# Thermal analysis of a solar-assisted pyrolysis drop tube reactor with radiation heat transfer

Amir Hossein Bashiri, Shervin Karimkashi, Vignesvar Krish Subramani, Sylvain Rodat, Stéphane Abanades, Mika Järvinen, Ville Vuorinen

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Amir Hossein Bashiri, Shervin Karimkashi, Vignesvar Krish Subramani, Sylvain Rodat, Stéphane Abanades, et al. Thermal analysis of a solar-assisted pyrolysis drop tube reactor with radiation heat transfer. Applied Thermal Engineering, 2026, 290, pp.130033. 10.1016/j.applthermaleng.2026.130033. hal-05512248

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Research Paper

Thermal analysis of a solar-assisted pyrolysis drop tube reactor with radiation heat transfer

Amir Hossein Bashiri a,*, Shervin Karimkashi a,†, Vignesvar Krish Subramani b,†, Sylvain Rodat b,†, Stéphane Abanades b,†, Mika Järvinen a, Ville Vuorinen a,†

ARTICLE INFO

Dataset link: https://github.com/Aalto-CFD/Pyrolysis-Reactor-Simulation

Keywords:

CFD

Drop tube reactor

Radiation heat transfer

Conjugate heat transfer

OpenFOAM

A B S T R A C T

Fast pyrolysis of biomass is recognized as a promising method for bio-oil production which can be further utilized in the production of sustainable fuels. Numerical simulations are carried out in order to verify the discrete ordinate method (DOM) implementation in OpenFOAM open-source software. 1D and 3D test problems are investigated including conjugate and radiative heat transfer in addition to a 2D test case. The main application of interest is a 3D solar-assisted drop tube reactor, which is studied from the viewpoint of heat distribution within the reactor. The main results of the research are as follows: (1) 1D modeling indicates that the heat fluxes are correctly implemented for transient and steady state configurations. (2) 2D rectangular enclosure study indicate correct heat flux implementation in OpenFOAM for the *viewFactor* and *fvDOM* methods. (3) 3D reactor studies in OpenFOAM and STAR-CCM+ indicate that the heat distributions are highly similar within 2% deviation in peak temperature values. The final 3D results are noted to be grid independent and independent on the angular discretization number when $N_φ ≥ 16$. As a conclusion, the validated/verified numerical approach offers a general open-source framework for coupled CHT-radiation problems with applicability beyond the studied drop tube reactor.

1. Introduction

Radiative heat transfer is a fundamental mode of energy transport that plays a crucial role in our daily life and in various engineering applications, ranging from heat exchangers to solar stills [1,2]. Radiation is particularly important in high temperature environments, where it often prevails over conduction and convection such as combustion systems [3] and furnaces [4]. Additionally, radiation is a key factor in climate and ventilation [5], solar receivers [6], and fire safety [7], where accurate modeling is essential to ensure system reliability and performance. Neglecting radiation can lead to significant inaccuracies in thermal predictions [8,9], as the main challenge lies in correctly solving the radiative transfer equation (RTE) alongside the fluid flow.

This study is part of the Circular Fuels project [10], which aims to convert waste biomass into sustainable aviation fuels (SAF) using concentrated solar heat, solar electricity, and thermochemical conversion. In this system, widely available feedstock, such as waste wood and agricultural residues, is processed through a solar-assisted fast pyrolysis reactor to produce bio-oil.

While biomass valorization is well-established, for example in the catalytic cracking of biomass tar for hydrogen production [11], the

direct integration of concentrated solar heat with fast pyrolysis presents a novel challenge. Harnessing solar energy instead of burning conventional fuels for heat, enhances efficiency, lowers emissions [12], and reduces operational costs [13,14]. High-powered dish receivers or solar concentrators quickly raise the reactor to the required temperature and allow control of the heating rate [12]. Additionally, solar pyrolysis generates valuable by-products like biochar. Similar to findings in desalination studies using wood charcoal fillers to increase the desalination efficiency [2], the effective utilization of these by-products can improve economic feasibility by providing additional revenue streams [15]. The overall efficiency of this method depends not only on the type of feedstock and reaction dynamics but also on the design of the solar thermal system and reactor configuration.

Numerical and experimental modeling of pyrolysis and related thermal applications have been studied significantly in recent years. Computational Fluid Dynamics (CFD) simulations, combined with experimental investigations, have provided deeper insights into the role of radiation in thermal conversion processes such as biomass pyrolysis,

<footnote>a Department of Energy and Mechanical Engineering, Aalto University, Espoo, 02150, Finland</footnote>

<footnote>b CNRS, Materials and Solar Energy Laboratory (PROMES-CNRS), 7 Rue du Four Solaire, Font Romeu, 66120, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: amir.bashiri@aalto.fi (A.H. Bashiri).</footnote>
<table><caption>**Table 1**
Summary of studies on numerical simulation of the pyrolysis reactor.</caption><thead><tr><td>Focus</td><td>Considerations</td><td>Methodology</td><td>Reference</td></tr></thead><tbody><tr><td>Solar-thermal reactor performance for biomass pyrolysis</td><td>- Energy efficiency<br/>- Reactor optimization<br/>- Radiation neglected<br/>- ANSYS FLUENT software</td><td>- Two-fluid approach<br/>- Evaporation model and species transport equations</td><td>[24]</td></tr><tr><td>Fast pyrolysis of biomass pellets</td><td>- Temperature-dependent thermal properties<br/>- Radiation neglected<br/>- COMSOL software</td><td>- Single particle model<br/>- Three-component, five-reaction stepwise kinetic scheme</td><td>[25]</td></tr><tr><td>Solar-driven pyrolysis reactor using concentrated radiation</td><td>- Reactor structure<br/>- Influence of gas velocity and lamp power<br/>- Effect of porous media<br/>- Chemical reaction neglected<br/>- COMSOL software</td><td>- Radiative heat transfer for different wavelengths</td><td>[26]</td></tr><tr><td>Coupling a fluidized bed reactor with a solar thermal receiver</td><td>- Design optimization<br/>- Annual operational stability<br/>- Radiation neglected<br/>- ANSYS FLUENT software</td><td>- Eulerian-Eulerian model<br/>- Detailed kinetic model</td><td>[27]</td></tr><tr><td>Biomass thermal conversion in pyrolysis kiln</td><td>- Heat distribution<br/>- Radiation neglected<br/>- ANSYS FLUENT software</td><td>- Eulerian-Eulerian model<br/>- Chemical reactions</td><td>[28]</td></tr><tr><td>Resolving particle-scale heat transfer in fixed-bed reactors</td><td>- Temperature distribution<br/>- Particle size changes<br/>- Radiation neglected<br/>- MFiX-DEM software</td><td>- CFD-DEM coupling with chemical reactions</td><td>[29]</td></tr><tr><td>Flow and heat transfer of biomass and heat carrier in a V-shaped down-tube reactor</td><td>- Mass ratio of carrier to biomass<br/>- Temperature distribution<br/>- Radiation neglected<br/>- ANSYS FLUENT software</td><td>- CFD-DEM modeling<br/>- Experimental study</td><td>[30]</td></tr><tr><td>Fast pyrolysis in a circulating fluidized bed</td><td>- Effect of specularity and restitution coefficients<br/>- ANSYS FLUENT software</td><td>- 3D Eulerian-Eulerian model<br/>- Heat Transfer + Hydrodynamics</td><td>[31]</td></tr></tbody></table>

gasification, and combustion. These studies have contributed to optimizing reactor designs, improving heat transfer efficiency, and refining numerical models for better predictive accuracy.

For a more comprehensive review of recent numerical advances, see [16–19], and for reviewing different types of the solar pyrolysis systems, see [12,20–23]. Table 1 summarizes selected recent studies on numerical simulations of pyrolysis reactors, highlighting their objectives, key considerations, and numerical methodologies. As summarized in Table 1, many studies simplify or neglect the role of radiative heat transfer. In contrast, solar-assisted reactor studies that account for radiation often do not consider conjugate heat transfer.

In particular, Bashir et al. [24] investigated biomass fast pyrolysis in a solar-thermal reactor heated by a parabolic trough concentrator using an Eulerian-Eulerian CFD model in Ansys Fluent. Their model accounted for multiphase flow, heat transfer, drying, and pyrolysis reactions. They applied a constant heat flux at the reactor walls to model the effects of the solar radiation. Optimal bio-oil production was achieved at 400°C, with gas residence times under 2s and minimal contact between pyrolysis gas and char.

Additionally, Soria et al. [32] explored solar fast pyrolysis of beech wood pellets using a parabolic trough concentrator to harness solar energy for biomass conversion. A 2D unsteady CFD model was developed to simulate the process, incorporating key conservation equations and pyrolysis kinetics. Temperature profile has been used at the walls to represent incoming radiation while neglecting radiation inside the reactor. The model, validated against experiments, showed that temperature and heating rate strongly influence product distribution. Higher temperatures significantly boosted gas lower heating value,

while faster heating rates further enhanced the gas yield. At extreme conditions, 50 °C s⁻¹, the highest gas energy content was achieved.

Conjugate heat transfer (CHT) analysis has been extensively studied and plays a crucial role in accurately predicting thermal interactions between solid and fluid domains, making it essential for various engineering applications [33–36].

Jin and et al. [37] investigated this phenomena using a new developed solver for OpenFOAM to simultaneously solve the combustion, radiation and conjugate heat transfer equations. Their objective was to find the temperature values at the walls, which provides initial temperature distribution for the engine block surfaces that are crucial for designing liners and estimating their lifespan. Moreover, Errera et al. [38] developed a predictive model for steady state CHT problems with radiative boundary condition. The destabilizing influence of radiation was demonstrated, and the approach was validated through a test case involving fluid and solid coupling.

Cintolesi et al. [39] conducted a study of radiative heat transfer with CHT using P1 radiation model in OpenFOAM. The model was validated against some benchmark cases, showing good accuracy. They also analyzed forced convection, natural convection, and radiation coupled with conduction, showing that the surface radiation affects the overall temperature behavior by increasing the fluid-solid interface temperature. Fureby [40] applied large-eddy simulation in OpenFOAM to study the effects of the bluff-body in stabilizing a flame focusing on the effects of the thermal radiation and CHT. In their study, P1 andfvDOM radiation models have been used showing that considering radiation and CHT improves the results agreement with experimental results for velocity, temperature and CO. According to the results, fvDOM provides slightly better accuracy than P1.
Based on our literature survey, there is no prior research specifically focusing on the numerical study of fluid flow, CHT, and radiation heat transfer in a 3D solar-assisted drop tube pyrolysis reactor using OpenFOAM. Only a limited number of studies have incorporated both radiation and CHT, and they primarily focus on other applications than pyrolysis reactors. Most earlier studies have ignored radiative heat transfer or represented incoming radiation through prescribed temperature boundary conditions, rather than solving the radiation directly. This highlights a significant general research gap in the modeling of coupled CHT-radiation problems in OpenFOAM. Furthermore, a particular research gap is the accurate prediction of heat distribution within solar-assisted reactors.

Considering the aforementioned research gaps, the primary objectives of this numerical study are as follows:

1. Develop and validate a robust numerical framework by comparing the open-source CFD package OpenFOAM with the commercial software STAR-CCM+ to numerically predict reactor performance.

2. Analyze the coupled effects of radiative, conductive, and convective heat transfer and fluid flow using the discrete ordinates method (DOM) for accurate radiation modeling.

3. Validate thefvDOM radiation model in OpenFOAM using analytical benchmarks across multiple test cases.

4. Establish the numerical framework for future research into feedstock particle characteristics and pyrolysis processes to improve reactor performance.

The paper is organized as follows. Section 2 describes the methodology, including the governing equations for fluid flow, heat transfer, and the radiative transfer equation, as well as the numerical methods, such as DOM radiation model andfvDOM implementation in OpenFOAM. The section ends with details on the validation cases in 1D and 2D, along with a full explanation of the 3D reactor geometry. Section 3 presents the results and discussions, starting with the validation outcomes for the 1D and 2D test cases, followed by analysis of the results for the drop tube geometry. Finally, Section 4 provides conclusions for this study.

## 2. Methodology

This section first introduces the governing equations necessary for simulating fluid flow and heat transfer. The mass conservation and Navier-Stokes equations for a single-phase flow are employed to solve for fluid motion, while the energy equation is used to compute the temperature distribution. Since this study consists of both solid and fluid regions, conjugate heat transfer principles are applied to capture the thermal interactions between these domains. Furthermore, to account for radiative effects, radiation heat transfer is incorporated as a source term in the energy equation. Next, numerical methods used in this study for simulating radiative heat transfer are mentioned. Finally, the pyrolysis drop tube reactor is introduced, and a set of test cases for the validation and better understanding of radiative heat transfer in such a system are investigated.

### 2.1. Governing equations

#### 2.1.1. Fluid flow and heat transfer equations

The continuity equation for compressible flow is given by Eq. (1) and the momentum equation is also mentioned in Eq. (2) [1]. In these equations $\rho, \vec{V}$, and $\tau$ represents the density, velocity vector, and shear stress tensor, respectively.

$$ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{V}) = 0 \qquad (1) $$

$$ \frac{\partial(\rho\vec{V})}{\partial t} + \vec{V} \cdot \nabla(\rho\vec{V}) = -\nabla p + \nabla \cdot \tau + \rho\vec{g} \qquad (2) $$

In conjugate heat transfer problems, the energy conservation equations for both fluid and solid domains are essential to accurately model the heat transfer process. The energy conservation equation accounts for the transport of energy due to conduction, convection, and, radiation for fluid and solid regions are given by Eqs. (3) and (5), respectively [41].

In fluid region:

$$ \frac{\partial(\rho_f c_{p,f} T_f)}{\partial t} + \vec{V} \cdot (\rho_f c_{p,f} \nabla T_f) = \nabla \cdot (k_f \nabla T_f) + \dot{q}_f - \nabla \cdot q_r \qquad (3) $$

$$ \nabla \cdot q_r = \kappa_f (4\sigma T_f^4 - G) \qquad (4) $$

In Eq. (3), $\rho_f$ is the fluid density, $c_{p,f}$ is the specific heat capacity, $T_f$ is the fluid temperature, $t$ is time, $\vec{V}$ is the velocity vector, $k_f$ is the thermal conductivity of fluid, $\nabla \cdot q_r$ represents volumetric radiation heat flux, and $\dot{q}_f$ is the volumetric heat sources. In Eq. (4), $\kappa_f$ is the absorption coefficient, and G denotes the total incident radiation for the medium. Further details can be found in [41].

In the solid region,

$$ \rho_s c_{p,s} \frac{\partial T_s}{\partial t} = \nabla \cdot (k_s \nabla T_s) + \dot{q}_s, \qquad (5) $$

where $\rho_s$ is the solid density, $c_{p,s}$ is the specific heat capacity of solid, $T_s$ is the temperature of solid, $k_s$ is the thermal conductivity of solid, and $\dot{q}_s$ represents the volumetric heat source.

Continuity of temperature, Eq. (6), and heat flux, Eq. (7), is maintained at the fluid-solid interface to ensure proper coupling between the two regions [42].

$$ T_f = T_s \qquad (6) $$

$$ k_f \nabla T_f \cdot \vec{n} = k_s \nabla T_s \cdot \vec{n}, \qquad (7) $$

where $\vec{n}$ is the unit vector normal to the interface. These equations collectively enable the analysis of heat transfer between fluids and solids, which is crucial in many engineering applications, such as heat exchangers, electronic cooling, and thermal management systems.

The turbulentTemperatureRadCoupledMixed boundary condition ensures proper coupling by linking temperature values at the interface while considering radiation heat transfer. Also, the turbulentTemperatureCoupledBaffleMixed boundary condition is based on Eqs. (6) and (7) without considering the radiative heat transfer at the interface. Both of these boundary conditions have been already implemented in OpenFOAM.

#### 2.1.2. Radiative Transfer Equation (RTE)

Radiative heat transfer governs the changes in radiation intensity within a participating medium due to absorption, emission, and scattering. For a non-gray, homogeneous medium with a constant refractive index, the variation of thermal radiation intensity in the quasi-steady form along the direction of the unit vector $\vec{s}$ is expressed as Eq. (8) [41].

$$ \dot{s} \cdot \nabla I_{\lambda}(\vec{r}, \hat{s}) = \frac{dI_{\lambda}}{d\hat{s}} = \kappa_{\lambda}(\vec{r}) I_{b,\lambda}(\vec{r}) - (\kappa_{\lambda}(\vec{r}) + \sigma_{s,\lambda}(\vec{r})) I_{\lambda}(\vec{r}, \hat{s}) \\ + \frac{\sigma_{s,\lambda}}{4\pi} \int_{4\pi} I_{\lambda}(\vec{r}, \hat{s}') \phi(\vec{r}, \hat{s}, \hat{s}') d\Omega, \qquad (8) $$

where $\kappa_{\lambda}$ is the spectral absorption coefficient. $\sigma_{s,\lambda}$ is the spectral scattering coefficient, $\phi$ is the phase function, and $d\Omega$ represents the differential solid angle. The first term on the right-hand side of Eq. (8), $\kappa_{\lambda} I_{b,\lambda}$, denotes intensity gain due to emission, where $I_{b,\lambda}$ is the black-body intensity at wavelength $\lambda$, defined by the local temperature. The second term, $-(\kappa_{\lambda} + \sigma_{s,\lambda})I_{\lambda}$, accounts for the intensity loss due to absorption and out-scattering. The third term represents intensity gain from in-scattering contributions.

In this work, we neglect in-scattering and define the extinction coefficient as $\beta_{\lambda} = \kappa_{\lambda} + \sigma_{s,\lambda}$, which simplifies Eqs. (8) as (9) [43].

$$ \frac{dI_{\lambda}}{ds} = \kappa_{\lambda} I_{b,\lambda} - \beta_{\lambda} I_{\lambda}. \qquad (9) $$
We further simplify Eq. (9) by eliminating wavelength dependency for radiation through a medium and consider that the medium is gray, which eventually leads to Eq. (10),

$$ \frac{dI}{ds} = \kappa I_b - \beta I. \qquad (10) $$

## 2.2. Numerical methods

### 2.2.1. Radiation models

The Radiative Transfer Equation, Eq. (8), is an integro-differential equation dependent on the spatial and angular variables, and solving it analytically is challenging. To address this challenge, numerical methods such as the view factor method [44], the spherical harmonics (P-N) method [45], the Discrete Ordinates Method (DOM), and the Photon Monte Carlo method (PMC) [46] are typically applied. In this work, we mainly use the DOM, while the view factor method is used only for the validation purposes.

The view factor method is used to calculate radiative heat exchange between surfaces in a domain based on geometric relationships, quantified by the view factor coefficients. This method does not account for scattering, absorption, or emission within a medium. After determining the view factor coefficients, it solves the energy balance for each surface to obtain the radiative heat flux, making it suitable for simple geometries.

To address these limitations, the DOM is used. DOM discretizes the angular domain into a finite set of directions called ordinates, enabling systematic computation of the Radiative Transfer Equation (RTE). This approach effectively models absorption, emission, and scattering phenomena, including anisotropic scattering, making it particularly suitable for semi-transparent media and cases involving both diffuse and specular reflections. DOM provides higher accuracy in cases with sharp radiation intensity variations and is applicable to both optically thin and thick media. Additionally, DOM is well-suited for radiation modeling in multi-phase flows, making it a valuable method for simulating heat transfer between gas and particulates, such as in combustion systems and fluidized beds. However, its accuracy depends on the number of discrete directions used, and higher resolutions increase the computational cost. Despite its accuracy, DOM has certain drawbacks, such as high computational costs and potential numerical diffusion due to angular discretization.

In the DOM, the angular dependence of intensity $I(\vec{r}, \vec{\Omega})$ is replaced by discrete intensities $I(\vec{r}, \vec{\Omega}_n)$ along specific directions $\vec{\Omega}_n$. Eq. (11) represents the angular discretized form of Eq. (10) [41],

$$ \vec{\Omega}_n \cdot \nabla I(\vec{r}, \vec{\Omega}_n) + \beta I(\vec{r}, \vec{\Omega}_n) = \kappa I_b(\vec{r}). \qquad (11) $$

### 2.2.2. Numerical solver

OpenFOAM 2312 has been employed, an open-source software widely used for engineering simulations, specifically utilizing the cht-MultiRegionFoam solver. This solver models unsteady, buoyant, and turbulent fluid flow while enabling conjugate heat transfer between fluid and solid domains. It also supports different radiation models to account for thermal radiation in the energy balance equation. OpenFOAM provides various radiation models, including P1, fvDOM, and solarLoad, making it a versatile tool for radiative heat transfer simulations. For cross-verification, the problem is also simulated in STAR-CCM+, and the results are compared against those from OpenFOAM.

It should be noted that a widely adopted extension of DOM is the Finite Volume Discrete Ordinates Method (fvDOM), which combines spatial discretization using the Finite Volume Method (FVM) with an angular discretization of the DOM. In the fvDOM solver, the computational domain is divided into control volumes, and the angular domain is discretized into a finite number of directions. The polar angle domain of $\pi$ and the azimuthal angle domain of $2\pi$ are divided into a specified number of equally spaced angles. The RTE is solved for each direction

and control volume, incorporating radiative properties of the medium, such as absorption and emission coefficients as well as the boundary conditions, like emissivity and absorptivity.

This method is appropriate for calculating the absorption and emission coefficients as well as the emission intensity of a medium with a low particle volume fraction [47]. A key limitation of the fvDOM method in OpenFOAM is its inability to model scattering effects in participating media that contain particle suspensions [48]. This restricts its applicability in scenarios where scattering plays a significant role in radiative heat transfer.

In the fvDOM solver, Eq. (10) is discretized in space and solid angle and it is shown as Eq. (12) for each Ray [49],

$$ \nabla \cdot (J_i I_{\lambda}) = -\kappa I_{\lambda} \Omega_i + \Omega_i \left( \kappa_g I_b + \frac{E}{4\pi} \right). \qquad (12) $$

The radiative intensity of the ith ray is denoted by $J_i$. In Eq. (12), the first term on the right-hand side represents absorption, while the second term accounts for emission, which consists of contributions from both the continuous and dispersed phases. Absorption in the continuous phase is given by $\kappa_g$, expressed as $(\kappa - \sigma_s)$, whereas emission in the dispersed phase is denoted by $E$.

Several advancements have improved radiative heat transfer modeling in OpenFOAM. Wang et al. [50] enhanced the fvDOM solver with improved discretization schemes, increasing accuracy and computational efficiency for large scale simulations. Sun et al. [51] developed a radiation solver integrating DOM and the P1 method with the Weighted Sum of Gray Gases (WSGG) model for non-gray gas properties, achieving error reductions below 5% in combustion simulations. Weng [50] implemented the full spectrum correlated k-distribution (FSCK) approach in OpenFOAM, improving radiation calculations in non-homogeneous, non-gray media. Sikic et al. [52] applied large eddy simulations (LES) to study pool fire phenomena using FireFOAM, incorporating radiation modeling for non-gray media. To address OpenFOAM's limitation in simulating radiation within solids, Fan et al. [53] introduced mcrtFOAM, a Monte Carlo ray tracing solver with a mesh-agglomeration technique to reduce computational time and memory usage. Kumar et al. [54] employed the chtMultiRegionFoam solver alongside fvDOM to study radiation and conjugate heat transfer in CO₂ flow over a forward-facing step. Their results emphasized the significant role of radiation in thermal analysis, with variations in absorption and emission having a more pronounced impact than scattering.

## 2.3. Computational cases

In this section, the pyrolysis drop tube reactor is described first, and then, for the sake of validating the OpenFOAM software, we solve two 1D case studies and one two-dimensional case study. Thereafter, we compare their results with available analytical solutions. These validation case studies are chosen in a way that they can represent simplified problems relevant to the main heat transfer mechanism in the studied reactor.

The proposed solar-assisted pyrolysis reactor is optimized by Subramani et al. [55], and it comprises of a cavity, insulation, and drop tube, as shown in Fig. 1(a). Biomass is fed vertically into the reactor, falling by gravity and carried by an inert gas like argon. Small biomass particles are suspended in the gas flow, improving heat and mass transfer as well as mixing efficiency [56]. To ensure efficient pyrolysis, the reactor should ideally maintain a uniform heating and a minimum temperature of 773 K along the tube.

The 3D pyrolysis reactor presents a complex geometry, making direct numerical simulations challenging. To ensure accurate modeling, we opted to decompose the 3D geometry into simpler 1D and 2D cases, which can be solved using analytical methods. This approach allows us to systematically verify our settings and boundary conditions used in the chtMultiRegionFoam solver by comparing numerical results with analytical solutions. By validating these simplified cases, we ensure the sanity of our implementation before extending the simulation to the full 3D reactor geometry.
<figure><img src="image_3.png" /><figcaption>Fig. 1. (a) Front section isometric view, (b) top-view, and (c) I-section view, of the pyrolysis drop tube reactor.</figcaption></figure>

### 2.3.1. Case 1: A 1D cavity with a solid layer

The first case study shown in Fig. 2(a) presents a one-dimensional segment of the pyrolysis reactor. It is comprised of the cavity, which is filled by air, and the drop tube wall thickness, which is made of Mullite and called solid layer, and the aperture, which is the inlet for radiative heat flux. This domain is named as “Case 1” annotated by region “i” in Fig. 1(c).

In this case, radiation heat flux enters from the right side of the cavity domain, serving as the dominant mode of heat transfer. Radiative heat is absorbed by the surface of the solid layer, shown as ‘Interface’ in Fig. 2(a), leading to an increase of the temperature inside the solid. The absorbed heat is then transferred through the solid layer via conduction, eventually reaching the left surface of the solid layer. At the left surface of the solid layer, the absorbed energy is emitted out via radiation.

To solve this simple case, appropriate boundary conditions should be assigned to each surface. The velocity boundary conditions at the walls and at the interface are defined as no-slip, enforcing zero velocity on both solid walls and the interface. A constant pressure boundary condition is imposed to ensure consistency with the velocity field and the governing equations. Table 2 summarizes the boundary conditions for temperature and radiation intensity. The cavity and the solid temperature are initially set to 300 K.

The left and right surfaces are considered as opaque and diffuse which is incorporated in the `greyDiffusiveRadiation` boundary condition for directional incident radiation intensity (`IDefault`). The `MarshakRadiation` is a mixed type boundary condition which calculates the incident radiation on the walls based on temperature and emissivity of the wall. At the fluid-solid interface, the `turbulent-TemperatureRadCoupledMixed` boundary condition is applied for coupling the temperature values.

<caption>Table 2</caption>

OpenFOAM boundary conditions applied in Case 1.

<table><thead><tr><th>Surface</th><th>T</th><th>IDefault</th><th>G</th></tr></thead><tbody><tr><td>Right</td><td>fixedValue<br>1000 K</td><td>greyDiffusiveRadiation</td><td>MarshakRadiation</td></tr><tr><td>Left</td><td>externalRadiation</td><td>greyDiffusiveRadiation</td><td>MarshakRadiation</td></tr><tr><td>Interface</td><td>turbulent<br>Temperature<br>RadCoupledMixed</td><td>greyDiffusiveRadiation</td><td>MarshakRadiation</td></tr></tbody></table>

For the analytical solution, the one-dimensional heat transfer equation is solved by applying the suitable radiation boundary condition for the left and right surfaces of the solid. The length of the solid domain is considered to be equal to the thickness of the drop tube in the target solar-assisted pyrolysis reactor, which is 3.5 mm. For a solid layer, which is both homogeneous and isotropic, the general form of the 1D transient conduction equation is given by Eq. (13), where $\alpha_s$ represents thermal diffusivity of the solid. The comparison between the OpenFOAM results and the analytical solutions are provided in Section 3.

$$ \frac{\partial T}{\partial t} = \alpha_s \frac{\partial^2 T}{\partial x^2} \qquad (13) $$

### 2.3.2. Case 2: 1D drop tube with the solid material

Case 2 is an extension of Case 1, focusing only on a 1D section of the drop tube, labeled as region “ii” in Fig. 1(c). In Case 1, conduction heat flux propagates throughout the solid, reaching its left surface, where thermal radiation is emitted out. The system eventually reaches a steady state condition, which serves as the temperature boundary condition for Case 2. The detailed schematic of case 2 is illustrated in Fig. 2(b).
<figure><img src="image_4.png" /><figcaption>Fig. 2. Schematics of 1D cases simulated in OpenFOAM, (a) cavity with Mullite thickness (case 1), and (b) drop tube with solid material (case 2).</figcaption></figure>

<table><caption>**Table 3**

Thermophysical properties of the solid material in case 2.</caption><thead><tr><td>Name</td><td>Material</td><td>ρ (kg m<sup>-3</sup>)</td><td>Cp (J kg<sup>-1</sup> K<sup>-1</sup>)</td><td>k (W m<sup>-1</sup> K<sup>-1</sup>)</td></tr></thead><tbody><tr><td>Solid material</td><td>Dry wood</td><td>600</td><td>1500</td><td>0.2</td></tr></tbody></table>

Case 2 is relevant to the heat transfer from the inner surface of the drop tube to the biomass particles moving within the tube. For simplicity, we consider a solid material with specified thickness. Thermal radiation is emitted from both right and left surfaces of the tube and subsequently, absorbed by the solid material. As a remark, pyrolysis is not considered in this work.

In the fluid region, only radiative heat transfer is considered and treated as a non-participating medium, meaning it does not absorb, emit, or scatter radiation and it is excluded from the simulation domain in this study. Additionally, all surfaces are assumed to be black and diffuse. In the solid region, the absorbed heat is transferred solely by conduction. The pressure and velocity boundary conditions remain the same as those applied in Case 1 for the interface and surfaces.

The left and right boundaries of the one-dimensional section are assigned an ambient temperature of 834 K using the externalWall-HeatFluxTemperature boundary condition in OpenFOAM.

Since the main purpose of the current reactor is wood pyrolysis, in this case study, thermal properties of dry wood are considered for the solid material, see Table 3. The comparison between the OpenFOAM results and the analytical solutions are provided in Section 3.

### 2.3.3. Case 3: 2D - Rectangular enclosure

This case evaluates a two-dimensional rectangular domain filled with a non-participating fluid, which is exposed to radiation, while neglecting convection heat transfer inside the domain. Its primary objective is to validate the viewFactor andfvDOM solvers in OpenFOAM for radiative heat transfer by comparing the computational results with the analytical solution provided by Júnior et al. [57]. The physical model represents a simplified two dimensional enclosure, as shown in Fig. 3. The domain consists of four surfaces, each 1 m in length. The top surface is maintained at 100 K, while the remaining three surfaces are held at 1 K and other boundary conditions are mentioned in Table 4. The enclosure is filled with air and considered

<figure><img src="image_5.png" /><figcaption>Fig. 3. Geometry of the rectangular enclosure in case 3.</figcaption></figure>

as non-participating media, while all surfaces are assumed to be black body, diffuse and opaque. The upwind scheme is used for the computation of radiation intensity *I*, while for *G*, the linear corrected scheme is applied. This approach ensures numerical stability and accuracy in solving the radiative transfer equations. The results are provided in Section 3.

### 2.4. Case 4: 3D - Pyrolysis drop tube reactor

Finally, after describing the benchmark test cases (1–3), in this subsection, the geometry of the pyrolysis reactor (considered as case 4, hereafter) and the relevant boundary conditions and numerical setups implemented in OpenFOAM are discussed. The reactor tube has an external diameter of 30 mm. It is insulated with Ultra-board material to prevent heat loss, and it includes an open-air receiver cavity with a 15 mm diameter aperture and a surface area of 177 mm². The reactor operates with a maximum power input generating an average irradiation flux of approximately 1.88 MW m⁻² at the aperture. It is worth noting
<table><caption>**Table 4**
Boundary conditions applied in case 3.</caption><thead><tr><th>Surface</th><th>Parameter</th><th>viewFactor</th><th>fvDOM</th></tr></thead><tbody><tr><td>Top</td><td>Velocity</td><td>noSlip</td><td>noSlip</td></tr><tr><td>Left</td><td>Pressure</td><td>10<sup>5</sup> Pa</td><td>10<sup>5</sup> Pa</td></tr><tr><td>Right</td><td>IDefault</td><td>—</td><td>greyDiffusiveRadiation</td></tr><tr><td rowspan="2">Bottom</td><td>q<sub>r</sub></td><td>greyDiffusiveRadiationViewFactor</td><td>—</td></tr><tr><td>G</td><td>MarshakRadiation</td><td>MarshakRadiation</td></tr></tbody></table>

<table><caption>**Table 5**
Thermophysical properties of the materials used for case 4.</caption><thead><tr><th>Section</th><th>Material</th><th>Density (kg m<sup>-3</sup>)</th><th>Cp (J kg<sup>-1</sup> K<sup>-1</sup>)</th><th>k (W m<sup>-1</sup> K<sup>-1</sup>)</th><th>μ (kg m<sup>-1</sup> s<sup>-1</sup>)</th></tr></thead><tbody><tr><td>Drop tube</td><td>Mullite</td><td>2700</td><td>900</td><td>5</td><td>—</td></tr><tr><td>Insulation</td><td>Ultra-board</td><td>400</td><td>800</td><td>0.15 (@T = 1073.15 K)<br>0.22 (@T = 1473.15 K)</td><td>—</td></tr><tr><td>Inert gas</td><td>Argon</td><td>1.622</td><td>520.6</td><td>0.016</td><td>2.13e-5</td></tr><tr><td>Cavity</td><td>Air</td><td>1.225</td><td>1006.4</td><td>0.024</td><td>1.79e-5</td></tr></tbody></table>

<table><caption>**Table 6**
Boundary conditions for the 3D CNRS geometry in OpenFOAM.</caption><thead><tr><th>Domain</th><th>Name</th><th>Parameter</th><th>Boundary Condition</th></tr></thead><tbody><tr><td rowspan="2">Cavity</td><td rowspan="2">Aperture</td><td>T</td><td>fixedValue, T = 2402 K</td></tr><tr><td>U</td><td>noSlip</td></tr><tr><td rowspan="3">Insulation</td><td rowspan="2">Exterior surfaces</td><td>T</td><td>externalWallHeatFluxTemperature<br>h = 10 W m<sup>-2</sup> K<sup>-1</sup>, T<sub>∞</sub> = 300 K</td></tr><tr><td>U</td><td>fixedValue, U<sub>z</sub> = -0.02 m s<sup>-1</sup></td></tr><tr><td>Inlet</td><td>T</td><td>fixedValue, U<sub>z</sub> = -0.02 m s<sup>-1</sup></td></tr><tr><td rowspan="3">Drop tube</td><td rowspan="2">Outlet</td><td>T</td><td>zeroGradient</td></tr><tr><td>U</td><td>inletOutlet</td></tr><tr><td>Exterior surfaces</td><td>T</td><td>externalWallHeatFluxTemperature<br>h = 10 W m<sup>-2</sup> K<sup>-1</sup>, T<sub>∞</sub> = 300 K</td></tr></tbody></table>

that on a clear, sunny day, the maximum solar irradiance reaching the Earth's surface is typically around 1 kW m⁻².

The thermal properties of the various materials used in the reactor are provided in Table 5.

The cavity is filled with air, which is treated as a non-participating medium and perfect gas. The radiation boundary conditions applied are as follows: the cavity region, the aperture, the interface between the drop tube thickness and cavity, and the interface between the insulation and cavity are all modeled as opaque and diffusely reflecting surfaces, by considering the *greyDiffusiveRadiation* for radiation intensity (I), *IDefault*.

Similarly, in the drop tube region, the interface along the drop tube thickness is treated as an opaque and diffusive surface.

The boundary condition considered for the cavity interfaces with the insulation and the drop tube is black body, no-slip for velocity, and for temperature, *turbulentTemperatureRadCoupled-Mixed*. The complete list of the boundary conditions implemented in OpenFOAM are described in Table 6. In addition, a constant temperature boundary condition is applied to represent the effect of a uniform radiative heat flux of 1.88 W m⁻² on the aperture surface, which is originated from outside of the computational domain (solar heat).

For thermal coupling of the different regions in this study in OpenFOAM, the *turbulentTemperatureRadCoupledMixed* boundary condition is applied at the insulation – cavity, drop tube – drop tube thickness, and cavity – drop tube thickness interfaces, while the remaining interfaces are treated with *turbulentTemperature-CoupledBaffleMixed*. In addition, a no-slip condition for velocity is imposed on all interfaces.

The working fluid inside the drop tube is argon and it is assumed to be non-participant and modeled as a perfect gas. The flow inside the drop tube is assumed to be laminar, with an inlet velocity of 0.02 m s⁻¹ and an inlet temperature of 300 K. We have ignored the pyrolysis process and particles in the current study, since our main objective is to study heat transfer within the system.

It is relevant to discuss the simulation procedure for case 4 in more detail. The computational meshes were created for three-dimensional geometry of the drop tube reactor domain. Special attention was paid to areas around the walls of the cavity and the drop tube. The simulations were performed in transient mode with constant time step (0.02 s) until a steady state condition was achieved. Convergence was evaluated using both residual reduction and the stabilization of monitored physical quantities. The residuals of velocity and temperature were required to decrease below 10⁻⁶. In addition, global mass and energy balances were monitored to ensure negligible temporal variation. The solution was considered converged when all residual criteria were satisfied and no further changes were observed in the monitored fields.

To ensure the reliability of the numerical results, mesh independence and angular discretization sensitivity studies were conducted for the radiative heat transfer calculations. Several mesh resolutions and radiation angular discretization numbers were tested, and their influence on the temperature field was evaluated, see Appendix B. With the mesh and angular discretization number selected, the main production runs were carried out. Post-processing was done to examine the temperature distributions, radiative and conductive heat fluxes, and the energy balance in the reactor.

### 3. Results and discussions

#### 3.1. Case 1: 1D - Cavity with solid layer

Next, the OpenFOAM simulation results are presented for Case 1 and the results are validated against the previously discussed analytical solution. As mentioned, the 1D cavity is filled with air, while the drop tube is made of mullite which in this section called solid layer. The thermal properties of air and mullite are provided in Table 5. As shown in Fig. 4(b), radiative heat is transferred from the right interface through the non-participating media to the mullite surface where it
<figure><img src="image_6.png" /><figcaption>Fig. 4. (a) Heat flux variation over time at the left and right surfaces, and (b) temperature distribution at steady state for the solid layer from OpenFOAM.</figcaption></figure>

is absorbed and conducted towards the cold left-end of the domain. Fig. 4(a) illustrates the heat transfer rate (W) over time for the mullite layer. As seen, a steady state condition is reached in approximately 3 min, indicating the time when the incoming heat equals the heat losses. The transient heat equation in the solid layer, solved using finite difference method, is expressed in Eq. (13). At steady state, the boundary condition on both the left and right surfaces of the solid layer satisfies $q''_{radiation} = q''_{conduction}$. At the right surface, the energy balance can be written as Eq. (14).

$$ \epsilon\sigma(T_{rad.}^4 - T_s^4) = -k_s \frac{dT_s}{dx}, \qquad (14) $$

where $T_{rad.}$ represents the incoming radiation temperature. On the left surface, $T_{rad.}$ is replaced by the ambient temperature.

The results show that the energy transfer predicted by the numerical solution closely matches the analytical model, which confirms the precision of the numerical solution by OpenFOAM. The particular observation from this 1D test case is that the heat balance condition from the non-participating media to the solid surface is correctly implemented.

### 3.1.1. Case 2: 1D - drop tube with solid material

Next, the transient simulation results are shown for a solid material placed at the center of a drop tube, exposed to radiative heat from both sides, along with a comparison to the corresponding analytical solution for validation. It is noted that this test problem is a simplified version of the real world 3D drop tube although the problem represents the essential physics regarding radiation heat transfer: solid walls radiate heat which is absorbed by a solid material (e.g. wood particles) and also conducted in the solid.

Fig. 5 illustrates the mesh independence study conducted for this case using the chtMultiRegionFoam solver. The figure shows the temperature profile inside the solid material at a given time. The relative error for temperature at the middle of the solid material between 8-cell and 16-cell grid size is 0.1%, while between 16-cell and 24-cell resolutions the error is almost negligible. Based on the results, the grid size of approximately 0.1 mm, corresponding to 16 cells, is sufficient to ensure solution accuracy in OpenFOAM.

Furthermore, the numerical results obtained from OpenFOAM were validated against the analytical solution. As shown in Fig. 5, the temperature distribution within the solid material exhibits excellent agreement with the analytical solution throughout the simulation domain. This consistency, noted for the present 1D problem, confirms the accuracy and reliability of the numerical setup and boundary conditions employed in the OpenFOAM simulation. Fig. 5 corresponds to the time

instant 5.15 s, at which both the left and right interfaces of the solid material approach 500 K.

Fig. 6 presents the temporal evolution of temperature on the left and right surfaces, as well as the mean temperature levels of the solid material until it reaches 773.15 K, a temperature range suitable for fast pyrolysis. The results obtained from OpenFOAM are compared with the analytical solutions to assess the accuracy of the model. As shown in Fig. 6, the average temperature of the solid material reaches 773.15 K at approximately 21.7 s. The results also demonstrates good agreement between the numerical and analytical results.

### 3.2. Case 3: 2D - Rectangular enclosure

In the present study the main focus was on investigating the performance of thefvDOM implementation in OpenFOAM as it was noted to be applicable in all the studied cases. As an example of another radiation model, the viewFactor radiation model is also readily available in OpenFOAM and it can be used to model thermal energy exchange between different surfaces based on the view factor coefficients and considering the geometric orientations. The model considers that the medium between the surfaces is non-participating which means that the medium does not absorb, emit or scatter the radiation.

Fig. 7(a) illustrates graphically how each surface of the enclosure exchanges radiation with all other surfaces. Each line in the figure represents a ray path, indicating how individual surface elements on one boundary are exposed to the elements on other surfaces. This visualization helps to understand how OpenFOAM calculates the view factor values, which are used to determine how much radiative energy is transferred between the surfaces.

Fig. 7(b) presents the results of the view factor calculations obtained using the viewFactor radiation solver in OpenFOAM, compared with the analytical solution. The results show that the view factors obtained with the solver closely match the corresponding analytical solutions. The largest deviation observed is negligible, indicating that the solver provides reliable predictions for radiative heat transfer calculations.

Next, grid sensitivity and sensitivity of the heat flux calculations to the angular discretization in case 3 has been investigated using $fvDOM$ to verify the model accuracy in the 2D domain shown in Fig. 3. In the $fvDOM$ solver for two-dimensional geometries, angular discretization is applied only in the azimuthal direction ($\phi$) and represented by $n_{\phi}$ and the total number of rays is calculated by $4 \times n_{\phi}$.

Fig. 8 presents the obtained heat flux distributions using different grid resolutions, while the angular discretization parameter $n_{\phi}$ is kept fixed at 16. Results are shown for the left and right surfaces, which yield identical values due to symmetry, as well as for the bottom surface. As
<figure><img src="image_7.png" /><figcaption>Fig. 5. Temperature distribution inside the solid material for different mesh resolutions in OpenFOAM compared with the analytical solution at time = 5.15 s.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6. Temperature variation with time within the solid material for case 2.</figcaption></figure>

shown in Fig. 8, the 30 × 30 mesh is chosen because it captures the heat flux near the bottom corner of the domain more accurately compared to the 10 × 10 and 20 × 20 meshes. As illustrated in Fig. 9, increasing $n_φ$ while using the 30 × 30 grid resolution leads to a smoother distribution of heat flux across all surfaces. The results show that the heat flux across all the surfaces remains unchanged when the $n_φ$ is increased from 16 to 32.

Figs. 8 and 9 show that increasing the angular resolution improves the representation of directional radiative intensity and reduces the ray effects, resulting in smoother wall heat flux distributions and better energy conservation, while insufficient angular resolution may lead to non-physical fluctuations and inaccurate radiative heat transfer predictions, particularly in configurations dominated by directional radiation. Table 7 shows the results of the mean radiative heat flux calculations

usingfvDOM at different grid resolutions and $n_φ$. The relative errors remain consistently low at different angular discretizations when a 30 × 30 mesh is used. Also, the errors between the results of different grid resolutions remain small. Our conclusion is that by choosing 16 for $n_φ$ and 30 × 30 for mesh size, the relative errors between the analytical solution and OpenFOAM computations are negligible across all the surfaces of the cavity, while choosing a higher $n_φ$ or grid resolution do not improve the errors significantly.

Finally, Table 8 presents the final results of the mean heat exchange rate by radiation obtained using the *fvDOM* and the *viewFactor* radiation models. An analysis of these results reveals that the numerical outcomes obtained from both the radiation models are in good agreement with the analytical solution. The maximum deviation between the analytical and numerical results is lower than 1%.
<figure><img src="image_9.png" /><figcaption>Fig. 7. The (a) corresponding surface-to-surface visibility, and (b) view factor values computed using the viewFactor solver and analytical solution in OpenFOAM for case 3.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 8. Effect of different spatial discretization values for case 3 usingfvDOM.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 9. Effect of different angular discretization values for case 3 usingfvDOM.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Fig. 10. The final mesh used in case 4 (the pyrolysis drop tube reactor): (a) front section isometric view, (b) top-view.</figcaption></figure>

<table><caption>**Table 7**
Mean radiation heat flux values from OpenFOAM at different spatial and angular discretization numbers for case 3 using *fvDOM*.</caption><thead><tr><th>Mesh</th><th>n<sub>φ</sub></th><th>Left/Right (W m<sup>-2</sup>)</th><th>Bottom (W m<sup>-2</sup>)</th></tr></thead><tbody><tr><td>Analytical</td><td>-</td><td>1.66</td><td>2.348</td></tr><tr><td>10 × 10</td><td>16</td><td>1.6674</td><td>2.3354</td></tr><tr><td>20 × 20</td><td>16</td><td>1.66831</td><td>2.33378</td></tr><tr><td>30 × 30</td><td>16</td><td>1.66721</td><td>2.33599</td></tr><tr><td>40 × 40</td><td>16</td><td>1.66629</td><td>2.33783</td></tr><tr><td rowspan="5">30 × 30</td><td>2</td><td>1.66637</td><td>2.33767</td></tr><tr><td>4</td><td>1.66602</td><td>2.33837</td></tr><tr><td>8</td><td>1.66875</td><td>2.33292</td></tr><tr><td>16</td><td>1.66721</td><td>2.33599</td></tr><tr><td>32</td><td>1.66561</td><td>2.33919</td></tr></tbody></table>

<table><caption>**Table 8**
A comparison of the mean radiation heat flux values at different surfaces using the *ViewFactor* and *fvDOM* models compared with the analytical solution for case 3.</caption><thead><tr><th>Surface</th><th>Analytical (W m<sup>-2</sup>)</th><th>ViewFactor (W m<sup>-2</sup>)</th><th>fvDOM (W m<sup>-2</sup>)</th></tr></thead><tbody><tr><td>Top</td><td>-5.67</td><td>-5.6702</td><td>-5.67041</td></tr><tr><td>Right/Left</td><td>1.66</td><td>1.6675</td><td>1.66721</td></tr><tr><td>Bottom</td><td>2.348</td><td>2.3446</td><td>2.33599</td></tr></tbody></table>

### 3.3. Cas4: 3D - Pyrolysis drop tube reactor

Finally, results of the OpenFOAM simulations using *fvDOM* for the pyrolysis drop tube reactor are presented in this section. These simulations were performed using the boundary conditions and numerical settings described previously in Section 2.4.

The final grid resolution and angular discretization were selected based on the mesh and angular discretization independence studies detailed in Appendix B. The mesh used for this simulation is shown in Fig. 10. The validation of *chtMultiRegionFoam* is presented in Appendix A using a 2D channel flow benchmark, showing agreement between the computed Nusselt number and the theoretical prediction.

In the absence of experimental data for validation, STAR-CCM+, a well-established and reliable commercial solver, was used to verify the OpenFOAM results. Fig. 11 illustrates the temperature distributions within the reactor obtained using OpenFOAM and STAR-CCM+. Both side and top cross sectional views are presented. The OpenFOAM results, shown in Fig. 11a, employ a constant wall temperature boundary

condition at the aperture. For STAR-CCM+, two boundary conditions are considered at the aperture: a constant wall temperature, Fig. 11b, and a diffusive radiative heat flux, Fig. 11c.

A qualitative comparison between two different solvers and two different boundary conditions indicate that the results are in overall good conformity. This outcomes validates the implementation of *fvDOM* in OpenFOAM as well as the relevance of the choice of constant temperature boundary condition in OpenFOAM instead of diffusive radiation heat flux. As a side note, we had limitations in using a diffusive radiation heat flux boundary condition in OpenFOAM. Therefore, we used an equivalent constant temperature boundary condition in OpenFOAM. To assess the impact of explained approximation, a comparison of the results using the two boundaries in STAR-CCM+ has been conducted for verification. More detailed analysis is provided later (see Fig. 12).

Fig. 11 shows how the radiative heat absorbed by the cavity/drop tube is transferred through the drop tube by conduction, increasing the temperature of the fluid side surface of the drop tube. Heat is then transferred to the inert gas inside the drop tube through a combination of radiation and convection. As a result, the gas temperature rises as it flows through the region surrounded by the cavity marked by (I).

Considering the side views (marked (1) in Fig. 11) the temperature within the insulated cavity shows spatial variations of approximately 200 K. In contrast, the temperature on a fixed mid-section cut-plane from the top view (marked (2)) is relatively uniform. This behavior is mainly due to the insulation, which limits heat losses to the surroundings and promotes thermal homogenization within the cavity region.

The spatial variation of temperature within the insulation material is influenced by its temperature-dependent thermal conductivity. As the local temperature increases, the thermal conductivity of the insulation increases, which reduces conductive thermal resistance through the wall. In addition, the incoming radiative heat is focused at the mid-section of the drop tube. When this localized heating is combined with the temperature-dependent behavior of the insulation, it causes spatially non-uniform heat distribution within the insulation material, as observed in Fig. 11.

For OpenFOAM, the results further show that the mid-section of the drop tube wall, which is directly exposed to incoming radiation, reaches the highest temperature levels. At steady state, the maximum wall temperature at this location is approximately 1235.6 K. Throughout the cavity region, the temperature at the interface between the drop
<figure><img src="image_13.png" /><figcaption>Fig. 11. Temperature distributions within the 3D drop tube reactor in (a) OpenFOAM with constant wall temperature, (b) STAR-CCM+ with constant wall temperature, and (c) STAR-CCM+ with diffusive radiation heat flux boundary conditions at the aperture. In each case, subfigures (1) and (2) show the side and top views at mid-section of the 3D reactor, respectively.</figcaption></figure>

tube and the cavity remains above 900 K. The inert gas enters the cavity region (marked as "II") in Fig. 11 with area-average of 791 K, and exits with area-average of 976 K, and a volume-averaged temperature of 1016 K within this region marked as "II".

Overall, the temperature distribution within the drop tube suggests that the reactor design is suitable for solar-assisted fast pyrolysis of biomass. Under the assumed incoming radiative heat flux, a drop tube reaches a volume-averaged temperature of 784 K. In addition, most of the drop tube volume reaches temperature 773 K, which is commonly reported as the minimum temperature required for fast pyrolysis and high bio-oil yields [58].

The results of the 3D drop tube reactor simulations along four longitudinal sampling lines (see Fig. 12(a)) are shown in Fig. 12. The results are shown for the three different case 4 simulations.

The results from the STAR-CCM+ simulations indicate that both the constant temperature boundary condition and the radiative heat flux boundary condition led to similar temperature distributions within the entire domain. Fig. 12(b) indicates that e.g. the peak temperature values deviate by approximately 2.0% as observed at the midsection of the interface between the cavity and the drop tube thickness, where the radiative heat flux is the strongest. Also, the maximum temperature within the drop tube thickness shows a relative difference of approximately 0.4%. This suggests that applying either of the two boundary conditions to model the incoming radiation yields nearly identical results in terms of temperature distribution within the cavity, and consequently, inside the drop tube.

The non-uniform temperature distribution along the cavity walls, shown in Fig. 12, is also connected to the reactor geometry and the nature of the radiative heat propagation. The asymmetric stadium-shaped

cavity, optimized by [55], receives concentrated radiation through a small aperture located on one side of the reactor. The cavity geometry and the placement of the drop tube opposite the aperture partially obstruct the direct radiation path. Consequently, the radiative heat is redistributed through diffuse reflections inside the cavity, leading to non-uniform temperature patterns along the cavity walls and drop tube, as observed in Fig. 12.

As illustrated in Fig. 12(b), the mid-section of the drop tube, which is directly exposed to the incoming radiative flux, absorbs a larger amount of radiative energy and reaches the highest temperature of 1235.6 K. Consequently, the maximum temperature decreases to 1164.54 K along the longitudinal line shown in Fig. 12(c), where the considered region is not directly exposed to the incident radiation and it is obstructed by the drop tube. Furthermore, the maximum temperatures along the two b1 and b2 longitudinal lines show in Fig. 12(d) are identical, with a local peak temperature value of 1188 K. Moreover, according to Fig. 12, the temperature distributions obtained from OpenFOAM and STAR-CCM+ are in very good agreement.

Finally, to ensure consistency, the energy balance within the reactor is checked. For this purpose, temporal variations of the incoming radiative heat and total heat loss for the OpenFOAM results are plotted in Fig. 13. The total heat loss accounts for all the thermal energy that escapes through the outer surface of the reactor by convection. As observed from Fig. 13, once steady state is reached, the incoming radiative heat balances the total heat losses from the reactor, indicating thermal equilibrium. The results show that most of the heat loss occurs through the insulation in regions directly exposed to the incident radiation and near the aperture, where wall temperatures are the highest, as illustrated in Fig. 11. The heat losses can be largely explained by exterior
<figure><img src="image_14.png" /><figcaption>Fig. 12. Temperature variation along four longitudinal lines in the 3D drop-tube reactor under various numerical configurations. Locations of the longitudinal sample lines are depicted on the top view (a); and the results are shown along line a1 (b), line a2 (c), and lines b1 and b2 (d).</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 13. Temporal variations of heat with time in the drop tube reactor (case 4), indicating the energy balance at steady state.</figcaption></figure>

surface boundary condition with prescribed convection coefficient ($h = 10 \text{ W m}^{-2} \text{ K}^{-1}$) leading to heat flux through the insulation material.

The net incoming flux of radiation at steady state is $315.88 \text{ W m}^{-2}$, but it is effectively balanced by the total exterior surface flux of $313.78 \text{ W m}^{-2}$. The remaining balance of roughly $2.1 \text{ W m}^{-2}$ is represented by the enthalpy increase of the working fluid as it leaves the drop tube. This value of internal energy increase was independently verified by solving the sensible heat transfer equation, $q'' = \dot{m}C_p\Delta T$.

The results yielded a net transfer of $2.047 \text{ W m}^{-2}$ based on the weighted average temperature difference between the outlet and inlet of the drop tube which shows a 2.6% difference.

## 4. Conclusions

In this study, a numerical framework for the simulation of coupled radiative, convective and conductive heat transfer in a solar-assisted drop tube reactor in OpenFOAM was developed and validated. Validation was carried out in three phases. The first two phases represented a simplified version of specific sections of the pyrolysis reactor, while the third phase considered the entire reactor:

1. **One-dimensional cases.** Two 1D cases were considered; a steady state and a transient case, in which combined conduction and radiation were solved in OpenFOAM and compared against analytical solutions. Errors remained below 1%, confirming the accuracy of the numerical schemes and boundary condition implementations. These cases demonstrated that thefvDOM reliably models radiative heat transfer in simple geometries.

2. **Two-dimensional enclosure.** A 2D radiative enclosure was simulated using both thefvDOM and viewFactor model. Comparisons with the analytical view factor values showed negligible deviations, indicating that both models can accurately capture radiative heat exchange inside enclosures.

3. **Three-dimensional reactor.** Finally, the validated OpenFOAM setup was applied to the full 3D drop tube reactor. In the absence of an analytical solution, OpenFOAM predictions were compared with well-known Star-CCM+ software results. Temperature distribution contours within the drop tube were compared between
<table><caption>**Table A.1**
Boundary conditions for parallel plate validation in OpenFOAM.</caption><thead><tr><th>Domain</th><th>Name</th><th>Parameter</th><th>Boundary Condition</th></tr></thead><tbody><tr><td rowspan="2">Fluid</td><td rowspan="2">Inlet</td><td>T</td><td>fixedValue, T<sub>in</sub> = 300 K</td></tr><tr><td>U</td><td>parabolic profile</td></tr><tr><td rowspan="2">Outlet</td><td>T</td><td>zeroGradient</td></tr><tr><td>U</td><td>zeroGradient</td></tr><tr><td rowspan="2">Fluid/Solid</td><td rowspan="2">Interface</td><td>T</td><td>turbulentTemperatureCoupledBaffleMixed</td></tr><tr><td>U</td><td>noSlip</td></tr><tr><td>Solid</td><td>Wall</td><td>T</td><td>fixedValue, T<sub>wall</sub> = 500 K</td></tr></tbody></table>

the outcomes of the two softwares, demonstrating that Open-FOAM can accurately model conjugate heat transfer in complex 3D geometries with the presence of radiative heat transfer.

The obtained results from the pyrolysis reactor simulations confirm that:

* The *fvDOM* solver available in OpenFOAM can accurately model coupled conduction, convection, and radiation for the complex geometry of a pyrolysis reactor.

* OpenFOAM and Star-CCM+ produce comparable results for the 3D drop tube reactor.

* Both constant temperature and radiative heat flux boundary conditions reach nearly identical temperature distributions within the reactor.

* Under the assumed solar radiation heat flux, the interface between the cavity and the drop tube thickness reaches a peak temperature of approximately 1235 K at the mid-section of the tube's length, while the minimum wall temperature remains well above 773.15 K, i.e. exceeding the threshold for fast pyrolysis.

For our future work, we will consider utilizing this validated framework for incorporating detailed particle-scale interactions, pyrolysis kinetics, and multiphase flow models. Such extensions could enable detailed optimization of the reactor geometry as well as the operating conditions in order to maximize the biomass pyrolysis yield and energy efficiency.

## Nomenclature

T Temperature, [K]

t Time, [s]

$C_p$ Specific Heat Capacity, [J kg$^{-1}$ K$^{-1}$]

k Thermal Conductivity, [W m$^{-1}$ K$^{-1}$]

I Radiation Intensity, [W m$^{-2}$]

G Incident Radiation, [W m$^{-2}$]

q Volumetric Heat Source, [W m$^{-3}$]

U Velocity, [m s$^{-1}$]

h Convection heat transfer coefficient, [W m$^{-2}$ K$^{-1}$]

Abbreviations

CFD Computational Fluid Dynamics

CHT Conjugate Heat Transfer

DEM Discrete Element Method

DOM Discrete Ordinate Method

fvDOM Finite Volume Discrete Ordinate Method

RTE Radiative Transfer Equation

Greek Letters

α Thermal Diffusivity, [m$^2$ s$^{-1}$]

β Extinction Coefficient, [m$^{-2}$]

κ Absorption Coefficient, [m$^{-2}$]

μ Dynamic viscosity, [Pa s]

σs Scattering Coefficient, [m$^{-2}$]

ρ Density, [kg m$^{-3}$]

Subscripts

b Black body

f Fluid

s Solid

r Radiation

λ Spectral parameter

Z Vertical Z-direction

## CRediT authorship contribution statement

**Amir Hossein Bashiri:** Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Investigation, Formal analysis, Data curation, Conceptualization. **Shervin Karimkashi:** Writing – review & editing, Supervision, Formal analysis, Conceptualization. **Vignesvar Krish Subramani:** Writing – review & editing, Resources. **Sylvain Rodat:** Writing – review & editing, Supervision, Project administration, Funding acquisition. **Stéphane Abanades:** Writing – review & editing, Supervision, Funding acquisition. **Mika Järvinen:** Writing – review & editing, Supervision, Project administration, Funding acquisition. **Ville Vuorinen:** Writing – review & editing, Supervision, Project administration, Funding acquisition, Conceptualization.

## Funding

The authors wish to gratefully acknowledge the support of the Horizon Europe Circular Fuel project 101118239, funded by the European Climate, Infrastructure and Environment Executive Agency(CINEA), which funds this research.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
<figure><img src="image_16.png" /><figcaption>Fig. A.1. Nusselt number calculations for the laminar parallel plate flow in OpenFOAM with constant surface temperature.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. B.1. Temperature distribution profiles at three longitudinal sample lines for various angular discretization values: (a) line a1, (b) line a2, and (c) line b1.</figcaption></figure>

## Appendix A. 2D conjugate heat transfer analysis in parallel plate flows

In this section, a fully developed laminar flow is considered to validate the conjugate heat transfer simulation using the `chtMultiRegionFoam` solver in OpenFOAM. The validation is carried out for steady, laminar flow between two parallel plates separated by a distance $D$, where each wall includes a solid region of thickness $0.1D$. The exterior walls are maintained at a constant temperature of $500$ K, while the inlet and outlet conditions are specified in detail in Table A.1. At the solid-fluid interfaces, the `turbulentTemperatureCoupledBaffleMixed` boundary condition is employed to ensure proper thermal coupling between the regions. This validation aims to assess the solver's accuracy in predicting heat transfer through the conjugate

interface. The analytical Nusselt number for this case is $7.54$ under steady-state conditions. As shown in Fig. A.1, the numerical results, $Nu \approx 7.58$, obtained from OpenFOAM, exhibit excellent agreement with the analytical solution, thereby, confirming reliability of the simulation setup for conjugate heat transfer problems.

## Appendix B. Grid and angular discretization sensitivity analysis for 3D pyrolysis reactor in OpenFOAM

The accuracy of the Navier-Stokes, energy, and radiative transfer equations depends critically on how the domain is discretized. In particular, the radiative transfer equation demands discretization not only in space but also in angle, since radiation is tracked along specific
<figure><img src="image_18.png" /><figcaption>Fig. B.2. Temperature distribution profile at three longitudinal sample lines for various mesh sizes: (a) line a1, (b) line a2, and (c) line b1.</figcaption></figure>

directions. To identify the most efficient mesh and angular discretization setup, we conducted a two-part independence study for the 3D pyrolysis reactor, while implementing the same boundary conditions mentioned in Table 6, except for the thermal conductivity value of the insulation material, which is assumed to be constant and homogeneous, independent of the temperature, with a value of 0.5 W m⁻¹ K.

### Angular discretization independence

In this study, in OpenFOAM, the number of division points used to discretize the polar angle direction, $n_\theta$, is always equal to $2 \times n_\phi$. We then varied the azimuthal discretization, $n_\phi$, within the first quadrant and evaluated its effect on temperature distribution on three different longitudinal lines shown in Fig. 12(a). The results for three different $n_\phi$ values for the Medium-size mesh is represented in Fig. B.1 along three sample lines. The relative change in maximum temperature is 0.33% between the first two azimuthal angle discretization values and 1.73% between the next two. Accordingly, we adopted an angular discretization value of $n_\phi$ equals 16.

### Spatial mesh independence

We tested four mesh resolutions by monitoring temperature along three predefined probe lines shown in Fig. B.2. The temperature difference between Medium-size mesh and Fine-size mesh was below 0.1% which is negligible, and between Coarse-size mesh and Medium-size mesh is approximately 0.4% for maximum temperature values. These small variations indicated convergence, and therefore, we chose the Medium-size mesh with approximately 1 million cells and $n_\phi = 16$ for the production simulations.

This two-step approach ensures both spatial and angular discretizations are sufficiently refined to yield reliable results without unnecessary computational expenses.

## Data availability

The data supporting the findings of this study are publicly available in a GitHub repository at the following link: https://github.com/Aalto-CFD/Pyrolysis-Reactor-Simulation The repository contains the numerical case setups and related files used to generate the results presented in the article.

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