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· Catalysis is central to modern fuel production, enabling the conversion of fossil resources, biomass, and CO₂ into valuable fuels through advanced thermochemical and electrochemical processes.

· Significant advancements in catalyst design and characterization have enhanced the efficiency of key processes like hydrocracking, Fischer-Tropsch synthesis, steam reforming, and CO₂ hydrogenation.

· Both heterogeneous and homogeneous catalysts play crucial roles in sustainable fuel development, driving innovations in conventional and green energy pathways.

Abstract

Numerical modeling has emerged as a cornerstone in the design, optimization, and control of modern energy systems. This review explores the evolution and application of numerical techniques, including computational fluid dynamics (CFD), finite element methods (FEM), and transient energy simulation tools, across various domains such as solar thermal collectors, fuel cells, wind turbines, and heat exchangers. The paper discusses governing equations, solution strategies, and model validation techniques. Recent advancements in coupling numerical models with artificial intelligence and optimization algorithms are highlighted. Results from selected case studies are analyzed to illustrate the impact of modeling assumptions, boundary conditions, and grid refinement on prediction accuracy. Finally, challenges such as numerical instability, computational cost, and data integration are discussed, along with future directions emphasizing real-time simulations and digital twins for energy systems.

Keywords

Numerical modeling, CFD, energy systems, renewable energy, thermal modeling

1. Introduction

Numerical modeling has significantly transformed the landscape of energy systems analysis and design over the past three decades. Its capability to simulate physical phenomena such as heat transfer, fluid flow, chemical kinetics, and multiphase reactions enables researchers and engineers to evaluate system performance under various operating scenarios without the need for expensive or time-consuming experiments. This transformation is largely driven by the advancement in computational power, numerical algorithms, and the availability of user-friendly simulation platforms [1]. The widespread deployment of computational fluid dynamics (CFD), finite volume method (FVM), finite element method (FEM), and other numerical techniques has allowed energy researchers to understand and optimize systems ranging from microscale heat exchangers to megawatt-scale renewable energy plants [2].

Energy systems are inherently complex due to their nonlinear, multi-physics, and often transient nature. For instance, the dynamic interaction between solar irradiation, ambient conditions, and system thermal inertia in solar collectors requires careful modeling of time-dependent energy balances [3]. Similarly, wind turbines experience turbulence, mechanical stress, and flow-induced vibrations, which can be only captured through 3D simulations coupled with structural mechanics [4]. Traditional empirical models fail to capture such intricacies, highlighting the essential role of numerical modeling in both design and operational strategies [5]. The importance of numerical modeling is also amplified in the current context of decarbonization and climate goals, where the need to evaluate and compare low-carbon technologies requires rigorous virtual testing under diverse climatic and operational conditions [6].

Modeling begins with the formulation of governing equations, such as conservation of mass, momentum, and energy. These partial differential equations (PDEs) are then discretized using numerical schemes such as finite difference, volume, or element methods [7]. The choice of solver, time step, convergence criterion, and meshing strategy greatly influences the simulation results [8]. In recent years, hybrid models that couple physics-based simulations with data-driven approaches such as machine learning have gained traction, offering predictive accuracy along with computational efficiency [9]. For example, surrogate models trained on high-fidelity simulations can be used to accelerate parametric sweeps and optimization tasks [10].

One of the most widely adopted numerical tools in energy research is ANSYS Fluent, which offers robust CFD capabilities including multiphase flow, species transport, radiation modeling, and user-defined functions [11]. Other notable tools include COMSOL Multiphysics for FEM-based coupled physics modeling, OpenFOAM for open-source CFD, TRNSYS for thermal system simulation, and EnergyPlus for building energy modeling [12]. These platforms have enabled detailed studies of energy storage systems [13], bioenergy reactors [14], thermal desalination units [15], and even combustion chambers in engines [16].

The validation of numerical models remains a critical aspect of the modeling workflow. Experimental data, either from laboratory setups or full-scale pilot plants, are required to benchmark and calibrate the model predictions [17]. Sensitivity analyses are also used to assess the impact of uncertain parameters on key performance indicators such as efficiency, temperature distribution, or pressure drop [18]. In this context, uncertainty quantification techniques and Monte Carlo simulations have been increasingly used [19].

2. Methodology

Numerical modeling in energy systems is structured around the formulation, discretization, and solution of governing physical equations representing thermodynamic, fluid, and structural phenomena. The methodology involves several key stages including the definition of geometry, boundary and initial conditions, mesh generation, equation selection, numerical solver setup, and post-processing of simulation outputs [1]. The process starts with a clear problem definition—whether it's optimizing a heat exchanger, simulating combustion inside a gas turbine, or modeling energy flow in a building envelope.

At the core of numerical simulation are the conservation equations, which represent mass, momentum, and energy balances. These are expressed as partial differential equations (PDEs) and solved using techniques such as the Finite Volume Method (FVM), Finite Element Method (FEM), or Finite Difference Method (FDM) [2]. For instance, the conservation of mass (continuity) is generally represented as:

∂ρ/∂t + ∇·(ρu) = 0 (1)

Where ρ is the fluid density, t is time, and u is the velocity vector. The momentum equation for incompressible, Newtonian fluids is given by:

ρ(∂u/∂t + u·∇u) = -∇P + μ∇²u + F (2)

Where P is pressure, μ is dynamic viscosity, and F represents external forces such as gravity or electromagnetic fields. The energy equation is typically formulated as:

ρCp(∂T/∂t + u·∇T) = ∇·(k∇T) + Q_gen (3)

Where T is temperature, Cp is specific heat capacity, k is thermal conductivity, and Q_gen is internal heat generation per unit volume.

Once equations are established, spatial discretization is applied. In FVM, for example, control volumes are constructed and fluxes are evaluated at the surfaces using numerical schemes like upwind or QUICK. In FEM, the domain is broken into elements with shape functions applied to interpolate values across nodes [3]. Selection of grid type (structured or unstructured), mesh size, and refinement zones critically affects solution accuracy and computational time. Table 1 presents a comparative overview of major discretization techniques.

Table 1. Validation Metrics for Selected Energy Simulations

System Modeled	RMSE (°C)	R²	Data Source
Solar flat plate	1.3	0.98	Experimental test rig
Fuel cell (PEM) stack	0.8	0.96	Stack operating data
Wind turbine rotor	2.1	0.93	SCADA field measurements
HVAC cooling coil	0.5	0.99	Laboratory wind tunnel

Sensitivity analysis is performed to determine how variation in one input affects model outputs. For example, changes in ambient temperature, material properties, or geometric parameters can significantly alter energy efficiency or heat loss [8]. Monte Carlo or Latin Hypercube sampling techniques are used when dealing with uncertainty propagation across multiple parameters [9]. Optimization algorithms such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO), or Simulated Annealing (SA) are employed to identify optimal designs or operating conditions [10].

Table 3 summarizes the applicability of numerical methods across key energy domains. It demonstrates the dominance of CFD and FEM in most thermal and fluid systems, while FDM remains limited to simple geometries or academic problems.

Table 2. Preferred Numerical Techniques Across Energy Domains

Energy System	Preferred Method	Reason
Solar Thermal Collectors	FVM	Convection + Radiation
PEM Fuel Cells	FEM	Multiphysics Coupling
Wind Turbines	CFD	Aeroelastic Flow
Combustion Chambers	LES	Turbulent Reactive Flows
HVAC Ducts	FVM	Airflow and Mixing
Buildings Energy Simulation	EnergyPlus	Integrated Load Analysis

Recent developments focus on the integration of artificial intelligence with numerical modeling to enhance predictive capabilities and reduce computation time. For instance, neural networks have been trained to emulate CFD solvers or to perform mesh refinement [11]. Another trend is digital twin deployment in which real-time sensor data is continuously fed into a numerical model to create a live virtual representation for operational optimization and fault diagnosis [12].

Moreover, open-source platforms like OpenFOAM and Modelica have democratized access to numerical modeling, allowing academic and industrial users to customize solvers or create domain-specific modules [13]. Cloud-based simulation services such as SimScale and OnScale also offer on-demand computing power for large simulations [14].

The methodological rigor and customization offered by numerical modeling make it indispensable in today’s energy research landscape. Its power lies in flexibility: from microscale modeling of nanoporous materials in adsorption systems to large-scale simulation of district heating networks, numerical models bridge the gap between theory and practice [15]. However, practitioners must remain vigilant about model validation, numerical stability, and computation cost, especially as systems grow in scale and complexity [16].

3. Results

The outcomes of numerical simulations in energy systems hinge critically on solver fidelity, mesh resolution, boundary condition accuracy, and physical modeling choices. Across applications such as heat exchangers, building envelopes, solar collectors, and internal combustion engines, key dependent outputs like temperature distribution, pressure drop, and flow fields reveal considerable sensitivity to numerical settings. This section presents synthesized results from benchmark studies in literature, simulations conducted across selected case models, and model performance metrics that demonstrate the capabilities and limitations of numerical modeling in energy.

Mesh resolution directly impacts the numerical accuracy and convergence behavior in energy simulations. A refined mesh captures steeper gradients in temperature or velocity, particularly in zones with high thermal or hydrodynamic activity, such as near walls or at interfaces of phase change. In Figure 1, a sensitivity study illustrates the effect of mesh element count on the outlet temperature of a compact fin-tube heat exchanger under steady-state flow conditions. The outlet temperature stabilizes beyond 20,000 elements, indicating mesh independence. A coarser mesh underpredicts the temperature due to poor resolution of boundary layer effects. Studies have reported similar trends in solar thermal absorber simulations, where numerical error reduced from 5.3% to under 1% after mesh refinement [1, 2].

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Fig.1. Influence of mesh refinement on the outlet temperature in a heat exchanger simulation. Results show stabilization after 20,000 elements.

Solver time is another important consideration, especially for transient or 3D simulations involving multiphysics coupling. Figure 2 compares the computation time for Finite Difference Method (FDM), Finite Volume Method (FVM), and Finite Element Method (FEM) solvers when applied to a 2D heat conduction problem in a composite wall. FEM showed faster convergence due to superior mesh adaptability and element stiffness formulation, while FDM was significantly slower due to its dependence on structured grid alignment. Similar results are documented in turbine blade cooling simulations where FEM reduced time by 35% compared to FVM while maintaining accuracy [3, 4].

Thermal field visualization is a hallmark strength of numerical modeling. Heat maps offer intuitive understanding of spatial gradients, crucial for energy systems involving heat exchange, combustion, or insulation analysis. Figure 3 shows the simulated temperature distribution across the interior wall of an insulated room exposed to variable external conditions. Localized hot zones near corners and edges highlight thermal bridging phenomena. Validated CFD studies have emphasized the need to incorporate such spatially resolved patterns into building energy efficiency calculations, especially in climates with high diurnal thermal swings [5, 6].

Fig.2. Solver time comparison for FDM, FVM, and FEM techniques. FEM demonstrates the fastest convergence for the studied case.

Model validation against experimental benchmarks is essential to ensure simulation reliability. Figure 4 presents a scatter plot comparing simulated and experimental outlet temperatures for a fluid flowing through a helical coil heat exchanger. The data shows excellent correlation with minimal deviation, underscoring the model's calibration quality. Regression analysis yielded an R² of 0.992, consistent with best practices for model validation published in energy modeling guidelines [7, 8]. However, certain deviations around high-temperature regions suggest limitations in turbulence closure models used during simulation. Studies on fuel reformer simulations also show 3–5% deviation due to radiation and mixing assumptions [9].

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Fig.3. Simulated wall temperature distribution in a thermally insulated enclosure showing hot and cold spots.

Numerical models are instrumental in evaluating flow behavior and turbulence intensity, particularly in systems such as HVAC ducts, engine manifolds, or wind turbine nacelles. Figure 5 illustrates a box plot of turbulence intensity distributions across three HVAC zones. The data, derived from transient LES simulations, reveals higher mean intensity in Zone B due to a duct bifurcation and recirculation pocket. These insights inform fan placement, duct shaping, and noise mitigation strategies. Several studies have integrated similar analysis in designing data center cooling layouts and hospital air filtration systems [10, 11].

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Fig.4. Comparison between simulated and experimental temperature values, illustrating high correlation with slight deviation

Tool usage trends across the reviewed literature indicate the widespread adoption of commercial and open-source numerical platforms for energy applications. Figure 6 presents the share of tools employed in 150 peer-reviewed modeling studies. ANSYS Fluent remains the dominant tool (35%), followed by COMSOL and OpenFOAM. Notably, EnergyPlus usage was restricted to building-scale simulations. These trends align with review articles highlighting the importance of tool selection in multi-domain modeling [12, 13]. Additionally, the emergence of Modelica-based co-simulation environments has facilitated system-level integration of thermal, electrical, and control domains [14].

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Fig.5. Distribution of turbulence intensity values across three spatial zones in an HVAC duct system.

Sensitivity studies further underline the influence of boundary conditions and model parameters. Varying inlet air velocity in a solar chimney model altered the natural convection rate and outlet air temperature by over 25%, confirming the strong dependency on external conditions. Similarly, modifying material conductivity in phase-change material walls resulted in different melting front positions, which impacted stored thermal energy and overall LCOE values [15, 16]. Such parametric insights are critical for design optimization and techno-economic assessments.

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Fig.6. Proportion of numerical simulation tools used in 150 reviewed energy modeling publications.

Transient modeling results reveal dynamic response characteristics in energy systems. A study of an integrated solar water heater with a thermosyphon loop demonstrated thermal lag of 8–15 minutes depending on irradiation profile and fluid flow inertia. These findings emphasize the need to move beyond steady-state approximations, especially for systems interacting with variable renewable energy inputs [17, 18].

Additionally, the coupling of AI and numerical modeling has shown promise in reducing computational time. In a study involving district cooling networks, surrogate models trained using neural networks were able to predict outlet fluid temperature with less than 2% error while reducing computation time by 90%. Similar machine-learning-accelerated models are now being developed for combustion chambers, wind turbines, and carbon capture reactors [19, 20]. Another critical observation in numerical modeling lies in the treatment of multiphase and reactive systems, where both computational complexity and physical modeling accuracy pose challenges. For example, in simulations of solar-assisted desalination units, phase change processes such as evaporation and condensation must be modeled accurately across solid–liquid–vapor interfaces. CFD studies on humidification–dehumidification (HDH) units have shown that mist flow patterns and droplet coalescence zones significantly affect heat and mass transfer coefficients. This insight, unobtainable through lumped-parameter models, was instrumental in redesigning the air channel geometry, which increased freshwater yield by 18% compared to the base design [21,22].

In wind energy applications, CFD modeling has enabled detailed performance mapping of turbine blades under dynamic stall conditions. Unsteady Reynolds-Averaged Navier-Stokes (URANS) and Large Eddy Simulation (LES) models accurately predicted lift and drag coefficients, enabling estimation of fatigue loads. Field-validation of these simulations using SCADA data indicated good agreement, with deviations below 5% for power output at wind speeds above cut-in velocity. In particular, blade tip vortices captured in LES simulations aligned with thermal imaging data from drone inspections [23,24]. Such predictive insights are pivotal for predictive maintenance and control system tuning.

The integration of radiation models with energy simulations is another dimension gaining traction. For example, in concentrating solar power (CSP) systems, ray-tracing algorithms are coupled with energy conservation equations to simulate flux distributions on receiver tubes. The combination of Monte Carlo ray tracing and finite volume solvers yields accurate spatial profiles of incident energy, wall temperatures, and heat loss patterns. Comparative studies between experimental and simulated receiver temperatures showed deviations of 1.5–2.2 °C under standard test conditions, validating the numerical model's fidelity [25,26].

Moreover, numerical modeling of fuel cells has matured significantly with the advent of multiphysics solvers. In PEM fuel cells, coupled simulations for fluid flow, electrochemical reactions, thermal transport, and species diffusion allow for accurate prediction of polarization curves and local hot spots. Simulations highlighted the importance of GDL (gas diffusion layer) porosity and water management in maximizing current density uniformity. Changes in operating pressure and relative humidity were observed to shift the peak power output by as much as 12%, showcasing how numerical tools can guide stack design [27,28].

In the domain of combustion modeling, species transport and chemical reaction kinetics present a unique modeling challenge. Advanced simulation platforms now allow for detailed modeling of NOx formation, soot particle tracking, and ignition delay time prediction. In a case study involving ammonia-hydrogen gas turbines, simulations using reduced chemical kinetics schemes showed that NOx formation was highly sensitive to premixed zone temperature and residence time. The validated model was used to predict emissions across a full load range, assisting in designing low-NOx combustors for carbon-free fuels [29,30].

Another notable application is in building-scale simulations for energy consumption and thermal comfort analysis. Tools like EnergyPlus, OpenStudio, and Modelica-based systems simulate hourly building loads, accounting for weather data, internal heat gains, occupancy patterns, and HVAC control logic. In a case involving a double-skin facade, numerical modeling showed that dynamic shading could reduce annual cooling energy by 23% in a hot-arid climate. Temperature and air velocity distributions from CFD models were used as boundary conditions for room-level thermal simulations, enabling highly resolved comfort mapping [31,32].

Digital twins are increasingly being implemented for real-time simulation, monitoring, and control. In a pilot study of a solar PV-integrated smart building, the digital twin modeled real-time irradiance, ambient temperature, and load patterns to optimize HVAC scheduling and battery storage dispatch. Numerical solvers continuously updated boundary conditions and fed outputs to a neural network-based controller. This resulted in a 12.6% energy cost reduction while maintaining thermal comfort within ASHRAE 55 standards [33,34].

Exergy analysis combined with numerical modeling further enhances the ability to identify inefficiencies and optimization potential. In a simulation of a combined solar and biomass heating system, exergy destruction was mapped across components including the gasifier, heat exchanger, and storage tank. The exergy-based model was able to recommend changes in operating temperature and flow rate, leading to a 9.8% increase in second-law efficiency over baseline. Such results are difficult to obtain experimentally due to the intricacies of isolating entropy-generation terms in real systems [35,36].

High-resolution transient models have also demonstrated utility in modeling district heating and cooling networks. These models incorporate pipe thermal inertia, varying user demand, and weather-driven boundary conditions. In a district cooling model developed for a Middle Eastern city, simulations demonstrated that modifying supply temperature scheduling based on load forecasts could reduce pumping energy by 14% and increase COP by 11% compared to a fixed-temperature strategy [37,38].

Finally, the use of cloud-based platforms for parallel numerical simulations is transforming the computational landscape. Simulation platforms such as SimScale and ANSYS Cloud have enabled researchers and startups to run multi-scenario simulations in parallel, significantly reducing project timelines. For example, in a parametric study involving 96 design variations of a solar chimney, cloud simulations reduced computation time from 7 days to under 18 hours while maintaining accuracy within 1.2% [39,40].

Hybrid energy systems, particularly those integrating renewable sources like photovoltaic (PV), wind, and thermal storage, have greatly benefited from multi-domain numerical modeling. For example, simulating a hybrid PV-thermal system using co-simulation between MATLAB/Simulink (for power electronics) and TRNSYS (for thermal storage) has allowed accurate prediction of total energy yield, thermal stratification in storage tanks, and PV output under shading scenarios. The results showed that hybrid models produced 18% more energy than traditional split systems under partial shading and fluctuating load conditions. These models captured interdependencies between solar irradiation, water temperature, and load demand which would otherwise be missed in single-domain simulations [41,42].

Numerical modeling also provides unique capabilities in modeling rare or extreme operating conditions which are hard to replicate experimentally. This includes emergency shutdown scenarios, component failure, or stress testing. In a molten salt tower receiver CSP plant, numerical simulation of a pipe rupture scenario helped engineers quantify the rate of salt solidification and propagation of thermal shock along the system. These simulations were used to design an automatic dump valve and auxiliary heaters that reduced failure risk by 62% during operational anomalies. Such predictive models have also been deployed for nuclear safety simulations and failure diagnostics in wind turbine gearboxes [43,44].

In gas turbine modeling, advanced CFD and FEM techniques have enabled integrated simulations of combustor–turbine interactions, where pressure oscillations, flame flashback, and thermal fatigue are tightly coupled. Simulations using coupled conjugate heat transfer (CHT) models revealed temperature oscillations at the stator blades due to pressure waves initiated in the combustor. This feedback loop, previously unobservable in uncoupled simulations, enabled the design of dampers and blade cooling strategies to reduce thermal stress amplitude by over 20%. In addition, fatigue life prediction based on transient thermal cycles extended component life by 15–18% in validated field deployments [45,46].

Battery energy storage modeling is another growing area where multiphysics simulations offer critical insights. Coupled models including electrical (Ohmic loss), thermal (heat generation/dissipation), and chemical (Li-ion concentration) fields allow for full-stack performance analysis under different charge/discharge cycles. Numerical models of Li-ion battery cells showed that high discharge rates lead to significant temperature gradients, which in turn result in non-uniform capacity fade. Optimizing cooling strategies based on these simulations extended battery cycle life by 25% and improved energy efficiency by 6.5%. Several electric vehicle OEMs have adopted similar models for battery pack thermal management [47,48].

Another high-impact domain of numerical modeling is hydrogen production and storage. CFD simulations of high-pressure electrolyzers have shown how bubble dynamics affect electrode surface coverage and ionic resistance, directly impacting hydrogen production efficiency. Simulations also revealed nonuniform current density distribution in PEM electrolyzers due to poor water management. Design modifications including flow field restructuring and localized humidification improved hydrogen yield by 11% in pilot systems. Similarly, numerical simulations of metal hydride storage tanks helped predict temperature spikes during absorption/desorption, enabling the design of embedded heat exchangers to mitigate performance drop [49,50].

Thermal energy storage (TES) systems using phase-change materials (PCMs) also benefit from numerical simulations that track solid–liquid interfaces and latent heat release. In a seasonal TES tank study, CFD-FEM coupling was used to predict phase boundary motion under diurnal heating. The simulation helped optimize insulation thickness and inlet configuration, which improved storage efficiency by 14% and reduced daily exergy losses. Similar studies have been applied in building-integrated TES and solar cookers, particularly in off-grid and developing regions [51,52].

In the domain of geothermal systems, subsurface heat transport and groundwater flow modeling require fine spatial and temporal resolution. Finite element modeling of a vertical borehole heat exchanger coupled with a heat pump showed that rock thermal conductivity and borehole spacing critically affect long-term thermal performance. Field calibration of models showed that predictive accuracy within 5% was achievable for outlet temperature under seasonal loading. These results were used to develop optimized spacing guidelines that reduce borehole interference in multi-building installations [53,54].

Power-to-X (P2X) systems, which convert electricity into fuels (e.g., hydrogen, methane, ammonia), rely heavily on process simulation and CFD modeling for reactor design and integration. Numerical simulations of solid oxide electrolysis cells (SOECs) demonstrated the importance of temperature uniformity for durability. FEM-based simulations helped identify current bottlenecks and regions of accelerated degradation. Optimizing flow channel geometry and thermal management based on numerical results led to a 12% increase in cell lifespan. Similar modeling work has been performed on Sabatier reactors and Fischer-Tropsch synthesis units integrated with DAC systems [55,56].

District energy systems are another area where numerical modeling is pivotal. Dynamic hydraulic and thermal simulations were used to optimize the control strategy for a 2.5 km district heating loop in a cold-climate city. Using a Modelica-based platform, hourly simulations of heat load, valve opening, and return temperatures helped avoid temperature overshoot and under-supply scenarios. Energy savings of 10% and reduction in thermal comfort complaints were reported during a one-year pilot [57,58].

Lastly, techno-economic assessments (TEA) and life cycle assessments (LCA) are increasingly being combined with numerical modeling to estimate sustainability and cost-effectiveness of energy technologies. For instance, a solar-assisted air conditioning system was simulated over an entire cooling season using TRNSYS. The output cooling energy, auxiliary electricity use, and collector efficiency were then fed into a TEA module. The resulting LCOE was $0.112/kWh, 14% lower than conventional systems. When integrated with LCA, the simulation also showed 37% lower CO₂ emissions and 22% lower water consumption per cooling kWh delivered [59,60].

4. Discussion

The advancement of numerical modeling in energy systems has unlocked profound opportunities to understand, design, and optimize processes that were previously too complex, too dangerous, or too expensive to investigate experimentally. From component-level optimization to full-system integration, numerical models serve as essential virtual laboratories, capable of simulating physical behavior under a wide range of conditions and geometries. These models not only accelerate innovation cycles but also guide the deployment of sustainable technologies by predicting performance and informing policy and investment decisions. The discussion that follows reflects on the insights extracted from the results, comparing the advantages and limitations of numerical techniques across energy domains, and highlighting future directions where numerical modeling could exert the most impact.

A key observation from the reviewed studies is the indispensability of mesh and solver sensitivity in ensuring model fidelity. The convergence of results with mesh refinement, as seen in heat exchanger and CSP receiver simulations, underscores the importance of mesh independence checks before accepting results as physically representative. Although finer meshes offer better spatial resolution, they also increase computational demand, necessitating a trade-off that must be assessed early in the model setup phase. Advanced adaptive mesh refinement techniques, which allocate finer mesh where gradients are steepest, can offer a solution, though they remain computationally intensive and are not yet universally supported across platforms [1,2]. In high-performance energy systems like fuel cells or turbines, where small-scale phenomena such as species diffusion or micro-scale turbulence have system-wide impacts, mesh quality directly influences predictive capability. The ability of numerical models to capture localized hot spots or stagnation zones is pivotal for thermal management and reliability, and improvements in meshing algorithms continue to expand their applicability.

The comparative solver performance highlighted in the results confirms that method selection significantly affects both accuracy and computational cost. While FDM remains suitable for simple, structured domains, its inflexibility in handling complex geometries makes it less favored in current research. FEM and FVM have emerged as the dominant schemes due to their capacity to handle irregular meshes and multiphysics couplings. FEM's strength in structural and thermal analysis complements FVM’s dominance in fluid dynamics, making their integration through co-simulation or hybrid solvers increasingly common [3,4]. However, solver selection is often constrained by the commercial tools used, where licensing cost and user interface design can skew preferences regardless of pure performance metrics. Open-source platforms like OpenFOAM offer extensive flexibility, but require significant learning curves, while cloud-based platforms democratize access but raise concerns about data confidentiality in industrial applications.

Validation of numerical models using experimental or real-world data remains a gold standard. As the results demonstrate, even highly refined models can deviate under dynamic or boundary-sensitive conditions, pointing to the intrinsic uncertainties in modeling assumptions such as turbulence models, boundary conditions, or material properties. The availability of high-fidelity experimental data is often a limiting factor in conducting thorough validation. For emerging technologies like DAC (Direct Air Capture) or SOECs (Solid Oxide Electrolysis Cells), access to benchmark datasets is sparse, requiring either scaled-down experimental setups or reliance on inferred boundary conditions. This limitation amplifies the importance of uncertainty quantification techniques, which can statistically evaluate the robustness of results even in the absence of extensive experimental benchmarks [5,6].

Thermal visualization via contour plots and heat maps, as shown in several figures, adds immense value to design feedback. These visual tools enable quick identification of bottlenecks, thermal leaks, or inefficiencies that would otherwise be concealed in lumped parameter analyses. The results show that spatial temperature profiles in systems like building envelopes or PCM storage tanks help optimize insulation layout, phase boundary movement, and even structural reinforcement. However, interpretation of such visual data demands strong domain knowledge, especially in multiphysics simulations where variables influence each other non-linearly. Future work could benefit from integrating augmented reality (AR) overlays that project simulation results onto physical systems in real-time, a feature currently being piloted in aerospace applications [7,8].

Turbulence modeling remains a major challenge, particularly in HVAC systems, combustion devices, and wind turbines. Despite improvements in RANS-based models, limitations in accurately capturing recirculation zones and transient eddies persist. LES and DNS (Direct Numerical Simulation) offer improved accuracy but are computationally prohibitive for large domains. Hybrid models like Detached Eddy Simulation (DES) provide a compromise, but their adoption is limited by solver support and validation data. The results demonstrate that even modest differences in turbulence modeling can shift predicted pressure drop or temperature by up to 8%, a difference significant enough to alter system designs or economic feasibility. Continued development of AI-augmented turbulence closure models could offer new pathways to combine accuracy and speed, especially when trained on high-fidelity datasets [9,10].

One of the most exciting developments is the integration of artificial intelligence and machine learning into the modeling workflow. As illustrated in the results, neural networks and surrogate models are increasingly being used to emulate expensive CFD runs or provide real-time control signals based on historical simulation data. These methods are especially valuable in district energy systems, battery thermal management, and building control where real-time optimization is required. However, challenges remain in ensuring that AI models generalize beyond their training data, particularly in systems prone to stochastic disturbances or nonlinear transients. Research efforts focused on physics-informed neural networks (PINNs) seek to embed governing equations directly into the training process, reducing the need for large datasets and increasing model robustness. Such hybrid approaches are likely to dominate future modeling frameworks where interpretability and generalization are both required [11,12].

Interfacing modeling platforms to enable co-simulation across domains is also gaining momentum. Thermal-hydraulic-electrical simulations, as shown in hybrid PV-thermal systems and P2X applications, benefit from co-simulation environments that synchronize multiple solvers. Tools like Modelica, FMI (Functional Mock-up Interface), and Simulink offer partial solutions, but full integration remains cumbersome, particularly when time scales and solver schemes differ significantly. For example, coupling a millisecond-scale combustion simulation with a minute-scale HVAC load forecast requires advanced buffering, interpolation, and solver coordination. Enhancing these interfaces will be essential for systems like net-zero buildings, microgrids, and integrated desalination–power–cooling networks [13,14].

One recurrent limitation identified across case studies is the treatment of boundary conditions and external drivers. Most simulations assume ideal or static boundary inputs, such as constant ambient temperature, fixed solar irradiation, or predefined occupancy. In reality, these parameters are dynamic and uncertain. Digital twins, as shown in the results, provide a solution by enabling boundary conditions to be updated in real-time using sensor data. This not only improves model accuracy but also supports predictive control and fault detection. However, digital twins demand high-fidelity calibration, continuous data acquisition, and cyber-physical system integration—factors that are still maturing in commercial implementations. Future research should focus on standardizing digital twin frameworks and validating their long-term performance in field deployments [15,16].

The coupling of numerical modeling with techno-economic and life-cycle analysis is another critical frontier. As seen in the solar cooling and biomass systems, numerical outputs like energy yield, temperature profiles, and flow rates can be directly fed into LCOE and carbon footprint calculators. This convergence of performance and sustainability metrics allows stakeholders to make informed decisions on trade-offs between cost, efficiency, and emissions. However, integrating TEA and LCA modules into numerical simulation platforms remains limited, often requiring export-import steps or post-processing in separate software. Developing unified modeling environments that natively support technical, economic, and environmental assessment would streamline workflows and make sustainability assessments more transparent [17,18].

Finally, numerical modeling has played an indispensable role in democratizing energy research. Through the use of open-source tools, cloud platforms, and digital learning environments, researchers from developing regions or underfunded institutions can contribute to innovation on a level playing field. The ability to simulate a 3D solar tower, model a hybrid energy system, or test fault scenarios without access to physical infrastructure is a testament to the power of numerical tools. This opens the door for collaborative, globally inclusive research ecosystems that accelerate the transition to sustainable energy [19,20]. Another important point that emerges from this comprehensive review is the evolving role of numerical modeling in addressing uncertainty and risk management in energy systems. While deterministic models provide insight into ideal behavior under fixed conditions, the stochastic nature of weather, user behavior, and equipment degradation necessitates probabilistic modeling approaches. The integration of Monte Carlo methods and Latin Hypercube Sampling with CFD or system simulations allows for the exploration of variability in inputs and its effects on outputs such as efficiency, temperature distribution, or emissions. This is particularly valuable in renewable energy applications like solar PV and wind farms, where fluctuating weather data plays a dominant role in performance predictions. As numerical platforms evolve, real-time stochastic modeling may become more feasible, offering robust system design frameworks that can withstand variability without significant performance degradation [1,2].

In addition, the rise of low-carbon and negative emission technologies demands highly flexible and scalable modeling frameworks. Technologies such as Direct Air Capture (DAC), Bioenergy with Carbon Capture and Storage (BECCS), and hybrid systems integrating solar thermal, hydrogen, and battery storage all require simulation models capable of adapting to rapidly evolving process configurations and chemical pathways. For example, modeling DAC systems under varying humidity and CO₂ concentrations has revealed significant sensitivity in adsorption behavior and regeneration energy demands, findings that would be difficult to obtain experimentally. The use of numerical tools to conduct such sensitivity analyses enables a deeper understanding of performance limits and trade-offs under realistic atmospheric and climatic conditions. Moreover, coupling these models with economic and policy simulations creates a powerful decision-making framework for climate strategy planning [3,4].

Another direction with growing attention is the use of reduced-order models (ROMs) derived from full-scale numerical simulations. These simplified models retain key system dynamics while enabling rapid evaluation of control strategies, scenario analyses, or real-time implementation in embedded systems. For instance, ROMs developed from 3D CFD simulations of heat exchangers or combustion chambers can be used in controller design without the computational burden of full-scale models. This has direct implications in the automotive and aerospace industries, where real-time feedback is essential for operational safety and efficiency. ROMs are also being applied in building energy management systems to optimize HVAC operation based on occupancy and weather forecasts [5,6].

Furthermore, the emergence of high-fidelity multiphysics simulation platforms has catalyzed interdisciplinary collaboration in energy research. Domains that were traditionally isolated—such as electrical and mechanical engineering, materials science, architecture, and environmental policy—are now converging through shared simulation environments. For example, the design of an energy-efficient smart building may involve CFD simulations of airflow, FEM analysis of structural response to wind loads, transient thermal simulations for insulation performance, and energy simulations for HVAC operation—all integrated into one workflow. This level of interdisciplinary modeling fosters innovation and reduces the likelihood of design oversights, enabling more resilient and high-performing systems to emerge [7,8].

It is also worth noting that the educational value of numerical modeling cannot be overstated. Simulation tools offer students and researchers an opportunity to explore system behavior under controlled and extreme conditions, developing intuition and a systems-level understanding. Virtual labs using modeling software are now a cornerstone of engineering education, especially in regions where physical labs are cost-prohibitive. As the energy transition accelerates, it is imperative to equip the next generation of engineers and scientists with the skills to design, analyze, and optimize sustainable systems through numerical methods. Open educational resources, shared simulation case libraries, and community-supported solver platforms can help reduce the access gap and enhance global competence in this critical field [9,10].

Looking ahead, the integration of numerical modeling into regulatory frameworks, design standards, and certification processes could bring significant efficiencies. Digital simulation data can be used to pre-certify system performance, evaluate compliance with energy codes, or conduct environmental impact assessments, thereby reducing reliance on slow and expensive physical prototyping. Several countries are already piloting digital submission of simulation results as part of building permitting or green certification programs. If properly validated and standardized, these processes could usher in a digital-first era for energy system design and compliance [11,12].

Moreover, the role of numerical modeling in emergency planning and resilience analysis is expanding. Simulations can help evaluate the response of energy systems to disasters such as floods, earthquakes, cyberattacks, or energy supply disruptions. For example, in district energy systems, transient hydraulic simulations can predict pressure waves resulting from sudden valve closure or pipe rupture, allowing for proactive design of relief systems. In power grids, coupled thermal–electrical simulations of substation components can help anticipate overheating during heatwaves or cascading failures during blackouts. These applications are crucial as climate change increases the frequency and severity of extreme events, demanding robust and adaptable energy infrastructure [13,14].

Finally, the future of numerical modeling will likely revolve around greater automation, intelligence, and integration. Automated meshing, AI-based solver selection, and self-learning models that improve with operation are all on the horizon. Such systems could autonomously refine their predictions based on field performance data, similar to the way autonomous vehicles learn driving patterns. Integration with blockchain technology could add transparency and security to simulation result storage and verification, particularly in high-stakes sectors such as aviation or nuclear energy. The convergence of these technologies will make numerical modeling not just a design tool, but a living, evolving system companion throughout the lifecycle of energy assets [15,16].

In summary, the trajectory of numerical modeling in energy systems is one of increasing relevance, capability, and interdisciplinarity. As modeling platforms become more powerful and accessible, they will continue to play a central role in designing the energy systems of tomorrow. The challenge lies in ensuring that these tools are used wisely—validated rigorously, integrated meaningfully, and interpreted cautiously—to support a just, efficient, and sustainable global energy transition [17,18]. It is imperative that future modeling work emphasizes transparency, reproducibility, and stakeholder engagement so that insights generated through simulations can translate into real-world impact [19,20].

5. Conclusion

Numerical modeling has firmly established itself as a foundational methodology in the research, development, and optimization of energy systems across all scales and domains. This review has presented a comprehensive exploration of the evolution, capabilities, and application of numerical modeling techniques including computational fluid dynamics (CFD), finite element method (FEM), and system-level transient simulation in the context of energy conversion, distribution, and consumption. From single-component analyses like heat exchangers and fuel cells to multi-domain simulations of hybrid renewable systems, numerical modeling has demonstrated unmatched versatility and depth.

The results synthesized from literature and modeled case studies confirm that accurate and insightful simulation hinges on appropriate model selection, mesh resolution, solver configuration, and robust boundary condition definition. The significance of mesh independence studies, solver sensitivity analyses, and boundary specification was evidenced by simulation deviations observed in temperature, flow distribution, and efficiency predictions. Tools like ANSYS Fluent, COMSOL Multiphysics, EnergyPlus, and OpenFOAM have enabled researchers to visualize phenomena ranging from turbulence eddies to solar irradiance flux profiles, often offering predictive accuracy within 1–5% of experimental data.

Key findings from this review highlight the growing utility of hybrid modeling approaches, where physics-based solvers are augmented with artificial intelligence to reduce computation time while retaining high fidelity. Applications in district cooling optimization, combustion chamber tuning, and battery thermal management have particularly benefited from surrogate models and physics-informed neural networks. These developments have opened new avenues for real-time simulation, predictive maintenance, and digital twin deployment.

The integration of numerical models with economic and environmental assessment tools also enhances their decision-making utility. Life Cycle Assessment (LCA) and Techno-Economic Analysis (TEA) linked to simulation outputs provide a more holistic view of sustainability and performance trade-offs. In solar-thermal and biomass-based systems, such combined modeling frameworks revealed opportunities to improve exergy efficiency while minimizing CO₂ emissions and water consumption, information crucial to policy and infrastructure development.

Furthermore, the review highlights the role of numerical modeling in resilience analysis, disaster planning, and low-carbon technology deployment. Modeling has become indispensable in simulating extreme conditions such as pipe ruptures in CSP systems, grid failures in hybrid microgrids, and hydrogen leak scenarios in electrolysis facilities. These simulations have informed the design of fault-tolerant systems, emergency shutdown protocols, and robust infrastructure able to withstand uncertain environmental and operational challenges.

However, the power of numerical modeling also demands responsibility. While simulations offer tremendous potential, their accuracy is bound by assumptions, model fidelity, and validation quality. Poorly calibrated or over-simplified models may yield misleading conclusions, especially when extrapolated beyond their calibration range. This makes experimental validation, uncertainty quantification, and transparent documentation essential practices in the responsible application of modeling.

Equally important is the democratization of numerical tools. Open-source platforms, cloud-based computing, and community-driven solver development are increasingly making high-quality modeling accessible to institutions with limited physical or financial resources. This democratization is critical to global climate action, as it enables developing nations and remote research groups to contribute to and benefit from the global energy transition through simulation-based innovation.

Looking forward, the future of numerical modeling in energy systems is poised to be shaped by several key trends. First is the further integration of AI and machine learning to create faster, more adaptive solvers. Second is the development of standardized co-simulation platforms that can seamlessly couple electrical, thermal, chemical, and structural domains. Third is the rise of real-time digital twins and cyber-physical systems, where simulations dynamically adapt based on live sensor data to optimize performance or detect anomalies. Fourth is the expansion of uncertainty-aware modeling, where probabilistic approaches guide robust and risk-informed decision-making.

To realize this future, several enablers must be strengthened: open-access data for validation, collaborative modeling repositories, modular solver libraries, and interdisciplinary training programs. In particular, fostering a generation of engineers and scientists proficient in both physical modeling and computational techniques will be crucial. Educational institutions and research organizations must prioritize simulation training as a core pillar of energy science curricula.

In conclusion, numerical modeling stands not only as a tool for understanding and predicting energy system behavior, but as a catalyst for innovation, efficiency, and sustainability. Its role will only deepen as energy systems grow more integrated, dynamic, and distributed. When used responsibly and creatively, numerical models will continue to bridge the gap between concept and reality, accelerating the path toward a resilient, equitable, and low-carbon energy future.

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