# A Numerical Case Study of Particle Flow and Solar Radiation Transfer in a Compound Parabolic Concentrator (CPC) Photocatalytic Reactor for Hydrogen Production

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## Abstract

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## Highlights

**What are the main findings?**

- A comprehensive simulation model including particle flow and radiation transfer was developed for a CPC photocatalytic reactor.
- The ray tracing method was utilized to determine the radiation reaching the surface of the receiving tube, while the discrete ordinates method (DOM) was also employed to solve the radiative transfer equation (RTE), which shows the complete process of solar energy transfer.

**What is the implication of the main finding?**

- Local volume radiative power absorption (LVRPA) and total radiative power absorption (TRPA) inside the receiving tube was obtained by this study, which is critical data for the photocatalytic reactor.
- Natural convection with intermittent disturbances is demonstrated to be effective operating mode for the CPC photocatalytic reactor.

## Abstract

## 1. Introduction

^{2}[22].

## 2. Results and Discussion

#### 2.1. The Distribution of Radiation on the Surface of Receiving Tube

^{−2}, which is approximately 7.6 times the incident solar radiation intensity (965.5 W·m

^{−2}). The average irradiation intensity on the tube wall is 750.07 W·m

^{−2}, which is 0.78 times the incident solar radiation intensity. This discrepancy arises due to the non-uniform solar radiation distribution on the surface of the tube wall, with certain areas such as the bottom of the receiving tube not receiving effective solar radiation. The total amount of irradiation received on the surface of the receiving tube is 188.51 W, while the total amount of solar radiation incident from the opening surface of the CPC concentrator is 596.46 W. Consequently, the light collection efficiency of the CPC concentrator is calculated to be 31.6%. The lower concentration efficiency in the output is primarily attributed to the use of a low-precision tracking system in the CPC system. In the experimental setup of this study, the adjustment rate is 5° per hour [22]. For more detailed information on this issue, please see the Supplementary Materials.

#### 2.2. Natural Circulation Flow in the Receiving Tube

#### 2.3. The Phase Volume Fraction of the Photocatalysts in the Receiving Tube

^{−4}, which is based on the actual load of photocatalyst [33]. The figure illustrates that over time, the distribution of the particles phase fraction inside the tube gradually changes from a uniform state to a segregation state. On the one hand, the variation in the particle phase fraction at cross-sections L = 0.2 m, 0.6 m, 1.0 m, and 1.4 m of the receiving tube appears to be quite similar over time. It is noticeable that the particle phase fractions at the top of the cross-sections tend to approach zero, while particle phase fractions at the bottom of the cross-sections increase gradually. This phenomenon suggests that the natural circulation flow inside the receiving tube is insufficient to ensure that the particles remain suspended. On the other hand, at the L = 1.8 m cross-section, after a period of time, the particle phase fraction notably decreases with time, almost reaching zero at t = 120 s. As the photocatalyst particles settle inside the tube, the changes in the distribution of the particle phase volume fraction greatly affect the absorption of light radiation via the suspended particle system inside the tube. The following section will focus on the changes in the radiation transfer characteristics within the receiving tube.

#### 2.4. Radiation Transfer Characteristics in the Receiving Tube

_{2}, which is a typical photocatalyst, and its major absorption spectrum are UVA radiation (about 6.3% of the solar spectrum energy) and UVB radiation (about 1.5% of the solar spectrum energy). Therefore, the actual radiation absorbed by the photocatalyst amounts to 12.25 W. Based on this value, the ratio of the TRPA to the total received UV spectrum of the solar radiation could be derived, as depicted by the red diagonal bars in Figure 6 It can be observed that the TRPA value gradually increases over time and reaches its maximum value at t = 120 s within the simulated period. Combining the particle phase volume fraction distribution obtained in Section 2.3 and the discussion in the previous paragraph, we hypothesized that this result is due to the partial settling of particles at 120 s, resulting in a larger particle phase volume fraction at the bottom of the receiving tube. Consequently, the absorption coefficient in this area significantly increases, leading to an increase in the absorption of the solar radiation. This result also indicates that a certain degree of photocatalyst segregation process contributes to the enhancement of solar radiation absorption in the CPC tubular photocatalytic reactor. Therefore, in the application of such photocatalytic reactors, employing an operating mode of natural circulation supplemented with periodic disturbances, such as forced circulation, is rational and efficient.

## 3. Mathematical Model

#### 3.1. Physical Model

#### 3.1.1. Ray Tracing Approach

**k**and the position vector

**q**, as outlined in Equation (3):

^{−2}according to the experiments reported in the literature regarding the average summer solar radiation in Xi’an City [22]. It should be noted that solar radiation intensity is significantly influenced by weather conditions, and, in this case, the analysis was conducted under relatively high solar radiation intensity in the summer, which can fully reflect the hydrogen production performance of the system. In relation to the boundary conditions, 90% of the light is set as specular reflection on the CPC surface, while 10% of the radiant energy is absorbed. For the receiving tube, all light entering the tube wall was set to be absorbed.

#### 3.1.2. Euler–Euler Two-Fluid Flow Model

**K**can be expressed in the following form based on the constitutive equation for Newtonian fluids:

_{m}is typically obtained through empirical formulas or theoretical analysis models. In this study, the Krieger viscosity model [35] is used, as shown in Equation (12).

_{d,max}represents the maximum solid phase volume fraction, which is commonly assumed to have a default value of 0.62.

**F**. Although many forces contribute to it, in two-phase liquid–solid flow, the most important force among the interphase interaction forces is drag force. Therefore, this term is written as the drag force term, as follows:

_{m}**u**

_{slip}denotes the slip velocity, defined as ${u}_{slip}={u}_{c}-{u}_{d}$; and the drag coefficient β can be expressed in the following equation:

**represents the particle diameter; C**

_{p}_{D}denotes the drag coefficient between fluid and particles. Regarding the particle diameter, as discussed in our previous research [24], photocatalyst particles tend to aggregate in the liquid, resulting in a particle size distribution. The average particle size D32 can be used as the hydraulic average particle size for calculation purposes, with a value of 3 μm. Since the volume fraction of the discrete phase in this study is relatively low, the Schiller–Naumann drag model was selected. This model is primarily derived from a rigid spherical single-particle derivation and is suitable for liquid–solid two-phase systems with low solid phase content.

_{p}is the Renolds number of photocatalyst particles, and the expression is

**q**represents the heat conduction term, for which the expression is $q=-k\nabla T$.

_{h}_{air}is the ambient temperature, with its value being 303 K in this study according to the average temperature in Xi’an in the summer.

_{src}represents the heating source term in the fluid energy equation, α

_{ht}denotes the medium’s optical–thermal efficiency in the receiving tube. Considering that the infrared light in solar radiation is mainly absorbed by water, this coefficient is set to 0.173. q

_{j}represents the energy carried by the j-th ray.

#### 3.1.3. Discrete Ordinates Method

**r**,

**s**) is the total energy scattered from any direction to the s-direction.

_{2}photocatalyst particles are listed in Table 2, which include two wavelength bands, namely UVA (320–400 nm) and UVB (275–320 nm), according to the literature [32]. It should be noted that these coefficients are all dimensionless coefficients, meaning they are multiplied by the concentration of particles to obtain the optical property parameters required in Equation (18). Since the absorption of TiO

_{2}photocatalyst primarily occurs in the ultraviolet region, these two wavelength bands cover all portions of solar radiation that TiO

_{2}can absorb.

#### 3.1.4. Simulation Method and Process

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Symbols | |

c | Speed of light |

c_{cat} | Concentration of the photocatalyst |

c_{p} | Specific heat capacity |

d_{R} | Diameter of the receiving tube |

Gap | Gap between CPC and receiving tube |

k | Heat conductivity coefficient |

l_{R} | Length of the receiving tube |

n | Refractive index |

p | Pressure |

r_{B} | Radius of the base circle of the CPC |

C_{D} | Drag coefficient |

I | Radiation density |

T | Temperature |

Q | Quantity of heat |

Abbreviations | |

CPC | Compound parabolic concentrator |

CFD | Computational fluid dynamics |

DOM | Discrete ordinates method |

LVRPA | Local volume radiative power absorption, W·m^{−3} |

RTE | Radiative transfer equation |

TRPA | Total radiative power absorption, W |

UV | Ultraviolet |

Vectors and tensors | |

F | Volume forces, N·m^{−3} |

Fm | Interphase interaction force, N·m^{−3} |

g | Gravitational acceleration vector |

I | Unit tensor |

k | Wave vector |

q | Position vector of the ray |

r | Position vector of the space |

s | Spatial angle vector |

u | Velocity vector |

K | Viscous stress tensor, Pa |

S | Strain rate tensor |

Greek letters | |

α_{ht} | Efficiency of solar-to-thermal conversion |

α_{p} | Thermal diffusion coefficient |

ε | Emission coefficient |

θ_{a} | Maximum acceptance half angle, ° |

θ_{t} | Truncation angle, ° |

κ | Absorption coefficient in aqueous medium, m^{−1} |

ρ | Density, kg·m^{−3} |

σ | Scattering coefficient, m^{−1} |

τ | Viscous stress tensor |

φ | Azimuth angle in spherical coordinates |

Ψ | Polar angle in spherical coordinates |

φ | Phase volume fraction |

φ_{S} | Scattering phase function |

Ω | $\mathrm{Solid}\mathrm{angle};\mathrm{its}\mathrm{differential}\mathrm{form}\mathrm{is}d\mathsf{\Omega}=\mathrm{sin}\psi d\psi d\phi $, sr |

Superscripts and subscripts | |

c | Continuous phase |

d | Discrete phase |

m | Mixed phase |

p | Particle |

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**Figure 1.**(

**a**) Top view, (

**b**) right view, and (

**c**) left view of the contour of the radiation distribution on the surface of the receiving tube, respectively.

**Figure 2.**Contour of temperature of L = 0.2 m, 0.6 m, 1.0 m, 1.4 m, and 1.8 m cross-sections in receiving tube at (

**a**) t = 30 s, (

**b**) t = 60 s, (

**c**) t = 90 s, and (

**d**) t = 120 s.

**Figure 3.**Contours, streamlines, and vectors of the natural convective flow of L = 0.2 m, 0.6 m, 1.0 m, 1.4 m, and 1.8 m cross-sections in receiving tube at (

**a**) t = 30 s, (

**b**) t = 60 s, (

**c**) t = 90 s, and (

**d**) t = 120 s.

**Figure 4.**Contours of the phase volume fraction of photocatalyst at L = 0.2 m, 0.6 m, 1.0 m, 1.4 m, and 1.8 m cross-sections in receiving tube at (

**a**) t = 30 s, (

**b**) t = 60 s, (

**c**) t = 90 s, and (

**d**) t = 120 s.

**Figure 5.**Contours of (

**a**) absorption coefficient of the UVA spectrum and (

**b**) absorption coefficient of the UVB spectrum. (

**c**) LVRPA in the receiving tube of L = 1.0 m cross-section in receiving tube at t = 30 s, t = 60 s, t = 90 s, and t = 120 s, respectively.

Parameters | Numerical Value |
---|---|

d_{R}/mm | 30 |

l_{R}/m | 2 |

r_{B}/mm | 15 |

Gap/mm | 25 |

θa/° | 5 |

θt/° | 60 |

**Table 2.**The absorption coefficient and scattering coefficient of titanium dioxide [32].

Coefficient | Numerical Value/m^{2}·kg^{−1} |
---|---|

κ^{*}_{UVA} | 189.9 |

σ^{*}_{UVA} | 1175.1 |

κ^{*}_{UVB} | 508.5 |

σ^{*}_{UVB} | 1016.1 |

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## Share and Cite

**MDPI and ACS Style**

Geng, J.; Wei, Q.; Luo, B.; Zong, S.; Ma, L.; Luo, Y.; Zhou, C.; Deng, T.
A Numerical Case Study of Particle Flow and Solar Radiation Transfer in a Compound Parabolic Concentrator (CPC) Photocatalytic Reactor for Hydrogen Production. *Catalysts* **2024**, *14*, 237.
https://doi.org/10.3390/catal14040237

**AMA Style**

Geng J, Wei Q, Luo B, Zong S, Ma L, Luo Y, Zhou C, Deng T.
A Numerical Case Study of Particle Flow and Solar Radiation Transfer in a Compound Parabolic Concentrator (CPC) Photocatalytic Reactor for Hydrogen Production. *Catalysts*. 2024; 14(4):237.
https://doi.org/10.3390/catal14040237

**Chicago/Turabian Style**

Geng, Jiafeng, Qingyu Wei, Bing Luo, Shichao Zong, Lijing Ma, Yu Luo, Chunyu Zhou, and Tongkun Deng.
2024. "A Numerical Case Study of Particle Flow and Solar Radiation Transfer in a Compound Parabolic Concentrator (CPC) Photocatalytic Reactor for Hydrogen Production" *Catalysts* 14, no. 4: 237.
https://doi.org/10.3390/catal14040237