\epsilon}^{IR}$ | Emissivity in the infrared spectral band (-) |

${\eta}_{rec}$ | Receiver efficiency (-) |

$\theta $ | Angle of the arc circle (rad) |

${\varphi}_{abs}$ | Absorbed power by the particles (W) |

${\varphi}_{loss,conv}$ | Convective losses of the receiver (W) |

${\varphi}_{loss,rad}$ | Radiative losses of the receiver (W) |

$\lambda $ | Thermal conductivity (W/mK) |

$\rho $ | Density (kg/m^{3}) |

$\sigma $ | Boltzman constant (W/m^{2}K^{4}) |

${\phi}_{inc}$ | Incident solar flux density (W/m^{2}) |

## Appendix A. Convective Losses Studies in Cavity Receivers

Reference | Topic of the Paper | Main Parameters | Results | Correlation |
---|---|---|---|---|

R.D. Jilte et al. [7] | Numerical 3D study of the combined natural convection and radiation losses associated to dish concentrators | T° = [250–650] °C. Cavity inclination: 0 to 90°. Cavity shapes: cylindrical, conical, etc and spherical | Convection losses decrease strongly with the inclination angle of the cavity for all the receiver shapes. | $Nu=0.122R{a}^{0.31}{\frac{{T}_{wall}}{{T}_{amb}}}^{0.066}\ast {\left(1+\mathrm{cos}\left(\frac{\pi}{2}-\alpha \right)\right)}^{0.38}$ For Rayleigh number between 2 × 10 ^{8} and 6 × 10^{8}.Standard deviation of 16%. |

C. Zou et al. [9] | 3D numerical simulation of cylindrical cavity receiver | Cavity inclination: 45° | The receiver efficiency varied sharply with the aperture size. The ratio of radiation to convection losses ranges from approximately 2 to 4 at low DNI and high DNI, respectively. | Ø |

J. Kim et al. [12] | Model of heat losses of Solar Power Tower receivers | External and cavity receiver with a 9 m^{2} absorberT° = [600–900] °C. ${v}_{wind}$ = [1,2,3,4,5,6,7,8,9,10] m/s, head-on and side-on. | For low wind velocity, calculated efficiency of Solar Two external receiver was 88%. For cavity receivers at 900 °C, the ratio of radiation to convection losses ranged generally between 2 and 7, except with head-on high wind condition (10 m/s) that result in a ratio around 1. | $\frac{{Q}_{conv}}{{Q}_{conv}+{Q}_{rad}}=a\ast ln\left(\frac{{S}_{ape}}{{S}_{rec}}\ast {T}_{rec}^{4}\ast {10}^{-12}\right)+b$ $a=-4.611\text{}\times \text{}{10}^{-4}{v}_{wind}^{2}+5.517\text{}\times \text{}{10}^{-3}{v}_{wind}-0.1071$ $b=-5.917\text{}\times \text{}{10}^{-4}{v}_{wind}^{2}+3.158\text{}\times \text{}{10}^{-2}{v}_{wind}+0.1190$ Standard deviation of 11.4%. |

A.M. Clausing [16] | Analytical approach of convective losses in cavity central receivers | ${P}_{rec}$ = 1 and 39 MW T° = 920 and 1020 °C ${H}_{ape}$ = 5.6 and 0.93 m | The influence of the wind on the convective losses at normal operating conditions are small. The inclination of the aperture is critical since it strongly influences the height of the convective zone within the cavity. | Ø |

A.M. Clausing [17] | $\alpha $ = 45 and 180° with the vertical Gr = [$1.39\times {10}^{9}$–$1.24\times {10}^{11}$]. | The wind and buoyancy driven bulk flow both appear to have secondary influences for the cubic cavity orientations. There is a strong evidence for the existence of the stagnation zone and the proposed low convective energy flux across the boundary of this zone. The comparison with experimental data resulted in a convective heat transfer coefficient of 7.2 and 9.7 W/m ^{2}K for the inactive and the absorber surfaces respectively. | $Nu=0.082R{a}^{\frac{1}{3}}\left[-0.9+2.4\frac{{T}_{wall}}{{T}_{amb}}-0.5{\frac{{T}_{wall}}{{T}_{amb}}}^{2}\right]z\left(\alpha \right)$ $z\left(\alpha \right)=\{\begin{array}{c}1si0\le \alpha \le 135\xb0\\ \frac{2}{3}\left[1+\frac{\mathrm{sin}\left(\alpha \right)}{\sqrt{2}}\right]\text{}\mathrm{sin}\alpha 135\xb0\end{array}$ Factor z added to account for all orientation. Without it, previous authors found absolute deviation of 1%. | |

J. Samanes et al. [18] | Comparison of correlations for convective losses | Ps-10 like cavity receiver (300 tubes in a 7 m radius and 12 m height, and ${P}_{rec}$ = 56.7 MW) Five models are compared and validated | Clausing’s model provides the heat losses on each surface of the cavity receiver, and its method is considered as the best choice. | Ø |

R. Flesch et al. [19] | Numerical analysis of convective losses | Head-on and side-on wind $\alpha $ = [0–90]° | Clausing’s model is a good approach for natural convection. When no wind is present, the losses decrease considerably with increasing the inclination angle of the receiver. | Ø |

## Appendix B. Calculation of Radiosities

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**Figure 1.**Schematic view of the receiver’s cavity with the different thermal fluxes. Red dashed lines indicate thermally insulated walls.

**Figure 2.**Scheme of a top view of a section of the (x,y) plane of a receiver for the following parameters: the circular arc of radius ≈ 11 m and spanning angle π/2, is discretized in M = 5 segments. The aperture geometry is defined by a length ${L}_{ape}$ = 4 m and a distance from the absorber ${d}_{ape}$ = 4 m.

**Figure 3.**Schematic side view of a section of the (y,z) plane of a receiver showing the aperture inclination $\alpha $ and the geometrical parameters associated with it.

**Figure 4.**Evolution of the tubes wall temperature as a function of the incident solar flux density, following equation 6 and with a targeted receiver efficiency of 85%.

**Figure 5.**Influence of the absorber geometry (i.e., of the arc) on the radiative (

**a**) and convective losses (

**b**), both normalized by the receiver power, and on the receiver efficiency (

**c**), for the following parameters: d

_{ape}= 7 m, L

_{ape}= 4 m, H

_{ape}= 5 m and α = 0.

**Figure 6.**Influence of the aperture—absorber distance on the radiative; (

**a**) and convective losses (

**b**), both normalized by the receiver power, and on the receiver efficiency; (

**c**), for the following parameters: M = 5 panels, L

_{ape}= 4 m, H

_{ape}= 5 m and α = 0.

**Figure 7.**Influence of the aperture inclination on the radiative (

**a**) and convective losses (

**b**), both normalized by the receiver power, and on the receiver efficiency (

**c**), for the following parameters: M = 5 panels, d

_{ape}= 7 m, L

_{ape}= 4 m and H

_{ape}= 5 m.

**Figure 8.**Influence of the aperture dimensions on (

**a**) the radiative losses—normalized by the receiver power—and (

**b**) on the receiver efficiency, for the following parameters: M = 5 panels, d

_{ape}= 7 m, α = 30° and θ = π/2.

**Figure 9.**(

**a**) Influence of the aperture area on the different losses—normalized by the receiver power—and efficiency, and (

**b**) Influence of the aperture shape on the receiver efficiency, for the same parameters as previously.

**Figure 10.**Maps of the domain of parameters that allows to reach the targeted receiver efficiency, with a convection coefficient h

_{air}of 10 W/m

^{2}K (

**a**) and 5 W/m

^{2}K (

**b**), for the following parameters: M = 5 panels and α = 30°.

$\mathbf{Absorptivity}\text{}{\mathit{\alpha}}^{\mathit{s}\mathit{o}\mathit{l}}$ | $\mathbf{Reflectivity}\text{}{\mathit{r}}^{\mathit{s}\mathit{o}\mathit{l}}$ | ${\mathit{\alpha}}^{\mathit{I}\mathit{R}}$ | ${\mathit{r}}^{\mathit{I}\mathit{R}}$ | $\mathbf{Emissivity}\text{}{\mathit{\epsilon}}^{\mathit{I}\mathit{R}}$ | $\mathbf{Density}\text{}\mathit{\rho}$
(kg/m^{3}) | $\mathbf{Thermal}\text{}\mathbf{Conductivity}\text{}\mathit{\lambda}$ (W/mK) | |
---|---|---|---|---|---|---|---|

Absorbent Surfaces | 0.9 | 0.1 | 0.85 | 0.15 | 0.85 | 8110 | 26.1 |

Reflective Surfaces | 0.22 | 0.78 | 0.95 | 0.05 | 0.95 | 315 | 0.1 |

**Table 2.**Calculations of the receiver efficiency with variations of the absorber temperature, for two aperture areas ${S}_{ape}$ and two convective coefficients ${h}_{air}$, with the following parameters: $\theta $ = 2π/3 radians, $\alpha $ = 30°, and ${d}_{ape}$ = 9 m.

$\mathbf{Aperture}\text{}\mathbf{Area}\text{}{\mathit{S}}_{\mathit{a}\mathit{p}\mathit{e}}$ | 20 m^{2} | 25 m^{2} | ||||||
---|---|---|---|---|---|---|---|---|

$\mathbf{Absorber}\text{}\mathbf{Temperature}\text{}{\mathit{T}}_{\mathit{p}}\text{}(\xb0\mathbf{C})$ | ${\mathit{h}}_{\mathit{a}\mathit{i}\mathit{r}}\text{}(\mathbf{W}/{\mathbf{m}}^{2}\mathbf{K})$ | ${\mathit{\eta}}_{\mathit{r}\mathit{e}\mathit{c}}\text{}(\%)$ | ${\mathit{h}}_{\mathit{a}\mathit{i}\mathit{r}}\text{}(\mathbf{W}/{\mathbf{m}}^{2}\mathbf{K})$ | ${\mathit{\eta}}_{\mathit{r}\mathit{e}\mathit{c}}\text{}(\%)$ | ${\mathit{h}}_{\mathit{a}\mathit{i}\mathit{r}}\text{}(\mathbf{W}/{\mathbf{m}}^{2}\mathbf{K})$ | ${\mathit{\eta}}_{\mathit{r}\mathit{e}\mathit{c}}\text{}(\%)$ | ${\mathit{h}}_{\mathit{a}\mathit{i}\mathit{r}}\text{}(\mathbf{W}/{\mathbf{m}}^{2}\mathbf{K})$ | ${\mathit{\eta}}_{\mathit{r}\mathit{e}\mathit{c}}\text{}(\%)$ |

800 | 10 | 90.6 | 5 | 92.9 | 10 | 89.6 | 5 | 91.9 |

850 | 10 | 89.4 | 5 | 92.0 | 10 | 88.3 | 5 | 90.9 |

900 | 10 | 88.2 | 5 | 91.0 | 10 | 87.0 | 5 | 89.8 |

950 | 10 | 86.9 | 5 | 90.0 | 10 | 85.6 | 5 | 88.5 |

1000 | 10 | 85.6 | 5 | 88.9 | 10 | 84.1 | 5 | 87.3 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gueguen, R.; Grange, B.; Bataille, F.; Mer, S.; Flamant, G. Shaping High Efficiency, High Temperature Cavity Tubular Solar Central Receivers. *Energies* **2020**, *13*, 4803.
https://doi.org/10.3390/en13184803

**AMA Style**

Gueguen R, Grange B, Bataille F, Mer S, Flamant G. Shaping High Efficiency, High Temperature Cavity Tubular Solar Central Receivers. *Energies*. 2020; 13(18):4803.
https://doi.org/10.3390/en13184803

**Chicago/Turabian Style**

Gueguen, Ronny, Benjamin Grange, Françoise Bataille, Samuel Mer, and Gilles Flamant. 2020. "Shaping High Efficiency, High Temperature Cavity Tubular Solar Central Receivers" *Energies* 13, no. 18: 4803.
https://doi.org/10.3390/en13184803