# Modelling of flow regimes in tubular concentrating direct absorption solar collectors

Miguel Sainz-Mañas, Françoise Bataille, Cyril Caliot, Gilles Flamant

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Miguel Sainz-Mañas, Françoise Bataille, Cyril Caliot, Gilles Flamant. Modelling of flow regimes in tubular concentrating direct absorption solar collectors. Applied Thermal Engineering, 2025, 279, Part C, pp.127716. 10.1016/j.applthermaleng.2025.127716. hal-05308337

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Submitted on 14 Oct 2025

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# Modelling of flow regimes in tubular concentrating direct absorption solar collectors

Miguel Sainz Mañas<fnref n="1" />, Françoise Bataille<fnref n="1" />, Cyril Caliot<fnref n="2" />, and Gilles Flamant<fnref n="1" />, *

## Abstract

Direct absorption solar collectors have gained attention in the last decades as a promising solution to enhance the performance of conventional thermal collectors. In this concept, the heat transfer fluid absorbs the concentrated radiation volumetrically, which optical properties can be enhanced by dispersing nanoparticles. While several works have reported the benefits of volumetrically absorbing the incident radiation, few studies have explored its effect on the fluid temperature distribution. The present paper offers a comprehensive numerical analysis of the optical and thermal behavior of a parabolic-trough direct absorption solar collector using a graphene nanoparticle dispersion as absorbing medium. A Monte Carlo based ray-tracing approach is coupled to a computational fluid dynamics analysis to offer a complete evaluation of the performance of such systems. The results reveal a trade-off between complete absorption inside the tube and strong absorption in the wall vicinity, which takes place at higher optical depths. Furthermore, the fluid dynamics simulations underscore the role of buoyancy forces in achieving homogeneous temperature distributions, especially at lower flow rates. Neglecting gravitational effects may lead to inaccurate predictions of the system thermal performance. The numerical predictions align closely with experimental campaigns conducted for a similar collector, with total collector efficiencies of 66.3 % and 71.3 % for 0.2 g/L and 0.3 g/L nanofluids respectively. While these results represent a first-order comparison, they suggest that the model is reliable for designing and optimizing PT-DASC systems for real-world applications.

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## Keywords

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* Direct Absorption Solar Collector, Graphene nanoparticles, Computational Fluid Dynamics, Parabolic Trough Collector, Ray-tracing simulation, Experimentations.

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<footnote n="1">¹ Processes, Materials and Solar Energy laboratory, PROMES - CNRS, 7 rue du Four Solaire, 66120 Odeillo, France and UPVD, Rambla de la Thermodynamique, Tecnosud, 66100 Perpignan, France.</footnote>

<footnote n="2">² LMAP, UPPA, CNRS, E2S, 1 allée du parc Montaury, 64600 Anglet, France.</footnote>

<footnote>* Corresponding authors: Gilles.Flamant@promes.cnrs.fr</footnote>
# 1 Nomenclature

<table><thead><tr><th>Variable</th><th>Definition</th><th>Greek letter</th><th>Definition</th></tr></thead><tbody><tr><td>A</td><td>Aperture area – m<sup>2</sup></td><td>β</td><td>Extinction coefficient – m<sup>-1</sup></td></tr><tr><td>C<sub>p</sub></td><td>Specific heat capacity – J.kg<sup>-1</sup>.K<sup>-1</sup></td><td>δ</td><td>Optical depth</td></tr><tr><td>E</td><td>Internal energy – W.h</td><td>γ</td><td>Thermal expansion coef. – K<sup>-1</sup></td></tr><tr><td>G</td><td>Incident radiation – W.m<sup>-2</sup></td><td>η</td><td>Efficiency</td></tr><tr><td>Gr</td><td>Grashof number</td><td>η̇</td><td>Maximum efficiency</td></tr><tr><td>ġ</td><td>Gravitational acceleration – m.s<sup>-2</sup></td><td>ρ</td><td>Fluid density – kg.m<sup>-3</sup></td></tr><tr><td>k</td><td>Thermal conductivity – W.m<sup>-1</sup>.K<sup>-1</sup></td><td>τ</td><td>Transmissivity</td></tr><tr><td>L</td><td>Optical path – m</td><td>Φ</td><td>Source term – W</td></tr><tr><td>L<sub>c</sub></td><td>Characteristic length – m</td><td>φ</td><td>Volumetric source term – W.m<sup>-3</sup></td></tr><tr><td>ṁ</td><td>Mass flow rate – kg.s<sup>-1</sup></td><td></td><td></td></tr><tr><td>N<sub>R,i</sub></td><td>Number of radial divisions</td><td></td><td></td></tr><tr><td>N<sub>θ,i</sub></td><td>Number of angular divisions</td><td>0</td><td>Initial condition</td></tr><tr><td>n</td><td>Refractive index</td><td>avg</td><td>Average</td></tr><tr><td>P</td><td>Heat power – W</td><td>col</td><td>Collector</td></tr><tr><td>p</td><td>Static pressure – W</td><td>exp</td><td>Experimental</td></tr><tr><td>Q̇</td><td>Fluid flow rate – L.h<sup>-1</sup></td><td>in</td><td>Inlet</td></tr><tr><td>q̇<sub>f</sub></td><td>Front-incoming radiative flux density – W.m<sup>-2</sup></td><td>N<sub>R,i</sub></td><td>Cell mesh radial position</td></tr><tr><td>q̇<sub>b</sub></td><td>Back-incoming radiative flux density – W.m<sup>-2</sup></td><td>N<sub>θ,i</sub></td><td>Cell mesh radial position</td></tr><tr><td>Re</td><td>Reynolds number</td><td>n</td><td>Cell indicator</td></tr><tr><td>Ri</td><td>Richardson number</td><td>out</td><td>Outlet</td></tr><tr><td>S</td><td>Surface – m<sup>2</sup></td><td>PT</td><td>Photo-thermal</td></tr><tr><td>T</td><td>Temperature – °C</td><td>rad</td><td>Radiative</td></tr><tr><td>ΔT</td><td>Temperature variation – K</td><td>tot</td><td>Total</td></tr><tr><td>t</td><td>Time – s</td><td>u</td><td>Useful</td></tr><tr><td>U</td><td>Fluid velocity – m.s<sup>-1</sup></td><td></td><td></td></tr><tr><td>V</td><td>Volume – m<sup>3</sup></td><td></td><td></td></tr></tbody></table>

<table><thead><tr><th>Acronyms</th><th>Definition</th></tr></thead><tbody><tr><td>BRDF</td><td>Bidirectional reflectance distribution function</td></tr><tr><td>CFD</td><td>Computational fluid dynamics</td></tr><tr><td>DASC</td><td>Direct absorption solar collector</td></tr><tr><td>DNI</td><td>Direct normal irradiance</td></tr><tr><td>HTF</td><td>Heat transfer fluid</td></tr><tr><td>PTC</td><td>Parabolic trough collector</td></tr><tr><td>PT-DASC</td><td>Parabolic trough-DASC</td></tr><tr><td>UDF</td><td>User-defined function</td></tr></tbody></table>
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# 1. Introduction

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Concentrated Solar Power (CSP) is one of the most promising sustainable solutions for industrial decarbonization, not only for generating electricity but also for producing and storing heat [1]. Parabolic trough collectors (PTC) are commonly used in CSP plants to provide dispatchable electricity generation. The performance of these collectors is limited to unavoidable optical and thermal losses intrinsic to their geometry and working conditions [2]. New concepts based on PTC have arisen in the last decades to increase its performance, among which direct absorption solar collectors (DASC) have attracted the attention of the scientific community. Parabolic trough DASC (PT-DASC) consists of replacing the metal absorber of conventional PTC receivers by a fully transparent tube, allowing a volumetric absorption of the concentrated radiation by the heat transfer fluid (HTF). Conventional HTF, as water or thermal oils, have poor absorption properties in the UV and visible spectral ranges. By dispersing a small number of nanoparticles, the absorption efficiency of HTF in this spectral range can be increased. These dispersions are called nanofluids. The interest of these collectors over conventional PTC have been highlighted in several studies, either with experimental or numerical approaches, which have been well-reviewed in the following states of the art [3–6].

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Direct absorption solar collectors have three main potential advantages over traditional surface collectors [6]. First, as no metallic surface absorber is needed, the mechanical constraints of metal-glass sealings found in conventional PTC receivers is no longer a problem. The second main advantage is related to the inherent semi-transparent behavior of DASC, which can offer versatility for hybrid photovoltaic-thermal collectors, photochemistry, and other applications. Finally, heat is absorbed volumetrically throughout the fluid, enhancing the internal heat distribution. This entails two sub-advantages: peak fluid temperatures are found far from the walls, thus reducing heat losses with the environment; and the system is not restricted to turbulent flow conditions, as heat exchanges with the inner tube do not need to be optimized. While the first two advantages can be assessed experimentally, the internal heat distribution within the HTF is significantly complex to evaluate in on-sun experiments using concentrating DASC. To overcome experimental difficulties, computational fluid dynamics (CFD) provides a solution for a deeper understanding of the thermal dynamics inside the PT-DASC receiver.

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Several studies considered CFD approaches to evaluate the thermal performances of nanofluids such as single or multi-wall carbon nanotubes (MWCNT) [7–11], CuO [11–13], silica [7,9] or plasmonic nanoparticles [14,15]. The most promising particles are MWCNT, which present higher thermal efficiencies than silica [7] or CuO [11] particle dispersions. Nevertheless, the geometry of the receiver has demonstrated a major influence on the thermal efficiency of concentrating DASC. Heat losses vary considerably depending on whether single glass, double glass, or evacuated double-glass tubes are used, as shown in the CFD analysis of Kasaeian et al. [8]. Additionally, Li et al. [16] underlined the benefits of reducing radiative losses by adding an indium tin oxide (ITO) coating over the entire perimeter of a glass tubular receiver. Dugaria et al. [10] proposed a novel planar PTC receiver with a transparent glass window, which offered the possibility of adding a reflective surface in the rear part of the receiver for increasing the total optical thickness, thus obtaining high photo-thermal efficiencies with low particle concentrations. While the proposed planar solution simplifies the numerical solution of the radiative transfer problem, the concentrated radiation is not distributed homogeneously throughout the fluid [10]. In this respect, tubular receivers benefit from an incident radiation normal to the surface, which entails a radiation cone gradually attenuated throughout the semi-transparent fluid. The angular and spatial distribution of the concentrated solar flux inside the tube depends on the PTC aperture and its focal length, the tube diameter, and the optical properties of the glass interfaces. In addition, this radiative intensity distribution is highly sensitive to the reflector
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bidirectional reflectance distribution function (BRDF), and the nanofluid absorption and scattering properties. Among the presented PT-DASC numerical studies using tubular receivers, few consider a complete radiative model to calculate the power absorbed by the fluid [12,14,15].

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The spatial distribution of the concentrated solar flux on receiver surfaces is usually modelled with a ray-tracing software based on Monte Carlo algorithms [17]. The resulting radiative flux distribution can be introduced as a volumetric source term in the energy equation within a model solving flow and combined heat transfer. Qin et al. [14,15] employed a ray-tracing approach with COMSOL Multiphysics® to assess the influence of using a rear reflective coating on the radiative contour inside the receiver. Their study highlights the importance of precisely calculating the radiative power throughout the semi-transparent fluid to obtain accurate temperature contours of the working fluid. However, no experimental validation was proposed in Qin et al. [14,15] studies. In a PT-DASC configuration, Xu et al. [12] analyzed the incoming radiation at the glass tube exterior surface using a Monte Carlo ray-tracing approach, thus not considering the influence of the glass interfaces. It must be noted that all previous three studies did not consider the spectral properties of the concentrating optics nor the scattering effects of the medium on the calculated volumetric source term. The numerical analysis presented by Xu et al. [12] was compared to an experimental PT-DASC and show accurate predictions of the real collector performances. The work of Xu et al. [12] brought out the effects of buoyancy-driven flows on homogenizing the internal heat at low Reynolds (Re) numbers and reducing heat losses, a potential advantage of working in laminar flow conditions instead of turbulent flows.

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Turbulent flows are usually privileged in heating and cooling applications, as heat exchanges are optimized at high Reynolds numbers (> 5000). However, some applications may require precise heating control, for which low flow velocities and hence low Reynolds numbers (< 3000) may be privileged. At low flow velocities, natural convection might create recirculation due to buoyancy forces driven by the fluid density gradient. This phenomenon stimulates fluid mixing, which homogenizes the fluid temperature distribution. In conventional PTC, the concentrated solar radiation is absorbed in the lower part of the tubular receiver surface, driving the described effect. However, conventional PTC operate at high Reynolds numbers to increase heat transfers with the steel tube, thus avoiding any buoyancy effect. On the contrary, DASC receivers are not constrained by the flow regime to maintain a high thermal efficiency, as the incident radiation is absorbed in the volume of the HTF. In this case, buoyancy-induced mixing might increase the overall efficiency by preventing high HTF temperatures near the glass surface, thereby reducing heat losses to the environment. The benefit of buoyancy-induced mixing was underlined by Xu et al. [12], but its variation with the Reynolds number and its influence on the collector performance was not further assessed. While several authors have addressed the influence of natural convection on the particle distribution homogeneity [18–21], no previous research has addressed the effects of buoyancy-driven flows under laminar flow conditions on the performance of concentrating DASC.

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Both the absorption of the concentrated radiative flux and the temperature distribution within the fluid are critical parameters determining the performance of concentrating DASC. However, CFD studies focusing on concentrating DASC often lack a comprehensive radiative analysis that includes the spectral properties of components involved in the concentrating optics. The volumetric absorption of the radiative flux enables DASC collectors working in non-turbulent flow regimes, thus providing versatility to meet the requirements of the desired application. However, the influence of the Reynolds number on the performance of a concentrating DASC remains unexplored. In this paper, both mechanisms are addressed by coupling a ray-tracing model feeds with wavelength dependent optical properties, and a CFD model including buoyancy-forces. This approach allows further investigation of
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the thermal dynamics of the fluid in such collectors and highlights future paths for performance

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optimization.

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The performance of a medium-size PT-DASC with an evacuated double-glass receiver is studied numerically. A stable graphene aqueous dispersion is selected as working fluid. To evaluate the collector's performance, first, a radiative transfer model is developed to accurately simulate the PT-DASC optical behavior under real working conditions. Secondly, a three-dimensional CFD model of the PT-DASC receiver is presented, integrating the previously calculated volumetric radiative source term.

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The influence of the Reynolds number on buoyancy-induced velocity and temperature contours is analyzed in detail. Finally, the numerical model results are compared to the experimental performance of a similar PT-DASC pilot employing the same particle dispersion. This study provides a complete analysis of the performance of a graphene-based PT-DASC, along with guidelines for future research and critical considerations for the development of such collectors.

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## 2. Methodology

The aqueous dispersion considered in this study, ref. CW-GLB02A1 (supplied by Carbon-Waters®), is composed of hydroxylated graphene particles of 0.1 – 6 µm flake lateral size and 1 – 3 nm thickness [22,23]. The initial graphene dispersion supplied has a particle concentration of 1 g/L, from which 0.2 g/L and 0.3 g/L dilutions were obtained by dilution with demineralized water. Their optical properties have been characterized with a spectrophotometer. The extinction coefficient, $\beta$, of each dilution has been determined (refer to Figure S1 in the supplementary material). In previous studies, the optical properties of various dilutions of the dispersion have been further analyzed [24] as well as their stability in real working conditions under concentrated sunlight [25]. In the present numerical analysis, graphene dispersions of 0.2 and 0.3 g/L are considered based on their respective absorption coefficient corresponding to average absorptances of 87.69 ± 0.03% and 94.75 ± 0.02% within the studied receiver (33.6 mm light path), respectively [25]. These values enable investigating the impact of nearly complete absorption (0.3 g/L) versus incomplete absorption (0.2 g/L) on the volumetric distribution of the radiative source term.

### 2.1. Radiative source term calculation

The concentrated solar radiative flux arriving to the fluid inside the receiver has been modelled using the Monte Carlo ray-tracing software Solstice® [17,26], a software capable of precisely calculating the radiative flux incident to solar receivers including concentrating optics [27]. The collector geometry considered for the radiative model is presented in **Figure 1**, based on the experimental pilot layout presented in our previous work [25], which dimensions are detailed in **Table 1**. All participating components are considered non-scattering and non-grey media and their optical properties are given in the supplementary material.

<figure><img src="image_2.png" /><figcaption>**Figure 1:** Schematic of the geometry of the concentrating PTC considered for the ray-tracing simulations.</figcaption></figure>

<table><caption>**Table 1:** Dimensions of the direct absorption parabolic trough collector modelled with the ray-tracing software.</caption><thead><tr><th>Concentrator dimensions</th><th>Value</th><th>Receiver dimensions</th><th>Value</th></tr></thead><tbody><tr><td>Aperture length</td><td>2 000 mm</td><td>Tube length</td><td>2 000 mm</td></tr><tr><td>Aperture width</td><td>960 mm</td><td>Outer diameter glass envelope</td><td>70 mm</td></tr><tr><td>Space between mirror sheets</td><td>20 mm</td><td>Inner diameter glass envelope</td><td>63.6 mm</td></tr><tr><td rowspan="2">Focal length</td><td rowspan="2">240 mm</td><td>Outer diameter glass tube</td><td>40 mm</td></tr><tr><td>Inner diameter glass envelope</td><td>33.6 mm</td></tr></tbody></table>
1 The sun is modelled as a disk, with a half angle of 4.66 mrad and a direct normal irradiance (DNI) of
2 1000 W/m². The spectral distribution of the incoming sunlight is also considered, taken here as the
3 AM1.5D solar spectrum (W/m²/nm¹) from the ASTM G173-03 standard from the American Society for
4 Testing and Materials (ASTM) [28].

5 A 2-axis tracking concentrator has been considered, assuming a perfect alignment of the aperture area
6 with the incoming radiation. The PTC reflective surface is divided in two parts along its width to leave
7 place for the receiver supports, as in the experimental design. The spectral properties of the Almeco®
8 VWR193 aluminum reflective sheet used in the PT-DASC pilot are considered. Additionally, a slope-
9 error of 0.5 mrad has been attributed to the reflecting material, value obtained from the BRDF
10 measurements of the aluminum sheet (Figure S3 in the Supplementary Material). The receiver is
11 assumed to be perfectly aligned with the concentrator focal line.

12 The receiver is composed of two borosilicate-glass concentric tubes, with a vacuum chamber in
13 between, and the absorbing medium flowing inside (see Table 1 for dimensions). All materials in the
14 receiver are fully characterized by their spectral refractive index (n) and their spectral extinction
15 coefficient (β), which spectral behavior is detailed in the Supplementary Material. For the presented
16 case study, the nanofluid particle concentration, 0.2 g/L and 0.3 g/L, is considered sufficiently low to
17 not affect the real part of the refractive index (n) of the base fluid. For this reason, the n of water is
18 attributed to the nanofluid absorbing medium together with the measured β corresponding to the
19 particle concentration. The transmitted radiation through the medium at a given optical path, L, is
20 calculated with the Beer-Lambert law: τ = e⁻⁻β·L.

21 In order to calculate the absorbed concentrated radiative flux, the medium inside the receiver, the
22 nanofluid, has been meshed following radial (N<sub>R,i</sub>) and angular (N<sub>θ,j</sub>) divisions (see Figure 2). No
23 longitudinal divisions have been imposed in the z-axis, as no variation of the incoming flux through the
24 collector length is expected (ideal 2-axis sun tracking without obstacles). The ray-tracing software
25 computes the radiative heat flux density at each face of the cell (subscript n) for a given direction,
26 either front-incoming (dot<sub>f,n</sub> - W/m²) or back-outgoing (dot<sub>b,n</sub> - W/m²) (Figure 2). A large number of
27 realizations were used to insure the 99 % confidence intervals of the Monte Carlo estimations are
28 below 0.3 % of the corresponding radiative heat fluxes. The energy balance between the incoming and
29 outgoing radiation leads to the volumetric radiative source term (φ<sub>rad</sub> - W/m³):

$$
\varphi_{rad}(N_{R,i}, N_{\theta,j}) = \frac{\sum (\dot{q}_{f,n}(N_{R,i}, N_{\theta,j}) - \dot{q}_{b,n}(N_{R,i}, N_{\theta,j})) S_{n,N_{R,i},N_{\theta,j}}}{V_{N_{R,i},N_{\theta,j}}} \quad 1
$$

30 with S<sub>n</sub> the surface of the n face of the cell, and V<sub>N<sub>R,i</sub>,N<sub>θ,j</sub></sub> the cell volume. By integrating φ<sub>rad</sub> over the
31 fluid volume inside the receiver, the total solar heat absorbed (Φ<sub>rad,tot</sub> - W) is obtained:

$$
\Phi_{rad,tot} = \int_V \varphi_{rad}(N_{R,i}, N_{\theta,j}) \, dV_{N_{R,i}, N_{\theta,j}} \qquad 2
$$

32 Accounting only for the PTC optical losses and assuming no thermal losses the ratio of the total Φ<sub>rad,tot</sub>
33 to the incident solar heat flux on the collector represent its maximum photo-thermal efficiency (η̂<sub>PT</sub>):

$$
\hat{\eta}_{PT} = \frac{\Phi_{rad,tot}}{A_{col}G_0} \qquad 3
$$

34 where A<sub>col</sub> is the concentrator aperture area, and G<sub>0</sub> is the solar direct normal irradiation (DNI).
The number of rays launched by the Monte Carlo ray-tracing software influences the precision of the calculated $\Phi_{rad,tot}$. Additionally, the refinement of the mesh also affects the precision of $\Phi_{rad,tot}$. The influence of the number of $N_{R,i}$ and $N_{\theta,j}$ divisions has been evaluated and reported in the supplementary material. In our study, the ray-tracing calculation was made with $10^8$ rays and a mesh discretization of 20 $N_{R,i}$ and 24 $N_{\theta,j}$ divisions (a total of 960 cells), resulting in a 99 % confidence interval on $\Phi_{rad,tot}$ of less than 0.3 %.

<figure><img src="image_3.png" /><figcaption>**Figure 2:** Detailed mesh of the absorbing medium inside the receiver with radial ($N_{R,i}$) and angular ($N_{\theta,j}$) divisions. In the right side, the front-incoming and back-outgoing radiation obtained from the ray-tracing software incident to each face of a given cell is represented.</figcaption></figure>

## 2.2. Fluid dynamics numerical model

The fluid dynamics inside the receiver detailed in **Table 1** are studied with the CFD software Ansys Fluent®. CFD plays a key role in resolving the coupled heat transfer and fluid flow phenomena within the volume of the receiver, where spatially varying radiation absorption may play an important role on velocity and temperature distributions inside the receiver tube. In this work, the effects of absorbed concentrated radiation at various fluid flow rates on the temperature and velocity contours of the working fluid are exhibited. In the current work, the HTF flow rates ($\dot{Q}$) are varied between 75 L/h and 225 L/h, corresponding to inlet velocities ($U_{in}$) ranging from 0.023 to 0.071 m/s. Under these conditions, Reynolds numbers remain below 3000, corresponding to a laminar flow regime for all cases.

In these configurations, buoyancy-driven forces could play a significant role in the establishment of the velocity and temperature fields justifying to account for gravity forces in all simulations. The Richardson number ($Ri$) quantifies the weight of buoyancy forces versus inertial forces, comparing the Grashof ($Gr$) and Reynolds ($Re$) number:

$$Ri = \frac{Gr}{Re^2} = \frac{g \gamma \Delta T L_c}{U^2} \qquad 4$$

where $g$ (m/s²) is the gravitational acceleration, $\gamma$ (K⁻¹) is the thermal expansion coefficient, $\Delta T$ (K) the temperature gradient, $L_c$ (m) the characteristic length (here the inner tube diameter), and $U$ (m/s) the fluid velocity. As $Ri$ increases, buoyancy forces dominate the flow regime, resulting in a natural convection driven flow rate. If $Ri$ remain close to unity, mixed convection occurs within the fluid, while low values of $Ri$ imply a forced convection dominant flow regime. In the current study, $Ri$ varies between 0.1 and 2, falling within the mixed convection range. Under these $Ri$ flow condition, a transient simulation is conducted to ensure convergence of residuals.
As a first approach, the gravity vector ($\vec{g}$) is assumed perpendicular to the concentrator aperture plane (sun at the zenith), following the -$\vec{y}$ axis direction, with a magnitude of 9.8 m/s². However, under real operating conditions, the direction of gravity varies with the sun's position. To account for this variability, an additional case was assessed with a sun elevation of 53.22°, corresponding to the mean sun elevation between 14h00-14h20 in April at Odeillo (France). This situation is representative of an experimental campaign conducted with the PT-DASC pilot. Both combined approaches allow for an assessment of the influence of the gravity vector on the thermal behavior of the HTF.

Heat losses related to the fluid temperature distribution were not addressed in this work. For this reason, the CFD domain only considers the HTF volume (3D geometry) with an adiabatic condition for all limiting faces (see Figure 3). At the inlet, a laminar Poiseuille velocity contour is attributed via a user-defined function (UDF), with a fixed inlet temperature ($T_{in}$) of 300 K. A pressure outlet boundary condition is considered with no variation of pressure (gauge pressure equals 0 Pa). A coupled approach is chosen for the pressure-velocity coupling, while the momentum and energy equations are discretized using a second-order scheme. For the transient formulation, a first-order implicit scheme is employed.

<figure><img src="image_4.png" /><figcaption>Figure 3: 2D schematic of the fluid domain studied in the CFD analysis. The main boundary conditions are detailed as well as the inlet velocity profile.</figcaption></figure>

The fluid motion is governed by the Navier-Stokes equations describing the conservation laws of momentum and internal energy. The momentum equation is defined as follows:

$$ \frac{\partial}{\partial t}(\rho \vec{U}) + \nabla \cdot (\rho \vec{U} \otimes \vec{U}) = \rho \vec{g} - \nabla p + \nabla \cdot \tau \qquad 5 $$

where $\rho$ is the fluid density, $\vec{U}$ is the velocity vector, $t$ is time, $\vec{g}$ the gravitational acceleration, $p$ is the static pressure and $\tau$ is the stress tensor. The total energy equation is solved in the following form:

$$ \frac{\partial}{\partial t}(\rho E) + \nabla \cdot (\vec{U}(\rho E + p)) = \nabla \cdot (k \nabla T) + \nabla \cdot (\tau \cdot \vec{U}) + \varphi \qquad 6 $$

where $E$ is the specific total energy, $k$ if the thermal conductivity, $T$ is temperature, and $\varphi$ represents any additional volumetric heat source (W/m³). In the presented numerical model, the radiative source term $\varphi_{rad}$ is inserted as heat source term into the energy equation of the CFD software through a UDF.

Owing to the low particle concentration in the dispersions studied, the thermo-physical properties of the nanofluid remain unchanged compared to that of the base fluid, as supported by the thermal conductivity measurements provided in the supplementary material. This observation is consistent with the findings of Pavía et al. [29], who reported that a concentration of at least 0.1 wt% graphene is typically required to achieve a 10% enhancement in thermal conductivity. Our study involved nanofluid with concentration lower or equal to 0.03 wt%. Consequently, the temperature-dependent thermophysical properties of water are used in the CFD simulations. The polynomial equations along with their numerical coefficients are provided in the supplementary material (see Eq. S3-S5).
1 A mesh sensitivity study, conducted on a reduced geometry due to computational limitations, is detailed in the Supplementary Material. Based on this analysis, the mesh selected for the CFD simulations (see Table 1 for dimensions) is composed of 30 million cells with a mesh refinement near the cylinder surface (Figure 4).

<figure><img src="image_5.png" /><figcaption>Figure 4: Mesh of the domain studied in the CFD approach.</figcaption></figure>

2.3. Experimental set-up for comparison

The current numerical analysis mimics the experimental study of a PT-DASC pilot with a similar geometry and identical graphene particle concentrations, 0.2 and 0.3 g/L [25]. The performance of the collector was assessed by comparing the solar heat flux incident to the concentrator and the useful heat power ($P_u$) gained by the absorbing HTF (equation 7 and 8).

$$P_u = \dot{m} C_p (T_{out} - T_{in}) \qquad 7$$

$$\eta_{exp} = \frac{P_u}{A_{col} G_0} \qquad 8$$

where $\dot{m}$ is the mass flow rate, $C_p$ is the isobaric specific heat of the HTF, $T_{in}$ is the inlet HTF temperature, $T_{out}$ is the outlet HTF temperature, $\eta_{exp}$ is the total collector efficiency of the experimental PT-DASC, $A_{col}$ is the concentrator aperture area, and $G_0$ is the measured direct normal irradiance (DNI). The experimental campaign was conducted with flow rates between 60 – 120 L/h, DNI varying between 900 – 1000 W/m² and $T_{in}$ close to ambient temperature (15 – 25 °C) [25]. Outlet temperatures remained low, below 50 °C, thus heat losses with the environment remained small. Under such conditions, the measured collector performances can be reasonably compared with the CFD model predictions and thus justifying the adiabatic wall assumption.
## 3. Results and discussion

### 3.1. Radiative source term

The volumetric radiative source term, $\varphi_{rad}$, is presented in **Figure 5** for both nanoparticle concentrations, 0.2 g/L and 0.3 g/L. The concentration cone, intrinsic to parabolic concentrators [14,15], is identified by their contours, with a gap between [-2, 2] mm along the x-axis resulting from the separation of the reflecting sheets (see **Figure 1**). $\varphi_{rad}$ follows a logarithmic trend within the first layers traversed by the concentrated flux, according to the logarithmic nature of the Beer-Lambert law. However, closer to the tube center, the concentration of the radiative power becomes dominant, leading to higher $\varphi_{rad}$ values. The size of the $\varphi_{rad}$ disk at the center is directly dependent on the sun's angular size and slope error defined in the ray-tracing setup.

<figure><img src="image_6.png" /><figcaption>**Figure 5:** Contours of the volumetric radiative source, $\varphi_{rad}$, throughout the fluid for (a) 0.2 g/L and (b) 0.3 g/L obtained from the ray-tracing calculations.</figcaption></figure>

The $\varphi_{rad}$ contours vary slightly depending on particle concentration (**Figure 5**). In both nanofluids, the incident radiation is not fully absorbed in the lower semi-cylinder, with absorptions of 62.1 % and 76.5 % for 0.2 and 0.3 g/L respectively. This is the result of the optical properties of the nanofluids, which were thoroughly analyzed in our previous study [24]. A notable portion of the incident radiation is absorbed in the upper part, which is best seen at low graphene concentration (0.2 g/L - **Figure 5.a**). High $\varphi_{rad}$ values are also present close to the wall (in the lower semi-cylinder), which are best seen at high particle concentration (0.3 g/L - **Figure 5.b**). Still, the radiative source term shows the highest values at the tube center where the solar irradiation converges in both scenarios. The optimal particle concentration must ensure complete absorption of the incident radiation while maintaining a distribution of $\varphi_{rad}$ spread in the volume rather than located in the wall vicinity. For the current receiver geometry, the most suitable particle concentration appears to be between 0.3 and 0.4 g/L, representing mean absorptions of 95 % and 98 %, respectively. Below this range, a significant fraction of the incident radiation is lost; while at higher particle concentrations, a substantial portion of $\varphi_{rad}$ is absorbed at the wall vicinity. These concentration ranges correspond to optical depths, $\delta = \beta \cdot L$, between 3 and 4.

The $\Phi_{rad,tot}$ for each particle concentration is 1281.42 ± 0.74 W and 1382.30 ± 0.58 W for 0.2 g/L and 0.3 g/L respectively. Compared to the incident power at the concentrator aperture area, a photo-thermal efficiency $\eta_{PT}$ of 66.7 ± 0.04 % and 71.9 ± 0.03 % is obtained for 0.2 g/L and 0.3 g/L respectively.
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## 3.2. Temperature and velocity contours

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The above presented $\phi_{rad}$ contours are incorporated into the energy equation in the CFD model as a volumetric radiative source term. The $\Phi_{rad,tot}$ obtained after volume-integration with the CFD mesh yields a relative error of less than 0.5 % for both particle concentrations when compared with that calculated with the ray-tracing simulations, which validates the interpolation between meshes.

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The absorbed radiation leads the average HTF temperature to rise along the length of the tube, with a slope depending on the particle concentration and fluid flow rate (Figure 6). The nanofluid temperature increases linearly throughout the pipe length, with a higher outlet temperature as the fluid flow rate decreases. For a 0.2 g/L particle concentration, a temperature increase of 14.6 K, 12.2 K, 10.5 K, 9.1 K, 7.3 K, and 4.9 K is obtained with a flowrate of 75 L/h, 90 L/h, 105 L/h, 120 L/h, 150 L/h and 225 L/h, respectively. When the graphene dispersion is increased to 0.3 g/L, the temperature differences result in 15.8 K and 9.9 K for the flowrate of 75 L/h and 120 L/h, respectively. A laboratory-scale concentrator (2 m long) is modelled in this work, which entails low temperature differences between the inlet and outlet of the volumetric receiver. At these absolute temperatures, heat losses with the environment are expected to be low considering the receiver geometry, which justifies the assumption of adiabatic boundary conditions. For larger concentrators, heat losses must be considered as their impact on the temperature increase and collector performance could be much higher.

<figure><img src="image_7.png" /><figcaption>**Figure 6:** Mass-weighted average temperature ($T_{avg}$) at different axial positions for varying flow rates and nanofluid concentrations: 0.2 g/L (solid lines) and 0.3 g/L (dashed lines).</figcaption></figure>

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The previous figure underlines the mass-weighted mean temperature increase when decreasing the fluid flow rate. However, the temperature distribution throughout the fluid must be studied as it is essential to understand the advantages and limitations of such collectors. The variation of the temperature contours at different longitudinal pipe sections is presented in Figure 7 considering a 0.2 g/L graphene dispersion and three flow rates: (a) 75 L/h (Re of 1074), (b) 150 L/h (Re of 2150), and (c) 225 L/h (Re of 3000). At 75 L/h, buoyancy-driven plume advection and recirculation tend to homogenize the temperature distribution within the pipe, a similar trend to that observed by Xu *et al.* [12]. Near the outlet (at 1990 mm), the minimum and maximum temperature increase are 11.1 K and 16.6 K respectively for 75 L/h; resulting in relative differences up to 35 %. At higher flow rates, the influence of buoyancy effects decreases and higher temperature differences are present: minimum and maximum temperature increase of 4.1 K and 9.6 K for 150 L/h; and 2.2 K and 7.4 K for 150 L/h. These gaps result in relative temperature differences up to 60 % and 70 % for 150 L/h and 225 L/h respectively (Figure 7.b and Figure 7.c).
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Simulation results are coherent with the *Ri* number trends, indicating a mixed convection flow regime in all three cases with less dominating natural convection as the fluid flow rate increases: *Ri* = 1.6 at 75 L/h, 0.2 at 150 L/h and 0.1 at 225 L/h. At low *Ri* numbers, the radiative source term spatial distribution slightly influences the fluid temperature distribution, with a hot spot at the tube center and wall vicinity (Figure 7.c). At higher *Ri* numbers, the influence of the radiative source term distribution vanishes as fluid mixing due to natural convection dominates the fluid temperature spatial distribution. Supplementary materials include animated velocity and temperature contour evolutions along the tube length, providing further insight into the fluid dynamics at different *Ri* numbers.

<figure><img src="image_8.png" /><figcaption>**Figure 7:** Temperature contours for the 0.2 g/L case-study at various sections along the tube length for three different flow rates: (a) 75 L/h, (b) 150 L/h, and (c) 225 L/h. The gravity vector (ĝ) is assumed perpendicular to the concentrator aperture plane following the ŷ axis.</figcaption></figure>
## 3.3. Influence of the PTC inclination

Although the results above highlight the significance of buoyancy forces, their influence is largely overlooked in most CFD analyses of PT-DASC found in the literature [7–11,13–15]. **Figure 8** evaluates the impact of including or neglecting gravity on the fluid temperature distribution near the outlet, considering a flow rate of 75 L/h with a 0.2 g/L nanofluid.

When buoyancy forces are not accounted for, high glass temperatures, up to 343 K, develop at the bottom glass surface, leading to a highly non-uniform temperature distribution (Figure 8.a). Such temperature gradients can compromise the mechanical integrity of the glass tube by inducing differential thermal stresses between the upper and lower semi-cylinders. However, this behavior is far from that expected in real working conditions, where buoyancy-driven fluid mixing promotes a homogeneous fluid temperature distribution resulting in low and uniform glass temperatures (around 316 K) (Figure 8.b). These findings underscore the importance of incorporating natural convection effects if working conditions lead to *Ri* numbers close or superior to 1. Neglecting buoyancy forces may significantly impact heat loss modeling, likely leading to overestimated losses.

<figure><img src="image_9.png" /></figure>

<caption>**Figure 8:** Temperature distribution contours at 1990 mm from the pipe inlet for three different cases: (a) gravity not considered ($\vec{g}=0$), (b) gravity in the vertical direction ($\vec{g}_y=-9.8 \text{ m/s}^2$), and (c) gravity with a $36.78^\circ$ inclination angle ($\vec{g}_x=5.87 \text{ m/s}^2$, $\vec{g}_y=-7.85 \text{ m/s}^2$). The represented cases consider a 0.2 g/L particle concentration at 75 L/h fluid flow rate. The mass-weighted average temperature remains identical, as expected (adiabatic walls), in all cases at 314.71 K.</caption>

Furthermore, in real operating conditions, the direction of the gravity vector relative to the concentrated radiative flux varies with the sun position. In **Figure 8.c**, the mean sun elevation between 14h00-14h20 on April at Odeillo (France) has been considered, which result in a $36.78^\circ$ inclination angle for the gravity vector to be considered. CFD simulations indicate minimal variation in the fluid temperature distribution near the outlet when comparing vertical and inclined gravity vectors, with similar surface temperatures. This behavior can be attributed to the important buoyancy-driven recirculation resulting from the high $\phi_{rad}$ values at the center line of the tube, which do not change with the modified gravity angle. At the studied flow rate (75 L/h), natural convection has a major influence on the velocity profiles, $Ri = 1.62$. At higher flow rates (above 150 L/h), buoyancy effects vanish, and gravity vector orientation may play a more significant role in the fluid temperature distribution.
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## 3.4. Comparison with experimental results

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The collector efficiency of the simulated PTC ($\eta_{num}$) was calculated with equations 7 and 8, as applied in the experimental approach. **Figure 9** compares the collector efficiency obtained from the experimental campaigns ($\eta_{exp}$) to the calculated ones from CFD simulations for various flow rates. For both particle concentrations, $\eta_{exp}$ and $\eta_{num}$ remain unchanged across different flow rates, within the measurements uncertainties for the experimental results. A significant decrease is observed only for a concentration of 0.2 g/L and a 120 L/h mass flow rate that was attributed to particle agglomeration due to a strong decrease in the suspension pH. The simulation slightly overestimates the collector efficiency for a 0.2 g/L graphene concentration, while underestimating it for a 0.3 g/L concentration. Still, both cases replicate the experimental results with acceptable accuracy. $\eta_{num}$ values of 66.3 % and 71.3 % were obtained for 0.2 g/L and 0.3 g/L respectively, while the mean $\eta_{exp}$ values were 62.3 ± 0.6 % and 74.3 ± 0.8 % for 0.2 g/L and 0.3 g/L respectively. The reported 3-4 % differences between $\eta_{num}$ and $\eta_{exp}$ results are attributed to environmental working conditions that cannot be modelled in the numerical approach (the reflector and glass cleanliness, parasite reflections, or small heat losses at the inlet and outlet collectors, among others).

<figure><img src="image_10.png" /></figure>

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<caption>**Figure 9:** Collector efficiency ($\eta$) obtained from the experimental campaigns (data points with error bars) and the CFD analysis (solid lines with circle data points). Both particle concentrations are studied: 0.2 g/L (green) and 0.3 g/L (red).</caption>

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The current collector dimensions induce working HTF temperatures close to ambient, thus reducing heat losses significantly. This allows the comparison between the presented numerical simulations and the experimental results described in [25].

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Excluding the optical losses of the components intervening in the ray-tracing simulations previous to the absorbing fluid, the absorber efficiency ($\eta_{num,abs}$) can be analytically estimated assuming that the collector efficiency is equal to the product of the optical efficiency with the absorber efficiency. Using the material properties detailed in the supplementary material, $\eta_{num,abs}$ of 77.9 % and 83.8 % are obtained for 0.2 g/L and 0.3 g/L respectively, considering an intercept factor equal to 1 (a reasonable assumption for a first-approach model of the DASC pilot [10,11]).

29
1 **4. Conclusions and perspectives**

2 This paper presents a comprehensive numerical analysis of the radiative and thermal behavior of a
3 direct absorption solar collector (DASC) using an aqueous graphene dispersion. Ray-tracing simulations
4 provided accurate predictions of the radiative source term obtained for various particle dispersion (0.2
5 and 0.3 g/L). The computed radiative source term was integrated into a CFD model to analyze the
6 thermal and fluid dynamics inside the volumetric receiver. A detailed analysis on the effects of
7 buoyancy-driven flows on the temperature distribution for varying Reynolds number has been
8 presented.

9 Through the ray-tracing simulations, the extinction coefficient associated to each particle
10 concentration was found to have a direct influence on the spatial distribution of the calculated
11 volumetric radiative source term ($\varphi_{rad}$). Simulations reported a not complete absorption in the lower
12 semi-cylinder with a notable portion absorbed in the upper part, best seen at low graphene
13 concentrations. The calculated source term shows the highest values at the tube center where the
14 solar irradiation converges, with high $\varphi_{rad}$ values are also present close to the wall in the lower semi-
15 cylinder. A trade-off is found between complete absorption inside the tube and strong absorption in
16 the wall vicinity. The optimal nanoparticle concentration is found between 0.3 and 0.4 g/L,
17 corresponding to optical depths, $\delta$, in the range 3 – 4 (with reference to the tube diameter). These
18 results highlight the necessity of accurately modelling the radiative source term for comparison with
19 real working devices, particularly for parabolic concentrating thermal collectors.

20 By introducing the obtained volumetric radiative source term to the CFD simulations, a direct effect on
21 the temperature distribution throughout the fluid was observed. The simulations confirmed a linear
22 variation between flow rate and temperature distribution, with higher outlet temperatures at low flow
23 rates due to prolonged residence time in the receiver. Additionally, buoyancy-driven flows play a
24 significant role in the homogeneity of the fluid temperature distribution, especially at low Reynolds
25 numbers, reinforcing the importance of mixed convection in such systems. Moreover, the studied
26 system proved high resilience to changes in concentrator inclination (affecting the gravity vector) at
27 low flow rates, as minimal variation in the temperature distribution contours was found between
28 inclination cases. Nevertheless, the present study underlines the importance of considering natural
29 convection effects. Neglecting gravity can lead to an underestimation of the collector performance.

30 The numerical results presented have been compared to experimental campaigns conducted with a
31 similar PT-DASC setup. The temperature rises along the receiver and the computed total efficiencies
32 align with the experimental results, supporting the assumptions made in the numerical model,
33 including the adiabatic boundary conditions. While these comparisons provide a first-order
34 assessment of the numerical model's accuracy, more advanced experimental techniques such as
35 radiative flux mapping or detailed flow diagnostics would be required to further assess the robustness
36 and predictive capability of the numerical approach.

37 The calculated total collector efficiency increased from 66.3% to 71.3% when the particle
38 concentration raised from 0.2 g/L to 0.3 g/L, which can be attributed to the increase of the nanofluid
39 absorption coefficient with graphene concentration. Increasing the particle concentration to 0.4 g/L is
40 expected to slightly increase the collector efficiency, as the nanofluid's mean absorption would
41 increase from 95% to 98% % in the selected absorber tube. Further increasing the graphene
42 concentration is unlikely to be beneficial, as the modest gains in absorption would come at the expense
43 of higher costs.

44 These performance results highlight the potential of NDASC for a wide range of thermal applications,
45 including heat supply for industrial processes. In particular, the ability to achieve high collector
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efficiencies under laminar flow conditions, unlike conventional solar thermal collectors, demonstrates

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the versatility of this volumetric receiver concept in terms of size. This feature broadens the

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applicability of the system to scenarios where low flow rates are preferred, such as compact solar

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thermal infrastructures. Since the absorption of the nanofluid can be tuned by changing the graphene

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concentration, cogeneration of heat and power can be envisioned using hybrid PVT (Photovoltaic-

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Thermal) collectors with PV cell integrated on the external glass tube. Apart from thermal applications,

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the concept can be used in photochemical processes using nano-size catalysts.

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While the present study provides valuable insights into the radiative and thermal dynamics of direct

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absorption collectors operating at low temperatures, real operating devices are expected to reach fluid

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temperatures above 150 °C, for which several aspects require further investigation. At higher fluid

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temperatures, the CFD model should incorporate radiative and convective heat losses to improve

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efficiency estimations. This would necessitate the exploration of low-emissivity or anti-reflective

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coatings on the glass tube, which may lead to increased collector performance. In addition, future

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research should investigate alternative nanofluid formulations that enhance optical absorption while

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maintaining stability at elevated temperatures. At such temperatures, the hydraulic system must be

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pressurized to prevent base fluid boiling, which may result in different optimal flow conditions that

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should be examined. Finally, high fluid temperatures may pose significant constraints for operational

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solar receivers that will operate under pressure, both in terms of mechanical stresses on the double-

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glass receiver and the stability of the nanofluid. Therefore, real-time temperature control strategies

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may become necessary and should be further explored [30,31].

21
1 Acknowledgements

2 This work was supported by the French "Investments for the future" ("Investissements d'Avenir") and

3 "France 2030" programs managed by the National Agency for Research (ANR) under contract ANR-10-

4 LABX-22-01 (labex SOLSTICE) and ANR-22-PESP-0005 (SHIP4D project), respectively.

5
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