## Abstract

Concentrated solar energy can be used as the source of heat at above 1000 °C for driving key energy-intensive industrial processes, such as cement manufacturing and metallurgical extraction, contributing to their decarbonization. The cornerstone technology is the solar receiver mounted on top of the solar tower, which absorbs the incident high-flux radiation and heats a heat transfer fluid. The proposed high-temperature solar receiver concept consists of a cavity containing a reticulated porous ceramic (RPC) structure for volumetric absorption of concentrated solar radiation entering through an open (windowless) aperture, which also serves for the access of ambient air used as the heat transfer fluid flowing across the RPC structure. A heat transfer analysis of the solar receiver is performed by means of two coupled models: a Monte Carlo (MC) ray-tracing model to solve the 3D radiative exchange and a computational fluid dynamics (CFD) model to solve the 2D convective and conductive heat transfer. Temperature distributions computed by the iteratively coupled models were compared with experimental data obtained by testing a lab-scale 5 kW receiver prototype with a silicon carbide RPC structure exposed to 3230 suns flux irradiation. The receiver model is applied to optimize its dimensions for maximum efficiency and to scale-up for a 5 MW solar tower.

## 1 Introduction

Key energy-intensive industrial processes that operate at high temperatures (>1000 °C), such as cement manufacturing and metallurgical extraction, contribute significantly to greenhouse gas emissions as they are predominantly driven by fossil fuel combustion. Concentrated solar energy offers a source of high-temperature process heat for the decarbonization of these energy-intensive industrial processes [1], as well as for the thermochemical production of solar fuels [2]. However, designing a solar receiver to efficiently convert concentrated solar radiation to high-temperature heat is challenging because of the re-radiation losses that are proportional to temperature to the fourth power and a limited choice of heat transfer fluids (HTFs) at the relevant high temperatures [3]. Using a porous ceramic structure as the absorber and air as the HTF can provide a viable solution because re-radiation losses are reduced due to the volumetric absorption of incident concentrated solar radiation, i.e., incident radiation penetrates the porous structure and is absorbed over the entire volume, while air as the HTF is stable at high temperatures, naturally abundant and nontoxic [4].

This work focuses on the heat transfer modeling of a volumetric solar receiver featuring an open cavity containing a reticulated porous ceramic (RPC) foam-type structure as the absorber and air as the HTF. The motivation for this receiver concept is threefold. First, the RPC structure serves simultaneously the functions of an efficient radiative absorber and convective air heater and has been extensively studied and applied [5–7]. Second, the cavity enclosing the RPC structure is directly exposed to the concentrated solar radiation entering through its aperture and incident on the RPC structure. Because of the cavity effect, a significant portion of the thermal emission from the hot RPC is re-absorbed within the cavity due to multiple internal reflections, thus approaching a blackbody and further reducing re-radiation losses [2,3]. Third, an open (windowless) receiver can draw ambient air into the cavity and across the RPC structure, thereby eliminating the need for fragile, troublesome windows or difficult high-rate heat transfer across opaque ceramic walls. Furthermore, open solar air receivers facilitate the integration of high-temperature thermal energy storage via a thermocline-based packed bed of rocks using air as the HTF [8]. This way, solar heat can be stored and dispatched round-the-clock to continuous industrial processes. Recent studies have proposed the use of solar receivers in such a configuration for various thermal and thermochemical processes [1,9,10].

The design of the solar air receiver modeled in this work originates from a previous experimental study in which a lab-scale prototype was built and tested [7]. This lab-scale receiver accepted 5 kW solar radiative power input and delivered air at above 1000 °C with thermal efficiencies—defined as the ratio of air enthalpy gain to solar radiative power input—exceeding 65% even at such a small scale, thus showing promise for this design. A scalable approach would use an array of solar receiver modules arranged side by side on top of the solar tower, each module being an optimized and scaled-up version of the lab-scale prototype and each being attached to hexagon-shaped secondary compound parabolic concentrators in a honeycomb configuration. The analysis presented in this work should guide the optimization and scale-up of the solar receiver module. Thus, the objectives of this work are to develop a numerical model for analyzing the heat transfer and fluid flow of the solar receiver, and to apply it for scale-up and optimization of the design. Validation is accomplished by comparing with experimental data obtained from the 5 kW prototype testing. The following sections describe the solar receiver model, experimental validation, representative model results, and model application for scale-up and optimization.

## 2 Model Description

The Monte Carlo (MC) ray-tracing method was selected to model the radiative heat transfer because it can accurately capture the directional and spectral phenomena within a radiatively participating medium of the RPC structure and provide a near-exact solution. The MC model outputs the divergence of radiative flux for each discrete volume of the RPC domain, which in turn is used as the energy source term in a computational fluid dynamics (CFD) model to solve for convection and conduction heat transfer. The interdependent MC and CFD models are coupled and solved iteratively until convergence.

### 2.1 Monte Carlo Model

#### 2.1.1 Domain and Assumptions.

The MC model domain of the solar receiver, assumed axisymmetric, is schematically shown in Fig. 1. The cavity's open aperture provides entry to concentrated solar radiation as well as ambient air, suctioned into the cavity by a downstream blower. The insulated cavity contains the RPC structure divided into a cylinder and a disk. The cylinder and disk RPC subdomains are discretized axially and radially to form ring-shaped volume elements $dVi,j$. Similarly, the cavity boundaries are discretized into boundary elements $dAi,j$. Air is assumed a radiatively nonparticipating medium. Table 1 lists the geometric dimensions of the domain, which closely correspond to the 5 kW lab-scale prototype used for experimental validation. Note that the octagonal cross section of the RPC bricks of the lab-scale prototype is simplified by the circular cross section of the axisymmetric model domain. The MC model performs 3D ray tracing on the domain.

Dimension | Symbol | Value |
---|---|---|

Aperture radius | $r$ | 0.020 m |

Half-angle of conical insulation | $\theta cone$ | 45 deg |

Length of conical insulation | $lcone$ | 0.025 m |

Inner radius of cylindrical RPC subdomain | $rcyl,i$ | 0.045 m |

Thickness of RPC | $tRPC$ | 0.024 m |

Outer radius of cylindrical RPC subdomain | $rcyl,o$ | 0.069 m (=$rcyl,i+tRPC$) |

Length of cylindrical RPC subdomain | $lcyl$ | 0.075 m |

Length of disk RPC subdomain | $ldisk$ | 0.024 m |

Radius of disk RPC subdomain | $rdisk$ | 0.069 m (=$rcyl,o$) |

Dimension | Symbol | Value |
---|---|---|

Aperture radius | $r$ | 0.020 m |

Half-angle of conical insulation | $\theta cone$ | 45 deg |

Length of conical insulation | $lcone$ | 0.025 m |

Inner radius of cylindrical RPC subdomain | $rcyl,i$ | 0.045 m |

Thickness of RPC | $tRPC$ | 0.024 m |

Outer radius of cylindrical RPC subdomain | $rcyl,o$ | 0.069 m (=$rcyl,i+tRPC$) |

Length of cylindrical RPC subdomain | $lcyl$ | 0.075 m |

Length of disk RPC subdomain | $ldisk$ | 0.024 m |

Radius of disk RPC subdomain | $rdisk$ | 0.069 m (=$rcyl,o$) |

#### 2.1.2 Material Properties and Boundary Conditions.

Table 2 summarizes the material properties used in the MC model. The RPC domain is treated as homogenous (without differentiating between solid and void regions) and radiatively participating (scattering–absorbing–emitting) with volume-averaged effective optical properties to enable faster computation compared to direct pore-level simulations. Effective properties were obtained by applying pore-level MC ray tracing on the exact 3D digital geometry of RPC foams obtained by computed tomography [11]. As the strut size of the RPC is significantly larger than nearly all wavelengths of incoming and emitted rays, diffraction can be neglected and the geometric optics regime can be applied. Thus, the scattering and absorption coefficients of the RPC are given by $\alpha RPC=(1\u2212rRPC)\u22c5\beta RPC$ and $\sigma RPC=rRPC\u22c5\beta RPC$, respectively, where $\beta RPC$ is the extinction coefficient and $rRPC$ is the surface hemispherical reflectivity of the RPC struts. The scattering albedo $\omega RPC$, which is the ratio of $\sigma RPC$ and $\beta RPC$ (effectively equal to $rRPC$ in this case), gives the probability of a ray being scattered in the MC method. $rRPC$ is wavelength dependent, hence so are $\sigma RPC$ and $\omega RPC$. SiSiC, the material used for manufacturing the RPC, has a nearly constant $rRPC$ of ∼0.1 across most wavelengths [12]. Similarly, a near-constant $\omega RPC$ of ∼0.7 for alumina RPC has been reported across a large range of wavelengths [13]. In contrast, $\beta RPC$ is a pure geometrical-dependent property and independent of wavelength. The cavity boundaries are assumed to be gray-diffuse, with a total hemispherical absorptivity $\alpha b$ of 0.70 for the ceria laminate and 0.28 for the alumina–silica insulation [14]. Following Kirchhoff's law, the total hemispherical emissivity, and absorptivity of the boundaries, $\epsilon b=\alpha b$. Note that multiple internal reflections within the RPC pores and on the inner cavity walls diminish the effect of directional optical properties. Hence, scattering within the isotropic RPC of constant porosity is assumed isotropic.

Property | Unit | Value or correlation | Ref. |
---|---|---|---|

RPC | |||

No. of pores per inch | – | PPI = 10, 20, 30 | Manufacturer |

Porosity | – | $\epsilon RPC=0.90,0.91,0.87$ for 10, 20, 30 PPI, respectively | Manufacturer |

Mean pore diameter | m | $dm,RPC=10\u22125\xd7(5.3022\u22c5\epsilon RPC+2.1549)\xd7(357/PPI)$ | [11] |

Extinction coefficient | m^{−1} | $\beta RPC=\u2212630.674\epsilon RPC2\u2212120.060\epsilon RPC+1229.361000dm,RPC$ | [11] |

Surface reflectivity of strut | – | $rRPC=0.1$ (SiSiC) | [12] |

Scattering coefficient | m^{−1} | $\sigma RPC=\beta RPC\xd7rRPC$ | [11] |

Absorption coefficient | m^{−1} | $\alpha RPC=\beta RPC\xd7(1\u2212rRPC)$ | [11] |

Scattering albedo | – | $\omega RPC=\sigma RPC\beta RPC=rRPC=0.1$ (SiSiC), $\omega RPC=0.7$ (alumina) | [11,13] |

CeO_{2} laminate insulation | |||

Total hemispherical absorptivity (and emissivity) | – | $\alpha b,CeO2=0.70$ | [14] |

Al_{2}O_{3}–SiO_{2} insulation | |||

Total hemispherical absorptivity (and emissivity) | – | $\alpha b,Al2O3\u2212SiO2=0.28$ | [14] |

Property | Unit | Value or correlation | Ref. |
---|---|---|---|

RPC | |||

No. of pores per inch | – | PPI = 10, 20, 30 | Manufacturer |

Porosity | – | $\epsilon RPC=0.90,0.91,0.87$ for 10, 20, 30 PPI, respectively | Manufacturer |

Mean pore diameter | m | $dm,RPC=10\u22125\xd7(5.3022\u22c5\epsilon RPC+2.1549)\xd7(357/PPI)$ | [11] |

Extinction coefficient | m^{−1} | $\beta RPC=\u2212630.674\epsilon RPC2\u2212120.060\epsilon RPC+1229.361000dm,RPC$ | [11] |

Surface reflectivity of strut | – | $rRPC=0.1$ (SiSiC) | [12] |

Scattering coefficient | m^{−1} | $\sigma RPC=\beta RPC\xd7rRPC$ | [11] |

Absorption coefficient | m^{−1} | $\alpha RPC=\beta RPC\xd7(1\u2212rRPC)$ | [11] |

Scattering albedo | – | $\omega RPC=\sigma RPC\beta RPC=rRPC=0.1$ (SiSiC), $\omega RPC=0.7$ (alumina) | [11,13] |

CeO_{2} laminate insulation | |||

Total hemispherical absorptivity (and emissivity) | – | $\alpha b,CeO2=0.70$ | [14] |

Al_{2}O_{3}–SiO_{2} insulation | |||

Total hemispherical absorptivity (and emissivity) | – | $\alpha b,Al2O3\u2212SiO2=0.28$ | [14] |

*C*is assumed over the receiver's aperture, defined by:

*r*and

*I*is the direct normal solar irradiation (assumed 1 kW/m

^{2}, i.e., 1 sun). The value of

*C*is commonly reported in units of “suns.” $Psolar$ is measured by a water calorimeter during experimental testing [7].

#### 2.1.3 Monte Carlo Algorithm.

A statistically meaningful number of stochastic energy bundles or “rays” are launched, and their interaction with the radiatively participating domain is tracked to approach the exact solution of the equation of radiative transfer. The collision-based MC method was applied, wherein the energy content of each ray is held constant irrespective of the distance traveled through the participating medium, until it is either scattered or absorbed. Figure 2 schematically shows the algorithm of the MC model, as described further. The number of rays launched is *n* = 10^{6}; each ray carries an equal amount of power $w=Psolar/n$. For simplification, $Psolar$ is assumed to have an axisymmetric and uniform distribution over a cone half-angle $\theta cone=45deg$ from the central axis of the cavity, which approximates the radiative input to the lab-scale prototype used for experimental validation. The position and the direction of each ray incident on the aperture are determined by the appropriate cumulative distribution functions $f(\u211c)$, $\u211c$ being a random number selected from a uniform set [0,1]. Thermal emission from the RPC medium and cavity boundaries is temperature controlled, i.e., the maximum number of rays that an element can emit ($NdVi,jemis,max$ for a volume element $dVi,j$ and $NdAi,jemis,max$ for a boundary element $dAi,j$) is determined by the temperature of the element $(TdVi,jandTdAi,j)$:

The MC model was implemented through a Fortran 90 script using the development environment Microsoft Visual Studio and the compiler Intel Visual Fortran.

### 2.2 Computational Fluid Dynamics Model

#### 2.2.1 Domain and Assumptions.

Figure 3 presents a scheme of the CFD domain, which includes regions where airflows freely (receiver entrance, cavity, air gaps, and air outlet), RPC (porous medium), and the impermeable solids enclosing the cavity (ceria laminate, alumina–silica insulation, steel shell, and water-cooled aluminum radiation shield). The air domain is extended upstream of the solar receiver to develop the flow before entering through the open aperture. In the RPC region, the temperature fields of the RPC and air are calculated by solving the energy conservation equation separately for each material. 2D axisymmetry and no gravity are assumed, as experimental data indicated negligible buoyancy effects even at low air flowrates of 2 kg/h [7].

#### 2.2.2 Material Properties.

Table 3 summarizes the properties of all materials used in the CFD model.

Property | Unit | Value or correlation^{a} | Ref. | |
---|---|---|---|---|

SiSiC and alumina RPC | ||||

No. of pores per inch | – | PPI = 10, 20, 30 | Manufacturer | |

Porosity | – | $\epsilon RPC=0.90,0.91,0.87$ for 10, 20, 30 PPI, respectively | Manufacturer | |

Mean pore diameter | $m$ | $dm,RPC=10\u22125\xd7(5.3022\xd7\epsilon RPC+2.1549)\xd7(357/PPI)$ | [11] | |

Specific surface area | $m2m3$ | $Asp=PPI/(357\xd710\u22125)(5.65595\xd7\epsilon RPC2\u22126.08569\xd7\epsilon RPC+4.49806)$ | [11] | |

Permeability | $m2$ | $\kappa RPC=\epsilon RPC3.7752/(5.4685\xd7Asp2)$ | [11] | |

Solid (bulk) density | $kgm3$ | $\rho s,SiSiC=2830$, $\rho s,alumina=3960$ | [15] | |

Solid specific heat capacity | $Jkg\u22c5K$ | c_{p}_{,s} = 0.94 (SiSiC), 0.75 (alumina) | [15] | |

Solid thermal conductivity | $Wm\u22c5K$ | T (°C) | k_{s}_{,SiSiC} | [6,16] |

20 | 110 | |||

200 | 85 | |||

500 | 60 | |||

1000 | 42 | |||

1500 | 38 | |||

$ks,alumina=85.868\u22120.22972\xd7T+2.607\xd710\u22124$$\xd7T2\u22121.3607\xd710\u22127\xd7T3+2.7092\xd710\u221211\xd7T4$ | ||||

Solid-fluid heat transfer coefficient | $Wm2\u22c5K$ | $hs\u2212f=Nu\xd7kair/dm,RPC$ | ||

Dimensionless numbers | – | $Nu=4.173+2.359\xd7\epsilon RPC$$+(0.3772\xd7\epsilon RPC2\u22120.7479\xd7\epsilon RPC+0.4849)$$\xd7ReRPC(1.0953\u22120.2239\u22c5\epsilon RPC)\xd7PrRPC(0.671\u22120.0213\u22c5\epsilon RPC)\u25b9$ | [11] | |

$ReRPC=\rho air\xd7ud\xd7dm,RPC/\mu air$ | ||||

$PrRPC=cp,air\xd7\mu air/kair$ | ||||

CeO_{2} laminate insulation | ||||

Density | $kgm3$ | $\rho CeO2laminate=504.4$ | [14] | |

Thermal conductivity | $Wm\u22c5K$ | $kCeO2laminate=2.2\xd710\u22127\xd7T2\u22122.8387\xd710\u22124\xd7T+0.176786$ | ||

Specific heat capacity | $Jkg\u22c5K$ | $cp,CeO2laminate=\u22120.000127\xd7T2+0.269765\xd7T+299.8,forT\u22641100K$ | ||

Al_{2}O_{3}–SiO_{2} insulation | ||||

Density | $kgm3$ | $\rho Al2O3\u2013SiO2=560.65$ | [14] | |

Thermal conductivity | $Wm\u22c5K$ | $kAl2O3\u2013SiO2=0.00012926\xd7T+0.019654$ | ||

Specific heat capacity | $Jkg\u22c5K$ | $cp,Al2O3\u2013SiO2=4\xd710\u22127\xd7T3\u22121.3797\xd710\u22123\xd7T2$$+1.5987289\xd7T+477.7,forT\u22641480K$$cp,CeO2laminate=444.27,forT>1100K$$cp,Al2O3\u2013SiO2=1118.44,forT>1480K$ | ||

Stainless steel shell | ||||

Density | $kgm3$ | $\rho SS=8470$ | [14] | |

Thermal conductivity | $Wm\u22c5K$ | $kSS=0.0158\xd7T+10.169$ | ||

Specific heat capacity | $Jkg\u22c5K$ | $cp,SS=0.2827\xd7T+327.29$ | ||

Total hemispherical emissivity | – | $\epsilon SS=0.8$ | ||

Aluminum radiation shield | ||||

Density | $kgm3$ | $\rho Al=2700$ | comsol material library. T-dependent, shown here are representative values at 25 ℃ | |

Thermal conductivity | $Wm\u22c5K$ | $kAl=237$ | ||

Specific heat capacity | $Jkg\u22c5K$ | $cp,Al=898$ | ||

Air | ||||

density | $kgm3$ | $\rho air=pRsp,air\u22c5T$; $Rsp,air=287J/kg\u22c5K$ | comsol material library | |

Thermal conductivity | $Wm\u22c5K$ | $kair=\u22120.00227583562+1.15480022\xd710\u22124$$\xd7T\u22127.90252856\xd710\u22128\xd7T2+4.11702505$$\xd710\u221211\xd7T3\u22127.43864331\xd710\u221215\xd7T4$ | ||

Specific heat capacity | $Jkg\u22c5K$ | $cp,air=1047.63657\u22120.372589265\xd7T$$+9.45304214\xd710\u22124\xd7T2\u22126.02409443$$\xd710\u22127\xd7T3+1.2858961\xd710\u221210\xd7T4$ | ||

Dynamic viscosity | $Pa\u22c5s$ | $\mu air=\u22128.38278\xd710\u22127+8.35717342\xd710\u22128$$\xd7T\u22127.69429583\xd710\u221211\xd7T2+4.6437266$$\xd710\u221214\xd7T3\u22121.06585607\xd710\u221217\xd7T4$ |

Property | Unit | Value or correlation^{a} | Ref. | |
---|---|---|---|---|

SiSiC and alumina RPC | ||||

No. of pores per inch | – | PPI = 10, 20, 30 | Manufacturer | |

Porosity | – | $\epsilon RPC=0.90,0.91,0.87$ for 10, 20, 30 PPI, respectively | Manufacturer | |

Mean pore diameter | $m$ | $dm,RPC=10\u22125\xd7(5.3022\xd7\epsilon RPC+2.1549)\xd7(357/PPI)$ | [11] | |

Specific surface area | $m2m3$ | $Asp=PPI/(357\xd710\u22125)(5.65595\xd7\epsilon RPC2\u22126.08569\xd7\epsilon RPC+4.49806)$ | [11] | |

Permeability | $m2$ | $\kappa RPC=\epsilon RPC3.7752/(5.4685\xd7Asp2)$ | [11] | |

Solid (bulk) density | $kgm3$ | $\rho s,SiSiC=2830$, $\rho s,alumina=3960$ | [15] | |

Solid specific heat capacity | $Jkg\u22c5K$ | c_{p}_{,s} = 0.94 (SiSiC), 0.75 (alumina) | [15] | |

Solid thermal conductivity | $Wm\u22c5K$ | T (°C) | k_{s}_{,SiSiC} | [6,16] |

20 | 110 | |||

200 | 85 | |||

500 | 60 | |||

1000 | 42 | |||

1500 | 38 | |||

$ks,alumina=85.868\u22120.22972\xd7T+2.607\xd710\u22124$$\xd7T2\u22121.3607\xd710\u22127\xd7T3+2.7092\xd710\u221211\xd7T4$ | ||||

Solid-fluid heat transfer coefficient | $Wm2\u22c5K$ | $hs\u2212f=Nu\xd7kair/dm,RPC$ | ||

Dimensionless numbers | – | $Nu=4.173+2.359\xd7\epsilon RPC$$+(0.3772\xd7\epsilon RPC2\u22120.7479\xd7\epsilon RPC+0.4849)$$\xd7ReRPC(1.0953\u22120.2239\u22c5\epsilon RPC)\xd7PrRPC(0.671\u22120.0213\u22c5\epsilon RPC)\u25b9$ | [11] | |

$ReRPC=\rho air\xd7ud\xd7dm,RPC/\mu air$ | ||||

$PrRPC=cp,air\xd7\mu air/kair$ | ||||

CeO_{2} laminate insulation | ||||

Density | $kgm3$ | $\rho CeO2laminate=504.4$ | [14] | |

Thermal conductivity | $Wm\u22c5K$ | $kCeO2laminate=2.2\xd710\u22127\xd7T2\u22122.8387\xd710\u22124\xd7T+0.176786$ | ||

Specific heat capacity | $Jkg\u22c5K$ | $cp,CeO2laminate=\u22120.000127\xd7T2+0.269765\xd7T+299.8,forT\u22641100K$ | ||

Al_{2}O_{3}–SiO_{2} insulation | ||||

Density | $kgm3$ | $\rho Al2O3\u2013SiO2=560.65$ | [14] | |

Thermal conductivity | $Wm\u22c5K$ | $kAl2O3\u2013SiO2=0.00012926\xd7T+0.019654$ | ||

Specific heat capacity | $Jkg\u22c5K$ | $cp,Al2O3\u2013SiO2=4\xd710\u22127\xd7T3\u22121.3797\xd710\u22123\xd7T2$$+1.5987289\xd7T+477.7,forT\u22641480K$$cp,CeO2laminate=444.27,forT>1100K$$cp,Al2O3\u2013SiO2=1118.44,forT>1480K$ | ||

Stainless steel shell | ||||

Density | $kgm3$ | $\rho SS=8470$ | [14] | |

Thermal conductivity | $Wm\u22c5K$ | $kSS=0.0158\xd7T+10.169$ | ||

Specific heat capacity | $Jkg\u22c5K$ | $cp,SS=0.2827\xd7T+327.29$ | ||

Total hemispherical emissivity | – | $\epsilon SS=0.8$ | ||

Aluminum radiation shield | ||||

Density | $kgm3$ | $\rho Al=2700$ | comsol material library. T-dependent, shown here are representative values at 25 ℃ | |

Thermal conductivity | $Wm\u22c5K$ | $kAl=237$ | ||

Specific heat capacity | $Jkg\u22c5K$ | $cp,Al=898$ | ||

Air | ||||

density | $kgm3$ | $\rho air=pRsp,air\u22c5T$; $Rsp,air=287J/kg\u22c5K$ | comsol material library | |

Thermal conductivity | $Wm\u22c5K$ | $kair=\u22120.00227583562+1.15480022\xd710\u22124$$\xd7T\u22127.90252856\xd710\u22128\xd7T2+4.11702505$$\xd710\u221211\xd7T3\u22127.43864331\xd710\u221215\xd7T4$ | ||

Specific heat capacity | $Jkg\u22c5K$ | $cp,air=1047.63657\u22120.372589265\xd7T$$+9.45304214\xd710\u22124\xd7T2\u22126.02409443$$\xd710\u22127\xd7T3+1.2858961\xd710\u221210\xd7T4$ | ||

Dynamic viscosity | $Pa\u22c5s$ | $\mu air=\u22128.38278\xd710\u22127+8.35717342\xd710\u22128$$\xd7T\u22127.69429583\xd710\u221211\xd7T2+4.6437266$$\xd710\u221214\xd7T3\u22121.06585607\xd710\u221217\xd7T4$ |

*T* is presented in K, unless specified otherwise.

#### 2.2.3 Governing Equations.

*p*. Mass conservation without mass source/sink is given by the continuity equation for both the free flow and RPC domains:

#### 2.2.4 Boundary Conditions.

#### 2.2.5 Numerical Solution.

The CFD model was implemented using the commercial software comsol multiphysics 5.6, which solves the governing equations using the finite element method. A separate set of differential algebraic equations were generated for the fluid flow, for heat transfer across the solids, and for heat transfer across air. The resulting sequence of linear systems is solved by an iterative method called the generalized minimal residual method.

### 2.3 Iterative Coupling of Monte Carlo and Computational Fluid Dynamics Models.

The MC model uses the emission temperatures of the RPC $(TRPC)$ and of the cavity boundary elements $(Tbdry)$ to solve for the volumetric and surface heat sources to be used in the CFD model. As these temperatures are unknown initially, a guess value is used to start the MC computation. The CFD model, in turn, uses the heat sources to solve for an updated temperature field. The MC and CFD models are coupled and solved iteratively until convergence, defined as a maximum relative change of ≤1% in the temperature of any RPC volume element $(\Delta TRPC,max)$ from one iteration to the next. Effectively, the MC-CFD model is considered converged when the temperature field used for thermal emission in the MC model results in the calculation of the same temperature field by the CFD model. Figure 4 shows the iterative coupling schematically. At the start of the simulation (first MC-CFD iteration), as the temperatures are unknown, an isothermal temperature field is provided to the MC model as a guess value for thermal emission by the $dVi,j$ and $dAi,j$ elements.

### 2.4 Comparison With Experimental Data.

In a previous work [7], a 5 kW-scale prototype of the solar receiver modeled in this work was experimentally tested using concentrated radiation delivered by a high-flux solar simulator. Steady-state air temperature at the receiver outlet $(Tair,out)$ and RPC temperatures across the cavity (*T*_{1} and *T*_{2}) were measured for a range of air mass flowrates $m\u02d9air$. The location of the thermocouples is shown in Fig. 5. Experimental runs using a receiver with an RPC structure made of SiSiC 10 PPI and exposed to mean solar concentration ratio *C* = 3230 suns were selected because of the approximated axisymmetry of incident radiation for the given experimental setup. Figure 5 compares the numerical modeled and experimentally measured temperatures. The modeled and experimental values of $Tair,out$ are within 5% (about ±50 ℃) at all $m\u02d9air$ values. The maximum error in thermocouple measurement (±8 ℃) is well below this difference. The model overestimates *T*_{1} and *T*_{2} by up to 14%. Differences are attributed to the uncertainties in the experiments, mainly uncertainty in $Psolar$ of ±5% and in the position of thermocouples, and to the simplifying assumptions used in the modeling, mainly assumption of steady-state conditions and axisymmetric domain. As expected, temperatures decrease with $m\u02d9air$. As an example, for the experiments at $m\u02d9air=8.0kg/h$, the RPC attained 1188 °C (*T*_{1}) and 1223 °C (*T*_{2}), and the airflow attained 1004 °C.

## 3 Results and Discussion

### 3.1 Representative Model Results.

This section describes model results for a representative case using a solar receiver with an RPC structure made of SiSiC 10 PPI and exposed to *C* = 2475 suns, corresponding to $Psolar=3.1kW$.

#### 3.1.1 Fluid Flow and Heat Transfer.

Figure 6 shows contour plots of the air velocity, relative pressure, RPC temperature, and air temperature for the entire 2D-axisymmetric domain, for $m\u02d9air=7.40kg/h$. Air velocity (Fig. 6(a)) is very low (<1 m/s) at the receiver inlet and across the RPC and rapidly increases by an order of magnitude at the narrow receiver outlet due to the decrease in density (increase in temperature) and the narrow flow cross section, which also explains the pressure loss occurring predominantly along the outlet (Fig. 6(b)). Flow circulation is seen at the interface between the air and the RPC at the rear lateral part of the cavity, where part of the air is deflected by the RPC, which is nonideal for heat transfer. This flow profile results from insufficient suction generated by the air gap between the lateral RPC section and the insulation. Air is primarily suctioned by the outlet ducts directly at the rear, thus poorly cooling the lateral section of the RPC (Fig. 6(c)). The RPC is coldest at the rear section where the incoming air at atmospheric temperature makes first contact with the RPC. RPC temperature being lower at the directly irradiated surface than in the interior was also observed in the experiments [7]. Air heats up from near-ambient temperature to 1200 C within the first few millimeters of the RPC owing to efficient heat transfer. Model results also indicate that the incident concentrated solar radiation is absorbed mostly within the first few millimeters of the 2.5 cm-thick RPC. This is due to the Bouguer's law exponential attenuation of the incident radiation for the isotropic topology of the RPC. Recent studies on hierarchically ordered porous topologies have shown a more efficient volumetric absorption, which in turn diminishes undesired temperature gradients within the porous structure [17].

The initial design of the receiver prototype, which was experimentally tested [7] and is modeled in this work, was based on a solar reactor [18], which did not require an outlet design for high fluid flowrates as it was operated under high vacuum with a windowed aperture. Flow circulation and high velocities (with associated high pressure drop) in the outlet section of the air receiver modeled here indicate the need to re-design the receiver geometry for better airflow, as discussed further.

#### 3.1.2 Energy Balance.

*c*(

_{p}*T*) is the temperature-dependent specific heat capacity of air. The receiver thermal efficiency, $\eta thermal$, is defined as the ratio of $Pair,out$ and $Psolar$:

$Pcond$ is calculated in the CFD model by integrating the normal conductive heat flux over the external surface of the receiver and over the surfaces of the cooling water channels. $Pre\u2212rad$ is calculated in the MC model by summing the power of all rays exiting the aperture. $Pre\u2212rad$ consists of the rays thermally emitted by the RPC and cavity boundaries, and the rays of incident solar radiation reflected back through the aperture by the RPC and cavity boundaries. The reflected component is <0.1% of $Psolar$ due to the high surface absorptivity of SiSiC (thus high absorption coefficient of the RPC). The effective absorptance is thus >0.999.

Figure 7 plots the three components of the receiver energy balance as a percentage of $Psolar$, and the mean air temperature at receiver outlet for three $m\u02d9air$ values of the case SiSiC 10 PPI, *C* = 2475 suns $(Psolar=3.1kW)$. With increasing $m\u02d9air$, the shares of $Pcond$ and $Pre\u2212rad$ decrease (Fig. 7(b), left axis) due to a decrease in RPC temperature, accompanied by a decrease in $Tair,out$ (Fig. 7(b), right axis) and consequently $Pair,out$ increases. The share of $Pair,out$ is equivalent to $\eta thermal$ in percentage. $Pre\u2212rad$ is low due to the small cavity aperture (4 cm diameter) and high *C*. $Pcond$ is significant due to high surface area-to-volume ratio of the lab-scale receiver. Scale-up analysis of the receiver, as discussed further, shows a significant decrease in the share of $Pcond$ with increasing receiver size.

### 3.2 Model Application for Scale-Up and Optimization.

The validated model was applied for scaling-up and optimization of the receiver. The lab-scale receiver design was modified, and key cavity dimensions were parameterized (Fig. 8(a)). Modifications included removal and replacement of the water-cooled radiation shield and the ceria laminate with additional insulation, addition of air extraction ducts downstream of the RPC sections to homogenize airflow, and implementation of a converging outlet zone to reduce pressure loss.

#### 3.2.1 Optimizing Cavity Dimensions.

The model is applied to study the influence of cavity radius *R* and depth *L* relative to the aperture radius *r* on the receiver efficiency and RPC temperature distribution. For assumed design conditions of $Psolar=100kW$ at *C* = 1000 suns (aperture radius *r* = 0.178 m) and $m\u02d9air=200kg/h$ (to obtain *T*_{air,out} ∼ 1000 ℃), the cavity width and depth are varied by changing $R/r$ from 1.5 to 3.5, and $L/R$ from 0.5 to 2, respectively (Fig. 8(a)).

Figure 8(b) plots the model results for the thermal efficiency $\eta thermal$ and re-radiation losses $Pre\u2212rad$ as a function of $L/R$ for different $R/r$ ratios. At low cavity depths, re-radiation losses $Pre\u2212rad$ are high due to limited internal absorption of reflected and emitted radiation from within the cavity. With the increasing cavity depth, $Pre\u2212rad$ initially decreases significantly and eventually tapers off. At the same time, conduction losses increase with the increasing cavity depth due to higher receiver surface area, resulting in a peak of $\eta thermal$ at a certain cavity length for every $R/r$. For the same reasons, as $R/r$ is increased, the associated $\eta thermal$ values first increase and then drop, as seen by comparing the five $R/r$ curves. While the spread of $\eta thermal$ is narrow (0.67–0.70), the results show that the cavity depth should be at least equal to the cavity internal width, i.e., *L*/*R* > 1.0, to achieve a significant cavity effect and minimal $Pre\u2212rad$. Peak $\eta thermal$ values of nearly 0.70 indicate that the optimum cavity dimension for the given conditions is at *R*/*r* = 2.5, and *L*/*R* = 1.1–1.6. Figure 8(c) plots the effective absorptance $\alpha eff$ of the cavity as a function of $L/R$ for different $R/r$. $\alpha eff$ is defined as the fraction of solar power incident on the cavity aperture that is absorbed by the cavity, i.e., $\alpha eff=1\u2212Prefl$. The benefit of the cavity effect is clearly demonstrated as $\alpha eff$ is >0.99 even for the shallow cavity with *R*/*r* = 1.5 and *L*/*R* = 0.58 due to the highly absorptive SiSiC. With increasing $R/r$ and $L/R$, $\alpha eff$ exceeds 0.999 and approaches 1. This parametric analysis demonstrates the application of the receiver model to identify optimal cavity dimensions for a given set of conditions.

*Scaling-up*: To investigate the influence of scaling-up on the thermal performance, simulations were performed by varying $Psolar$ over four orders of magnitude, namely, 5, 50, 500, and 5000 kW. *C* was fixed at 2000 suns, thus determining the aperture radius *r* and rest of the cavity dimensions, assuming *R*/*r* = 2.5 and *L*/*R* = 1.6. Thickness of the SiSiC 10 PPI RPC was kept at 2.5 cm. A mean $Tair,out$ of 1200 ℃ ±1.5% was obtained at each scale by adjusting $m\u02d9air$ as the model input. Figure 9 plots the heat balance partition as a percentage of $Psolar$ on the left axis and the mean cavity temperature $Tcav,mean$ on the right axis as a function of $Psolar$. With the increasing receiver size, the surface area-to-volume ratio decreases, resulting in a significant decrease in the share of $Pcond$. Consequently, a lower value of $Tcav,mean$ is required to achieve the same mean $Tair,out$, which also reduces the share of $Pre\u2212rad$. The share of $Pair,out$, which is equivalent to $\eta thermal$ in percentage, reaches 87% at the 500 kW size and remains unchanged when scaled up further to 5000 kW size.

#### 3.2.2 Optimizing Aperture Radius.

In the preceding two analyses, $Psolar$ and *C* were set independently. However, for a receiver mounted on the top of a solar tower, both $Psolar$ and *C* are strongly dependent on the solar flux density distribution delivered by a given heliostat field on the plane of the receiver's aperture. With the increasing aperture radius, $Psolar$ increases, *C* decreases due to the Gaussian distribution of concentrated solar flux, and $Pre\u2212rad$ increases. Thus, an optimal radius exists where, for a given target $Tair,out$, the receiver delivers maximum $Pair,out$ out of the total available solar radiative power on the plane of the aperture, defined here as $Psolar,tot$ [19]. In this section, the receiver model is applied to optimize the aperture radius for a simulated solar flux density distribution at the THEMIS solar tower (France), generated using Monte Carlo ray tracing, which was published in Ref. [20]. The flux distribution was recreated for this analysis by interpolating between the contours of the published flux distribution, obtaining the same peak flux density of 3800 kW/m^{2} at the center and the same $Psolar,tot$ of 5.3 MW integrated over the 3 m × 3 m target area (Fig. 10).

$Psolar$ is redefined as $Psolar,aper$ (solar power over aperture of radius $r$) to distinguish from $Psolar,tot$ (solar power over 3 m × 3 m area on aperture plane). Accordingly, a new receiver thermal efficiency $\eta thermal,tot$ is also defined as the thermal power delivered by the receiver $(Pair,out)$ as a fraction of $Psolar,tot$. Simulations were performed for five values of *r* spanning the entire flux map (0.39, 0.53, 1.00, 1.22, and 1.50 m), by setting as model inputs the associated *C* and $Psolar,aper$ at each *r*, *R*/*r* = 2.5 and *L*/*R* = 1.6. Mean $Tair,out$ of 1350 ℃ ±1.5% was obtained in each case by adjusting $m\u02d9air$. RPC thickness was kept at 2.5 cm.

Figure 11 plots the partition of the heat balance as a percentage of $Psolar,tot$ (=5.3 MW) on the left axis for the five *r* values and the mean concentration ratio *C* on the right axis. With increasing *r*, more $Psolar,aper$ is intercepted by the aperture, resulting in more $Pair,out$ being delivered at the same mean $Tair,out$. At the same time, $Pre\u2212rad$ also increases with *r* due to increasing aperture area for re-radiation losses to the environment. Power not intercepted by the aperture is termed $Pspillage$, which decreases with increasing *r*. In an industrial setup, $Pspillage$ can be absorbed and further used for preheating purposes. $Pair,out$ peaks at *r* = 1.0 m as beyond this radius, the marginal gain in $Psolar,aper$ is offset by the significant loss due to $Pre\u2212rad$. Thus, the optimal *r* for maximum utilization of available solar power for this flux density distribution would be 1.0 m, where the receiver delivers nearly 67% of $Psolar,tot$ (equivalent to $\eta thermal,tot$ in %) at mean *T*_{air,out} = 1350 ℃. Such an optimization could be useful to dimension a solar receiver for a given heliostat field and solar tower.

## 4 Conclusion

We have developed a coupled MC-CFD heat transfer model of a solar open air receiver lined with an RPC structure directly exposed to high-flux solar irradiation. Modeling data were compared to experimental data obtained from experimental testing of a 5 kW lab-scale prototype. Model results of the lab-scale receiver indicated that the incident concentrated solar radiation is absorbed mostly within the first few millimeters of the 2.5 cm-thick RPC. This is due to the Bouguer's law exponential attenuation of the incident radiation for the isotropic topology of the RPC; hierarchically ordered topologies can provide a more efficient volumetric absorption. The incoming air at ambient temperature makes first contact with the directly irradiated surface of the rear RPC section, resulting in temperatures lower than the RPC interior. Energy balance of the lab-scale prototype shows significant conduction heat losses of up to 25% due to the high surface area-to-volume ratio. The model was applied for scale-up and optimization in three cases. First, a parameter study on the cavity internal radius $(R/r)$ and depth $(L/R)$ revealed optimal parameters of *R*/*r* = 2.5 and *L*/*R* = 1.6 at conditions of *P*_{solor} = 100 kW and *C* = 1000 suns. The optimal cavity dimensions maximize $\eta thermal$ and minimize temperature gradients across the cavity. The cavity effect and high surface absorptivity of SiSiC result in an effective absorptance of the cavity of nearly 1 and limits re-radiative losses. Second, upon scaling-up the receiver from 5 kW to 5 MW (*C* = 2000 suns, target mean *T*_{air,out} = 1200 ℃), conduction losses to the environment diminish to less than 1% of $Psolar$ due to decreasing surface area-to-volume ratio. Third, the receiver's aperture radius was varied over a simulated flux distribution at a solar tower (target mean *T*_{air,out} = 1350 ℃). A trade-off was observed between solar power intercepted by the receiver and re-radiative losses, leading to an optimal aperture radius of 1.0 m, where the receiver delivers nearly 67% of the total solar radiative power of 5.3 MW available on the aperture plane. The presented analyses demonstrate the capability of the parameterized model for designing and sizing the receiver for a solar tower configuration.

## Funding Data

The Swiss State Secretariat for Education, Research and Innovation (Grant No. 16.0183).

The European Union's Horizon 2020 Research and Innovation Program (Project INSHIP—Grant No. 731287).

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable upon reasonable request.

## Nomenclature

- $n$ =
total number of solar rays incident on receiver's aperture

- $r$ =
radius of cavity aperture (m)

- $w$ =
power content of each ray in MC model (kW)

*C*=mean solar concentration ratio on aperture, relative to 1 kW/m

^{2}(sun)- $I$ =
direct normal solar irradiation (kW/m

^{2})- $L$ =
length (depth) of the cavity from aperture to the rear RPC surface (m)

- $R$ =
internal radius (width) of the cavity till the lateral RPC surface interface (m)

- $cp,air$ =
specific heat capacity of air (J/kg/K)

- $cp,s$ =
specific heat capacity of RPC solid material (J/kg/K)

- $dAi,j$ =
discrete boundary (area) element of the cavity boundaries (m

^{2})- $dVi,j$ =
discrete volume element of the RPC domain (m

^{3})- $dm,RPC$ =
mean pore diameter of RPC (m)

- $hs\u2212f$ =
solid–fluid heat transfer coefficient (W/(m

^{2}· K))- $kair$ =
thermal conductivity of air (W/(m · K))

- $ks$ =
thermal conductivity of solid (W/(m · K))

- $lb$ =
distance to closest cavity boundary in direction of ray entering the RPC domain (m)

- $lcone$ =
length of conical insulation section (m)

- $lcyl$ =
length of cylindrical RPC section (m)

- $ldisk$ =
length of disk RPC section (m)

- $lext$ =
extinction length of ray upon entering the RPC domain (m)

- $qbdry$ =
surface heat source in CFD model (kW/m

^{2})- $qdAi,j$ =
net radiative heat flux on a discrete cavity boundary element $(dAi,j)$ (kW/m

^{2})- $rcyl,i$ =
inner radius of cylindrical RPC section (m)

- $rcyl,o$ =
outer radius of cylindrical RPC section (m)

- $rdisk$ =
radius of disk RPC section (m)

- $rRPC$ =
total hemispherical reflectivity of RPC strut surface

- $tRPC$ =
thickness of RPC (m)

- $m\u02d9air$ =
mass flowrate of air across the receiver (kg/h)

- $q\u02d9RPC$ =
volumetric heat source in CFD model (kW/m

^{3})- $Asp$ =
specific surface area of RPC (m

^{2}/m^{3})- $LRPC$ =
axial length of the lateral cylindrical RPC section (m)

- $Pair,out$ =
share of air enthalpy gain across the receiver in $Psolar$ (%)

- $Pcond$ =
share of conductive heat losses through the cavity aperture in $Psolar$ (%)

- $Pemis$ =
share of thermally emitted radiative losses through the cavity aperture in $Psolar$ (%)

- $Prefl$ =
share of solar reflected radiative losses through the cavity aperture in $Psolar$ (%)

- $Pre\u2212rad$ =
share of re-radiative heat losses through the cavity aperture in $Psolar$ (%)

- $Psolar$ =
solar radiative power incident on receiver's aperture (kW)

- $Psolar,aper$ =
total solar radiative power over the receiver's aperture, in the context of a given flux distribution (kW)

- $Psolar,tot$ =
total solar radiative power over the whole target area, in the context of a given flux distribution (kW)

- $Pspillage$ =
radiative power not intercepted by the receiver's aperture for a given flux distribution (kW)

- $PrRPC$ =
Prandtl number

- $ReRPC$ =
pore-scale Reynolds number

- $Tair$ =
air temperature (°C)

- $Tair,in$ =
air temperature at receiver inlet (°C)

- $Tair,out$ =
air temperature at receiver outlet (°C)

- $TdAi,j$ =
temperature of discrete cavity boundary element $dAi,j$ (°C)

- $TdVi,j$ =
temperature of discrete RPC volume element $dVi,j$ (°C)

- $Nexit$ =
counter of total number of rays exiting through the cavity aperture

- $Nexit,emis$ =
counter of number of thermally emitted rays exiting through the cavity aperture

- $Nexit,refl$ =
counter of number of scattered/reflected solar rays exiting through the cavity aperture

- $NdAi,jabs$ =
counter of number of rays absorbed by a discrete cavity boundary element $(dAi,j)$

- $NdVi,jabs$ =
counter of number of rays absorbed by a discrete volume element of the RPC $(dVi,j)$

- $NdAi,jemis$ =
counter of number of rays emitted by a discrete cavity boundary element $(dAi,j)$

- $NdVi,jemis$ =
counter of number of rays emitted by a discrete volume element of the RPC $(dVi,j)$

- $NdAi,jemis,max$ =
maximum number of rays that a discrete cavity boundary element $(dAi,j)$ can emit

- $NdVi,jemis,max$ =
maximum number of rays that a discrete volume element of the RPC $(dVi,j)$ can emit

- Nu =
Nusselt number

- $T1,T2$ =
RPC temperatures at the positions of thermocouples in the experiments

- $\u2207\u22c5q\u02d9dVi,j$ =
divergence of radiative flux for a discrete volume element of the RPC $(dVi,j)$ (kW/m

^{3})

### Greek Symbols

- $\alpha b$ =
total hemispherical absorptivity of cavity boundaries

- $\alpha eff$ =
effective absorptance of cavity

- $\alpha RPC$ =
effective absorption coefficient of RPC (m

^{−1})- $\beta RPC$ =
effective extinction coefficient of RPC (m

^{−1})- $\epsilon b$ =
total hemispherical emissivity of cavity boundaries

- $\epsilon RPC$ =
porosity of RPC

- $\eta thermal$ =
receiver thermal efficiency relative to $Psolar$

- $\eta thermal,tot$ =
solar receiver thermal efficiency relative to $Psolar,tot$, in the context of a given flux distribution

- $\theta cone$ =
half-angle of conical insulation section (deg)

- $\kappa RPC$ =
permeability of RPC (m

^{2})- $\mu air$ =
dynamic viscosity of air (Pa·s)

- $\rho air$ =
density of air (kg/m

^{3})- $\rho s$ =
solid (bulk) density of RPC material (kg/m

^{3})- $\sigma $ =
Stefan–Boltzmann constant (= 5.6704 × 10

^{−8}W/(m^{2}K^{4}))- $\sigma RPC$ =
effective scattering coefficient of RPC (m

^{−1})- $\omega RPC$ =
scattering albedo of RPC

### Abbreviations

## References

*Recommended Values of the Thermophysical Properties of Eight Alloys, Major Constituents and Their Oxides*, Purdue University, Lafayette, p.

_{2}Using 3D-Printed Hierarchically Channeled Ceria Structures

_{2}Into Separate Streams of CO and O

_{2}With High Selectivity, Stability, Conversion, and Efficiency