Graphical Abstract Figure
Graphical Abstract Figure
Close modal

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 [57]. 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.

Fig. 1
Scheme of the MC axisymmetric model domain of the solar receiver. Geometric dimensions are listed in Table 1.
Fig. 1
Scheme of the MC axisymmetric model domain of the solar receiver. Geometric dimensions are listed in Table 1.
Close modal
Table 1

Geometric dimensions of the solar receiver (Fig. 1) used in the MC model

DimensionSymbolValue
Aperture radiusr0.020 m
Half-angle of conical insulationθcone45 deg
Length of conical insulationlcone0.025 m
Inner radius of cylindrical RPC subdomainrcyl,i0.045 m
Thickness of RPCtRPC0.024 m
Outer radius of cylindrical RPC subdomainrcyl,o0.069 m (=rcyl,i+tRPC)
Length of cylindrical RPC subdomainlcyl0.075 m
Length of disk RPC subdomainldisk0.024 m
Radius of disk RPC subdomainrdisk0.069 m (=rcyl,o)
DimensionSymbolValue
Aperture radiusr0.020 m
Half-angle of conical insulationθcone45 deg
Length of conical insulationlcone0.025 m
Inner radius of cylindrical RPC subdomainrcyl,i0.045 m
Thickness of RPCtRPC0.024 m
Outer radius of cylindrical RPC subdomainrcyl,o0.069 m (=rcyl,i+tRPC)
Length of cylindrical RPC subdomainlcyl0.075 m
Length of disk RPC subdomainldisk0.024 m
Radius of disk RPC subdomainrdisk0.069 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 αRPC=(1rRPC)βRPC and σRPC=rRPCβRPC, respectively, where βRPC is the extinction coefficient and rRPC is the surface hemispherical reflectivity of the RPC struts. The scattering albedo ωRPC, which is the ratio of σRPC and βRPC (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 σRPC and ωRPC. SiSiC, the material used for manufacturing the RPC, has a nearly constant rRPC of ∼0.1 across most wavelengths [12]. Similarly, a near-constant ωRPC of ∼0.7 for alumina RPC has been reported across a large range of wavelengths [13]. In contrast, βRPC is a pure geometrical-dependent property and independent of wavelength. The cavity boundaries are assumed to be gray-diffuse, with a total hemispherical absorptivity αb 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, εb=αb. 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.

Table 2

Material properties relevant to radiative heat transfer used in the MC model

PropertyUnitValue or correlationRef.
RPC
No. of pores per inchPPI = 10, 20, 30Manufacturer
PorosityεRPC=0.90,0.91,0.87 for 10, 20, 30 PPI, respectivelyManufacturer
Mean pore diametermdm,RPC=105×(5.3022εRPC+2.1549)×(357/PPI)[11]
Extinction coefficientm−1βRPC=630.674εRPC2120.060εRPC+1229.361000dm,RPC[11]
Surface reflectivity of strutrRPC=0.1 (SiSiC)[12]
Scattering coefficientm−1σRPC=βRPC×rRPC[11]
Absorption coefficientm−1αRPC=βRPC×(1rRPC)[11]
Scattering albedoωRPC=σRPCβRPC=rRPC=0.1 (SiSiC), ωRPC=0.7 (alumina)[11,13]
CeO2 laminate insulation
Total hemispherical absorptivity (and emissivity)αb,CeO2=0.70[14]
Al2O3–SiO2 insulation
Total hemispherical absorptivity (and emissivity)αb,Al2O3SiO2=0.28[14]
PropertyUnitValue or correlationRef.
RPC
No. of pores per inchPPI = 10, 20, 30Manufacturer
PorosityεRPC=0.90,0.91,0.87 for 10, 20, 30 PPI, respectivelyManufacturer
Mean pore diametermdm,RPC=105×(5.3022εRPC+2.1549)×(357/PPI)[11]
Extinction coefficientm−1βRPC=630.674εRPC2120.060εRPC+1229.361000dm,RPC[11]
Surface reflectivity of strutrRPC=0.1 (SiSiC)[12]
Scattering coefficientm−1σRPC=βRPC×rRPC[11]
Absorption coefficientm−1αRPC=βRPC×(1rRPC)[11]
Scattering albedoωRPC=σRPCβRPC=rRPC=0.1 (SiSiC), ωRPC=0.7 (alumina)[11,13]
CeO2 laminate insulation
Total hemispherical absorptivity (and emissivity)αb,CeO2=0.70[14]
Al2O3–SiO2 insulation
Total hemispherical absorptivity (and emissivity)αb,Al2O3SiO2=0.28[14]
A uniform solar flux concentration ratio C is assumed over the receiver's aperture, defined by:
(1)
where Psolar is the incident solar radiative power on the receiver's circular aperture of radius r and I is the direct normal solar irradiation (assumed 1 kW/m2, 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 = 106; 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 θcone=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(), 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):

Fig. 2
Algorithm of the MC model
Fig. 2
Algorithm of the MC model
Close modal
(2)
(3)
As the temperatures are unknown at the start of the simulation, a guess value for all TdVi,j and TdAi,j is provided. Subsequently, the temperature distribution given by the solution of the CFD model is used as the new TdVi,j and TdAi,j in a next iteration of MC-CFD simulation until convergence, as explained further. When a solar ray enters the cavity, and subsequently the RPC domain, and the extinction length is calculated by lext=1/βRPClog and is compared to the distance to the closest boundary lb in the ray direction to determine if the ray is extinct by scattering/absorption inside the medium or if it is intercepted by a boundary. If extinction occurs within the medium, comparison of the scattering albedo ωRPC=σRPC/βRPC with determines whether it is scattered or absorbed. If scattered, the ray is sent into a new random direction with a new lext, and extinction inside the medium is checked again. If absorbed, the absorption counter of the relevant volume element (NdVi,jabs) is incremented. If the updated absorption counter is less than the emission limit of that volume element (NdVi,jemis,max), a new ray is emitted from the centroid of that element with a new random direction and lext, the element's emission counter (NdVi,jemis) is incremented, and the ray is traced further. However, if the updated absorption counter has surpassed the emission limit, no new ray is emitted and the ray history is terminated. This ensures energy conservation. If the ray does not get extinct inside the medium, it either intersects a boundary or exits the system through the cavity aperture. If the latter is the case, the counter of exiting rays Nexit is incremented and the ray history is terminated. If the ray intersects a boundary, αb is compared to to determine if it is reflected or absorbed. If reflected, similar to scattering within a volume element, the ray is redirected to a new random direction characteristic for diffuse reflection (given by Lambert's cosine law), with its residual lext and is checked for extinction inside the medium. If absorbed, similar to absorption within a volume element, the absorption counter of the relevant boundary element is incremented (NdAi,jabs), and if the emission limit of the element (NdAi,jemis,max) has not been reached, a new ray is emitted from the center of the element with a new random direction and lext, the element's emission counter (NdAi,jemis) is incremented, and tracing is continued. Once all incoming solar and thermally emitted rays are traced until termination of their history, every element accumulates a nonnegative difference between the number of rays absorbed and emitted—divergence of radiative flux for a volume element q˙dVi,j and net heat flux for a boundary element qdAi,j—which serve as heat sources q˙RPC and qbdry in the energy equations of the CFD model.
(4)
(5)
(6)
(7)
Another model output of importance is re-radiation losses Prerad based on the Nexit counter, which includes solar rays scattered/reflected (Nexit,refl) and thermally emitted rays by the RPC and cavity boundaries (Nexit,emis) out through the aperture.
(8)
(9)

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].

Fig. 3
Scheme of the CFD model domain. The boundary conditions are indicated.
Fig. 3
Scheme of the CFD model domain. The boundary conditions are indicated.
Close modal

2.2.2 Material Properties.

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

Table 3

Material properties relevant to heat and mass transfer used in the CFD model

PropertyUnitValue or correlationaRef.
SiSiC and alumina RPC
No. of pores per inchPPI = 10, 20, 30Manufacturer
PorosityεRPC=0.90,0.91,0.87 for 10, 20, 30 PPI, respectivelyManufacturer
Mean pore diametermdm,RPC=105×(5.3022×εRPC+2.1549)×(357/PPI)[11]
Specific surface aream2m3Asp=PPI/(357×105)(5.65595×εRPC26.08569×εRPC+4.49806)[11]
Permeabilitym2κRPC=εRPC3.7752/(5.4685×Asp2)[11]
Solid (bulk) densitykgm3ρs,SiSiC=2830, ρs,alumina=3960[15]
Solid specific heat capacityJkgKcp,s = 0.94 (SiSiC), 0.75 (alumina)[15]
Solid thermal conductivityWmKT (°C)ks,SiSiC[6,16]
20110
20085
50060
100042
150038
ks,alumina=85.8680.22972×T+2.607×104×T21.3607×107×T3+2.7092×1011×T4
Solid-fluid heat transfer coefficientWm2Khsf=Nu×kair/dm,RPC
Dimensionless numbersNu=4.173+2.359×εRPC+(0.3772×εRPC20.7479×εRPC+0.4849)×ReRPC(1.09530.2239εRPC)×PrRPC(0.6710.0213εRPC)[11]
ReRPC=ρair×ud×dm,RPC/μair
PrRPC=cp,air×μair/kair
CeO2 laminate insulation
Densitykgm3ρCeO2laminate=504.4[14]
Thermal conductivityWmKkCeO2laminate=2.2×107×T22.8387×104×T+0.176786
Specific heat capacityJkgKcp,CeO2laminate=0.000127×T2+0.269765×T+299.8,forT1100K
Al2O3–SiO2 insulation
Densitykgm3ρAl2O3SiO2=560.65[14]
Thermal conductivityWmKkAl2O3SiO2=0.00012926×T+0.019654
Specific heat capacityJkgKcp,Al2O3SiO2=4×107×T31.3797×103×T2+1.5987289×T+477.7,forT1480Kcp,CeO2laminate=444.27,forT>1100Kcp,Al2O3SiO2=1118.44,forT>1480K
Stainless steel shell
Densitykgm3ρSS=8470[14]
Thermal conductivityWmKkSS=0.0158×T+10.169
Specific heat capacityJkgKcp,SS=0.2827×T+327.29
Total hemispherical emissivityεSS=0.8
Aluminum radiation shield
Densitykgm3ρAl=2700comsol material library. T-dependent, shown here are representative values at 25 ℃
Thermal conductivityWmKkAl=237
Specific heat capacityJkgKcp,Al=898
Air
densitykgm3ρair=pRsp,airT; Rsp,air=287J/kgKcomsol material library
Thermal conductivityWmKkair=0.00227583562+1.15480022×104×T7.90252856×108×T2+4.11702505×1011×T37.43864331×1015×T4
Specific heat capacityJkgKcp,air=1047.636570.372589265×T+9.45304214×104×T26.02409443×107×T3+1.2858961×1010×T4
Dynamic viscosityPasμair=8.38278×107+8.35717342×108×T7.69429583×1011×T2+4.6437266×1014×T31.06585607×1017×T4
PropertyUnitValue or correlationaRef.
SiSiC and alumina RPC
No. of pores per inchPPI = 10, 20, 30Manufacturer
PorosityεRPC=0.90,0.91,0.87 for 10, 20, 30 PPI, respectivelyManufacturer
Mean pore diametermdm,RPC=105×(5.3022×εRPC+2.1549)×(357/PPI)[11]
Specific surface aream2m3Asp=PPI/(357×105)(5.65595×εRPC26.08569×εRPC+4.49806)[11]
Permeabilitym2κRPC=εRPC3.7752/(5.4685×Asp2)[11]
Solid (bulk) densitykgm3ρs,SiSiC=2830, ρs,alumina=3960[15]
Solid specific heat capacityJkgKcp,s = 0.94 (SiSiC), 0.75 (alumina)[15]
Solid thermal conductivityWmKT (°C)ks,SiSiC[6,16]
20110
20085
50060
100042
150038
ks,alumina=85.8680.22972×T+2.607×104×T21.3607×107×T3+2.7092×1011×T4
Solid-fluid heat transfer coefficientWm2Khsf=Nu×kair/dm,RPC
Dimensionless numbersNu=4.173+2.359×εRPC+(0.3772×εRPC20.7479×εRPC+0.4849)×ReRPC(1.09530.2239εRPC)×PrRPC(0.6710.0213εRPC)[11]
ReRPC=ρair×ud×dm,RPC/μair
PrRPC=cp,air×μair/kair
CeO2 laminate insulation
Densitykgm3ρCeO2laminate=504.4[14]
Thermal conductivityWmKkCeO2laminate=2.2×107×T22.8387×104×T+0.176786
Specific heat capacityJkgKcp,CeO2laminate=0.000127×T2+0.269765×T+299.8,forT1100K
Al2O3–SiO2 insulation
Densitykgm3ρAl2O3SiO2=560.65[14]
Thermal conductivityWmKkAl2O3SiO2=0.00012926×T+0.019654
Specific heat capacityJkgKcp,Al2O3SiO2=4×107×T31.3797×103×T2+1.5987289×T+477.7,forT1480Kcp,CeO2laminate=444.27,forT>1100Kcp,Al2O3SiO2=1118.44,forT>1480K
Stainless steel shell
Densitykgm3ρSS=8470[14]
Thermal conductivityWmKkSS=0.0158×T+10.169
Specific heat capacityJkgKcp,SS=0.2827×T+327.29
Total hemispherical emissivityεSS=0.8
Aluminum radiation shield
Densitykgm3ρAl=2700comsol material library. T-dependent, shown here are representative values at 25 ℃
Thermal conductivityWmKkAl=237
Specific heat capacityJkgKcp,Al=898
Air
densitykgm3ρair=pRsp,airT; Rsp,air=287J/kgKcomsol material library
Thermal conductivityWmKkair=0.00227583562+1.15480022×104×T7.90252856×108×T2+4.11702505×1011×T37.43864331×1015×T4
Specific heat capacityJkgKcp,air=1047.636570.372589265×T+9.45304214×104×T26.02409443×107×T3+1.2858961×1010×T4
Dynamic viscosityPasμair=8.38278×107+8.35717342×108×T7.69429583×1011×T2+4.6437266×1014×T31.06585607×1017×T4
a

T is presented in K, unless specified otherwise.

2.2.3 Governing Equations.

The fluid flow in the freely flowing air domain is described by the Navier–Stokes equations and the flow in the RPC porous domain is described by the Brinkman equations, which are solved for the velocity vector u and the pressure field p. Mass conservation without mass source/sink is given by the continuity equation for both the free flow and RPC domains:
(10)
where ρair is the air density. Momentum conservation in the free flow domain is given by:
(11)
where I is an identity matrix, K is the viscous stress tensor, and F is a vector to account for the influence of volume forces,
(12)
where μair is air dynamic viscosity. Momentum conservation in the RPC domain is given by:
(13)
where the viscous stress tensor in the RPC is given by:
(14)
Energy conservation is solved separately for the RPC and air. Temperature field of the RPC, TRPC, is obtained by solving:
(15)
where q˙RPC is the heat source in the RPC originating from the solution of radiation heat transfer in the MC model, and q˙sf is the interfacial heat transfer from the RPC to the air on account of local temperature difference.
(16)
where Asp is the specific surface area of the RPC. Temperature field of air, Tair, is obtained by solving:
(17)
where qbdry is the heat flux imposed on the cavity boundaries, originating from the solution of radiation heat transfer in the MC model.

2.2.4 Boundary Conditions.

On the boundaries of the air domain extended in front of the receiver, a normal velocity uin is imposed such that it results in the desired mass flowrate m˙air:
(18)
where Ain is the surface area of the extended air domain and ρair,in is the density of air at 1 atm and 20 ℃. At the receiver outlet, the pressure is set at 1 atm. This inlet–outlet combination of velocity and pressure boundary conditions results in a stable numerical solution while maintaining control over the resulting m˙air. On the external surface of the receiver, empirical correlations were applied to model convective heat loss to the surroundings. The receiver surface can reach up to 200 ℃ at some locations, as observed from infrared camera images, and hence, radiative heat losses to the ambient were also modeled by assuming a surface emissivity of the steel shell of 0.8 and an ambient temperature of 20 ℃. Inside the aluminum radiation shield, a constant temperature of 20 ℃ was set on the internal walls of the cooling water channel.

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 (ΔTRPC,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.

Fig. 4
Flowchart of iterative coupling of the MC and CFD models
Fig. 4
Flowchart of iterative coupling of the MC and CFD models
Close modal

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 (T1 and T2) were measured for a range of air mass flowrates m˙air. 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˙air values. The maximum error in thermocouple measurement (±8 ℃) is well below this difference. The model overestimates T1 and T2 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˙air. As an example, for the experiments at m˙air=8.0kg/h, the RPC attained 1188 °C (T1) and 1223 °C (T2), and the airflow attained 1004 °C.

Fig. 5
Left: Solar receiver schematic showing positions of thermocouples in experiments relevant for comparison with the model. Right: Comparison of experimentally measured and numerically modeled values of Tair,out, T1, and T2 as a function of the air mass flowrate for a 5 kW lab-scale solar receiver with an RPC structure made of SiSiC 10 PPI and exposed to C = 3230 suns.
Fig. 5
Left: Solar receiver schematic showing positions of thermocouples in experiments relevant for comparison with the model. Right: Comparison of experimentally measured and numerically modeled values of Tair,out, T1, and T2 as a function of the air mass flowrate for a 5 kW lab-scale solar receiver with an RPC structure made of SiSiC 10 PPI and exposed to C = 3230 suns.
Close modal

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˙air=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].

Fig. 6
Field contour plots for the 2D-axisymmetric domain: (a) air velocity and streamlines, (b) relative pressure, (c) RPC temperature, and (d) air temperature. Simulation case: SiSiC 10 PPI, C = 2475 suns (Psolar = 3.1 kW), and m˙air=7.40kg/h.
Fig. 6
Field contour plots for the 2D-axisymmetric domain: (a) air velocity and streamlines, (b) relative pressure, (c) RPC temperature, and (d) air temperature. Simulation case: SiSiC 10 PPI, C = 2475 suns (Psolar = 3.1 kW), and m˙air=7.40kg/h.
Close modal

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.

At a steady state, the solar radiative power entering the aperture Psolar is converted to three components: (1) gain in enthalpy of air (Pair,out), (2) conduction losses to the ambient and to the cooling water through the insulating walls (Pcond), and (3) re-radiation losses to the environment through the aperture (Prerad):
(19)
Pair,out is calculated in the CFD model by integrating the product of the air mass flowrate and the temperature-dependent specific heat capacity of air over the cross section of the receiver exit, with integration limits from ambient temperature (Tair,in) to the local air temperature on the outlet cross section (Tair,out):
(20)
where m˙air is the air mass flowrate, Tair,in and Tair,out are the inlet and outlet air temperature, respectively, and cp(T) is the temperature-dependent specific heat capacity of air. The receiver thermal efficiency, ηthermal, is defined as the ratio of Pair,out and Psolar:
(21)

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. Prerad is calculated in the MC model by summing the power of all rays exiting the aperture. Prerad 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˙air values of the case SiSiC 10 PPI, C = 2475 suns (Psolar=3.1kW). With increasing m˙air, the shares of Pcond and Prerad 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 ηthermal in percentage. Prerad 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.

Fig. 7
(a) Schematic representation of the receiver energy balance components and (b) components as a percentage of Psolar (left axis) and mean Tair,out (right axis) at m˙air = 4.85, 7.40, and 9.31 kg/h. The curve enclosing the Pair,out region is equivalent to ηthermal in %. Parameters: SiSiC 10 PPI, C = 2475 suns (Psolar = 3.1 kW).
Fig. 7
(a) Schematic representation of the receiver energy balance components and (b) components as a percentage of Psolar (left axis) and mean Tair,out (right axis) at m˙air = 4.85, 7.40, and 9.31 kg/h. The curve enclosing the Pair,out region is equivalent to ηthermal in %. Parameters: SiSiC 10 PPI, C = 2475 suns (Psolar = 3.1 kW).
Close modal

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.

Fig. 8
(a) Geometric parameters of the cavity, (b) receiver efficiency and re-radiation losses as a function of L/R for different R/r, and (c) cavity effective absorptance as a function of L/R for different R/r. Each simulation case is SiSiC 10 PPI, Psolar = 100 kW at C = 1000 suns (r = 0.178 m), m˙air=200kg/h.
Fig. 8
(a) Geometric parameters of the cavity, (b) receiver efficiency and re-radiation losses as a function of L/R for different R/r, and (c) cavity effective absorptance as a function of L/R for different R/r. Each simulation case is SiSiC 10 PPI, Psolar = 100 kW at C = 1000 suns (r = 0.178 m), m˙air=200kg/h.
Close modal

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˙air=200kg/h (to obtain Tair,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 ηthermal and re-radiation losses Prerad as a function of L/R for different R/r ratios. At low cavity depths, re-radiation losses Prerad are high due to limited internal absorption of reflected and emitted radiation from within the cavity. With the increasing cavity depth, Prerad 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 ηthermal at a certain cavity length for every R/r. For the same reasons, as R/r is increased, the associated ηthermal values first increase and then drop, as seen by comparing the five R/r curves. While the spread of ηthermal 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 Prerad. Peak ηthermal 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 αeff of the cavity as a function of L/R for different R/r. αeff is defined as the fraction of solar power incident on the cavity aperture that is absorbed by the cavity, i.e., αeff=1Prefl. The benefit of the cavity effect is clearly demonstrated as αeff 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, αeff 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˙air 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 Prerad. The share of Pair,out, which is equivalent to ηthermal in percentage, reaches 87% at the 500 kW size and remains unchanged when scaled up further to 5000 kW size.

Fig. 9
Influence of varying Psolar from 5 to 5000 kW: heat balance partition as a percentage of Psolar(leftaxis) and mean cavity temperature Tcav,mean (right axis) as a function of Psolar at C = 2000 suns and mean Tair,out=1200∘C . The curve enclosing the Pair,out area is equivalent to ηthermal in %. Cavity dimensions: R/r = 2.5, L/R = 1.6.
Fig. 9
Influence of varying Psolar from 5 to 5000 kW: heat balance partition as a percentage of Psolar(leftaxis) and mean cavity temperature Tcav,mean (right axis) as a function of Psolar at C = 2000 suns and mean Tair,out=1200∘C . The curve enclosing the Pair,out area is equivalent to ηthermal in %. Cavity dimensions: R/r = 2.5, L/R = 1.6.
Close modal

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 Prerad 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/m2 at the center and the same Psolar,tot of 5.3 MW integrated over the 3 m × 3 m target area (Fig. 10).

Fig. 10
(a) Simulated solar flux density distribution at the receiver plane of the THEMIS solar tower, recreated from Ref. [20], and (b) mean concentration ratio C and integrated power Psolar over a circle of radius r at the center of the flux distribution
Fig. 10
(a) Simulated solar flux density distribution at the receiver plane of the THEMIS solar tower, recreated from Ref. [20], and (b) mean concentration ratio C and integrated power Psolar over a circle of radius r at the center of the flux distribution
Close modal

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 ηthermal,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˙air. 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, Prerad 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 Prerad. 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 ηthermal,tot in %) at mean Tair,out = 1350 ℃. Such an optimization could be useful to dimension a solar receiver for a given heliostat field and solar tower.

Fig. 11
Influence of varying the aperture radius r over the solar flux density distribution at the receiver plane of the THEMIS solar tower to deliver mean Tair,out = 1350 ℃: heat balance components as a percentage of Psolar,tot (=5.3 MW) on left axis and C on right axis. The curve enclosing the Pair,out area is equivalent to ηthermal,tot in %. Cavity dimensions: R/r = 2.5, L/R.
Fig. 11
Influence of varying the aperture radius r over the solar flux density distribution at the receiver plane of the THEMIS solar tower to deliver mean Tair,out = 1350 ℃: heat balance components as a percentage of Psolar,tot (=5.3 MW) on left axis and C on right axis. The curve enclosing the Pair,out area is equivalent to ηthermal,tot in %. Cavity dimensions: R/r = 2.5, L/R.
Close modal

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 Psolor = 100 kW and C = 1000 suns. The optimal cavity dimensions maximize ηthermal 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 Tair,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 Tair,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/m2 (sun)

I =

direct normal solar irradiation (kW/m2)

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 (m2)

dVi,j =

discrete volume element of the RPC domain (m3)

dm,RPC =

mean pore diameter of RPC (m)

hsf =

solid–fluid heat transfer coefficient (W/(m2 · 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/m2)

qdAi,j =

net radiative heat flux on a discrete cavity boundary element (dAi,j) (kW/m2)

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˙air =

mass flowrate of air across the receiver (kg/h)

q˙RPC =

volumetric heat source in CFD model (kW/m3)

Asp =

specific surface area of RPC (m2/m3)

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 (%)

Prerad =

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

q˙dVi,j =

divergence of radiative flux for a discrete volume element of the RPC (dVi,j) (kW/m3)

Greek Symbols

αb =

total hemispherical absorptivity of cavity boundaries

αeff =

effective absorptance of cavity

αRPC =

effective absorption coefficient of RPC (m−1)

βRPC =

effective extinction coefficient of RPC (m−1)

εb =

total hemispherical emissivity of cavity boundaries

εRPC =

porosity of RPC

ηthermal =

receiver thermal efficiency relative to Psolar

ηthermal,tot =

solar receiver thermal efficiency relative to Psolar,tot, in the context of a given flux distribution

θcone =

half-angle of conical insulation section (deg)

κRPC =

permeability of RPC (m2)

μair =

dynamic viscosity of air (Pa·s)

ρair =

density of air (kg/m3)

ρs =

solid (bulk) density of RPC material (kg/m3)

σ =

Stefan–Boltzmann constant (= 5.6704 × 10−8 W/(m2K4))

σRPC =

effective scattering coefficient of RPC (m−1)

ωRPC =

scattering albedo of RPC

Abbreviations

CFD =

computational fluid dynamics

HTF =

heat transfer fluid

MC =

Monte Carlo

PPI =

pores per inch

RPC =

reticulated porous ceramic

SiSiC =

silicon-infused silicon carbide

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